Paper Title: North Polar Spur: gaseous plume(s) from star-forming regions at $\sim$3-5 kpc from Galactic Center?.





 


Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany 
Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany
Ioffe Institute, Politekhnicheskaya st. 26, Saint Petersburg 194021, Russia
Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya str., Moscow 119017, Russia


NRC `Kurchatov Institute', acad. Kurchatov Square 1, Moscow 123182, Russia
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov str., Nizhny Novgorod 603950, Russia






















We argue that the North Polar Spur (NPS) and many less prominent structures are formed by gaseous metal-rich plumes associated with star-forming regions (SFRs). The SFRs located at the tangent to the 3-5 kpc rings might be particularly relevant to NPS.  A multi-temperature mixture of gaseous components and cosmic rays rises above the Galactic disk under the action of their initial momentum and buoyancy. 
Eventually, the plume velocity becomes equal to that of the ambient gas, which rotates with different angular speed than the stars in the disk.  As a result, the plumes acquire characteristic bent shapes. 
An ad hoc model of plumes' trajectories shows an interesting 
resemblance to the morphology of structures seen in the radio continuum and X-rays. 


North Polar Spur


    
North Polar Spur : gaseous plume(s) from star-forming regions at ∼3-5 kpc from Galactic Center?
    
Eugene Churazov 1,2 
Ildar I. Khabibullin 3,1,2 
Andrei M. Bykov 4 
Nikolai N. Chugai 5 
Rashid A. Sunyaev 2,1 
Victor P. Utrobin 5,6 
Igor I. Zinchenko 7

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






§ INTRODUCTION











North Polar Spur <cit.> is an elongated bright structure first discovered in all-sky radio images along with other similar structures dubbed Loops <cit.>. Unlike other Loops, NPS (the northern part of Loop I) is visible in soft X-rays across its full extent <cit.>.    

Given the NPS's large angular size, ≳90, the first proposed explanation <cit.> was a shell of a very nearby, d≲100 pc, supernova remnant. Alternatively, a scenario in which NPS represents a distant, d≳10 kpc, Galaxy-scale structure has been put forward, with the possible energetic outburst in the Galactic center being considered as the primary source behind it <cit.>. The latter might be caused by an intense star-formation episode in the vicinity of the Galactic Center or by the activity of the supermassive black hole Sgr A* <cit.>. Both scenarios suggest that NPS is powered by a shock wave propagating in the hot medium (either interstellar or circumgalactic). This shock is responsible for the acceleration of radio-emitting relativistic electrons, as well as the heating and compression of the X-ray-emitting hot gas. 


Although these two scenarios involve very different physical sizes, time scales and required energetics, it turned out to be very difficult to unequivocally distinguish between them based on the available data, with contradicting conclusions having been reached from, e.g., the Faraday Rotation and X-ray absorption measurements <cit.>. Recent discoveries of the giant structures in gamma-rays <cit.> and soft X-rays in the southern Galactic sky <cit.>, the "bubbles" appear to be more symmetric with respect to the Galactic center, strengthening the case for the central energy release as a viable solution <cit.>.  

In this Letter, we consider another scenario for NPS formation motivated by the morphological and spectral properties of the soft X-ray emission measured in the course of the eROSITA all-sky survey <cit.>. We propose that NPS is produced by a break-out of the massive star formation regions associated with the end of the Galactic bar, which locates its base at d∼5 kpc from us <cit.>. Flows of enriched hot gas move upward from the Galactic disk and get entrained in the relative rotation of the circumgalactic medium above the disk, resulting in the spiral-like structure, appearing like a loop in the sky. In this model, it is the advection of the enriched gas and relativistic particles, rather than a shock, that is responsible for the appearance of NPS, predicting the high metal abundance of X-ray emitting gas and lack of evolutionary signatures in the direction perpendicular to the NPS edge.















§ X-RAY PICTURE




For X-ray analysis, we use the data accumulated by eROSITA <cit.> telescope onboard the SRG  X-ray observatory <cit.> in the course of its all-sky survey in the Eastern Galactic hemisphere (i.e. the Galactic longitude range 0^∘<l<180^∘). Details on the data preparation and analysis are given in Appendix <ref>. Images extracted from the eROSITA all-sky survey in the 0.52-0.61 keV and 0.7-0.96 keV bands are shown in Fig. <ref> (left and middle panels, respectively). The former band includes the triplet of He-like oxygen O VII near 574 eV, while the bright lines of helium-like neon Ne IX (at 905 and 921 eV) and neon-like iron Fe XVII (between 725 and 826 eV) fall into the latter. For plasma in collisional ionization equilibrium (CIE), the peak line emissivities in these bands are at temperatures ∼ 0.2  keV and  ∼ 0.6  keV, respectively.

Rather complex X-ray morphology of NPS is obvious from these images and the comparison between them. One can readily see that the correlation length of X-ray structures is larger in the direction along NPS for both bands but a simple combination of concentric spherical shells is not a good description of NPS.
For example, the surface brightness in the 0.52-0.61 keV band dominated by the O VII line, has a sharp outer edge but a rather flat surface brightness across the inner region, unlike an edge-brightened shell. Similarly, the 0.7-0.96 keV surface brightness is composed of a fainter diffuse emission surrounding a bright central "filament". 

More quantitatively this is illustrated via profiles of the X-ray emission in several narrow bands. The white dashed lines in Fig. <ref> outline a wedge that is aligned with the NPS "outer" boundary. Radial profiles extracted from this wedge are shown in Fig. <ref>. The steep rise, corresponding to the "edge" of NPS, is visible in all bands. However, further inside NPS, the profiles do not follow a typical edge-brightened behavior characteristic for a spherical shock (see, e.g., profiles at energies below 0.61 keV). This suggests that in 3D the volume occupied by the plasma emitting at these energies resembles a flattened sheet rather than a spherical shell. In harder bands, the profiles feature a steep jump, followed by a gradual increase of intensity before reaching a peak at x≈ 35  degrees in this plot. The center of the wedge is at (l,b)=(338^∘,32^∘) in Galactic coordinates. 
  

An alternative to the local SNR or GC-related scenarios, which associate the steep rise with a shock, is the assumption that NPS is a multi-temperature gas, where a hotter (>0.3 keV) component, responsible for the emission in iron lines above 0.7 keV, coexists with a warm (e.g., 0.15-0.2 keV) plasma, producing the bulk of the emission below 0.7 keV and most prominently in oxygen lines. 

In what follows, we consider a scenario in which such multi-temperature gas is venting from the active star-forming regions in the central parts of the Galactic disk. 
























§ MORPHOLOGICAL MODEL



Here we consider a simple morphological model that is based on a broad-brush picture of the evolution of massive star-forming region in the Galactic disk <cit.>. The first shock waves of supernova explosions originating from the most massive newborn stars in such a region produce hot and dense gas strongly enriched with metals of the ejecta. Initially confined by the surrounding dense and cold gas, the hot gas manages to break through and form an (adiabatically cooled) plume that moves away from the disk and interacts with the interstellar (ISM) and circumgalactic (CGM) media forming some sort of chimneys and plumes (see Appendix <ref>). The trajectories of rising plumes might be complicated, especially when the gas is multi-phase and magnetized. They are also affected by the pressure, magnetic fields, and velocities in the ambient ISM and/or CGM <cit.>.   



For illustration purposes, we consider an ad hoc model that encapsulates all the complicated physics in three parameters. Namely, the initial (vertical) velocity of the plume _z,0 and two effective scale heights z_ϕ and z_z. Here z_ϕ describes how the plume that initially moves together with stars and gas close to the disk plane, decelerates and joins the pressure-supported hot halo gas, which is either non-rotating or rotating slower than the stellar disk. In turn, the z_z parameter controls the decline of the vertical (perpendicular to the disk) velocity component. As a result, the two velocity components have the following dependence on z:     


    _z(z)=   _z,0  e^-z/z_z
    _ϕ(z)=   _ rot  [1- e^-z/z_ϕ ].




    _z(z)=_z,0  e^-z/z_z    and  _ϕ(z)=_ rot  [1- e^-z/z_ϕ ].


A flat rotation curve with _ rot = 220   km s^-1 is adopted for the sake of simplicity, the effective scale heights are set to z_ϕ=2  kpc and z_z=5  kpc, and _z,0= 200   km s^-1. The choice of these parameters is rather arbitrary, but it suffices for illustration purposes and the final answer is not dramatically sensitive to it.

For the source of the plume, e.g. a star-forming region, we assume a circular trajectory in the disk plane with the velocity _ϕ, src. In our illustrative model, the following two versions are considered:


    _ϕ, src=   _ rot    ( a) 
    _ϕ, src=   Ω R     ( b),



    ( a)   _ϕ, src=_ rot   and    (b)  _ϕ, src=Ω R  ,

where R is the distance from the Galactic Center. Version (a) corresponds to the source moving with the same velocity as stars, while version (b) mimics a pattern motion. The former case is relevant when the active phase of a given star-forming region associated with the dense gas is very long and the active region always moves together with the gas (with the velocity _ rot). The latter case corresponds to a situation when the spatial distribution of the plume sources is linked to the Galactic bar and/or spiral arms. For illustration of this case, we adopted the value Ω=33  km s^-1 kpc^-1 derived for the bar pattern speed <cit.>. 

As a further simplification, we assume that none of these parameters depend on the position of the plume within the galaxy. 

The above relations (<ref>-<ref>) can be trivially integrated to derive a position of the gas lump released some time t ago relative to the current position of the "source". By design, plume trajectories appear in 3D as spirals rising from the disk from the current location of the plume source (see Appendix <ref> and Fig. <ref>). 
We note here that for a strongly underdense lump, the buoyancy force is solely set by pressure gradients in the ambient gas. These gradients themselves depend on the rotation velocity of the halo gas and can have both radial and vertical components. In the above model, we explicitly neglect the former component, although in real systems it can force the plume to move radially. In particular, for slow (fast) rotation of the halo gas, underdense lumps can move to larger (smaller) radii (see Fig. <ref>).

As a final step, we project the trajectory onto the sky as seen from the position of the Sun. This modifies the appearance of the plume due to different distances from the Sun to various segments of the plume 
and the Sun's motion relative to the source. An example of plume trajectories for a set of "sources" chosen by hand is shown in Fig <ref>. A table with the positions of sources is given in the Appendix <ref>. The trajectories have been integrated for ∼ 140  Myr, which is approximately one rotation period around the Galactic Center near the Sun for the adopted value _ rot.

Figure <ref> illustrates the key morphological properties of the plumes. Naturally, the trajectories remain confined to the regions defined by the radial distance to the source from the Galactic Center, while the notable left-right asymmetry is imposed by the rotation direction of the Galaxy. All trajectories have rising parts close to the Galactic Plane, with the direction of the curvature set by the mutual position of the source and the Sun and the direction of the Galaxy rotation. This is clearly illustrated by the family of trajectories for sources located at the 5 kpc ring around GC. The morphological difference is even more pronounced for sources located at larger distances from GC (cases marked at Cygnus X and Vela in Table <ref>). For the Cygnus X region, the plume's apparent trajectory rises to the Galactic Poles before forming a spiral. For the Vela region, the plume is bent much earlier and follows an almost horizontal line at |b|∼ 20-25 degrees.  

If NPS is indeed a gaseous plume, its base should be associated with the star formation regions that are currently at Galactic longitudes l∼ 20^∘-40^∘ (see Fig. <ref>). Incidentally, this range of longitudes corresponds to the tangential direction to the 3-5 kpc rings around the Galactic Center and indeed hosts the most active star-forming regions in the Galaxy, including a prototypical mini-starburst complex W43 (see Appendix <ref>). Below we discuss this scenario in more detail.



§ DISCUSSION



Motivated by the morphological model presented above, we assume that the leading edge of NPS is located at a tangent to the 5 kpc ring. In this case, the distance of the NPS base from the Sun is D_ Sun,NPS≈ 6.6  kpc. Of course, some parts of the plumes can be closer to the Sun, while some further away. 

Given the apparent size of NPS on the sky, one can assume that its physical size (the "depth" along the line of sight) is of the same order as the distance from the Sun, i.e. l∼ D_ Sun,NPS. Based on that, we can estimate the hot gas density from the observed peak surface brightness I_X in the 0.7-1.05 keV band. This yields a proton number density 

    n_p∼ 7× 10^-4( Z/Z_⊙ )^-1/2( l/6  kpc )^-1/2  cm^-3,

where Z is the metal abundance with respect to the Solar value <cit.>. In this derivation, we assumed that the gas is in collisional ionization equilibrium with temperature 0.6  keV and used  model <cit.> to predict the emissivity. The metallicity dependence in the above expression is valid for Z/Z_⊙≳ 0.1.   The corresponding cooling time of the gas with temperature kT≈ 0.6  keV can be estimated using the cooling function of <cit.> as t_ cool∼  2× 10^9( n/10^-3  cm^-3 )^-1( Z/Z_⊙ )^-1/2  yr.




While the 3D geometry of NPS is uncertain, we assume that its volume is ≈ l^3 (i.e. its line-of-sight extension is comparable to its transverse size) to get an estimate of the total energy (enthalpy) E∼ l^3 (n_p+n_e) 5/2 kT∼ 2× 10^55  erg



for l=6  kpc.
Assuming that the integrated energy release associated with the formation of 1 solar mass of stars is ∼ 10^49  erg, it needs ∼ 2× 10^6 M_⊙ of gas to be converted into stars. Assuming the star-formation rate of 0.1  M_⊙  yr^-1, it takes ∼ 20 Myr to generate enough energy for powering the entire NPS. The long cooling time and preferential accumulation of trajectories in the NPS region imply that it could be a result of cumulative contributions from many sources.  


In the simplest version of the above scenario, the gas in NPS is in pressure equilibrium with the ambient medium. Assuming that the temperature of the ambient medium is ∼ 0.15  keV, the regions filled with ∼ 0.6  keV plasma should have ∼ 4 times lower density. Yet, NPS appears much brighter than the diffuse emission of the Galaxy. There are two plausible reasons for that. One is that for lines of Fe XVII, Ne IX, and Ne X, the gas in the halo is simply too cool to produce a bright emission. The second reason is that the plumes are plausibly much more metal-rich than the halo gas. This is particularly important for O VII lines, which are present in the spectrum of the halo. The gas in the plumes could easily be a factor of 10 or more metal-rich and even in the absence of shock, it can shine prominently (see Fig. <ref>). In Appendix <ref> we argue that the plume metallicity is not much larger than the Solar one. This implicitly suggests that the abundance in the halo is ≲ 0.1. 



An important implication of the model is the global asymmetry of the Milky Way diffuse X-ray and radio emission. The asymmetry is present even if the distribution of plumes' sources is itself symmetric. In 3D, the plumes can form a symmetric pattern, but the direction of the Galaxy rotation produces the apparent asymmetry when viewed from the Sun's position. Some extra asymmetry could come from global gas motions in the Galactic halo <cit.>, e.g. caused by interaction with the satellite galaxies, but it comes on top of the "natural" East-West asymmetry. The North-South asymmetry, which is clearly visible in both radio and X-ray sky, in this model is attributed to the asymmetry of the primary sources of the plume. When the hot gas finds its way of least resistance through the dense cold gas in the disk, it may preferentially go to one side rather than forming a symmetric structure on both sides.




The morphological model considered above, treated trajectories of individual gas plumes as independent, neglecting the hydrodynamic nature of the flows and, in particular, possible gas mixing instabilities that would naturally arise in such a situation. The sharpness of the NPS's outer edge as well as its overall filamentary appearance might be a manifestation of the suppression of these instabilities by the well-ordered and sufficiently strong magnetic field. 

Indeed, the synchrotron radio emission from the NPS region is known to be polarized <cit.>. 
The polarization implies that the magnetic field is ordered. In the model discussed above, these structures can be especially prominent plumes located at different distances from the Galactic Center. The ordering is caused by stretching the field lines during certain phases of plume evolution and/or by the halo gas motions rather than by shocks (as illustrated in Appendix <ref>). 

On a more speculative side, we note that in this model the gas vented from the disk tends to accumulate high above the plane in general direction of the Galactic Center. From this point of view, a question arises whether structures like the eROSITA bubbles <cit.> could be produced by the same mechanism. In this case, these structures can evolve on a longer time scale than implied by the shock-driven scenario. In fact, different regions across NPS could come from gas lumps having different ages (and different star-forming regions) and, therefore, can have different properties.  

In the proposed scenario, the NPS might be a Galactic analog of the magnetic structures directly observed to reach significant distances above the disks in many nearby galaxies, e.g. NGC 4217 <cit.>. On the other hand, NPS might also be a case similar to the off-disk parts of the so-called anomalous spiral arms in NGC 4258, which are believed to be powered by the interaction of the relativistic jet with the galactic disk <cit.>. In all these cases, we might witness signatures of the disk-halo interaction via intense feedback episodes resulting in complex multi-phase and magnetized interface region <cit.>.



One observational test that can falsify the plume scenario is the measurements of the non-equilibrium ionization (NEI)  signatures in the gas, which are pertinent to the shock scenario <cit.>. Indeed, for the gas electron density ∼ 10^-3  cm^-3 and the downstream plasma velocity ∼ 200   km s^-1, the ionization parameter τ=n_e× t∼ 1.5× 10^11  cm^-3 s is expected at a distance of 1  kpc downstream of the shock. Such scales are easily resolved for any distance to NPS (for the "nearby" shock scenario the ionization parameter is even lower at the same angular distance from the shock) and can be derived from the spectra. The complication here is that NPS does not have a simple "layered" structure that allows for clean separation of shells at different 3D distances from the edge. As a result, the lines of O VII, O VIII, Fe XVII, Ne IX, and Ne X are always present in a proportion that is difficult to predict. We discuss the NEI scenario in a forthcoming publication.    





If the model outlined above is correct, NPS and similar structures offer a possibility to probe the rotation pattern of the hot gas in the Milky Way. Future large grasp microcalorimetric missions like Line Emission Mapper <cit.> capable of mapping the entire NPS region in the course of its all-sky survey <cit.> will be instrumental in measuring abundances and velocities using emission lines of elements from carbon to iron across the 0.2-2 keV energy band to narrow the range of plausible models. 






§ CONCLUSIONS

While the North Polar Spur is usually attributed to a shock front associated with the activity of our Galactic Center or a local SNR, we discuss an alternative scenario. It posits that metal-enriched plumes rise above the disk from active star-forming regions. Interactions with the hot halo gas give these plumes the appearance of bent spirals. Their shapes sensitively depend on the rotation pattern of the hot gas above the disk. These plumes are mostly hotter than the ambient gas and can be metal-rich compared to the hot gas in the Milky Way halo. 
In this model, the NPS itself is associated with the star formation within a 3-5 kpc distance from the Galactic Center. Fainter plumes might be associated with other star-forming regions in the Galaxy. 







§ ACKNOWLEDGMENTS


This work is partly based on observations with the eROSITA telescope onboard SRG space observatory. The SRG observatory was built by Roskosmos in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI) in the framework of the Russian Federal Space Program, with the participation of the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The eROSITA X-ray telescope was built by a consortium of German Institutes led by MPE, and supported by DLR. The SRG spacecraft was designed, built, launched, and is operated by the Lavochkin Association and its subcontractors. The science data are downlinked via the Deep Space Network Antennae in Bear Lakes, Ussurijsk, and Baikonur, funded by Roskosmos. 

The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data were processed using the eSASS/NRTA software system developed by the German eROSITA consortium and analyzed using proprietary data reduction software developed by the Russian eROSITA Consortium.


