Paper Title: Neutron star kicks plus rockets as a mechanism for forming wide low-eccentricity neutron star binaries.


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0000-0002-8032-8174]Ryosuke Hirai (平井遼介)
Astrophysical Big Bang Laboratory (ABBL), Cluster for Pioneering Research, RIKEN, Wako, Saitama 351-0198, Japan
School of Physics and Astronomy, Monash University, Clayton, VIC3800, Australia
OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery, Clayton, VIC3800, Australia

0000-0002-8338-9677]Philipp Podsiadlowski
University of Oxford, St Edmund Hall, Oxford OX1 4AR, UK

0000-0002-3684-1325]Alexander Heger
School of Physics and Astronomy, Monash University, Clayton, VIC3800, Australia
Argelander-Institut für Astronomie, University of Bonn, Auf dem Hügel 71, 53121 Bonn,	Germany

0000-0002-7205-6367]Hiroki Nagakura (長倉洋樹)
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan



§ ABSTRACT 
Recent neutron star surface observations corroborate a long-standing theory that neutron stars may be accelerated over extended periods after their birth. We analyze how these prolonged rocket-like accelerations, combined with rapid birth kicks, impact binary orbits. We find that even a small contribution of rocket kicks combined with instantaneous natal kicks can allow binaries to reach period–eccentricity combinations unattainable in standard binary evolution models. We propose these kick + rocket combinations as a new channel to form wide low-eccentricity neutron star binaries such as Gaia NS1, as well as inducing stellar mergers months to years after a supernova to cause peculiar high-energy transients. 




§ INTRODUCTION


Neutron stars (NSs)ns, the compact remnants of massive stars, are known to be born with high spatial velocities of a few×100 km s^-1 <cit.>. In the current paradigm, this velocity is given at birth due to asymmetries in the sn explosion, driven by hydrodynamical instabilities <cit.> and asymmetric neutrino emission <cit.>. Since this velocity is related to the core-collapse sn explosion mechanism itself, the acceleration occurs on dynamical timescales of τ_kick≲10 s and the resulting velocity is often called a “natal kick”.

On the other hand, some kick mechanisms have been proposed in the past that involve post-natal processes. A famous example is the “electromagnetic rocket” model, where the new-born ns is propelled along the spin axis by asymmetric spin-down radiation from off-centred or higher-order magnetic fields <cit.>. 
Recently, the NICER X-ray telescope revealed the non-antipodal orientation of hot spots on a neutron star surface, strongly supporting the existence of such higher-order magnetic fields <cit.> and therefore it is very likely that the electromagnetic rocket mechanism contributes to the ns kick to some degree. The total magnitude of the velocity achieved through this mechanism scales as

    v_roc≈260 km s^-1ϵ̅/0.1(R/10 km)^2(P_spin,ini/1 ms)^-2,

where ϵ̅ expresses the degree of asymmetry of the outgoing Poynting flux, R and P_spin,ini are the ns radius and birth spin period respectively <cit.>. In the most optimal magnetic field configuration, this can reach up to v_roc∼1,400 km s^-1[It should be noted that the strength of the magnetic field only influences the acceleration timescale in this model. The final velocity is determined by the reservoir of rotational energy and asymmetry of the radiation, hence the field strength is irrelevant.]. Another post-natal model is the so-called “neutrino rocket” mechanism, where neutrinos are asymmetrically emitted via neutrino cyclotron radiation from neutrons circulating superfluid vortices in the ns interiors <cit.>. Under favourable assumptions, this mechanism can accelerate the ns to velocities reaching v_roc≲1,000 . Although some difficulties have been pointed out for these “rocket” mechanisms, such as the requirement of a high birth spin (P_spin∼1 ms) that is in contradiction with observationally inferred values <cit.>, these models have not been firmly ruled out <cit.>. In particular, there are no arguments against the rocket mechanisms making minor contributions to the total kick.


A distinct property of the rocket mechanisms is its long acceleration timescale. Since it is driven by spin-down, the acceleration timescale can be τ_roc∼0.1–100 yr (≫τ_kick). Whereas this is still negligible compared to the lifetime of pulsars (∼10^7-8 yr), meaning it will be difficult to detect ongoing acceleration, it is longer than typical orbital periods of binary systems containing ns. Traditionally, ns kicks in the context of binary evolution have been treated as an impulsive process, in which the post-sn orbit can be analytically derived. The resulting orbital properties depend on the magnitude and direction of the kick, but in most cases it causes the orbit to become wider and eccentric or disrupt it completely. However, when the kick cannot be treated impulsively but instead the acceleration timescale is comparable to, or longer than, the orbital period, these impulsive solutions no longer hold. Instead, we need to consider the opposite extreme where the ns acceleration is a small perturbation compared to the orbital acceleration.

The two-body problem with a constant acceleration on one or both of the bodies is sometimes called the “accelerated Kepler problem” <cit.>, which is analogous to the Stark effect in quantum mechanics. Due to the secular nature, the orbit can be altered in ways that are qualitatively different from instantaneous kicks. The impact on ns binaries, however, has not been well quantified and the existing studies only deal with special cases where the rocket mechanisms act on initially circular orbits <cit.>. 

In this paper, we explore the implications of rocket-like acceleration in tandem with classical impulsive kicks, in particular, how it influences our understanding of the formation channels for various ns binaries. 
One of the main consequences is that it provides a channel to form the wide low-eccentricity ns binary Gaia NS1 <cit.> and other similar systems <cit.> that cannot be understood in standard binary evolution models. This may provide direct testable evidence for the rocket mechanisms in ns binaries. 

This paper is structured as follows. In Section <ref>, we introduce the secular accelerated Kepler problem and derive the analytic solution. The combined effect of rapid ns kicks and the rocket effect are demonstrated in Section <ref>, and its astrophysical implications are discussed in Section <ref>. We summarize our conclusions in Section <ref>.



§ THE ACCELERATED KEPLER PROBLEM


In this section we introduce the accelerated Kepler problem (or classical Stark problem), which is a two-body problem with a fixed acceleration on one of the bodies. The equation of motion of our two-body problem is

    m_1r̈_1   =F_12+m_1a_roc,
    
        m_2r̈_2   =F_21,

where m_i are the masses, r_i are the position vectors where subscripts i=1,2 denote the primary and secondary, and a_roc is the rocket acceleration acting on the new-born ns. F_12=-F_21 is the gravitational force between the two stars. Dots denote time derivatives. To solve for the orbit, we separate the equations into the centre of mass and relative motion by defining

    r_com≡m_1r_1+m_2r_2/m_1+m_2,
       r≡r_1-r_2.

We rewrite the gravitational force as

    F_g≡F_12=-F_21=-Gm_1m_2/r^2r̂=-k/r^2r̂,

where we have defined k≡ Gm_1m_2 and the unit vector r̂=r/|r|. The equation of motion for the centre of mass and relative position can hence be expressed as

    r̈_com=m_1a_roc/m_1+m_2,
       r̈=F_g/μ+a_roc,

where μ≡ m_1m_2/(m_1+m_2) is the reduced mass. The same expression for Eq. (<ref>) will hold even for cases where both stars have separate accelerations, by replacing a_roc with the difference of the two accelerations.





 §.§ Deriving the analytical solution

The problem has been known to be integrable for more than two centuries, but it was only in the work by <cit.> that the general closed-form solution to this problem was shown using elliptical coordinates. Under the secular approximation (|a_roc|≪|F_g/μ|), the semi-major axis (or orbital period) is preserved whilst the orbit is torqued by the rocket acceleration. The general behaviour is that the orbital plane librates around the direction of the rocket and the eccentricity oscillates up and down, while the binary as a whole is also accelerated. In this section, we derive an alternate analytical solution to the secular accelerated Kepler problem in vector form. Our much simpler form allows for an intuitive understanding and makes it suitable for statistical investigation. 

Our solution can be summarized as follows. The energy and angular momentum of the orbit are defined as

    E   ≡1/2μṙ^2-k/r=-k/2a,
    L   ≡μr×ṙ,

where a is the semi-major axis.
The energy gives an average orbital velocity

    v̅_orb=√(-2E/μ)=√(k/μ a).

Let us define a dimensionless angular momentum vector l≡L/L_0 where L_0≡ k/v̅_orb is the angular momentum of a circular orbit (e=0) with the same energy. The magnitude of this vector is |l|=√(1-e^2).
By taking the time derivative of the normalized angular momentum and using Eq. (<ref>), the torque on the orbit by the rocket acceleration can be expressed as

    l̇=μ/L_0r×a_roc.

Notice that l̇·a_roc=0, so the angular momentum in the direction of the rocket is strictly constant over time, whereas the transverse components are torqued by the rocket acceleration. Making the secular approximation, this becomes

    <l̇>   =μ/L_0<r>×a_roc,
       =-3/2v̅_orbe×a_roc,

where the angle brackets represent quantities averaged over an orbit 

    <x>≡1/P_orb∫_0^P_orbx(t)dt,

and the eccentricity vector (or Laplace-Runge-Lenz vector) is defined as

    e≡1/kṙ×L-r̂.

We have also used that <r>=-3ae/2 and k/(μ a)=v̅_orb^2.

Similarly, if we take the time derivative of the eccentricity vector, we obtain

    ė=1/k(a_roc×L+ṙ×L̇),

and the orbit-averaged value is

    <ė>   =1/v̅_orb(a_roc×l+<ṙ×l̇>),
       =-3/2v̅_orbl×a_roc.

Note here that <ė>·a_roc=0, so the component of the eccentricity vector parallel to the rocket is constant under the secular approximation.

We now define two support vectors

    h_±≡l±e,

There are several properties to these vectors. First, |h_±|=1, which is trivial from |e|=e, e·l=0 and |l|=√(1-e^2). Secondly, <ḣ_̇±̇>·a_roc=0, because l̇·a_roc=<ė>·a_roc=0, meaning that the component of the h_+ and h_- vectors parallel to the rocket are constant over time. Using Eqs. (<ref>) & (<ref>), the secular evolution of these vectors can be expressed as

    <ḣ_̇±̇>=∓3/2v̅_orbh_±×a_roc.

This implies that as the ns is accelerated by the rocket mechanism, both h_+ and h_- precess about the direction of the thrust a_roc in opposite circular motions. Let us take the direction of the rocket a_roc as the z axis. If the cumulative rocket velocity up to some time t is

    Δ v_roc(t)≡∫_0^t a_rocdt,

 the post-rocket state can be described by 

    h_±'   =R_z(±3Δ v_roc/2v̅_orb)h_±,

where R_z(θ) is a rotation matrix about the z-axis for an angle θ. We can see that the two vectors will circulate around the z-axis in opposite directions. The post-rocket orbit can then be completely described by

    l'   =1/2(h_+'+h_-'),
    e'   =1/2(h_+'-h_-').

The post-rocket eccentricity is simply e'=|e'|.

Although we have assumed that a_roc is constant, our above solution still holds with variable acceleration magnitudes as long as it does not change significantly over an orbital period. In the rocket mechanism, the magnitude should slowly decline over time as the ns spins down. Eventually, a_roc should become negligible and the eccentricity oscillation would cease. The asymptotic value of the accumulated velocity Δ v_roc determines the final eccentricity.

There are some complications if the rocket direction (or the spin axis) precesses. Our solution would still likely hold if the precession timescale is much shorter than the orbital period. On average, the rocket acceleration should be pointed along the axis of precession with a slightly lower magnitude, and the orbital evolution can be computed using this averaged acceleration. When the precession timescale is comparable to or longer than the orbital period, the above solution no longer holds. It has been shown in previous studies that in such cases the precessing rocket acceleration can lead to resonant eccentricity excitations <cit.>. For even longer precession timescales, the rocket acceleration ceases before the spin direction changes appreciably, so again our above solution may be applicable.




 §.§ Numerical validation





We verify that our analytic solution describes the true solution well by comparing against direct numerical integrations of the accelerated Kepler problem.
Figure <ref> compares an example of the evolution of individual components of l and e based on Eqs. (<ref>)–(<ref>) (dashed curves) against direct numerical integration of Eqs. (<ref>)–(<ref>) (solid curves) with a low acceleration magnitude (|a_roc|=|F_g/μ|/50). We start the integration from a binary with an eccentricity e=0.3 and the rocket direction is tilted from l by 60^∘ in the x direction. We can see that the xy components of the vectors l and e both oscillate sinusoidally with respect to the accumulated rocket velocity Δ v_roc. We show the trajectory of the vectors l and e projected on the xy-plane in Figure <ref>. Both vectors follow elliptic trajectories around the z-axis with the same semimajor and semiminor axes. When converted to the h_± vectors, we can see they follow circular trajectories moving in opposite directions. As a result of these circulations, the eccentricity oscillates, although not as a sine curve. The analytic solution is in perfect agreement with the numerical result.



The amplitude of the eccentricity oscillation depends on the initial eccentricity and rocket angle. For example, when the initial orbit is circular and the rocket is aligned with the orbital angular momentum (h_+=h_-=â_roc), there is no orbital evolution. On the other hand, if the initial orbit is circular and the rocket is perpendicular to the orbital angular momentum, the eccentricity can reach unity e=1.




To assess when the secular approximation breaks down, we perform numerical integrations with various acceleration timescales τ_acc≡v̅_orb/|a_roc|. The results are shown in Figure <ref>. As we demonstrated above, when the acceleration timescale is taken to be sufficiently longer than the orbital period τ_acc/P_orb≳10, the evolution of the eccentricity closely follows our secular solution (dashed curve). At τ_acc/P_orb≲1, the secular approximation completely breaks down and hence the binary is dissociated within a single orbit. For the intermediate acceleration timescales where it is larger than but comparable to the orbital period (2≲τ_acc/P_orb≲10), the numerical results slightly deviate from the analytic solution while maintaining the periodic oscillation behaviour. We conduct similar tests for other orbital configurations and find similar results; the secular solution describes the numerical results quite well as long as τ_acc/P_orb≳2.





§ COMBINED EFFECT OF MASS-LOSS, NATAL KICK AND ROCKET


When sn occur in binaries, the orbit will be altered by a combination of various effects. First, there is the immediate recoil due to the mass loss by sn ejecta <cit.>, which causes the orbit to widen and change its eccentricity. Another effect is the instantaneous kick on the ns associated with core collapse. The rocket mechanisms should then act on binaries that have already been perturbed by these rapid effects. The pre-rocket perturbations are crucial, as the rocket does not change the orbit unless the acceleration is tilted from the orbital angular momentum and/or there is initially non-zero eccentricity. Therefore, it is important to consider the full combined effect of the various forms of ns acceleration.


Figure <ref> illustrates an example of how an initially circular orbit is altered by different combinations of rapid and rocket-like kicks. Initial system parameters are set to m_1=3 , m_2=1 , P_orb,0=1 d, which has an average orbital velocity v̅_orb,0∼338 [Note that this is not the v̅_orb relevant for the post-rocket orbit calculation]. The sn progenitor is assumed to have lost its hydrogen envelope through binary interactions and has become a helium star <cit.>. We assume the primary star instantaneously loses m_ej=1.5 in the sn. After mass loss, the Blaauw effect causes the binary to jump both in period and eccentricity from the open circle to the filled circle. We further apply either an instantaneous kick with Δ v_kick=0.5 v̅_orb,0 (blue dots), a rocket-like kick with Δ v_roc=0.5 v̅_orb,0 (cyan dots), or a mixture of both where we apply a Δ v_roc=0.1 v̅_orb,0 rocket kick after the Δ v_kick=0.5 v̅_orb,0 rapid kick (red dots). For each model, we sample 10^4 systems assuming an isotropic probability distribution for the kick/rocket direction. For the mixture model, we independently sample both the kick and rocket directions from isotropic distributions. For the kick-only model (blue dots), the distribution spreads out diagonally with a strong correlation between the post-sn period and eccentricity. In the rocket-only model (cyan dots), the eccentricity is altered whereas the orbital period is unchanged. It is clear that the instantaneous and rocket-like kicks impact the orbit in qualitatively different ways.

In the kick+rocket mixture model (red dots), the period distribution is purely determined by the rapid component (see histograms on the top panel). The rocket effect then spreads the distribution out in the eccentricity direction. This enables the binary to cover a much wider area in period–eccentricity space from a given progenitor system, despite the rocket magnitude being rather small (Δ v_roc=0.1 v̅_orb,0∼34 ). As the rocket effect depends on the average orbital velocity (Eq. (<ref>)), its effects are most pronounced in the longer period regions where it can allow the binary to reach low eccentricities, down to e=0.







§ IMPLICATIONS


There are many implications of this wide phase space coverage by the kick+rocket mixture model, but in this paper we highlight a few select interesting applications.



 §.§ Wide low-eccentricity binaries


An important distinct feature of the rocket effect is that the periastron distance can widen after the sn (see in Figure <ref> how dots are distributed to the right of the black curve). Note that it is impossible to widen the periastron distance beyond the pre-sn separation with only rapid kicks. This effect is enhanced when the rocket acts on orbits already strongly perturbed by the rapid kick because the systems that are kicked to wider orbits are more susceptible to the rocket effect and can be circularized more easily. Therefore, the kick+rocket combination can in principle widen the periastron distance by arbitrary amounts as long as the secular approximation holds in the rocket phase. 

Recent observations by the Gaia satellite have discovered a number of wide (P_orb∼200–1000 d) ns binaries with ∼1 stellar companions <cit.> (see black crosses in Figure <ref>)[Gaia is only sensitive to long-period systems. The majority of systems in Figure <ref> would be the progenitors of low- or intermediate-mass X-ray binaries with much shorter orbital periods.]. Most notable is Gaia NS1, which has an orbital period of P_orb∼731 d and the lowest eccentricity e∼0.122 <cit.>. These systems are considered to be progenitors for symbiotic X-ray binaries, which also tend to be on wide and low-eccentricity orbits, e.g. GX 1+4 (P_orb=1160.8±12.4 d, e=0.101±0.022) <cit.>, 4U 1700+24 (P_orb=404±3 d, e=0.26±0.15) <cit.> and IGR J16194-2810 (P_orb=192.73 d with a red giant companion, implying a rather large periastron distance) <cit.>. It is challenging to devise formation channels for such systems in the traditional binary evolution framework, as the orbital periods are too short to avoid binary mass transfer from the sn progenitor that will shrink their orbits down to much tighter periastron distances. Even if the post-mass-transfer separations can be kept wide enough, the rapid kick needs to be Δ v_kick≲30 km s^-1 in order to keep the eccentricity low enough, whereas typical observed kick velocities are Δ v_kick≳100 km s^-1 <cit.>. Dynamical assembly channels in dense cluster environments have also been ruled out due to their low eccentricity <cit.>.

With rocket mechanisms, these systems can be formed more naturally (see Figure <ref>). The system can experience mass transfer or common envelope phases to strip the envelope of the primary and dramatically shrink the orbit. At this point the ejecta mass is small and the orbit is tight enough so that the sn mass loss and natal kick tend to not disrupt the binary but instead send it on a wide and eccentric orbit. Once the orbit has widened, the average orbital velocity is now slow enough that even small rocket-like kicks (Δ v_roc≲30 km s^-1) can induce large changes to the eccentricity. Although calculating detailed formation rates of low-eccentricity ns binaries is out of the scope of this letter, we note that classical binary evolution scenarios will need to be stretched to extremes whereas our scenario does not require any non-standard assumptions other than the rocket mechanism. Based on simple Monte Carlo simulations of kicks+rockets, we expect ∼0.05–0.1 % of ns + 1 binaries to end up with wide (P_orb≥100 d) and low-eccentricity (e≤0.3) orbits if their pre-sn periods are 1 d≤ P_orb≤30 d. See Appendix <ref> for details of the Monte Carlo calculations.








 §.§ Induced mergers

Whereas some systems lower their eccentricities due to the rocket effect, some systems will increase their eccentricities. If the rocket acceleration is directed at an appropriate angle, the eccentricity can reach very close to e=1, inevitably causing a tidal disruption of the companion or merger between the two components. These interactions may explain the origin of some superluminous sn or tzo if the companion is a non-degenerate star <cit.>.

In cases where the companion is also a compact object, it may induce a binary merger event that emits gravitational waves and/or electromagnetic signatures like short-duration grb, kilonovae or other peculiar transients. The resulting transient will be separated from the sn that gave birth to the ns by a span of weeks to years, as the merger is induced by a secular effect. If, for example, a binary ns merger occurs with a sufficiently short delay time, the merger will occur with unique conditions (e.g. relatively hot ns, dense circumstellar material due to sn ejecta), possibly giving rise to peculiar grb or late-time bumps in the sn light curve. Additionally if the delay time is long enough so that the grb occurs after the peak of the associated sn light curve, it may cause the accompanying sn to be undetected like in some known long grb <cit.>.





 §.§ Rockets in triples

There are possibly even more interesting effects in the context of triple star evolution. For example if the sn progenitor was in the inner binary of a hierarchical triple, the mass loss and natal kick not only alters the orbital period of the binary but also the outer orbit. In particular, the change in the outer orbit strongly depends on the orbital phase at which the explosion occurs, as it determines the direction of the mass-loss recoil. The subsequent rocket effect will cause the inner orbit to librate and change its eccentricity. As long as the ns acceleration timescale is longer than the outer orbital period, the same libration will occur on the outer orbit too, as the inner binary as a whole will also be accelerated. It is not trivial which libration will be more significant. Both the inner and outer orbits have different initial configurations and angle with respect to the rocket. Also, the degree of libration and eccentricity change depends on Δ v_roc/v̅_orb, which is also different between the inner and outer orbits. The multitude of scales involved can lead to a wide variety of possible outcomes.

Once the kick and rocket effects have sufficiently tilted the inner and outer orbits, the mutual inclination may exceed the critical inclination required for secular evolution like Kozai-Lidov oscillations <cit.> and higher-order effects <cit.>. Or it could take the triple all the way to an unstable configuration <cit.>, which should eventually terminate through stellar mergers or ejection of one of the stars <cit.>. Therefore, even if the rocket effect itself cannot modify the orbits enough to cause interesting phenomena, it can still be used as an efficient mechanism to hand over the system to where triple effects will finish off the job. Alternatively, the Kozai-Lidov oscillations may start to operate before the rocket effect has ceased. It may cause further interesting modulations and/or resonances if the oscillation timescales of the triple and rocket effects are comparable. We leave exploration of such phenomena for future work.






§ CONCLUSION


Motivated by recent observations of non-axisymmetric magnetic fields on ns, we explored the implications of combinations of rapid and rocket-like kicks in binary systems. We derive an analytic solution for the secular accelerated Kepler problem in vector form. This simple-formed solution is useful for other applications such as planetary science and artificial satellite maneuvering <cit.>.

We demonstrate how the combined effect of mass loss, rapid natal kicks, and rocket-like kicks impact the post-sn binary orbits. We find that even with only small contributions from the rocket component, the resulting binary orbit can be qualitatively different from what is achievable with purely rapid kicks. This can transform our understanding of the formation of many astronomical objects.

The wide low-eccentricity ns binaries such as Gaia NS1 and GX 1+4 are challenging to form in classical binary evolution models. Our proposed new scenario based on rapid kick+rocket mixtures can more robustly form these systems without requiring drastic changes to how we understand binary interactions. We also expect that there could be many types of peculiar transients from mergers induced by the rocket effect between ns and their companions. By studying these observables in detail, we may in turn gain insight into the physics that drive the rocket mechanism, such as the ns birth spins, configuration of magnetic fields in new-born ns or efficiency of cyclotron neutrino emission from superfluid vortices.


There are potentially numerous other applications where the rocket effect may be relevant, for example the formation of X-ray binaries, gravitational-wave sources, and triple star evolution. We plan to explore many of these other avenues in future studies.



RH thanks Ilya Mandel, Shoichi Yamada, Yongzhong Qian and Andrew Melatos for useful discussions during the early stages of this project. HN is supported by Grant-in-Aid for Scientific Research (23K03468) and also by the NINS International Research Exchange Support Program.
AH is supported by the Australian Research Council through grants DP240101786 and DP240103174.








gnuplot <cit.>
          






§ MONTE CARLO SIMULATION OF FORMATION RATES


Given that the magnitude of rocket-like kicks are still poorly constrained both observationally and theoretically, it is tricky to estimate formation rates of the various objects formed in our kick+rocket scenario. Here, we conduct a simple Monte Carlo simulation to provide ballpark estimates of the formation probabilities of Gaia NS1-like systems from given pre-sn binary configurations. We define any system with P_orb≥100 d as “wide” and e≤0.3 as “low-eccentricity” binaries. 

We first start from a system with a 3 sn progenitor and a 1 companion star on a circular (e=0) orbit. We assume that 1.5 is lost in the sn and the primary now becomes a 1.5 ns. The progenitor is assumed to have undergone some form of binary interaction to lose a large fraction of its envelope so that the ejecta mass is within the typical range of observed stripped-envelope sn ejecta masses <cit.>. The Blaauw mechanism causes the binary to become wider and more eccentric after this. We then randomly draw the rapid kick velocity Δ v_kick from a Maxwellian distribution with an average velocity σ_kick=265 <cit.> and a random direction from a spherically isotropic distribution. After applying this rapid kick to the ns, we check whether the binary is still bound and has an orbital period satisfying the wide condition (P_orb≥100 d). If the period is long enough we then draw a random rocket velocity Δ v_roc from a Maxwellian distribution with an average velocity σ_roc and a random direction from a spherically isotropic distribution. We then compute the post-rocket orbit and check if the resulting eccentricity satisfies the low-eccentricity condition (e≤0.3).


Several models are computed with varying rocket velocity distributions (σ_roc=0, 30, 100 km s^-1) and initial orbital periods P_orb,0∈[0.1,1000] d. For each model we draw 10^6 samples to obtain the probability of forming wide and low-eccentricity systems. 

Firstly, we see that the fraction of systems that stay bound after the sn decreases with increasing initial period (dashed). Among the surviving systems, only a fraction of them achieve long orbital periods satisfying the wide condition when the pre-sn orbital period is not wide. Out of the total number of samples, the fraction of wide systems ranges 0.1–1 % (dotted), peaking at around P_orb,0∼4 d. 
When there is no subsequent rocket phase (solid black), it is basically impossible to form wide low-eccentricity systems unless the initial period is already long (P_orb,0≳50 d). On the other hand, with a rocket phase there is finite probability to form such binaries even if the pre-sn period is significantly shorter than the threshold (solid red and pink). Among the wide binaries, the fraction that end up having low eccentricities only weakly depends on the initial period, ranging ∼7–10 %.




Our main conclusion is that as long as the binary can achieve long orbital periods after rapid kicks, there is almost a fixed probability (∼7–10 %) of forming low-eccentricity binaries through the rocket mechanism as long as the rocket magnitude is comparable to the post-kick orbital velocity. This is in contrast to pure-kick models where it is impossible to create wide low-eccentricity binaries unless the pre-sn orbit is already wide. Orbital periods around P_orb,0∼100 d is still too short to avoid any binary interactions between the massive sn progenitor and the low mass companion. Typically, such interactions would tighten the orbit down to much shorter orbital periods. Therefore, if we integrate our distribution in Figure <ref> weighted by the distribution of post-binary-interaction periods, pure-kick models have a near-zero chance of forming wide low-eccentricity systems, especially Gaia NS1. Given that dynamical assembly in clustered environments have been proven to be difficult to create Gaia NS1-like systems <cit.>, triple evolution or our kick+rocket scenario may be the only plausible explanations for the formation of Gaia NS1-like binaries.


In this simple analysis we have ignored various possible effects that may influence the probabilities. For example, we have assumed that the kick, rocket and orbit are independent whereas there may be correlations between the direction of these components. There may be correlations between the pre-sn orbital periods and the rocket magnitudes due to tidal spin-up of the sn progenitor. Different forms of kick velocity distributions may impact the overall survival and wide-orbit fractions. The thresholds we use for “wide” and “low-eccentricity” are arbitrary choices. We leave detailed exploration of formation rates of Gaia NS1-like binaries to future work.



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aasjournal





 Paper Title: Magnetically Arrested Circumbinary Accretion Flows.


UTF8gbsn






0000-0002-0491-1210]Elias R. Most
emost@caltech.edu
TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
0000-0001-7167-6110]Hai-Yang Wang(王海洋)
haiyangw@caltech.edu
TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA





§ ABSTRACT 
    Binary systems with comparable masses and a surrounding accretion disk can accrete gas through spiral accretion streams penetrating the central cavity formed by tidal interactions.  Using three-dimensional Newtonian magnetohydrodynamics simulations, we investigate the possibility of a magnetically arrested accretion flow through the cavity.  Rather than solely continuously feeding the binary through spiral accretion streams, the accretion is regulated by the strong magnetic field inside the cavity.  Transport of mass and angular momentum onto the binary then proceeds largely periodically in magnetic flux eruption episodes.  The ejected flux tubes carry angular momentum outwards and away from the binary, inject hot plasma into the disk and can launch flares.  This likely intermittent scenario could have potential implications for the emission signatures of supermassive black hole binaries, and potentially shed light onto the role magnetic fields play in the binary's orbital evolution.








§ INTRODUCTION


Binary systems are a prime focus of study in astronomy and astrophysics,
including protoplanetary disks <cit.>, binary stars <cit.>, and
supermassive black hole binaries <cit.>.
Accretion onto the binary has the potential to power electromagnetic
transients and alter the dynamics of the orbit (see, e.g., <cit.> for a recent review).
In the case of supermassive black hole binaries, accretion onto the system
may help explain hardening the binary beyond the parsec separation scale
<cit.>. As such, understanding gas accretion onto a supermassive black
hole binary is of crucial importance for understanding the merging black
hole population seen with the Pulsar Timing Array (PTA) <cit.> and the Laser Interferometer Space Antenna (LISA)
<cit.>.

For binaries surrounded by an accretion disk, the time-varying
gravitational potential of the binary exerts a torque on the surrounding circumbinary disk, balancing viscous torques and opens up a cavity
at 2-3 times the binary separation (near the location of the inner Lindblad resonance <cit.>).
One of the main features of circumbinary accretion is that accretion proceeds mainly along two spiral arms penetrating the low-density cavity.
A main driver behind circumbinary disk research is to understand the amount of
angular momentum transferred onto the binary,and in particular whether the binary hardens or
widens due to gas accretion (see, e.g.,  and references therein), as well as to unravel potential electromagnetic
counterparts (e.g., ).

Over the past decade, several numerical studies have clarified many of the hydrodynamical
aspects of circumbinary accretion, including the dependence on binary properties (mass ratio <cit.>, 
orbital eccentricity <cit.>) and the properties of the disk (aspect ratio <cit.>, viscosity <cit.>, cooling <cit.>, self-gravity of the gas <cit.>
). 
The stage when gravitational wave emission is non-negligible has also been studied by considering relativistic precession <cit.> and decoupling on the final orbits <cit.>. 

Due to the long integration times required to reach a quasi-steady state of angular momentum transfer between the binary and the disk <cit.>,
most simulations have been carried out in two-dimensions focusing on the orbital plane.
Few studies have investigated the hydrodynamial torque evolution in
full three dimensions <cit.>.

On black hole horizon scales several works
have also investigated the dynamics of the black hole-gas interaction in 
numerical or post-Newtonian relativity <cit.>, including jet launching and mini-disks <cit.>, which form around the black holes within
their respective Hill spheres <cit.>.
On arbitrary – and in the supermassive black context (sub-)parsec – scales
few simulations have been carried out focusing mainly on the outer disk, and
its magnetic torque evolution <cit.>. 
Qualitatively, the inner cavity was either excised or remained in a state resembling broadly previous hydrodynamical works.

On the other hand, magnetic fields in single accreting black holes have
been shown to cause drastic changes to the accretion flows <cit.>.
In particular, inferences from observations with the Event Horizon
Telescope indicate, that black holes such as M87* and Sgr A* <cit.> might
be in a magnetically arrested (MAD) state <cit.>. In contrast to
advection dominated standard and normal accretion (SANE) flows, magnetically arrested flows are halted
by a strong magnetosphere around the black hole, essentially establishing
a cavity stabilized by magnetic pressure, which the flow can only cross via interchange-like instabilities
pushing the field locally into the accretion disk <cit.>. 
Accretion then proceeds in episodic bursts, with the initial development of Rayleigh-Taylor fingers <cit.> ( see
also, e.g.,  for similar dynamics in T Tauri stars), before full magnetic flux tubes are ejected into the disk,
and cause an asymmetric burst of accretion onto the black hole <cit.>. This inflow of magnetic flux re-establishes a magnetically dominated zone, pushing out
the disk and resetting the magnetically arrested accretion cycle <cit.>.
Most importantly, angular momentum transport onto the black hole then mainly happens during the
flux eruptions, and itself is episodic, magnetically driven, and in parts outward-directed
<cit.>, and can spin down the black hole significantly <cit.>. At the same time, it has been suggested that magnetic flux eruptions
into the disk are mainly loaded with non-thermal electron-positron pairs and
can potentially power X-ray transient emission <cit.> with implications for X-ray flares observed in the galactic center <cit.>. 
It is important to point out that the picture painted here has been
mainly investigated for accreting black holes, although it is generic and has also
been found for accreting young proto-stars <cit.> and for low-mass neutron star X-ray
binaries <cit.> (see also ).


Motivated by recent progress in supermassive black hole binary formation
(e.g., ) and findings that large-scale accretion flows on parsec scales may well be very strongly magnetically dominated <cit.>, it seems possible that a strongly magnetized state inside the circumbinary disk cavity could form, with accretion properties similar to magnetically arrested accretion flows on black hole horizon scales.


In this work, using massively parallel GPU-enabled simulations, we
demonstrate that the picture of magnetically arrested accretion flows around black holes naturally carries over to circumbinary disks, albeit with modifications introduced by the strong tidal potential of the binary.
In particular, we show that the circumbinary disk cavity can be rapidly filled
with vertical magnetic flux, establishing a MAD-like accretion state.
We further identify Rayleigh-Taylor fingers, flares, and flux eruptions, which we
find to reshape the cavity size and transport most of the angular momentum between the binary inside
the cavity and the outer circumbinary disk.



§ METHODS

We study the dynamics of a circumbinary accretion disk
endowed with a strong magnetic field.  To this end, we model the binary as
Newtonian point masses on a fixed Keplerian orbit, and the dynamics of the gas
flow using ideal magnetohydrodynamics <cit.>. The fluid is described by a
mass density, ρ, velocity, v, internal energy density
e, pressure P, and magnetic field, B. 
The evolution of the system then follows <cit.>,

    ∂ρ/∂ t+∇·(ρv)    =s_ρ , 
    ∂(ρv)/∂ t+∇·(ρvv + P̅𝕀 - BB)   =s_p-ρ∇Φ , 
    ∂ E/∂ t+∇·[(E+P̅) v - (B·v) B]   =s_E-ρv·∇Φ ,
    ∂B/∂ t-∇× [ v×B ]   =0 ,

where we have defined the conserved energy, E, and effective pressure
P̅,

    E = e + 1/2ρ v^2 + 1/2B^2 ,
    P̅ = P + 1/2B^2 .


The gravitational potential of the binary is modeled as 

    Φ = 
      - GM_1/(|r-r_1|^2 + ϵ^2)^1/2
      - GM_2/(|r-r_2|^2 + ϵ^2)^1/2 ,

where M_i and r_i are the location and mass of the binary
constituents respectively, G is the gravitational constant, r
is the radial coordinate to the center of mass of the binary, and ϵ is a small number. We further adopt a unit convention of G M=a=1 in our simulations, and focus in an equal mass system, M_1=M_2.

Since we model the binary dynamics on scales of the orbital separation,
a, which are much larger than the physical extent of the binary
constituents (e.g., the horizon scale in the case of supermassive black hole binaries), we require an effective boundary/sink prescription.
While the type of sink can impact the formation of mini-disks and their
dynamics (see, e.g., <cit.> for a detailed discussion), we here mainly focus on the evolution of the cavity, and adopt a
simple prescription removing matter at the rate of the local Keplerian rate,
 s_p = - Ω_K,i ρ(v - v_i), 
 s_ρ = - Ω_K,i ρ,  
 within a sink radius
 of r_ sink= 0.07 a, where Ω_K,i =
 (GM_i)^1/2/(|r-r_i|^2 +ϵ^2)^3/2
is the Keplerian frequency of each sink, 
ϵ=0.05a is a small number acting as Plummer gravitational softening, and v_i the velocity of the sink
on a fixed Keplerian orbit.
The form of s_E follows from the density and momentum sink using the evolution equations.
 We further model the system as being locally isothermal,
 with the temperature being fixed to T= c_s^2 = |Φ|/ℳ^2,
 where c_s^2 is the speed of sound and ℳ=10 is the Mach number (or having disk aspect ratio h=1/ℳ=0.1).
 We initialize the magnetic field as purely poloidal by adopting a vector potential
 A_ϕ = R^2 A_0 max((ρ -
 0.04ρ_max)^2, 0), where R is the cylindrical radius, and A_0 is chosen to ensure an initial disk
 magnetization parameter β = 2 P /B^2 = 20. We further adopt a numerical floor value of β_ floor = 10^-3 . 
 


 We numerically integrate the above system using the public GPU-enabled
 version, [<https://gitlab.com/theias/hpc/jmstone/athena-parthenon/athenak>], of the  code <cit.> based on the  library
 <cit.>. 
 In this work, we adopt
  to solve the Newtonian MHD system using piecewise parabolic reconstruction <cit.>, an HLLD Riemann solver <cit.>, and a
 constraint transport algorithm <cit.> for the divergence-free magnetic field evolution.

 To perform the simulation, we adopt a Cartesian grid with 6 levels of
 static mesh refinement. The outer boundary of the domain is located at ± 80 a, and we adopt a finest resolution of Δ x ≃ 0.0065 a. We then numerically integrate
 the system on 1,260 V100 GPUs on the DOE OLCF Summit system for 190 binary orbits. 
  Overall, the cost of the simulation presented here is about 110,000 V100 GPU-hours.












§ RESULTS

We present our results on magnetically arrested circumbinary disk accretion in the following way.
First, we discuss the initial build-up of magnetization inside the cavity and its transition to a magnetically arrested state. We then describe the accretion cycle by means of magnetic flux eruptions, before concluding with a discussion on angular momentum transport.




 §.§ Evolution towards a magnetically arrested state

We begin by detailing the overall evolution of the structures inside the cavity, which is shown in Fig. <ref>. 
Initially, the cavity is empty and begins to fill rapidly through tidal accretion streams, as can be seen after eight orbital periods. This phase is not unlike the early onset of two-dimensional circumbinary accretion (e.g., ), or recent three-dimensional simulations with an outer accretion flow <cit.>. In this stage, rapid mini-disk formation sets in, with the outer disk subsequently becoming unstable to the magnetorotational instability (MRI) <cit.> and turbulent. The magnetization, β = 2 P / B^2 ≫ 1, in most parts of the cavity including inside the mini-disks. Overall, despite the more complex effective spatially varying viscosity (see <cit.> for a detailed discussion), this stage of our evolution can be characterized as mainly hydrodynamical.

As the evolution proceeds, accretion from the circumbinary disk advects substantial amounts of magnetic flux into the cavity, leading to a build-up of net magnetic flux. We can see that already after 51 orbits, some parts of the cavity are entirely magnetically dominated, with β≪ 1. While the tidal accretion streams are still predominantly present, the mini-disks start to exhibit turbulent features, with their magnetization level approaching equipartition values, β≃ 1. The outer disk has become strongly turbulent at the inner cavity wall, with strong magnetically driven outflows present above the sink cells. These jet-like features are consistent with strong net vertical flux inside the cavity.
In this transitional state, the flow is still largely hydrodynamical.



After about 100 orbits, however, the flow structure inside the cavity has drastically changed. The continued inflow of magnetic flux into the cavity has lead to a regime that is largely magnetically dominated, with β≃ 10^-3 almost everywhere inside the cavity. Rather than having a turbulent accretion flow at the inner boundary wall, we can see a very clear and sharp delineation between the cavity and the accretion flow in the meridional plane. This state is reminiscent of a magnetically arrested accretion flow onto a single black hole <cit.>. There, the magnetic field pushes out the disk, with mass accretion in the quiescent state being confined to a thin current sheet in the equatorial plane of the black holes <cit.>. Different from hydrodynamical simulations in which the radial location of the cavity is fully determined by the viscous and gravitational torque balance, here the joint torque from the magnetic field and gravitational interaction balances the torque from MRI turbulence in the circumbinary disk, causing a smaller cavity size.
We observe the presence of a similar equatorial current sheet in our simulations (Fig. <ref>, top row). 



 §.§ Magnetic flux eruption cycle 


One characteristic feature of magnetically arrested accretion flows around single black holes is an intermittent phenomenon regarding the exchange of mass and angular momentum <cit.>.
The system features two accretion states: a quiescent state and an eruption state. We show this cycle for our circumbinary disk simulation in Fig. <ref>.



In the quiescent state, strong magnetic fields prevent the cavity from accreting mass except through tidal accretion streams. 
However, our simulations indicate that the mass and angular momentum carried by these accretion streams might be insufficient to provide for permanent mini-disk formation.
We caution that the size of the mini-disks (if any during this accretion phase) that we find in our simulation is subject to resolution effects and flux accumulation inside the sink, which likely will not influence the cavity wall dynamics we investigate here. 

At the end of the quiescent state, mass accumulates at the inner cavity wall on a viscous timescale of the outer accretion flow. 
Eventually the growth of an interchange instability <cit.> leads to the inward accretion of Rayleigh-Taylor fingers into the cavity, which does not coincide with the tidal accretion streams.
This appearance of Rayleigh-Taylor fingers is not unlike accretion states that have been found for T Tauri stars <cit.>, low mass X-ray binaries <cit.>, and magnetically arrested black holes <cit.>. 
These flux eruptions lead to outward-moving vertical magnetic flux tubes that get ejected into the accretion disk (Fig. <ref>, left and center column). In analogy with the accretion cycle for magnetically arrested black hole flows <cit.>, these low-density, high magnetic pressure flux eruption regions are asymmetric and will lead to net torques on the system, as we will detail in Sec. <ref>. 
In this state, mass accretion is enhanced, leading to a transient formation of a mini-disk as magnetic pressure support on the outer cavity wall is released.
Concurrently, the rapid accretion of magnetic flux leads to reconnection events that can trigger the ejection of large plasmoids/flares (see Fig. <ref>, top center panel). These have also been found in magnetically arrested   black hole disks, and have been suggested as an explanation for X-ray flares in the galactic center <cit.>.