IK acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679.


aa

 






















§ X-RAY DATA ANALYSIS



 Data from all four consecutive scans are combined together after filtering for the periods of enhanced solar activity, which causes a strongly elevated level of the instrumental background. For the imaging analysis, the data taken with all seven Telescope Modules (TMs) are combined, while for the spectral analysis, only five TMs protected by the on-chip filter (i.e. TMs 1-4, and 6) are used. Data reduction, filtering, vignetting-correction, and background subtraction are performed in the same way as was done in the previous studies exploring Galactic diffuse X-ray sources <cit.>.


In particular, the energy-dependent contribution of the instrumental background is modeled and subtracted based on the calibration data accumulated via observations with the "Closed Filter Wheel" configuration, while corrections for exposure time and vignetting are conducted so that the data are characterized by Field-of-View averaged response matrices. 








§ PLUMES IN 3D


In this section, we outline two simple scenarios for plume trajectories. In the first scenario, a plume is produced by a continuous source. The plume initially moves together with stars/gas in the disk and eventually joins the motion of the halo gas. In the second scenario, a trajectory of a massless buoyant bubble is considered.



 §.§ Plumes from a moving source

The first case corresponds to the model described by equations <ref> and <ref>. Fig. <ref> illustrates the shapes of the "plumes" in a stationary 3D frame. The blue curve shows the case when the source of the plume moves with the same velocity as the gas in the disk. As the plume rises, it gradually joints the slowly rotating gas in the disk. The red curve shows modest modifications that appear when the source of the plume moves slower than the gas and stars in the disk. By design, the rising plumes retain the same distance from the Galactic Center in projection to the disk.   












 §.§ Trajectory of a massless buoyant bubble

Here we consider the trajectory of a single (massless) bubble moving in a stratified atmosphere of the Galaxy. 
Similarly to galaxy clusters <cit.>, the velocity of the bubble is set by the balance of pressure gradients and the drag force acting on the bubble, i.e.

    u^2u⃗/u1/2ρ A C_d=-∇ P V,

where u⃗ is the 3D velocity vector of the bubble velocity with respect to the halo gas (u= | u⃗ |), ρ and P are the halo gas density and pressure, respectively, A and V are the area and volume of the bubble, and C_d is the appropriate drag coefficient <cit.>. The bubble size is assumed to be smaller than the pressure scale height and the velocity remains subsonic. For an atmosphere in hydrostatic equilibrium, 1/ρ∇ P=-∇ϕ and, therefore,   u⃗ can be directly calculated from the gravitational potential ϕ, which in our case can include a contribution from halo gas rotation. 
To this end, we use the approximation of the MW potential from <cit.> to which we add a centrifugal term (-V_h^2ln R) for a cylindrical rotation, where R is the distance from the MW center in the disk plane and V_h is the gas rotation velocity. The gas rotation velocity of the gas is added to u⃗ to get the bubble velocity in the non-rotating frame. The effects of rotation on the bubble trajectory are illustrated in Fig. <ref>. This figure shows that different configurations are possible depending on the gas rotation pattern. We note here, that unlike Fig. <ref>, the trajectories of individual "bubbles" are shown rather than the "plumes" that are formed by many bubbles released at different times.











§ ENHANCED METALLICITY OF PLUMES




The enhanced metallicity of the NPS hot material enriched by supernovae is important for the proposed scenario. 
The thermal energy of the NPS E_ th∼ 2 × 10^55 erg (see <ref>) suggests the explosion of N_ sn∼ 2× 10^4 
  core-collapse supernovae (CCSN) with a typical kinetic energy of 10^51 erg (lifetime of a star-forming region is short for SNe Ia to explode). 
We rely on two sets of oxygen nucleosynthesis calculations  for CCSN 
 progenitors in the range of 11 - 40M_⊙ by <cit.>
 (WW95) and 
<cit.> 
  (K06).
In K06, the  data on 11M_⊙ progenitors are lacking. We, therefore, adopt the ejected mass of oxygen from WW95 for this particular progenitor.  The abundance of other elements relative to oxygen is assumed to follow (approximately) the solar composition. 

 The average mass of ejected oxygen per CCSN (m_ O) is inferred assuming Salpeter initial mass function   dN/dm ∝ m^-2.35 in the mass range of 0.1 – 100M_⊙. We find comparable values of m_ O, 2.2M_⊙ and 2.6M_⊙,  
  for WW95 and K06 data, respectively.
The average m_ O=2.4M_⊙ multiplied by the CCSN number N_sn  provides us with the total amount of oxygen synthesized by supernovae of the star-forming region M_ O≈ 5× 10^4M_⊙.
The synthesized oxygen mixed with the gas of the solar composition (solar abundance X()_⊙ = 0.01) produces enhanced overall metallicity. 
The total mass of the mixture (M_ hot) is fixed by the thermal energy of the X-ray-emitting gas 
 E_ th = (5/2)kT_ xM_ hot/(μ m_p). 
For the average molecular weight μ = 0.61 and temperature kT_ x =0.6 keV one obtains  
 M_ hot = 4×10^6M_⊙. 
The expected oxygen abundance of the NPS hot gas is then X() = X()_⊙ + M_ O/M_ hot≈ 0.02, twice the solar value.
  



§ THE CHIMNEY MODEL





The clustering of core-collapse supernovae produced at the end of the evolution of massive stars in OB associations and compact clusters has a profound effect on the interstellar medium 
and the galactic ecology <cit.>. Multiple correlated supernovae and the powerful winds of OB stars create superbubbles and eventually (for powerful enough systems) produce the chimney-type structures <cit.>. We consider such superbubbles as potential sources of the plumes. 
 To study the effect of the galactic magnetic field on the superbubble breakthrough from the disk to halo <cit.> performed 3D MHD simulations for different assumptions on the large-scale magnetic field structure. The magnetic field with the strength of ∼ 5 μG with a broad ∼ kpc scale height can confine a superbubble of a modest kinetic luminosity 3× 10^37  for about 20 Myr within the scale height of | z |∼ 300 pc from the disk, while in a model with the field that follows the scaling B ∝ρ^1/2 (with the mid-plane field magnitude of 5 μG) the superbubble blows out to the halo. The evolution of superbubbles in density stratified disks that blow out into galactic halo may be a subject of Rayleigh-Taylor instabilities <cit.>. 
 A parsec-resolution multi-phase simulation of the local star-forming galactic disk with the account for SNe feedback effects revealed that the hot galactic outflows (with gas temperatures above 10^6 K) may carry about 10%-20% of the energy and 30%-60% of the metal mass injected by supernovae <cit.>. These values are broadly consistent with the energy requirements needed to produce the X-ray plumes discussed in the text.  
 
 Non-thermal particles accelerated by shocks from SNe and stellar winds at the active phase of the evolution of superbubble which is about 10 Myrs <cit.>. Then relativistic particles will be blown out to the low halo with the frozen-in magnetic fields of the chimney-type plasma outflow. Relativistic electrons of energy ∼ 100 GeV would have in the halo a lifetime of about 10 Myrs due to synchrotron - Compton losses and will radiate in magnetic fields of a few μG at frequencies of  50 GHz. The synchrotron-Compton cooling time for the relativistic electron of energy E_ GeV (measured in GeV) in the galactic halo can be estimated as 
 
    t_ syn∼  10^9E_ GeV^-1/ 1 + 0.1 B_μ G^2  yr.


 

























§ THE ROLE OF W43


Below, we discuss in brief the mini-starburst complex W 43 as a prototype 
powerhouse located in the Galactic molecular ring assuming that an ensemble of such complexes 
may supply the plumes over a long period of ∼ hundred million years. 
The molecular complex W 43 with the estimated 
mass ∼ 7 × 10^6 is presumably located at the connecting point
of the Scutum-Centaurus Galactic arm and the Galactic bar <cit.>. W 43 excited a giant H II region dubbed NGC 3603. The equivalent diameter of the coherent complex of molecular clouds is ∼ 140 pc and it is surrounded by an atomic gas envelope of about two times larger diameter. <cit.> noted that the transition from circular to elliptical orbits
in the spiral arm and bar potentials might cause high-velocity streams and thus efficient star formation episodes of current rate at least ∼ 0.01 yr^-1 as estimated from 8 μm luminosity measurements but it is likely increasing to ∼ 0.1 yr^-1. On longer time scales ∼ 100 Myrs of interest for this paper multiple episodes of high star formation rate can be expected from colliding molecular clouds entrained by the large scale streams.   

The giant HII region NGC 3603 which is associated with the complex produces the Lyman continuum photon rate of about  4 × 10^51 photons s^-1 is the most powerful HII region in the Galaxy <cit.>. A compact dense cluster of young massive stars HD 97950 (also known as W 43 cluster)  powering NGC 3603 has an estimated dynamic stellar mass of 18,000  including   
more than 70 O and 3 WR stars. The distance to the powerful starburst region NGC 3603 
of 7.2 ± 0.1 kpc was estimated by <cit.> from GAIA DR2 stellar parallax measurements (see also <cit.> for a recent discussion). From the VLBA trigonometric parallax measurements of masers  toward W43 <cit.> derived a distance of 5.49_-0.34^+0.39 kpc to the complex <cit.>.  The origin and ages of 
molecular clumps and stellar population in NGC 3603 are still the subject of debate <cit.>. ALMA observations revealed that the core mass function in W43-MM1 is much shallower in the high-mass range than the standard initial mass function <cit.>.  

<cit.> discussed the formation of cluster W 43  due to a collision of two molecular clouds about 1 Myr ago. This age is consistent with that of the highest mass stellar population in the cluster. 
The suggestion by <cit.> about the origin of the star-forming complex W 43/NGC 3603 due to the collision of molecular clouds following their circular orbits in the Scutum-Centaurus arm and the clouds at the elliptical orbits of the Galactic bar makes the complex W 43 as well as other complexes along the Molecular Ring where the Galactic arms meet the bar to be favorable locations for the intense star formation episodes in the past.   
 
Very high energy emission source HESS J1848-018 <cit.> can possibly be associated with W 43. From  Fermi-LAT detection of extended high energy gamma-ray source source in the direction of W 43 <cit.> estimated the total cosmic ray energy within the source to explain the detected flux to be (2.3 ± 0.3) × 10^48 erg.

















 Paper Title: Constraints on the $γ$-parameter for the vacuum solution of Cotton gravity with geodesics and shadows.







	 ednaldobarrosjr@gmail.com
Faculdade de Física, Universidade Federal do Pará, Campus Universitário de Tucuruí, CEP: 68464-000, Tucuruí, Pará, Brazil

     
    tarcisojunior17@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill

	 fslobo@ciencias.ulisboa.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     esialg@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

   drubiera@ucm.es
Departamento de Física Téorica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

      
        fc53497@alunos.fc.ul.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     henriquefisica2017@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill



	





§ ABSTRACT 

We consider a recently introduced extension of General Relativity dubbed as Cotton gravity (CG), based on the use of the Cotton tensor, to estimate the size of a new constant γ appearing within a spherically symmetric, vacuum solution of the theory. Taking into account its non-asymptotically flat character, we use the inferred size of the central brightness depression of the supermassive object at the heart of the Milky Way galaxy (Sgr A*) by the Event Horizon Telescope to constrain at 2σ the CG parameter as γ M ≈ 3.5 × 10^-12. We study the potential observational consequences from the smallness of such a value using exact and numerical expressions for the deflection angle, optical images from optically and geometrically thin accretion disks, isoradials, and instability scales (Lyapunov index) of nearly bound geodesics associated to photon rings. Our results point towards the impossibility to distinguish between these two geometries using current and foreseeable techniques in the field of interferometric detection of optical sources.






Constraints on the γ-parameter for the vacuum solution of Cotton gravity with geodesics and shadows

    Henrique A. Vieira
    August 5, 2024
====================================================================================================







§ INTRODUCTION



Despite the undeniable success of Einstein's General Relativity (GR) in explaining a large number of gravitational phenomena in different regimes <cit.>, there are several drawbacks of the theory. On the one hand, the accelerated expansion of the Universe, as found by Type-Ia supernovae observations  <cit.>, can only be accounted for, in the frame of GR, via a hypothetical fluid known as dark energy <cit.>, whose properties can be assimilated to that of a cosmological constant term <cit.> yet there is no direct proof of its existence. Similarly, the explanation of galactic rotation curves demands more than 90% of the total mass of most galaxies to be dominated by a non-baryonic form of matter dubbed as dark matter <cit.>, which we have been unable to detect in any terrestrial observatory yet. On more theoretical grounds, the problems with the unavoidable development of space-time singularities inside black holes and in the early cosmological evolution and the loss of predictability they bring about <cit.>, the Hawking process and the associated information loss paradox <cit.>, and the development of a framework unifying gravity and quantum mechanics (quantum gravity <cit.>), are challenges that still haunt the gravitational community.

In response to these challenges, a large avenue of research has emerged in the last few decades arranged around alternative approaches to GR, generally dubbed as modified theories of gravity (see e.g. <cit.> for some reviews). Within this context, recently Harada presented a theory called “Cotton Gravity" (CG) <cit.>. In this proposal, the effects describing gravity are attributed to the Cotton tensor, and the corresponding field equations contain third-order derivatives. The main interest in this proposal comes from the fact that any solution of the Einstein equations, with or without a cosmological constant term, also satisfies the field equations of CG. Furthermore, the cosmological constant itself emerges as an integration constant out of the field equations. In addition, in <cit.> Harada used the post-Newtonian limit of CG to solve the field equations numerically, interpreting the rotation curves of several galaxies without employing dark matter sources. Note that the possibility that dark matter may be geometric effects of modified theories of gravity has been explored in the literature <cit.>.


Regarding new solutions within CG, the first non-trivial one with static, spherical symmetry was found in <cit.>, which extends the Schwarzschild solution via a new parameter γ. Other non-trivial solutions were found in <cit.>. On the other hand, in <cit.> it was emphasized that the Codazzi tensor can be derived from the equations of CG, which implies that Harada's formulation can be understood as a modification of the GR equations due to the presence of this residual tensor. This way, with the introduction of the Codazzi tensor, it became possible to express CG through a new formulation, which allowed for the representation  of conformally flat cosmological solutions without a vacuum in CG <cit.>. 

However, there is ongoing debate about the viability of CG as a potential extension of GR. Some researchers see CG as promising for addressing GR's limitations, while others doubt its theoretical consistency <cit.>. Further research is needed to determine whether CG can replace or complement GR, or if it will remain a theoretical curiosity.


In order to set constraints in the new parameter γ of CG, one can resort to the gravitational lensing of light. This is the deflection of light trajectories as they pass close to massive bodies (stars, black holes, galaxies, etc) and can act like a natural telescope by amplifying the source brightness and even creating multiple images. 
Recent developments related to this effect can be divided into two groups:  microlensing (i.e. the measurement of the collective magnification of various images) and strong lensing <cit.>. For the sake of this work we are (mostly) interested in the strong field regime, given the fact that it offers the possibility of testing GR beyond the weak-field limit tests carried out so far. In such a limit,  Bozza <cit.> proposed a formalism to calculate the deflection angle for every spherically symmetric metric in terms of a logarithmic expansion. Such a formalism has been very influential in advancing analytical methods to compare theoretical predictions with real images <cit.>, including alternative spherically symmetric metrics <cit.>.


The black hole shadow and photon ring heuristics are two such examples of visual phenomena caused by strong gravitational lensing <cit.>. The former consists in the dark silhouette of a black hole that is projected in the celestial plane, with a critical boundary separating backwards-traced geodesics that are scattered towards infinity from those that are captured by a black hole <cit.>. The latter appears as an infinite series of discrete and stacked, exponentially-damped in luminosity, lensed images of the emission surrounding a black hole, under the assumption of an optically thin and non-spherical accretion disk <cit.>. The recent progress in high-frequency very-long-baseline interferometry (VLBI) provides a direct way of comparing such heuristics with actual images, by the Event Horizon Telescope (EHT) Collaboration, of the super-heated overflow of the plasma around the supermassive central objects of M87* <cit.> and Sgr A* <cit.>. In turn, this provides an unprecedented opportunity to perform tests of new gravitational Physics, which has been put to good use very recently, see e.g.  <cit.>.

The main aim of this work is therefore to make use of this opportunity to study the optical appearances of the CG black hole above. In particular, we shall employ the reported bounds on Sgr A*'s shadow by the EHT Collaboration to constrain the CG black hole's parameter space. Next, we generate images of this geometry within this restricted parameter space when illuminated by an optically and geometrically thin accretion disk, modeled by three semi-analytic emission profiles developed by Gralla-Lupsasca-Marrone (GLM) <cit.>, and compare its optical appearance with that of a Schwarzschild black hole. Furthermore, we study the Lyapunov exponents of nearly bound orbits to elaborate on the differences between both geometries. In this work we consider units G=c=1. 

This paper is outlined in the following manner: In Sec. <ref>, we briefly present the field equations of CG and the static, spherically symmetric metric used throughout this work. In Sec. <ref>, we obtain the null geodesic equation and the effective potential of the CG black hole, using the above-mentioned spherically symmetric metric.
Section <ref> addresses gravitational lensing in CG, where we first derive the exact analytical expression for the deflection angle using elliptic integrals, followed by an approximate expression for the weak field regime.
In Sec. <ref>, we generate the images of the Cotton black hole using the restricted γ-space parameter obtained in the previous sections,  and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter.
In Sec. <ref>, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. In Sec. <ref>, we explore the Lyapunov exponent of nearly bound orbits, which is a quantity that measures the instability scale of these orbits.
Finally, in Sec. <ref>, we summarize our results and conclude.
 



§ A SPHERICALLY SYMMETRIC SOLUTION OF COTTON GRAVITY




Recently, as mentioned above, a generalization of the gravitational field equations of GR, dubbed as “Cotton Gravity" (CG) <cit.>, was proposed. The field equations read as follows

    C_αμν=κ^2∇_βΘ_βαμν^β,

where κ^2=8π, is  Newton's constant in suitable units, ∇_β is the usual covariant derivative  built with the Christoffel symbols of the metric, C_αμν  is the Cotton tensor defined as

    C_αμν≡∇_μR_να-∇_νR_αμ-1/6(g_αν∂_μ-g_αμ∂_ν)R,

where R_μν stands for the Ricci tensor and R  is the Ricci scalar, and finally the energy-momentum tensor  Θ_βαμν is defined as

    Θ_βαμν   ≡   1/2(g_βμΘ_αν-g_αμΘ_βν-g_βνΘ_αμ+g_ανΘ_βμ) 
        
          -1/6(g_βμg_αν-g_αμg_βν)Θ ,

with Θ_βμ=g^ανΘ_βαμν and Θ=g^βμΘ_βμ. 

The most remarkable feature of CG gravity is that every solution of the Einstein equations with a cosmological constant term G_μν+Λ g_μν=κ^2Θ_μν satisfies Eq. (<ref>) <cit.>, while the opposite is not true. In contrast to GR, however, the cosmological constant in CG is determined by integration of the field equations. Furthermore, the conservation law (∇_βΘ_μν^β=0) is a natural consequence of the theory, just like in GR, as can be verified by contracting Eq. (<ref>) by g^αν to yield

    g^ανC_αμν=κ^2∇_βΘ_μν^β=0.


For the sake of this work we use the exact vacuum solution obtained by Harada in <cit.>. Similarly to GR, where R_μν=0, this solution is obtained in Cotton Gravity when C_αμν=0 is considered. Such a solution belongs to the static, spherically symmetric class of metrics with line element

    ds^2=-A(r)dt^2+dr^2/A(r)+C(r)(dθ^2+sin^2θ dϕ^2).

whose metric function reads as <cit.>

    A(r)=1-2M/r-Λ/3r^2+γ r  ,       C(r)=r^2  .

This is the first non-trivial solution found in CG theory, which contains a new parameter γ, not present in the Schwarzschild case of the GR, and which breaks the asymptotically flat character of the space-time. Moreover, the cosmological constant is obtained naturally as an integration constant no matter the value of the cosmological constant Λ, which we take to be vanishing in this work. We draw attention to the fact that solutions containing a linear γ r term have been obtained in other theoretical contexts (e.g. <cit.>).






§ GEODESICS AND EFFECTIVE POTENTIAL





 §.§ Gravitational lensing


In this section, we obtain the null geodesic equation and the effective potential of the CG black hole, using the spherically symmetric line element (<ref>). To this end, we consider the point-particle Lagrangian density given by

    L=1/2g_μνẋ^μẋ^ν=ϵ,

where ẋ^μ is the four-velocity of the photon,  ϵ=(-1,0,1) indicates the time-like, null, or space-like character of the geodesic, respectively, and an overdot denotes the derivative with respect to the affine parameter λ. Taking into account the metric (<ref>) yields the following form for the Lagrangian

    L=1/2[-A(r)ṫ^2+ṙ^2/A(r)+C(r)θ̇^2+C(r) sin^2 θϕ̇^2]=ϵ .



Given  the spherically symmetric character of the space-time, we can consider the motion in the equatorial plane, θ=π/2, without any loss of generality. In addition, since the Lagrangian (<ref>) does not directly depend on t and ϕ, the Euler-Lagrange equations imply the conservation of two quantities:


    E   =A(r)ṫ,
     
    L   =C(r)ϕ̇.,


where E is the energy of the particle and L the angular momentum, respectively (per unit mass). Since we are interested in null geodesics, ϵ=0, Eq. (<ref>) with the conserved quantities (<ref>) and (<ref>), leads to

    ṙ^2=1/b^2-V_ eff(r).

where we have conveniently re-absorbed a factor L in the affine parameter λ, and introduced the impact parameter b ≡ L/E. In this equation, the effective potential reads as

    V_ eff(r)=A(r)/C(r).

which for the  metric function considered in this work, Eq. (<ref>), reads

    V_ eff(r)=1/r^2-2 M/r^3+γ/r.

The maximum of the effective potential determines the region of unstable orbits, dubbed as the photon sphere, whose radius is denoted by r=r_p, and the corresponding impact parameter as the critical impact parameter, dubbed by b_c. The latter separates those orbits that are captured by the black hole, b<b_c, from those that are scattered back to asymptotic infinity, b>b_c. Those that have b=b_c may linger indefinitely at the photon sphere, turning several times around the black hole. To determine these quantities, we note that circular light orbits obey ṙ=0 and r̈=0 simultaneously. Under these conditions, r_p and b_c are expressed for the metric function (<ref>) as


    r_p   =√(1+6 γ  M)-1/γ,
    
        b_c   =3√(6)  M/√((1+ √(1+6 γ   M)+6 γ   M √(1+6 γ  M)+9 γ  M)),


which for γ=0 it recovers the critical b_c=3√(3)M and the photon sphere radius r_p=3M of the Schwarzschild geometry.

With the expressions (<ref>) and (<ref>) we can develop the equation of motion which describes the trajectory of the photon as

    (dr/dϕ)^2=C^2(r)(1/b^2-V_ eff).

These are the general expressions for a given spherically symmetric metric. 





 §.§ Shadow's radius and constraints to the γ parameter



The shadow of a black hole is a silhouette, a  region marked by a strong decrease in the observed luminosity, projected onto the celestial plane of a distant observer. This projection arises from photons originating from a background source, which are absorbed due to its intense gravity. To calculate the angular size and the shadow of the black hole, we adopt the formalism described by <cit.>. In this approach, they consider a static observer located at (r_0, θ_0=π/2), projecting an angle θ relative to the radial coordinate r, as given by

    tanθ=√(C(r)/A^-1(r))dϕ/dr|_r=r_0.

The shadow of the black hole, understood as the region which borders (on the observer's plane image) the critical curve, is thus constructed by the light rays emitted from the observer back to the source.

To proceed with our derivation, we conveniently re-write Eq.(<ref>) into

    (dr/dϕ)^2=C(r)/A^-1(r)(C(r)/A(r)E^2/L^2-1).

A light ray with impact parameter b>b_c will be scattered off the black hole at a radial distance r_0>r_m, while  the photon sphere (i.e. the locus of unstable bound geodesics, r_p) can be found via the condition dr/dϕ|_r_p=0  which allows to express the impact parameter b as

    1/b^2=A(r_p)/C(r_p).

Now, by introducing the definition

    h^2(r)=C(r)/A(r),

we can rewrite Eq. (<ref>) as

    (dr/dϕ)^2=C(r)/A^-1(r)(h^2(r)/h^2(r_p)-1).


Substituting Eq.(<ref>) into Eq.(<ref>), and after proper manipulations, allows to find the expression

    sin^2θ=h^2(r_p)/h^2(r_0),

which yields the angular size of the black hole's shadow θ_s as

    sin^2θ_s=b_c^2/h^2(r_0)=b_c^2A(r_0)/C(r_0).

Using the expressions worked out above, for CG the angular shadow's size reads as 

    sin^2θ_s   =   2/r_0^2γ^2 (γ +8γ^2M)(-1+6M√((1+6γ M))
       +   √((1+6γ M))-9γ M )(1-2M/r_0+γ r_0).








Similarly, we can obtain an expression for the shadow radius as seen by a far-away observer, a regime in which we approximate sin(θ) ∼θ and θ_s=b_c/r_0
which gives us the following relation (assuming θ=π/2)

    r_s=r_p√(g_tt(r_0)/g_tt(r_p)).

For CG this reads as

    r_s=(√(a)-1)/γ√((√(a)-1)[r_0(γ r_0+1)-2M]/r_0(1-√(a)+4γ M)) ,

where a=1+6γ M. In Fig. <ref> we depict the behaviour of the shadow's radius in CG with the parameter γ, considering the black hole distance and mass estimates reported in Table <ref>.




We now aim to restrict the parameter space of γ by assessing the compatibility of the CG geometry's corresponding shadow radius r_s with the EHT's inferred shadow size of Sgr A* using the data above. This is so because, as argued in <cit.>, it is reported that the observed shadow size of Sgr A* can be inferred by utilizing the size of the bright emission ring as its proxy, subject to a calibration factor that multiplicatively accounts for the theoretical and observational uncertainties. Such a correlation becomes possible thanks to priors on Sgr A*'s mass-to-distance ratio obtained via orbital dynamics measurements of the stars closest to the galactic center (i.e. dubbed S-stars), and which are reported in Table <ref>.  These measurements allow the EHT collaboration to estimate the fractional deviation δ between Sgr A*'s inferred shadow size r_sh and that of a Schwarzschild black hole r_sh,Sch= 3 √(3) M, with angular radius ω_g = M/D  via the formula

    r_sh/M=(δ +1) 3 √(3) ,

which, under the assumption of a normal distribution of the estimation uncertainties, yields the following shadow size bounds <cit.>

    4.55 ≲ r_sh/M ≲ 5.22  ,

at 1σ deviation and

    4.21 ≲ r_sh/M ≲ 5.56  ,

at 2σ deviation.



We note that, to quantify the theoretical biases in the calibration factor utilized in the inference of the shadow's size of Sgr* A, the EHT collaboration performs simulations of black hole images to study a wide range of effects concerning the accretion flow properties and the underlying space-time. In this regard, the collaboration assumes the Kerr metric by default, although they also cover black hole metrics that either deviate from the Kerr space-time or describe alternative geometries, such as wormholes or boson stars. Because their analysis does not include the Cotton geometry, by performing tests with the above constraints we introduce a theoretical bias in our own analysis, since we assume these results to be valid for the CG spacetime. The extent to which the Cotton geometry could modify the above fractional deviation, and the constraints stemming from it, could be evaluated by performing a similar analysis to that by the EHT collaboration. However, such an effort lies well beyond the purpose of this work. 

To perform tests on the CG metric with its shadow radius, we compare the evolution of r_s as a function of the logarithm of γ M, in Fig. <ref>. We restrict our analysis to positive values of γ, which are required to describe the large-distance gravitational acceleration. 
Because we are dealing with a non-asymptotically flat space-time, one must take into account that its asymptotic Arnowitt-Deser-Misner (ADM) mass <cit.> is undefined for an observer at radial infinity and cannot be associated with the mass parameter appearing in the metric.
To circumvent this issue, we shall take a similar approach to <cit.> and report the the shadow radius in units of the local mass M, based on the reasoning that the stellar orbital dynamics of the S group of stars probe this quantity, since they are influenced by the local gravity of Sgr A*. 
Additionally, since the CG black hole's shadow radius depends explicitly on the observer's radial position, recall Eq.(<ref>), we compute the shadow radius considering an observer located at r_O ∼ 8000  kpc <cit.>. 




Upon inspection of Fig. <ref>, we verify that r_sh starts to increase around values of γ M ∼ 10^-12. Below this value, the CG black hole shadow radius coincides with that of a Schwarzschild black hole. We note that this is qualitatively the same behavior that is evidenced in <cit.> for the Rindler metric. Actually, the contribution of γ to the shadow radius becomes significant when the term γ r ≫ 1, while the upper 2σ bound in Eq.(<ref>) becomes saturated at a value of approximately γ M ≃ 3.525 × 10^-12.
For the values of γ M predicted by galaxy radial acceleration rotation curves in <cit.>, or the recent constraints obtained by analyzing CG's influence on CMB power spectrum and large scale structures <cit.> (i.e. both estimating γ M ∼ 10^-17), the shadow radius of the Cotton black hole lies well within the constraints imposed by Sgr A*'s shadow bounds, though there is no appreciable difference between it and its Schwarzschild counterpart. This result is not surprising, given that γ encodes large-distance gravitational effects within the CG framework. 
We have also found that the Cotton black hole shadow radius remains within the 2σ bounds for γ M ∼ 10^20. 
Such values describe extremely large and unphysical gravitational acceleration effects, exceeding any other current estimates by several orders of magnitude. For this reason, we have opted for excluding them from the γ parameter range in Fig. <ref>.










§ GRAVITATIONAL LENSING



This section deals with the phenomenon of gravitational lensing for CG. First, we will obtain the exact analytical expression for the deflection angle in terms of elliptic integrals. Then we will calculate an approximate expression valid for the weak field regime.





 §.§ Exact deflection angle 


To proceed with our derivation, it is convenient to make the following variable change in the radial coordinate u = 1/r in Eq. (<ref>). This allows us to re-write the photon equation of motion as

    ( du/d ϕ)^2 = 2 M G(u),

where the function G(u) reads, for CG, as

    G(u) =u^3 - u^2/2M  - u γ/2M + 1/2Mb^2.


The general expression for the deflection angle at a finite distance has been worked out in <cit.> as

    α = √(1/2M)( ∫_u_R^u_0du/√(G(u))+∫_u_S^u_0du/√(G(u))) + Ψ_R - Ψ_S,

where u_0 = 1/r_0 is the inverse of the distance of closest approach, while u_R = 1/r_R and u_S are the inverse of the radius of the observer and the source, respectively, and the indices refer to the position of the observer (R) and the source (S). 
The function Ψ encodes a correction term accounting for the finite distance, and is given by

    Ψ = arcsin( bu √(A(u))) .

Note that if we take the limit of  γ→ 0, and consider also that observer and source are located at infinity we will have

    α_SC = √(2/M)∫_0^u_0du/√(G(u)) - π,

which is the well-known deflection angle formula for the Schwarzschild solution.  This change in the definition of the deflection angle is necessary because the Cotton metric is not asymptotically flat. Moving on, the integrals of Eq. (<ref>) can be solved by rewriting the function G(u) as

    G(u) = (u-u_1)(u-u_2)(u-u_3),

where u_1,u_2 and u_3 are the roots of the cubic polynomial. We can check that u_0 = 1/r_0 is one of the roots of this polynomial. We have chosen (for reasons that will become clear later) u_2 = 1/r_0. Now we can write

    u^3 -u^2(u_1 +u_3 +1/r_0) +u (u_1 u_3 +u_1/r_0 + u_3/r_0) 
       
    
         - u_1 u_3/r_0  = u_3 - u^2/2M - u γ/2M + 1/2Mb^2.

After comparing the coefficients in Eq. (<ref>), we get

    u_1 +u_3 + 1/r_0 =   1/2M, 
        u_1u_3 + u_1/r_0 + u_3/r_0 =  
            - γ/2M.

Solving the above system we find that u_1 and u_3 are  given by

    u_1 = -2 M
       +r_0 ∓√(-(12 M^2 -8 γ  M r_0^2-4 M r_0-r_0^2))/4 M r_0 , 
       u_3 = -2 M+r_0 ±√(- (12 M^2-8 γ  M r_0^2-4 M
       r_0-r_0^2))/4 M r_0,

respectively. Note that we have two different solutions, one of which is u_1> u_3 and another that u_3> u_1.
Since our goal is that these roots fulfill the condition u_3> u_2> u_1, we choose the negative sign in the expression u_1 and the positive sign in the expression u_3. Now both integrals of Eq. (<ref>) can be put into the form

    I_1(u^') =  ∫_u^'^u_2du/√((u-u_1)(u-u_2)(u-u_3))

with u_3>u_2>u^'>u_3 and u^' being any finite distance. The result of the above integral is 

    I(u^') = 2/√(u_3-u_1) F(δ,k^2),

where F(δ,k^2) is an incomplete first-order elliptic integral of the parameters

    δ   =   arcsin√((u_3-u_1)(u_2-u^')/(u_2-u_1)(u_3-u^')), 
    
        k^2    =   u_2-u_1/u_3-u_1.



 To get the result of the integrals given in (<ref>) we just need evaluate I(u_R)+I(u_S). The complete result will also depend on the correction given by the functions Ψ(u_R) - Ψ(u_S). 
For the Cotton metric, the function Ψ is 

    Ψ(u) = sin ^-1(b u √(1-2 M u+γ/u)),

while we have for the parameters

    δ(u^') = sin ^-1{ 2 √(2)[ 
        (M r_0 (1-u^' r_0) √(r_0^2 c)/√(r_0^2 c)  +6 M  r_0- r_0^2) ×
    ×(√( r_0^2 c)-2 M
        r_0 (2 R  r_0+1)+ r_0^2)^-1]^1/2},












where c=-12 M^2+4 M r_0 (2 γ  r_0+1)+r_0^2

    k^2= √( r_0^2-12 M^2+4 M r_0 (2 γ 
      r_0+1))+6 M -r_0/2√(r_0^2 -12 M^2+4 M
       r_0 (2 γ  r_0+1)).

which closes all the expressions we needed to characterize the deflection angle in CG. It is also interesting to re-write these expressions in terms of the impact parameter by inverting its definition, in CG, as 

    r_0    =   b^2 γ/3+2/3 b √(b^2 γ ^2+3)×
          ×cos[1/3cos^-1((2 b^4 γ ^3 +9 b^2 γ
       -54 M)/2 b (b^2 γ^2+3)^3/2)].









For the sake of the graphical representation of the deflection angle we prefer to use the variable b^' = 1-b_c/b. This variable is limited to a value between 0 ≤ b^'≤ 1, where zero represents the strong field limit and one the weak field one. In Figs. <ref> and <ref> we show the deflection angle as a function of the variable b^' for values of γ in the range from 10^-16 to 10^-9. We see that α decreases with increasing γ and that there is no significant change in the values of α from γ <10^-13. If we increase the value of the parameter γ beyond the order of magnitude of 10^-9, the deflection angle begins to give negative values. Observationally, this value diverges for b^' =0 and tends to zero for b^' =1. We expect our theoretical result to reproduce this behaviour.
In other words, for the phenomenon of light deflection, Cotton's parameter is limited to values smaller than γ = 10^-9.










§ IMAGES OF BLACK HOLES WITH THIN ACCRETION DISKS



Having restricted the γ-space parameter, we now proceed to generate the images of the Cotton black hole (by pushing the value saturating the upper 2σ bound of the Sgr A* shadow) and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter. We consider the setting of a black hole illuminated by an optically (i.e. fully transparent to its own radiation) and geometrically thin accretion disk, when observed at both a face-on inclination (i.e. θ_o = 0^∘) and at a nearly edge-on inclination (i.e. θ_o = 80^∘).  

We begin our considerations from Eq. (<ref>), which we conveniently re-write in the form

    dϕ/dr=±b/C(r)√(1/1-b^2V_ eff),

that describes the variation in a photon's azimuthal angle as a function of its radial coordinate. By numerically ray-tracing the previous equation for a bunch of trajectories within some range of the impact parameter b, we compute the null geodesics from an observer's screen backwards towards their point of origin. Figure <ref> displays the geodesics in the Schwarzschild space-time (left panel) for M=1, and in the CG space-time (right panel), the latter with a value of γ M = 3.525 × 10^-12. Upon close observation of the figure, one highlights the absence of any significant qualitative differences between the near-horizon lensing properties of the two geometries, under the considered parameter values. In addition to larger distances, the effects of γ become more noticeable at much higher orders of magnitude than those considered here, as is discussed in Section <ref> below. 





To simulate the emission from an accretion disk we resort to the semi-analytic models provided by Gralla, Lupsasca and Marrone in Ref. <cit.>, which describe the radial emission profile of an accretion disk based on a Standard Unbound (SU) Johnson distribution function given by

    I(r;γ,μ,σ)=e^-1/2[γ+arcsinh(r-μ/σ)]^2/√((r-μ)^2+σ^2),

where the γ, μ and σ parameters modify the skewness, peak location and kurtosis of the distribution, respectively. Such models are designed to reproduce the most prominent image features of time-averaged General Relativistic Magneto-Hydrodynamic (GRMHD) simulations of a thick accretion flow around a Kerr black hole, providing a less computationally taxing alternative towards the generation of black hole images <cit.>.
These models describe an infinitesimally thin, axisymmetric, radially dependent emission profile, assumed to be monochromatic in the disk's frame of reference and optically thin at the emitting frequency. In the absence of absorption, the relativistic radiative transfer equation <cit.> implies that the specific intensity I_ν/ν^3 is conserved along a light ray such that I_ν_o=g^3 I_ν_e, where the subscripts o and e denote the observer and emitter frames, respectively. The quantity g is defined as the redshift factor g=ν_o/ν_e = √(A(r)/A(r_∞)). Since the disk is geometrically thin, a light ray gains additional brightness for each time it intersects the accretion disk while on its way to a distant observer (i.e. every n half-orbit). Thus, considering a monochromatic emitter I_ν_e≡ I(r), the integrated (bolometric) intensity I=∫ I_ν_o d ν_o in the observer's plane is computed according to

    I_o(r)=∑_n(A(r)/A(r_o))^2.I(r)|_r=r_n.



For the purpose of generating black hole images we restrict our analysis up to the n=2 photon ring, as higher-order photon rings will provide a negligible contribution to the net luminosity (as shall be explained in the next section). We shall suitably adapt the three models introduced in <cit.> to a spherically symmetric scenario, to simulate the accretion flow around the Schwarzschild and Cotton black holes, with the parameters

    GLM1   :   γ=-3/2  ,   μ=0   ,      σ=M/2 , 
    GLM2   :   γ=0   ,     μ=0   ,      σ=M/2 , 
    GLM3   :   γ=-2  ,    μ=17M/3  ,  σ=M/4 .

whose corresponding emission profile is depicted in Fig. <ref>.