After the magnetic flux tubes got ejected into the disk and sufficient magnetic flux has been re-accreted into the cavity, the cavity restores its magnetically dominated state, resetting the cycle and suppressing Rayleigh-Taylor instabilities, at least intermittently. Meanwhile, the ejected flux tubes get sheared and mixed into the disk (Fig. <ref>, right column).


We can also quantitatively study the dynamics of the cavity. To this end, we compute the mass accretion rate, Ṁ, into the cavity, which we approximate by a sphere around the binary's center of mass with radius r≃ 0.6 a. 
The cavity we observe is generally smaller than in hydrodynamic simulations <cit.>, although we caution that our accretion state and cavity size may not yet be fully converged.
As shown in Fig. <ref>, the magnetically arrested flow leads to periodic mass accretion patterns, which vary by a factor of 3-4, when normalized to the mass accretion rate, Ṁ_ out, at larger distances (r≃ 3.5 a).
Indeed, we can spot clear accretion cycles matching the flux eruptions with a period of 3-5 orbits as seen in Fig. <ref>.

In the black hole accretion community, commonly the dimensionless flux
ϕ_B = Φ_B / √(|Ṁ|) is used to characterize a magnetically
arrested state, where Φ_B is the magnetic flux computed over the horizon <cit.>. In a magnetically arrested state, a large value of ϕ_B ≫ 1 implies that energy in excess of the accreted mass energy is extracted from the black hole <cit.>.  
Since the Newtonian sinks in our simulation cannot act like actual horizons, this diagnostic cannot be used. 
Even more so, the varying cavity size and presence of tidal accretion streams complicates a simple analysis, despite the obvious features of a magnetically arrested state, including copious flux eruptions. Instead, we opt to compare the net flux of electromagnetic energy Ė_ EM into the cavity (Fig. <ref>). Correlating this with the mass accretion rate, we find that spikes in the electromagnetic energy flux usually precede eruption events, consistent with the necessary requirement of replenishing the magnetic flux inside the cavity after a successful eruption.




 §.§ Angular momentum transport via flux eruptions


One important aspect of circumbinary disk accretion is the question of angular momentum transport, or conversely, whether interactions with the gaseous disk make the binary orbit spin up or shrink for equal mass systems <cit.>.

Magnetically arrested accretion flows can feature yet another angular momentum transport channel. As has been shown by <cit.>, flux eruptions themselves are able to transport angular momentum outwards, potentially leading to a spin down of the system <cit.>.
This process is intrinsically intermittent and quasi-periodic, mainly influenced by the magnetic accretion cycle.
As we will show in the following, this picture seems to largely carry over to magnetically arrested accretion flows.
Conversely, the periodicity of angular momentum transport presented in hydrodynamical simulations is governed by tidal accretion streams and a slowly precessing eccentric cavity.

We quantify the amount of angular momentum transport by considering the transport equation for angular momentum in cylindrical coordinates, which can be obtained from the Euler equations (see the discussion of vertically integrated angular momentum transfer in <cit.>, and the three-dimensional version in <cit.>). The conserved angular momentum density, j = R^2 ρ v_ϕ, is then given by,

    ∂_t j + ∂_R ( j v_R - R^2 B_R B_ϕ) + ∂_ϕ( j - R B_ϕ B_ϕ + R P̅)
       + ∂_z (j v_z  - R^2 B_z B_ϕ) + R ρ ∂_ϕΦ = 0 ,

where R as the cylindrical radius, z as the height coordinate, and ϕ the azimuthal coordinate. 
 
We compute the effective angular momentum fluxes as integrals on cylindrical surfaces located at the binary's center of mass, i.e.,

    ∂_t <j>_z,ϕ + ∂_R ( <J̇_ adv+ J̇_ mag+ J̇_ grav>_z,ϕ) = 0 ,

where we have defined the <…>_z,ϕ as the z- and ϕ-integral over the cylinder, and 

    J̇_ adv   = R^2  ρ v_ϕ v_R ,
    J̇_ mag   = -R^2  B_R B_ϕ ,
    J̇_ grav   =  ∫_0^R  d r'  r' ρ ∂_ϕΦ ,

as the advective, magnetic and gravitational angular momentum flux, respectively. We further define the total angular momentum flux as J̇ = J̇_ adv + J̇_ mag + J̇_ grav. 
Following <cit.>, we define the sign of the angular momentum flux such that J̇ > 0 corresponds to an outward flux of angular momentum, whereas J̇ < 0 corresponds to an inward one.

Using this decomposition, we can analyze the torque carried by a flux eruption event (Fig. <ref>). Focusing on the same accretion cycle shown in Fig. <ref>, we can see that the torque inside the flux eruption is entirely magnetically dominated. In magnetically dominated regions of the erupted flux tube, magnetic angular momentum fluxes are up to 100-times larger than the advected hydrodynamical fluxes. Outside of the eruption region, and also inside the Rayleigh-Taylor fingers, the angular momentum fluxes are largely hydrodynamical. 
The ejected flux tube carries angular momentum into the disk, which will eventually drive a wind/outflow due to the strong vertical net flux. This behavior is consistent with flux eruptions around single black holes <cit.>.

We can further quantify the flux transport behavior throughout the evolution of the system over several eruption cycles. We show the resulting temporal evolution of the angular momentum flux in Fig. <ref>, focusing on the evolution after the first 100 orbits when the system has become magnetically arrested. We can see that consistent with the appearance of magnetically arrested accretion cycles (over several binary orbits) the system features periodic outbursts of radial angular momentum flux. This flux starts roughly at or slightly outside the stationary cavity radius, r≃ a, and extends outward to several binary distances. These fluxes are also outward-pointing. Indeed, when comparing the strength of the magnetically driven to the advected angular momentum flux (Fig. <ref>, right panel), we find that these outbursts are entirely magnetically driven, consistent with the flux eruption picture (Fig. <ref>), and with a similar analysis in magnetically arrested black hole disks <cit.>.

It can be shown that for an equal mass non-eccentric accreting binary, such as the one consider here, the evolution of the orbital separation, a, should follow <cit.>,

    ȧ/a = 8 (ℓ_0 - 3/8) <Ṁ>/M ,

where ℓ_0 ≡<J̇>/(<Ṁ> a^2 Ω_b^2), and Ω_b is the Keplerian frequency of the binary. From this equation, we can see that for ℓ_0 < 3/8, the binary separation would shrink. Hydrodynamical simulations in two-dimensions imply ℓ_0 ≃ 0.68 <cit.>, whereas three-dimensional simulations find ℓ_0 = 0.55 <cit.>.
Both these values lead to an eventual growth of the binary orbit.

On the other hand, we find that during flux eruptions ℓ_0 can instantaneously drop below the threshold value, ℓ_0 < 3/8, or even become negative (Fig. <ref>, bottom panel). Correlating these patterns with those of the mass accretion rate, Ṁ, we find that this behavior correlates directly with the magnetically arrested accretion cycle. In the quiescent phase, accretion of angular momentum onto the binary appears to be hydrodynamical and approaches approximately the equilibrium value, ℓ_0≃ 0.68, found in two-dimensional studies.
During flux eruptions when transport is entirely magnetically driven, angular momentum can move outwards.

We caution, that while our findings qualitatively match results for single black hole accretion flows <cit.>, our overall flow is very likely not yet converged, and the cavity could still develop strong eccentricities <cit.>. A determination of which effect ultimately dominates total angular momentum transfer and budget in this system, will likely require substantially longer simulations than the ones presented here.




§ CONCLUSIONS


In this work, we have numerically investigated a magnetically arrested accretion state of a circumbinary disk.

Our simulations indicate, that just as for single accreting black holes, circumbinary accretion flows can become magnetically arrested. 
In this state, the inner cavity is filled with a strong magnetic flux regulating mass and angular momentum transport onto the binary <cit.>.
Magnetically arrested accretion states have been well studied in the black accretion literature <cit.>.
Different from these works, we find that the presence of tidal forces from the binary induces constantly present tidal streams, 
which, however, are periodically disrupted and reformed during the magnetically arrested accretion cycle.


Periodic magnetic flux eruptions through magnetic interchange instabilities <cit.>, will then enhance mass transport inwards, and transiently drive angular momentum outwards. 
At the same time, this process triggers reconnection and the ejection of plasmoids and flares, akin to models for X-ray flares in the galactic center <cit.>. The ejection of flux tubes into the disk is likely also accompanied with potential non-thermal emission <cit.>, which we plan to investigate in a follow-up work.
The mixing of the ejected flux tubes with strong vertical net flux with the disk may also affect wind launching from the system <cit.>.

Most importantly for our context, magnetically arrested accretion flows around black holes have been shown to transport angular momentum outwards in flux eruption episodes <cit.>, leading to a spin-down of the black hole <cit.>. We similarly find that flux eruption episodes carry angular momentum outwards, which instantaneously can satisfy the criterion for the binary orbit to shrink. 
In this sense, magnetically arrested flows may offer a way to provide the necessary angular momentum loss for the binary's orbit to shrink. However, we caution that our results are very likely not yet in a converged regime, with longer integration times being necessary to render a verdict on the actual net angular momentum transport rate.

While we find the period of the magnetically arrested circumbinary accretion cycle to be between 3-5 orbits, the precise duration will likely depend on the amount of magnetic flux inside the accretion flow. Here we model the initial accretion disk as having largely poloidal flux, enabling the disk to easily replenish the amount of magnetic flux inside the cavity. 
Recent works on large-scale feeding onto supermassive black holes
indicate that the magnetic field at large distances may be toroidal, with the disk being largely magnetically and radiation dominated (e.g., ; see also ).
When accreting such magnetic fluxes, further dynamo action either inside the circumbinary or mini-disks may likely be necessary to produce the required net vertical fluxes for the cavity to make the flux eruption happen (e.g., ).
This will potentially prolong the timescale between magnetically arrested cycles, likely making this feature intermittent. 
The cavity structure and mini-disk formation could also be affected by changes in the magnetic field topology or large-scale fluxes.
Thermal effects in such disks may also naturally alter the circumbinary accretion state <cit.>.
We plan to study the role of large scale magnetic fluxes in a future work.

Besides uncertainties in the (effectively initial) magnetic flux distribution, the development of an eccentric cavity <cit.> may likely alter the onset of the interchange instability and the periodicity of the flux eruption.
Quantifying this effect will require simulations such as the one presented here, over likely 10-times longer timescales, which is beyond the scope of this initial work, but well within in reach of modern GPU-based simulation approaches. 



§ ACKNOWLEDGMENTS

The authors are grateful for insightful discussions with X. Bai, A. Dittmann, P.F. Hopkins, E.S. Phinney, and B. Ripperda.
The simulations were performed on DOE OLCF Summit under allocation AST198. 
Additional simulations were done on DOE NERSC supercomputer Perlmutter under grant m4575.
HYW thanks for the support from B. Thomas Soifer Chair's Graduate Fellowship.
Part of this project was completed during the second LISA Spring meeting at Caltech. The meeting was supported by the Jet Propulsion Laboratory Astronomy and Physics Directorate.
This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP).

AthenaK <cit.>,
      Kokkos <cit.>,
	  matplotlib <cit.>,
	  numpy <cit.>,
	  scipy <cit.>












 Paper Title: X-ray Emission of Nearby Low-mass and Sun-like Stars with Directly Imageable Habitable Zones.



X-ray Emission of Nearby Stars]X-ray Emission of Nearby Low-mass and Sun-like Stars with Directly Imageable Habitable Zones


Breanna A. Binder
babinder@cpp.edu

0000-0002-4955-0471]Breanna A. Binder
Department of Physics and Astronomy, California State Polytechnic University, Pomona, CA, USA

0000-0002-1046-025X]Sarah Peacock
University of Maryland, Baltimore County, Baltimore, MD, 21250, USA
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0002-2949-2163]Edward W. Schwieterman
Department of Earth and Planetary Sciences, University of California, Riverside, CA, USA
Blue Marble Space Institute of Science, Seattle, WA, USA

0000-0002-0569-1643]Margaret C. Turnbull
SETI Institute, Carl Sagan Center for the Study of Life in the Universe, Off-Site: Madison, WI 53713, USA


Department of Physics and Astronomy, California State Polytechnic University, Pomona, CA, USA

0000-0002-7084-0529]Stephen R. Kane
Department of Earth and Planetary Sciences, University of California, Riverside, CA, USA

0000-0003-0944-2334]Alison Farrish
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0001-6398-8755]Katherine Garcia-Sage
Heliophysics Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA


§ ABSTRACT 
Stellar X-ray and UV radiation can significantly affect the survival, composition, and long-term evolution of the atmospheres of planets in or near their host star's habitable zone (HZ). Especially interesting are planetary systems in the solar neighborhood that may host temperate and potentially habitable surface conditions, which may be analyzed by future ground and space-based direct-imaging surveys for signatures of habitability and life. To advance our understanding of the radiation environment in these systems, we leverage ∼3 Msec of  and  observations in order to measure three fundamental stellar properties at X-ray energies for 57 nearby FGKM stellar systems: the shape of the stellar X-ray spectrum, the luminosity, and the timescales over which the stars vary (e.g., due to flares). These systems possess HZs that will be directly imageable to next-generation telescopes such as the Habitable Worlds Observatory and ground-based Extremely Large Telescopes (ELTs). We identify 29 stellar systems with / ratios similar to (or less than) that of the Sun; any potential planets in the habitable zones of these stars therefore reside in present day X-ray radiation environments similar to (or less hostile than) modern Earth, though a broader set of these targets could host habitable planets. An additional 19 stellar systems have been observed with the  X-ray Telescope; in total, only ∼30% of potential direct imaging target stars has been observed with , , or . The data products from this work (X-ray light curves and spectra) are available via a public Zenodo repository (doi: 10.5281/zenodo.11490574).








§ INTRODUCTION
 

In the coming decades, ground- and space-based direct imaging will afford the best potential for detecting and spectrally characterizing terrestrial planets in the habitable zones of our nearest neighbors <cit.>. While the James Webb Space Telescope (JWST) has and will continue to characterize some rocky exoplanets <cit.>, its ability to spectrally examine temperate terrestrial—and therefore potentially habitable—exoplanets is limited primarily to target systems with transiting planets and late M dwarf host stars. This is due to the lack of coronagraphic instrumentation on JWST to image rocky planets around more massive, Sun-like stars, and because the characterization of terrestrial planets through transit observations is most favorable for M dwarf hosts, which have a combination of favorable transit probabilities, planet-star size ratios, and planet-star separations that allow for the probing of low atmospheric altitudes when accounting for the effects of refraction <cit.>. Moreover, some key candidate biosignature molecules like O_2 cannot be detected directly at Earth-like abundances with JWST via transit spectroscopy even for the most favorable targets <cit.>, but could be more easily detected by reflected light observations of planets in nearby systems <cit.>.

The occurrence rates derived from Kepler indicate that terrestrial planets are likely present within the habitable zones (HZs) of a large fraction of nearby stars <cit.>. The Astro2020 Decadal Survey has identified the discovery and characterization of these worlds as the highest priority for NASA's investments <cit.>, and the Habitable Worlds Observatory[<https://habitableworldsobservatory.org/home>] will build on the LUVOIR and Habitable Exoplanets Observatory concepts <cit.> to realize this goal. Due to detection biases, the vast majority of nearby temperate rocky planets amenable for direct imaging have not yet been discovered, but will be revealed by HWO in its survey phase <cit.>, precursor science with extreme precision radial velocity (EPRV) measurements <cit.>, or direct imaging with upcoming 30-m class telescopes <cit.>.

A detailed understanding of the target stars themselves is a strong prerequisite for maximizing the science yield of exoplanet imaging missions, including extracting the statistical features of the overall planet population <cit.> as well as characterizing the climates, atmospheric composition and stability <cit.>, and potential biosignatures of individual planets <cit.>. Evaluating planetary habitability, specifically, must consider myriad factors beyond circumstellar location including the planet’s bulk composition <cit.> and stellar X-ray and extreme ultraviolet (XUV; 1-912 Å; corresponding to an energy range of 0.01-12 keV) emission <cit.>. Planets orbiting in or near their star’s HZ can experience high levels of XUV radiation, which directly influences atmospheric survival, composition, and long-term evolution <cit.>. In some cases, planetary atmospheres and surface volatiles can be entirely driven away by XUV irradiation <cit.>. Further, the atmospheric states of planets orbiting M dwarf stars may be profoundly different than for those orbiting Sun-like FGK stars due their to extended super-luminous pre-main sequence phase <cit.> during which XUV radiation could ablate away planetary atmospheres or lead to atmospheric oxidation via massive H escape <cit.>. Outgassing could replenish these atmospheres <cit.> though the retention and photochemistry of the atmosphere would continue to be controlled by the host star spectrum and activity <cit.>. 

While X-ray detections have been made for some HWO target stars <cit.> and more detailed studies of the X-ray properties of low-mass nearby stars have been performed <cit.>, no comprehensive and uniform X-ray analysis of potential direct imaging target stars has yet been done. In this paper, we present a comprehensive analysis of archival X-ray observations for all FGKM stellar systems with HZs that will be directly-imageable with upcoming telescopes such as HWO and ground-based Extremely Large Telescopes (ELTs). A follow-up paper yielding panchromatic XUV–IR spectra (including modeled EUV wavelengths) for each one of these systems is forthcoming (Peacock et al., in prep). We describe our sample and the archival data in Sections <ref> and <ref>. Our analysis procedure is detailed in Section <ref>. We extract light curves for stars detected at high significance and identify periods of count rate variability (such as flaring events or epochs of elevated count rates). We then extract X-ray spectra for stars, separating variable- from non-variable epochs, and fit the resulting spectra with one-, two-, or three-component thermal plasma models. Finally, the average best-fit spectral models are used to convert observed count rates (or count rate upper limits, in the case of non-X-ray-detected stars) to luminosities. In Section <ref>, we discuss the evolution of the / ratio with effective temperature and stellar age, and present an extensive appendix describing the X-ray observations and a summary of the known stellar physical parameters and exoplanet systems from the literature for each stellar system in our sample.



§ THE SAMPLE


To identify the highest priority targets for “deep dive” investigations of nearby stars and their planetary systems, we have used the predicted imaging and spectroscopy capability of (1) NASA’s next generation space-based Great Observatories such as HWO, and (2) ground-based ELTs. To first order, the performance of these facilities is described in terms of two specifications: the inner working angle (IWA) — the smallest planet-star angular separation that can be imaged (expressed in milliarcseconds, or mas), and the minimum planet to star flux ratio (or fractional planet brightness, FPB) — where starlight suppression technology and postprocessing algorithms can separate the planet signal from stellar photons and systematic noise. Intriguingly (and somewhat vexingly), for HZ planets the IWA requirement favors stars that are more luminous and therefore have wider HZs, while the limiting flux ratio favors target stars that are less luminous and therefore outshine their HZ planets by a smaller factor <cit.>. A third specification, planet limiting magnitude (V_ lim), determines which of these targets planets would be bright enough to obtain spectra in a reasonable amount of observing time – and this requirement simply favors the nearest stars.

To form our preliminary target list, for space-based missions we chose stars for which the HZ (0.95 - 1.7 AU scaled by √(L_ bol)) falls at least partially outside an inner working angle of 58 mas, and for which Earth-sized planets of albedo ∼0.3 would be brighter than V_ lim < 30, with a fractional planet brightness (FPB) > 4×10^-11. For ground-based observatories, we examined the 25 stars within 10 pc having the most favorable FPB whose HZs are at least partially visible outside an IWA of ∼30 mas, consistent with the literature <cit.>. This results in a well-defined list of 175 target stars whose HZs fall at least partially within the detection space of upcoming exoplanet direct imaging observatories. We compared our target list with the potential target stars for HWO <cit.>, who used slightly different cut-off requirements for V_ lim, FPB and IWA. Our list contains more M-dwarfs (more favorable targets for ELTs) while the <cit.> list includes more F stars (more favorable for space-based observations). <cit.> also excludes stars in close binaries or higher-order multiple systems, while we produce a more inclusive X-ray catalog of nearby stars, assuming technical challenges posed by binaries may eventually be surmounted. There are 229 unique stars when combining the two lists, which we summarize in Table <ref>. 

cccccc[!htbp]
Potential Target Stars for Future HZ Imaging Surveys


    
    HD     Common     Distance     Spectral     From     Potential 

    ID     Name        (pc)         Type       This Work?        HWO Targeta 
    
    166     V439 And         13.77     G8V           

    693     6 Cet 	       18.89    	F8V	   		    

    739     θ Scl     21.72    	F5V	   		    

    1326     GX And		    3.57    	M1.5V	   	     

    1581     ζ Tuc	    8.61    	F9.5V	   	    

    ⋮    ⋮    ⋮    ⋮    ⋮    ⋮

    ^aFrom <cit.>. Select entries are shown to illustrate the table form and content. The full machine-readable table (229 rows) is available online from the journal. 




Using this input list of 229 viable targets for direct imaging campaigns, we searched the  archives of the XMM-Newton Observatory and the Chandra X-ray Observatory for publicly available data. Many M stars (and some K stars) in this initial target list were previously studied as part of the Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary systems (MUSCLES) and Mega-MUSCLES Hubble Space Telescope Treasury programs, which characterized the X-ray/UV emission from K and M exoplanet host stars. We incorporate these results, presented in <cit.>, into our work here (see Section <ref>). We retrieve 192  and  observations for 57 stellar systems containing Sun-like and low-mass stars from the X-ray archives, amounting to ∼3 Msec of observing time (not including the MUSCLES/Mega-MUSCLES observations). An additional 19 stellar systems have been observed by . Figure <ref> shows a map of the 3D spatial distribution of these nearby stars and indicates which stars have X-ray observations available.

There are three M-dwarfs highlighted in <cit.> as strong potential HWO targets (Lacaille 8760, Lacaille 9352, and Lalande 21185). Both Lacaille 8760 and Lacaille 9352 were observed by  and are included in this work. A detailed study of the panchromatic spectrum and habitability potential of Lalande 21185 is in preparation (Schwieterman et al., in prep) and is not included in the sample here. Roughly half of the target stars contained in our study are F- and G-type stars. Table <ref> provides a summary of the stellar physical parameters for all stars included in our study, including the effective temperature (), spectral type <cit.>, stellar mass (M), bolometric luminosity (), and approximate age (in Gyr). All data were compiled using the NASA Exoplanet Archive[See <https://exoplanetarchive.ipac.caltech.edu/>]. Right ascension (R.A.), declination (Dec.), and distances were taken from Gaia DR3 <cit.>. More detailed discussion of each stellar system is provided in Appendix <ref>.







§ X-RAY DATA
 
We retrieved all publicly available imaging observations for our target stars from the [See <https://nxsa.esac.esa.int/nxsa-web/#home>] and [See <https://cda.harvard.edu/chaser/>] archives. Table <ref> summarizes the number of observations and the approximate exposure times utilized in this work. Below, we provide a more detail description of how the observations were processed and the methods used to extract light curves and spectra for each facility.



 §.§ XMM-Newton

We retrieved  observations from the  Science Archive and reprocessed the data from the raw  files using SAS 18[<https://www.cosmos.esa.int/web/xmm-newton/sas>]. The European Photon Imaging Camera (EPIC) on board  carries a set of three X-ray CCD arrays: one array uses pn-CCDs (referred to as PN) and two utilize metal oxide semi-conductor CCDs (referred to as MOS1 and MOS2). The PN data were reprocessed with  and the MOS data were reprocessed using  and filtered on an energy range of 0.2-15 keV. Due to the brightness of many of the stars, pile-up (when two or more photons fall on a pixel in less than the readout time) is a significant concern. The PATTERN parameter records the number and pattern of CCD pixels that are triggered for a given event; we use the recommended PATTERN=0 to mitigate pile-up effects[See the  ABC Guide at <https://heasarc.gsfc.nasa.gov/docs/xmm/sl/epic/image/sas_cl.html>]. Background light curves were extracted using  and inspected for background flares, and good time intervals (GTIs) were generated to reject observation times exhibiting strong background flaring events. 

We inspected the reprocessed, cleaned images for evidence of an X-ray source at the location of the target star. If the target star was detected at ≳500 counts in the PN image, we extracted source light curves (binned to 100 s) to search for count-rate variability. We used  to extract both PN and MOS1/2 spectra for stars with ≳2000 net counts,  to generate redistribution matrix files (RMFs), and  to generate ancillary response functions (ARFs) for all available observations. All spectra were binned to contain at least 25 counts per bin.

The majority (∼83%) of the observations analyzed in this work are from the  archive, containing ∼75% of the total exposure time. However, with a spatial resolution of ∼15^'', only stellar binaries with angular separations larger than ∼30^'' can be resolved with the PN camera. Figure <ref> (left panel) shows a 10 ks /PN image of the binary system GL 412, where the two components (and a third, unidentified X-ray source) are resolved.

ccccccccccc
Properties of Sample Stars

    
    
    
    Star     Alternate     R.A.     Dec.     Spectral     Distance     Mass     T_ eff          Age^*      
 3-4
    Name     Name(s)        2c(J2000)          Type^*       (pc)        ()     (K)          ()     (Gyr)       References
    
    η Crv     HIP 61174     12:32:03.75     -16:11:46.55     F2V     17.96±0.17     1.48^+0.29_-0.19     6854^+124_-99     4.62^+0.22_-0.25     1.5^+0.2_-0.4     1,2,50 

    ξ Oph     HIP 84893     17:21:00.68     -21:06:49.81     F2V     17.06±0.10     1.44^+0.30_-0.20     6756^+110_-127     4.13^+0.17_-0.18         1,2 

    υ And A     HD 9826     01:36:47.60     41:24:13.54     F8V     13.41±0.06     1.29±0.04     6157±112     3.35^+0.10_-0.14     3.12     1,2,29 

    ι Hor     HR 810, HD 17051          02:42:34.03     -50:47:57.54     F9V          17.32±0.02     1.17^+0.17_-0.15     6146^+106_-184     1.74^+0.04_-0.06     0.47; 2.72     1-4,9 

    ν Phe       HIP 5862           01:15:12.13     -45:31:51.14       F9V     15.17±0.04     1.15^+0.18_-0.14     6119^+130_-154     2.00±0.06     2.88     1,2,3,9 

    γ Pav     HIP 105858     21:26:26.81     -65:21:45.50       F9V     9.27±0.04     1.15^+0.18_-0.14     6109^+110_-176     1.47±0.05     1,7.25     1,3,44,45 

    β Vir     HIP 57757     11:50:42.51     01:45:48.67     F9V     11.12^+0.07_-0.06     1.13^+0.16_-0.14     6071^+120_-90     3.77^+0.11_-0.14     2.96     1,2 

    GJ 1095     HIP 35136     07:15:50.19     +47:14:20.89     F9V     16.86^+0.19_-0.20     1.12     5866     1.54±0.06     1, 7     1,2,10 

    LHS 237     GJ 288, HIP 37853     07:45:34.63     -34:09:53.86     F9V     15.20±0.13      1.11     5744     1.51±0.06         1,2 

    ι Per     HIP 14632     03:09:06.10     +49:36:46.35     F9.5V     10.51±0.03     1.08^+0.18_-0.10     5938^+141_-169     2.23^+0.04_-0.10     4.20±0.48     1,2,9 

    β Hyi     HIP 2021     00:25:55.80     -77:15:10.06     G0V     7.47±0.03     1.14^+0.01_-0.02     5839     3.70±0.05     6.32     1,2,9 

    β Com     HIP 64394                 13:11:51.43     +27:52:55.58     G0V          9.18±0.03     1.10^+0.14_-0.13     5969^+188_-100     1.41^+0.05_-0.04      <1.12, 1.7     1,2,9 

    LHS 208         HD 39091, π Men       05:37:11.89     -80:27:52.05     G0V          18.27±0.02     1.094±0.039     6037±45     1.44±0.02     2.98^+1.40_-1.30     5 

    ρ CrB     GJ 9537, HIP 78459      16:01:02.41     +33:18:00.28     G0V     17.47±0.02     1.05^+0.12_-0.15     5833^+141_-124     1.81^+0.06_-0.04     11.04     1,2,9 

    GL 672     HIP 84862     17:20:39.74     +32:27:47.22     G0V     14.54±0.02     1.05^+0.16_-0.10     5817^+120_-84     1.29^+0.04_-0.03     12.04     1,2,9 

    χ^1 Ori     HIP 27913, GJ 222        05:54:22.76     +20:16:33.07     G0V          8.66±0.08      1.028^+0.03_-0.028     5822     1.17±0.06     0.35; 4.32     1,2,6,9 

    GL 788     HIP 100017     20:17:32.60     +66:51:18.04     G0V     17.57±0.52     1.02±0.03     5886      1.18±0.10     3.76^+1.88_-1.92     2,9 

    47 UMa          GJ 407, HD 95128     10:59:27.53     +40:25:49.80     G0V          13.80±0.03     1.005±0.047              5829±95               1.58±0.04     6.48^+1.44_-1.04     9,29 

    GL 620.1A       HD 147513                 16:24:01.39     -39:11:34.68     G1V          12.903±0.02     1.06^+0.12_-0.14     5873^+110_-103     1.032^+0.003_-0.004     2; <1; 0.4     1-4,9,19 
 
    GL 311          π^1 UMa, HD 72905      08:39:11.64     +65:01:16.67     G1.5V        14.27^+0.14_-0.15     1.01     5920     1.03±0.05     0.2     2,6,8 

    α Cen A     GJ 559A, LHS 50        14:39:36.49     -60:50:02.37     G2V          1.347±0.003     1.06^+0.05_-0.04     5801     1.61±0.07     7.84^+1.08_-1.28     2,9 

    44 Boo A     ι Boo, HIP 73695 A     15:03:46.74     +47:39:15.89     G2V     12.8±0.2     1.04±0.1     5877     1.552     1.5     46,47,48 

    18 Sco          GJ 616, HIP 79672         16:15:37.52     -08:22:17.91     G2V          14.13±0.02     1.04^+0.12_-0.14     5791^+117_-93     1.09^+0.04_-0.02     5.84^+1.88_-1.96     1,9,10 

    51 Peg     HIP 113357, GJ 882     22:57:28.22     +20:46:08.78     G2V     15.46±0.03     1.03^+0.17_-0.09     5758^+102_-120     1.37^+0.02_-0.05     6.76     1,2 

    ζ^1 Ret     LHS 171, HIP 15330     03:17:49.26     -62:34:20.76     G2V     12.12±0.08     0.96^+0.03_-0.06     5699     0.80±0.05     1.56±0.44     2,9 

    HD 136352       GJ 582, ν^2 Lupi      15:21:45.55     -48:19:07.88     G2V          14.68±0.02     0.87±0.04        5664±61     1.04±0.06     12.3^+1.2_-2.9     11,12 
 
    GL 327     HIP 43726     08:54:17.50     -05:26:03.56     G3V     17.42^+0.44_-0.47     1.09±0.04     5790     1.09^+0.05_-0.04     1.32     2,9 

    μ Ara       HD 160691                 17:44:08.68     -51:50:05.65     G3IV/V       15.60±0.03      1.03^+0.15_-0.11     5772^+119_-84     1.90^+0.04_-0.06     5.7±0.6     1,2,7 

    κ^1 Cet     HD 20630               03:19:21.98     +03:22:14.22     G5V          9.14±0.03      1.02^+0.16_-0.10     5712^+161_-124     0.86^+0.02_-0.03     0.35; 0.7; 2.2     1,2,6,20,21 

    HD 140901      GL 599     15:47:28.54     -37:55:02.15     G7IV/V     15.25±0.01     1.06±0.05     5587^+153_-125     1.22±0.03     3.2±2.8     1,49 

    GJ 777A     LHS 3510, HD 190360     20:03:38.25     +29:53:40.08     G7IV/V     16.01±0.02     0.99±0.08     5552±11     1.20±0.06     13.4     1,2,9 

    ξ Boo A     37 Boo, HD 131156     14:51:23.53     +19:06:00.97     G7V     6.71±0.16     0.90±0.04     5551±21     0.55±0.04     <4     38 

    δ Pav     GJ 780                   20:08:46.81     -66:11:13.53     G8IV         6.09±0.01       0.99^+0.14_-0.10     5590^+156_-140     1.25^+0.02_-0.04     6.2     1,2
 
    LHS 2156     HIP 47080     09:35:38.55     +35:48:32.33     G8IV/V     11.20±0.02     0.96^+0.16_-0.09     5499^+156_-136     0.80^+0.02_-0.04         1,2 

    82 Eri          GJ 139, HD 20794          03:20:00.08     -43:03:59.59     G8V          6.003±0.009      0.94±0.12      5413^+91_-118     0.65±0.01     5.76; >12     1,2,9,13 
 
    τ Ceti     HD 10700, GJ 71       01:44:02.17     -15:56:01.25     G8.5V       3.603±0.007          0.92^+0.12_-0.10     5333^+124_-75     0.51±0.01     >12     1,2,9 

    44 Boo Ba         15:03:46.74     +47:39:15.89     K0V     12.8±0.2     0.98     5300     0.51         46,47,48 

    55 Cnc A     ρ^1 Cnc, HIP 43587     08:52:35.22     +28:19:47.22     K0IV/V     12.59±0.01     0.90^+0.14_-0.09     5252^+123_-172     0.64±0.02     9.5     1,2 

    70 Oph A        LHS 458, HIP 88601A       18:05:27.47     +02:29:42.81     K0V          5.15±0.03      0.89±0.02      5300±50     0.65±0.02     6.2±1.0     1,22 
 
    40 Eri A        GJ 166A, HD 26965         04:15:13.91     -07:40:05.08     K0.5V        5.04±0.01      0.85^+0.11_-0.10     5092^+140_-149     0.43±0.01     6.9±4.7     1,14,23 
 
    δ Eri     HIP 17378     03:43:14.80     -09:45:36.30     K1III/IV     9.04±0.07     1.19     5095     3.37^+0.07_-0.06     6.28     2 

    GL 451A         Groombridge 1830      11:53:04.16     +37:41:34.18     K1IV         9.16^+0.16_-0.17       0.66     4950     0.23±0.04     5.0±0.3     2,6 

    GL 117          HIP 13402, HD 17925        02:52:32.56     -12:46:14.00     K1.5V        10.355±0.005     0.89^+0.09_-0.13     5225^+91_-136     0.40±0.01     1.5;<1.2;0.1     1,2,6,9 
 
    α Cen B     GJ 59 B, LHS 51       14:39:35.06     -60:50:15.10     K2IV          1.347±0.03     0.87±0.07     5234±63     0.52±0.05     8, >11     2,9,30 

    GL 783 A         HD 191408A, LHS 486       20:11:12.54     -36:06:29.58     K2.5V        6.05±0.03      0.74     4922     0.28±0.10     7.05±0.65     1,2,6,25 
 
    GL 183          HD 32147, HIP 23311       05:00:49.59     -05:45:30.96     K3V          8.845±0.003      0.81^+0.11_-0.09     4931^+133_-134     0.29±0.01     2     1,3,6 
 
    GL 892     HD 219134     23:13:21.06     +57:10:10.80     K3V     6.531±0.004     0.79±0.03     4817±62     0.29±0.01     11.0±2.2,12.46     1,2,36 

    GJ 667 A        HIP 84709                  17:18:56.53     -34:59:25.40           K3V         6.97±0.83     0.59              4810              0.13^+0.17_-0.09         2,3,31 

    LHS 1875     GL 250, HIP 32984     06:52:17.47     -05:10:25.42     K3.5V     8.747±0.004     0.76^+0.09_-0.10     4719^+157_-129     0.22^+0.02_-0.01     3.1,<0.48     1,2,9 

    GL 570 A     HIP 73184      14:57:29.18     -21:25:23.31     K4V     5.881±0.003     0.75^+0.08_-0.10     4681^+165_-126     0.22^+0.01_-0.02     <0.6;3     1,2,9 

    ξ Boo B          14:51:23.19     +19:06:03.97    K4V     6.71±0.16     0.66±0.07     4350±150     0.10±0.02     <4     38 

    44 Boo Bb         15:03:46.74     +47:39:15.89     K4V     12.8±0.2     0.55     5035     0.24         46,47,48 

    GJ 667 B        WDS J17190-3459 B                   K4V                             31 

    70 Oph B        LHS 459       18:05:27.82     +02:29:39.12     K5V     5.15±0.03     0.73±0.01     4390±200     0.15±0.02     6.2±1.0     1,22 

    61 Cygni A       HIP 104214               21:06:59.64     +38:45:49.90     K5V          3.497±0.001       0.67^+0.08_-0.09      4304^+161_-128     0.15±0.01     6     1,2,26  
 
    61 Cygni B      HIP 104217                 21:07:00.88     +38:45:21.86     K7V          3.497±0.001       0.61^+0.08_-0.09     3949^+156_-137     0.104^+0.012_-0.009     6     1,2,26 
 
    GL 412A        HIP 54211                 11:05:22.09     +43:31:51.41     M1V          4.83±0.03       0.38     3688     0.020^+0.090_-0.014     3     2,15,28 

    GL 570 B     (HIP 73182)     14:57:27.70     -21:25:07.94     M1.5V     5.86±0.06     0.55±0.05     3345     0.13±0.09         1,2,37 

    GJ 832      LHS 3685, HD 204961       21:33:33.90     -49:00:45.47     M1.5V     4.964±0.001     0.44±0.02     3601     0.03^+0.09_-0.02     9.24     1,2,4 

    GJ 667 C                                    17:19:00.30     -34:59:52.06           M1.5V      7.244±0.005     0.33±0.02     3350±50     0.0137±0.0009     >2     31 

    Kapteyn's Star     GJ 191                 05:11:50.38     -45:02:37.73     M2V        3.9331±0.0004      0.28±0.02     3742±157     0.016±0.004     11.5^+0.5_-1.5     1,2,16 
 
    Wolf 1055     HD 180617, GL 752 A     19:16:55.26     +05:10:08.04     M2.5V     5.912±0.02     0.48±0.02     3534±51     0.032±0.001         40 

    AD Leo       GJ 388     10:19:35.70     19:52:11.30     M3V     4.964±0.001     0.423±0.012     3477±23     0.0236±0.0001     0.4±0.1     39 

    GL 570 C     HIP 73182     14:57:27.70     -21:25:07.94     M3V     5.86±0.06     0.39±0.03         0.018         37 

    Wolf 1061           GJ 628, LHS 419     16:30:17.96     -12:40:04.26     M3.5V     4.306±0.001     0.30±0.02     3309±157     0.011^+0.003_-0.002         1,2 

    Luyten's Star     GJ 273     07:27:25.11     +05:12:33.78     M3.5V     5.92±0.02     0.29±0.02     3382±49     0.0087^+0.0063_-0.0065         1,2,43 

    55 Cnc B           08:52:40.28     +28:18:54.91     M4.5V     12.48±0.02     0.26±0.02     3187±157     0.008±0.002         1,35 

    GL 783B         HD 191408B, LHS 487       20:11:12.80     -36:06:32.17     M4.5V     6.05±0.03     0.24         0.0009     7.05±0.65     25 

    GJ 777B     LHS 3509     20:03:37.57     +29:53:45.49    M4.5V     15.97±0.01     0.21±0.02     3099±157     0.0047±0.0012         1,34 

    40 Eri C        LHS 25          04:15:19.12     -07:40:15.25     M4.5V     5.04±0.01     0.204±0.006     3100     0.008      6.9±4.7     1,14,23,24 
 
    υ And B          01:36:50.16     41:23:25.98     M4.5V     13.47±0.02     0.19±0.02     3159±157     0.0044^+0.0012_-0.0011         29 
 
    Proxima Centauri     GJ 551               14:29:34.16     -62:40:33.87     M5.5V        1.3012±0.0003       0.13±0.02     2900±100     0.0016^+0.0004_-0.0006     4.85     1,17,18 

    GL 412B         WX UMa, LHS 39        11:05:24.50     +43:31:33.27     M6.6V     4.83±0.03     0.095     2700     0.00095     3     27,28 

    VB 10     GL 752 B     19:16:57.61     +05:09:01.59     M8V     5.912±0.02     0.0881^+0.0026_-0.0024     2508^+63_-60     0.000499±0.000004     1     41,42 

    1: <cit.>, 2: https://nexsci.caltech.edu/missions/EXEP/EXEPstarlist.htmlExoCat-1, <cit.>, 3: <cit.>, 4: <cit.>, 5: <cit.>, 6: <cit.>, 7: <cit.>, 8: <cit.>, 9: <cit.>, 10: <cit.>, 11: <cit.>, 12: <cit.>, 13: <cit.>, 14: <cit.>, 15: <cit.>, 16: <cit.>, 17: <cit.>, 18: <cit.>, 19: <cit.>, 20: <cit.>, 21: <cit.>, 22: <cit.>, 23: <cit.>, 24: <cit.>, 25: <cit.>, 26: <cit.>, 27: <cit.>, 28: <cit.>, 29: <cit.>, 30: <cit.>, 31: <cit.>, 32: <cit.>, 33: <cit.>, 34: <cit.>, 35: <cit.>, 36: <cit.>, 37: <cit.>, 38: <cit.>, 39: <cit.>, 40: <cit.>, 41: <cit.>, 42: <cit.>, 43: <cit.>, 44: <cit.>, 45: <cit.>, 46: <cit.>, 47: <cit.>, 48: <cit.>, 49: <cit.>, 50: <cit.> 

    ^*Significant disagreements in literature are discussed in Appendix <ref>.


cccccc[!htbp]
    Summary of X-ray Observations
    
    
    
                          2c        2c 
 2-3 5-6
    Star     # Obs.     Exp. Time (ks)         # Obs.     Exp. Time (ks) 
    
    
    η Crv       0            4     38.5 

    ξ Oph        0            1     19.8 

    υ And     1     5.3           4     58.9 

    ι Hor      32     211.3          0     

    ν Phe        1     16.0            0     

    γ Pav     1     15.9            0     

    β Vir      1     41.1            0     

    GJ 1095          0            1     96.2  

    LHS 237          1      15.6           0     

    ι Per      0            1     4.9  
















































    ⋮    ⋮    ⋮       ⋮    ⋮

    Total           151     2235.7        41     815.0 

    Select entries are shown to illustrate the table form and content. The full machine-readable table (57 rows) is available online from the journal.