The GLM1 and GLM2 models are utilized to simulate an accretion disk with an emission profile extending all the way towards the event horizon, being distinguished by the radial emission decay near this region. In contrast, the GLM3 model represents an accretion disk with its emission peak located near the radius of the innermost stable circular orbits (r_ISCO), which is 6M for a Schwarzschild black hole. The latter allows us to separate the individual luminosity contributions of the n=1 and n=2 rings from the direct emission, thereby facilitating the comparison of these features between the Schwarzschild and Cotton black holes, whereas in the former models the photon rings appear as a discrete set of boosts of luminosity superimposed on top of the disk's direct emission <cit.>. In our simulations, the observed luminosity is normalized with respect to its total value, to ensure both geometries are compared in identical settings. We resort to the Geodesic Rays and Visualization of IntensiTY profiles (GRAVITYp) ray-tracing code, developed by some of us.



By inspection of Fig. <ref>, we report no appreciable visual differences between a Schwarzschild and a Cotton black hole of the same mass, with γ M ≃ 3.525 × 10^-12, regardless of the emission conditions. At the considered distance, the Cotton black hole's corresponding event horizon radius, photon sphere radius and critical impact parameter, are nearly indistinguishable from those of a Schwarzschild black hole of the same mass M,  differing only by a factor of the same order of magnitude of γ. Since for a non-asymptotically flat geometry the shadow radius does not coincide with the critical impact parameter, the geometries appear identical. This similarity persists onto the photon ring(s), where the gravitational lensing features play a significant role. In the literature, however,  various black holes with a similar critical impact parameter have been found to be distinguished by the properties of their corresponding photon rings <cit.>. These differences occur when, within the parameter space compatibility with Sgr A* shadow radius, the additional parameter in a given geometry modifies significantly the effective potential (i.e. its shape, location, concavity) seen by photons on their approach to a black hole, from the canonical Schwarzschild case. For some geometries, however, the constraints to the parameter space are too restrictive, to the point that there are virtually no changes to the effective potential, which is our case. Finally, we point out that although Eq. (<ref>) suggests the observed intensity depends on the observer's radial coordinate, for a non-asymptotically flat space-time, there are no visual differences between the Cotton and Schwarzschild black holes since the intensities are normalized with respect to their total value. Without normalizing the intensities, the observed intensities of the accretion disk emission in the Cotton spacetime would decrease, as in this case A(r) →∞ for an observer at infinity.





§ ISORADIALS



In this section, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. We solve the isoradial equatorial curves corresponding to the n=0 emission, following the analytical approximation provided by <cit.>, which states that the arrival point of a light ray may be mapped onto an observer's plane with Cartesian coordinates (X',Y') as 


    [ X'; Y' ]
        = R/√(A(R))√(1-(1-x)^2)[  cos(φ); sin(φ). ]

Here, R denotes a given radial distance from the origin, while φ corresponds to the polar angle in the observer's plane[For a better understanding of this coordinate representation, the reader is referred to Fig. 7 of <cit.>.]. The value of x may be computed via the following approximation


    x    =    A(R)y {1
     -e/100(1-A(R))y[ ln(1-y/2) +y/2] 
     
           +(315 I_1^2 -135 I_2 + 180 A^2(R) +120 I_1 +32)/720y^2
        
                   + (I_1/2 + 1/6)y
     },









with y being expressed by the trigonometric relation between the polar angle in the observer's plane and the observer's inclination angle θ_o via

    y=1+sin(φ)/√(sin^2(φ)+cot^2(θ_o)),

and where the quantities I_1 and I_2 are calculated as

    I_1=∫_0^1 P(u)   du  , 
    
     I_2=∫_0^1 P^2(u)   du,

with P(u)=-4u^2A(u) + A(R), and where the variable change r=R/u was considered to obtain A(u).

By solving the above equations for all polar angles, for a light ray with its emission point located at a constant r=R, one obtains the isoradial curves of the n=0 emission. In Fig. <ref>, we display isoradial curves for a Schwarzschild black hole and for a Cotton black hole with γ M ≃ 3.525 × 10^-12, as seen by an observer with inclination angle θ_o = 80^∘. 



This result emphasizes the similarity in the horizon-scale lensing phenomena between these two geometries, as seen by the complete overlap of the isoradial curve of both space-times; something which is in agreement with the analysis in the previous section. 




§ THEORETICAL LYAPUNOV EXPONENT




In this section, we explore the Lyapunov exponent of nearly bound orbits, a quantity that measures the instability scale of such orbits. Since these instabilities take place near the photon sphere, the Lyapunov exponent tracks a local effect and is therefore independent of the observer's distance to the black hole. 

We begin our analysis by considering the effect of a small radial perturbation near the photon sphere (i.e. r=δ r + r_p  , δ r ≪ r_p) and apply it to Eq. (<ref>). We perform a second-order Taylor series expansion of the effective potential V_ eff around r_p

    V_ eff(r)≃ V_ eff(r_p)+V'_ eff(r_p)δ r+1/2V”_ eff(r_p)(δ r)^2,

where the first term of the expansion corresponds to the critical impact parameter b_c and the second term disappears, since unstable circular null geodesics correspond to critical maxima of the effective potential. The prime notation above denotes a derivative with respect to the radial coordinate. Substituting this result into Eq. (<ref>), leads to the following expression 

    (d δ r/dϕ)^2 = -1/2C^2(r_p)V”_ eff(r_p)(δ r)^2.

By writing the differential equation in the form

    πd δ r/dϕ = λδ r,

whose solutions are of an exponential form, δ r_n=δ r_0e^λ n, we obtain the Lyapunov exponent λ which quantifies the instability scale of a small perturbation. Computing the second derivative of the effective potential and utilizing the critical curve equation C'(r_p)A(r_p)-C(r_p)A'(r_p)=0 <cit.>, yields the following expression for the theoretical Lyapunov exponent

    λ=π(AC”-A”C/2)^1/2.


For the Schwarzschild geometry γ = π, indicating that an initial perturbation grows (in radial distance) by a factor e^γ with each half-orbit around the black hole. For the Cotton geometry, the Lyapunov exponent reads

    λ=π(1+γ r_p)^1/2.

This result supports the qualitative analysis done in the previous sections. The λ exponent of the CG black hole, differs from the Schwarzschild black hole by an order of magnitude proportional to γ.  Consequently, despite the exponential behaviour driven by the Lyapunov exponent, the deviations between these two geometries are captured by photon rings of a similarly order, requiring many half-orbits around the black hole before becoming significantly different. The gravitational lensing causing the n=1 and n=2 photon rings is far from being sensitive to the contribution of γ. Thus, one cannot hope to be able to distinguish between the Schwarzschild and CG geometries via their n=1 and n=2 ring features (i.e. relative intensities, relative widths), which are the ones that could be within reach of future interferometric detectors <cit.>.





§ CONCLUSION


In this work we investigate the phenomenon of gravitational lensing and its associated observables within an extension of GR recently proposed in the literature and dubbed as Cotton gravity. We estimated the size of the new parameter γ of Cotton gravity and which arises with a newly found spherically symmetric vacuum (Schwarzschild-like) solution, by making a numerical comparison with bound on the shadow's size of the supermassive black hole at the center of our galaxy (Sgr A*) reported by the Event Horizon Telescope Collaboration. Such a bound  is saturated, for CG, for values as small as γ M = 3.525 × 10^-12 (at 2σ in such bounds), where we took into account some subtleties associated to the non-asymptotic flatness of CG geometry to identifying the mass parameter of it with the one of the Schwarzschild case. 

To study the impact of this result in geodesic effects, we first computed the exact expression for the deflection angle in the form of elliptic integrals. There are two important considerations about this result. Firstly, in order to obtain the result of the Schwarzschild solution, one must first assume γ=0 and additionally u_R and u_S=0 (which corresponds to the consideration of infinite distances). This is because the result for Schwarzschild is usually determined using the definition for infinite distances, which is not possible in the case of Cotton gravity given its non-asymptotically flat character. Secondly, the deflection angle can be written as a function of the impact parameter b, though under an explicit cumbersome form. Furthermore, we computed the deflection angle using a Taylor series approximation to solve the integrals, which yields a result which is valid for the weak field regime where M/b ≪ 1. We have considered a linear approximation for the parameter γ in addition to the series with respect to the mass (we expect that γ has in reality a much smaller order of magnitude than mass and distances), and checked the consistence of the analytical approximation with the numerical integration of the deflection angle. Overall, we find that the deflection angle remains unchanged for values of the CG parameter γ < 10^-14.


The smallness of such a parameter of CG drove us to consider the chances of distinguishing between Schwarzschild and CG black holes using observables associated to gravitational lensing. To this end, we simulated the appearance of a CG black hole, with such a value of the γ parameter, surrounded by a thin accretion disk, modelled by three semi-analytic radial intensity profiles. We found the constraints placed on γ by Sgr A*'s shadow bounds to be too tight for any significant visual differences to appear. In this regard, we note that although in CG the observed intensity depends on the observer's radial distance, our images do not show any difference in this regard due to having normalized intensity profiles. 

In looking for further settings to search for the presence of any differences, we studied the isoradial curves of a CG black hole against those of a Schwarzschild black hole, utilizing a generalization of the Poutanen formula. We found that there is a full overlap of the isoradial curves between the CG and Schwarzschild geometries, confirming the absence of any visual differences in this regard. We also considered the theoretical Lyapunov exponent for a Cotton gravity black hole, associated to the instability scale of photon rings. We derived its exact expression by perturbing the photon equation of motion about the photon sphere radius. The CG theoretical Lyapunov exponent was found to differ from Schwarzschild's by a factor proportional to CG's γ parameter, and thus photon rings do not help in discriminating between both geometries, at least for those rings (n=1 and n=2) that could be expected to be measured in the last decades via interferometric means.

To conclude, the analysis carried out in this work points that the CG space-time recently found in the literature is very unlikely to be distinguishable from the expectations of the canonical Schwarzschild black holes with the devices we have at this moment and in the foreseeable near future.



MER thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, for partial financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FSNL acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT) Scientific Employment Stimulus contract with reference CEECINST/00032/2018, and funding through the research grants UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/0054/2021.
DRG is supported by the Spanish Agencia Estatal de Investigación Grant No. PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, UE (European Union), ERDF (European Regional  Development Fund) A way of making Europe.


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 Paper Title: Measuring the Spin of the Galactic Center Supermassive Black Hole with Two Pulsars.








Department of Astronomy, School of Physics, Peking University, 
Beijing 100871, China
Kavli Institute for Astronomy and Astrophysics, Peking University, 
Beijing 100871, China

lshao@pku.edu.cn
Kavli Institute for Astronomy and Astrophysics, Peking University, 
Beijing 100871, China
National Astronomical Observatories, Chinese Academy of Sciences, 
Beijing 100012, China

§ ABSTRACT 
As a key science project of the Square Kilometre Array (SKA), the discovery and
timing observations of radio pulsars in the Galactic Center would provide
high-precision measurements of the spacetime around the supermassive black hole,
Sagittarius A* (Sgr A*), and initiate novel tests of general relativity. The
spin of Sgr A* could be measured with a relative error of ≲ 1% by
timing one pulsar with timing precision that is achievable for the SKA. However,
the real measurements depend on the discovery of a pulsar in a very compact
orbit, P_b≲0.5  yr. Here for the first time we propose and
investigate the possibility of probing the spin of Sgr A* with two or more
pulsars that are in orbits with larger orbital periods, P_b∼ 2- 5 
yr, which represents a more realistic situation from population estimates. We
develop a novel method for directly determining the spin of Sgr A* from the
timing observables of two pulsars and it can be readily extended for combining
more pulsars. With extensive mock data simulations, we show that combining a
second pulsar improves the spin measurement by 2-3 orders of magnitude in some
situations, which is comparable to timing a pulsar in a very tight orbit.


Measuring the Spin of the Galactic Center Supermassive Black Hole with Two Pulsars
    Lijing Shao
    August 5, 2024
==================================================================================



Introduction.—Black holes (BHs) are the most intriguing objects for
testing general relativity (GR). While the detection of gravitational waves from
stellar-mass BH mergers <cit.> provided abundant knowledge of the
highly dynamical, strong-field regime of gravity, the supermassive BHs (SMBHs)
existing at the center of most massive galaxies <cit.> represent another ideal laboratory for BH physics.  The SMBH
dwelling in our Galactic Center (GC) <cit.>, Sagittarius A* (Sgr A*), with the largest mass
to distance ratio among the known BHs, is the most promising target for
precision observations and unique tests of strong-field
gravity <cit.>.

Previous studies have shown that timing observations of radio pulsars in compact
orbits around Sgr A* would allow us to probe the spacetime around this SMBH to 
unprecedented accuracy and provide us with unique tests of the no-hair theorem 
for Kerr BHs <cit.>. Regular timing
of a normal pulsar in a highly eccentric orbit (e≳ 0.8) tightly around
Sgr A* (P_b≲ 0.5  yr) will measure the spin and quadrupole moment
of Sgr A* to a relative precision of about 1% within five years for a timing
precision of 1  ms, which is achievable for future radio
telescopes <cit.>.
However, the existence and detectability of such pulsars in very close orbits
around Sgr A* is still unclear. The large dispersion measures and the scattering
caused by highly turbulent interstellar medium in the GC make observations favor
high radio frequencies <cit.>, which then render pulsars
flux-limited because of the steep spectra of emission. Nevertheless, theoretical
models suggest that there can be 10^2-10^3 pulsars orbiting around Sgr A* with
P_b≲ 100  yr <cit.>, and present
observations including the discoveries of a magnetar <cit.>, a millisecond pulsar <cit.> and several normal
pulsars <cit.> in that region all suggest that a
large population of pulsars are awaiting for discoveries in the GC for future
high-frequency surveys.

Measuring the BH spin is crucial for gravity tests.  Determining the spin of
Sgr A* based on the timing of a single pulsar suffers from parameter degeneracy
at the leading order <cit.>.  The measurable secular
changes of the pulsar orbit in a short time, caused by the frame-dragging
effect <cit.>, are the linear rate of the periastron precession,
ω̇, and the linear change of the projected semimajor axis of the
pulsar orbit, ẋ. However, these two quantities cannot fully determine
the BH spin which has three components. To break the degeneracy, one needs at
least one accurate measurement of  higher-order time derivatives such as 
ω̈ or ẍ <cit.>, which in general requires a
much longer observation time span or a pulsar in an orbit with very short 
orbital period whose relativistic effects are large. 

There are still two possible ways to break the leading-order degeneracy and 
provide a better spin determination without requiring a pulsar in a rare orbital
situation. One is to combine the measurement of the proper motion of the pulsar
from astrometric observations <cit.>.  One can infer the secular
change in the longitude of the ascending node, Ω̇—which is not
observable in timing observation—from the proper motion of the pulsar and
provide an additional constraint on the spin of the central BH. However, an
expected astrometric accuracy in the order of 10 μ
as <cit.> is still not enough and contributes little when 
combining with the timing data <cit.>. The other possibility is to
combine the observation of another pulsar with a different orbital inclination
i, as the leading-order degeneracy mainly depends on i <cit.>.
Considering a potentially large pulsar population in the GC, finding two or more
pulsars orbiting around Sgr A* in the future may provide us a more likely way in
measuring the spin of Sgr A*.

In this Letter, we develop a novel method for determining the spin of
Sgr A* by combining  observations of two or more pulsars. We demonstrate that
for a pulsar with orbital period P_b∼ 2-5  yr, combining a second
pulsar can improve the spin measurement significantly. While previous studies
focused on timing one pulsar in a very tight orbit (P_b≲ 0.5 
yr), we show that finding two or more pulsars with larger orbital periods are
more likely to be the case for future observations.

Combining two pulsars.—When the changes in the Keplerian parameters
of a pulsar orbit are slow over the observation time span, the direct way to
characterize  secular effects is to fit the timing data with timing parameters
varying linearly in time, such as  ω=ω_0+ω̇ t and 
x=x_0+ẋ t <cit.>. In a pulsar-SMBH system, the
leading-order derivatives will be measured soon after the start of observation
and have a rather high measurement precision. Here we use the two combinations
introduced by <cit.> for the leading-order derivatives and other
timing observables, which are 𝒳≡ -ẋs_i^2
(xΩ̂)^-1 and 𝒲≡(ω̇-
ω̇_M)s_i^2Ω̂^-1, where s_i=sin i,
Ω̂=4πβ_O^3/P_b(1-e^2)^3/2 and 
ω̇_M=6πβ_O^2/P_b(1-e^2), with β_O=(2π GM/c^3P_b
)^1/3 the orbital velocity parameter introduced by <cit.>.
Note that s_i and M in these combinations can be measured precisely from the
Shapiro delay <cit.> that is large enough here even for face-on
orbits <cit.>. 

To combine the timing results of two pulsars and determine the spin of the 
central BH, one has to translate the measurement constraints of both pulsars in
a common parameter space. In this case, the natural choice is the
χ-χ_λ plane, where χ_λ is the projection of the
dimensionless spin χ of the BH in the line of sight direction. For each
pulsar, timing observations give one constraint represented by a hyperbola,


    𝒳^2/s_i^2(1-s_i^2)+𝒲^2/s_i^2(1+3s_i^2)=
        χ^2-1+3s_i^2/(1-3s_i^2)^2(χ_λ+3
        𝒲c_i/1+3s_i^2)^2 ,


where c_i=cos i.  If one  observes two or more pulsars, all these curves in
the χ-χ_λ plane should intersect at one point that gives χ
and χ_λ. 





Figure <ref> shows an illustration of using the timing of two pulsars to
determine the spin of Sgr A* in the χ-χ_λ plane. The system 
parameters are randomly chosen to avoid the accidental degeneracy which is
discussed later.  For each pulsar, the measurements of ( X,  W)
give a hyperbola shown by the solid curve, and the i↔π-i
ambiguity gives another hyperbola shown by the dashed line. In general, there
can exist at most eight different solutions for (χ, χ_λ).  A rough
measurement of any second-order derivatives, for example, the ω̈ 
for one of the pulsar, as shown by the vertical red line, or timing of a third 
pulsar can reduce this degeneracy into two solutions whose χ has the same
value and χ_λ has opposite signs.

The methodology described above can be easily extended for combining more
pulsars.  For each additional pulsar, it provides an independent test of GR as
all these curves should intersect at one point within their measurement
uncertainties. On the other hand, if one assumes that GR is correct, timing more
pulsars can be a unique probe for us to explore and model the (stellar and gas)
mass distribution near the GC <cit.>. As
only the leading-order time derivatives and a rough value of a higher-derivative
parameter are used, one is expected to obtain a high-precision measurement after
a short time observing two pulsars.