 §.§ Chandra

We retrieved  ACIS imaging observations from the  archive[<https://cda.harvard.edu/chaser/>] and reprocessed all  data using the CIAO v4.15 <cit.> task  and standard reduction procedures. We ran the point source detection algorithm  on each individual exposure to generate a list of X-ray source positions and error ellipses. The major and minor axes of these error ellipses were increased by a factor of five, and all X-ray sources were masked so that a background light curve could be extracted for each observation. We inspected these background light curves for background flaring events. We created good time intervals (GTIs) using  and filtered all event data on these GTIs. Finally, we restricted the energy range of our resulting “clean”  files to 0.5-7 keV.



We re-ran  on the cleaned image to identify significant X-ray sources and obtain first-pass estimates of their source properties. Due to its exquisite spatial resolution (0.496^'' pixel size) and a typical on-axis PSF size less than 1^'' for the ACIS instruments,  is the instrument of choice for resolving crowded fields. For stellar systems with multiple components and angular separations of less than a few arcseconds,  is currently the only telescope that can be used to definitively associate X-ray emission with a given stellar component. Figure <ref> (right panel) illustrates the spatial resolving power of  with a 5 ks exposure of the triple star system 40 Eri, where the brightest X-ray source is clearly associated with the position of 40 Eri C, a fainter X-ray source is found at the location of 40 Eri A, and 40 Eri B is undetected.

If the target star was detected in the cleaned  image with ≳50 net counts, we extracted source light curves (binned to 100 s) to search for evidence of X-ray variability. For sources with ≳500 net counts, the CIAO tool  was used to extract spectra and generate RMFs and ARFs. All spectra were binned to contain at least 10 counts per bin. 



 §.§ Swift

The X-Ray Telescope (XRT) onboard the Neil Gehrels  Observatory additionally observed some nearby stars.  is optimized to quickly slew to observe transient events, and typically obtains short “snapshot” X-ray observations (∼2 ks or less) with XRT of relatively bright X-ray sources. These observations are useful for providing flux estimates (or flux upper limits) for stars of interest, but often cannot provide the longer-duration light curves or higher signal-to-noise spectra that can be obtained from  or . We therefore use the High Energy Astrophysics Science Archive Research Center (HEASARC)[<https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/swift.pl>] to determine which stars were observed by  but not  or , and we use the  data products generator[<https://www.swift.ac.uk/user_objects/index.php>] to create stacked images of these additional stars and measure count rates (or count rate upper limits). The 19 stars that were observed only by /XRT, but not by  or , are not considered part of our primary stellar sample, but we still provide updated provide X-ray luminosity estimates in Appendix <ref>.




§ ANALYSIS PROCEDURE

We first inspected all X-ray observations for evidence of X-ray emission at the location of the target stars. For stars that were detected, we first extract light curves (binned to 100 s) to search for rapid variability within the observation. For brighter stars (i.e., those with ≳2000 net counts with  or ≳500 net counts with ), changes in observed count rate are readily apparent by visual inspection; for fainter stars, a more quantitative assessment of variability is needed in addition to visual inspection. 



 §.§ Assessing Count Rate Variability

We use the Anderson-Darling test, which is a non-parametric statistical test that is more sensitive to short duration variations than the more commonly used Kolmogorov-Smirnov (KS) test <cit.>. The Anderson-Darling statistic, A^2, is computed as <cit.>:


    A^2 = -N-1/N∑_i=1^N [(2i-1) ln(t_i)+(2N+1-2i)ln(1-t_i)],


where N is the number of photons observed and t_i is the arrival time of the i-th bin, scaled to the interval (0,1). We use 's  routine to compute A^2 and the critical values for the 5% and 1% significance levels when compared to a normal distribution. An A^2 value that is larger than a critical value for a given significance level is evidence of variability; the typical 5% critical value is ∼0.75 and the typical 1% critical value is ∼1.05 for the observations considered here. We additionally compare the observed light curve to a constant count rate and compute a standard reduced χ^2_r (χ^2 divided by the degrees of freedom) statistic.

Variability in observed count rate is, to first order, driven by a change in X-ray flux incident on the detector. In our case, this change in flux is due to changes in the intrinsic X-ray luminosity of the target star. However, changes in the underlying spectra of the star can additionally impart more subtle changes on the light curve, as the sensitivity of  and  to X-ray photons is energy-dependent. We therefore “flag” segments of each light curve according to the type of variability observed, hereafter referred to as the VarFlag, so that the spectral properties can be measured and an accurate count rate-to-luminosity conversion factor can be established as a function of time during the observation. The VarFlags are defined as follows:


    
  * Q: quiescent periods, when the count rate is low and constant within the uncertainties
    
  * F: strong flaring events, where the count rate is above ∼80% the quiescent maximum count rate
    
  * D: decaying periods following flares or descending count rates that follow highly elevated count rates; these periods are roughly defined as immediately following an obvious flare where the count rate is below 80% the maximum flare count rate but still above ∼150% of the quiescent count rate
    
  * E: elevated count rate periods, when count rates are approximately 50% higher than the quiescent count rate but not obviously associated with a strong flaring event
    
  * R: rising count rate periods that precede a strong flare


Dividing a single observation into a series of sub-exposures is a subjective process: we wish to define sub-exposures that are short enough to capture unique variations seen in the light curve (such as rapid flaring events, some lasting only 200-300 s) but long enough so that a sufficient number of X-ray counts are available for adequate spectral modeling (the ∼2000 and ∼500 count limits for  and , respectively). Our main goal is to accurately measure the X-ray luminosity of the star as a function of time.

Figure <ref> shows an example light curve of Proxima Centauri ( observation number 0801880501) and a cumulative distribution function (CDF) of photon arrival times compared to a constant count rate. The A^2 statistic, critical values, and χ^2_r are shown, and the light curve has been color-coded according to the VarFlag assigned to each segment of the light curve. Table <ref> summarizes the variability metrics for every X-ray observation of every star available in the archives (the full machine-readable table is available online from the journal). While some stars in our sample have been the targets of extensive observing campaigns by  and/or  for many years, observational data is quite sparse for other stars. For stars with many observations, long-term periodic X-ray variability has been detected and is likely the result of coronal activity cycles <cit.>. Similar long-term X-ray variability cycles may exist for sparsely observed stars in our sample, but will not be detectable in single snapshot observations. Additional observations are required to constrain the long-term coronal activity cycle and/or flaring activity for many stars in the present sample.



clcccc
    X-ray Variability Metrics
    
    
                              Exp.     Net         

     Star       Observation ID     Time (ks)      Counts     χ^2_r     A^2 
    
    
    





    ι Hor        XMM/0673610201     5.3      4230±70     1.5     0.49  
































    β Vir       XMM/0044740201     41.1     18450±160     3.1     0.73 

    υ And A     XMM/0722030101     5.3     940±40     2.1     0.36 








    44 Boo           XMM/0100650101     18.8     287110±550     3.1     0.64 

    GL 620.1A        XMM/0822070201     7.8     12220±110     1.9     0.41  



































    κ^1 Cet     Chandra/22327      14.0     6600±80     1.7     3.26 
 






































































































    Wolf 1061        Chandra/20163      38.3     250±40     0.9     29.78 


 







    Prox Cen         XMM/0801880501     20.0     66790±260      35.2     17.55  



    GL 412B           XMM/0742230101     10.3     32030±210     243.1     3.43 

    VB 10        XMM/0504010101     24.2     660±110     1.1     11.01 

    ⋮    ⋮    ⋮    ⋮    ⋮    ⋮

    Select entries are shown to illustrate the table form and content. The full machine-readable table (200 rows) is available online from the journal. Net counts are reported for 0.2-15 keV for  observations and 0.5-7 keV for  observations.






 §.§ Spectral Modeling

All X-ray spectral modeling was performed with XSPEC v12.11.1 <cit.> using standard χ^2 statistics. Depending on the number of counts and the overall complexity of the spectra, we used one-, two-, or three-temperature apec models to parameterize the X-ray spectra. Since all of the stars are located very close to the Sun (within ∼20 pc), the intervening hydrogen column density is expected to be low; we adopt a fixed absorbing column N_ H (using the tbabs model) of 10^19 cm^-2 for all stars <cit.>. We assume abundances from <cit.> for both the plasma emission and interstellar absorption models. The plasma temperature and normalizations were left as free parameters during the fit. The normalization of each apec component is defined as (neglecting cosmological terms):


    𝒩=10^-14/4π d^2∫ n_e n_ H dV,


where d is the distance to the source, n_e and n_ H are the electron and hydrogen column densities (in units of cm^-3), respectively, and dV is the volume element (in units of cm^3). The integral ∫ n_e n_ H dV is the volume emission measure (VEM). Once the best-fit model is obtained, we use the cflux convolution model to measure the flux on each plasma component. 

While the spectra of some stars required only one or two apec components to achieve a statistically acceptable fit, most spectra required a third apec component. We initialized all spectral models with the default (solar) abundance. For some stars (e.g., ι Hor, κ^1 Cet) the best fit model was insensitive to the exact choice of abundance; for these stars the abundance was kept fixed at the solar value. If a statistically acceptable fit was not obtained with solar abundances, we allowed the abundance to be a free parameter of the fit, subject to the constraint that the abundance be the same for all plasma components.

For stars that exhibited count rate variability, spectra were extracted and modeled for each sub-exposure. Table <ref> provides full best-fit parameters for all stars with enough counts to perform spectral modeling for all observations and sub-exposures (the full machine-readable table is available online from the journal). For stars with multiple observations or multiple sub-exposures with the same VarFlag, we present the average best-fit model for a given VarFlag in Table <ref>. For uniformity, we refer to the coolest component of a three-component model as “APEC #1," the hottest component as “APEC #3," and the intermediate-temperature component as “APEC #2." The two-temperature thermal plasma models have their individual components matched to the three-temperature component with the most similar temperature.  Figure <ref> shows an example of the averaged spectrum for ι Hor, which did not exhibit significant count rate variability in any of the publicly available 38  observations and was therefore assigned a VarFlag of Q (quiescent). The best-fit models for the individual exposures are shown in black, with the average best-fit spectrum superimposed in red.



cccccccccccccc
All Best-Fit Parameters from X-ray Spectral Modeling

    
    
                  2cAPEC #1        2cAPEC #2        2cAPEC #3 
 4-5 7-8 10-11
        Star     ObsID     VarFlag     kT_1     𝒩_1        kT_2     𝒩_2        kT_3     𝒩_3     logL_ X     abund     <χ^2/ dof> 

                   (keV)     (10^-4)        (keV)     (10^-4)         (keV)     (10^-4)     [erg s^-1]         
    
    
    η Crv     14474     Q     0.16^+0.07_-0.14     3.3^+10.0_-1.6        0.50±0.08     4.6^+5.0_-2.9                28.43±0.06     0.32^+0.56_-0.16     1.14 






































    β Vir     0044740201     Q     0.09±0.01     4.7^+1.2_-0.9        0.37±0.01     2.4±0.1                28.08±0.02     1 (fixed)     2.48 

    χ^1 Ori     0111500101/0-10ks     E     0.10±0.02     39.7^+55.7_-19.9                0.49±0.03     52.9^+10.5_-9.4                  0.84^+0.10_-0.09     12.1^+7.8_-4.8      28.88^+0.03_-0.04     0.31^+0.07_-0.05     1.31 




    β Com     0148680101/0-5ks     Q     0.07±0.02     23.5^+114.8_-15.1                0.39±0.02     3.4±0.3                          28.11±0.02     1 (fixed)     1.55 











    44 Boo     0100650101/0-6ks     D     0.25±0.02     23.8^+4.4_-6.6        0.50^+0.13_-0.12     9.8^+5.4_-3.6        0.91^+0.03_-0.02     275.3^+13.8_-15.7     28.67±0.01     0.09±0.01     1.45 




    GL 620.1A     0822070301     Q     0.11^+0.02_-0.01     9.5^+5.3_-3.6                0.46±0.03     1.4^+2.5_-2.3                  0.77^+0.05_-0.04     6.3^+1.9_-1.7      28.82±0.03     0.43^+0.09_-0.06     1.60 














































































































































































































































































































    ⋮    ⋮    ⋮    ⋮    ⋮        ⋮    ⋮       ⋮    ⋮    ⋮    ⋮    ⋮

    Select entries are shown to illustrate the table form and content. The full machine-readable table (360 rows) is available online from the journal.



Figure <ref> shows the VEM for each star (calculated from the best-fit normalization, defined in Equation <ref> above) as a function of temperature for the coolest APEC component (circles), the intermediate APEC component (squares), and the hottest APEC component (diamonds) using the measurements summarized in Table <ref>. The points are color-coded by stellar spectral type: F and G stars are shown in yellow, K stars in orange, and M stars in red. In general, the temperatures and VEMs of M stars exhibit the largest spread in coronal temperatures relative to more massive stars. The coronal temperatures of FG and K stars are similar for the coolest and hottest APEC components, although VEMs of FG stars are roughly an order of magnitude higher than those of K stars at similar temperatures. The VEMs do not significantly change within a given spectral type group, but the intermediate-temperature APEC component of FG stars is on average hotter (by ∼2 MK) than the intermediate APEC component of K stars.

The three-component thermal plasma model that generally describes the X-ray emission from F and G stars is described by plasma temperatures of ∼1.3 MK, 5.1 MK, and 9.4 MK, with corresponding (log) VEMs of 55.2 [cm^3], 55.4 [cm^3], and 55.0 [cm^3], respectively. For K stars, the temperatures become 1.2 MK, 3.3 MK, 10 MK, with (log) VEM values of 54.1 [cm^3], 53.9 [cm^3], and 53.6 [cm^3], respectively. For these hotter stars, periods of elevated count rates or flaring events different from the quiescent periods in that VEMs increased, while the plasma temperatures remained mostly constant. This result is in sharp contrast to the model fits for M stars, which showed more significant variations in both temperature and VEM during flaring events. 


ccccccccccccc
Average Best-Fit Parameters from X-ray Spectral Modeling

    
    
               2cAPEC #1        2cAPEC #2        2cAPEC #3 
 3-4 6-7 9-10
        Star     VarFlag     kT_1     𝒩_1        kT_2     𝒩_2        kT_3     𝒩_3     logL_ X     abund     <χ^2/ dof> 

                (keV)     (10^-4)        (keV)     (10^-4)         (keV)     (10^-4)     [erg s^-1]         
    
    
    η Crv          Q     0.20^+0.12_-0.09     2.3^+4.3_-1.1        0.55^+0.09_-0.07     3.1^+2.2_-1.7                28.41^+0.05_-0.10     0.27^+0.61_-0.18     1.31 

    υ And A     Q     0.29±0.03     0.8±0.1                           27.57±0.03     1 (fixed)     0.91 

    ι Hor         Q     0.09±0.03     7.7±1.1                            0.42±0.08     2.9±0.2                  0.71±0.14     1.6±0.2      28.75^+0.10_-0.14     1 (fixed)     1.27 

    β Vir     Q     0.09±0.01     4.7^+1.2_-0.9        0.37±0.01     2.4±0.1                28.08±0.02     1 (fixed)     2.48 

    χ^1 Ori        Q     0.08±0.03     258±228                            0.42±0.07     28.6±9.2                 0.69±0.06     21.3±8.2     28.94^+0.07_-0.10     0.20±0.12     1.55 

    ⋮    ⋮    ⋮    ⋮       ⋮    ⋮       ⋮    ⋮    ⋮    ⋮    ⋮








































































    Select entries are shown to illustrate the table form and content. The full machine-readable table (76 rows) is available online from the journal.









 §.§ X-ray Luminosity

In addition to insights about the temperatures and VEMs of coronal structures associated with our target stars, the best-fit X-ray spectral models provide us with a count rate-to-luminosity conversion factor that mitigates against instrumental effects from the extracted light curves and gives us a clearer view of the luminosity evolution of the target star. Furthermore, we can “unfold” our X-ray spectra (i.e., de-convolve the observed spectrum and the energy-dependent detector response function) to assess the intrinsic energy emission from the target star as a function of energy (or wavelength). 

Figure <ref> shows the continuum luminosity of each star (in units of ) as a function of wavelength during quiescent times. All spectra have been corrected for minor foreground absorption, and we use the XSPEC command  to extend our best-fit model to a wavelength range of ∼1-200 Å. For nearly all stars, flaring events are driven by (sometimes dramatic) enhancements in X-ray luminosity at wavelengths ≲10 Å (energies ≳1.2 keV). Figure <ref> shows quiescent and flaring spectra for three representative stars (κ^1 Cet, GJ 570A, and Prox Cen) from 1-1000 Å (top panel), as well as the ratio of the flaring and quiescent spectra (bottom panel). For both κ^1 Cet (G5V) and GJ 570A (K4V), the flux at ∼1 Å (∼12 keV) during flaring times is a factor of ∼30 larger than during quiescent times, but the fluxes at longer wavelengths (≳100 Å; energies ≲0.12 keV) are similar. In the case of Prox Cen (M5.5V), however, the flaring spectrum is brighter than the quiescent spectrum by at least an order of magnitude at all wavelengths, with short-wavelength (≲ 3 Å) emission enhanced by more than a factor of 10^3. This finding is unlikely to be the result of the reduced soft energy responses of  or , as the loss of effective area due to contamination build-up occurs most significantly at energies below 1 keV <cit.>. Nevertheless, we use detector response functions specific to the orbital cycle in which the individual observations were obtained to de-convolve the detector response as accurately as possible. Plots of individual stellar X-ray spectra, color-coded by VarFlag, are discussed in Appendix <ref>.





A question that our analysis can elucidate is: For what fraction of time is a star observed to be emitting at a given X-ray luminosity? While this is related to the flaring frequency and the typical peak luminosity and duration of flares, our goal in this work is not to directly characterize flaring events or the physical mechanisms that cause them. Rather, we wish to predict the X-ray environments in which HZ planets reside. 

To do this, we use our best fit spectral models to first convert the observed count rates to luminosities. For stars that are too faint to have their spectra directly modeled, we use the spectral model of the star with the closest spectral type and physical parameters to convert the observed count rates into luminosities. For stars that are detected but too faint for reliable light curve extraction, we simply estimate their time-averaged X-ray luminosity; for un-detected stars we calculate 3σ luminosity upper limits. We then apply Astropy's <cit.>  algorithm to the light curves; the “fitness” parameter is used to define the measurement uncertainties. The Bayesian block algorithm <cit.> is a segmentation technique, which aims to optimally split (in this case) time series data into “blocks” such that each block is statistically different from its neighboring block. An example Bayesian blocked light curve for κ^1 Cet is shown in Figure <ref>.



The output of the  routine is a list of time bins and the luminosity the star was emitting at during each bin. From these lists, we measure the fraction of the time that each star was observed to be fainter than a given  or / value. The results are shown in Figure <ref>. For comparison,  of the modern-day Sun is 6.31×10^26  (log/ = -6.78) when quiet and 8.04×10^27  (log/ = -5.68) when active <cit.>, while ∼4 Gyr ago the  of the Sun was ∼75% the modern value and its  was a factor of ∼2-3 higher <cit.>. On this plot, a perfectly non-variable star would be a straight vertical line, while stars that experience significant changes in X-ray luminosity (i.e., Proxima Centauri) will show a pronounced horizontal extent. We measure the luminosities at which the stars spend less than 25%, 50%, 70%, and 90% of the observed time (which we refer to as L_25, L_50, L_75, and L_90, respectively). These values are summarized in Table <ref>; we also include the time-averaged luminosities (with uncertainties) of faintly detected stars and the 3σ upper limits for non-detected stars in the L_50 column. 







§ DISCUSSION

We compared the 25th and 90th percentile / ratios for stars in our sample to the current minimum and maximum ratios for the Sun <cit.>, and find that eight stars that were detected in the X-ray observations at high significance exhibit / ratios similar to that of the modern Sun: ξ Oph (F2V), β Vir (F9V), 47 UMa (G0V), β Com (G0V), 55 Cnc A (G8V), GL 892 (K3V), GL 783 (K3V), and GL 183 (K3V). Three of these stars (β Vir, β Com, and GL 183) are younger than the Sun, while four are older (47 UMa, 55 Cnc A, GL 892, and GL 783) and one (ξ Oph) does not have an age estimate available in the literature. Twelve stars have / ratios consistent with those of the early Sun: η Crv (F2V), β Com, GL 311 (G1.5V), ζ^1 Ret (G2V), GL 327 (G3V), HD 140901 (G7IV), the unresolved binary 70 Oph (K0V+K5V), LHS 1875 (K3V), GL 570A (K4V), 61 Cyg A (K5V) and B (K7V), and the unresolved GL 570 BC pair (M1.5V+M3V). We note that one star, β Com, exhibits a range in / that fits within the narrow overlap region between the modern and early Sun (log/≈ -6.4 to -5.7). An additional 11 stars were not detected or only marginally detected and have / upper limits lower than the modern day minimum solar /: ν Phe, LHS 237, LHS 208, ι Per, ρ CrB, GL 672, 51 Peg, GJ 777A, 82 Eri, GL 451A, and Kapteyn's Star. For roughly a dozen stars in our sample, the observed  (or  upper limit) translates into X-ray surface fluxes below the ∼10^4  minimum observed in the quiet Sun and other “coronal hole” stars <cit.>. A closer look at these very X-ray faint stars will be the subject of an upcoming study (Binder et al., in preparation). In Table <ref>, we summarize these X-ray-detected Solar-like stars (their spectral types and approximate ages are taken from Table <ref>) and the relevant solar / comparison (modern or early Sun). We additionally provide the “tier” each star was assigned in <cit.>, indicating the priority which each star should be considered for an HWO direct imaging survey (with A indicating the highest priority group, and tiers B and C requiring further study).



Any HZ planets that may exist around the stars listed in Table <ref> are currently experiencing high energy radiation environments that are similar to or less hostile than the modern Earth. Out of the 229 unique stars we identified as potential targets for future direct imaging surveys, only ∼25% had ever been imaged with either  or  (this fraction increases to ∼30% when  imaging is included), and out of the 57 stellar systems with high quality X-ray observation we find that 28 systems (∼50%) have / ratios similar to or less than that of either the modern or early Sun. These stars span the full range of spectral types and ages considered here, and represent all three tiers of the <cit.> ranking system. It is therefore likely that there are an additional ∼100 potential direct imaging target stars with HZs with current X-ray radiation environments similar to or even less hostile than that of the Sun. We note that threshold /ratios below that of the modern or early Sun could be a conservative condition for finding habitable planets. Although, the past evolution of the star would also have to be taken into account. Detailed coupled geological and atmospheric evolution analyses would be most informative for predicting the targets with the best chances of maintaining potentially habitable conditions <cit.>. A systematic X-ray survey of these as-of-yet unobserved nearby stars is needed to inform the selection of the best targets for direct imaging surveys.

In general, M stars exhibit significantly more X-ray variability than earlier type stars and spend a significant amount of time at / ratios dramatically larger than that of the Sun. We can supplement our sample of Sun-like and low mass stars with X-ray observations of an 23 additional M- and K-type stars from the MUSCLES/Mega-MUSCLES survey <cit.>. <cit.> found a major decrease in the X-ray-to-bolometric luminosity ratio (/) with increasing effective temperature. We compute / (or upper limits, in the case of X-ray non-detections) for all stars in our sample using L_50 from Table <ref> and  from Table <ref>. In Figure <ref>, we plot the (logarithmic) / ratio as a function of T_ eff (also from Table <ref>) for our full sample (colored points), with the MUSCLES/Mega-MUSCLES sample shown in gray. Our sample extends this inverse relationship between / and T_ eff by ∼1000 K in T_ eff, with / ratios that are about an order of magnitude lower than observed for K and M stars. 

There is a small clustering of stars with T_ eff∼5200-6200 K with significantly higher / ratios than the majority of comparable stars, likely due to young stellar ages. Chromospheric activity in solar-type stars is observed to decrease rapidly after ∼2-3 Gyr <cit.>, after which activity levels remain mostly constant. We indicate the approximate age of each star in Figure <ref>: young stars (≲2 Gyr) are shown in red, older stars (≳2 Gyr) are shown in purple, and stars without age estimates or ambiguous ages are shown in dark blue. Most of these high-/, high-T_ eff stars are indeed believed to be consistent with the younger ages where each spectral type emits elevated high energy emission <cit.>. There are three stars with ambiguous ages that lie within this region of the diagram (χ^1 Ori, 55 Cnc A, and GL 451A); the observed X-ray emission of these stars suggests that younger age estimates may be preferred for these stars.



In Figure <ref> we again show the / ratio as a function of T_ eff for stars in our sample, but we use L_25, L_50, L_75 and L_90 to compute the / ratio. This allows us to assess the degree to which intrinsic X-ray variability contributes to the scatter in / observed in Figure <ref>. The ratio of L_75/L_25 provides an estimate of how much a star varies “normal” about the median . We find that the variability of the youngest stars in our sample exhibit the smallest dynamic range, with L_75/L_25∼1.3 for stars with T_ eff≳5000 K (for the coolest young star in the plot, AD Leo, L_75/L_25∼1.8). The L_75/L_25 ratio for older stars shows more dispersion, varying by up to a factor of ∼3.5.



 


§ CONCLUSIONS

We have analyzed ∼3 Msec of high quality  and  observations of 57 nearby stellar systems, as well as snapshot  images of an additional 19 stellar systems, to provide a uniform catalog of the X-ray properties of potential target stars for future direct imaging surveys. All X-ray light curves and spectroscopic data is publicly available on Zenodo (DOI: 10.5281/zenodo.11490574). The X-ray environments in the HZs around F, G, and early K stars are more likely to be similar to those that the Earth has evolved in, and we identify 29 stars with current / ratios similar to or less than that of the Sun. Our results are consistent with <cit.>, who found that the X-ray fluxes of late M stars can be a factor of ∼3-15 times larger than for K stars. In addition to high average / ratios, M stars exhibit more frequent and dramatic flares than earlier-type stars. We note that, of the 229 candidate target stars for direct imaging, only ∼30% have been observed by , , or . 




This work is supported by NASA Exoplanets Research Program (XRP) award #80NSSC23K0039 (PI Turnbull). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. B.B. is grateful for the warm hospitality of the University of California, Santa Barbara Kavli Institute for Theoretical Physics, where the majority of this manuscript was written; this research was supported in part by grant no. NSF PHY-2309135 to the Kavli Institute for Theoretical Physics. S.P. acknowledges support from NASA under award number 80GSFC21M0002. E.S. and S.P. acknowledge support from the CHAMPs (Consortium on Habitability and Atmospheres of M-dwarf Planets) team, supported by the National Aeronautics and Space Administration (NASA) under grant nos. 80NSSC21K0905 and 80NSSC23K1399 issued through the Interdisciplinary Consortia for Astrobiology Research (ICAR) program. This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We would like to thank Eric Mamajek for useful discussions related to this work, and the anonymous reviewer for helpful comments that improved our paper. 



CXO, XMM, Swift (XRT), Exoplanet Archive

astropy <cit.>
          




§ DISCUSSION OF INDIVIDUAL STARS

In this appendix we provide additional context for each stellar system (e.g, when there is a notable discrepancy in the published literature with regards to a given physical parameter) and summarize the known or candidate exoplanets associated with each star. We then present the results of our X-ray analysis for each stellar system. Table <ref> summarizes the observations of faint/non-detected stars, including if the star was detected in a given observation or not, the count rate (or upper limit on the count rate) and detection significance, and the “match star” that was used to convert count rates to luminosities.

Figure <ref> shows X-ray images of resolved and unresolved binary star systems included in our study. Individual images are discussed further for each star below. Figure Set <ref> shows the light curves extracted for all stars from all observations in which they were detected, with sub-exposures colored by VarFlag. Figure Set <ref> shows the average best-fit spectra for all sufficiently bright stars. For stars that exhibit flaring events, we also show the ratio of the flaring-to-quiescent spectra as a function of wavelength.














 §.§ η Crv

There are no significant discrepancies in the stellar physical parameters of η Crv. Although there are no known or candidate exoplanets, the system is known to host two debris disks: one at a distance of ∼160-180 AU, and a hotter disk at ∼3-7 AU <cit.>.

η Crv was observed four times with . All observations were taken with ACIS-S in 1/4 subarray mode for ∼10 ks. In all observations, η Crv was detected with ∼1000 net counts (0.5-7 keV), but no significant variability was observed in the X-ray light curves (see Figure Set <ref>). The spectrum of η Crv is well-described by a two-component thermal plasma model, with no significant changes in the temperatures or normalizations across the different observations.


 §.§ ξ Oph

There are no significant discrepancies in the stellar physical parameters of ξ Oph, and the star does not host any no known or candidate exoplanets. ξ Oph was observed once with /ACIS-S for 19.8 ks and robustly detected with ∼160 counts (8σ significance). There are hints of low-level variability in the X-ray emission of ξ Oph, but the relatively low number of net counts prohibits spectral modeling.


 §.§ υ And

υ And A is an F8V dwarf in a presumed binary system with a proper motion companion υ And B, an M4.5V dwarf (also referred to as υ And D). The two stars have an angular separation of ∼0.9^' (corresponding to ∼750 AU). There are no significant discrepancies in the physical parameters of either star. υ And A hosts three confirmed exoplanets (υ And b, c, and d) which were discovered via RV measurements. The masses of υ And b, c, and d are  ∼0.67 M_ Jup, ∼0.67 M_ Jup, ∼2 M_ Jup, and ∼4 M_ Jup, respectively, with orbital periods (orbital distances) of ∼4.6 days (0.06 AU), ∼241 days (0.82 AU), and ∼1282 days (2.5 AU), respectively <cit.>.

The υ And system was observed four times with /ACIS-S in 1/8 subarray mode (for ∼15 ks per observation) and once with  for 5.3 ks. υ And A was detected in all five observations with sufficient counts to extract light curves (see Figure Set <ref>) and with enough counts in  observations 10976 and 10977 to enable spectral modeling. There is no evidence for strong X-ray variability in any observation, and both  spectra were consistent with a single-temperature thermal plasma model with a temperature of ∼0.3 keV (∼3.4 MK; see Table <ref>). υ And B was out of the field of view of the  observations. However, the star was within the field of view of the  PN detector but fell just on the gap between two CCDs and was not detected (see Figure <ref>). 


 §.§ ι Hor

ι Hor is characterized either as a late F-type <cit.> or early G-type <cit.> star. There is additional disagreement on the age of the system: <cit.> quotes an age of 2.72 Gyr, while <cit.> suggest a much younger age of ∼0.47 Gyr <cit.>. We adopt a spectral type of F9V and an age of 0.47 Gyr in this work. ι Hor hosts one confirmed exoplanet that was first detected via radial velocity (RV) measurements by <cit.>. It is a giant planet with ∼2.3  and an orbital period of ∼300 days <cit.>.

ι Hor is one of the most thoroughly observed stars in our sample, with 32 publicly available  observations taken between 2011 and 2018. A detailed study of the coronal activity and structure of the star is presented in <cit.>, who combined the  monitoring with observations from TESS and the STIS instrument on the Hubble Space Telescope. They corroborate a 1.6 year X-ray activity cycle for the star, although ι Hor does not exhibit the same type of dramatic changes in luminosity as seen in other stars in our sample (see Figure <ref>). Our X-ray analysis of ι Hor is consistent with <cit.>, although we do not undertake a more extensive study of the abundances of the star. Light curves of ι Hor are shown in Figure Set <ref>, and the best-fit spectrum is described in Table <ref>.


 §.§ ν Phe

ν Phe is characterized either as a late F-type <cit.> or early G-type <cit.> star. There are no confirmed or candidate exoplanets in the system, and the star was undetected in a 16 ks  observation.


 §.§ γ Pav

γ Pav is a known F-type star, with the specific spectral type ranging from F9V <cit.> to F7V <cit.>. There are significant discrepancies in the estimated age of γ Pav: <cit.> quote an age of ∼1 Gyr based on photometric observations and comparison to stellar evolution models, while and asteroseismic analysis by <cit.> yields an age of 7.25 Gyr. There are no known or candidate exoplanets orbiting γ Pav. The star was only marginally detected (2.8σ) in a 25.9 ks  observation. The low X-ray luminosity of γ Pav is consistent with the older age estimate of <cit.>.


 §.§ β Vir

There are no significant discrepancies in the stellar physical parameters for β Vir, and the star does not host any currently known or candidate exoplanets. β Vir was observed once with  (observation ID 0044740201) and was detected with ∼18,450 net counts on the PN camera (thick filter). The X-ray light curve does not show evidence of significant variability (see Figure Set <ref>). The spectrum is well described by a two-temperature thermal plasma model (see Table <ref>).


 §.§ GJ 1095

The spectral type of GJ 1095 is given as F9V by <cit.> or G0V by <cit.>. There are no known or candidate exoplanets orbiting GJ 1095. The star was only marginally detected (2.9σ) in a 96.2 ks  observation.


 §.§ LHS 237

There are no significant discrepancies in the stellar physical parameters of LHS 237, and no age estimate is currently available in the literature. LHS 237 is a triple-star system: the inner spectroscopic binary has an orbital period of ∼10 years <cit.>, and the third component is one of the coolest known white dwarfs (NLTT 18141 = GJ 288B) located ∼14.5^' from the central binary <cit.>. There are no known or candidate exoplanets in the system.

LHS 237 was serendipitously imaged by  for 15.6 ks in observation 0840210501 (the intended target for the observation was a Wolf-Rayet star). The inner spectroscopic binary is not detected, and the white dwarf is outside the  field of view.


 §.§ ι Per

There are no significant discrepancies in the stellar physical parameters for ι Per. The spectral type is either F9.5V <cit.> or G0V <cit.>. There are no known or candidate exoplanets in the system. ι Per was observed for 4.9 ks with  in ACIS-S 1/4 subarray mode, but the star was not detected.


 §.§ β Hyi

There are no significant disagreements in the stellar physical parameters of β Hyi, and the star does not host any currently known or candidate exoplanets. β Hyi was observed once with /ACIS-S in 1/4 subarray mode and once with . The star was detected in both observations with sufficient counts to extract light curves, but the star is too faint for accurate spectral modeling to be performed. There is no strong evidence for X-ray variability in either the  or  light curves of β Hyi (see Figure Set <ref>).


 §.§ β Com

There are only minor differences in the reported age of β Com: <cit.> report an age <1.12 Gyr, while <cit.> report an age of 1.7 Gyr. All other spectral parameters are taken from <cit.>. There are currently no known or candidate exoplanets orbiting β Com.

β Com was robustly detected (with ∼38,000 net counts) in a 57.2 ks observation by . We extracted a light curve from this observation and found marginal evidence for count rate variability (see Figure Set <ref>). Given the large number of counts available, we split the observation into 5 ks-long sub-exposures and extracted spectra from each sub-exposure to search for evidence of underlying spectroscopic variability. Throughout the observation, a two-temperature thermal plasma model was sufficient to describe the X-ray spectrum of β Com, with the best fit temperatures and normalizations remaining constant within the uncertainties. All best-fit sub-exposure spectral parameters for β Com are provided in Table <ref>, and the average best-fit spectral parameters are listed in Table <ref>. The best-fit spectral models are shown in Figure Set <ref>.



 §.§ LHS 208 (π Men)

All spectral parameters for LHS 208 are taken from <cit.>. LHS 208 hosts three confirmed exoplanets. HD 390931 b is a giant planet that was detected via RV measurements <cit.>; it has ∼12  and an orbital period of ∼2090 days <cit.>. π Men c is a super-Earth that was discovered via transits <cit.>; it has an ∼3.5  with an orbital period of ∼6 days <cit.>. π Men d was detected via RV measurements with ∼13  and an orbital period of ∼125 days <cit.>. LHS 208 not detected in a 36.9 ks observation with  or a 19.6 ks observation with .


 §.§ ρ CrB

Some minor discrepancies, mass ranges from 0.95 <cit.> to 1.05  <cit.> and spectral type from G0V <cit.> to G2V <cit.>. ρ CrB hosts three confirmed exoplanets (ρ CrB c, d, and e), with the existence of a fourth exoplanet controversial <cit.>. The masses of ρ CrB c, d, and e are  ∼28 , ∼22 , and 3.8 , respectively, with orbital periods of ∼102 days (∼0.4 AU), ∼282 days (∼0.8 AU), and ∼13 days (∼0.11 AU), respectively. ρ CrB is also approaching the end of its main sequence lifetime, with a prognosis of planetary engulfment for most of the known planets in the system <cit.>. ρ CrB was observed once with /ACIS-S for 9.8 ks, but was not detected.


 §.§ GL 672

There are no significant discrepancies in the stellar physical parameters for GL 672, and there are no known or candidate exoplanets in the system. GL 672 was observed for 9.9 ks by /ACIS-S but was not detected.


 §.§ χ^1 Ori

χ^1 Ori is a G0V-type star in a ∼14 year elliptical (e∼0.45) orbit with a low-mass companion. RV measurements compiled by <cit.> were used to infer a secondary mass of ∼0.15  with a likely spectral type of M4V or M5V. The estimated angular semimajor axis of the orbit is ∼0.668^'', unresolvable by  but potentially spatially resolvable by  (see Section <ref>). The inferred spectral parameters may be biased due to the presence of this low-mass companion. There is significant disagreement in the literature about the age of the system. <cit.> and <cit.> estimate an age of 4.32 Gyr, while <cit.> use activity-rotation diagnostics to estimate an age of 300-400 Myr. There are no confirmed or candidate exoplanets in the system.

Since only one  observation is available for χ^1 Ori, we are unable to identify the source of the X-ray emission – whether it be from the G0V star, the low-mass companion, or both. There is one potential, minor flaring event during the observation, starting ∼21 ks after the beginning of the observation and lasting for ∼4 ks, where the peak count rate exceeds the median count rate by ∼40% (see Figure Set <ref>). While this may not be a bona fide flare event, we classify this period as flaring so that its spectrum can be compared to the remainder of the observation. The light curve otherwise shows a count rate that is either constant within the uncertainties (which we classify as quiescent) or somewhat elevated above the median count rate level by ∼20% (which we classify as elevated).

Figure Set <ref> shows the quiescent, elevated, and potential flare spectra for the χ^1 Ori system. The elevated and flare spectra are nearly indistinguishable, and both show excess emission over the quiescent spectrum at wavelengths ≲10Å. This difference in short-wavelength emission is driven primarily by the hottest thermal plasma component in our model, which has a temperature of ∼0.8 keV for the elevated and flaring spectra but ∼0.7 keV in the quiescent spectrum (see also Table <ref>). The integrated X-ray luminosity of χ^1 Ori is constant within the uncertainties across the three spectra.



 §.§ GL 788

There are no significant discrepancies in reported stellar parameters of GL 788. The system does not host any known or candidate exoplanets. GL 788 was detected in a 9.6 ks  observation with ∼4470 net counts. There was no evidence of count rate variability apparent in the X-ray light curve (see Figure Set <ref>). The spectrum was well-described with a single, cool thermal plasma (kT∼0.2 keV).


 §.§ 47 UMa

There are no significant discrepancies in the stellar physical parameters reported in the literature for 47 UMa. The star hosts three confirmed giant exoplanets that were detected via radial velocity measurements. 47 UMa b, c, and d have masses of ∼2.4 M_ Jup, ∼0.5 M_ Jup, and 1.5 ∼2.4 M_ Jup, respectively, and have orbital periods (distances) of ∼1077 days (∼2 AU), ∼2287 days (∼7.8 AU), and ∼19000 days (∼12.4 AU), respectively. All planetary parameters were taken from <cit.>.

47 UMa was detected at ∼12σ significance in a 6.2 ks  observation. We extracted a light curve from the observation but found no evidence of variability (see Figure Set <ref>), and the star was too faint to enable spectroscopic modeling.


 §.§ GL 620.1A

GL 620.1A is either co-moving with <cit.> or in a wide binary with <cit.> the DA-type white dwarf WD 1620-391. The angular distance between the two components is ∼5.8^', which would correspond to a physical separation of ∼4500 AU at the distance of GL 620.1A. GL 620.1A is classified as a barium star, and its over-abundance in s-process elements has been attributed to pollution due mass transfer during the white dwarf progenitor's AGB phase <cit.>. There is some disagreement in the literature over the spectral type and age of GL 620.1A. Spectral types range from G1V <cit.> to a later G3/5V <cit.>; we adopt a spectral type G1V. While <cit.> quote an age of 2 Gyr, other estimates of the age of GL 620.1A are lower: <cit.> and <cit.> derive upper limits of <1 Gyr and <0.68 Gyr, respectively, while <cit.> adopt an age of 0.4 Gyr. One giant exoplanet is confirmed via RV measurements of GL 620.1A <cit.>; the planet has ∼1.2  and an orbital period of ∼528 days (orbital distance of ∼1.3 AU).

GL 620.1A is detected in both publicly available  observations (Figure <ref>) with sufficient counts to enable spectral modeling (see Figure Set <ref> and Table <ref>). There is no significant variability present in the count rate light curves of GL 620.1A (Table <ref>, see Figure Set <ref>), so we do not attempt to divide the observations into sub-exposures.




  §.§.§ The White Dwarf WD 1620-391

While GL 620.1A is robustly detected in both publicly available  observations, WD 1620-391 is not (see Figure <ref>a). The 3σ upper limits on the count rates of 0.014 ct s^-1 (observation ID 0822070201) and 0.0236 ct s^-1 (observation ID 0822070301). We use WebPIMMs[See <https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] to convert the observed /PN count rates (both observations were taken with the thick filter) to a flux assuming an APEC thermal plasma model with kT≈0.5 keV <cit.>. We find flux (luminosity) upper limits of 2.0×10^-14  (4×10^26 ) and 3.4×10^-14  (6.8×10^26 ) for observations 0822070201 and 0822070301, respectively. These luminosity limits are a factor of ∼5-7 higher than the observed luminosity of G 29-38; targeted observations with a more sensitive X-ray instrument are needed to determine if WD 1620-391 is emitting X-rays at a levels comparable to other X-ray-detected white dwarfs.