Numerical simulation.—To further explore the method introduced above
and give a quantitative description of the expected measurement precision in
measuring the spin of Sgr A* by timing two pulsars,  we perform mock data
simulations based on the numerical timing model for pulsar-SMBH systems
developed earlier <cit.>. In this timing model, we numerically
integrate the pulsar orbital motion based on the post-Newtonian (PN) equation of
motion, r̈=-GMr̂/r^2+r̈_ 1PN
+r̈_ SO+r̈_ Q, where r is the relative
separation pointing from the SMBH to the pulsar, r̈_ 1PN is
the first PN correction, r̈_ SO and r̈_ Q
are the spin and quadrupole contribution from the central SMBH
respectively <cit.>, M is the mass of the SMBH,
r≡|r|, and r̂≡r/r. Taking advantage of the PN
expansion, we  treat the dimensionless spin χ and the dimensionless
quadrupole moment q of the SMBH as independent parameters, which renders a
test of the no-hair theorem q=-χ^2 for Sgr A* possible <cit.>.
As the mass ratio m_ PSR/M≲ 10^-6, we treat the pulsar as a test
particle in the BH's spacetime. We take into account the Römer delay, the
leading-order Shapiro delay and the Einstein delay in the timing
model <cit.> and ignore the effects
caused by the proper motion of Sgr A* <cit.>. More
details can be found in Ref. <cit.>.

The full parameter set, Θ=Θ_ BH∪Θ_
PSR, in the timing model  for a single pulsar includes the parameters of the
SMBH, Θ_ BH={M,χ,q,λ,η}, where λ and
η describe the direction of the spin, and the parameters of the pulsar,
Θ_ PSR={P_b,e,f,i,ω,N_0,ν,ν̇}, where f is the
initial orbital phase and {N_0,ν,ν̇} relate the pulsar's rotation
number N and pulsar's proper time T via N(T)=N_0+ν T+ν̇T^2/2.  We
have set the longitude of the ascending node of the pulsar orbit, Ω, to
be zero as it is not an observable for timing observation. When combining the
timing of two or more pulsars, one should have one set of Θ_ PSR for
each pulsar while Θ_ BH is common for different pulsars. In
addition, the difference between the longitudes of the ascending node, ΔΩ=Ω_2-Ω_1, should also be included in the timing
model <cit.>. As we can have an overall rotation of the system
around the line of sight direction, we fix Ω_1=0 in the following
discussions and we denote the pulsar with a smaller orbital period as PSR1 or
the inner pulsar.

We use the Fisher matrix method for parameter estimation <cit.>. 
The log-likelihood function ℒ in the covariance matrix C_αβ= (∂^2ℒ/ ∂Θ^α∂Θ^β)^-1 is simply the sum from two pulsars,
ℒ=ℒ_1+ℒ_2, and 


    ℒ_i=1/2ν_i^2∑_a=1^N^(i)_ TOA[N^(i)_a(Θ)-N^(i)_a(Θ̅)]^2/(σ^(i)_ TOA)^2 ,  i=1,2


where N^(i)_a(Θ)=N (Θ;t_a^(i)  TOA) is the pulsar
rotation number calculated by the timing model corresponding to the a-th pulse
arrived at t_a^(i)  TOA. Θ̅ denotes the true system
parameters and σ_ TOA^(i) is the timing precision for pulsar i. 
As we do not consider the interaction between the two pulsars, the properties of
the Fisher matrix method and the timing model allow one to construct the full
Fisher matrix F≡ C^-1 though the combination of the Fisher matrices
F_i of each single pulsar. This simplification is important for combining more
pulsars.



We assume a timing precision σ_ TOA=1  ms and an observation 
time span T_ obs=5  yr for both pulsars, which are realistic for 
future instruments such as the Square Kilometre Array (SKA) and the 
next-generation Very Large Array (ngVLA) <cit.>.
In the upper panel of Fig. <ref> we show the measurement precision of the
spin of Sgr A* from combining the timing of two pulsars (solid curves) compared 
to the results obtained from the inner pulsar only (dashed line). Here we fix 
the orbital period of the inner pulsar to be P_b1=2.3  yr and orbital 
eccentricity e_1=0.8 while varying the orbital period P_b2 and 
eccentricity e_2 of the second pulsar. The lower panel of this figure shows 
the improvement factor F_ imp=δχ_ single / δχ_
combine. For this case, the inner pulsar with an orbital period P_b1=2.3
  yr can only constrain δχ / χ to ∼ 15%. Combining a 
second pulsar even with a large orbital period can provide an improvement factor
of ∼ 10 because of the breaking of the leading-order degeneracy.  The
improvement factor from combining a second pulsar will be smaller if the inner
pulsar has a tighter orbit for which itself can break the leading order
degeneracy. Another case study of P_b1=0.5  yr shows an improvement
factor of 2∼ 3 when combined with a second pulsar whose orbital period
P_b2≲ 3  yr.



As the second pulsar has a larger orbital period than the inner one, the 
constraint of the spin from it is in general looser than that from the inner
pulsar <cit.>.  Thus the
improvement of the spin measurement by combining two pulsars is mainly caused by
the breaking of the leading-order degeneracy.  Figure <ref> shows an
example of the dependence of the degeneracy on the most relevant orbital
parameter, the orbital inclination i. In this example we set P_b1=2.3 
yr, e_1=0.8, P_b2=4.2  yr, and e_2=0.6. The orbital
inclination i_1 of the inner pulsar is fixed to be 60^∘ and we vary the
orbital inclination of the second pulsar, i_2.  One can clearly see that the
improvement factor varies over two orders of magnitude and it is not dominated
by  the second pulsar's measurement precision. 

The simulation results are consistent with the prediction from the analysis we
presented before.  From Eq. (<ref>), it is obvious that the shape of the
hyperbola of each pulsar is controlled by their inclination angles, which
determine the leading-order degeneracy. Moreover, from Eq. (<ref>) one
can also predict the position of the three degeneracy peaks for the combined
results in Fig. <ref>, where the improvement from combining a second
pulsar is the smallest. The central peak at i=90^∘ originates from the 
denominator of the first term in Eq. (<ref>), while the other two peaks 
are attributed to the situation where two hyperbolas are tangent to each other
at the point they intersect. We show the predicted peak positions from
Eq. (<ref>) in Fig. <ref> with vertical lines.

Pulsars in the GC.—We have shown the large potential of measuring the
spin of Sgr A* with two or more pulsars. Compared to the measurement with a
single pulsar, the key advantage of the approach we proposed here is that it
does not require a pulsar in an extremely compact orbit around Sgr A*. Although
several pulsars including a magnetar and a millisecond pulsar have been found in
the GC region <cit.>, none of them are close enough
to the Sgr A* for doing the measurement described above. Here we give an
estimation of the probability of finding suitable pulsars in future
high-frequency radio surveys.

We define two possibilities P_1 and P_2 as follows. P_1 represents the 
probability of finding the innermost pulsar with an orbital period P_b≤
0.5  yr  via future instruments like SKA, which then can be used to
measure the spin of Sgr A* by timing itself alone, while P_2 represents the
probability of finding two innermost  pulsars both with orbital period P_b≤
5  yr. By combining the timing observations of these two pulsars, one may
constraint the spin of Sgr A* to ∼ 1% within five years according to our
simulations.  We assume a simple power-law distribution of pulsars near the
Sgr A*, p(r) ∝ r^-α, where p(r) is the probability density of
finding a pulsar in a unit volume at a distance r from the Sgr A*. 
Considering the large uncertainties in the GC pulsar population, the power-law
index α can vary largely depending on the detailed
models <cit.>. 

Under these assumptions, one can calculate P_1 and P_2 via

    P_1   =   1-exp[-N_4(r_0.5  yr/R_4)^3-α]
         ,
    
        P_2   =   1-[1+N_4(r_5  yr/R_4)^3-α]exp[-N_4(r_5  yr/R_4)^3-α] ,

where r_0.5  yr∼ 100  AU and r_5  yr∼ 475 
AU are the semimajor axes of the pulsar orbits with P_b = 0.5   yr and
P_b = 5  yr respectively. N_4 is the expected total number of pulsars
that can be observed within R_4=4000  AU around Sgr A*.  P_1 and P_2
for different α and N_4, motivated from current observations and
simulations, are listed in Table <ref> <cit.>. In the reasonable parameter space of α and N_4, one
always has P_1≲ P_2,  indicating that it is more likely to find two
pulsars that are suitable for measuring the spin of Sgr A*.




Discussions.—In this Letter, we proposed to measure the spin of
Sgr A* with timing observations of two or more pulsars, which does not require a
pulsar with an extremely small orbital period but can still provide similar
measurement precision by breaking the leading-order degeneracy of spin
parameters. Although the GC pulsar population has a large uncertainty at
present, it is expected to find a lot of new pulsars in future high-frequency
radio surveys towards the GC and we give a rough estimation of the probability
of finding proper systems.  It seems that we are more likely to find the systems
that are suitable for the procedure proposed here. Even if we can find a pulsar
with an orbital period P_b≲ 0.5  yr, combining a proper second
pulsar can still provide an improvement factor  ∼ 2 for the SMBH spin
measurement.

In this work we have ignored the complex environments near the Sgr A*, which
will complicate the measurement <cit.>. In general, pulsars with larger orbital periods
will suffer from more perturbations caused by the mass distributions around
Sgr A*, which may even spoil the measurement. It is suggested that one may only
use the timing data around the periastron passages and treat each periastron
passage incoherently <cit.>. This will  enlarge the measurement
uncertainty of the spin from timing a single pulsar, where second-order time
derivatives are required but hard to  measure during the short periastron
passage. However, as only the leading-order time derivatives are required in the
method we proposed here, it is expected to have a good measurement even only the
timing data around the periastron passages are used. On the other hand, by
combing the timing observations of more pulsars, one will have the ability to
model the mass distribution around Sgr A* that will be of vastly useful for
studies of the GC region.


We thank Norbert Wex, Ziri Younsi, and Fupeng Zhang for helpful discussions.
This work was supported by the National SKA Program of China (2020SKA0120300),
the National Natural Science Foundation of China (11991053), the Beijing Natural
Science Foundation (1242018), the Max Planck Partner Group Program funded by the
Max Planck Society, and the High-performance Computing Platform of Peking
University. 






 Paper Title: Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles.







liuning@njnu.edu.cn
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China



arvindm215@gmail.com
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China


		

	

§ ABSTRACT 


We explore the multi-scatter capturing of the massive dark matter (DM) particle inside the neutron star via a momentum-dependent dark matter-nucleon scattering cross-section. We find that the capturing enhanced for the positive velocity and momentum transfer dependent DM-nucleon scattering in comparison with the constant cross-section case. Further, a large capture of the DM particles can be thermalized and lead to black hole formation and, therefore, destroy the neutron star. Using the observation of the old neutron star in the DM-dominated region, we obtain strong constraints on massive DM parameters.  

		

		
		
		Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles
    
	Arvind Kumar Mishra 
	
	https://orcid.org/0000-0001-8158-6602
    < g r a p h i c s > 

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



	


§ INTRODUCTION


The various observations suggest that dark matter (DM) exists from its gravitational interaction with the baryons  <cit.>). The ongoing DM searches experiment assumes some form of the DM-baryon interactions; however, irrespective of current technological progress, the micro-physical properties of the DM particles remain an enigma <cit.>. 
The direct DM-detection experiments are sensitive for 𝒪(1-10^4) GeV DM masses. However, if the DM particles are massive, i.e., m_χ> 10^6 GeV, or lighter, m_χ< GeV, then during DM-nucleon collision, the energy transfer to the nucleus becomes small; therefore the detection experiment becomes insensitive to these masses.


In this work, we aim to explore the constraints on the massive dark matter, m_χ> 10^6 GeV, from the neutron star observations. The neutron star provides a lucrative place to study the DM-nucleon interaction, thanks to their strong gravity and density, see a recent review <cit.>. NS situated in the DM-dominated regions can be used to investigate the constraints in two ways: First, after capturing, DM particles become thermalized in the NS (via DM-nucleon scattering or DM-annihilation to the visible particles), which leads to heating of the NS and, therefore, observation of isolated of NS with high temperature can constraint the DM parameters <cit.>. Second, after thermalization, the DM particles can form a self-gravitating core, which can collapse, form a black hole, and ultimately destroy the NS; therefore, observation of the old neutron star in a DM-dominated region (central region of galaxies) can used to probe the DM microphysics <cit.>. However, most of the previous work has been done using a constant cross-section, without form factor, and the rest frame of the DM halo. Furthermore, incorporating the above effects into account, we obtained a strong constraint on the low-mass DM particles from the survival of NS <cit.>.


The constraint on the massive DM has been explored from heating for both a single and multiscatter capture formalism <cit.>. In previous works, <cit.>, the single scatter capture formalism has been applied to put constraints on the massive DM particles from the survival of NS, specifically using constant DM-nucleon cross-section without form factor. However, for actual estimation, one needs to consider the multiscatter capture formalism for massive DM particles <cit.>.  For other methods to explore massive DM particles, please see Refs <cit.>. 


In this work, we explore the constraint on the massive DM particles from the neutron star survival. To explore the capture of massive DM particles on the neutron star, we adopt from the multiscatter capture approach. Here, we focus on the velocity-dependent and momentum-transfer dependent DM-nucleon cross-section, which, to our knowledge, was not explored for massive DM particles in the context of black hole formation (see, for low mass DM constraint in our previous work <cit.>). In addition, we have also considered the general relativistic effect, the finite size effect of the nucleon, and the relative motion of the neutron star and galaxy frame in the capture rate. 


Further, considering the power law parametrization of momentum and velocity dependent DM-nucleon cross-section, we found that the capture rate is enhanced with a positive power of velocity and momentum. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel, where v_rel is the relative velocity of DM and nucleon.
Using this model, pulsar observation strongly constrain bosonic DM mass range, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. We also report that although the DM accretion is higher for σ∝ v^4_rel model, a large thermalization time puts weak constraints on the DM parameters. Interestingly, a cross-section model with a large momentum dependency, i.e., σ∝ q^4 (q represent the momentum transfer), provides no constraint on the DM parameters. 


The arrangement of our work is as follows: In section <ref>, we present a brief discussion on the DM particle capture rate using a multiscatter capture process. In Section <ref>, we discuss the black hole formation from DM accretion. Further, we present DM-Nucleon cross-section and estimate the captured particles <ref>. In Section <ref>, we present and discuss our main results. Finally, in Section <ref>, we conclude our work.



§ MULTISCATTER CAPTURE OF DM PARTICLES ON NEUTRON STAR 



We assume that the neutron star is situated in a dark matter-rich environment and explore the capture of the massive DM particles to the neutron star. In the case of massive DM particles, an energy transfer via a single interaction between the DM particles and nucleons will be insufficient to capture the DM particles. Thus, a multiple scattering will be required <cit.>. Inside the NS, the number of times by which DM scatter with the nucleons can be obtained as the ratio of the typical diameter of the NS, 2R_NS with the mean free path of the DM particles, λ_MFP, given by  N∼ 2R_NS/λ_MFP, where R_NS is neutron star radius. Assuming λ_MFP∼ 1/(n_Nσ), one can obtain, N∼ 2n_Nσ R_NS, where σ is DM-nucleon scattering cross-section, and n_N is the nucleon density <cit.>. 


Further, the probability of DM particles, p_N(τ) participating in N collisions with an optical depth, τ  can be defined via Poisson distribution (τ,N) and given by <cit.>

    p_N(τ)=2∫^1_0 y e^-yτ(yτ)^N/N! dy  ,

where 
 y is related to the incidence angle subtended by DM particles while entering the star.
The optical depth is defined as τ=3σ/2σ_sat, where σ is the DM-nucleon cross-section and σ_sat is the saturation cross-section allowed during the scattering between the DM-nucleon interaction. In the context of NS, a saturation limit is dictated via geometrical cross-section, given by  σ_geo=π R^2_NS/N_B where, R_NS is the radius of NS, and N_B is total number of the baryons, respectively. For NS, considering, R_NS=10.6 km and N_B=1.7× 10^57, we obtain σ_sat=2× 10^-45 cm^2, which we assumed to remains a constant with DM mass    [Although we have assumed the saturation limit to be a constant quantity throughout the DM mass range, in general, it depends on the DM mass, see Refs. <cit.>. Therefore our estimation are somewhat conservative.].





In the multiscatter capture process, the capture rate for N scatter, C_N is given by <cit.> [The derivation has been done  is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>]

    C_N=  π R^2_NS p_N(τ)∫^∞_v_escf(u)/u^2  w^3 g_N(w) dw  .

where g_N(w) corresponds to the probability of DM particles reducing their velocity lower than the escape velocity, v_esc after N collisions, and f(u) is the DM velocity distribution function.  In Eq. (<ref>), u is the DM velocity at very large distance, and w=√(u^2+v^2_esc). 
Therefore, the total capture rate after the N collision is obtained as

    C_total=∑^∞_N=1 C_N .

However, summation can be cut off at some value N_max corresponding to condition, p_N_max(τ)≈ 0. In our calculation, 
we approximate, g_N(w)=Θ[v_esc(1-⟨ z_i⟩β_+/2)^-N/2-w], where, β_+=4m_N m_χ/(m_N+m_χ)^2 in which, m_χ and m_N, represents the mass of the DM particles and nucleons, respectively <cit.>. Here we assume, ⟨ z_i⟩=1/2, <cit.> however for more general estimation of g_N(w), see Ref. <cit.>. 
Furthermore, the DM particle velocity distribution function is considered to be Maxwellian, given in the neutron star frame as <cit.> 

    f_0(u) = (3/2)^3/24/√(π)ρ_χ/m_χu^2/u^3_0e^-3/2u^2/u^2_0 ,

 where ρ_χ, and u_0 represent energy density, mass, and average velocity of the DM particles, respectively.


Furthermore, as on the surface of the neutron star, the DM velocity is high, so during the interaction, DM imparts large energy to the nucleons, see Ref.<cit.>. Therefore, in this analysis, we consider the finite-size effect of the nucleon during the DM-Nucleon interaction. Here, we adopt a conservative approach and assume the dipole form factor for DM-nucleon interaction (considered in the next subsection), given by <cit.>

    F(q^2)=Λ^4/(q^2+Λ^2)^2

where q≃√(2)(m_χm_N/m_χ+m_N)v_esc is momentum transfer during the DM-nucleon collision and m_N is the nucleon mass. Further, we assume Λ≃ 0.25 GeV throughout our analysis <cit.>. Furthermore, in our computation, we use the general relativistic (GR) correction, which modifies the capture rate and escapes velocity. So, using the GR correction, the capture rate and escape velocity is modifies via <cit.>

    C_N→C_N/(1-v^2_esc)  and    v_esc→√(2[ 1-(1-v^2_esc)^1/2]).

 





§ BLACK HOLE FORMATION FROM DM ACCRETION



After the accretion, the DM particles start interacting with the neutron star material and may get thermalized. In the context of the multi-capture process, the interaction of the DM particles with the neutron star is complex because, at each step, the DM particle transfers less momentum, and hence, the estimation of the thermalization time is complex.  Now, we consider the simplified approach and assume the form of the thermalization time scale as <cit.>

    t_th≃   10700μ/(1+μ)^2(10^5K/T_NS)^2(10^-45cm^2/σ)   years.

where, μ=m_χ/m_N, and σ is DM-nucleon scattering cross-section. When the thermalization time scale is smaller than the lifetime of the NS (t_th<t_NS), DM particles will be thermalized inside the NS and lie inside the  thermal radius given by 

    r_th=
    ( 9T_NS/8π G_Nρ_B m_χ)^1/2  .

where G_N is the Newton's gravitational constant,  and ρ_B is the baryon energy density. For a given density and temperature of the NS, r_th∝√(1/ m_χ), which implies that the thermal radius is smaller for massive DM particles and vice versa. 
Further, DM particles start self-gravitating whenever

    ρ_χ>ρ_B  for    r ≤ r_th.