 §.§ GL 311 (π^1 UMa)

There are no significant discrepancies in the stellar physical parameters reported in the literature for GL 311. The system does not host any currently known or candidate exoplanets.

The variability metrics for GL 311 (see Table <ref>) suggest that count rate variability is observed in the available  observation (observation number 0111400101; see Figure Set <ref>). There are two minor flare-like events in the light curve. The first begins at ∼21 ks after the beginning of the observation, and reaches a peak count rate that is a factor of ∼2 higher than the median count rate. This event lasts for approximately 8 ks and it immediately followed by a second count rate increase, which exceeds the median count rate by a factor of ∼1.6 and lasts for approximately 5 ks. The post-flare count rate is elevated by ∼15% relative to the pre-flare count rate. Spectroscopic modeling before, during, and after the flaring event show that the temperatures of all three thermal components increased in temperature during the flaring event, although a modest decrease in the normalization (related to the VEM) during the flare led to a constant (within the uncertainties) luminosity (Figure Set <ref>; see also Table <ref>). 

Thus, although GL 311 clearly exhibits a degree of rapid X-ray variability (on a timescale of a few hours), this activity does not have a significant affect on the overall X-ray luminosity output of the star. The relative stability of the star's X-ray luminosity is also apparent in Figure <ref>, where the luminosity curve for GL 311 is near-vertical.


 §.§ α Cen

α Centauri A (G2V) and B (K1V) are members of the extensively studied triple system that also includes Proxima Centauri (M5.5V; see Section <ref>). α Cen A and B are in a ∼80 year elliptical orbit <cit.>. One candidate exoplanet, α Cen B b, was reported <cit.> but was later found to be a false positive signal <cit.>. There are currently no other known or candidate exoplanets in the α Cen AB system. 

The semi-major axis of the α Cen system is ∼18^'', but due to the brightness of the stars there is considerable blending in the  images (see Figure <ref>). We therefore do not attempt to extract X-ray products (light curves, spectra) for each star individually. The α Cen system exhibits significant X-ray variability (see Figure Set <ref>). The system is well-described by a three-component thermal plasma model, and increases in the observed count rates are driven largely by increases in the hottest plasma temperature (from ∼0.6 keV during quiescent periods to ∼0.8-0.9 keV during times associated with flaring events) and normalization (which increases by a factor of ∼12). The spectra of α Cen are shown in Figure Set <ref> (see also Table <ref>). Due to the significant blending of the two components, we do not use the quiescent α Cen spectrum as a baseline for inferring other stellar luminosities.


 §.§ 44 Boo

44 Boo (also called ι Boo) is a triple star system with a G0V primary (44 Boo A) in ∼210 year orbit with a spectroscopic binary composed of a K0V (44 Boo Ba) dwarf and a K4V (44 Boo Bb) dwarf <cit.>. 44 Boo A is separated from the spectroscopic binary by ∼0.2^''. The 44 Boo B binary has an orbital period of only ∼6.4 hours and is a possible contact binary <cit.>. There are no known or candidate exoplanets in the 44 Boo system. 

44 Boo is robustly detected with  in an 18.8 ks observation, with count rate variations clearly seen in the light curve (see Figure Set <ref>). We do not include the final ∼5 ks in our spectral analysis due to a significant background flaring event during the observation. Our spectral modeling revealed no significant differences between the minor flaring event from ∼6-10 ks and the relatively constant count rate period from 15-20 ks. The spectrum of 44 Boo did change during periods of declining count rate (the first 6 ks and from 10-15 ks) as the result changes in the hottest thermal plasma component: kT_3 decreased from ∼2.1-2.2 keV during quiescent/flaring times to ∼1.5 keV during periods of secular decline, while the normalization of this component (related to the VEM) more than doubled. These changes to APEC #3 result in a decrease in X-ray flux at the shortest wavelengths (≲10 Å); see Table <ref> and Figure Set <ref>.


 §.§ 18 Sco

There are minor differences in the reported spectral type of 18 Sco: <cit.> quote a spectral type of G5V, but <cit.> classifies the star as a G2V. We adopt a spectral type of G2V. 18 Sco hosts one candidate super-Earth with  ∼6.8 M_⊕ and an orbital period of ∼20 days <cit.>.  The star is detected in both a 20.1 ks  observation and a 62.2 ks observation with /ACIS-S. However, the 18 Sco is not bright enough in either observation for timing or spectroscopic analysis to be performed. 


 §.§ 51 Peg

There are no significant discrepancies reported for the stellar parameters of 51 Peg. The star is famous for hosting the first discovered exoplanet <cit.>, the hot Jupiter 51 Peg b with  ∼0.46 M_ Jup and an orbital period of ∼4.2 days <cit.>. 51 Peg was observed with both  and , but was not detected in either observation.


 §.§ ζ Ret

The wide binary ζ Ret contains ζ^1 Ret, which is listed as a G3V/G5V by <cit.> but G2V as <cit.>, and ζ^2 Ret, a G2V star <cit.>. The two stars are separated by ∼5.2^'. The system does not hot any known or candidate exoplanets.

Although this separation would be easily resolvable by , only ζ^1 Ret is contained with the fields of view of the  cameras. The star is robustly detected (with ∼7700 net counts) in a 20.9 ks  observation (ID 0404920101). No significant variability is observed in the X-ray light curve of ζ^1 Ret (see Figure Set <ref>), and the spectrum is well-described by a two-component thermal plasma model (with temperatures of ∼0.1 keV and ∼0.4 keV; see Figure Set <ref> and Table <ref>).


 §.§ HD 136352

There are no significant discrepancies in the stellar physical parameters reported in the literature for HD 136352. The star hosts three confirmed exoplanets, all of which have masses in the super-Earth/sub-Neptune regime. HD 136352 b, c, and d have  of ∼4.7 , ∼11 , and ∼8.8 , respectively, with orbital periods (distances) of ∼11.6 days (∼0.1 AU), ∼28 days (∼0.17 AU), and ∼107 days (∼0.43 AU), respectively. The inner two planets were found to transit the host star by <cit.>, providing bulk densities for the planets. Subsequent observations by <cit.> revealed that the third, outer planet also transits the host star. HD 136352 is marginally detected (∼2.7σ) in a 16.9 ks  observation with a count rate of 0.0041±0.0015 ct s^-1 with the PN camera and medium filter (Table <ref>).


 §.§ GL 327

There are no significant discrepancies in reported stellar parameters of GL 327 in the literature. The system does not host any currently known or candidate exoplanets. GL 327 was robustly detected with ∼10,500 net counts in a 23.5 ks exposure with  (observaton ID 0404920201). Count rate variability was observed in the X-ray spectrum as shown in Figure Set <ref>, so we split the observation into three sub-exposures and modeled the three spectra independently. In all three sub-exposures, a single-temperature thermal plasma component was sufficient to describe the observations. During the quiescent period from ∼6-21 ks, the plasma temperature was ∼0.4 keV, which increased to ∼0.5 keV during the descending and flaring periods. The normalization during periods of increased X-ray count rate additionally increased by ∼20%. The spectra of GL 327 are shown in Figure Set <ref> (see also Table <ref>.


 §.§ μ Ara

No significant discrepancies in stellar parameters are present in the literature for μ Ara, although there is some uncertainty about the evolutionary stage; we adopt a spectral type of G3IV/V <cit.>. μ Ara hosts four known (giant) exoplanets, all detected via RV measurements. HD 160691 b, c, d, and e have masses of  ∼1.7 , ∼2.4 , ∼10.2 , and ∼0.5 , respectively, and the planets have orbital periods (orbital distances) of ∼646 days (∼1.5 AU), ∼4470 days (∼5.5 AU), ∼10 days (∼0.09 AU), and ∼307 days (∼0.9 AU), respectively. All planetary data were taken from <cit.>. μ Ara is not detected in either of the two publicly available  observations.


 §.§ κ^1 Cet

There is some discrepancy in the reported age estimates of κ^1 Cet: <cit.> lists an age of 2.2 Gyr and <cit.> derive an upper limit of <2.76 Gyr, but many studies report significantly younger ages. For example, <cit.> derive an age of 300-400 Myr, while <cit.> estimated an age of 750 Myr and and <cit.> estimate an age of 650 Myr. There are no currently known or candidate exoplanets orbiting κ^1 Cet.

κ^1 Cet is relatively well-observed in X-rays, with five  observations and two  observations available in the archives, and the star is easily detected in all exposures. It was also observed twice with  using ACIS-S7 in 1/8 subarray mode with a 0.44 s readout time in both observations. While there are clear indications of variability in the light curves of κ^1 Cet the maximum observed count rate only ever exceeds the median count rate by a factor of ∼2 (see Figure Set <ref>). However, despite the only modest changes in count rate, the spectra during these minor flare-like events is quite distinct from the lower count rate quiescent periods, with significant excess emission a wavelengths shortward of 10 Å compared to quiescent times. The change in count rate is being driven by a significant increase in the temperature (by ∼0.2 keV, or ∼2.3 MK) and normalization (a factor of ∼2.2 increase) of the hottest component in a three-thermal plasma component model (Figure Set <ref>; see also Table <ref>).


 §.§ HD 140901

There are no significant discrepancies in the published stellar physical parameters of HD 140901. Two planets have been identified via RV measurements. HD 140901 c has  ∼1.8 M_ Jup with an orbital period of ∼14390 days <cit.>. The existence of HD 140901 b is controversial; if the planet exists, it has a mass of  of ∼16M_⊕ with an orbital period of ∼9 days (orbital distance 0.085 AU), but the RV data of HD 140901 can be equally well described by a retrograde orbital inclination of ∼138^∘ <cit.>. 

HD 140901 was observed once by /ACIS-I for 24.7 ks. The star was robustly detected with ∼220 net counts. The light curve shows clear evidence for variability (see Figure Set <ref>), but the star is not bright enough to enable spectral modeling.


 §.§ GJ 777

GJ 777 is a binary star system composed of a G6IV primary (GJ 777A) and an M4.5V secondary (LHS 2509), which are separated by ∼3000 AU (∼3^').
GJ 777A hosts two currently known exoplanets, detected via RV measurements: HD 190360 b has  ∼1.8 M_ Jup and an orbital period of ∼2854 days <cit.>, and HD 190360 c has  ∼19 M_⊕ and an orbital period of ∼17 days <cit.>. No X-ray emission is detected coincident with the GJ 777 binary system in either of two snapshot  observations.


 §.§ ξ Boo

ξ Boo is a visual binary <cit.> with an angular separation of ∼4.9^'' <cit.>, the stars are unresolvable by  (see Figure <ref>). No discrepancies in stellar physical parameters in the literature. No known or candidate exoplanets. ξ Boo A is a known BY Draconis variable <cit.>.

The ξ Boo binary is robustly detected by , although the two components cannot be resolved. The system exhibits dramatic X-ray variability, as shown in the light curve in Figure Set <ref>. The X-ray spectra of ξ Boo are described by three-component thermal plasma models in all variability states. The flaring spectrum requires hotter temperatures for all three plasma components than the quiescent spectrum, with the normalization of the hottest component increasing by a factor of ∼3.9 during flaring times compared to quiescent times. Figure Set <ref> shows the best fit spectral models for ξ Boo (see also Table <ref>).


 §.§ δ Pav

There are no significant discrepancies in the stellar physical parameters reported in the literature for δ Pav. The star does not host any currently known or candidate exoplanets.

δ Pav was detected in a 32.6 ks  observation (ID number 0780510401) ∼1000 net counts. The A^2 statistic provides evidence that the star may be variable on short timescales, however there are not a sufficient number of counts to split the observation into sub-exposures (see Figure Set <ref>). We attempted to model the spectrum of δ Pav despite the low number of counts; the spectrum required two thermal plasma components to achieve a statistically acceptable fit (see Table <ref>). The best-fit spectrum in shown in Figure Set <ref>. 



 §.§ LHS 2156

There are no significant discrepancies in the stellar physical parameters reported in the literature for LHS 2156. There are no known or candidate exoplanets in the system.

LHS 2156 was detected with ∼60,000 net counts in a 68.4 ks  observation with the PN camera (thick filter). Figure Set <ref> shows the X-ray light curve, which exhibits count rate variability over the duration of the observation. The spectra of LHS 2156 are well described by three-component thermal plasma models. The best-fit temperatures and normalizations of two cooler plasma components remain constant within the uncertainties in all sub-exposures. The hottest thermal plasma component increases in temperature by ∼0.1 keV during flaring periods compared to quiescent periods, an the normalization increases by ∼25%. The best-fit spectra for LHS 2156 are shown in Figure Set <ref> (see also Table <ref>).


 §.§ 82 Eri

There is some discrepancy in the literature about the age of 82 Eri: <cit.> and <cit.> quote ages >12 Gyr, but a much younger age of 5.76±0.66 Gyr was found by <cit.>. 82 Eri hosts four confirmed exoplanets, all of which were discovered via RV measurements <cit.>. HD 20794 b, c, d, and e have  values of ∼2.8 , ∼2.5 , ∼3.5 , and ∼4.8 , respectively, with orbital periods (distances) of ∼18 days (∼0.13 AU), ∼43 days (∼0.23 AU), ∼89 days (∼0.26 AU), and ∼147 days (∼0.5 AU), respectively. All planetary parameters were taken from <cit.>. 82 Eri has been observed by both  and  (see Table <ref>), but was not detected in either observation. 


 §.§ τ Ceti

There are no significant discrepancies in the stellar physical parameters of τ Ceti reported in the literature. τ Ceti hosts four exoplanets, all confirmed via radial velocity measurements <cit.>, with some evidence of additional planets in or near the τ Cet HZ <cit.>. τ Ceti e and f both have masses  ∼3.9 M_⊕ with orbital periods of ∼163 days (∼0.55 AU) and ∼636 days (∼1.3 AU), respectively, while τ Ceti g and h both have masses of  ∼1.8 M_⊕ and orbital periods of ∼20 days (∼0.13 AU) and ∼49 days (∼0.24 AU), respectively.

τ Ceti was observed by  ACIS-S for 45.1 ks in 1/8 subarray mode and detected at ∼7σ significance with ∼110 net counts. We extracted a light curve from this observation and find an A^2 statistic that is strongly suggestive of variability (see Table <ref> and Figure Set <ref>), but due to the relatively low number of counts we did not attempt to extract a spectrum. τ Ceti was marginally detected in an 11.9 ks  observation, but with insufficient counts for light curve extraction or spectral modeling to be performed. The luminosity estimates for τ Ceti suggest that the star's X-ray luminosity changes by a factor of ∼2.


 §.§ 55 Cnc

55 Cnc is a binary star system containing a G8V star (55 Cnc A) and a M4.5V dwarf (55 Cnc B). No significant discrepancies in stellar physical parameters. The two components are separated by ∼1,065 AU <cit.>. Five exoplanets are known to orbit 55 Cnc A. 55 Cnc b was the first to be discovered <cit.> with a mass of  ∼0.8 M_ Jup and an orbital period ∼14.7 days <cit.>. 55 Cnc c, d, e, and f has masses of  ∼0.16 M_ Jup, ∼3.9 M_ Jup, ∼8 M_⊕, and ∼0.15 M_ Jup, respectively, with orbital periods of ∼44 days (∼0.24 AU), ∼4870 days (∼5.5 AU), ∼0.7 days (∼0.02 AU), and ∼260 days (∼0.8 AU), respectively.

Cnc 55 was observed twice by /ACIS-S in 1/8-subarry mode (for 10.7 ks and 18 ks) and once for 10.4 ks by . 55 Cnc A was not detected by , but was detected (at ∼5.6σ) significance by . No evidence for strong variability was observed in the  light curve of 55 Cnc A (Table <ref>, see also Figure Set <ref>). Despite the relatively low number of net counts, we attempted to model the X-ray spectrum of 55 Cnc A. We find that a single-component thermal plasma model is sufficient to describe the X-ray spectrum (see Table <ref> and Figure Set <ref>). 55 Cnc B was not detected in the  observation or one of the  observations (the star was not within the field of view of the second  observation).


 §.§ 70 Oph

70 Oph is a visual and spectroscopic binary containing a K0V (70 Oph A) star and a K5V (70 Oph B) star with an orbital period of ∼88 years <cit.>. The orbit is highly elliptical, with an eccentricity e=0.50 that causes the orbital separation of the stars to vary from ∼11.4 AU to ∼34.8 AU <cit.>. These orbital separations correspond to angular separations of ∼2.3-6.8^'', unresolvable by  but potentially spatially resolvable by  (see Section <ref>). <cit.> find a best-fit age of 6.2±1.0 Gyr, in broad agreement with the 6.8-7.0 Gyr found by <cit.>. These ages (which where derived using astroseismic observations) are in contrast to the significantly younger age (∼680 Myr) predicted by <cit.>. There are no currently known or candidate exoplanets in the system.

There are three  observations available for the 70 Oph system. The individual stars are not resolvable in the  observations (see Figure <ref>); we therefore cannot separate the X-ray emission from the K0V star (70 Oph A) from the K5V star (70 Oph B) with the currently available observations. 

Despite the changes in X-ray count rate observed in the light curves (see Figure Set <ref>), the best-fit spectral models of 70 Oph across different VarFlags are nearly indistinguishable (Figure Set <ref>; see also Table <ref>), and the X-ray luminosity of the two stars changes only by ∼40% between the flaring and quiescent periods. Since we cannot separate the X-ray emission of the binary components individually, we do not use the best-fit spectrum of the 70 Oph system as template for inferring luminosities of other stars in our sample.


 §.§ 40 Eri

The 40 Eri triple system contains a K0V star (40 Eri A) that is either in a wide orbit or co-moving with a white dwarf (40 Eri B)-M4.5V dwarf (40 Eri C) pair. The projected angular distance between 40 Eri A and 40 Eri BC is ∼1.3^'. While the orbital period of the wider A-BC pair is very long, the orbital period of the B-C pair is measured to be ∼230 years <cit.>. The angular separation between 40 Eri B and 40 Eri C is ∼6.9^'', easily resolvable by  (see Figure <ref>). 40 Eri C has a visual magnitude of ∼11, which we use to estimate , and <cit.> estimate  to be ∼3,100 K. There is one confirmed planet orbiting 40 Eri A, which was discovered via RV measurements. HD 26965 b has  of ∼8.5  with an orbital period of ∼42 days <cit.>. No currently known or candidate planets are associated with the 40 Eri BC pair.

The first soft (0.1-2.0 keV) X-ray detection of the 40 Eri system was obtained by HEAO-1, which measured a luminosity of (9.6±3.2)×10^28  <cit.>. Due to the poor spatial resolution of HEAO-1, the binary components could not be resolved. Different scenarios for X-ray production are explored in <cit.>, and the X-ray emission was attributed to 40 Eri C. 

The  image of the system clearly shows X-ray sources coincident with both 40 Eri A and C (although not the white dwarf, 40 Eri B; see discussion below). 40 Eri C is the brighter of the two sources, with roughly an order of magnitude more net counts than observed for 40 Eri A. There is no evidence for strong variability in either star (see Figure Set <ref>). While the coolest thermal plasma component in the 40 Eri C spectrum is similar in temperature to other late-type stars in our sample, the hotter two thermal plasmas are considerably hotter (see Table <ref> and Figure Set <ref>). Despite the lack of obvious variability in 40 Eri C, the best-fit X-ray spectral parameters most closely resemble those of WX UMa in its flaring and post-flaring states. It is possible this star indeed exhibits significant coronal variability, but it was not detectable in the short snapshot  observation available in the archive.




  §.§.§ The White Dwarf 40 Eri B

The white dwarf 40 Eri B is not detected in the 5 ks /ACIS-S observation of the system. The 3σ upper limit on the count rate is 0.0033 ct s^-1. We again use WebPIMMs to convert the observed count rate upper limit to a flux (and luminosity) upper limit, assuming the same spectral model as for WD 1620-391 (Section <ref>). We find a flux upper limit of 1.6×10^-13 , which corresponds to a luminosity of ∼5×10^25 . This luminosity upper limit is below the measured X-ray luminosity of G 29-38 <cit.>, strongly suggesting a lack of a cold accretion disk around this white dwarf.



 §.§ δ Eri

There are no significant discrepancies in the reported physical parameters for δ Eri, and the star does not host any known or candidate exoplanets. δ Eri was observed once with  (observation observation) for 56.2 ks. There was marginal evidence for significant variability in the X-ray light curve of δ Eri (see Figure Set <ref>), but the star was too faint for spectral modeling.


 §.§ GL 451A (Groombridge 1830)

There are no significant discrepancies in the stellar physical parameters reported in the literature for GL 451A. The system does not host any currently known or candidate exoplanets. GL 451A was observed once with /ACIS-S (observation ID 9931) for 32.8 ks in 1/8-subarray mode across ACIS detectors 5-7 with a readout time of 0.54 s. No X-ray source is found coincident with the star's position.


 §.§ GL 117

<cit.> report an age of 1.5 Gyr for GL 117 and <cit.> derive an upper limit of <1.2 Gyr. These ages are in contrast with the very young age (∼100 Myr) predicted by <cit.>. There are no currently known or candidate exoplanets orbiting GL 117.

GL 117 is robustly detected in  observation 0203060501, and there are low-level count rate variations observable in the light curve (see Figure Set <ref>). The largest of these variations reaches a count rate level that is ∼30% above the median count rate. Given the ample number of X-ray counts recorded by , we divide the light curve into sub-exposures but we find no significant differences in the best fit spectral parameters nor the predicted luminosities (see Figure Set <ref> and Table <ref>). The bright X-ray luminosity of the star, particularly when compared to other K dwarfs in our sample, are consistent with a young age for GL 117.


 §.§ GL 783

GL 783 is a binary system containing a K3V star (A) and an M4.5V star <cit.>. The two stars were stated to be separated by ∼8^'' by <cit.> but Gaia DR3 positions indicate a separation of ∼2.6^'' <cit.>, both of which are unresolvable by  but potentially spatially resolvable by  (see Section <ref>). GL 783B has an apparent visual magnitude of ∼12.5 <cit.>, which we use to estimate . At the time of this writing there were no published values of  for GL 783B. The system does not host any currently known or candidate exoplanets.

There is one  observation available for the GL 783 system, and the individual stars are not resolvable (see Figure <ref>). The therefore cannot separate the X-ray emission from the K3V star (GL 783A) from the M4.5V star (GL 783B) with the currently available observations. The A^2 statistic provides evidence that one or both stars may be variable on short timescales (see Figure Set <ref>), however there are not a sufficient number of counts to split the observation into sub-exposures. We attempted to model the spectrum of the GL 783 system despite the low number of counts; the spectrum required three thermal plasma components to achieve a statistically acceptable fit (see Figure Set <ref> and Table <ref>). The coolest thermal plasma component (∼0.1 keV) is similar to the coolest components of the other FGK stars in our sample, while the temperatures of the hottest (∼1 keV) and intermediate (∼0.3 keV) components more closely resemble components found in the later-K and M dwarfs in our sample. 

Since we cannot separate the X-ray emission of the binary components individually, we do not use the best-fit spectrum of the GL 783 system as template for inferring luminosities of other stars in our sample. Follow-up observations with higher spatial resolution are needed to associate the observed X-ray emission with one or both binary components.



 §.§ GL 183

There is some discrepancy in the reported spectral type of GL 183: <cit.> report a spectral type K3V, while K4III is reported by <cit.>. We adopt a spectral type K3V. There is additional tension in the age estimates, with a younger age (∼2 Gyr) reported by <cit.> and older ages reported by <cit.> and <cit.>. There are no currently known or candidate exoplanets orbiting GL 183.

GL 183 was detected in a 18.4 ks  observation (ID number 0780510301) ∼650 net counts. The A^2 statistic provides evidence that the star may be variable on short timescales (see Figure Set <ref>), however there are not a sufficient number of counts to split the observation into sub-exposures. We attempted to model the spectrum of GL 183 despite the low number of counts and found the spectrum required two thermal plasma components to achieve a statistically acceptable fit (Figure Set <ref>; see also Table <ref>), and the fit parameters were reasonably well-constrained.


 §.§ GJ 667

The GJ 667 is a triple star system wherein the two more massive components, A and B (both of which are early/mid K dwarfs) orbit one another in a highly elliptical orbit (e≈0.58). The average angular separation of the two stars is ∼1.8^'', possibly resolvable by . The third star in the system, the M1.5V dwarf GJ 667 C, is separated from the AB pair by ∼0.5^' (which would be easily resolvable with both  and ). There is no published age estimate for the system. GJ 667C has been studied previously in X-rays by <cit.>, and is known to host two confirmed super-Earth exoplanets: GJ 667C b, with  ∼5.6  and an orbital period of ∼7.2 days <cit.>, and GJ 667C c, with with  ∼3.8  and an orbital period of ∼28 days <cit.>. An additional three planets in wider orbits were reported by <cit.> but were unable to be confirmed by <cit.>; their existence remains controversial.

GJ 667 was observed twice with /ACIS-S in 1/8-subarray mode. GJ 667 A and B were not within the field of view of either observation. GJ 667 C was detected in both observations. We extracted an X-ray light curve from  ObsID 17317 (with an exposure time of 18.2 ks, see Figure Set <ref>), but the star was too faint for spectral modeling to be performed. There were insufficient counts in the second  observation (ObsID 17318, with an exposure time of 9.1 ks) for light curve or spectral analysis.


 §.§ LHS 1875

There is some discrepancy in the literature about the age of LHS 1875: <cit.> quote an age of ∼3.1 Gyr, while <cit.> provide a significantly younger upper limit of <0.48 Gyr. There are no known or candidate exoplanets orbiting LHS 1875. LHS 1875 was detected with ∼9260 net counts in a 21.4 ks /PN observation. No evidence for significant X-ray variability was observed (see Figure Set <ref>), and the spectrum was well-described by a three-component thermal plasma model (see Table <ref>).


 §.§ GL 570

There is a significant age discrepancy in the literature for the GL 570 quadruple-star system: <cit.> report a very young age of <0.6 Gyr, while <cit.> adopt an ade of ∼3 Gyr. There are no other significant discrepancies in the stellar physical parameters for the stars in this system. There are no currently known or candidate exoplanets in the system.

GL 570A (K4V) orbits a binary red dwarf pair (GL 570B, an M1.5V dwarf, and GL 570C, an M3V dwarf) with a semi-major axis of ∼190 AU (corresponding to an angular separation of ∼0.5^'). GL 570A and the GL 570BC pair are both detected in X-rays and are easily resolvable with  (see Figure <ref>). The red dwarfs the make up the BC pair are separated by ∼0.2^'' <cit.> and cannot be resolved. The fourth component, GL 570D, is a brown dwarf in a very wide orbit (∼4^') from the GL 570 ABC triple system <cit.>. GL 570D is not detected in X-rays.

We extract X-ray light curves for both GL 570A and the GL 570BC pair. There is clear evidence for variability in both light curves (see Figure Set <ref>), although there are not enough net counts from GL 570BC to enable spectroscopic modeling on sub-exposures. We therefore divide observation of GL 570A into three distinct sub-exposures (the first ∼20 ks, 20-30 ks, and 30-40 ks) and independently model the spectra, but only extract one spectrum of GL 570BC from the entire observation. All spectra are well described by a two-component thermal plasma model. The flaring event seen from 20-30 ks in the GL 570A light curve is characterized by changes in plasma temperatures (∼0.1 keV for the cooler component and ∼0.2 keV for the hotter component) and an increase in the hot-component normalization by ∼170%. The different best-fit spectral models for GL 570A, as well as a comparison of the flaring-to-quiescent spectra, are shown in Figure Set <ref>. The GL 570A model is overall hotter than the GL 570BC model (see Table <ref>).



 §.§ 61 Cyg

The 61 Cygni AB system is a wide binary containing a K5V star (A) and a K7V star (B), with an angular separation of ∼12^'' (the orbital period of the system is ∼659 years). Most age estimates suggest the system is old <cit.>. The system is resolved by  <cit.>. The system does not host any currently known or candidate exoplanets.

61 Cyg has been observed 38 times with , making it the best X-ray-observed star system in our sample. The observations are spaced ∼6 months apart and span ∼18 years, and the two stars are well-resolved in all available images (see Figure <ref>). Both stars are observed to flare (see Figure Set <ref>), and all the spectra extracted from both stars are well describe by a three-component thermal plasma model. The three best-fit temperatures in the models of 61 Cyg A remain constant within the uncertainties during different variability epochs. The changes in count rate observed during flaring events in 61 Cyg A are driven by changes in the normalizations of the two hotter thermal plasma components (and hence the VEMs): the hottest component normalization increases by an order of magnitude, and the intermediate temperature component normalization increases by a factor of ∼2.5. The spectra of 61 Cyg B show similar changes in normalization during flaring events, but the two cooler plasma components also increase in temperature during flaring events (by ∼0.1 keV). The best-fit spectra for 61 Cyg A and B are shown in Figure Set <ref> (see also Table <ref>). 


 §.§ GL 412

The GL 412 AB system contains a M1V primary (A) and a UV Ceti-type flaring M6.6V secondary <cit.>. X-ray flares were previously observed from the system with ROSAT <cit.> and attributed to GL 412B; GL 412A was not believed to be a significant source of X-rays. The system does not host any currently known or candidate exoplanets.

The two stars have an angular separation of ∼31^'' and are resolved in the  image (observation number 0742230101; see Figure <ref>). The image clearly suggests that the observation caught a significant WX UMa flaring event and that the M1V primary, while fainter, emits a detectable quantity of X-ray radiation. A third X-ray source is detected ∼37^'' from WX UMa, but is not obviously associated with the GL 412 binary. We measure a count rate of 0.057±0.002 ct s^-1 in the PN image (medium filter) for GL 412A. 

The light curve of GL 412B is shown in Figure Set <ref>. Spectral modeling shows that the X-ray luminosity increases by a factor of ∼30 between quiescent and flaring times. The quiescent spectrum (the first ∼2 ks of the observation) is well-described with only two thermal plasma components. The steep, ∼0.5 ks flare, the elevated count rate (from ∼3-7 ks), and subsequent declining period all require three components to adequately describe the X-ray spectra. The flaring spectrum exhibits a very hot thermal plasma component, with a temperature of ∼3.5 keV (∼40 MK), which cools as the flare subsides (it has a temperature of ∼2.2 keV during the “elevated” period and ∼1.4 keV during the “descending” count rate period; see Figure Set <ref> and Table <ref>).


 §.§ GJ 832

No significant discrepancies in stellar physical parameters. One confirmed exoplanet, GJ 832 b with  ∼0.6 M_ Jup and an orbital period of 3853 days <cit.>.

GJ 832 was observed twice by , once during a 8.9 ks (observation 0748010201) exposure during which the star was detected but too faint to enable timing or spectroscopic modeling, and once during a 28.6 ks exposure (observation 0860303301) when the star was overall brighter and underwent a short flaring event (see Figure Set <ref>). The quiescent spectrum is well-described by two thermal plasma components, while the lower signal-to-noise during the short flaring period allows the flare spectrum to be modeled with a single thermal plasma component. Overall, the luminosity of GJ 832 increases by a factor of ∼2.5 during the flare. The best-fit spectral models and a comparison of the flaring-to-quiescent spectrum are shown in Figure Set <ref> (see also Table <ref>). The spectrum extracted during the short flaring time during the observation shows an excess of X-ray flux at ∼2-30 Å, but shows weaker X-ray emission (compared to the quiescent spectrum) at wavelengths ≲2 Å.



 §.§ Kapteyn's Star

There are no significant discrepancies in the stellar physical parameters reported in the literature for Kapteyn's Star. Although the detection of sub-Neptune mass planet was reported by <cit.>, this was later refuted by <cit.> who argued that the observed RV variations were artifacts of stellar activity and rotation. There are not currently any additional candidate exoplanets around Kapteyn's Star.

Kapteyn's Star has been observed numerous times by , , and  (see Table <ref>), but is not detected in any of archival X-ray observations. We explored whether Kapteyn's Star would be detected in a stacked  X-ray image (the  background is lower and significantly less affected by spurious flaring events than ). The stacked  image has an effective exposure time of 72.5 ks. Kapteyn's Star is still not detected in the stacked image, with a count rate upper limit of <1.24×10^-4 ct s^-1. Assuming the best fit quiescent spectrum of Wolf 1055, this count rate corresponds to a luminosity upper limit of <1.1×10^24 .


 §.§ Wolf 1055

There are no significant discrepancies in the stellar physical parameters of Wolf 1055 or its binary companion VB 10 <cit.>. There is one confirmed exoplanet orbiting Wolf 1055: HD 180617 b (also identified as GL 572 Ab), with  ∼12 M_⊕ and an orbital period of ∼106 days (∼0.34 AU). 

Wolf 1055 and VB 10 are separated by ∼1.2^' and are easily resolvable with both  and . Both Wolf 1055 was observed once with  (for 24.2 ks) and twice with  (with exposure times of 12.2 ks and 29.2 ks). Both Wolf 1055 and VB 10 are detected in all three observations. A comparison of the  observation and the longer  observation (ObsID 7428) is shown in Figure <ref>. Wolf 1055 is bright enough in two out of these three observations to enable spectral modeling. Although VB 10 is bright enough to be significantly detected in all three observations and we are able to extract light curves from all three observations. The light curves of both stars obtained during  observation 0504010101 are shown in Figure Set <ref>.

The quiescent spectrum of Wolf 1055 is well-described by a three-component thermal plasma model, while the lower signal-to-noise during the flaring period in the second half of the  observation 0504010101 allows the flare spectrum to be modeled with a single thermal plasma component. Overall, the luminosity of Wolf 1055 increases by ∼80% during the flare. The best-fit spectral models and a comparison of the flaring-to-quiescent spectrum are shown in Figure Set <ref> (see also Table <ref>). The spectrum extracted during the flaring period during the second half of observation shows an excess of X-ray flux at ∼2 Å, and shows weaker X-ray emission (compared to the quiescent spectrum) at wavelengths ≲2 Å.


 §.§ AD Leo

There are no significant discrepancies in the stellar physical parameters reported for AD Leo; all stellar parameters are taken from <cit.> and references therein. There are no confirmed or candidate exoplanets orbiting AD Leo.

AD Leo is a known flaring star, and exhibits strong X-ray flaring both times the star was observed by . Both light curves are shown in Figure Set <ref>. High count rates enable spectroscopic modeling over multiple sub-exposures. The X-ray spectra during quiescent periods are generally well-described with a two-component thermal plasma model, while periods of flaring and enhanced variability require three thermal plasma components to adequately describe the observed spectra. Increases in X-ray count rate are driven almost entirely by the appearance of this third, hot (∼1.5 keV) thermal plasma component, which drives a significant increase (by three- to five-orders of magnitude) in X-ray emission below ∼2 Å (see Figure Set <ref>).


 §.§ Wolf 1061

There are no significant discrepancies in stellar physical parameters of Wolf 1061. No age estimate is available for the star. Wolf 1061 hosts three currently known exoplanets <cit.>: Wolf 1061 b, c, and d have masses of  ∼1.9 , ∼3.4 , and ∼7.7 , respectively, with orbital periods of ∼4.9 days (∼0.04 AU), ∼17.9 days (∼0.09 AU), and ∼217 days (∼0.47 AU), respectively. All planetary data taken from <cit.>.

Wolf 1061 was detected in one 38.3 ks observation with Chandra/ACIS-S with ∼240 net counts. The light curve shows evidence for variability (see Figure Set <ref>), but the star is not bright enough to enable spectroscopic modeling.


 §.§ Luyten's Star

There are no significant discrepancies in the reported stellar parameters for Luyten's Star in the literature. No age estimate is available. There are two confirmed exoplanets orbiting Luyten's Star: GJ 273b and GJ 273c, which were detected via RV measurements by <cit.>. GJ 273b and GJ 273c have masses of ∼2.9 M_⊕ and ∼1.2 M_⊕, respectively, and the planets have orbial periods of ∼18.6 days (∼0.09 AU) and ∼4.7 days (∼0.03 AU), respectively.

Luyten's star is detected at ∼5σ significance in a 28.4 ks observation with /ACIS-S. The light curve shows evidence of X-ray variability (see Figure Set <ref>), but are are unable to extract X-ray spectra due to the low number of net counts.


 §.§ Proxima Centauri

Proxima Centauri is a member of an extensively studied triple system along with α Cen A and α Cen B. There is currently one confirmed Earth-like exoplanet, Proxima Centauri b, which has  ∼1  and an orbital period of ∼11 days <cit.>. Two candidate exoplanets, Proxima Centauri c and d, are awaiting follow-up observations. If present, Prox Cen c <cit.> and d <cit.> are estimated to have  values of ∼6  and ∼0.26 , respectively, with orbital periods (distances) of ∼1900 days (∼1.5 AU) and ∼5 days (∼0.029 AU).

Proxima Centauri exhibits significant flaring events, such as those illustrated in Figure <ref> (see also Figure Set <ref>). The X-ray spectrum of Prox Cen is well-described by a two-component thermal plasma model (with temperatures of ∼0.25 keV and ∼0.85 keV) during quiescent times and periods of elevated or rising count rates. A third thermal plasma component is required to describe the spectra during flaring periods and during periods of descending count rates following a flare. During the flare, this third component is hot (∼1.5 keV), and cools to ∼0.6 keV as the count rate decreases (see Figure Set <ref> and Table <ref>). The flaring spectrum drives dramatic increases in both X-ray luminosity. Although the flaring spectrum of Prox Cen exhibits a significant increase in the X-ray flux at wavelengths <10 Å, the spectrum remains elevated (by an order of magnitude) compared to the quiescent spectrum out to long wavelengths (see Figure <ref>). This is in contrast to earlier-type stars, which generally show a similar X-ray emission between quiescent and flaring periods at wavelengths >100 Å.


cccccccccc
Stars with Available /XRT Imaging



    
    Star     Alternate     R.A.     Dec.     Spectral     Distance     Mass     T_ eff          log 
 3-4
    Name     Name(s)        2c(J2000)          Type^*       (pc)        ()     (K)          ()     ([]) 

    

    71 Ori     HIP 29650     06:14:50.77     +19:09:20.29     F6V     20.87     1.35     6533     2.94     <27.59 

    γ Lep A     GL 216A     05:44:27.45     -22:27:00.08     F6.5V     8.88     1.22     6258     2.34     27.53 

    ϕ^2 Cet     GL 37     00:50:07.34     -10:38:43.26     F7V     15.77     1.17     6250     1.85     <26.94 

    η CrB     GL 584A     15:23:12.23     +30:17:17.7     G2V     18.82     1.14     6029     1.64     <27.75 

    61 Vir     GL 506     13:18:24.97     -18:18:31.0     G7V     8.50     0.91     5585     0.84     <26.78 

    GL 567     LHS 5279     14:53:23.27     +19:09:13.54     K0V     11.54     0.93     5258     0.54     28.28 

    GL 68     LHS 1287     01:42:29.41     +20:15:55.87     K1V     7.60     0.88     5190     0.45     <26.29 

    μ Cas     GL 53A     01:08:22.74     +54:54:47.53     K1V     7.55     0.82     5316     0.46     <26.77 

    GL 785     LHS 488     20:15:18.88     -27:02:01.61     K2V     8.79     0.85     5071     0.4     <26.50 

    AK Lep     GL 216B     05:44:26.19     -22:25:24.26     K2V     8.89     0.80     4869     0.3     28.00 

    GL 688     HD 160346     17:39:16.72     +03:33:17.32     K2V     10.71     0.76     4982     0.34     <26.47 

    Lacaille 8760     AX Mic     21:17:10.80     -38:52:20.84     K9V     3.97     0.56     3599     0.1     26.89 

    Wolf 1453     GL 205     05:31:28.21     -03:41:11.50     M1V     5.70     0.56     3690     0.07     27.53 

    Lacaille 9352     GL 887     23:06:00.94     -35:50:49.79     M1V     3.29     0.54     3676     0.04     26.71 

    BR Pis     GL 908     23:49:13.59     +02:23:48.90     M1V     5.90     0.41     3685     0.029     <26.80 

    GL 229Ab     LHS 1827     06:10:34.46     -21:52:04.16     M1.5V     5.76     0.54     3912     0.06     27.55 

    GL 1     HD 225213     00:05:31.99     -37:22:03.88     M1.5     4.36     0.38     3696     0.02     26.75 

    Kruger 60 A     GJ 860A     22:27:58.11     +57:41:38.52     M3V     4.01     0.34     3344     0.01     27.45 

    Struve 2398Ba     GL 725B     18:42:43.94     +59:38:18.09     M3V     3.52     0.25     3345     0.021     27.57 

    Struve 2398Aa     GL 725A     18:42:43.94     +59:38:06.52     M3.5V     3.52     0.33     3401     0.015     27.57 

    ^aUnresolved binary. ^bUnresolved binary with a T7 brown dwarf.







§ UPDATED X-RAY LUMINOSITIES FOR /XRT-IMAGED STARS

We searched the /XRT archive for all nearby stars that did not have available  or  imaging. Nineteen of these stars (mostly K and M dwarfs; one system, Struve 2398, is a binary that is unresolved in the  imaging) have fluxes or flux upper limits available from , which we summarize in Table <ref>. The R.A. and Decl. coordinates for each star is taken from <cit.> and the stellar physical parameters are from <cit.>. We use the quiescent spectrum from Table <ref> of the star that mostly closely matches each  star's stellar physical parameters and the  response files[See <https://swift.gsfc.nasa.gov/proposals/swift_responses.html>] to convert  0.3-10 keV count rates to luminosities. Figure <ref> shows the / ratio as a function of T_ eff (as in Figure <ref>) with the nineteen  stars added.




aasjournal




 Paper Title: The large-scale anisotropy and flux (de-)magnification of ultra-high-energy cosmic rays in the Galactic magnetic field.












teresa.bister@ru.nl
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Nijmegen, The Netherlands
Nationaal Instituut voor Subatomaire Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands



gf25@nyu.edu
Center for Cosmology and Particle Physics, New York University, New York, NY 10003, USA



michael.unger@kit.edu
Institute for Astroparticle Physics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Institutt for fysikk, Norwegian University of Science and Technology (NTNU), Trondheim, Norway




§ ABSTRACT 
We calculate the arrival direction distribution of ultra-high-energy cosmic rays (UHECRs) with a new suite of models of the Galactic magnetic field (GMF), assuming sources follow the large-scale structure of the Universe. We find a significantly reduced dipole amplitude compared to previous GMF models, and trace the change to the accidental position of the peak of the extragalactic UHECR flux, which falls at the boundary of the strong flux de-magnification due to the GMF toward the central region of the Galaxy.
This serendipitous sensitivity of UHECR anisotropies to the GMF model will be a powerful probe of the source distribution as well as Galactic and extragalactic magnetic fields.  Demagnification by the GMF also impacts visibility of some popular source candidates.