Therefore, with the help of the above condition, we define the DM particles dictated via N_self, necessary for the self-gravitation of the DM particles inside the NS as

    N_self≃  4.8× 10^41( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^5/2( T_NS/10^5K)^3/2  .

So the condition for DM self-gravitation suggest, N_χ≥ N_Self.



Furthermore, when the number of DM particles becomes higher than those defined by the Chandrashekhar limit, then self-gravitating DM particles collapse and form a black hole. In the case of fermionic dark matter (FDM) particles, gravitational collapse requires gravity to surpass the fermi pressure. For non-interacting fermionic DM particles, the Chandrashekhar  limit is obtained as <cit.>

    N^Fermions_Chandra≃ 1.8× 10^51( 100 GeV/m_χ)^3 .

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa.
In this case, the BH formation occurs when DM particles start self-gravitating, i.e., N_χ≥ N_Self, and captured DM particles are higher than the Chandrashekhar limit,i.e.,N_χ>N^Fermions_Chandra. The mass of formed BH is obtained as M_BH∼ m_χN^Fermions_Chandra, and given by

    M_BH≃    1.8× 10^53( 100 GeV/m_χ)^2GeV.

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa. 
Furthermore, to form a BH in the bosonic dark matter (BDM) particle case, gravity must overcome pressure originating from the uncertainty relation. For non-interacting bosons, the Chandrashekhar limit is given by <cit.>

    N^Bosons_Chandra≃ 1.5× 10^34( 100 GeV/m_χ)^2 .
 
In bosonic DM case, BH forms as soon as DM starts self-gravitating, 

    N_χ≥ N_Self,  for  m_χ≤ 10^21( ρ_B/1.4× 10^14gm/cm^3)  ( 10^5K/T_NS)^3 GeV  .

In this case, the mass of formed BH is obtained as M_BH∼ m_χN_Self, and explicitly given by

    M_BH≃ 4.8× 10^43( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^1/2( T_NS/10^5K)^3/2  GeV.

From the above equation, we find that the massive DM particles and low neutron star temperature form low-mass BH and vice versa. 


After its formation, BH starts growing via accreting the surrounding material and also reducing its mass via Hawking evaporation. The BH mass evolves  as <cit.>

    dM_BH/dt≃ 4πλ_s( G_NM_BH/v^2_s)^2ρ_B v_s+(dM_BH/dt)_DM -1/1560π G^2_N M^2_BH  ,

where λ_s and v_s are the accretion eigenvalue and sound speed. In the r.h.s, the first and second term corresponds to the accretion rate obtained from baryon (Bondi-Hoyle accretion rate) and DM particles, and the last term represents Hawking evaporation. Assuming the DM accretion contribution is small, Eq. (<ref>) suggests an initial critical BH mass given as M^crit_BH=1.2× 10^37 GeV. Whenever formed BH with initial mass,M^ini_BH is larger than the critical mass, i.e., M^ini_BH>M^crit_BH, BH will grow via accretion and destroy the host NS. However, in the opposite case, i.e., M^ini_BH<M^crit_BH, black hole will evaporate and decrease its mass with the time. 


Furthermore, the value of the critical mass of the formed BH can be applied to estimate an upper limit on the mass of DM particles, which can be probed from the BH formation in the core of NS. The initial formed BH mass formed by fermionic DM particles,M_BH∼ m_χN^Fermions_Chandra, and bosonic DM particles via M_BH∼ m_χN_Self. Therefore assuming that BH grows via accreting the NS materials, we obtain the m_χ<2.5× 10^8 GeV (for T_NS=10^7 Kelvin) for BDM particles, and m_χ<10^10 GeV for the FDM particles. We emphasize that these constraints are obtained using the assumption that formed BH is growing via accreting the NS material and not considered DM accretion. Therefore, when DM accretion is taken into account (or NS temperature is changed for the BDM case), the constraint will be modified.    


 

§ DM ACCRETION FROM NON-CONSTANT DM-NEUCLEON ELASTIC SCATTERING

 

In this work, we explore the DM-Nucleon interaction, which is dependent on relative velocity and momentum transfer during the collision <cit.>, which are investigated in contest of the DM capturing on sun <cit.>, white dwarfs <cit.> and neutron stars <cit.>. 
 In this work, we parameterize the relative velocity-dependent cross-section (VDCS) as a power law form, given by <cit.>

    σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α≃σ_χN(w/u_0)^2α,

where, v_rel is the DM-nucleon relative velocity, and u_0 represent the normalized velocity. Here α depends on the DM models, and we specifically focus on the positive velocity dependence cases, i.e., α=1, and 2.  This kind of interaction is called p-wave (α=1), and d-wave (α=2) can be originated for the cases when initial state particles have relative angular momentum of 1 or 2 units, see Refs. <cit.>,<cit.>.  Here we consider the typical value, u_0=220 km/sec <cit.>. 
  

Further, we parameterize the momentum-dependent cross-section (MDCS) in the power law form as<cit.>

    σ^md_χN(q)=σ_χN(q/q_0)^2β,

where q is the momentum transfer and  q_0 is a normalized momentum. The momentum dependent cross-section can originate from various mechanisms such as dipole interaction between the DM and baryons, parity violating DM-baryon interaction, see Refs. <cit.>. Further, for calculation purposes, we assume, q_0=40 MeV <cit.>). 



Having been equipped with the form of the DM-baryon interaction, we will now calculate the probability and DM capture rate using the multicapture process.  This can be obtained by using Eq.(<ref>) and Eq.(<ref>). We have estimated the multiplescatter capture rate using constant, MDCS and VDCS in Appendices <ref>,<ref>, and <ref>, respectively. Below, we will briefly summarize the  results:


Capture rate in constant cross-section case: For constant cross-section case, the expression of probability is obtained after using Eq.(<ref>) as

    P^cont_N(τ_0)=2/τ^2_0 N![Γ(2+N)-Γ(2+N,τ_0)]

where, τ_0=3σ_χN F(q^2)/2σ_sat, and Γ(a,b) represent the incomplete Gamma function. Therefore using the above probability, C_N can be obtained as
 
    C^cont_N(τ_0)=   R^2/1-v^2_escn_χ/6u_0P^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering <cit.>. Further, the total capture rate after N collision is given by C^cont_tot(τ_0)=∑^∞_N=1C^cont_N(τ_0). In this case, an analytic expression of the total capture rate is obtained using various analytic approximations, see Refs. <cit.>. Further, to calculate the total collected DM particles, we considered DM particles to be asymmetric (neglecting DM annihilation). In this assumption, the number of collected DM particles by neutron star throughout its lifetime is given by N_χ(t_NS)=C^cont_tot(τ_0) t_NS. Here, we will not apply the analytical approximations and calculate N_χ(t_NS) numerically.  


Capture rate in MDCS case: For MDCS, the optical depth is given by τ_md=τ_0(q/q_0)^2β. So, the probability after N collision is obtained as

    P^MDCS_N(τ_0,q)=2/τ^2_0 N!( q/q_0)^-4β[Γ(2+N)-Γ(2+N,τ_0 ( q/q_0)^2β)] .

The form of C_N in the MDCS case is given by 

    C^MDCS_N(τ_0,q)= R^2/1-v^2_escn_χ/6u_0P^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)

  Further, the total capture rate after N collision is given by C^MDCS_tot(τ_0,q)=∑^∞_N=1C^MDCS_N(τ_0,q). Further, assuming the DM particles to be asymmetric, the total captured DM particles in the MDCS case is given by N_χ(t_NS)=C^MDCS_tot(τ_0,q) t_NS. In this case, we will calculate N_χ(t_NS) numerically.  

Capture rate in VDCS case: For VDCS, the optical depth is given by τ_vd=τ_0(w/u_0)^2α. So, the probability after N collision is obtained as

    P^VDCS_N(τ_0,w)=2/τ^2_0 N!( w/u_0)^-4α[Γ(2+N)-Γ(2+N,τ_0 ( w/u_0)^2α)] .

Using the above probability after simplification (see appendix <ref>), we obtain an analytic expression for the capture rate as
  
    C^VDCS_N(τ_0,w_0)= R^2/1-v^2_escn_χ/6u_0P^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)

In this case, the total capture rate after N collision is obtained as C^VDCS_tot(τ_0,w_0)=∑^∞_N=1C^VDCS_N(τ_0,w_0). Further, in the assumption that the DM particles are asymmetric, the total captured DM particles in this case is given by N_χ(t_NS)=C^VDCS_tot(τ_0,w_0) t_NS. In this case, we will calculate N_χ(t_NS) numerically. 






Furthermore, the estimation of total accreted DM particles using the general expression of capture rate and probability is tedious. Therefore, we try to restrict our calculation up to a certain accuracy as follows:  We calculate the total sum up to N=N_max, where N_max≤τ <cit.>. Further, we use a simplified expression of probability in some extreme cases of optical depth. For a single scatter (τ≪ 1) and multiscatter (τ≫ 1), the probability is approximated as  <cit.>

P_N (τ)≈ 
2τ^N/N!(N+2)+O(τ^N+1),   if   τ≪1

2/τ^2(N+1) Θ(τ-N),   if   τ≫1 

Therefore, using the capture rate expression estimated above, we calculate the accreted DM particles for constant, MDCS, and VDCS cases numerically.





§ RESULTS AND DISCUSSIONS



In this Section, we will present our results. For calculation purposes, we assume the NS parameters as M_NS=1.44 M_⊙, R_NS=10.6 km, N_B=1.7× 10^57, and v_esc=1.8× 10^5 km/sec <cit.>, otherwise specified explicitly. 



 §.§ Effect of DM-Nucleon interactions on the multiscatter probability


In Fig. <ref>, we plot the probability of the DM multiscatter capture as a function of a number of scattering, N, (left panel) and cross-section (right panel). It can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value of N, say N_max, i.e., p_N_max(τ)≈ 0. Further, fixing the DM parameters, m_χ=10^6 GeV,ρ_χ=0.3GeV/cm^3 and N_max takes values 1 and 2, and 3 corresponding to σ_χN=10^-47cm^2, σ_χN=10^-46cm^2, and σ_χN=10^-45cm^2, respectively. 
 Also, increments in the cross-section lead to an enhanced probability and, therefore, enlarge N_max. 

Further, the right panel of Fig <ref> suggests that
depending on the N, the probability becomes max at some σ_crit and vanishes on low DM-Nucleon scattering cross-section, σ_min such that σ_min<σ_crit . Nevertheless, increasing the number of scattering will shift the peak probability position and σ_min on a comparably higher cross-section.  Furthermore, we also find that increasing the DM masses decreases the probability.






In Fig. <ref>, we plot the probability of the DM capture using MDCS, σ∝ q^2β (Left panel) and VDCS, σ∝ v^2α_rel (Right panel) cases.  From the Left panel of Fig. <ref>, it can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value N_max as discussed previously. However, β=2 shows an interesting feature that probability increases first and decreases sharply after a large scattering. For our parametrization and fixing the other DM parameters, m_χ,ρ_χ and σ_χN, N_max takes values 6,30 and 1000 corresponding to β= 0,1, and 2, respectively.   Further, in the VDCS case (Left panel of Fig. <ref>), DM capturing inside the NS requires more scattering, N_max. For example, keeping DM parameters fixed, m_χ=10^6 GeV, ρ_χ=0.3GeV/cm^3 and σ_χN=10^-45cm^2, DM capturing in α=2 case required ∼ 10^9 scattering. Further, for constant cross-section, the probability decreases promptly, leading to a smaller N_max value in comparison with the MDCS and VDCS cases. It is important to emphasize that the value of N_max depends on the DM parameters; therefore, changing it will modify the N_max value.






 §.§ Accreted DM particles via multi-capture process

 Fig. <ref> shows the accreted DM particles via multi-scatter capture for MDCS (Left panel), and VDCS (Right panel) cases. We found that for both cases, the number of the accreted DM particles, N_χ, enhanced with the DM scattering, but decreased with the DM mass. For the MDCS case, N_χ increases with the positive power of cross-section, β>0 and number of scattering, N. Here we see that for β=1, N_χ is same for N=1 and N=10. This is because for such a small DM-neutron scattering and also parameters used in the calculation (see Left panel of Fig. <ref>), DM capturing happens through single scattering, and therefore  N_χ remains constant for N>1. For the β=2 case, DM capturing becomes possible via multiscattering, and therefore, N_χ increases with N until it gets saturated at N_max=1000.  Furthermore, in the VDCS case (Right panel of Fig. <ref>) and for α=1, the number of captured DM particles is equal for N=1 and N=1000 (same reason as previously). For the strong VDCS case, α=2, DM particles get captured via multiscattering, and therefore, N_χ increases till its saturation at N_max=5000. From  Fig. <ref>, we also point out that for positive MDCS and VDCS cases, the number of captured DM particles is higher than the constant cross-section case.








 §.§ Thermalization time scale


To estimate a DM thermalization time for non-constant DM-nucleon interaction, we assume as follows: In VDCS and MDCS case, we replace σ via Eq.(<ref>), and Eq.(<ref>), and the relative velocity is assumed to be the thermal velocity inside the NS, i.e., v_rel≃ v_th=√(3T_NS/m_χ). With this simplification, we plot the thermalization time scale for our various parameterizations in Fig. <ref>. We find that for constant and σ∝ v^2_rel, σ∝ q^2, t_th is remains constant but increases linearly with mass, i.e., t_th∝ m_χ. This behavior is evident from Eq.(<ref>), and Eq.(<ref>). It can be seen that for a few cases, i.e. 
σ∝ v^2_rel,σ∝ v^4_rel and σ∝ q^2, DM quickly thermalized inside the NS, and therefore, these cases are interesting in the context of the BH formation. However, for the σ∝ q^4 case, the thermalization scale is quite large; therefore, DM does not thermalize in the typical lifetime of the NS. 






 §.§ Constraining massive DM from NS collapse





Now, we will constrain the DM-nucleon parameter space represented via σ_χN -m_χ using the survival of the neutron star situated in the DM-dominated region. We consider the cold and old neutron stars whose distance from the galaxy center, central temperature, and age are estimated accurately. For this purpose, we consider pulsar J2124-3858 situated at a distance 270 pc with surface temperature, T_e<4.6× 10^5K (central temperature, T_c=2.5× 10^7K [The central temperature can be estimated using the surface data from the analytic formula discussed in Ref. <cit.> .]), t_age∼ 7.81×10^9 years. As this pulsar is close to the sun, we may approximate u_s≃ u_⊙= 230 km/sec <cit.>, and ρ_χ=0.3GeV  cm^-3, where u_⊙ represent the sun's speed relative to galactic halo. 



Fig <ref> shows a constraint on the DM-nucleon scattering cross-section for a constant cross-section using the pulsar data J2124-3858. Here, orange and red colors are excluded from the BH formation via bosonic and fermionic DM particles. The olive and brown color regions represent the BH evaporation regions for bosonic and fermionic DM particles, and in these regions, the constraint is relaxed. Further, sky-blue regions are excluded via non-thermalization of the captured DM particles, i.e., t_th>t_NS. The black dashed line is the saturation limit of the DM-Nucleon scattering cross-section, and above this line, the constraints are not valid.   

 
From Fig <ref>, we see that for BDM, the number of the accreted DM particles is high enough to form the BH across all the DM mass ranges. The higher DM particles form low mass BH, so for sufficiently large DM masses, formed BH masses become smaller than the critical mass, i.e., M^ini_BH>M^crit_BH, so BH start evaporating. Due to evaporation, constraints are relaxed; hence, the collapse of the neutron star may not constrain quite massive DM particles, i.e., m_χ>6× 10^8 GeV for bosonic DM. 
Therefore, most of the DM parameter space constraints are relaxed from the evaporation constraint. The BH formation from BDM  constrains very limited parameter spaces 10^-45 cm^2≤σ_χN≤ 4× 10^-46 cm^2 of BDM masses 5× 10^7 GeV≤ m_χ≤ 5× 10^8 GeV. Further, we find that for FDM particles, BH formation happens on high DM masses, i.e. m_χ>10^9 GeV. This is expected because overcoming the degeneracy pressure of the FDM requires more mass. Furthermore, BH formation in FDM does not constrain the DM microphysics as all the parameter space is relaxed from the saturation limit and evaporation constraints.       


 Fig. <ref> shows the constraint on the DM-nucleon scattering cross-section for the MDCS, case from the pulsar observation. The Left and Right panel corresponds to σ∝ q^2 (N_max=10) and  σ∝ q^4 (N_max=1000) cases, respectively. In this case, the DM accretion is enhanced (in comparison to the constant cross-section); therefore, the pulsar data constrain the massive DM particles. For σ∝ q^2, the pulsar data constraint  on cross-section 10^-48 cm^2≤σ_χN≤ 10^-45 cm^2 for BDM masses 10^6 GeV≤ m_χ≤ 6× 10^8 GeV and 
 2× 10^-47 cm^2≤σ_χN≤ 10^-45 cm^2 for FDM masses 10^8 GeV≤ m_χ≤ 10^10 GeV. 
 Although the pulsar data also constrains more massive DM particles, those parameter spaces are relaxed by Hawking evaporation. Furthermore, for σ∝ q^4 case, DM accretion is even higher than  σ∝ q^2 case.
 However, in this model, the thermalization time is higher than the lifetime of the pulsar, which makes the BH formation highly unlikely, and therefore, this model does not constrain the DM parameters.  






Furthermore, we constrain the velocity dependent DM-Nucleon scattering cross-section from the pulsar observation in Fig. <ref>. The Left and Right panel corresponds to σ∝ v^2_rel (N_max=1000) and  σ∝ v^4_rel (N_max=5000) cases, respectively.  For σ∝ v^2_rel, the captured DM particles are higher than the constant case; therefore, it constrains the DM parameter space more strongly. This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Furthermore, due to large positive power, the number of accreted DM particles is highest for model σ∝ v^4_rel (among all models discussed previously).  Therefore we expect that this model will put the tightest constraint. However, as seen in Fig. <ref>, the thermalization constraint strongly restricts the DM parameter space and limits the cross-section, 10^-50 cm^2≤σ_χN≤ 10^-47 cm^2 for the BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 3× 10^-46 cm^2 for the FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. 

From the above discussion, we find that the pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 






§ CONCLUSION



 A neutron star situated in a DM-dominated environment can efficiently capture the dark matter particles, and for large capturing, dark matter forms a black hole and destroys the host neutron star. In this work, we constrain the massive dark matter particles from the survival of the neutron star.

 To estimate the DM capture rate of massive dark matter particles m_χ>10^6 GeV, we adapted the multiscatter capture formalism to a non-constant DM-nucleon scattering cross-section. Adapting a model-independent approach, we assumed a power law parametrization of the velocity dependent and momentum dependent DM-Nucleon scattering cross-section, σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α and σ^md_χN(q)=σ_χN(q/q_0)^2β.
  Further, using the form factor and incorporating the general relativistic correction, we obtain analytic expressions for the capture rate in the neutron frame.
 Our estimation shows that the capture rate enhanced for the positive velocity and momentum dependent cross-section, and a number of the DM-Neucleon scattering, which leads to a black hole formation in a typical lifetime of the neutron star.    

 Further, applying the observation of pulsar data J2124-3858, we explored the constraint on the dark matter parameter space, i.e.,  σ_χN -m_χ for non-constant cross-section. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 
This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for bosonic DM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Interestingly, we also found that although for model σ∝ v^4_rel, the DM accretion is higher, but due to a large thermalization time, it put weak constraints on the DM parameters. Furthermore, a large momentum dependent cross-section model, i.e., σ∝ q^4 provides no constraint on the DM parameters. 