The large-scale anisotropy and flux (de-)magnification of
 ultra-high-energy cosmic rays in the Galactic magnetic field
    Michael Unger
    August 5, 2024
=======================================================================================================================








§ INTRODUCTION
 
Ultra-high-energy cosmic rays (UHECRs) are the highest energetic particles measured at Earth, with energies from 10^18 eV to beyond 10^20 eV. Their origin remains unclear, mainly because UHECRs are charged nuclei which are deflected by cosmic magnetic fields during their propagation from the sources to Earth. Hence, the directions of sources can only be reconstructed from the UHECR arrival directions when magnetic field deflections are appropriately accounted for. A large part of the effect comes from the Galactic magnetic field (GMF) of the Milky Way, which has a field strength of order μG extending over tens of kpc.
The mass composition of UHECRs becomes heavier with increasing energy E <cit.>, with a relatively narrow range of rigidities ℛ≡ E/Ze ≈ 5 EV <cit.>, for energies E≳8 EeV relevant for this work. The Larmor radius of UHECRs is ∼5.5 kpc (ℛ/5EV)/(B/μG) - hence the GMF has a sizable impact on the propagation of UHECRs.

In recent years, the Jansson-Farrar GMF model from 2012 () <cit.> has been used to test hypotheses about the sources of UHECRs from irregularities in the UHECR arrival directions (e.g. <cit.>),
but the robustness of the conclusions of these studies was difficult to assess due to the absence of realistic alternative GMF models which also fit the full data. Several Galactic magnetic field models fit only Faraday rotation measures (RMs)  but not polarized synchrotron emission, and some models only fit for the disk field, see <cit.> and references therein; see also <cit.> for a recent study of the GMF halo component using high-latitude RMs and polarized synchrotron emission. For comparisons of UHECR anisotropy predictions by some of those models and  see, e.g. <cit.>.

Many of the references given above aim at modeling the dipole - the only currently significant anisotropy in the arrival directions of UHECRs at E>8 EeV <cit.>. It has a magnitude of ∼7.3% and a current significance of 6.9σ <cit.> in the field of view of the Pierre Auger Observatory <cit.>. All higher multipole moments, however, are compatible with isotropy, according to the joint analysis of the Pierre Auger and Telescope Array (TA) collaborations <cit.>.
Using the  model for the Galactic magnetic field, it was verified by  that the dipole amplitude including its energy dependence can be explained rather well if UHECR sources follow the extragalactic matter distribution and hence the large-scale structure (LSS) of the Universe, while the measured dipole direction is only roughly right.  Additionally, constraints on the source number density and the extragalactic magnetic field smearing were derived by requiring that all higher multipoles are compatible with isotropy.

Recently, new modeling of the GMF has become available <cit.> (hereafter ). In addition to being based on the latest astronomical data,  provides a suite of models using a variety of improved functional forms for the field and for the thermal and cosmic-ray electron densities which are needed to predict the observables (RMs and polarized synchrotron emission), intended to encapsulate the uncertainty in the coherent GMF. In this work, we discuss the predictions of the large-scale distribution of arriving UHECRs according to the new  GMF models.  We show in particular the important influence of the magnification and de-magnification effect of the Galactic magnetic field.

The relevance of the anisotropic (de)magnification due to the GMF is amplified by the fact that due to energy losses in propagation, the “UHECR illumination" of the Galaxy is quite inhomogeneous. If the source density is high enough that the source distribution reflects the distribution of matter (and UHECRs are not magnetically trapped within nearby Galaxy clusters <cit.>), the flux of UHECRs above 8 EeV arriving at the Galaxy will be considerably enhanced in the direction of the Virgo cluster and Great Attractor (see Fig. 2 of <cit.>). As we show, the alignment of the quite concentrated illumination map with the boundary of GMF demagnification has a strong impact on the predicted dipole magnitude and direction, enabling greater sensitivity in probing the various contributing factors.





§ DEPENDENCE OF THE ANISOTROPY ON THE GALACTIC MAGNETIC FIELD
 

We follow the analysis of , which refined and extended the work of  <cit.> where the source distribution follows the extragalactic matter distribution based on CosmicFlows 2 <cit.> within 350 Mpc and
a uniform source distribution is assumed for larger distances [We checked that varying the source evolution outside of the CosmicFlows volume from (1+z)^0 to (1+z)^± 3 leads to relative differences of the predicted dipole amplitude ≤± 20%, i.e. smaller than those from varying the coherent model of the GMF within the  suite, see Fig. <ref>.]. The UHECR emission spectrum was fitted to the measured cosmic ray energy spectrum <cit.> and composition <cit.> at Earth, and the sources were assumed to be identical.
For calculating the Galactic magnetic field deflections, we use the eight new  models of the coherent field <cit.> and adopt the Planck re-tune <cit.> of the  random field <cit.> (“Planck" or “Pl" in the following) as the baseline choice.  
The Planck-tuned random field scales down the amplitude of the  random field, as well as refits some other parameters like the amplitude of the Perseus spiral arm, to take into account the improved component separation in WMAP7 and Planck with respect to WMAP5 upon which  was based. We take l_c=60 pc as a benchmark coherence length and evaluate the influence of “Galactic variance" from the particular realization of the random field by using a second realization of that field. Additionally, we consider a model with l_c=30 pc.
As a further set of comparisons, we also show predictions for the two  models used by : the  model with the original  random field <cit.> (), and the solely coherent version (). To be able to judge the effect of the Planck-tuned random field against the original  random field, we also show  with the two realizations of the Planck random field with l_c=60 pc ().




Dipole direction


In Fig. <ref>, the predicted directions of the dipole are shown for the 8 different  models, as well as for the three tested  models for different energy intervals. The  models differ by up to ∼50^∘ from each other, considering differences over all energy bins. In general, all models predict the dipole direction relatively close to the indicated 1σ and 2σ contours of the measured dipole direction, consistent with the origin of the dipole being the anisotropic extragalactic source distribution following the LSS.
The dipole directions of the  models are in general more south and further away from the Galactic center than found with the  models, especially for lower energies. Additionally, it is visible that the differences between the two random field realizations and coherence lengths (30 pc and 60 pc) are around 𝒪(15^∘) at lower energies and 𝒪(5^∘) for E>32 EeV and are thus subdominant to the differences between models.
Because the dipole direction hardly differs between the three  models with entirely different random fields (Planck random field,  random field and no random field), the dipole directions of the  models will probably be reasonably stable in regard to updates of the random field model.

The dipole direction depicted in Fig. <ref> is calculated from the model for the idealized continuum case of infinite source number density. For a more realistic treatment where sources are discrete and randomly distributed following the LSS, variations of the dipole direction are expected due to cosmic variance. These variations increase strongly with decreasing source number density <cit.>. The regions encompassing 68% of all dipole directions are shown in blue in Fig. <ref>, for the  model and for source number density of n_s=10^-3 Mpc^-3 [In this publication, n_s is the density of contributing sources. It may be smaller than the actual density of sources if the source emission is strongly beamed <cit.>. For transient sources, it is n_s ≃Γτ where Γ is the volumetric rate of the transients and τ is the mean arrival time spread, which depends on the extragalactic magnetic field, see <cit.> and references therein.] (for reference, the density of Milky Way-like galaxies is ∼10^-2 Mpc^-3 <cit.>).
The uncertainty due to cosmic variance in source locations is significantly greater than that from the variations between the different coherent models or realizations of the random field. This means that the systematic uncertainty in reconstructing the origin of the UHECR dipole is not dominated by the uncertainty on the GMF, within variations of the  models. Even though subdominant, there are subtle differences between the individual  models regarding how well they reproduce the measured dipole direction. This is discussed further in the Appendix, see Fig. <ref>.







Dipole and quadrupole amplitudes

In addition to the dipole direction, the dipole amplitude and its energy evolution are important observables that should be reproduced.
We choose the quadrupole moment as a representative of all higher multipoles as it is the first to be outside isotropic expectations (see Fig. 10 of ), and its measured value including uncertainties is reported in <cit.>.  Figure  <ref> depicts the dipole and quadrupole moments for the different  GMF models, for various source densities.  As the source density decreases, cosmic variance from one realization to the next increases, greatly expanding the variation in predictions relative to the case of high source density and increasing the mean values of the dipole and quadrupole amplitudes.

It is noteworthy how similar the dipole and also quadrupole amplitudes are for all  models. The variations between different  coherent field models are nearly as small as the variation between the tested random field realizations.  The predictions of the  and the  models are also similar for the quadrupole amplitude, when the same random field model is used. While we did not explore the sensitivity to random field model for the  suite, we did check for the three  models covering a range of random magnetic field strengths and coherence lengths that the dipole and quadrupole moments decrease almost linearly with increasing amplitude of the turbulent field part – as expected since the random field smoothes out the structure.

However, the dipole amplitude is distinctly smaller in the continuum limit for all  models than for even the  model, which has the strongest random field. As elucidated below, the reason for the reduced dipole amplitude with the  models relative to the  models can be traced to the interplay between the peaks in the extragalactic flux distribution and the region of strong de-magnification from the Galactic magnetic field. On account of this intricate relation, conclusions on the compatible range of source number densities are presently subject to large uncertainties and may change in the future when updated models for the random part of the Galactic magnetic field and better constraints on its coherence length become available. Other uncertainties come from the extragalactic magnetic field, which is largely unknown, and the details of the large scale structure. In the Appendix (see Fig. <ref>), we discuss the possibility of a non-negligible extragalactic magnetic field.





In Fig. <ref> we show how the 1σ regions of dipole and quadrupole amplitudes evolve with the effective source number density, n_s, for the  model. Details and a visualization of the dipole and quadrupole moments for all models and energy bins, including uncertainties, is given in Fig. <ref> in the Appendix.
Cosmic variance leads to larger variations of the dipole and quadrupole amplitudes than the variations between the different  models already for n_s=10^-2 Mpc^-3. Therefore, conclusions on the source number density can be drawn with little sensitivity to the specific  model.
For large source densities, the dipole amplitude in the >8 EeV energy bin is smaller than the measured one for all  models [The mean source distance decreases with the energy on account of composition evolution and energy losses during propagation (see Fig. 1 in <cit.>). Thus, the too-low model dipole amplitude at lower energy cannot be increased by a local source in a suitable direction without increasing even more the already large amplitude for >32 EeV - at least if that source follows the same emission as all others <cit.>.]. Hence, contrary to the findings of  using the  model which showed compatibility for n_s≥10^-3.5 Mpc^-3, the present analysis using the  models is incompatible with the continuous case and fits best for smaller number density. For the dipole and quadrupole amplitudes of the  models to be compatible with the measured ones for E≥ 8 EeV within 1σ, in at least a fraction of realizations, the source density has to be n_s∼10^-4 Mpc^-3 (see also Figs. <ref> and <ref>).
We stress that, as will be discussed in the next section, this conclusion is potentially sensitive to the random field.







§ (DE)MAGNIFICATION BY THE GALACTIC MAGNETIC FIELD
 


To understand why the dipole amplitude predicted by the  models in the continuum limit is so much lower than with the  model, we show in Fig. <ref> the logarithm of the magnification for the  model and for the  model, for ℛ≡ E/eZ = 5 EV (the mean charge of UHECRs increases with the energy in such a way that the rigidity stays almost constant <cit.> at ℛ≈(5±3) EV over the whole energy range discussed here; see Fig 4 in <cit.>).  The magnification is defined to be the flux from a standard source in the respective direction, relative to the flux in the absence of the Galactic magnetic field.
Cosmic rays from some directions – notably from sources behind the central region of the Galaxy – are demagnified: they are deflected strongly and simply never reach the solar system. Since energy losses of UHECRs in their passage through the Galaxy are negligible, Liouville's theorem implies that the flux integrated over 4 π radians is preserved, hence the existence of demagnified directions implies directions with magnification >1; see <cit.> for more discussion of the mechanism and also  <cit.>.  Corresponding maps to Fig. <ref> for all  models including also variations of the random field are displayed in Fig. <ref> in the Appendix.

Comparing the magnification maps to the extragalactic flux distribution according to the LSS source model (illumination) shown in Fig. <ref>, one sees that the peak flux is in a demagnified region for the  models. By contrast, the  model is neutral or even magnifies the flux from those directions. The difference in magnification thus explains the significantly smaller dipole amplitude for the  models compared to the  model. Also, the invisible parts of the extragalactic illumination in the Galactic North explain why the direction of the dipole seen on Earth is displaced more towards the Galactic South for the  models.  That is to say, the fact that the LSS model for the extragalactic UHECR illumination is quite concentrated in the Virgo direction rather than being a smooth dipole, has a large impact on the predicted arrival directions when the (de-)magnification of some directions by the GMF is taken into account.

The systematic difference between the magnification maps of the  models and  in the region of the peak of the LSS illumination map that lead to the deviation in dipole amplitude can be traced to differences in the respective toroidal halo models. This will be discussed in a separate publication.






To extract the most robust predictions for (de)magnification in the  model suite, we display in Fig. <ref> the combined magnification maps for all of the  coherent and random field models studied, where the colored regions are regions of unanimity among the models and in the white regions there is no consensus (for comparison, we also show the respective maps for the  model in Fig. <ref> in the Appendix).
It is visible that the variations between models are not very large, and that they all agree regarding the large central demagnified area. To demonstrate the implications of this, we also depict the directions of popular source candidates often used in the literature (e.g. <cit.>). Several of those candidates like M83, M87, Mkn421, Mkn501, Cen A, NGC4945 and NGC253 lie in the demagnification area for some or most rigidities ℛ≲5 EV or in some cases for all rigidities shown.


One further important consequence of the sensitivity of the dipole amplitude to the interplay between the illumination and GMF magnification maps is that the predicted dipole amplitude and direction differ substantially when the illumination is replaced by an idealized dipole with the same amplitude and direction – for example the “2MRS dipole" which is often used in the literature, e.g. <cit.>. The amplitude of the predicted dipole is typically a factor-2 larger with the idealized dipole than with the LSS model for the  models, and its direction differs by 𝒪(20^∘-60^∘); see Figs. <ref> and <ref> in the Appendix. Since GZK energy losses in propagation create inhomogeneities in the UHECR arrival flux (as captured at least approximately in the LSS model), predictions for the UHECR arrival direction anisotropy using a smooth dipole approximation for the illumination map based on the very large-scale dipole in the galaxy distribution averaged over distances much larger than that of contributing UHECR sources, can be expected to be misleading.




§ CONCLUSIONS

We have investigated the sensitivity of the predicted large-scale anisotropy of ultra-high-energy cosmic rays to the coherent part of the Galactic magnetic field, and also made a preliminary study of the dependence on the random part of the field. We find that the measured dipole which has been detected in the UHECR arrival flux <cit.> can be described reasonably well by a model where the sources follow the large scale structure of the universe and UHECRs are deflected by any of the suite of  GMF models <cit.> (as is also the case for the  model <cit.>).

The variations among the predicted dipole and quadrupole amplitudes, and among the dipole directions, when using different  models including different setups of the random field are small and subdominant to cosmic variance from random source positioning. Hence, the uncertainty on UHECR arrival directions from the Galactic magnetic field modeling, within the  family, will likely not obstruct conclusions about the sources of UHECRs based on their large-scale anisotropies. At the same time, we find enough differences among models, such that in the future with refined treatment of the random field, composition sensitivity, and LSS source modeling, it should be possible to disfavor or prefer some of the models.

An important discovery of our work, which goes beyond specific models of the GMF, is the unanticipated sensitivity of the dipole amplitude to the coherent field model. This results from the delicate interplay between demagnification of flux from sources behind the central portion of the galaxy and the direction of strongest extragalactic illumination from the Virgo cluster and Great Attractor. This sensitivity of the dipole amplitude will be a powerful tool to probe not only the GMF, but also the UHECR source distribution and potentially even hadronic interaction models which impact the charge assignment. The pattern of extragalactic illumination changes with UHECR energy, which should help in discriminating different contributing factors in the future.

The area of the sky where the flux is severely demagnified in the  model suite includes popular source candidates like M87, M83 and NGC 253, which are thus not expected to contribute many UHECRs at Earth, except for rigidities ℛ >5 EV; see Fig. <ref>. Another consequence of the demagnification is that using an idealized extragalactic dipole with the same direction and strength, but neglecting the intermediate-scale anisotropy due to GZK energy losses, gives misleading results.


The delicate relationship between the direction and amplitude of the peak extragalactic flux and the blind directions resulting from GMF demagnification implies that conclusions about the GMF model and source number density are sensitive to details of the source distribution as well as the random part of the Galactic magnetic field and the possible influence of the extragalactic magnetic field.  Thus conclusions about the UHECR source density and the relevance of cosmic variance in the source distribution must be left to the future when these aspects of the problem are better understood.






Acknowledgments: We thank Foteini Oikonomou for useful feedback on our analysis. The work of T.B. is supported by a Radboud Excellence fellowship from Radboud University in Nijmegen, Netherlands and that of G.R.F. is supported by National Science Foundation Grant No. PHY-2013199. T.B. and M.U. thank the Center for Cosmology and Particle Physics of New York University for its kind hospitality and T.B. acknowledges the support from the Alfred P. Sloan Foundation facilitating this research and the production of this paper.














 §.§ Agreement of the  models with measured dipole directions and multipole amplitudes
 
Even when we cannot make strong judgments regarding the quality of the dipole direction prediction for the different models due to the large influence of cosmic variance and present uncertainties on the random field and the LSS distribution, it is still possible to determine which models lead to a better agreement with the data than others.
Fig. <ref> (left) shows the number of realizations of source locations (out of 10,000 total, for each n_s) for which the predicted dipole direction is within the 2σ uncertainty of the measured dipole direction, simultaneously in all mutually exclusive energy bins E=(8-16) EeV, E=(16-32) EeV, and E>32 EeV.
At large source number density, when cosmic variance between realizations is subdominant to the differences between models, there are substantial variations between the different  models and between the different random field realizations. This can be compared to Fig. <ref> which shows the dipole directions in the continuous limit.

With decreasing source density, the number of compatible realizations decreases as expected since the variance of dipole directions increases. For source densities n_s≲ 10^-3 Mpc^-3 (which gives better agreement with the measured dipole and quadrupole amplitudes than higher densities for the  models as discussed in the main text), the differences between the models decrease and it is clearly visible that the bulk of  models gives a better fit to the dipole direction than the predecessor  model. Comparing the eight  models, no best model can be unambigously identified due to the relatively large fluctuations between different random field setups (large Galactic variance) and the variation of the number of compatible simulations with the source number density, but it can be seen that the  model consistently presents a worse fit to the dipole direction than the other models.







We show in Fig. <ref> (right) the number of simulations out of 10,000 total where the dipole direction and the dipole amplitude and the quadrupole amplitude are all within 2σ of the measured value. From Fig. <ref> (left) it is clear that because of cosmic variance, the number of simulations agreeing with the measured dipole direction decreases strongly with the source density. This is as expected, and thus does not necessarily exclude small source densities. But, from Fig. <ref> we know that densities n_s≲10^-5 Mpc^-3 are disfavored (for negligble extragalactic magnetic field) as they lead to too large quadrupole amplitudes. Large source densities n_s≳10^-2 Mpc^-3 on the other hand generally lead to too small dipole amplitudes for the  models. In general, agreement with all three observables is most often reached for n_s=10^-3 Mpc^-3, and especially for the , ,  and  models. These models can even reach a compatible dipole amplitude for large source densities n_s=10^-2 Mpc^-3. The random field realization plays a very large role in that case, and a compatibility with all three observables is only reached for few specific models.
The , , and  models, even though being a fair fit for the dipole direction as visible in Fig. <ref> (left), are less often compatible with the multipole amplitudes than the other models and are thus not favored according to Fig. <ref> (right). The  model is also disfavored according to Fig. <ref> (right), but that is due to the dipole direction not fitting as well as explained above.






§ INFLUENCE OF THE EXTRAGALACTIC MAGNETIC FIELD
  
Another important impact on the anisotropy of the UHECR flux comes from the extragalactic magnetic field (EGMF), which can dampen the multipole moments significantly.  modeled this effect by a smoothing of the arrival flux of the following form <cit.>:

    δθ = 2.9^∘B/nG10 EV/E/Z√(D L_c)/Mpc = 2.9^∘β_ EGMF10 EV/E/Z√(D/Mpc),

with the EGMF field strength B, the EGMF coherence length L_c, and the mean source distance D. In the second equality, the combination β_ EGMF≡ B / nG √(L_c / Mpc) is introduced to isolate the quantity that can be constrained.

Using the  model for the GMF,  it was investigated by  which combination of the EGMF parameter β_ EGMF and the source density n_s can produce a large enough dipole moment while keeping all higher multipole moments small enough to agree with the data. Here, we update these findings using now the  models for the GMF. For comparibility we use the same definition of two criteria (note that these criteria are slightly different than the ones used above in the main text:

    
  * The dipole moment in the Auger field of view at E>8 EeV should be d_8 EeV>5%. This value is around 2.5σ below the value measured by Auger <cit.>, so it constitutes approximately a 99% C.L. lower limit on the dipole amplitude.
    
  * All the higher multipole moments C_l>1 must be within the 99% isotropic expectation for all energy ranges (E=(8-16) EeV, E>8 EeV, E=(16-32) EeV, and E>32 EeV).





In Fig. <ref>, the number of simulations that fulfill both criteria simultaneously is visualized, both for the  and a selection of the  models. Here we choose to show the , , , and  models where the latter three exhibit the smallest and largest dipole amplitudes and the largest quadrupole amplitude, respectively (Fig. <ref>).
While for the  model negligible field strengths and very large source densities were preferred, for all shown  models smaller source densities and larger extragalactic magnetic fields are favored. This is as expected due to the smaller dipole amplitudes with the  models that have to be compensated by smaller source densities as seen above. The parameter space that is compatible with the data is very broad and extends over multiple orders of magnitude in source density. The region where most simulations fulfill both criteria stated above is at very small source densities of around n_s∼10^-8 Mpc^-3 in combination with a sizeable EGMF of around β_ EGMF∼1 nG Mpc^1/2. Comparing this to the densities of different source classes as indicated in Fig. <ref>, the favored region overlaps with the density of blazars <cit.>. More abundant source classes like starburst galaxies, other types of AGNs or even Milky-Way like galaxies are however also all compatible with the  models. The same is true for all transient source classes indicated in Fig. <ref>. Note, as shown above, the dipole direction is almost completely random for these small source densities.


As in the case without EGMF described above, the region of the parameter space of n_s and β_ EGMF is very sensitive to the LSS model and the random part of the Galactic magnetic field and may hence be subject to change once updated models of that become available. Additionally, a more accurate treatment of the EGMF deflections than the simplified smearing we employ here could lead to changes of the compatible values for the source density and EGMF parameters.




§ DIPOLE AND QUADRUPOLE AMPLITUDES

In Fig. <ref>, we show the dipole and quadrupole amplitudes and their 1σ uncertainties for various source number densities, for the different GMF models and no extragalactic magnetic field, as a function of energy.







§ ADDITIONAL MAGNIFICATION MAPS

In Fig. <ref>, the magnification maps for all  models with l_c=60 pc, as well as for the  models with different random field models are shown.
To determine how much the uncertainty on the random field influences the magnification map, we show in the second row the magnification maps for the  model with the three tested random fields (two realizations with l_c=60 pc and one with l_c=30 pc). The particular realization of the random field makes almost no difference to the magnification map, while the coherence length can have a visible impact. Especially the exact form and size of the demagnified region are sensitive to the coherence length.




Fig. <ref> displays the combined magnification maps (see main text for explanation) for all combinations of  and random field models used 
in this work. It is visible that the areas of magnification and de-magnification are distinctly different as for the  models. Many of the depicted source candidates like Mkn501 and NGC253 again lie in the demagnification area for rigidities ℛ≤5 EV – but, sources in the equatorial North like Mkn421 or M82 are significantly magnified with  in contrast to .











§ IDEALIZED EXTRAGALACTIC DIPOLE

In Fig. <ref> we show the dipole amplitude predicted by the LSS model, compared to a model where the flux at the edge of the galaxy (the “illumination map", Fig. <ref>) is replaced by an idealized dipole with the same amplitude and direction as for the LSS model. The amplitude of the LSS dipole at the edge of the galaxy is 6.2% for the E=(8-16) EeV energy bin, 8.1% for E>8 EeV, 11.9% for E=(16-32) EeV, and 21.5% for E>32 EeV <cit.>.
It is visible that the dipole amplitude is highly sensitive to the inhomogeneities in the extragalactic flux. For all shown  models, the amplitude of the LSS model is significantly smaller (around a factor 2) than for the model with idealized extragalactic dipole, while this relation is the other way around for the  model. This is due to the intricate relation between the illumination and the magnification of the GMF, which differs significantly between the  and  models as explained in the main text.




In Fig. <ref>, we show the dipole directions calculated when replacing the flux at the edge of the galaxy predicted by the LSS model by the idealized dipole. The direction of the dipole calculated at Earth differs substantially – by 𝒪(20^∘  to 60^∘) – between this simplification and the realistic model where the sources follow the LSS. Also, the direction predicted using the idealized dipole is systematically displaced towards the North, especially for lower energies, and moves significantly less with the energy than for the LSS model. Comparing the prediction for the  model with idealized dipole to the uncertainty contour from cosmic variance for n_s=10^-3 Mpc^-3 (Fig. <ref>), it is visible that the predicted dipole direction of the idealized dipole model is even outside that sizable uncertainty for lower energies.




This test demonstrates that simply replacing an extragalactic source catalog by its dipole component (the “2MRS dipole" which is often used in the literature, e.g. <cit.>) and ignoring higher multipoles of the distribution, can only give a rough idea of the deflection direction and expected dipole amplitude, but should not be expected to produce accurate predictions of the expected anisotropy at Earth.




 Paper Title: Constraining the AGN formation channel for detected black hole binary mergers up to z=1.5 with the Quaia catalogue.



firstpage–lastpage
A τ Matrix Based Approximate Inverse Preconditioning for Tempered Fractional Diffusion Equations
[This work was supported by the National Natural Science Foundation of China (No. 12001022).

^∗ Corresponding author: Chaojie Wang.


E-mail address: wangcj2019@btbu.edu.cn ]

    Xuan Zhang^a, Chaojie Wang^a,∗, Haiyu Liu^b
    August 5, 2024
=================================================================================================================================================================================================================================================================================




§ ABSTRACT 
Statistical analyses based on the spatial correlation between the sky maps of
Gravitational Wave (GW) events and the positions of potential host environments
are a powerful tool to infer the origin of the black hole binary mergers that have
been detected by the LIGO, Virgo, and KAGRA instruments. In this paper, we
tighten our previous constraints on the fraction of detected GW events that may
have originated from Active Galactic Nuclei (AGN). We consider 159 mergers detected
not later than June 1st, 2024, and the all-sky quasar catalogue Quaia. We
increase by a factor of 5.3 and 114 the number of considered GW sources and AGN
respectively, also extending our analysis from redshift 0.3 to 1.5. This is
possible thanks to the uniformity of the AGN catalogue and its high level of
completeness, which we estimate as a function of redshift and luminosity. We
find at a 95 per cent credibility level that un-obscured AGN with a bolometric
luminosity higher than 10^44.5 erg s^-1 (10^45 erg s^-1)
do not contribute to more than the 21 (11) per cent of the detected GW events.





gravitational waves - transients: black hole mergers - galaxies: active - methods: statistical






§ INTRODUCTION



With the only exception of the Gravitational Wave (GW) event GW170817
<cit.>, a confident one-to-one association between the mergers
of compact objects detected by the LIGO-Virgo-KAGRA (LVK) collaboration
<cit.> and their host galaxy is currently out
of reach.

This is primarily caused by the typical size of the uncertainties that are
associated to the sky position and to the luminosity distance of each GW
event. The difficulty of associating each merger to its host environment
hampers our ability to pin down the physical origin of the binary systems
of which the coalescences have been directly measured. In one class of the
possible formation scenarios that have been proposed
<cit.> binaries of compact objects like
binary black holes, binary neutron stars, and neutron star-black hole binaries
are efficiently assembled and driven to merger in dense environments like
globular clusters <cit.> or nuclear stellar clusters
<cit.>. In such hosts, binaries can have their semi-major axes
shrunk by interactions with single objects. This binary hardening from
single-binary encounters is important for the formation of systems that are
able to merge within a Hubble time due to loss of orbital energy, emitted in
the form of GWs. The least massive object of the three is expected to be ejected
from the location of the single-binary interaction <cit.>,
meaning that the hardened remnant binary will be composed by the two heaviest
elements that partake in the encounter.

Another reason why dynamically dense environments might be the hosts of the
most massive mergers detected by the LVK collaboration is the fact that their
escape velocities can be large enough to retain the recoil-kicked remnant
of a coalescence, turning it into a potential component of a subsequent GW event.
This “hierarchical merger” scenario might be a physical interpretation
for the existence of stellar-mass Black Holes (sBHs) in the pair-instability
mass gap <cit.>. This discontinuity is
predicted between ≈50 M_⊙ and ≈120 M_⊙
in the sBH mass spectrum as a consequence of the complete disruption of the
core during the supernova event at the end of the life cycle of stars with
very high masses, that are therefore expected to leave no remnant
<cit.>. However, the astrophysical mass distribution
of sBHs predicted by the LVK collaboration shows evidence for the presence
of objects in the pair-instability mass gap <cit.>. This
challenges the hypothesis that all the merging systems detected by the LVK
collaboration have originated from an isolated stellar binary.

Accretion discs of AGN are a unique type of potential host environment for the
assembly and the merger of binaries of compact objects. The reason is that in
this so-called “AGN formation channel” the binaries and their components
are not expected to interact only with other stellar objects, but also with
the gas the accretion disc consists of. Thanks to this interaction, the disc
might for example capture compact objects that have orbits moderately inclined
with respect to its plane <cit.>. These disc captures
have the effect of increasing the number density of compact objects in the disc,
where the binary formation can be gas-assisted <cit.>.
Another process which, as far as the formation of merging binaries of compact
objects is concerned, is typical of the AGN scenario is migration: the radial
motion of compact objects orbiting around the central supermassive black hole.
Inward migration takes place in AGN discs where the net torque exerted by the
gas onto the orbiting compact object is negative <cit.>,
and can increase the number density of sBHs and neutron stars in the inner
part of the accretion disc, facilitating the formation of binaries.

One possible way to put constraints on the fractional contribution of the AGN
channel to the total merger rate of compact object binaries is through the
investigation of the spatial correlation between the sky maps of the events
detected by the LVK collaboration and the positions of observed AGN. This approach
has been first suggested in <cit.>, where it was estimated that 300
GW detections are needed to statistically prove, with a significance of 3σ,
the GW-AGN connection if half of the mergers happened in an AGN. In
<cit.> it was later found that the number of required detections
decreases by a factor ≈3 if it is assumed that the inclination of the
binary angular momentum with respect to the line of sight is known with an
uncertainty of 5^∘. In the simulated GW detections used in
<cit.> and <cit.> the interferometers of LIGO and
Virgo have been assumed to be working at design sensitivity. This level of
instrumental sensitivity is a factor ≳2 better with respect to the one reached
by the interferometers during the first three observing runs <cit.>.

In <cit.> the analysis presented in <cit.> was repeated
assuming a realistic distribution of the sizes of the 90 per cent Credibility Level (CL) localisation
volumes (V90) of the mock GW detections. This distribution has been created by
using the sensitivity curves that characterised the LIGO and Virgo interferometers
during their third observing run (O3). It was found that the amount of data
collected during O3 would be enough to prove with a 3σ significance that
the AGN channel contributes to half of the total merger rate only if the GW
events originate form a rare host population with a number density lower than
10^-7 Mpc^-3. Integrating the the AGN luminosity function of
<cit.>, we find that this density corresponds to objects in the
local Universe with a bolometric luminosity greater than approximately
10^46 erg s^-1.

By building on this previous works, the first observational constraint on the
efficiency of the AGN channel was put using the spatial-correlation approach
in <cit.>, where the completeness of the quasar catalogues used
during the cross-match with GW sky maps and the exact position of the
potential host environments have been taken into account for the first time.
By using catalogues of AGN with a spectroscopic estimate of redshift of z≤0.3
obtained from Milliquasv7.7b <cit.> and the sky maps of the 30 GW
events detected in the same redshift range during the first three observing runs
of the LVK collaboration, it was found that the fraction of the detected mergers
that have originated in an AGN more luminous than 10^45.5 erg s^-1
(10^46 erg s^-1) is not expected to be higher than 0.49 (0.17) at a CL of 95 per cent.
A similar statistical investigation has been conducted also in <cit.>.
In this case the sky maps of the GW events detected during O3 have been cross-matched
spatially and temporally with the 20 unusual AGN flaring activities detected by
the Zwicky Transient Facility <cit.> that have been identified
in <cit.> as potential transient electromagnetic counterparts. We find no
evidence for a correlation the GW events and the unusual flares. A similar result
has been found in <cit.> analysing the spatial correlation between the
event GW190521 and an unusual flaring activity of the AGN J124942.3+344929.

In this work we present new observational constraints on the fractional contribution
of the AGN channel to the total merger rate, f_ AGN. We use a
spatial-correlation-based method similar to the one presented in <cit.>,
and apply it to a larger dataset. In particular we use the sky maps of the
GW events detected by the interferometers of the LVK collaboration up to June 1st,
2024, therefore including data coming from the fourth observing run (O4). This
brings the total number of used sky maps to 159, which is more than five times
the amount of GW events used in <cit.>. These are cross-matched with
the AGN of the all-sky catalogue Quaia <cit.>, which is derived
from the catalogue of extra-galactic quasar candidates published by the Gaia mission
<cit.>. The remarkable uniformity and completeness up to z=1.5 of Quaia
make it a relevant tool for all-sky spatial correlation analyses like the one
presented in this work.

In Section <ref> we present the main characteristics of our
dataset, which includes the different GW detections and the Quaia
catalogue. In Section <ref> we then describe the likelihood
function that we maximise in order to obtain our new constraints on
f_ AGN, which are presented in Section <ref>. Finally, in
Section <ref>, we draw the main conclusions regarding the AGN
channel that can be inferred from the results of our analysis, and discuss
about future developments of statistical spatial-correlation-based approaches.
We adopt the cosmological parameters of the Cosmic Microwave Background
observations by Planck <cit.>:
H_0=(67.8±0.9)  km  s^-1 Mpc^-1,
Ω_ m=0.308±0.012, and n_ s=0.968±0.006.







§ DATASETS


In this section, we first present the main characteristics of the GW events
the sky maps of which are used in our analysis. We then describe the main
properties of the Quaia catalogue and how we estimate its completeness
as a function of redshift for different sub-samples of it, characterised
by different thresholds of bolometric luminosity, L_ bol.





 §.§ GW events


In this work we use the 159 sky maps of mergers of compact object binaries detected
by the interferometers of the LVK collaboration up to June 1st, 2024.
In particular, all the GW events from the first three observing
runs are included, with the only exclusion of GW200308_173609 and of 
GW200322_091133. These two events are not taken into consideration because
they are poorly localised, and the corresponding value of V90 cannot be
estimated from the currently available posterior samples. Moreover, we use
all the events from the first half of O4 (O4a) as well as all the events
from its second half (O4b) that have been detected not later than June 1st,
2024. For this currently ongoing observing run, we select the detections
that have a probability of being of terrestrial origin smaller than 1 per
cent.

The sky maps of the events detected during the first three observing
runs are downloaded from the Gravitational Wave Open Science Center
<cit.>. We use the posterior samples obtained
using the IMRPhenomXPHM waveform model <cit.>
for all these events but GW190425_081805, GW191219_163120,
GW200105_162426, and GW200115_042309. For these four events
we use the Mixed posterior samples.

The sky maps of the 71 events of O4 are downloaded from the Gravitational-Wave
Candidate Event Database [<https://gracedb.ligo.org/>], operated
by the LIGO Scientific Collaboration. For each event, we use the most recent
version of its sky map, which is either obtained from the Bilby localisation
algorithm <cit.> or from the Bayestar one <cit.>.

Figure <ref> shows the luminosity distance of all the 159 GW
events used in this work as a function of V90. The median value of the
luminosity distance (2012.3 Mpc) and of V90 (7.67·10^8 Mpc^3)
are indicated by a horizontal dashed line and by a vertical one, respectively.






 §.§ Quaia AGN catalogue

In order to obtain tight constraints of f_ AGN with methods based
on spatial correlation like the one used in this work, is important to use
an AGN catalogue that has the highest possible value of completeness. This
property of a catalogue is in general a function of the redshifts, the
sky positions, and the luminosities of the objects it contains.

In the analysis here presented, the Quaia AGN catalogue <cit.>
[All the data concerning the Quaia catalogue, and its modeled
selection function are publicly available at
<https://zenodo.org/records/8060755.>] is cross-matched with the sky maps
of the GW events detailed in Section <ref>. This is done once the
completeness of such a catalogue has been estimated. Quaia is an all-sky
catalogue is based on sources identified as quasar candidates in the
third data release of the Gaia mission <cit.>. The final version of
this catalogue is obtained by selecting the objects that have an infrared
counterpart in the unWISE catalogue <cit.> and by performing
cuts in colours and proper motion in order to decrease the amount of
contaminants <cit.>. The result
is a catalogue containing 1,295,502 quasar candidates with a magnitude in the
Gaia G-band mag_ G<20.5. The redshifts of all these objects are
estimated using a k-Nearest Neighbors model trained on the AGN present in
Quaia of which the spectra are in the 16th Data Release (DR16Q) of the the
Sloan Digital Sky Survey (SDSS) <cit.>.

Figure <ref> shows the mollweide projection of the sky distribution
of the AGN contained in the Quaia catalogue. The colour of each pixel depends
on the amount of objects in the correspondent position in the sky. The size
of each pixel is ≈2.557·10^-4 steradians. Its remarkable uniformity
outside the region containing the Milky Way Galactic plane makes Quaia a
useful tool for spatial correlation analyses between AGN and GW events.



The AGN here used during the cross-match with the GW sky maps are the 660,031
objects present in Quaia that have a redshift estimate not larger than
z=1.5 and that are in the region of the sky where the galactic latitude
b is either higher than 10^∘ or smaller than -10^∘. The redshift
selection is performed since all the GW events used in this work are
expected to have taken place in the 0≤ z≤1.5 range (see Figure
<ref>). We choose not to use the objects with a small absolute
value of galactic latitude for simplicity. We consider the density of objects
in the catalogue to be independent from their Right Ascension or Declination,
as long as they are outside of the region of the Galactic plane.

The selection function marginalised over redshift that has been modeled
for Quaia highlights density variations in regions of the sky outside the
Galactic plane <cit.>. In this work thes
selection effects are not taken into account, and we obtain estimates of the
completeness of Quaia using the sky-averaged number density of objects with
the only exclusion of the Galactic plane (see Section <ref>).
This choice is not expected to have a significant effect on our final results,
because during the likelihood-maximisation process we use the average value
of the catalogue's completeness inside the 90 per cent CL localisation volume
for each GW event, and the sky projection of such regions is usually much
greater than the typical angular size of the modeled number density variations.

The exclusion of the Galactic plane from Quaia removes only ≈ 1.6
per cent of all the objects within z=1.5 and therefore is not expected to
decrease notably the constraining power of our analysis.



 §.§ Completeness estimation


The likelihood maximisation method presented in this work requires an estimate
of the completeness of the AGN catalogue that is used. We therefore estimate
this property of our Quaia AGN sample as a function of redshift and luminosity.
We do so in three steps:

    
  * Estimation of the bolometric luminosity of each object in Quaia with
    a redshift estimate not higher than z=1.5. This sub-section of Quaia
    is later referred to as Quaia_ z<1.5, and it differs from
    our sample used during the cross-match with GW sky maps only because in the
    latter the objects in the Galactic plane region have been removed. In
    Section <ref> we describe how these luminosities are
    evaluated starting from the magnitudes in the Gaia G_ RP band
    (mag_ G_ RP);
    
  * Comparison of the bolometric luminosities obtained from the Gaia
    measurements with the ones calculated from SDSS data and presented in
    <cit.>. We do this for all the objects that are both in our sample
    of Quaia and in SDSS DR16Q, and then use the results of this comparison
    to correct the luminosity of each object in our catalogue. This step is
    done in order to have estimates of the luminosities that are compatible
    with the ones of SDSS, which is the survey used during the fit of the
    luminosity function we compare the number of objects in our catalogue with
    to calculate its completeness of our catalogue. The comparison between the
    luminosity estimates obtained from Gaia data and the ones from SDSS is
    detailed in Section <ref>.;
    
  * Comparison between the number of objects present in our quasar sample
    in a specific redshift bin and the ones predicted by the AGN luminosity
    function presented in <cit.>. This is done five times,
    considering each time a different threshold in bolometric luminosity. The
    comparison with the quasar luminosity function and the final calculation of the
    completeness of our catalogue are delineated in Section <ref>.