To the best of our knowledge, this is the first constraint on the massive DM particles from the neutron star survival using the velocity and momentum dependent cross-section (in the multiscatter capture formalism). Nevertheless, we also point out that our calculation of capture rate has been done under certain assumptions, such as non-relativistic scattering of DM-nucleon, saturation cross-section is constant with DM mass (however, varies for massive DM particle), nucleon density  (also escape velocity) is constant. After inclusion, these effects will change the constraint, and we leave these interesting aspects for the future. 




§ ACKNOWLEDGEMENT


This work is supported by the National Natural Science Foundation of China (NNSFC) under grants No. 12275134, No. 12335005, and No. 12147228. AKM would like to thank Prof. Lei Wu and Prof. Chih-Ting Lu for the useful discussions. 




§ CALCULATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA CONSTANT CROSS-SECTION



After having equipped with the probability expression (see Section), we now estimate the multiscatter capture rate using Eq.(<ref>). The derivation of Eq.(<ref>) has been done is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>. To estimate the capture rate, we apply the Maxwell distribution function in the neutron star frame, Eq. (<ref>).


In a constant cross-section case, the probability, P^cont_N(τ_0) is independent of the DM velocity. So, applying Eq.(<ref>), Eq.(<ref>), and Eq.(<ref>) the analytic expression of C_N is obtained as
 
    C^cont_N(τ)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

 where 

    f(v_N,η)= 1/η[e^-3(v^2_N-v^2_esc)/2u^2_0+2√(v^2_N-v^2_esc) η/u_0 +η^2]
    ×[-√(6π)u_0( -1+2√(v^2_N-v^2_esc)η/u_0)√(v^2_N-v^2_esc)
       (1+e^√((v^2_N-v^2_esc))η/u_0-2e^-3/2u^2_0[(v^2_N-v^2_esc)+2√((v^2_N-v^2_esc)u_0η])u_0η 
    +π(3v^2_esc +u^2_0(1+3η^2))e^3/2u^2_0[v^2_N-v^2_esc+2√(v^2_N-v^2_esc)u_0η+u^2_0η^2]
       { ( 2 Erf√(3/2η) + Erf(√(3/2)(√(v^2_N-v^2_esc)-u_0η)/u_0) 
     + Erf(√(3/2)(√(v^2_N-v^2_esc)+u_0η)/u_0)}] ,

 where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering, and erf(z)=2/√(2)∫^z_0e^-y^2dy represent the error function.
In the above equation, ρ_χ is the DM energy density surrounding the NS, and N_B is the total number of baryons inside the NS. We checked that in the rest frame of DM halo, i.e., η→ 0, and point-like structure of nucleon (neglecting the form factor), our expression of C^const_χN matches with the Ref. <cit.>.  





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA MDCS



In this case, σ^md_χN∝ q^2β, where q≃√(2)(m_χm_N/m_χ+m_N)v_esc.  So, P^MDCS_N(τ_0,q) is independent of the initial DM velocity. Using this, we obtain the analytic form of the capture rate after N scattering for the arbitrary momentum dependence cross section. The form of C_N in the MDCS case is given by

    C^MDCS_N(τ_0,q)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA VDCS



In this case, σ^md_χN∝ w^2β, where w=√(u^2+v^2_esc). Therefore the probability P^VDCS_N(τ_0,w) depends on the DM velocity, see Eq. (<ref>). Using this probability expression, no reliable analytic (or numerical) solution was obtained. Therefore, to calculate the capture rate, we follow a conservative approach and approximate  w≃ w_0=√(u^2_0+v^2_esc) in the probability expression. After applying this approximation, we can easily integrate and obtain an analytic expression for the capture rate as

    C^VDCS_N(τ_0,w_0)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)



utphys

 


 Paper Title: Anti-correlation between Flux and Photon Index of Hard X-ray Emission from The Crab.







Koothodil Abhijith Augustine
abhijithaugustine007@gmail.com

0009-0008-7888-7584]Koothodil Abhijith Augustine
abhijithaugustine007@gmail.com
Institute of Astronomy, National Tsing Hua University, Hsinchu 300044, Taiwan

0000-0002-5617-3117]Hsiang-Kuang Changhkchang@mx.nthu.edu.tw
Institute of Astronomy, National Tsing Hua University, Hsinchu 300044, Taiwan
Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan
Center for Theory and Computation (CTC), National Tsing Hua University, Hsinchu 300044, Taiwan




§ ABSTRACT 

Using Swift Burst Alert Telescope (BAT) event-mode data during Gamma Ray Burst (GRB) occurrences, we conducted spectral analysis for the Crab system. From 38 good observations, which spans over a period of 18 years from 2006 to 2023, we found that the Crab's X-ray flux does not only flicker, but also significantly anti-correlates to its spectral power-law photon index. Since emission contribution of the Crab pulsar in this energy range is small, this anti-correlation is mainly about the emission of the Crab nebula. We suggest that this anti-correlation is an observational supporting evidence for the long-standing notion that the nebula emission is due to synchrotron radiation of shocked pulsar winds in the nebula.   
































§ INTRODUCTION 




The Crab is one of the most widely studied `stable' and bright astrophysical sources in the high energy regime. Its central pulsar, PSR B0531+21, is a rapidly rotating neutron star that emits radiation across the electromagnetic spectrum (see <cit.> and <cit.> for reviews on the Crab). The Crab has been used to calibrate multiple instruments in the X-ray and soft γ-ray regime <cit.>. 

The emission from the Crab is composed of two components: from the pulsar wind nebula (PWN) and from the pulsar. The pulsar drives electromagnetic radiation through energetic flows of electron-positron pairs, which produce the pulsar component of the emission when they are still in the magnetosphere and wind zone of the pulsar. They also create a PWN shock (forward shock) and a termination shock (reverse shock) upon interacting with the interstellar medium, leading to synchrotron emission <cit.>. The synchrotron radiation from electron-positron pairs in the PWN, which manifests itself as the nebula's glow from radio to gamma-ray bands, provides insights into particle acceleration, magnetic fields, and shock processes in the PWN (see, e.g., <cit.> and <cit.>).  These two components, from the pulsar and from the nebula, can be distinguished through imaging or phase-resolved timing analyses, even in challenging bands like hard X-rays (e.g.,<cit.>). It is generally believed that, in the X-ray band, the pulsar component contributes less than 20% of the total flux, with the majority coming from the PWN <cit.>.

While the Crab pulsar is quite stable except for a slow, secular decrease in flux <cit.>, the nebula was found to vary at a level of a few percents and at a seeming 3-year time scale with data taken by instruments on board RXTE, Swift, INTEGRAL and Fermi in X-ray and soft gamma-ray bands from 1999 to 2010 <cit.> and Suzaku from 2005 to 2012 <cit.>. What causes this variation is still not clear. It should be noted that the flux and spectral power-law photon index of the Crab based on NuSTAR measurements in 2015 and 2016 <cit.> are in agreement with the values measured 42 years ago by <cit.>. It suggests that the observed variability of the Crab at time scales of years is fluctuations around a steady mean. There is no indication of a long-term change up to date.

We report in this paper our study of the Crab's variability using Swift/BAT event-mode data from 2006 to 2023. Besides the flux variation at a few percents, we found this variation is negatively correlated to its spectral power-law photon index at a statistically discernible level. Such an anti-correlation may shed some light on the cause of the observed variation. We also found that the variation time scale seems to be longer than that before year 2010. 

This paper is structured as follows. First we describe the data (Section <ref>) that is used for this study and then how the data is analysed (Section <ref>). We proceed to discuss the results obtained from the analysis in Section <ref>. Finally, we conclude the paper in Section <ref>.








§ DATA
 


 §.§ Data Selection 

The Swift Burst Alert Telescope (BAT) is a highly sensitive, large field-of-view coded-aperture telescope designed to monitor a significant portion of the sky (2.2 steradians at a 10% coding fraction) for gamma-ray bursts (GRBs) <cit.>. When a trigger, usually a GRB, is detected, the spacecraft autonomously slews to point its two narrow-field instruments, the X-ray Telescope <cit.> and the Ultraviolet Optical Telescope <cit.>, for follow-up observations. For simplicity, we do not use data during the slew period, as it requires accounting for changes in instrumental responses during this time. Since there is ample data outside the slew period, we avoid using slew data. BAT automatically records about 1000 seconds of event data during onboard triggers. We utilize these event data <cit.>[<https://swift.gsfc.nasa.gov/results/batgrbcat/>] for our analysis. We selected GRB observations by BAT that fell within a 25^∘ field of view (FOV) of the Crab Nebula to maximize the p-code fraction. We identified 39 unique observations spanning from February 2006 to August 2023 (approximately 18 years).The BAT’s localization accuracy is determined by the instrument’s partial coding fraction (p-code), the fraction of operational detectors exposed to a source at any given time and sky position. This p-code fraction is directly proportional to the instrument's sensitivity and has a maximum value of 1, indicating that all available detectors are illuminated by the desired source. The importance of the partial coding fraction lies in its impact on the counts from a source (see <cit.> for more details on p-code fraction).





 §.§ Data Analysis 


For data analysis, we adhered to the standard multi-step data reduction process using the HEASARC (version 6.30.1) sub-package FTOOLS, as outlined in the latest (version 6.3) BAT Guide[<https://swift.gsfc.nasa.gov/analysis/bat_swguide_v6_3.pdf>]. Background subtraction for BAT was achieved through a process called “mask weighting". We utilized the Crab's position (RA: 83.633114, Dec: +22.01446667) for this process <cit.>, resulting in background-subtracted light curves for the Crab. During a BAT trigger, the spacecraft will often automatically slew to the trigger location. As discussed earlier, we avoid data during the slew period and instead use data from the post-slew period, which is usually longer than the pre-slew period, increasing total source counts as the Crab moves into the BAT's FOV from an off-axis position. This rigorous process ensured that our GTI was uncontaminated by the corresponding GRB or burst that triggered the telescope to slew and record the event data. The partial coding fraction (refer Table <ref>) of the Crab in all our data ranges from 0.58 to 1, indicating the Crab never lies close to the edge of the FOV in these observations. However, one observation (ID: 00234516000) did not yield any photons from the Crab after ray-tracing and was thus excluded, leaving a total of 38 observations. This careful selection process ensured that the data we analyzed was of the highest possible quality, free from significant contamination by other sources.



The spectrum of each observation was then constructed and fitted with a simple power-law model using XSPEC <cit.> version 12.13.1. 
Due to BAT's limited spatial resolution (point spread function of ∼ 19.5 arcmin <cit.>), the Crab is not resolved.
The photon index, flux, and their chi-square values for the fitting are provided in Table <ref>. All errors are quoted at a 90% confidence level. 




§ RESUTS
 

The flux and  photon index obtained from the spectral fitting  for all the 38 observations are plotted in Figure <ref>. The mean flux is 1.98_-0.09^+0.04× 10^-8 ergs/cm^2/s and the mean photon index is 2.15^+0.06_-0.06. Because of the large uncertainty, variation in the photon index cannot be confirmed statistically (fitting with a constant yielding χ^2_ν=0.58). The flux variation is more significant with a constant fit yielding χ^2_ν=2.39. 
However, they seem to vary in an opposite way. 
We therefore plot the flux against photon index for all observations in Figure <ref> to examine their possible anti-correlation.

Apparently in Fig. <ref> one can see two outliers, at the lowest and highest flux among the 38 data points respectively.These two outliers are from observations of ID 01104692000 (outlier 1) and ID 00180977000 (outlier 2). The Spearman correlation coefficient of this distribution, when outlier 1 is excluded, is -0.70 with a p-value of 1.63×10^-6. It indicates a quite strong anti-correlation. When outlier 2 is also excluded, the Spearman correlation coefficient becomes -0.72 with a p-value of 6.29×10^-7.

The Crab count rate taken from the publicly available `Swift BAT 157-Month Hard X-ray Survey'[<https://swift.gsfc.nasa.gov/results/bs157mon/287>] is also plotted in Fig. <ref>. It shows the variation time scale changed after year 2010, consistent with the flux we derived from Swift/BAT event data. To describe the variation time scale after 2010, we use a sinusoidal function to fit the flux and photon index, excluding outlier 1. The best-fit period for flux is 5.92 year (χ^2_ν = 1.33 with a p-value of 0.12) and that for the photon index is 5.94 year (χ^2_ν = 0.37). The time scale is about two times longer than that before 2010 
<cit.>. 
These two best-fit curves also clearly show the anti-correlation.



 


§ DISCUSSION AND CONCLUSION
 


Electromagnetic radiation of the Crab nebula, from radio to GeV gamma-ray bands, is usually considered to be due to synchrotron radiation because of its non-thermal spectral shape and high degree of polarization. That beyond GeV energy is attributed to coming from inverse Compton scattering between those synchrotron radiation photons (plus cosmic microwave background) and relativistic pair plasma in the pulsar winds. The synchrotron component, however, has a photon index increasing with energy, that is, softer towards higher energy bands. That, in turn, requires the radiating electrons to have corresponding power indices in their energy distribution, also increasing with energy. There are still many unsolved issues as for how to accelerate electrons to that kind of distribution <cit.>. Nonetheless, no matter whether pulsar winds are energized on the way before reaching the termination shock by magnetic reconnection <cit.> or turbulent magnetic relaxation <cit.> in the striped pulsar winds, shock acceleration at the termination shock should still be at work to form non-thermal populations of relativistic electrons, which emit the observed synchrotron radiation.

The major finding of this study, that is, the anti-correlation between hard X-ray flux of the Crab and its spectral photon index, is likely a manifestation of synchrontron radiation due to particles accelerated by diffusive shock acceleration (DSA) in the PWN. On one hand, relativistic electrons of a power-law energy distribution, N_E, e∝ E^-p can lead to a power-law synchrotron radiation of F_ν∝ν^-(p-1/2), or, put in the photon number flux density format, N_E, γ∝ E^-(p-1/2)-1=E^-(p+1/2)=E^-Γ, where Γ=p+1/2 is the photon index. Moreover, DSA may create a power-law energy distribution of electrons described as
(Eq. (17.30) in <cit.>)

    N_E,  e∝ E^-(1-ln P/lnβ)   ,

where P is the probability of particle remaining in the acceleration region after one collision and β is the multiplicity for energy gain after one collision. We therefore have the power index p in the energy distribution of electrons to be p=1-ln P/lnβ=1+|ln P|/lnβ, noting that P<1 and β>1. On the other hand, change in the synchrotron radiation flux may be due to change in the radiating particle flux or in the magnetic field strength. The latter actually provides a possible link to the change of the photon index.  When field strength increases, not only radiation flux increases but also the remaining probability P in Eq.(<ref>) increases because of magnetic irregularity or turbulence  with increased field strength. The energy multiplicity factor β is probably not sensitive to the magnetic field strength. With a larger P, the power index p of the electron energy distribution is smaller, and so is the radiation photon index Γ. The synchrotron radiation flux is therefore anti-correlated with its photon index if the energetic electrons are accelerated by DSA and the change is due to magnetic field strength change.

The magnetic fields around PWN shock fronts are compressed interstellar fields in the course of the nebula expansion. These fields may have been modulated in some way by the Crab supernova ejecta, which are considered to have dispersed into the interstellar medium. The details of this modulation is not clear, but we noticed that the X-ray flux variation shows a roughly 3-year timescale before April 2010, when a deeper drop in flux occurred and after which the variation timescale seems to lengthen (Figure 8 in <cit.>). The new timescale is about 6 years, as can be seen in Figure <ref>. These timescales might be relic of earlier ejecta-ISM interactions.

The anti-correlation of X-ray flux and photon index in 15 keV – 150 keV we report here has actually already revealed itself in the upper panel of Figure 2 in <cit.> with a positive correlation between spectral hardness and count rates using Suzaku data. This anti-correlation supports the idea that the hard X-ray emission from the Crab is synchrotron radiation of energetic electrons accelerated by DSA and the variation is due to magnetic field strength change around the shock front. The flux and spectral variation in the emission from the Crab nebula probably does not appear only in the hard X-ray regime.  At higher energies, the Crab is not spatially resolved and the pulsar contribution to the whole (nebula plus pulsar) emission increases with energy <cit.>. It is therefore not easy to examine this variation. On the other hand, in soft X-rays or lower energy bands,  the Crab can be resolved and the nebula dominates the whole emission. Similar studies in other energy bands are very much desired. In addition, the polarization degree of these emissions may also vary in a way positively correlated to the variation of the photon index. 
This may be verified in soft X-rays by IXPE in the near future <cit.>. 


This work is supported by the National Science and Technology Council (NSTC) of the Republic of China (Taiwan) under grant 112-2112-M-007-053. The results presented here are obtained by using the data obtained by the BAT on the Swift observatory by NASA with the participation of Italy and the UK. 
We are very much grateful to Amy Lien of the University of Tampa for her help in data processing of this work.


aasjournal












 Paper Title: Einstein Probe discovery of a super-soft outburst from CXOU J005245.0-722844: a rare BeWD binary in the Small Magellanic Cloud.






A. Marino, H. N. Yang
0000-0001-5674-4664]A. Marino
marino@ice.csic.es, hnyang@nao.cas.cn
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain
Institut d'Estudis Espacials de Catalunya (IEEC), Esteve Terradas 1, RDIT Building, Of. 212 Mediterranean Technology Park (PMT), 08860, Castelldefels, Spain 
0000-0002-7680-2056]H. N. Yang
National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, D-85748 Garching bei München, Germany
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China
0000-0001-7611-1581]F. Coti Zelati
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain
Institut d'Estudis Espacials de Catalunya (IEEC), Esteve Terradas 1, RDIT Building, Of. 212 Mediterranean Technology Park (PMT), 08860, Castelldefels, Spain 
0000-0003-2177-6388]N. Rea
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain
Institut d'Estudis Espacials de Catalunya (IEEC), Esteve Terradas 1, RDIT Building, Of. 212 Mediterranean Technology Park (PMT), 08860, Castelldefels, Spain 
0000-0002-6449-106X]S. Guillot
IRAP, CNRS, 9 avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
0000-0002-6789-2723]G. K. Jaisawal
DTU Space, Technical University of Denmark, Elektrovej 327-328, DK-2800 Lyngby, Denmark
0000-0002-0766-7313]C. Maitra
Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, D-85748 Garching bei München, Germany
0000-0002-0107-5237]F. Haberl
Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, D-85748 Garching bei München, Germany
0000-0002-5790-7290]E. Kuulkers
European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk,
The Netherlands

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China

Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 
Beijing 100049, People's Republic of China

Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 
Beijing 100049, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China
0000-0002-9615-1481]H. Sun
National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China
 
Anton Pannekoek Institute for Astronomy, University of Amsterdam, PO Box 94249, 1090 GE Amsterdam, The Netherlands

0000-0002-4622-796X]R. Soria
INAF – Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy
Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia
College of Astronomy and Space Sciences, University of the Chinese Academy of Sciences, Beijing 100049, People's Republic of China
0000-0002-9725-2524]B. Zhang
Nevada Center for Astrophysics, University of Nevada Las Vegas, NV 89154, USA
Department of Physics and Astronomy, University of Nevada Las Vegas, NV 89154, USA

Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, People's Republic of China

School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, 519082, People's Republic of China

Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China


National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China

 
National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China
 
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 
Beijing 100049, People's Republic of China
 
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 
Beijing 100049, People's Republic of China
School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China
0000-0003-0440-7193]J.-U. Ness 
European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, E-28692 Villanueva de la Cañada, Madrid, Spain

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People's Republic of China

0000-0001-7115-2819]K. Gendreau
Astrophysics Science Division, NASA's Goddard Space Flight Center, Greenbelt, MD 20771, USA

0000-0002-0940-6563]M. Ng 
MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
0000-0001-7681-5845]T. Strohmayer 
Astrophysics Science Division, NASA's Goddard Space Flight Center, Greenbelt, MD 20771, USA
Joint Space-Science Institute, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA










July, 2024




ApJL


§ ABSTRACT 
On May 27 2024, the Wide-field X-ray Telescope onboard the  (EP) mission detected enhanced X-ray emission from a new transient source in the Small Magellanic Cloud (SMC) during its commissioning phase. Prompt follow-up with the EP Follow-up X-ray Telescope, the  X-ray Telescope and  have revealed a very soft, thermally emitting source (kT∼0.1 keV at the outburst peak) with an X-ray luminosity of L∼4×10^38 erg s^-1, coincident with . This super-soft outburst faded very quickly in a week time. Several emission lines and absorption edges were present in the X-ray spectrum, such as the Oxygen (0.57 keV) and Neon (0.92 keV) He-like emission lines, and deep Nitrogen (0.67 keV) and Oxygen (0.87 keV) absorption edges. The X-ray emission resembles typical nova outbursts from an accreting white dwarf (WD) in a binary system, despite the X-ray source being historically associated with an O9-B0e massive star exhibiting a 17.55 days periodicity in the optical band. The discovery of this super-soft outburst nails down  as a BeWD X-ray binary: an elusive evolutionary stage where two main-sequence massive stars have undergone a common envelope phase and experienced at least two episodes of mass transfer. In addition, the very short duration of the outburst and the presence of Ne features hint at a rather massive, i.e., close to the Chandrasekhar limit, Ne-O WD in the system.