  §.§.§ Evaluation of bolometric luminosity from mag_ G_ RP:


To obtain an estimate of the bolometric luminosity of all the objects contained
in Quaia_ z<1.5, we first calculate the flux density in
photo-electrons per second using the following standard relation:

    f_ν_ G_ RP[e^-/s]=10^-m_ G_ RP,VEG- ZP_ RP,VEG/2.5  ,

where ZP_ RP,VEG=24.7479 is the photometric zero-point in the
Vega system for the Gaia G_ RP band. The flux density in Jansky
is then calculated by multiplying f_ν[e^-/s] by the
conversion factor c_ν=3.299·10^-36 Jy· s/e^-. The
values of ZP_ RP,VEG and of c_ν are taken from the online
documentation regarding the calibration of Gaia data
[<https://www.cosmos.esa.int/web/gaia-users/archive>].

The intrinsic luminosity of each AGN emitted at a rest-frame frequency of
ν_ G_ RP=10^14.588 Hz [this value has been
calculated assuming a fiducial wavelength for the G_ RP band
of 7750Å.] is then calculated as follows:

    ν_ G_ RPL_ν_ G_ RP=ν_ G_ RPf_ν_ G_ RP(4π d_ lum^2)(1+z)^-0.657  ,

where d_ lum is the luminosity distance correspondent to the
redshift z of each object. The last term of Equation <ref>
is used to take into account the shape of the typical Spectral Energy Distribution
(SED) of the objects in Quaia. The value of the exponent is calculated from
a linear fit of the mean SED of all SDSS quasars presented in <cit.>.
In particular,
in the range between ν_ G_ RP and
(1+z_ max)ν_ G_ RP (where, for
Quaia_ z<1.5, z_ max=1.5), we find that:

    log(ν L_ν)∝log(ν)·0.657  ,

and therefore

    ν_ em L_ν_ em= ν_ G_ RPL_ν_ G_ RP(1+z)^0.657  ,

where ν_ em L_ν_ em is the luminosity emitted in the rest-frame
frequency that is observed at ν_ G_ RP.

The bolometric luminosity is then calculated by multiplying
ν_ G_ RPL_ν_ G_ RP by the value that the
frequency-dependent bolometric correction presented in <cit.> has
at the frequency ν_ G_ RP. This value is 11.004.



  §.§.§ Comparison with the bolometric luminosity estimates
of <cit.>


After computing the bolometric luminosities from the Gaia RP magnitudes
following the procedure described in Section <ref>,
we can now compare to the ones listed in the catalogue presented in
<cit.>. This catalogue contains continuum and emission-line properties
of the 750,414 broad-line Quasars of SDSS DR16Q. Among these properties
there is the bolometric luminosity of each AGN, estimated from the continuum
luminosity at the rest-frame wavelengths of 5100, 3000, and 1350 Å. We
perform a cross-match between this catalogue and Quaia_ z<1.5,
adopting to arcsec as matching radius. A total of 136,368 matches are found.
For each of these AGN it is possible to calculate the difference between the
logarithm of the bolometric luminosity estimated from mag_ G_ RP
and the logarithm of the bolometric luminosity taken from the catalogue of
<cit.>. The mean value of the distribution of this differences is 0.073,
and its standard deviation is 0.213. The bolometric luminosities obtained from
mag_ G_ RP are on average slightly over-estimated. We then
proceed to correct this difference.

We divide the AGN that are both in Quaia_ z<1.5 and in the
catalogue presented in <cit.> into five bolometric luminosity bins.
This partition is performed using the estimates calculated starting from
mag_ G_ RP. Each of these sub-samples is then divided
in redshift bins, using the Quaia estimates of redshift. For each sub-sample
in luminosity we choose a different number of linear redshift bins and a
different value of the maximum redshift to consider, to ensure that in each
of these bins there are at least 10 objects. In Table <ref> we
list the minimum and the maximum bolometric luminosity for each of the five
sub-samples, the number of redshift bins in which it is divided, and the maximum
redshift considered in this division.

    
For each bin in luminosity and redshift, we perform a linear fit between
the logarithm of the bolometric luminosities estimated starting from
mag_ G_ RP, and the one of the bolometric luminosities
taken from the catalogue of <cit.>. We therefore obtain one set of
best-fit parameters for each bin in luminosity and redshift. We finally
use these best-fit parameters to correct the bolometric luminosity estimated
from mag_ G_ RP of each object contained in our quasar
sample. These adjusted estimates are the ones used in the rest of the
analysis presented in this paper. The distribution of the difference between
the logarithm of the original bolometric luminosities obtained from the Gaia
magnitudes and the logarithm of their corrected version has a mean of
0.073 and a standard deviation of 0.027.



  §.§.§ Comparison with the AGN luminosity function of <cit.>


In order to have an estimate of the completeness of our Quaia sample as a
function of redshift we compare the number of objects it contains with the
expectation value calculated from the AGN luminosity function presented in
<cit.>. In particular we use the best-fit double power law
function in which the parameters evolve as a function of redshift according to
Model 1 <cit.>.

We compare the observed number of AGN with the predicted one for five different
sub-samples of the catalogue. Each of these sub-samples is characterised
uniquely by a different threshold in bolometric luminosity. We divide all the
sub-samples into 8 linear redshift bins between z=0 and z=1.5. We calculate
the expected number of objects in each bin by integrating the luminosity function
of <cit.>. We do this for each value of luminosity threshold.
The luminosity function is expressed as a function of the absolute monochromatic
AB magnitude at a rest frame of 1450Å. For this reason, in order to obtain the
expected number of objects as a function of redshift for a specific threshold of
bolometric luminosity, we first convert this luminosity in the UV rest frame
magnitude. We do this using the magnitude-dependent bolometric correction function
presented in <cit.>.

The dashed lines in Figure <ref> show linear interpolations
of the number of expected AGN in all the different redshift bins, obtained through
the integration of the luminosity function. Different colours correspond to different
bolometric luminosity thresholds. The round markers correspond to the number of
objects brighter than the same threshold luminosities that are contained in
our quasar sample in the different redshift bins, divided by
1-sin(10^∘) to take into account our cut in galactic latitude.
For comparison, the crosses show the number of objects in SDSS DR16Q for each
redshift bin and luminosity threshold, divided by A_ SDSS/A_ sky,
where A_ SDSS=9,376  deg^2 is the area of the footprint of SDSS,
and A_ sky=41,253  deg^2 is the total area of the sky. We take the
bolometric luminosities of the AGN in SDSS DR16Q from the catalogue presented
in <cit.>.



We can now compute the completeness for each redshift bin and for each
different cut in bolometric luminosity dividing the sky-area-corrected
number of observed AGN by the expected one. Whenever the former is greater
than the latter, the completeness is set to one. Because of the selection
in galactic latitude we perform, the completeness in the region where
|b|<10^∘ is zero. The values of the estimated completeness for all
the different redshift bins and the different bolometric luminosity thresholds
are listed in Table <ref>.








§ METHOD


Once the bolometric luminosity has been calculated for each AGN of
our catalogue and the completeness of the latter has been estimated
as a function of redshift and luminosity, we calculate the posterior
probability distribution on f_ AGN. To do so, we use a likelihood
function similar to the ones used in <cit.> and in <cit.>.
The general analytical expression of this function is the following:

    ℒ   (f_ AGN)=∏_i=1^N_ GWℒ_i(f_ AGN) 
       =∏_i=1^N_ GW[c_i·0.90· f_ AGN·𝒮_i
        +(1-c_i·0.90· f_ AGN)ℬ_i]  ,

where N_ GW=159 is the total number of mergers of binaries of compact
objects we consider in the analysis, and c_i is the average completeness of
the catalogue in the region occupied by the 90 per cent CL localisation volume
of the i-th GW event. The 0.9 factor that multiplies f_ AGN is used
to take into account that we use 90 per cent CL localisation volumes, therefore
only in the 90 per cent of the time the true sources are expected to be within
them.

For every GW event we calculate one value of c_i for each different bolometric
luminosity threshold we use in our analysis. Figure <ref>
shows the Cumulative Distribution Function (CDF) of c_i for all these different
sub-samples. Different colours indicate different luminosity thresholds.

The sub-samples with a threshold luminosity of 10^46 erg s^-1 and
of 10^46.5 erg s^-1, and that therefore contain only the most
luminous and rare AGN, have a null completeness at low redshift. For this
reason ≈10 per cent of the GW events have c_i=0 when those two
catalogues are considered.

In Equation <ref>, 𝒮_i is the signal probability density
and is calculated as follows:

    𝒮_i=1/ V90_i∑_j=1^N_ AGN,V90_ip_j/n_ AGN,j  ,

where N_ AGN,V90_i is the number of AGN located within the 90 per cent CL
localisation volume of the i-th GW event, which has a size of V90_i,
and p_j, measured in Mpc^-3, is the probability density representing
how likely it is that the i-th merger has happened in the exact position of
the j-th AGN. Finally, n_ AGN,j is the number density of the AGN catalogue
in the redshift bin where the j-th AGN is. The value of this number density
for each AGN has been obtained by dividing the total number of objects in the
redshift bin in which it is located by the comoving volume enclosed in such a bin,
excluding the region in which |b|<10^∘.

The background probability density is calculated as follows:

    ℬ_i=0.9/ V90_i  ,

where, in analogy to what has been done in <cit.> and in
<cit.>, the 0.9 factor ensures that ℬ_i and
𝒮_i have the same normalisation.

We cross-match the sky maps of the 159 GW events with the 5 different
sub-samples of the AGN catalogue separately, using the
postprocess.crossmatch function of the package ligo.skymap.
The results of these cross-matches are used to evaluate 
ℒ(f_ AGN) using Equation <ref>.
We then calculate the posterior probability distribution normalising
the likelihood function and assuming a uniform prior on f_ AGN
in the [0,1] range.







§ RESULTS


The posterior probability on f_ AGN peaks at f_ AGN=0
independently on which bolometric luminosity threshold we consider. In
Figure <ref> is shown in blue the region of the
investigated parameter space that our analysis rejects with a credibility
of 95 per cent.

We show the comparison with previous results, obtained in <cit.>
using a more limited dataset. This consisted of three catalogues, characterised
by three different cuts in bolometric luminosity, of spectroscopically
identified AGN with redshift z≤0.3, that have been cross-matched with
the 30 GW events detected in the same redshift range during the first three
observing runs of the LVK collaboration. These quasar catalogues are selected
from the version 7.7b of the Milliquas catalogue <cit.>.

The increase of a factor ≈ 5 in the number of sky maps used in the
analysis here presented and the high level of completeness of our Quaia
sample in the redshift range we consider are the causes of the increase in
the constraining power with respect to our previous work.



In Table <ref> we list how many objects are considered during the
cross-matches, what fraction of our catalogue they consist of, and the
upper limits we put on f_ AGN at 68, 90, and 95 per cent credibility.


The 95 per cent CL upper limit on the fraction of the detected GW events
that originated in an un-obscured AGN brighter than 10^44.5 erg s^-1
is 21 per cent. Considering rarer, brighter quasars, the constraints become
tighter. In fact we find with the same level of credibility that no more
than 11 per cent of the mergers come from an AGN brighter than
10^45 erg s^-1. This increase of constraining power is primarily
caused by the fact that the AGN catalogue with the higher luminosity
threshold has a higher level of estimated completeness (See Figure
<ref>). Further increasing the value of the luminosity
threshold does not lead to significantly more stringent constraints.

Figure <ref> shows the logarithm of the ratio between the
single-event likelihoods (ℒ_i) calculated at f_ AGN=1 and
the ones calculated
at f_ AGN=0, as a function of V90_i. Different panels correspond
to different bolometric luminosity thresholds. In each plot the markers are coloured
according to the average completeness inside the localisation volume of the
corresponding merger. The dashed horizontal lines indicate where the logarithm
has a null value, therefore where
ℒ_i(f_ AGN=1)=ℒ_i(f_ AGN=0).
The markers above these lines correspond to GW events for which the hypothesis
of random-chance association between merger sky maps and AGN is disfavoured.


One candidate GW event in particular, S240511i, detected during the second
half of O4, corresponds to a value of
ℒ_i(f_ AGN=1)/ℒ_i(f_ AGN=0)
which is greater than one, independently from the luminosity threshold.
The corresponding marker is particularly evident in top part of the first
two panels of Figure <ref>. In the case of the cross-match
with the sub-sample of our catalogue with AGN brighter than
10^46.5 erg s^-1 (10^46 erg s^-1), for this GW event
log(ℒ_i(f_ AGN=1)/ℒ_i(f_ AGN=0))≈1.32 (0.72).
In particular, two AGN brighter than 10^46.5 erg s^-1 are found
within the 90 per cent CL localisation volume, which in the case of this GW
event has a size of V90≈10^7.64 Mpc^3. Rounding to the
second decimal digit, the right ascensions of the two quasars are 167.22^∘
and 175.58^∘, their declinations are -23.64^∘ and -13.35^∘, and
their redshift estimates from Quaia are 0.37 and 0.40. The values of the
number density of the sub-sample of the AGN catalogue with
L_ bol≥10^46.5 erg s^-1 in the two different redshift bins
that contain the two matching quasar candidates are
≈10^-8.86 Mpc^-3 and ≈10^-8.15 Mpc^-3. The
total probabilities within the 3D credible regions of the GW sky map containing
them are ≈0.53 and ≈0.77.

The value of the single-event likelihood evaluated at f_ AGN=1 is
much higher than the one evaluated at f_ AGN=0 because two AGN are
found within the 90 per cent CL localisation volume while due to random
chance 0.06 (0.31) would be expected in the redshift bin with a number
density of n_ AGN≈10^-8.86 Mpc^-3
(n_ AGN≈10^-8.15 Mpc^-3), considering the V90 of the
GW detection.

While the single-event likelihood function for S240511i favours the hypothesis
of an AGN origin, the statistical framework used in this work is focused on
analysing the entire population of GW events. The results concerning this
specific event are to be considered as hints, not as statistically significant
conclusions. Follow-up analyses, conducted especially when the full catalogue
of mergers detected during O4 will we published, will be necessary to confidently
assess whether or not this merger has an AGN origin.







§ DISCUSSION AND CONCLUSION


In this work we present new observational constraints on the fractional contribution
of the AGN channel to the total observed merger rate of binaries of compact objects, f_ AGN.
These constraints are obtained using the same spatial-correlation-based approach
used in <cit.>. With respect to our previous work, we make use of a new,
larger dataset, which consists of 159 GW events detected by the interferometers of
the LVK collaboration not later than June 1st, 2024, and of the all-sky AGN catalogue
Quaia. In particular we use all the AGN contained in this catalogue that have
a redshift estimate z≤1.5 and an absolute value of the galactic
latitude |b|≥10^∘.

We estimate the bolometric luminosity of every AGN using the magnitudes in the
Gaia G_ RP band. We also estimate the completeness of our catalogue
as a function of redshift for different values of bolometric luminosity threshold.
The average value of the completeness within the 90 per cent CL localisation
volume of each GW event is used during the likelihood maximisation process.

We calculate the posterior probability function on f_ AGN for different
thresholds on the bolometric luminosity. We find that this function always peaks
at f_ AGN=0. We calculate the upper limits of the 68, 90, and 95 per cent
credibility intervals on f_ AGN.
The main results of this work are summarised in Figure <ref>
and in Table <ref>. In particular we estimate that no more than the
21 per cent of the detected GW events used in our analysis originated from an
un-obscured AGN with a bolometric luminosity higher than 10^44.5 erg s^-1
with 95 per cent credibility. Objects brighter than such threshold consist of
almost the entirety of our Quaia sample. Fainter objects are not included due to
the flux limitations of Gaia. Tighter constraints are obtained when higher luminosity
thresholds are considered. In particular we find that f_ AGN is not
greater than 11 per cent at a CL of 95 per cent when un-obscured quasars brighter
than 10^45 erg s^-1 are considered.

Thanks to the increase of a factor ≈5 with respect to <cit.>
in the number of used GW events, and to the high level of completeness of 
the quasar catalogue, we are able to put much tighter constraints on the
efficiency of the AGN channel (see Figure <ref> for a comparison
with our previous work).

Our results have been obtained under some assumptions, which are inevitable
when inferring the properties of AGN in large catalogues. In order to calculate
the bolometric luminosities of the objects of Quaia we have to assume a shape of
the typical SED and a value for the bolometric correction. We take the values of
these AGN properties from <cit.> in order to be consistent with the
dataset presented in <cit.>, which we use to adjust the bolometric
luminosities obtained from Gaia magnitudes. Different assumptions on the SED and
on the bolometric correction might lead to different estimates of the bolometric
luminosities.

To calculate the completeness of our catalogue, we compare the observed number of
AGN with the expected one, which is obtained from the integration of a luminosity
function. We choose to use the one presented in <cit.> because our
bolometric luminosities are calibrated on their SDSS-based estimates from <cit.>,
and in the redshift range we consider that luminosity function has been fitted
on AGN from SDSS, which is a survey that contains un-obscured AGN detectable in the
optical band, just like the ones contained in Quaia.

An accurate estimation of the uncertainty on the completeness of the AGN catalogue
for each redshift bin and luminosity threshold is non-trivial. Factors that would
have to be taken into account are the uncertainty on the measurements of the
magnitude in the Gaia G_ RP band, the potential contamination of the
catalogue with stars or galaxies, the uncertainties on the redshift
estimates, on the SED shape, and on the bolometric correction. Moreover,
the luminosity function we compare the observed number of AGN with has an intrinsic
scatter. With the only exclusion of the first redshift bin of the two sub-samples
characterised by the highest luminosity thresholds, the numbers of observed objects
used for the calculation of the completeness (see Figure <ref>)
are high enough for the systematics mentioned above to be considered the dominant
source of uncertainty, since we are in general not limited by low number counts.

As far as our analysis is concerned, taking into consideration the uncertainty
on the completeness is not expected to change the results significantly. This
is because, as described in Section <ref>, we use the average value
of the completeness within the localisation volumes of the GW events to weight
the contribution of each of them to the total likelihood; the constraining power
of our method scales linearly with the completeness of the AGN catalogue, and
changes in this parameter are not expected to lead to any significant variation
on the value of the best estimate for f_ AGN. For this reason, we choose
to use the values of completeness listed in Table <ref> throughout
our analysis.

Another important caveat to mention is that our analysis and the constraints we
are able to put concern un-obscured AGN, which are visible both in the optical
and in the infrared band. In order to extend our conclusions to the entire population
of AGN, one should take into account what fraction of them is not visible in the
wavelengths observed by Gaia and by WISE, which are the two surveys from which
Quaia has been created from. This obscuration fraction is in general expected to
increase as a function of redshift and to decrease as a function of luminosity <cit.>.

The observational constraints presented in this work consist of a generalisation
of the ones obtained in <cit.>. Here we investigate a wider range of AGN
luminosities and the entirety of the redshift range reached by the interferometers
of the LVK collaboration. For this reason, in order to obtain in the future even
more general results one will need an all-sky AGN catalogue with a lower threshold in
flux with respect to Quaia, to be able to extend the analysis to the faint-end of the
un-obscured AGN population. We estimate that no more than one GW event out of
five has originated in an un-obscured AGN with a bolometric luminosity higher than
10^44.5 erg s^-1, but a larger fraction might still come from
fainter objects, or from obscured ones.

While the results of this work demonstrate that the efficiency of the AGN channel
can already be investigated with spatial-correlation analyses using the currently
available data, in the next years it will be possible to put even tighter constraints
using all the events that will be detected in the rest of O4 as well as the ones that
will be detected during the fifth observing run of the LVK collaboration, O5. Using more
data could either reduce the upper limits on f_ AGN, or it could cause a shifting
of the value of such parameter that maximised the posterior distribution, moving
it away from zero.

Future developments of the statistical method used in this work involve also the
introduction of physically-motivate priors on the intrinsic binary properties. Different
binary formation channels are indeed expected to produce different features in the
distributions of the masses and the spins of the merging systems, as well as on
their eccentricity. The analysis here presented has been kept purposely agnostic as
far as the physics of the formation mechanism is concerned. Introducing
physically-motivated changes in the likelihood function might result in different
constraints on f_ AGN and will inform us on which are the intrinsic binary
parameters that are able to add more information to the analysis.









§ ACKNOWLEDGEMENTS

The authors thank Elia Pizzati for the stimulating discussions
regarding quasar luminosity functions and bolometric corrections.
EMR acknowledges support from ERC Grant “VEGA P.", number 101002511.
This research has made use of data or software obtained from
the Gravitational Wave Open Science Center (gwosc.org), a service
of LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo
Collaboration, and KAGRA. LIGO Laboratory and Advanced LIGO are
funded by the United States National Science Foundation (NSF) as
well as the Science and Technology Facilities Council (STFC) of
the United Kingdom, the Max-Planck-Society (MPS), and the State
of Niedersachsen/Germany for support of the construction of Advanced
LIGO and construction and operation of the GEO600 detector.
Additional support for Advanced LIGO was provided by the Australian
Research Council. Virgo is funded, through the European Gravitational
Observatory (EGO), by the French Centre National de Recherche Scientifique
(CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN)
and the Dutch Nikhef, with contributions by institutions from Belgium,
Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal,
Spain. KAGRA is supported by Ministry of Education, Culture, Sports,
Science and Technology (MEXT), Japan Society for the Promotion of
Science (JSPS) in Japan; National Research Foundation (NRF) and
Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and
National Science and Technology Council (NSTC) in Taiwan.
Software: 
 <cit.>; 
 <cit.>; 
 <cit.>;
 <cit.>;
 <cit.>;
 <cit.>;
 <cit.>.





§ DATA AVAILABILTY

The AGN catalogue used in this work, the results of the cross-matches,
and related data products are publicly available at
<https://zenodo.org/records/12805938>. The code used to produce the
results presented in this paper will be shared on reasonable request to
the corresponding author.




mnras







 Paper Title: Stochastic gravitational wave background from cosmological neutrino-dominated accretion flows.





    Tong Liutongliu@xmu.edu.cn0000-0002-9130-2586]Yun-Feng WeiDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China0000-0001-8678-6291]Tong LiuDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China§ ABSTRACT 
    We investigate the stochastic gravitational wave background (SGWB) from neutrino-dominated accretion flows (NDAFs) based on the results of our fallback core-collapse supernova (CCSN) simulations. We find that the predicted SGWB is mainly determined by the typical CCSN initial explosion energy and progenitor metallicity. For the optimistic cases in which the typical initial explosion energy is low, the SGWB from NDAFs without disk outflows might be detected by next-generation space-based interferometers such as Decihertz Interferometer Gravitational wave Observatory (DECIGO) and Big Bang Observer (BBO). In the low-frequency regime ∼10^-3-10^-1 Hz, this background is comparable to that expected from standard inflationary models. Therefore, the SGWB from NDAFs may become a foreground for searches of the SGWB generated in the inflationary epoch. Combining the diffuse NDAF neutrino background and SGWB from NDAFs, one may constrain the properties of the CCSNe and NDAFs.
    
    
    § INTRODUCTION
    
    
    The stochastic gravitational wave background (SGWB) is expected to be created by the superposition of gravitational waves (GWs) from many uncorrelated sources. Such a background could be produced in many processes during cosmological and astrophysical evolutions. A cosmological SGWB may be produced by inflation <cit.>, cosmic strings <cit.>, alternative cosmologies <cit.>, and a variety of other phenomena. Astrophysical SGWB could have various possible sources (see <cit.> for a review ), including inspiral and coalescence of compact binaries <cit.>, white dwarf binaries <cit.>, rotating neutron stars <cit.>, core-collapse supernovae <cit.>, gamma-ray bursts <cit.>, Population III (Pop III) stars <cit.>, stellar core collapse <cit.>, and so on.
    
    The next-generation space GW detectors, such as Decihertz Interferometer Gravitational wave Observatory <cit.> and Big Bang Observer <cit.>, targeting 0.1-1 Hz, might be possible to detect SGWB from both cosmological and astrophysical origins. Notably, these experiments are expected to detect the primordial SGWB that arises in the very early universe during the inflationary epoch <cit.>. Such SGWB may serve as a powerful tool for studying the extremely early universe. However, astrophysical foreground sources could be a significant problem for searches of the inflationary SGWB. The SGWB from CCSNe <cit.> and Population III stars <cit.> are expected to be comparable to or mask the inflationary SGWB in some range of frequencies. Thus, a good understanding of astrophysical foregrounds around the deci-Hertz band is essential for DECIGO and BBO detecting the inflationary SGWB. In this paper, we focus on the low-frequency GW generated by anisotropic neutrino emission of neutrino-dominated accretion flows (NDAFs) and estimate the SGWB spectrum from NDAFs.
    
    NDAFs around rotating stellar-mass black holes (BHs) are one of the plausible candidates of GRB central engines in massive collapsars and compact object mergers. In the collapsar model <cit.> for long-duration GRBs, the core collapse of a massive star can produce a BH hyperaccretion system via the fallback process. For the very high accretion rate, the disk might be in a state of NDAFs. In the inner region of the disk, photons are trapped, and only neutrinos are emitted from the disk surface. These neutrinos annihilate in the space out of the disk and then form the primordial fireball to power a GRB. The properties of NDAFs have been widely studied in recent decades <cit.>, and for a review see <cit.>. In particular, the GW signals generated by the anisotropic neutrino emission from NDAFs have been investigated by some previous works <cit.>. The neutrino-induced GWs are detectable for ∼10 Mpc by DECIGO/BBO <cit.>. Of course, the BZ mechanism <cit.> could coexist even be dominated in the BH hyperaccretion process, but it has no GW radiation.
    
    An earlier estimation of the SGWB produced by NDAFs was given by <cit.>. They adopted some typical GW energy spectra of NDAFs to calculate the SGWB. Meanwhile, they used the GRB formation history to estimate the event rate of NDAFs. The results show that the SGWB from NDAFs is below the detection limit of DECIGO/BBO. However, the effects of progenitor properties and the initial explosion energy on the GW spectrum of NDAF are not included in their work. Besides, the event rate of NDAFs might be underestimated. In the collapsar model, an observable GRB is only triggered when jets launched by the central engine can break out from the envelope and circumstellar medium in the prompt emission phase <cit.>. If the jet is chocked in the envelope or the observer is not in the jet's line of sight, no GRB would be detected. Therefore, the event rate of GRBs should be much less than that of NDAFs. Of course, the GWs from NDAFs are partly anisotropic. In this work, we estimate the SGWB from NDAFs based on fallback CCSN simulations. In <cit.>, we revealed how the GW emission of NDAFs depends on the initial explosion energies, masses, and metallicities of the progenitor stars. Here, we aim to improve the estimation of the SGWB from NDAFs by including those dependencies.
    
    In this work, we mainly focus on the SGWB from NDAFs around BHs. Note that NDAFs around NSs may also operate in some GRBs and CCSNe <cit.>. Actually, for the newborn NSs in the center of massive collapsars, their strong magnetic fields <cit.> will restrain the accretion process and destroy the inner region of the disk. Besides, since the rapidly decreasing fallback accretion rate, NDAFs might only exist for the very initial stage of the NS hyperaccretion process. Therefore, the contribution of NDAFs around NSs to the SGWB is not considered here.
    
    The paper is organized as follows. In Section 2, we describe the setup of our CCSN simulations and discuss the effects of the initial explosion energies and the masses and metallicities of the progenitor stars on the GW emission of NDAFs. In Section 3, we estimate the amplitude of the SGWB from NDAFs. We discuss how the SGWB depends on the initial explosion energy and progenitor metallicity. We also explore the effect of initial mass function (IMF) on the SGWB from NDAFs. The conclusions and discussion are presented in Section 4.
    
    
    
    § MODEL
    
    
     §.§ CCSN simulations
    
    
    Here, we briefly review the setup of our CCSN simulations. In this paper, we adopt the presupernova (pre-SN) models with initial mass in the range of 20-40M_⊙ as progenitor models <cit.>. For those models, the models with zero metallicity (Z/Z_⊙ = 0) and solar metallicity (Z/Z_⊙ = 1) are referenced from <cit.> and <cit.>, respectively, as well as the ones with metallicity Z/Z_⊙ =0.01 are provided by Prof. Alexander Heger in private communication, where Z_⊙ represents the metallicity of the Sun. The stellar collapse and explosion simulations <cit.> are performed in a series of 1D simulations with the Athena++ code <cit.>. We use the piston approach <cit.> to carry out spherically symmetric explosion simulations. For each progenitor star, a piston was initially located at the outer edge of the iron core. When the star collapses, the piston firstly moves inward for 0.45 s and then moves outward with an initial high velocity and decelerates smoothly until coming to rest at 10^9 cm. We follow this approach to determine the initial explosion condition at the inner boundary.
    
    The simulation was divided into two steps to reflect the initial collapse and the subsequent explosion. In the first step, the numerical grid has an inner boundary at 10^9 cm, and a unidirectional outflowing inner boundary condition was used at the inner boundary to mimic the suction effect resulting from the hypothetical piston moving inwards. The outer boundary is set at the surface of the progenitor star. In this step, the grid has 10^4 cells with a logarithmic uniform interval for the radial direction. The simulation is run to 0.45 s, and then the piston turns outwards, corresponding to the outward propagation of the blast.
    
    In the second step, we use the same outflowing inner boundary condition, which is set at 10^9 cm. The outer boundary is set at 10^16 cm, and the medium outside the star is maintained in a constant state with a pressure and density three orders of magnitude lower than the corresponding ones on the star surface. In this step, the grid has 2,000 logarithmic cells. At the beginning of this step, we map the results of the first step to the new grid for the second step and inject energy into the innermost cell adjacent to the inner boundary to mimic the outward blast passing through the inner boundary. Here, the injected energy is the setting energy, which is assumed to have three values for each case, i.e., 2, 4, and 8 B (1 B=10^51  erg). All simulations were run until the remnant growth ceased. For more details of simulations, see <cit.> and <cit.>.
    
    For each simulation, we record the evolution of the fallback mass supply rate at the inner boundary. Ignoring the disk outflows, we roughly consider the mass supply rate as the mass accretion rate of the disk. If the mass accretion is high, the hyperaccretion disk would be in the state of NDAFs. Meanwhile, the mass and spin of the BH will violently evolve within tens of seconds. According to the conversion of the energy and angular momentum, the evolution equations of a spinning BH can be expressed as <cit.>,
    dM_BH/dt=Ṁ/√(3x_ms) ( 4-3a_*/√(x_ms) ),
    
    and
    da_*/dt=2√(3)Ṁ/M_BH ( 1- a_*/√(x_ms) )^2,
    
    where M_BH and Ṁ are the mass of the BH and the mass accretion rate, respectively, a_* is the dimensionless spin parameter of the BH, and x_ms=3+Z_2-√((3-Z_1)(3+Z_1+2Z_2)) is the dimensionless radius of the marginally stable orbit <cit.>, where Z_1=1+(1-a_*^2)^1/3[(1+a_*)^1/3+(1-a_*)^1/3] and Z_2=√(3a_*^2+Z_1^2) for 0< a_*< 1.
    
    In this work, the initial spin parameter is set as a_*=0.9, and the starting time is set at the time when the initial BH mass (core mass) reaches 2.3 M_⊙.
    
    
    
     §.§ GWs from NDAFs
    
    
    Neutrino emission of NDAFs is mainly determined by the mass accretion rate and the properties of central BH <cit.>. In the relativistic global solutions of NDAFs  <cit.>, the detailed neutrino physics, the nuclear statistical equilibrium, and the conditions of the ignition and neutrino trapping are all considered. For the case of the viscosity parameter α = 0.1, we can derive the fitting formula for the neutrino luminosity as a function of the BH mass and spin and the mass accretion rate, which is expressed as
    
        log L_ν (erg s^-1) =    52.80-0.03m_BH+1.01a_*
           +1.08logṁ,
    
    where m_BH=M_BH/M_⊙ and ṁ=Ṁ/M_⊙  s^-1 are the dimensionless BH mass and accretion rate, respectively. It should be noticed that the ignition and trapping accretion rates are roughly ∼ 0.001 and ∼ 5  M_⊙  s^-1, respectively, which also be significantly affected by the BH mass and spin and disk viscosity parameter <cit.>. But ∼ 0.01 M_⊙  s^-1 can be considered as the minimum accretion rate for detectable neutrino radiation from BH-NDAF systems in the center of massive collapsars <cit.>.
    
    
    
    Here, we estimate the GWs from anisotropic neutrino emission, which is firstly proposed by <cit.>. We follow the formalism developed by
    previous works <cit.> to calculate the neutrino-induced GW signals. For a geometrically thin disk model for NDAFs, the local energy flux of GWs can be written as (for details, see <cit.>)
    dE_GW/dA   =G/36π c^5D^2(1+2cosθ)^2tan^4(θ/2)
       ×∫_-∞^∞dt L_ν(t)^2,
    
    where dA=D^2dΩ is the surface element, D is the distance between the observer and the source, θ is the viewing angle. Here, θ=π/2 corresponds to the case that the observer is located in the equatorial plane. Then, integrating over a sphere surrounding the source, one can calculate the total energy E_GW. For θ=π/2, the E_GW is calculated as
    
    E_GW=β G/9c^5∫_-∞^∞dt L_ν(t)^2,
    
    where β∼ 0.47039. In order to calculate GW spectrum, one can write L_ν(t) in terms of the inverse Fourier transform as
    
    L_ν(t)=∫_-∞^+∞L̃_ν(f)e^-2π iftdf;
    
    then, the GW energy spectrum can be deduced as
    dE_GW(f)/df=2β G/9c^5 |  L̃_ν(f) |^2.
    
    In this work, we assume the orientation of disks is random and calculate the angle average GW energy spectrum per NDAF as
    
    <dE_GW(f)/df>=∑_i∫_△θ _i dθ/∫_0^π/2 dθdE_GW,i(f)/df
    where △θ_i is the angle range of angle bin i, and dE_GW,i(f)/df is the observed GW energy spectrum at the corresponding viewing angle. In the subsequent calculations, we all adopt angle-averaged GW energy spectra.
    
    
    
    The effects of the progenitor mass and metallicity on the GW energy spectrum of NDAFs are displayed in Figure 1. Panels (a), (b), and (c) correspond to the cases of progenitors with metallicity of Z/Z_⊙=0, 0.01, and 1, respectively. The initial explosion energy is 2 B. In Panels (a) and (c), we only show GW energy spectra of NDAFs from progenitors with a mass greater than 30 M_⊙. In our simulations, for progenitors with metallicities of Z/Z_⊙=0 and 1, the final remnant of progenitors with a mass less than 30M_⊙ are NSs rather than BHs. The GW amplitude is mainly determined by the neutrino luminosity, which is related to the mass accretion rate. Most of the fallback and neutrino emission come from the core of the star. Therefore, the GW emission of the NDAF is determined by the compactness of the pre-SN star core. Some previous works <cit.> studied the dependences of progenitor masses and metallicities on the structural characteristics of pre-SN stars and found a non-monotonic behavior for compactness as a function of progenitor mass and metallicity. As a result, for the same initial explosion energy, the GW emission of NDAFs is not strictly dependent on progenitor mass and metallicity. Besides, previous stellar evolutionary studies <cit.> found that the core compactness parameters of solar metallicity stars are commonly smaller than those of low metallicity stars. Thus, solar metallicity is unfavorable for neutrino emission and GW emission of NDAFs. Moreover, although the accretion rate is still large for massive progenitors, the central BH mass is more massive which will influence the neutrino luminosity of NDAFs <cit.>.
    
    Figure 2 shows the effects of the initial explosion energy on the GW energy spectra of NDAFs. The metallicity is set to Z/Z_⊙=0.01. The black, red, and blue curves correspond to the initial explosion energies of 2, 4, and 8 B, respectively. For different progenitor stars, the amplitudes of the spectral lines increase with decreasing initial explosion energy because the weaker explosion energy corresponds to a more powerful fallback and neutrino emission. In Figures 2(a) and 2(b), the case of 8 B is not shown because the final remnants of these progenitors are both NSs.
    
    
    
    § SGWB FROM NDAFS
    
    
    A prediction of the SGWB requires a good understanding of the average GW energy spectrum and the event rate of NDAFs. The properties of progenitors and the initial explosion energy would determine whether the fallback process produces NDAFs and GW emission of NDAFs. Here, we investigate the effects of metallicities and initial explosion energies of progenitors on SGWB from NDAFs.
    
    
    
     §.§ Cosmic NDAF history
    
    
    The progenitor stars of CCSNe have relatively short lifetimes (≲ 10^8 years) compared to cosmic timescales <cit.>. Thus, the rate of NDAFs can be calculated using the star formation rate and IMF. The cosmic NDAF rate at a redshift of z is calculated as
    
    R_NDAF (z)=R_SFR(z)∫_M_min^M_maxΨ (M)dM/∫_0.1^125 MΨ (M)dM,
    
    where R_SFR(z) is the cosmic star formation rate in units of Mpc^-3  yr^-1, which can be deduced from observations <cit.>. Here, we adopt the continuous broken power law description by <cit.>,
    
    R_SFR (z)= ρ̇ _0 [ (1+z)^αη +(1+z/C )^βη + (1+z/D )^γη ]^1/η,
    
    where α =3.4, β =-0.3, γ =-2, η =-10, C≃ 5100, D≃ 14, and ρ̇ _0=0.014  Mpc^-3  yr^-1. Here, we adopt a Salpeter IMF <cit.> with Ψ (M)∝ M^ζ with ζ  =-2.35 in the mass range of 0.1-125 M_⊙, but explore a liberal range from -2.15 to -2.45<cit.> in our final calculations. M_max and M_min are the maximum and minimum masses of progenitors that produce BH-NDAF systems, respectively. Due to the influence of the metallicity of the progenitor and initial explosion energy on the fallback process of CCSNe, not all massive stars can produce NDAFs. Especially, M_min depends on the metallicity of progenitor and initial explosion energy <cit.>. According to the results of our simulations, we gave M_min for different metallicity and initial explosion energy, and the results are listed in Table <ref>. Of course, we also checked that the fallback accretion rates of these progenitors are suitable for NDAFs until they decrease lower than the ignition accretion rates. The SGWB prediction depends weakly on the M_max and we set M_max∼ 50 M_⊙, which is the approximately upper limit of the progenitor mass for NDAFs because the new-born BH mass is too larger to effectively ignite NDAFs <cit.>.
    
    
    
     §.§ IMF-weighted average GW energy spectra
    
    
    The average GW energy spectrum for a population of progenitors is computed by weighting each progenitor by
    dE(f)/df =∑_i∫_△ M _iΨ (M)dM/∫_M_min^M_maxΨ (M)dM <dE_GW(f)/df>_i,
    
    where Ψ (M) is once again the IMF, △ M _i is the mass range of mass bin i, and <dE_GW(f)/df>_i is the angle-averaged GW energy spectrum of the NDAF with progenitor mass of M_i.
    
    
    
     §.§ Estimations of SGWB
    
    
    It is usually to characterize the SGWB by the energy density parameter <cit.>, i.e.,
    Ω_GW(f)=1/ρ_cdρ _GW/d ln f,
    
    where ρ_GW is the gravitational energy density and ρ_c=3H_0^2/8π G is the critical energy density needed to close
    the universe. Then, following <cit.>, the sum of the energy densities radiated by a large number of independent NDAFs at each redshift is given as
    Ω_GW(f)   =1/ρ_cc^2
       ×∫_0^∞dzR_NDAF(z)/1+z| dt/dz|f(z)dE(f_z)/df,
    
    where f_z≡ f(1+z) and |dt/dz  |=[H_0(1+z)√(Ω_Λ+Ω _m(1+z)^3)]^-1. Here, we ignore the anisotropy of GWs from NDAFs and adopt the cosmological parameters  Ω _m=0.3, Ω_Λ=0.7, and H_0= 100 h_0 km s^-1 Mpc^-1 with h_0=0.72.
    
    Figure 3 shows the effect of the metallicity of the progenitor on the spectrum of SGWB from NDAFs. The initial explosion energy of all progenitors is set as 2 B. The black, red, and blue curves correspond to Z/Z_⊙=0, 0.01, and 1, respectively. The gray dashed lines represent the sensitivity curves of DECIGO <cit.>, BBO <cit.>, and the ultimate-DECIGO <cit.>. Solar metallicity is not beneficial for the detection of the SGWB from NDAFs. This is because the solar metallicity is unfavorable for neutrino emission and GW emission of NDAFs. Moreover, if the solar metallicity is universal, the event rate of NDAFs would decrease. The horizontal orange dashed line shows the SGWB generated in the inflationary epoch. In low-frequency region (∼10^-3-10^-1 Hz), our estimate for the SGWB from NDAFs can be comparable to the most optimistic SGWB from slow-roll inflation.
    
    In Figure 4, we display the effects of the initial explosion energy on the spectrum of SGWB. We assume all progenitors have the same metallicity, Z/Z_⊙=0.01. The black, red, and blue curves correspond to the initial explosion energy of 2, 4, and 8 B, respectively. The weaker explosion energy corresponds to a more powerful fallback, which enhances the GW emission and the event rate of NDAFs. As a result, the initial explosion energy will significantly affect the detection of the SGWB from NDAFs.
    
    
    
    In Figure 5, we explore the effect of IMF on the spectrum of SGWB from NDAFs. The black curve corresponds to the Salpeter IMF with ζ =-2.35. The red and blue curves correspond to ζ =-2.15 and ζ =-2.45, respectively. A shallower IMF (i.e., ζ =-2.15) is beneficial for the detection of SGWB from NDAFs. The SGWB differs by almost a factor of 3 between the case of ζ =-2.15 and the case of ζ =-2.45. As shown in Figure 1, for the same progentior metallicity and initial explosion energy, relatively higher mass stars correspond to the higher accretion rate, then the more powerful GW emission of NDAFs until the effects of the central BH mass dominate the ignition of NDAFs. Shallower IMFs have more high-mass stars which contribute to the SGWB.
    
    
    
    § CONCLUSIONS AND DISCUSSION
    
    
    In this paper, we estimate the SGWB produced by NDAFs based on fallback CCSN simulations. The effects of progenitor properties and initial explosion energies on SGWB from NDAFs are studied. These factors can affect the GW energy spectra and event rates of NDAFs. The GW emission of NDAFs is determined by the fallback process in CCSNe. Lower initial explosion energy is beneficial for producing a more powerful fallback, resulting in stronger neutrino emission and GW emission of NDAFs. Moreover, low initial explosion energy may enhance the event rates of NDAFs. Therefore, if low initial explosion energy is universal CCSNe, even for failed CCSNe, the SGWB from NDAFs may be detected by interferometers such as DECIGO and BBO. The influence of metallicity on SGWB from NDAFs is not monotonic because the compactness of the pre-SN star core is not strictly dependent on metallicity. However, solar metallicity is not beneficial for the detection of SGWB. Another uncertainty in our models is the IMF. A shallower IMF would increase the amplitude of SGWB from NDAFs. Therefore, the SGWB could be a valuable tool to investigate the IMF.
    
    In NDAF models, we adopt α=0.1 as a typical viscosity parameter. Actually, a disk with low α is denser and has a higher neutrino luminosity <cit.>. Thus, the GW emission of NDAFs with low α might be more powerful. Besides, in our calculations, we do not consider the disk outflows, which plays an important role in critical accretion systems <cit.>. The neutrino luminosity of NDAFs with strong disk outflows would be at least one order of magnitude lower than that of NDAFs without outflows <cit.>. As a result, powerful disk outflow will decrease the GW emission of NDAFs. If powerful disk outflows are universal in NDAFs, the amplitude of SGWB from NDAFs would decrease.
    
    In summary, the uncertainties of the SGWB from NDAFs are large. The detection of the corresponding neutrino background will lead to an improved prediction of the SGWB from NDAFs. In <cit.>, we investigated the diffuse NDAF neutrino background (DNNB) in detail. Metallicity and initial explosion energy have similar effects on both DNNB and SGWB from NDAFs. For the optimistic cases where the typical initial explosion energy is low, the DNNB might be detected by the upcoming larger neutrino detectors such as Hyper-Kamiokande, Jiangmen Underground Neutrino Observatory (JUNO), and Deep Underground Neutrino Experiment (DUNE). The detection of DNNB would help constrain the average neutrino emission and event rate of NDAFs. By combining the DNNB and SGWB, we would better understand CCSNe and NDAFs.
    
    It is noted that the SGWB from NDAFs might mask the SGWB generated in the inflationary epoch at low frequency. Besides, the SGWB from cosmological CCSNe and Pop III stars are expected to mask the inflationary GWs in some range of frequencies. These astrophysical foreground sources could be a significant problem for searches of the inflationary SGWB. It is necessary to disentangle these foreground sources from the inflationary SGWB in future detection.
    
    
    We thank Prof. Alexander Heger for providing us pre-SN data. We also thank Xing-Jiang Zhu for helpful discussion. This work was supported by the National Key R&D Program of China (Grant No. 2023YFA1607902), the National Natural Science Foundation of China (Grant Nos. 12173031, 12221003, and 12303049), the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (Grant No. GZC20231424), and the science research grants from the China Manned Space Project.
    
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 Paper Title: Investigating the crust of neutron stars with neural-network quantum states.




Lifelong Person Search
    Jae-Won Yang, Seungbin Hong, and Jae-Young Sim, Member, IEEE

    August 5, 2024
=================================================================





empty



§ INTRODUCTION

Low-density nuclear matter found in the crust of neutron stars exhibits a complex and fascinating structure that varies significantly with density. In the outer crust, nucleons are bound to fully ionized nuclei. As density increases within this region, these nuclei become increasingly neutron-rich, and consequently, terrestrial experiments can only directly determine the dominant nuclei species that are present at lower densities. When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons. This leads to the formation of an unbound neutron component, marking the boundary between the outer and inner crust of the star. As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. Eventually, upon reaching nuclear saturation density, these nuclei coalesce, transitioning into a uniform distribution of nucleons <cit.>.


Modeling inner crust material poses significant challenges due to the inherent complexity of clustering phenomena, the emergence of superfluidity, and the existence of both free neutrons and neutron-rich nuclei. Various phenomenological approaches have been developed to address some of these challenges, including the compressible liquid-drop model <cit.> and self-consistent mean-field models <cit.>. In the former, the energy is parameterized as a function of global properties, and the nucleons within clusters are treated separately from the free neutrons, all of which are assumed to be uniformly distributed. While this treatment has a low computational cost, it neglects quantum mechanical shell effects, which are critical for determining the equilibrium composition of the crust. In contrast, self-consistent mean-field models are fully quantum mechanical, constructing the many-body state in terms of single-particle or quasi-particle wave functions. However, since they neglect correlations beyond the mean-field approximation, they have limited applicability for low-density nuclear matter, where strong correlations can lead to significant deviations from the mean-field behavior.

Ab initio methods offer a more fundamental approach, as they solve atomic nuclei and neutron-star matter starting from the individual interactions among nucleons <cit.>. Most of them, however, suffer from significant limitations when modeling the transition from neutron-rich nuclei to uniform matter <cit.>. Their applicability is presently limited to the uniform case—a notable exception in this regard is the lattice effective field theory approach <cit.>. Even continuum diffusion Monte Carlo methods, which are well known for their accuracy across different physical systems, are limited to the uniform phase of neutron-star matter because of the fermion-sign problem <cit.>. 

We overcome the limitation of diffusion Monte Carlo approaches by leveraging a variational Monte Carlo method based on neural-network quantum states (NQS) <cit.>. Owing to their flexibility, NQS enable an accurate description of the ground-state of quantum many-body systems. This feature is particularly relevant for modeling the different phases of matter comprising the crust of neutron stars, which is inaccessible by conventional ansätze. This approach also circumvents the fermion-sign problem, as Monte Carlo techniques are only employed to sample the modulus squared of the wave function, which is positive even for systems of fermions. 

In a recent application <cit.>, NQS based on the hidden-nucleon architecture have demonstrated success in modeling the normal and superfluid phases of dilute neutron matter within a unified framework. A more compact and flexible neural Pfaffian-Jastrow architecture, featuring a message-passing neural network to efficiently capture pairing and backflow correlations, was shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ground-state properties of ultracold Fermi gases <cit.>. Chiefly, the neural Pfaffian-Jastrow approach is designed to capture strong pairing effects, which characterize both cold atoms and the matter found in the crust of neutron stars. 

In this work, we put forward a generalization of the neural Pfaffian-Jastrow ansatz and adapt it to nuclear potentials, which strongly depend on the spin-isospin degrees of freedom of the interacting particles. We provide evidence that this ansatz is capable of modeling the different phases matter found in the crust of neutron stars, including the self-emergence of nuclear clusters and their dissolution. In particular, we find dramatically lower energies than those obtained with state-of-the-art auxiliary-field diffusion Monte method in the low-density regime. In addition to ground-state energies, we compute two-body distribution functions and the proton fraction of beta-equilibrated matter. Our findings underscore the viability of utilizing neural Pfaffians to achieve an ab initio description of the crust of neutron stars from the nuclear physics perspective. 




§ RESULTS




 §.§ Simulation setup


We model infinite nuclear matter using periodic boundary conditions and N=28 nucleons in a cubic simulation cell with side length L, where L is determined by the baryon density n_B=N/L^3. 
For each density investigated, we run three separate simulations with different proton fractions x ≡ n_p/n_B = N_p/N, where n_p is the proton density and N_p is the number of protons in the simulation. Specifically, we run simulations with N_p = 0, 4, and 14 at each density, allowing us to study pure neutron matter (PNM), isospin-symmetric nuclear matter (SNM), and an intermediate isospin-asymmetric case. The simulated nucleons are confined within the cell, with potential contributions from image particles considered up to a distance of 3L/2. Each nucleon has a continuous three-dimensional position vector and discrete spin and isospin projections on the z-axis. As common in quantum Monte Carlo calculations of nuclear systems <cit.>, we make the approximation that protons and neutrons have equal masses. 

The dynamics of neutrons and protons is modeled according to a non-relativistic Hamiltonian based on a leading-order pionless effective field theory expansion that contains two- and three-nucleon interactions. This “essential” Hamiltonian has been shown to reproduce properties of various nuclei across the nuclear chart, including ground-state energies, radii, and magnetic moments, within few-percent from their experimental values <cit.>. It also reproduces well the equation of state (EOS) of dilute neutron matter, as the predicted values agree with highly realistic phenomenological and chiral-EFT interactions <cit.>. 
Additionally, we assume the repulsive Coulomb interactions among finite-size protons to be completely screened by the electrons present in the system. 




 §.§ Energy per particle




We begin our analysis by comparing the energy per particle of SNM obtained from NQS with that obtained from state-of-the-art auxiliary-field diffusion Monte Carlo (AFDMC), as shown in Figure <ref>. Details on the AFDMC calculations are provided in the Auxiliary-field diffusion Monte Carlo subsection of the Methods section. For all simulated densities, n_B = 0.001 fm^-3, 0.01 fm^-3, 0.04 fm^-3, and 0.08 fm^-3, the NQS energies are lower than those predicted by AFDMC, especially at the lowest densities. Additionally, the curves exhibit distinctly different characters, with the NQS energies reaching a plateau at lower densities, while AFDMC energies continue to rise. This behavior highlights an important limitation of the AFDMC calculations, as they fail to capture the formation of nuclear clusters. This is ultimately due to the fermion sign problem, which is exacerbated by using a variational ansatz that is designed for the uniform phase. 
Therefore, the zero-density limit of the AFDMC results corresponds to an infinitely dilute gas of protons and neutrons, whose ground-state energy vanishes. As an additional point of reference for the zero-density ground state in the fully clustered system, we present the NQS results for ^28Si with no electromagnetic contribution to the potential, to be consistent with the SNM case. 
The similarity between the NQS energies of ^28Si and SNM at n_B=0.001 fm^-3 suggests that all nucleons in low-density SNM coalesce to form ^28Si. We stress that ^28Si self-emerges as the minimum of the quantum-mechanical energy of the system and is not a priori encoded in the Pfaffian ansatz. 

In the outer crust of neutron stars, nucleons are bound to fully ionized nuclei. At higher densities within the outer crust, these nuclei become increasingly neutron-rich. 
While terrestrial experiments can determine the dominant nuclei species at lower densities, at higher densities these nuclei are too neutron-rich to be stable and cannot be directly measured. Nonetheless, the dominant nuclei species in this regime can be reliably inferred through extrapolations from experiments using semiempirical nuclear mass formulae<cit.>.

When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons, leading to the formation of an unbound neutron component. This marks the boundary between the outer and inner crusts of the star. 
As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. 
Modeling matter at these densities presents several challenges. Not only are the extremely neutron-rich nuclei impossible to measure directly, but they also coexist with a sea of unbound neutrons. A commonly used approach at this density is the compressible liquid drop model (CLDM). However, being a classical model, CLDM does not account for the shell effects within nuclei<cit.>. 

Check my understanding, they only mention this for outer crust material but CLDM seems obviously inferior there. However it seems to be commonly used for the inner crust and I would imagine the shell effects still matter... Probably need a more relevant citation too. The end of the inner crust and beginning of the core of the star is marked by the transition to uniform matter from the full dissociation of the nuclei, around half nuclear saturation density.

mention superfluid. mention core modeling is different and its difficult to have a consistent model over the crust-core boundary<cit.>, maybe mention this method can do that but we don't here? Even more thorough description of inner crust densities as that's what we have, especially as it relates to the plots I show.








Figure <ref> shows the ground-state energy of nuclear matter as a function of the baryon density n_B and proton fraction x. To generate this surface, the NQS energies for SNM and PNM, E(n_B, 1/2) and E(n_B, 0) respectively, were fitted to the following function based on a cluster expansion of the energy<cit.>,

    E(n_B, x) = a_0 
        + a_2/3( n_B/n_0)^2/3
        + a_1 ( n_B/n_0)
        + a_2 ( n_B/n_0)^2.

Here, n_0=0.16 fm^-3 represents the nuclear saturation density and the constant a_0 was fixed at zero for PNM. These fits enabled the computation of the symmetry energy S(n_B) as a function of baryon density, defined as the energy difference between PNM and SNM, 

    S(n_B)=E(n_B, 0)-E(n_B, 1/2).

The symmetry energy is an important quantity for characterizing nuclear matter, as it directly affects its neutron richness.
Finally, the energy surface between the two extremes of x=0 and x=1/2 was approximated through a well-known expansion in neutron richness, 1-2x, around symmetric matter<cit.>.
The energy computed from this expansion is given by

    E(n_B,x)=E(n_B, 1/2)+(1-2x)^2 S(n_B),

where we have already assumed that higher-order terms (which would come in as a fourth power of neutron richness) are negligible, hence the quadratic coefficient is simply the symmetry energy S(n_B) from Eq. (<ref>). Note that other expansions have been proposed <cit.>, and found to be more accurate in the low-density regime. We plan to employ them in future NQS calculations of isospin-asymmetric matter, as controlling finite size effects at small x requires using more than N=28 nucleons — see the Appendix on Finite-size effects for additional details. 

From Figure <ref>, we observe that the energy of nuclear matter decreases as the proton fraction increases, regardless of density. As expected, the change in energy from x=0 to x=1/2 is more pronounced in the high-density region compared to the low-density region. However, the onset of nuclear clustering makes the energy per particle of SNM at low-density negative by about -10 MeV per particle, as shown in Figure <ref>. Thus,
even at very low densities, the curvature of the energy per particle as a function of x is positive.
Finite-size effects are discussed in more detail in the Finite-size effect corrections subsection of the Methods section. 





 §.§ Beta-equilibrated nuclear matter


Cold nuclear matter at densities found in the inner crust of neutron stars is mostly composed of neutrons, protons, and electrons. Assuming the matter is neutrino transparent, the relative abundance of these particles at a specific density can be determined using a model for the energy as a function of proton fraction, under the well-supported assumptions of charge neutrality, n_p = n_e, and beta equilibrium, μ_e=μ_n-μ_p <cit.>. We will use this model of beta-equilibrated nuclear matter and our simulation results for SNM and PNM to construct an equation of state for our Hamiltonian. 



The proton fraction in beta-equilibrated matter as a function of baryon density is displayed in Figure <ref>. The NQS predictions align closely with the phenomenological Skryme parameterizations of the equation of state (Sly4 EOS) for the inner crust of neutron stars <cit.>. Figure <ref> underscores the flexibility of the NQS: by leveraging identical neural network architectures across all densities, we successfully capture the complexity of nuclear structures relevant to the inner crust of neutron stars. In contrast, the AFDMC yields proton fractions that tend towards those of non-interacting matter. This behavior is related to the inability of AFDMC to capture the onset of nuclear clusters at low density, making the presence of protons less energetically favorable compared to more accurate NQS calculations.

Due to the self-formation of nuclei in the low-density region, the NQS energy per particle of SNM obtained for N=28 would likely be even lower if we could carry out NQS calculations with N=40 particles or more. For example, the experimental energy per particle of ^28Si is 8.45 MeV, while that of ^40Ca is 8.55 MeV. These differences would be even larger if Coulomb interactions were turned off. 
On the other hand, the PNM simulations, once corrected for finite-size effects arising from the kinetic energy<cit.>, are expected to provide results close to the infinite case <cit.>. Thus, finite-size effects might underestimate the symmetry energy, yielding too small a proton fraction.




 §.§ Clustering

The Pfaffian NQS ansatz allows us to capture the self-emergence of nuclei and nuclear clusters from first principles, when their formation corresponds to a lower minimum of the variational energy than the uniform case. To better quantify the impact of clustering, we compute the two-body pair distribution functions, which provide the probability of finding two nucleons with defined isospin at a given distance from each other. The analytical expressions for the distributions are given by

    g_α(r) = 1/2π r^2 n_B N∑_i < j^N ⟨Ψ | δ (r_ij-r) P_α| Ψ⟩/⟨Ψ | Ψ⟩,

where P_α denotes the projector onto either neutron-neutron (nn), proton-neutron (pn), or proton-proton (pp) pairs. 





Instead of showing g_α(r) for SNM, we find it more informative to focus on an isospin-asymmetric case, specifically x = 4/28, as it yields simpler scenarios. The first possibility is the uniform case, in which the distributions should resemble those of the free Fermi Gas (FG), as the input Hamiltonian is relatively soft. A second possibility is the formation of four deuterons, which would yield a peak in the pn distribution but not in the pp one, as the small clusters would float in the simulation box. Finally, if at least one ^4He cluster is formed, the pp distribution should exhibit a peak at short distances. The two-body distributions displayed in Figure <ref> indeed seem to favor the latter case. In fact, compared to the non-interacting NQS case, obtained turning off the two- and three-body potentials in the input Hamiltonian, the distributions show a significant enhancement of pairs at short distances across all channels. For comparison, Figure <ref> also displays the FG distributions in the thermodynamic limit, whose expressions can be derived analytically. 

Unfortunately, the analysis of two-body distribution functions alone does not reveal whether heavier nuclei are present, including ^8Be or more neutron-rich isotopes. To overcome this limitation, we show a three-dimensional snapshot of Monte Carlo configurations for n_B =0.01 fm^-3 and x = 4/28 in the left panel of Figure <ref>. Clearly, the four protons (represented by solid orange circles) are always close to each other, indicating the presence of ^8Be. While at least four neutrons (represented by solid blue circles) are consistently close to the protons, there is also evidence of a free neutron component. This behavior is a microscopic indication of the neutron-drip, which is expected to occur in neutron-star matter at these densities. We note that these features are not specific to the Monte Carlo configurations displayed in Figure <ref>; rather, analogous structures are evident in all the samples we examined. 


In the right panel of Figure <ref>, we visualize Monte Carlo configurations for SNM at n_B = 0.01 fm^-3. Consistent with the energy per particle plotted in Figure <ref>, the 14 protons and 14 neutrons coalesce to form ^28Si. As a critical improvement compared to the state-of-the-art AFDMC method, the NQS effectively accommodates the self-emergence formation of a medium-mass nucleus in dilute isospin-symmetric matter. In addition, since the NQS does not require the a priori specification of free neutrons, it provides a more realistic and unbiased representation of the physical system as determined by our Hamiltonian.








The summation portion of this calculation is accomplished in our Monte Carlo simulation by a histogram over pair distances. This requires that the spherical shell area is replaced by the volume of a thin shell with thickness corresponding to the histogram bin size, 4π r^2 → (4/3)π (r_o^3-r_i^3).






§ DISCUSSION

In this work, we introduce a highly flexible Pfaffian NQS that leverages backflow transformations through a message-passing neural network to solve the nuclear quantum many-body Schrödinger equation with high accuracy. Notably, fermion anti-symmetry, parity, and translation invariance are respected by construction. As a crucial improvement over conventional variational wave functions used in state-of-the-art AFDMC calculations, a single Pfaffian ansatz is able to capture the different phases of matter across a wide range of densities corresponding to the crust of neutron stars. These phases, including extremely neutron-rich nuclei surrounded by superfluid neutrons and a uniform gas of protons and neutrons, are extremely difficult to model within conventional nuclear quantum many-body approaches, with the exception of lattice effective field theory <cit.>.

We demonstrate how the formation of nuclei dramatically decreases the energy of SNM in the low-density region, thereby lowering the value of the symmetry energy. This lowering has a tangible consequence on the proton fraction in beta-equilibrated, neutrino-transparent matter. The Pfaffian NQS yields a proton fraction that agrees well with the one computed using the phenomenological SLy4 functional. The predicted fraction is considerably larger than that obtained from AFDMC calculations in the low-density regime, while it agrees with this method in the uniform case, corresponding to n_B = 0.08 fm^-3 and beyond.

To probe the formation of nuclei heavier than the deuteron, we computed isospin-dependent two-body distribution functions at n_B = 0.04 fm^-3, simulating 4 protons and 24 neutrons with periodic-box boundary conditions. The appearance of a peak at short distances in all channels is a clear indication of the presence of nuclei heavier than the deuteron, which is corroborated by Monte Carlo samplings of the many-body wave functions at n_B = 0.01 fm^-3. The latter indicate the formation of neutron-rich Beryllium isotopes, surrounded by free neutrons. The flexibility of the ansatz is such that when simulating SNM at n_B = 0.01 fm^-3, the 14 protons and 14 neutrons self-organize to form ^28Si. 

In future calculations, we will leverage leadership-class GPU clusters to simulate a larger number of particles. Reducing finite-size effects will allow us to accurately simulate systems with proton fractions between SNM and PNM, which currently suffer from large systematic uncertainties. Concurrently, we plan to include screened Coulomb repulsion between protons, a fundamental component for investigating signatures of the “pasta” phase. A first approach in this direction involves treating the Coulomb lattice of nuclei using the Wigner-Seitz approximation <cit.>.

Finally, we are incorporating high-resolution Hamiltonians, both phenomenological and derived from chiral effective field theory, into our variational Monte Carlo method based on NQS. This step is crucial for achieving a unified description of neutron-star matter, from the crust to the inner core.



§ METHOD




 §.§ Neural-network quantum states

NQS have proven effective in solving the continuous-space Schrödinger equation for nuclear many-body systems, ranging from the deuteron to ^20Ne <cit.>. NQS based on the hidden-nucleon architecture have also successfully modeled both the normal and superfluid phases of dilute neutron matter using a single unified ansatz, demonstrating their flexibility in capturing a wide range of quantum many-body phenomena from first principles. Additionally, a more compact and flexible neural Pfaffian-Jastrow (PJ) architecture, which utilizes a message-passing neural network (MPNN) to efficiently encode pairing and backflow correlations, has been shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ultracold Fermi gases <cit.>. This ansatz is particularly effective at capturing strong pairing effects relevant to both cold atoms and neutron star crust matter.

In this work, we build upon the PJ+MPNN ansatz introduced in Ref. <cit.>,

    Ψ_PJ(X) = 
        e^J(X)× pf[           0 ϕ(x_1, x_2)           ⋯ ϕ(x_1, x_N); ϕ(x_2, x_1)           0           ⋯ ϕ(x_2, x_N);           ⋮           ⋮           ⋱           ⋮; ϕ(x_N, x_1) ϕ(x_N, x_2)           ⋯           0;             ].

To ensure skew-symmetry, the pairing orbital ϕ(x_i, x_j) is defined as 

    ϕ(x_i, x_j) 
    = ν(x_i, x_j) 
    - ν(x_j, x_i),

where ν(x_i, x_j) is a dense feedforward neural network (FNN). 
Consistent with Ref. <cit.>, we enforce translational invariance and parity in the ansatz. However, to encompass nuclear potentials, the MPNN is modified to take single-nucleon isospin coordinates through the one-body stream and isospin coordinates for each pair of nucleons through the two-body stream. Additionally, the pairwise spin inputs are modified to use the individual spins of each pair instead of their spin products.

We make a notable improvement to the PJ+MPNN ansatz by allowing the Pfaffian and Jastrow correlator to be complex, while keeping the MPNN real-valued. Generalizing the architecture proposed in Ref. <cit.> for spin systems, we make the Pfaffian complex-valued by feeding the outputs of the MPNN into two separate, real-valued FNNs, which represent the phase and amplitude of ν(x_i, x_j). Similarly, the MPNN outputs are fed into two real-valued, permutation-invariant Deep Sets <cit.>, representing the phase and amplitude of the Jastrow correlator J(X). 


To help mitigate runaway instabilities <cit.>, the logarithms of the real-valued amplitudes of all the neural networks, generally denoted as x, are regularized as

    x ↦ a tanh(x/a),

where a > 0 is a constant that sets the minimum and maximum allowed values of x. For all of our simulations, a was fixed to 8. All the individual FNNs in our ansatz have two hidden layers, each with a width of 16 nodes. The output dimension of the FNNs is either 1 or 16 nodes, depending on whether the FNN maps to a latent space or the output space. The activation function used throughout this ansatz is hyperbolic tangent and the pooling operation is logsumexp. In total, this NQS consists of 10,772 trainable parameters.



 §.§ Optimization

The optimization of the NQS is performed using a version of the stochastic reconfiguration (SR) algorithm, enhanced with RMSProp-like regularization <cit.>. For a complex wave function that depends on real parameters, the SR algorithm is modified such that only the real parts of the energy gradient g≡∇_pE and the quantum Fisher information matrix S are used to update the parameters p <cit.>, 

    p_t = p_t-1 - η(S_t-1)^-1(g_t-1).

Here, η is the learning rate and t is the optimization step. The RMSProp-inspired gradient accumulation is computed as

    v_t = βv_t-1 + (1-β) (g_t-1)^2,

with β = 0.99. Then, the quantum Fisher information matrix is regularized as

    S_t ↦ S_t + ε diag(√(v_t) + 10^-8),

where the small constant ε > 0 is typically set to 10^-3. 

All simulations are executed in parallel across 8 NVIDIA A100 GPUs, with a total of 16,000 configuration samples from 800 independent chains. Each optimization step, including Metropolis sampling and observable computation, takes approximately 80 seconds, with convergence generally requiring between 1,000 and 3,000 iterations. For the lowest baryon density examined, n_B = 0.001 fm^-3, transfer learning was employed to improve convergence and stability. Unlike the other densities, which were initialized with random parameters, the optimization for this density utilized parameters previously trained for n_B = 0.01 fm^-3 with the same number of protons and neutrons, significantly reducing the convergence time. 



 §.§ Finite-size effects

To mitigate finite-size effects, we adjust the NQS energies for a given n_B and x obtained with N=28 nucleons by subtracting the free kinetic energy of the finite system and adding the energy of the free FG in the thermodynamic limit <cit.>. These corrections, listed in Table <ref>, apply to the energies shown in Figure <ref>, Figure <ref>, and Table <ref>, as well as those used for calculating the proton fractions in Figure <ref>. 



It should be noted, however, that the above procedure is a rather crude one, as finite-size effects are not limited to the kinetic energy expectation value alone <cit.>. Furthermore, leveraging the free FG in the thermodynamic limits is only applicable when clustering is not as prominent as we found. Illuminating in this regard are simulations with low but nonzero proton fractions, as the number of simulated protons may be insufficient to form clusters large enough to significantly lower the energy. To better understand this effect, consider the hypothetical formation of stable nuclei in a sea of free neutrons at rest for a proton fraction of x=4/28 in the zero-density limit. Using experimental data for ground-state energies, we can infer that the binding energy per particle for the formation of a ^8Be cluster is 2.02 MeV, which increases to 2.28 MeV for ^16O, and further increases to 2.44 MeV for ^40Ca. This effect becomes even more amplified if the formed nucleus is neutron-rich. For instance, the binding energy is 2.45 MeV for a ^12Be cluster, 3.02 MeV for an ^24O cluster, and 3.21 MeV for a ^56Ca cluster. Therefore, while we can extract useful information to investigate clustering by simulating of N_p=4 protons, it is more reliable to employ Eqs. (<ref>) and (<ref>) to interpolate the NQS energies between PNM and SNM.




 §.§ Beta-equilibrated matter solution

To model neutrino-transparent nuclear matter in beta equilibrium, 
we apply the charge neutrality condition, n_p = n_e, and the beta equilibrium condition, μ_e = μ̂, where μ̂≡μ_n - μ_p represents the difference between neutron and proton chemical potentials. Using the well-known expansion of the energy per particle in neutron richness from Eq. (<ref>), the difference in chemical potentials can expressed in terms of the symmetry energy as

    μ̂ = - ∂ E(n_B, x)/∂ x = 4 (1-2x) S(n_B).

Next, by calculating the relativistic electron density at zero temperature and applying the charge neutrality condition, we obtain

    x≡n_p/n_B = n_e/n_B = (μ_e^2-m_e^2)^3/2/3π^2 n_B.

Finally, the beta equilibrium condition couples Equations (<ref>) and (<ref>), enabling the numerical solution for the proton fraction x. 




 §.§ Auxiliary-field diffusion Monte Carlo



The AFDMC solves the ground-state component of a given Hamiltonian by performing an imaginary-time propagation upon a starting variational ansatz <cit.>

    |Ψ_0⟩ = lim_τ→∞ |Ψ(τ)⟩ = lim_τ→∞ e^-(H-E_V)τ |Ψ_V⟩  .

In the above equation, E_V is a normalization constant, which is chosen to be close to the true ground-state energy E_0. The variational ansatz is assumed to be of the Slater-Jastrow form: |Ψ_V⟩ = F̂ |Φ⟩. To keep a polynomial cost in the number of nucleons, only linear terms in the spin-isospin correlation operators F̂ are included <cit.>. The optimal values of the variational parameters defining the correlation operator are found by employing the same stochastic reconfiguration method used to optimize NQS <cit.>. 



The mean-field part | Φ⟩ of the starting variational ansatz is the ground state of the uncorrelated Fermi gas, built from a Slater determinant of plane waves. The infinite system is simulated with N nucleons in a cubic periodic box of volume L^3. The momentum vectors in this box are discretized as 

    𝐤 = 2 π/L{ n_x, n_y, n_z}  ,

where n_x, n_y, and n_z are integer numbers. 
In this work, we carry out AFDMC calculations of SNM, simulating 14 protons and 14 neutrons as these “magic” numbers fill the discretized momentum shells for each species. AFDMC calculations of PNM with 28 particles in box would require either employing multiple Slater determinants in |Φ⟩ or twist–averaged boundary conditions <cit.>, which goes beyond the scope of this work. On the other hand, there is excellent agreement between NQS and AFDMC calculations of the energy per particle of PNM carried out with N=14 particles within a periodic box <cit.>. Thus, to obtain the symmetry energy results labeled as “AFDMC” in Fig. <ref>, we use the PNM energies obtained with NQS. 



 §.§ Data

The ground-state energies of obtained from NQS simulations at different n_B and x are listed in Table <ref>. For the isospin-symmetric case, we also report the corresponding AFDMC values. 



§ ACKNOWLEDGMENTS

We would like to thank Omar Benhar and Anthea Fantina for illuminating discussions. AL and BF are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contracts DE-AC02-06CH11357, by the 2020 DOE Early Career Award program, and by the NUCLEI SciDAC program. JK is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under grant No. DE-SC0021152. MHJ is supported by the U.S. National Science Foundation Grants No. PHY-1404159 and PHY-2013047.



§ AUTHOR CONTRIBUTIONS

B. F., J. K., and A. L. developed the code, designed the Pfaffian-Jastrow ansatz. B. F. and J. K. performed the numerical simulations. A. L. and M. H. J. provided overall supervision and guidance throughout the project. All authors contributed significantly to writing the manuscript.


 






 Paper Title: General Relativistic Magneto-Hydrodynamic Simulations with BAM: Implementation and Code Comparison.
















^1 Institut für Physik und Astronomie, Universität Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany
^2 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
^3 Theoretical Physics Institute, University of Jena, 07743 Jena, Germany
^4 Dipartimento di Fisica G. Occhialini, Università di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

^5 INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
^6 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy
^7 CFisUC, Department of Physics, University of Coimbra, Rua Larga P-3004-516, Coimbra, Portugal
^8 GRAPPA, Anton Pannekoek Institute for Astronomy, Institute of High-Energy Physics, and Insitute of Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
^9 Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan



§ ABSTRACT 
Binary neutron star mergers are among the most energetic events in our Universe, with magnetic fields significantly impacting their dynamics, particularly after the merger. While numerical-relativity simulations that correctly describe the physics are essential to model their rich phenomenology, the inclusion of magnetic fields is crucial for realistic simulations. For this reason, we have extended the  code to enable general relativistic magneto-hydrodynamic (GRMHD) simulations employing a hyperbolic `divergence cleaning' scheme. We present a large set of standard GRMHD tests and compare the  code to other GRMHD codes, , , and , which employ different schemes for the evolution of the magnetic fields.  Overall, we find that the  code shows a good performance in simple special-relativistic tests. In addition, we find good agreement and consistent results when comparing GRMHD simulation results between  and . 


General Relativistic Magneto-Hydrodynamic Simulations with BAM: Implementation and Code Comparison
    Masaru Shibata^2,9
    August 5, 2024
==================================================================================================





§ INTRODUCTION



Magnetic fields play a crucial role in various astrophysical high-energy scenarios. In particular, they are important in binary neutron star (BNS) mergers because the magnetohydrodynamical effects can influence the lifetime of the merger remnant and shape the matter outflow in the post-merger phase. In the multi-messenger event GW170817 <cit.>, the first observation of a BNS system by gravitational waves (GWs) accompanied by electromagnetic (EM) counterparts including the gamma-ray burst (GRB) GRB170817A <cit.>, strong evidence has been found for BNS merger to be sources of GRBs triggered by a relativistic jet, e.g., <cit.>. The launch of such a relativistic jet is most likely powered by magneto-hydrodynamic processes, e.g., <cit.>. 

A key in the study of BNS systems are numerical-relativity (NR) simulations that solve Einstein's field equations together with magneto-hydrodynamics equations. Due to the high complexity and dynamics of the merger process, an accurate description of the spacetime and hydrodynamics is crucial to allow for realistic predictions and to enable us to study details of the physical processes. Furthermore, we can extract the GW signal and the matter outflow from NR simulations to inform GW models as well as models for the EM counterparts, which are fundamental for the correct interpretation of multi-messenger observations. Among others, this allows for studies on the behavior of matter at supranuclear densities, e.g., <cit.>, the expansion rate of the Universe, e.g., <cit.>, and the synthesis of heavy elements, e.g., <cit.>. In the last decade, there has been rapid progress towards more realistic representations of microphysics in NR simulations, such as using tabulated equations of state (EOS) based on nuclear physics calculations to describe the interior of neutron stars (NSs), including neutrino emission and transport, as well as including magnetic fields.

In this work, we focus on the latter, i.e., the implementation of general relativistic magneto-hydrodynamics (GRMHD) routines in our NR code  <cit.>, but also refer to previous works concentrating on employing tabulated EOSs <cit.> and on the implementation of an advanced multipolar first-order (M1) neutrino transfer scheme <cit.> to highlight recent progress in performing more accurate simulations with . Magnetic fields, although negligible for the inspiral, are amplified during and after the merger due to the Kelvin-Helmholz instability (KHI), e.g., <cit.>, the Rayleigh-Taylor instability, e.g., <cit.>, and the magneto-rotational instability (MRI), e.g., <cit.>. Accordingly, they influence in particular the evolution of the remnant system and the outflow of matter in the post-merger phase due to magnetically driven winds and jet formation, making their incorporation crucial for correct ejecta models and predictions of EM signals, e.g., <cit.>. At the same time, they significantly increase the complexity of the simulations and pose new numerical challenges. 

One challenge in GRMHD is to fulfill the divergence-free condition for the magnetic field. There are various approaches in the literature to ensure that no magnetic monopoles are formed. Among the most popular schemes used in the NR community nowadays are the constraint transport scheme as initially developed by <cit.>, e.g., <cit.>, vector potential methods, e.g. <cit.>, and divergence cleaning approaches, e.g.,<cit.>. Constraint transport uses the induction equation and Stokes theorem to integrate the magnetic field (see <cit.> for a detailed discussion of constraint transport methods). In vector potential methods, one exploits the fact that the magnetic field can be expressed by a vector potential, which is then evolved instead. For both methods, the divergence of the magnetic field is by construction zero at round-off accuracy. However, both methods require a staggered grid, as the magnetic field components are usually defined on the surface of a grid cell, which complicates implementation in codes using mesh refinement. Also, vector potential methods do not guarantee magnetic flux conservation across the refinement boundary when mesh refinement is employed. 

We apply in  the hyperbolic divergence cleaning approach following <cit.>. In this approach, a new field variable is introduced to damp and advect the divergence of the magnetic field. Hyperbolic divergence cleaning of this type originated in <cit.> and was introduced to NR in <cit.> for magnetized rigidly rotating NSs. <cit.> showed very promising results for standard tests of special relativistic MHD, while divergence cleaning encountered some preliminary issues for GRMHD, so their focus was on constraint transport for GRMHD. 

Divergence cleaning offers the advantage of a relatively simple implementation, but has limitations in maintaining an exact zero divergence of the magnetic field. In fact, we expect to find an order of 10^-3 for the divergence of the magnetic field normalized to the magnetic field strength <cit.>. On the other hand, divergence cleaning is directly compatible with high-order schemes for fluxes, and thus, high-resolution shock capturing schemes can be applied directly without large code changes. In codes with constraint transport or vector potential, the magnetic field needs to be interpolated from the cell surfaces to the cell center, which is usually done linearly. 

In the present work, the goal is to analyse how well divergence cleaning actually performs compared to the other methods. For this reason, we conduct a comprehensive comparative analysis of our implementation with other, well-established GRMHD codes using different treatments for the divergence-free constraint: the  code <cit.>, the sacra variant of <cit.> henceforth called , and the  code <cit.>. The latter uses a vector potential method, while the other two codes use constraint transport. Such direct code comparisons are still rare, but are crucial to identify how dependent results are on the method used and how serious errors due to small divergences of the magnetic field are in the evolution also in relation to other numerical inaccuracies. In <cit.>, a comparison for different open-source GRMHD codes showed discrepancies in BNS merger scenarios, particularly in the merger times and remnant lifetimes. 

Our code comparison includes relativistic shock tests in one, two and three dimensions as well as BNS merger simulations. While the basic structure of the codes is relatively similar, we have to consider in the comparison not only differences in the magnetic field treatment but also in several other numerical implementations, such as high-resolution shock capturing (HRSC) schemes or algorithms for flux conservation at refinement boundaries. By using higher and lower orders for the reconstruction methods of fluid variables and different grid setups, we can estimate their effects and try to separate them from differences caused by the divergence-free constraint treatments. In fact, the tests show that the latter are negligible compared to the effects caused by using different reconstruction methods.
For the BNS merger simulation, we run the setup also at three different resolutions to carefully analyze the resolution effects and to determine whether the differences become larger or smaller at higher resolutions. 

The article is structured as follows: Section <ref> summarizes the relevant GRMHD evolutions equations, describes the divergence-free constraint treatment, and new numerical methods implemented in the  code. In Sec. <ref>, we present a series of relativistic tests performed with  and compare them with results of the GRMHD codes , , and . We then discuss results of BNS merger simulations performed with  and , and compare them in Sec. <ref>. A summary of our main results follows in Sec. <ref>. In this article, we apply a metric with (-,+,+,+) signature and geometric units with G=c=M_⊙=1, unless otherwise specified. 



§ RELEVANT EQUATIONS AND NUMERICAL METHODS



We summarize in this section our new GRMHD implementation in the  code <cit.>. Below, we briefly outline the framework of the NR code, including spacetime evolution and grid structure. We then discuss the relevant evolution equations and applied divergence-free constraint treatment. Further, we highlight new numerical techniques that we implemented in  to address the increased complexity of the systems due to magnetic fields. 



 §.§ Spacetime Evolution and Grid Structure


 evolves the gravitational field in time using the methods-of-line approach and employing finite difference stencils for spatial discretization. For this, Einstein's field equations are solved in a 3+1 decomposition, in which the line element is:

    ds^2 = -α^2dt^2 +  γ_ij(dx^i + β^i dt)(dx^j + β^j dt),

with α as lapse function, β^i as shift vector, and γ_ij as spatial part of the metric tensor g_μν induced on a three-dimensional, spatial hypersurface. The latter is given by:

    γ_αβ = g_αβ + n_αn_β,

where n^α is the timelike, normal vector to the three-dimensional hypersurface defined by: 

    n^μ = 1/α(1, -β^i) ,  n_μ = (-α,0,0,0).


In this work, we apply a fourth-order Runge-Kutta integration and a Courant-Friedrichs-Lewy (CFL) coefficient of 0.25 in all  simulations. We use the BSSN reformulation <cit.> for the tests presented in Sec. <ref> and the Z4c reformulation with constraint damping terms <cit.> for the BNS simulations presented in Sec. <ref>, combined with 1+log slicing <cit.> and gamma-driver shift conditions <cit.>. 


The infrastructure of  includes adaptive mesh refinement (AMR). The grid consists of a hierarchy of L refinement levels labeled by l=0,...,L-1 with one or more Cartesian boxes. Using a 2:1 refinement strategy, the grid spacing on each level is given by h_l = h_0/2^l. For inner refinement levels with l ≥ l_ mv, the Cartesian boxes can move and adjust dynamically during the evolution to track the compact objects. Thereby, each box consists of n grid points per direction or respectively n_ mv grid points per direction for inner, moving boxes. The points are cell-centered and staggered to prevent division by zero at the origin. As a result, the grid points of the successive levels l and l+1 do not coincide. 

Furthermore, we apply in  the Berger-Oliger scheme <cit.> for local-time stepping (see <cit.>) and the Berger-Colella scheme <cit.> to ensures flux conservation across refinement boundaries (see <cit.>).



 §.§ General Relativistic Magneto-Hydrodynamics



We use the Valencia formulation of the GRMHD equations <cit.>. We note that we apply the ideal GRMHD approximation assuming infinite conductivity and zero resistivity. Although a resistive approach would be more accurate, it leads to more complications and stiff source terms in the final evolution equations, which is why it is rarely used in the literature; see <cit.> for some examples. 


In GRMHD, the full electromagnetic field is described by the Faraday tensor:

    F^μν = n^μ E^ν - n^ν E^μ + ϵ^μνκλ n_κ B_λ,

where ϵ^μνκλ is the four-dimensional Levi-Civita symbol and where E^ν and B^ν are, respectively, the electric and magnetic field as measured by an Eulerian observer. Accordingly, we can obtain the electric and magnetic field by the Faraday tensor and its dual ^* F^μν = 1/2ϵ^μνκλ F_κλ via:

    E^μ = F^μν n_ν B^μ = ^*F^μν n_ν.


In the case of ideal GRMHD with infinite conductivity, there is no charge separation and the electric field vanishes in the fluids rest-frame. Thus, ideal GRMHD corresponds to imposing u_μ F^μν=0 where u_μ is the fluid four-velocity. Following this condition, the electric field can be expressed by:

    E^ν = -1/Wϵ^μνκλ n_κ B_λ u_μ,

where W = α u^0 = 1/√(1-v^2) is the Lorentz factor. It is therefore sufficient to evolve the magnetic field only, which greatly simplifies the final evolution equations. 

We introduce the magnetic four-vector b^μ = u_μ^* F^μν describing the magnetic field for a comoving observer. The components are given by:

    b^0 = W B^i v_i/α,  b^i = B^i/W + b^0 (α v^i - β^i),

with v^i = u^i / W + β^i / α as the spatial components of the fluid velocity measured by an Eulerian observer. 

The full stress-energy tensor for a perfect fluid is then defined by:

    T^μν = ( ρ h + b^2 ) u^μ u^ν + (p + b^2/2) g^μν - b^μ b^ν,

with h = 1 + ϵ + p/ρ being the specific enthalpy, ρ the rest-mass density, p the pressure, and ϵ the internal energy density. b^2 = b^μ b_μ represents twice the magnetic pressure with p_ mag = b^2/2. We note that in our definition of the magnetic field a factor of 4 π is absorbed. 


The evolution equations for ideal GRMHD are derived from the conservation laws of baryon number and energy-momentum:

    ∇_μ(ρ u^μ) =0, ∇_μ T^μν = 0,

as well as from the Maxwell's equations:

    ∇_μ^*F^μν = 0,

with ∇_μ being the covariant derivative. Furthermore, an EOS is required to close the evolution system. In order to write the evolution system in the form of a balance law with:

    ∂q/∂ t + ∂F^i/∂ x^i = s,

we define the primitive variables w=(ρ, v^i, ϵ, p, B^i ) and the following conservative variables q:

    D    =   √(γ)ρ W, 
    
        S_j  =   √(γ)((ρ h +b^2) W^2 v_j- α b^0 b_j), 
    τ =   √(γ)((ρ h +b^2) W^2 - (p+p_ mag). 
       .- α^2 (b^0)^2 )- D, 
    B̂^k  =   √(γ) B^k ,
     
where γ is the determinant of γ_ij.
Finally, we obtain the following evolution equations in the form of Eq. (<ref>):

    q   = [D, S_j, τ, B̂^k], 
    F^i    = α[                                       D ṽ^i; S_j ṽ^i + √(γ)(p+p_ mag) δ_j^i - b_j B̂^i/W;   τṽ^i + √(γ)(p+p_ mag) v^i - α b^0 B̂^i/ W;                         B̂^k ṽ^i - B̂^i ṽ^k ], 
    s   = α√(γ)[                                    0; T^μν( ∂ g_ν j/∂ x^μ - Γ^λ_ μν g_λ j);    α(T^μ 0∂ln (α)/∂ x^μ-T^μνΓ_ μν^0);                                  0^k ],
 
with ṽ^i = v^i -β^i/α and the Christoffel symbols Γ_ μν^λ.



 §.§ Divergence-Free Constraint Treatment



While the system of equations derived above is generally sufficient to perform simulations and evolve the magnetic field in time, there is no guarantee that the time component of Maxwell's equations (<ref>) is satisfied. Even when starting from constraint-satisfying initial data, numerical errors can accumulate leading to magnetic monopoles and thus to violations of Eq. (<ref>). Approaches typically used to address this issue include constraint transport schemes as implemented in  and , vector potential methods as implemented in , and divergence cleaning techniques. We refer to <cit.> and <cit.> for implementations of constraint transport and vector potential methods in , , and , respectively. 

Our GRMHD implementation follows the hyperbolic divergence cleaning scheme from <cit.>. In particular, a new field variable ζ is introduced to damp and advect divergences of the magnetic field. We use a modification of the Maxwell's equations:

    ∇_μ( ^*F^μν + g^μνζ) = κ n^νζ,

with the damping rate κ. We set κ = 1 in this work. For ζ→ 0, the equation reduces again to the Maxwell's equations (<ref>). From the time component of Eq. (<ref>), we obtain an evolution equation for ζ:

    ∂_t ζ   + ∂_i ( α/√(γ)B̂^i - ζβ^i ) 
       = ζ (-κα - ∂_i β^i) + B̂^i ∂_i (α/√(γ)).


The spatial part of Eq. (<ref>) leads to the following modification for the evolution equation of the magnetic field:

    ∂_t B̂^j    + ∂_i [ (α v^i - β^i ) B̂^j - α v^j B̂^i + α√(γ)γ^ijζ] 
       = - B̂^i ∂_i β^j + ζ∂_i ( α√(γ)γ^ij).


Although divergences of the magnetic field are damped, this approach does not prevent the occurrence of constraint violations, unlike the constraint transport or vector potential methods. This is an obvious disadvantage of the method, leading to finite errors. Nonetheless, these errors should converge to zero in a reliable and controlled manner with the scheme employed, similar to the convergence of other constraints, e.g., the Hamiltonian constraint. On the other hand, the implementation is much simpler as there is no staggering of the magnetic field necessary and available high-order HRSC schemes can be directly employed. While  already uses cell-centered fields for both the metric and fluid variables, additional work would be required since the magnetic field is defined at the cell faces and not in the cell centers. In particular, the AMR restriction and prolongation algorithms would have to be changed and would become more complex. For a magnetic field defined at the cell surface, it would first have to be interpolated to the cell centers. In most codes using constraint transport and vector potential methods, this is often done linearly and, thus, the magnetic field relies on second-order methods. For a cell-centered magnetic field, however, we can use the same methods as for other variables, typically using fourth-order schemes in our implementation. Apart from questions of complexity and convergence rate, a key question is how a given physical system depends on numerical errors in the divergence constraint, which we investigate in the numerical examples that follow.



 §.§ Numerical Schemes




  §.§.§ Conversion to Primitive Variables


In order to write the evolution equations in form of a balance law, we use the conservative variables q as defined in Eqs. (<ref>)–(<ref>) which are analytical functions of the primitive variables w. Nevertheless, we need to know the primitive variables, for instance, to compute the flux terms F^i at each time step. However, the inversion of Eqs. (<ref>)–(<ref>) to obtain the primitive variables is non-trivial and must be solved numerically. We refer to <cit.> for a summary and discussion of different approaches commonly used in NR codes. The scheme we use follows the approach implemented in the RePrimAnd library <cit.>. More precisely, we use its master-function (see Eq. (44) of Ref. <cit.>) and use a Brent-Dekker scheme to find its root. In the numerical tests as well as in the BNS merger simulations, which we discuss in Secs. <ref> and <ref>, this method has proven to be robust and provides reliable results.



  §.§.§ Reconstruction



 contains HRSC methods for the treatment of shocks and discontinuities in the hydrodynamic variables. In order to evaluate the fluxes F at the individual grid cell interfaces and to solve the Riemann problems, the variables have to be reconstructed to the cell surfaces from either side. Since all field variables are cell-centered in our GRMHD implementation, we can use the same reconstruction schemes for the magnetic field variables and ζ that are already implemented in . These include, for example, a third-order convex-essentially-non-oscillatory (CENO3) scheme <cit.>, a fifth-order weighted-essentially-non-oscillatory (WENOZ) scheme <cit.>, or a fifth-order monotonicity preserving (MP5) scheme  <cit.>. 

While higher order schemes are generally more accurate and less dissipative, they can exhibit stronger oscillations, as we show and discuss in some relativistic shock tests in Sec. <ref>. Enhanced Oscillations can lead to unphysical values. For this reason, in the BNS merger simulations with  presented in Sec. <ref>, we apply the following procedure in lower density regimes to ensure the physical validity of reconstructed variables and, in particular, to maintain the positivity of the density and pressure:


    
  * We use a high-order reconstruction scheme to compute the GRMHD variables at the interface between grid cell i and i+1. 
    
  * If the rest-mass density ρ drops below a threshold density, we examine the oscillation of all variables by comparing the reconstructed values at the interface with the cell-centered ones of i and i+1. 
    
  * Whenever one reconstructed variable is below or above a value in the grid cells i and i+1, we change to a low-order reconstruction and recompute the values of all variables at the interface.
    
  * Finally, we examine the physical validity of the reconstructed values by demanding a positive rest-mass density and a positive pressure. If this is not given, we use a linear reconstruction method.


For high-order reconstruction schemes, we typically use WENOZ or MP5 as fifth-order methods or CENO3 as third-order method. We then fall back on a linear total variation diminishing (TVD) method with “minmod” slope limiter <cit.> as low-order reconstruction scheme.



  §.§.§ Riemann Solver



For the GRMHD simulations with  in Secs. <ref> and <ref>, we use the Harten, Lax, and van Leer (HLL) Riemann solver <cit.>. HLL uses a two-wave approximation via estimates for the fastest left- and right-moving signal speeds:

    F^ HLL   = λ_ maxF(q^+) - λ_ minF(q^-)/λ_ max - λ_ min
       - λ_ maxλ_ min/λ_ max - λ_ min(q^+-q^-),

where q^+ and q^- are the variables respectively reconstructed from the left and right side of the cell face, and F(q^+) and F(q^-) are the according flux terms. The fastest left- and right-moving signal speeds are determined by:

    λ_ max   = max(λ_i(q^+),λ_i(q^-),0), 
    λ_ min   = min(λ_i(q^+),λ_i(q^-),0),

with λ_i being the characteristic wave speeds. 

While there are three characteristic modes in pure general relativistic hydrodynamics, there are seven independent characteristic waves in ideal GRMHD: the entropy wave, two Alfven waves, and four magnetosonic waves (two slow and two fast modes). For a detailed discussion of the individual waves and derivation of the eigenvalues, we refer to <cit.>. Since the exact wave speeds require the solution of a non-trivial quartic equation for the magnetosonic waves, we use the commonly applied approximation proposed in <cit.>. We note that by using divergence cleaning, two additional modes with characteristic velocities equal to the speed of light must be taken into account. For the magnetic field variables and ζ, we therefore set λ_ max=1 and λ_ min=-1 in the Riemann solver. 


In addition to HLL, we started to implement the HLLD, a five-wave relativistic Riemann solver, following <cit.>, with minor changes to adapt to the divergence cleaning formulation. The implementation is still in test phase. For this reason, we use it only in some of the special relativistic tests presented in Sec. <ref>.



  §.§.§ Atmosphere



In grid-based NR simulations, the vacuum region surrounding the compact objects is usually modeled with an artificial atmosphere. The reason is that too low rest-mass densities are numerically very demanding and can lead to errors in the recovery of the primitive variables and in HRSC schemes. Therefore, we set in  a grid cell to atmosphere when its density falls below a density threshold ρ_ thr <cit.>. We assume a cold, static atmosphere: the density is reduced to a fraction of the star's original central density ρ_ atm = f_ atmρ_c, pressure and internal energy are determined according to the zero temperature part of the EOS, and the velocity of the fluid is set to zero. The magnetic field remains unchanged. The density threshold is defined as the fraction of the atmosphere value with ρ_ thr = f_ thrρ_ atm to avoid fluctuations around the density floor. In the BNS merger simulations presented in Sec. <ref>, we use f_ atm = 10^-11 and f_ thr = 100.



§ SPECIAL RELATIVISTIC TESTS



For validating our new implementation, we have performed a series of well-established special relativistic magneto-hydrodynamics tests in one-, two-, and three-dimensions. In particular, it is important to analyze the reliability of the code with respect to shocks. In each test, we compare our results and performance with the codes  <cit.>,  <cit.>, and  <cit.>.



 §.§ One-Dimensional Tests








Relativistic shock tube problems are simple, one-dimensional tests that can be used to demonstrate a code's ability to capture a variety of different shock wave structures. By setting different states of the fluid to the left and right of an interface, a shock is generated and evolved. Among the most popular and well-known shock tube problems in magneto-hydrodynamics are the Balsara tests <cit.>. We summarize the initial data for the Balsara tests I–V in Tab. <ref> by specifying the respective fluid variables of the left and right states. The tests are performed along the x-axis with the initial fluid discontinuity at x=0. In all Balsara tests, an ideal-gas law EOS with p=(Γ-1)ρϵ, where Γ is the adiabatic constant, is assumed: in Balsara test I with Γ = 2.0 and in Balsara test II–V with Γ = 5/3. We evolve the shock waves until t=0.4 for Balsara tests I–IV and t=0.55 for Balsara test V. 

The results are shown in Fig. <ref> for each Balsara test at three different resolutions with 1000, 2000, and 4000 grid points in x ∈[-0.5, 0.5]. We use MP5 reconstruction. For each test, we show the profiles of the density ρ and the B^y component of the magnetic field at the respective final time. Additionally, we show the exact solution of <cit.> as comparison. Overall, all tests are in good agreement. As expected, higher resolutions increase the accuracy and are better able to capture the shock fronts. This is particularly evident in the Balsara tests II, III and IV, where a high resolution is required to capture the sharp edges of the shock waves in ρ. Some oscillations are visible, e.g., in B^y of Balsara tests III and IV, that we attribute to the use of a reconstruction scheme with relatively high order. 

In order to evaluate how our GRMHD implementation performs in these tests with respect to other NR codes, we compare the results obtained with  with those obtained with , , and . Figures <ref> and <ref> show respectively the Balsara tests I and III for all four codes. The resolution is set to 2000 grid points in x ∈[-0.5, 0.5]. Although we try to choose the technical configurations rather similar, each working group using different codes has preferred numerical methods that also differ in the individual implementations. For instance, the HLLD Riemann solver as described in <cit.> is used for both tests by . This solver is more advanced than the HLL solver used by , , and . Furthermore, different methods are used to reconstruct the fluid variables: While  uses a fifth-order WENO5 scheme,  applies a third-order piecewise parabolic method (PPM).  also employs PPM reconstruction in Balsara test I. However, since Balsara test III is more demanding due to the large jump in the initial pressure,  uses here a second-order TVD method with “minmod” limiter. To assess the differences caused by different reconstruction algorithms, we show for  results obtained with a linear TVD method, labeled as `low order' in Figs. <ref> and <ref>, additionally to those with MP5 reconstruction. 

All codes demonstrate their ability to reliably capture and evolve the shock waves with small differences at the shock fronts. For Balsara test III, , , and  (with MP5) seem to be superior in modeling the exact solution, despite showing small oscillations. We explain this by the usage of higher orders for the reconstruction.  uses a second-order scheme in this test. In fact, the solution of  with linear TVD method perfectly matches the results of  when the same reconstruction method and Riemann solver are used. Also in Balsara test I, the largest differences appear for  using different reconstruction methods. This suggest that the differences in Figs. <ref> and <ref> for the individual codes are due to different HRSC algorithms rather than different methods for the magnetic field or the divergence-free constraint treatments.






 §.§ Two-Dimensional Tests




  §.§.§ Cylindrical Explosion







The first two-dimensional shock problem we perform in this test series is the cylindrical explosion, also known as cylindrical blast wave, a standard problem for GRMHD codes, e.g., <cit.>. We consider a dense medium within a cylindrical radius of r_ in = 0.8 in a less dense ambient medium outside of r_ out =1. The denser medium has an overpressure and therefore expands. The initial values for density ρ and pressure p are: ρ_ in  = 10^-2 and p_ in  = 1.0  for the inner medium, and ρ_ out  = 10^-4 and p_ out  = 3 × 10^-5 for the outer medium. For the transition region 0.8 ≤ r ≤ 1, we apply:

    ρ   = exp( (r_ out -r ) lnρ_ in + (r - r_ in) lnρ_ out/r_ out-r_ in), 
    
        p    = exp( (r_ out -r ) ln p_ in + (r - r_ in) ln p_ out/r_ out-r_ in),

with cylindrical radius r = √(x^2 + y^2) in the x-y plane. In this test, an ideal-gas EOS with adiabatic index Γ = 4/3 is used and we set a uniformly directed magnetic field along the x-axis with B^x = 0.1 and B^y = B^z = 0. We evolve this configuration until t=4. 

The resulting blast wave at the final time is presented in Fig. <ref>. We show snapshots of ρ, p, B^x, and B^y in the x-y plane of a simulation with  using CENO3 reconstruction and a grid consisting of two refinement levels with 200 × 200 grid points each, covering a range of [-6, 6] in x and y directions. A fast forward shock and a reverse shock bounding the inner region are observable. While there is no exact solution available for this tests, the structures shown in our results are consistent with those in <cit.>. 

For a more quantitative comparison of our results with , , and , we show one-dimensional slices of the final blast wave for ρ, p, and B^x along the x- and y-axes in Fig. <ref>. To avoid discrepancies due to different algorithms for flux conservation at the refinement boundaries, we use here a grid consisting of a single refinement level with grid spacing Δ x = 0.03 and 400 × 400 points. For , we apply the reconstruction as described in Sec. <ref> with CENO3 as higher order scheme and linear TVD as lower order method, examining oscillations in the fluid variables in regions with ρ < 2× 10^-4. Additionally, we run  once using a pure linear TVD method, labeled as `low order' in Fig. <ref>. Both,  and , apply here a second-order TVD scheme, while  results are shown for third-order PPM reconstruction. Lacking a well-known analytical solution for this test problem, we perform additionally one high-resolution test run with  using 8000 × 8000 grid points, i.e., a spacing of Δ x = 0.0015, using CENO3. Except for , that applies the HLLD Riemann solver, the results are obtained with HLL. The results for , , and  overlap with the ones obtained with  using the linear TVD reconstruction method. The higher-order reconstruction leads to stronger oscillations at the shock fronts, which are mainly visible in the ρ and p profiles along the y-axis. Nevertheless, all codes converge approximately to the same solution. As before in the Balsara tests, we conclude that the differences are primarily due to different HRSC schemes.



  §.§.§ Magnetic Rotor






As a second two-dimensional standard test, we consider the magnetic rotor, originally described for classical magneto-hydrodynamics <cit.> and later generalized to the relativistic case <cit.>. For this test, we initialize a dense, fast rotating fluid in the center of a non-rotating ambient medium. The central rotating fluid is set inside an area with cylindrical radius r=0.1 with ρ_ in = 10 and a uniform angular velocity of Ω = 9.95, resulting in a maximum velocity of the fluid of v_ max = 0.995. The density of the surrounding medium is ρ_ out = 1. We apply in this test an ideal-gas EOS with Γ = 5/3 and consider a uniform pressure with p=1 and magnetic field with B^x = 1, and B^y=B^z=0. The test is simulated until t=0.4. 

We perform this test with  on a grid with two refinement levels and 200 × 200 points covering the range [-0.5, 0.5 ] for x and y using CENO3 reconstruction. During the simulation, the rotation leads to a twisting of the magnetic field lines in the central region, which in turn slows down the rotor. Figure <ref> presents maps of the density ρ, the pressure p, and the magnetic field variables b^2, which can be considered as twice the magnetic pressure, and the B^x component at the final time. The results are in good agreement with those in <cit.>. 

Like for the cylindrical explosion, we compare our results with , , and  for one-dimensional slices at the final time. Figure <ref> shows profiles of ρ, p, and B^x along the x- and y-axes. As for the cylindrical explosion, we choose for the comparison a grid without mesh refinement consisting of 400 × 400 grid points with a spacing of Δ x = 0.0025. The results with  are obtained with the HLLD Riemann solver in combination with a third-order PPM reconstruction, while  and  apply HLL and a second-order TVD reconstruction. For  we show the results using the HLL Riemann solver once with CENO3 and once with linear TVD, labeled as `low order' in Fig. <ref>. Again, we perform additionally one high-resolution test run for  with HLL and CENO3 using 8000 × 8000 grid points and a spacing of Δ x = 0.000125. 



Overall, the results of the individual codes agree well. Mostly at the shock fronts, i.e., along x between -0.2 and -0.35 and along y between 0.1 and 0.25, some codes show stronger oscillations than others, which is strongly depending on the HRSC method used. The largest differences are present for  for the B^x profile. However, the differences decrease for higher resolutions. As shown in Fig. <ref>, with increasing resolution from Δ x = 0.0025 to Δ x = 0.00125 up to Δ x = 0.000625 the B^x profile for  converges against the same solution. The shock along the y-axis is well captured even at the lowest resolution, while the shock along the x-axis needs a higher resolution. Since the main difference between  and the other codes is the Riemann solver, we attribute to the HLLD solver being more suitable for different types of shock waves than for others.



  §.§.§ Kelvin Helmholz Instability




The last two-dimensional special relativistic magneto-hydrodynamics test we consider in this work addresses the KHI as proposed in <cit.>. For this test, we are focusing the comparison on the  and  codes. We use the same setup as in <cit.> and specify a tanh-shaped velocity profile with:

    v^x = -v_ shtanh(y/a).

Here, v_ sh determines the shear velocity and a the thickness of the shear layer. As in <cit.>, we set v_ sh = 0.25 and a = 0.02. Further, we initialize a uniform density of ρ = 1 and pressure of p=20, and apply an ideal-gas EOS with Γ = 4/3. The components of the magnetic field are B^x = √(2σ p) and B^y = B^z = 0. We set σ = 0.01. 

To disturb the shear layer, movement along the y direction is introduced by:

    v^y = A_0 v_ shsin(2 π x) exp(-100 y^2),

where A_0 = 10^-4 determines the strength of the perturbation. The v^z component remains zero. 

Figure <ref> presents snapshots of the density ρ for  at t=10.9 and for  at t=10.0. In both cases, the grid consists of one refinement level with 100 × 200 points covering a range for x ∈[-0.5, 0.5] and y ∈[-1, 1]. We use in  MP5 and in  PPM reconstruction. Both codes demonstrate their ability to resolve the vortex formed in the KHI test. The observable structure is very similar, although the vortex forms later for  than for . We attribute this to the usage of different Riemann solver:  applies HLL and  HLLD. A similar behavior for different solvers is also found in <cit.>. 





To be able to compare the simulations quantitatively, the perturbed velocity difference Δ v^y = 0.5 (v^y_ max-v^y_ min) is shown as a function of time in Fig. <ref>. We show the results for  additionally for using the HLLD solver that is still under development. As in <cit.>, Δ v^y grows exponentially until nonlinear saturation is reached. The growth rate is higher and nonlinear saturation is reached faster for  and  using HLLD than for  using HLL, whereas the saturation amplitude is about the same for all cases. This demonstrates that the exponential growth for the KHI test depends strongly on the Riemann solver used, while the nonlinear saturation is only weakly dependent.



 §.§ Three-Dimensional Tests




  §.§.§ Spherical Explosion





As a demanding three-dimensional shock problem for GRMHD codes, we perform the spherical explosion test with  and . We follow the description in <cit.>, which corresponds to a three-dimensional version of the cylindrical explosion test in Sec. <ref>, where the cylindrical radius is substituted for a spherical radius: A dense medium with ρ_ in  = 10^-2 and p_ in  = 1.0 is set inside a spherical radius of r_ in = 0.8. The ambient medium outside of r_ out = 1 has ρ_ out  = 10^-4 and p_ out  = 3 × 10^-5. In the transition region between r_ in = 0.8 and r_ out=1, we set:


    ρ   = exp( (r_ out - r ) lnρ_ in + (r - r_ in) lnρ_ out/r_ out-r_ in), 
    
        p    = exp( (r_ out - r ) ln p_ in + (r - r_ in) ln p_ out/r_ out-r_ in).

We apply again an ideal-gas EOS with Γ = 4/3 and initialize a uniform magnetic field with B^x = B^y = 0.0 and B^z = 0.1. The results obtained with  are shown in Fig. <ref> at the final time t=4. For this test, we apply CENO3 reconstruction and use a grid with two refinement levels and 200 × 200 × 200 points ranging from [-6,6] in each direction. The grid spacing is accordingly Δ x = 0.06 on l=0 and Δ x = 0.03 on l=1. 

We compare in Fig. <ref> our results for one-dimensional profiles along the x- and z-axes at the final time for ρ, p, and B^x. In this case, we focus on the comparison with  and choose a grid with two refinement levels as already described above. We include in Fig. <ref> lines for the profiles on level l=0 and l=1. Overall, the differences between  and  are small: In the center, differences are of the order 10^-8 for the rest-mass density, 10^-7 for the pressure, and 10^-5 for the magnetic field component B^x. The highest differences are found at the shock front of the p profile along the x-axis.  shows here larger values than  by a factor of almost ten. However, these larger differences are only outside the inner refinement level, where the resolution is quite low.

This test also enables us to access the conservation of the magnetic flux across refinement levels. By integrating the magnetic flux over the refinement boundary on levels l=1 and l=0, we can compare the magnetic flux violations. Figure <ref> shows the time evolution of the magnetic flux conservation across the positive z refinement boundary, i.e., the area with z=3, x ∈ [-3,3] and  y ∈ [-3,3], and presents the relative difference between ∫_z+ B_l_0 d A and ∫_z+ B_l_1 d A. We present results for , , and additionally for  using a lower resolution with Δ x = 0.12 on l=0 and Δ x = 0.06 on l=1 (labeled as `low res.'). While both codes use methods to correct fluxes across refinement boundaries, it is evident that the magnetic flux is better conserved in the simulation with  than in the simulations with . For , the violations are of the order of 10^-14, i.e., at round-off accuracy. For , the difference between the fluxes is initially zero, but increases significantly once matter crosses the refinement boundary at t ≈ 1.5, reaching orders of 10^-5. Using divergence cleaning in , the evolution equation for the magnetic field contains additional source terms. Although  incorporates algorithms to correct the fluxes across refinement boundaries, the source terms are not matched and may marginally differ, resulting in slightly different values. Figure <ref> shows small improvements for increased resolution.





§ BINARY NEUTRON STAR MERGER SIMULATIONS







 §.§ Configurations


We perform simulations of BNS merger with magnetic fields and compare the simulation results with . The publicly-available, spectral code fuka <cit.> is used to construct initial data for an equal-mass BNS system with gravitational masses m_A = m_B = 1.35 M_⊙ and baryonic masses m_b,A = m_b,B = 1.49 M_⊙. As  has an implemented fuka reader, the same initial data is used to directly compare the evolution and results of the BNS simulations. We set an initial coordinate separation of ∼ 28.4 km to ensure a short inspiral, focusing the simulation on the merger and post-merger phases. The initial ADM mass is M_ ADM =2.673 M_⊙. For the EOS, a piece-wise polytropic fit of SLy <cit.> is applied following <cit.> with four pieces, three for the NS core and one for the crust. In order to include thermal effects, the zero-temperature EOS is extended by a thermal pressure P_ th = (Γ_ th - 1) ρϵ_ th <cit.>, where ϵ_ th is the thermal part of the specific internal energy and we set Γ_ th = 1.75. We note that a slightly different value is used for the simulations with , namely Γ_ th = 1.8. 

We initialise a poloidal magnetic field for each star by a vector potential:

    A_x    = -y A_b max(p - p_ cut,0)^2, 
    
        A_y    =  x A_b max(p - p_ cut,0)^2, 
    
        A_z    =  0,

with x = x - x_ NS and y = y - y_ NS as distances from the center of the NS along x and y, respectively. We set A_b = 1000 and p_ cut=0.004 × p_ max, where p_ max is the maximum pressure in the NS at t=0 ms. In this way, we obtain a maximum magnetic field strength of the order ∼ 10^15 G inside the stars. 


For the grid, we use in  eight refinement levels where the three outermost are non-moving and the five innermost are moving levels. The simulations are performed on three different resolutions: R1 with n = 128 and n_ mv = 64 grid points, R2 with n = 256 and n_ mv = 128 grid points, and R3 with n = 512 and n_ mv = 256 grid points in each direction. The grid spacing on the finest box covering the NS are Δ x_ min≈ 368.6 m for R1, Δ x_ min≈ 184.3 m for R2, and Δ x_ min≈ 92.1 m for R3. We use the HLL Riemann solver as described in Sec. <ref> and the reconstruction method as described in Sec. <ref> with MP5 as high-order reconstruction and CENO3 as low-order reconstruction, i.e., we examine oscillations and fall back to CENO3 or further to linear reconstruction if necessary. We employ this only for low density regions: in the R1 and R2 simulations for ρ < 100 ×ρ_ atm and in the R3 simulation for ρ < 10^6 ×ρ_ atm[At the highest resolution, we encountered issues when using MP5 due to larger oscillations leading to unphysical behavior of the matter outflow along the grid axes. This problem could be solved with a higher threshold for the rest-mass density using our scheme to detect enhanced oscillations and maintain the positivity of the reconstructed density and pressure.].

 uses a similar grid configuration with ten refinement levels and the same resolutions R1, R2 and R3 as in . However, the box sizes of the refinement levels are larger as  uses a fixed mesh refinement (FMR). The half size of the finest level is set to L_ fin =35.5  km. For the inner part of the finest refinement level, a constant atmosphere is set with ρ_ atm = 10^3  g/cm^3. Outside a power-law profile is applied with ρ_ atm = 10^3 (L_ fin/r)^3   g/cm^3.  introduces a restriction for high magnetic-pressure regions with b^2/ρ≥ 10^3 by artificially suppressing the specific momentum. The reason is that the numerical accuracy is not good in those regions and, as a result, the primitive recovery procedure often fails. However, in the current work, in which the post-merger evolution is not followed over a long timescale, we do not find such a highly magnetized region. As in Sec. <ref>,  applies the HLLD solver with third-order PPM reconstruction.



 §.§ Evolution and Magnetic Field Amplification






The evolved BNS system merges after about four orbits and forms a hypermassive neutron star (HMNS), collapsing into a BH within a few tens of milliseconds. The lifetime of the HMNS varies for different resolutions because the HMNS just prior to the collapse is marginally stable, and hence, small perturbations including numerical errors can trigger the collapse. In Fig. <ref>, we present snapshots of the rest-mass density and the magnetic field lines for the simulation with the highest resolution. The snapshots show the two NSs at the beginning of the simulation, at the merger (at about 9.6  ms), and the remnant before and after the collapse. The initial magnetic dipole field inside the NSs is clearly visible, as well as the winding of the magnetic field lines during the merger, forming a helicoidal structure. 

We show the time evolution for the magnetic energy and maximum magnetic field strength in Fig. <ref> for the R1, R2, and R3 simulations with . The magnetic energy is defined by:

    E_ mag = 1/2∫ u^0 √(-g) b^2 d^3x.

Additionally, the toroidal and poloidal contribution are shown. We note that they are defined with respect to the coordinate center, which is only approximately correct after the merger assuming the remnant stays in the center. During the merger, the magnetic field is amplified. We observe here a strong resolution dependency: For R1, there is almost no amplification during the merger itself. Only on later time scales of about 10  ms after the merger the magnetic energy and field strength rise. For R2, the maximum magnetic-field strength doubles, and for R3, it increases tenfold, leading to energies up to 10^49  erg. The amplification is triggered by a variety of instabilities. For example, the shear layer between the two NS induces a KHI during the merger, which forms vortices that twist the magnetic field and lead to exponential growth, e.g., <cit.>. Another mechanism is driven by large-scale differential rotation in the post-merger, which winds up the magnetic field lines. While the magnetic winding can already be resolved with lower resolutions, capturing the KHI and also the MRI requires higher resolutions. In order to fully capture these instabilities and the associated magneto-hydrodynamic turbulence, even higher resolutions are required as illustrated in <cit.> (see also <cit.>). 

Although the magnetic field is initially purely poloidal, the toroidal component becomes dominant after the merger as shown in Fig. <ref>. For a qualitative analysis of the magnetic field in the remnant system, Fig. <ref> presents snapshots in the x-z plane after the merger. We show the magnetic field strength and the ratio between the toroidal and poloidal component before and after the collapse for the R3 simulation. Before the collapse, the magnetic field in the polar region is already predominantly poloidal. This becomes even clearer after the collapse. The disk of the remnant has rather toroidal magnetic fields. After the collapse, the magnetic field in the polar region decreases from an order of magnitude of 10^12  G to an order of magnitude of 10^11  G. The magnetic field is strongest inside the disk with up to 10^16  G. 

Additionally, we show in Fig. <ref> the divergence of the magnetic field. As already mentioned, a limitation of the implemented divergence cleaning method is that the divergence of the magnetic field is not exactly zero and cannot prevent the formation of magnetic monopoles. To assess how large the divergence actually is, we present maps of the remnant after the merger. It is largest in the interior of the disk, where the magnetic field is also strongest, with values in the order of 10^8  G / cm and smallest in the polar regions with values in the order of 10^3  G / cm. 

One important point is to compare errors introduced by the magnetic field divergence violation with other uncertainties, e.g., due to finite resolution, flux computation, or shock treatments. Following Gauss's law, we can translate the divergence into an error on the magnetic field within a grid cell by multiplying with its volume dV_i and dividing by its surface area dS_i. With E_ mag∝ B^2, we obtain an estimate of the relative error on the magnetic energy by normalizing with the magnetic field strength and squaring the quantity. In Fig. <ref>, we compare this relative error of the simulations using R1, R2, and R3 with the relative difference of the magnetic energy between resolution R2 and R3. The error in the magnetic energy due to the magnetic field divergence increases during the merger. Overall, the relative error is between 10^-8 and 10^-4 and decreases with higher resolution. The relative difference between the magnetic energy at R2 and R3 resolution is about 10^-2 and increases to 10 after the merger. We note that, in agreement with other GRMHD simulations shown in previous studies, even our highest resolution is well outside the convergence regime, explaining the large differences here. Nevertheless, this indicates that the errors due to the divergence of the magnetic field when using the divergence cleaning scheme are minor if not negligible compared to the errors caused by other resolution-dependent effects. 





 §.§ Comparison



Finally, we compare the results of the BNS simulations performed with  and  in Figs. <ref> and <ref>. The time evolution of the total baryonic mass, ejecta mass, and the central density inside the NSs is shown in Fig. <ref> and the magnetic energy and maximum field in Fig. <ref> for both codes at resolution R1 to R3. We use the geodesic criterion for the ejecta. Accordingly, matter is considered unbound if the condition u_t < -1 with a positive radial velocity v_r >0 is satisfied. We find that the merger time in the simulations with  is about 1.4  ms earlier than in the simulations with . This is due to the different gauge conditions used in . We discuss this in more details in Appendix <ref>. 



The ejecta mass is considerably larger in the simulations performed with  than in the ones performed with , i.e., by a factor of ∼ 1.9 for resolution R1, ∼ 4.1 for resolution R2, and ∼ 2.4 for resolution R3. Thereby, the variation for different resolutions is greater for  ranging between 0.015 M_⊙ and 0.022 M_⊙ than for  ranging between 0.0055 M_⊙ and 0.0064 M_⊙. A possible reason for the larger variations could be attributed to using different reconstruction schemes with different orders in  and . Another reason that could explain the strong differences in ejecta mass is the treatment of the vacuum region. As described in Sec. <ref>,  uses an artificial atmosphere with f_ atm = 10^-11, resulting in atmosphere densities of the order of 10^4  g/cm^3, while  uses lower atmospheric values of at most 10^3  g/cm^3, which further decreases with increasing radius. This can lead to differences in the ejecta as the outward flowing matter travels more freely. The lower atmosphere densities can therefore explain the higher ejecta masses in the simulations with . 





The values for the final ejecta mass, disk mass, remnant BH mass, and energy radiated by GWs evaluated at the end of the simulations at t ≈ 50  ms are listed in Tab. <ref>. Since resolution R1 is too low to give reliable results after the BH formation, we stopped the simulation for  after the collapse. For , there was no BH formation within the simulation period for R1. Results for R1 simulations are therefore not listed in Tab. <ref>. Indeed, the remnant BH masses and disk masses are more similar for both codes. Comparing the values for R3 resolution,  predicts a slightly larger BH mass of ∼ 2.57 M_⊙ than  with ∼ 2.54 M_⊙. On the other hand, the disk masses in the simulations with  are greater than the one with  by about ∼ 0.005 M_⊙ for R2 and ∼ 0.002 M_⊙ for R3. In order to assess the error size in the conservation of the energy, we compute Δ E:=M_ ADM - M_ BH - M_ disk - M_ eje-E_ GW. For R2 and R3 in , we get respectively Δ E ≈ 0.0504  M_⊙ and 0.0188  M_⊙, while for R2 and R3 in , we get Δ E ≈ 0.0121  M_⊙ and 0.0025  M_⊙. The values of Δ E for R3 are much smaller than in R2 for both codes, showing good convergent behavior. The error size for R3 in  is slightly smaller than that in , which we suggest comes from the different settings of the artificial atmosphere influencing the measured M_ eje.



Considering the violation of the divergence constraint for the magnetic field, we find larger violations for  than for the  simulations, caused by the employed evolution schemes. The norm of the magnetic field divergences normalized to the norm of the magnetic field strength lies for  between 10^-9  cm^-1 and 5 × 10^-8  cm^-1 and for  between 10^-19  cm^-1 and 2 × 10^-18  cm^-1. 

In addition, we analyze the conservation of the magnetic flux across refinement boundaries. For this purpose, we integrate the magnetic field across the refinement boundaries at level l and its adjacent coarser level l-1, and compute the relative difference of the magnetic flux similarly as in the spherical explosion test. We show the results along the positive z direction for  at the refinement boundary of level l=10 and for  at level l=6. The comparison here is not straightforward since the refinement boxes for 's FMR are much larger than for . The refinement boundary of the finest level of  lies between the refinement boundaries of l=6 and l=5 of , whereby the boxes move dynamically within . Nevertheless, it is visible that the relative differences for  are orders of magnitude larger than for : for  at orders of  10^-2 and 10^-1, whereas for  at orders of 10^-15 to 10^-14, i.e., machine accuracy. 

However, for comparing the magnetic energy and the maximum field strength in Fig. <ref>, both codes predict overall similar values for the respective resolutions. In the R1 simulations, there is almost no amplification during the merger in both cases. The magnetic field only increases after the merger, which happens slightly earlier for  than for , but with a similar slope in the magnetic energy. With R2, both codes show an amplification of the magnetic field during the merger up to 10^48  erg. Despite the different merger time, the collapse time coincides at this resolution for both codes, and the lines for the magnetic energy almost overlap between 17  ms and 25  ms. Thereafter,  predicts slightly higher energies than . In the highest-resolution simulation, both codes reach a maximum magnetic field strength of ∼ 10^16  G shortly after the merger. The simulation performed with  reaches higher magnetic energies than  by a factor of two, which could be caused by the higher-order reconstruction in  that possibly resolves small-scale effects slightly better. The collapse time is a few milliseconds earlier with  than with . In general, both codes predict similar magnetic energies and field strengths at the end of the simulation.



§ CONCLUSIONS



We successfully extended the infrastructure of the  code for ideal GRMHD simulations employing a hyperbolic divergence cleaning scheme. Additionally, we compared results for well-known special-relativistic tests with established GRMHD codes: , , and . 
Overall, the tests showed good agreement between all codes. In the one-dimensional Balsara test, we observed only minor differences at the shock fronts, which can be attributed to the fact that the reconstruction methods used in the individual codes are either more diffuse or more oscillatory.
Similarly, we found minor differences in the two-dimensional cylindrical explosion and magnetic rotor tests, which we attribute to different reconstruction schemes for the fluid variables. The largest differences occur for , whereby higher resolution converges to the same results. The  code is the only one using the HLLD Riemann solver for these tests. The other codes use the HLL solver, which could explain the larger differences here.
We also compare results for the Kelvin-Helmholz instability test between  and . Both codes show that they are able to capture the vortex. In , the vortex forms earlier than in  and the perturbation grows faster. However, we show that we achieve the same growth rate in  when using the same Riemann solver as in .
In the three-dimensional spherical explosion test, we compare the conservation of magnetic flux over a refinement boundary for  and . Both codes apply schemes to conserve the flux. Still,  proves to be superior in magnetic flux conservation. Applying divergence cleaning in  leads to additional source terms in the evolution equation for the magnetic field, which are not matched and could explain the worse performance. 

We performed first BNS simulations with our new  implementation and compare our results with simulations performed with  using the same initial data. Although we obtain large differences in ejecta mass, with  having more than twice the amount of ejecta than , both codes predict similar values for the magnetic field. The amplification of the magnetic field during the merger is stronger with increasing resolution, as capturing KHI and MRI that trigger this amplification requires high resolution. Note that none of the simulations performed are in the convergent regime, because resolving the KHI requires an extremely high grid resolution, which is not feasible for full three-dimensional GRMHD simulations. 

We demonstrate that our divergence cleaning implementation in  is able to perform reliable simulations, including the magnetic field. The scheme is simpler than other divergence-free treatments, but it does not prevent the formation of magnetic monopoles, even though this corresponds to a constraint violation near zero. However, the finite resolution error associated with other variables is larger than the errors due to divergence cleaning in all cases considered here, and all these errors should converge to zero. Our results show that uncertainties stemming from different methods for the fluxes, shock capturing methods, or Riemann solvers are more severe. 



§ ACKNOWLEDGEMENTS


We thank I. Markin for implementing the fuka data reader in the  code and F. M. Fabbri for initiating the GRMHD implementation. KK and MS also thank K. Van Aelst for the fuka data reader in the .
TD and AN acknowledge support from the Deutsche Forschungsgemeinschaft, DFG, project number DI 2553/7. Furthermore, TD acknowledges funding from the EU Horizon under ERC Starting Grant, no. SMArt-101076369.
The simulations with  were performed on the national supercomputer HPE Apollo Hawk at the High Performance Computing (HPC) Center Stuttgart (HLRS) under the grant number GWanalysis/44189, on the GCS Supercomputer SuperMUC_NG at the Leibniz Supercomputing Centre (LRZ) [project pn29ba], and on the HPC systems Lise/Emmy of the North German Supercomputing Alliance (HLRN) [project bbp00049]. This work is in part supported by the Grant-in-Aid for Scientific Research (grant Nos. 23H01172, 23K25869 and 23H04900) of Japan MEXT/JSPS.





§ COMPARISON GENERAL RELATIVISTIC HYDRODYNAMIC SIMULATIONS





As benchmark, we run the BNS simulation with R1 once without magnetic field. From these simulations, we analyze the different merger times between  and . Additionally, we run the same set-up with the sacra code <cit.>. On the one hand,  implements an original Shibata-Nakamura version of the BSSN formulation <cit.> and employs the gauge condition based on the Nakamura variable F_i≡δ^jkγ̃_ij,k <cit.>. On the other hand, the sacra code implements the BSSN formulation employing the gauge conditions in a very similar way as . Also, the AMR implementation based on the box-in-box in sacra follows  <cit.>. Therefore, in the sacra run, we employ the same grid configuration as . In the  run, the grid configuration is a nested grid as described in the main text. 

We show the time evolution of the central rest-mass density in Fig. <ref>. The results from the corresponding simulations with magnetic fields are added as dashed lines. The different gauge condition changes the merger time in the  simulation and indeed the differences become smaller. Since the merger time is a gauge-dependent quantity, a fair comparison between our different codes with different settings is difficult. Therefore, we focus our comparison in Sec. <ref> on gauge-independent quantities such as the ejecta mass, magnetic energy, and magnetic field strength.