§ INTRODUCTION
 

Be-white dwarf X-ray binary systems (BeWDs) are a subclass of X-ray binaries where a white dwarf (WD) accretes matter from a Be-type main-sequence star. Binary evolution models predict that BeWDs should be about seven times more common than Be-neutron star (BeNS) systems <cit.>. Population synthesis models <cit.> indicate that BeWDs originate from binaries initially composed of a Be star and a subdwarf O or B star. A substantial fraction of these systems can evolve either into red giants (via merging of the WD with a non-degenerated star; 60–70% of the cases), or into double WDs detectable by the LISA mission as sources of gravitational wave emission (30–40%). However, BeWDs have been relatively elusive, with only a few such systems identified during X-ray outbursts <cit.>. The emission from these sources in outburst is typically characterized by very soft X-ray spectra with no detectable X-ray emission above >2 keV, indicative of nuclear burning on the WD surface. BeWDs are indeed classified as Super-Soft Sources (SSS), a category that includes various types of X-ray systems powered by thermonuclear burning onto the surface of an accreting WD <cit.>. This category also encompasses novae during their SSS phase <cit.>. The lack of sensitive wide-field soft X-ray monitors has made the detection of outbursts from BeWDs rather challenging, especially during the rising phase.
On May 27th 2024, the Wide-field X-ray Telescope (WXT) onboard the  (EP) mission detected enhanced X-ray emission from a new transient source, initially labelled EP J0052.9-7230, in the SMC during its commissioning phase <cit.>. This source was promptly identified as  ( hereafter), following a serendipitous 52-s observation of the field by the Neil Gehrels Swift Observatory () X-ray Telescope (XRT) a few hours later <cit.>, as part of the  Small Magellanic Cloud Survey (S-CUBED; ). Two follow-up NICER observations on May 28th and 29th revealed a rapid decrease in the X-ray count rate between the two observations <cit.>. The source has a bright optical counterpart (average magnitude of G∼14.9), which has been classified as an O9-B0e star <cit.>. The long-term optical light curve shows a periodicity at ∼17.55 days, which is likely related to the orbital period of the system <cit.>.

This Letter reports on the identification of this source as a BeWD following the discovery of its super-soft outburst by EP and its X-ray evolution. In the following, we adopt the position listed in the  Source Catalog Release 2.1 <cit.>, R.A. = 00^h52^m4513, decl. = –72^∘28^'4387 (J2000.0) and a distance of 62 kpc <cit.>. All parameter uncertainties are quoted at a 1σ confidence level unless otherwise stated.








§ OBSERVATIONS AND DATA REDUCTION

In this paper, we have used observations from various X-ray telescopes, including the WXT and Follow-up X-ray Telescope (FXT) onboard EP, ,  and , as well as archival  observations (see Table <ref> for details). 



 §.§  

As a space observatory to discover and characterize X-ray transients, EP <cit.> carries two instruments, the WXT and FXT. EP was launched on January 9, 2024 into a low-Earth circular orbit of 592 km height with an orbital period of 97 minutes.
The WXT employs Lobster-eye micro-pore optics (MPO) to provide an instantaneous field-of-view of ∼3600 square degrees. The instrument comprises 12 modules, each covering about 300 square degrees. Each module includes a mirror assembly with 36 MPO optics, an optical baffle, and a focal-plane detector array of CMOS sensors.
The in-orbit calibrations of WXT have been completed, whose results are in excellent agreement with those of the on-ground calibrations (Cheng H., in preparation).
 was simultaneously detected by the CMOS detectors 14 and 37 of the WXT during observations on 2024 May 27 from 08:41:28 to 12:47:16 UTC. Data were processed using the  tool version 0.1.0. The source region was determined as a circle with a radius of 67 pixels (1 pixel ∼ 0.136) centered on the source. In the field of view of WXT, there are multiple bright objects together with their cross-arms. Therefore, we carefully selected a circle of radius 268 pixels nearby as the background region to minimize the influence of other bright objects.

The FXT was configured in partial window mode and full frame mode for the observations performed on May 28 and June 4, respectively. Data were processed using the  tool available within the FXT Data Analysis Software (). Source and background photons were extracted using a circle of radius 70 centered on the source and a closeby circle of radius 128, respectively. 



 §.§ 

The XRT observations were processed using , employing the latest version of CALDB (v. ). We extracted photons using a 20 circular region centered on the source. In the first observation (WT mode), we estimated the background with a similar-sized region far from the source. In the second observation (PC mode), we used an annular region with inner and outer radii of 45 and 150 for background estimation. We then extracted light curves and spectra through the  pipeline. For the last four observations, we combined the data (totaling 8 ks of exposure) and set a 3σ upper limit on the net count rate of 0.002 counts s^-1 (calculated over the 0.3–10 keV energy range using the  command). 
In the first two  observations considered in this work, the UVOT was operated in image mode, using the UVW1 filter in the first one and cycling through all available filters in the second one. In the last four observations, the UVOT was operated in event mode using the UVM2 filter. We used a circle with a radius of 5 radius for the source and a nearby circle with a radius of 10 devoid of optical sources for the background. For image mode data, we performed aperture photometry (properly corrected to infinite aperture) with .  
The optical brightness in the UVM2 filter varied from ∼12.6 to 13.2 mag (Vega) between May 27 and June 19, without showing any clear monotonic trend. For event mode data, we processed the files with  and  and searched for periodic signals using the unbinned event files (see Sect. <ref>).



 §.§ 

  data were processed using the  pipeline with default screening settings <cit.>. Several  light curves showed irregular count rate increases during the outburst. To minimize charged particle contamination, we reprocessed the data with , setting  to 0–5 and  to >1.5 GeV/c. Any observation segment still affected by non X-ray flares after this procedure was neglected from the following analysis. The  pipeline was used to produce spectra and background files. In particular, the  model[<https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_threads/scorpeon-overview/>] with default settings was adopted for the creation of the background file.



 §.§ 

The EPIC-pn onboard  was configured in the small window mode, while both EPIC-MOS cameras operated in timing mode.
Data were processed using the Science Analysis System (SAS). We applied intensity filters to the time series of the entire field of view in the EPIC-pn data to remove several background flaring episodes (which were also visible at low energies), resulting in a net on-source exposure time of ∼15 ks. No X-ray emission was detected at the source position. Using the  tool and assuming a circular region with a radius of 10 arcsec for the source photons along with an annulus with inner and outer radii of 10 and 20 respectively for the background, we estimated a 3-σ upper limit on the net count rate of 0.007 counts s^-1 over the 0.3–10 keV energy range. Given the non-detection in the EPIC-pn data, we did not perform further analysis on the EPIC-MOS and RGS data.



 §.§ 

The field of  was observed with the ACIS-I instrument on board the  X-ray Observatory three times on July 20, 2002 (7.7 ks), April 25, 2006 (49.3 ks), and April 26, 2006 (47.4 ks). 
We processed the data using the  software package. In 2002,  was detected at a count rate of (2±1)×10^-3 counts s^-1  (0.3–8 keV), whereas in the stacked observations from 2006 it was detected at a smaller count rate of 1.3_-0.7^+1.0×10^-4 counts s^-1  (0.3–8 keV). In this last case, about 70% of the photons were detected at high energies, within the 2–8 keV range.




§ DATA ANALYSIS AND RESULTS



 §.§ The EP/WXT light curve

The WXT light curve is shown in Figure <ref>. After removing spurious drops in count rates caused by instrumental artifacts, the analysis of the observations before and after the source detection revealed a rapid increase in emission starting around MJD 60456.8, with a linear rise at ∼0.3 counts s^-1 day^-1, peaking around MJD 60457.5 after approximately 0.6 days. This was followed by a slow linear decay at ∼0.02 counts s^-1 day^-1 until around MJD 60458.8, and then a faster decay at ∼0.23 counts s^-1 day^-1.








 §.§ X-ray spectral analysis


The X-ray flux and spectral shape of  varied significantly over hours. Therefore, we extracted time-resolved spectra over intervals with approximately constant count rates. For  observations 7204550101, 7204550102, 7204550103, and 7204550104, we divided them into 4, 6, 4, and 2 epochs, respectively, extracting one spectrum for each. The signal was largely or completely background-dominated in all , EP/FXT and /XRT observations performed after June 2, so we excluded these spectra from further analysis. The final spectral sample consists therefore of 1 FXT, 1 XRT and 19  time-resolved spectra, hereafter referred to as epochs (see Table <ref> for a summary). We grouped all the spectra ensuring a minimum of 25 counts per bin, allowing the use of the χ^2-statistics. Figure <ref> (a) shows the evolution of the extracted spectra of the source. We included data only where the source emission was above the background, which varied from spectrum to spectrum but was always within the 0.3–1.3 keV range. We note that despite the not-optimal calibration of  data below ∼0.4 keV[<https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_threads/cal-recommend/>], we decided to retain data between 0.3 and 0.4 keV due to the source bright soft emission.









We performed the spectral analysis with  v.12.14.b <cit.>. A systematic error of 1% was included, as recommended by the  team[<https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_threads/spectrum-systematic-error/>]. The interstellar absorption was modelled through the  component, with abundances from <cit.> and cross sections from <cit.>. We derived the unabsorbed bolometric fluxes between 0.001 and 10 keV using the  convolution model. We initially adopted a simple absorbed blackbody model () to fit the spectra. Although it matched the continuum reasonably well, the overall fit was poor, with significant unmodeled residuals (see, e.g., Figure <ref>, panel c). For , artifacts like fluorescence lines from Earth's atmosphere can result from improper background modeling. To check for these residuals, we re-extracted the spectrum from the brightest  snapshot using the  pipeline with the  option set to “script” instead of “file”, allowing a proper background fit with the continuum. The residuals in the best-fit model were identical regardless of the method used, indicating that the observed spectral features are real and not due to the background. 

We significantly improved all the fits by including a set of absorption edges (modelled with the  component in ) and emission lines (described with  models). 
The edges at 0.32 keV in  [See <https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_threads/arf-rmf/>] and at 0.47 keV in FXT are probably due to calibration issues and are not discussed here. Source intrinsic absorption edges were found at energies of ∼0.63, ∼0.88 and ∼1.1 keV, although they almost never appeared all together in a single spectrum. Two localised excesses at about 0.5–0.7 keV and 0.9 keV were visible in Epochs 1-14 and 1, 3-6, respectively, and modelled with  components. The inclusion of these discrete features improved the fits and flattened the residuals in all spectra. However, due to the correlation with other best-fit parameters, we could not constrain the width σ_l of either feature in each observation. Therefore, we fixed the widths to the best-fit values: 0.07 keV for the broad complex at ∼0.6 keV and 0.03 keV for the narrower line at ∼0.9 keV. 

We consistently obtained N_ H values within the range of (1–2)×10^21 cm^-2. This range is significantly larger than the expected value for the SMC, which is only ∼6×10^20 cm^-2 <cit.>. The discrepancy between our obtained and the expected N_ H values might indicate the presence of local absorption along the line of sight, as has been suggested for other SSS in the SMC <cit.>. From the blackbody normalization we estimated the radius of the blackbody emitting regions in each time-resolved spectrum, finding a wide range from about 10^3 to 3×10^4 km. The best-fit values obtained for both the continuum and the discrete features are listed in Tables <ref> and <ref>, respectively. The simultaneous evolution of the best-fit parameters can be observed in Figure <ref>. 



 §.§ Pre-outburst detections and post-outburst upper limits

The upper limit on the net count rate measured in the last four stacked /XRT observations and in the /EPIC-pn observation converts to unabsorbed bolometric fluxes of <6×10^-12 and <7×10^-13 , respectively, assuming the same parameters measured in the latest  spectrum (see Table <ref>). On the other hand, the conversion from count rate to flux in the archival  observations is more uncertain because the energy distribution of photons measured in 2006 (see Sec. <ref>) appears to be consistent with a slightly harder spectrum than that recorded during the outburst. For the flux conversion, we conservatively assume an absorbed power-law model with a photon index of Γ=2, resulting in an unabsorbed bolometric flux of ∼8×10^-15 . We emphasize that this value should be treated with great caution, as the exact spectral shape of this source in quiescence is not known.



 §.§ Searches for periodic signals


To search for periodic signals, we used the data collected when the source was brightest and corrected the photon times of arrival to the barycenter of the Solar System. We considered data of the first two  observations and the first EP/FXT observation, all restricted to the 0.3–2 keV energy band to maximize the signal-to-noise ratio. /UVOT data taken in event mode during the longest observation on June 17 were also searched. We then extracted power spectra from the unbinned event files from each dataset separately.
No statistically significant signals were detected in any of the data. The 3σ upper limits on the pulsed fraction were calculated <cit.>, resulting in: <4% up to 2 kHz in the  data; between 10-16% up to 250 Hz in the EP/FXT data; between 7-10% up to 50 Hz in the /UVOT data. For the  data, which had the highest signal-to-noise ratio, we also employed Fourier-domain acceleration search techniques using [<https://github.com/scottransom/presto>] <cit.> to search for signals up to 2 kHz. This analysis was conducted on the entire dataset of the first two observations as well as on 300-s data chunks. No statistically significant signal was detected. These results are consistent with those reported by <cit.>. 



§ DISCUSSION AND CONCLUSIONS


In this paper, we have presented the EP/WXT discovery of an X-ray outburst from the BeWD binary  <cit.> in the SMC and the follow-up X-ray observing campaign performed with EP/FXT, , /XRT, and /EPIC. The source experienced a rapid increase in bolometric luminosity from ∼4×10^33  before the onset of the outburst to ∼4×10^38  at the outburst peak in less than two days. A slower decay over one order of magnitude in luminosity was then observed in about six days, after which the system was no longer detectable by , likely due to being too faint, too soft, or both. An  /EPIC-pn observation set a stringent upper limit of <5×10^35  on the bolometric luminosity of the source 12 days after the last  detection. This indicates a reduction in luminosity by at least three orders of magnitude during that period, marking the end of the outburst. We therefore conclude that the total duration of the outburst from  ranged between 6 and 12 days, making it remarkably short. Throughout the entire outburst, the source consistently exhibited a soft spectral shape, with no emission detected beyond ∼1.5 keV. This characteristic identifies the source as a new SSS in the SMC. The lack of any additional hard component could be intrinsic, possibly due to the low density of surrounding material cleared by the companion star's wind, or it might simply be too faint given the source's large distance. The source reached a temperature of ∼125 eV in about one day since the EP/WXT detection and then exhibited a rapid decay to ∼50 eV in less than a week. The blackbody radius showed an almost opposite trend, going from a minimum of ∼2000 km coincident with the kT peak to a maximum of ∼16000 km after about 2 days. The observed trends in kT, R_ bb and L_ bol are reminiscent to what was observed in the SSS phase of several novae, analysed as well with blackbody models <cit.> and in outbursting BeWD binaries <cit.>. The initial temporary decrease in the temperature observed between epochs 1 and 3 could also be due to variability in the early SSS phase as reported by <cit.> or could be an artefact due to intercalibration issues between FXT, XRT and . However, we caution that blackbody models are only a crude approximation of the actual physics in SSS. Therefore, while the trends observed in the parameters are reliable, the specific values obtained for these parameters are less so. 

Almost all the spectra analysed in this paper showed a dense and intricate array of discrete spectral features, including four edges and two emission lines. The 0.63 keV and 0.87 keV edges can be confidently associated[We mostly refer to <https://xdb.lbl.gov/Section1/Sec_1-8.html> for the identification of these lines.] with NVII and OVIII absorption edges. The broad emission line at ∼0.63 keV could instead be due to an unresolved blend of discrete features associated with ions of both Oxygen and Nitrogen, such as NVII, OVII, and OVIII <cit.>. The emission line at 0.92 keV and the 1.2 keV absorption edges might instead be associated with ions of neon, respectively He-like and H-like. Nitrogen, oxygen and neon features are often found in the X-ray spectra of novae and SSS <cit.> and they can be considered as a signature of a WD accretor. In addition, as discussed above, the evolution of the luminosity, effective temperature, and size of the heated region also point towards the WD nature for the compact object in . However, the duration of the  outburst, i.e., about a week, stands out with respect to most novae in the SSS phase, which typically lasts tens to hundreds of days (e.g. , but see  for exceptions). The duration of an SSS phase depends primarily on the mass and chemical composition of the WD. WDs near the Chandrasekhar limit have shorter SSS phases since less accreted material is needed for a thermonuclear runaway. Conversely, higher hydrogen abundance in the accreted material slows the process <cit.>. The combination of a ∼10 days duration, a peak effective temperature of ∼120 eV and a peak luminosity above 10^38 , suggests the presence of a very massive object <cit.>. However, we caution that the existing literature may not be specifically tailored to the scenario of a WD accreting from a Be star. In addition, the presence of O and Ne features could be the signature of an Ne-O WD in the system, which is among the most massive WDs, typically exceeding 1.1 M_⊙ <cit.>. Future high-resolution optical and/or X-ray spectra might shed light on the chemical composition of the WD and potentially confirm the accreting star in  as a Ne-O WD.





The ensemble of the X-ray spectral clues gathered in this work strongly indicates that the compact object in  is a massive, possibly Ne-O, WD. We conclude that   is a BeWD binary. Such probably common but very elusive systems provide valuable insights into the evolution of binary stars, showcasing observable effects of the common envelope (CE) phase and mass transfer in massive star binaries. 
Specifically,   likely originated from a binary system initially composed of two main-sequence stars of masses of about 8M_⊙ and 6M_⊙ with an orbital period of ∼3 days. Following conservative mass transfer as computed by <cit.>, the system evolves into a 2.1M_⊙ He-star and a 11.9M_⊙ Be main-sequence star with an orbital period of about 21 days. According to the computations by <cit.>, such a He-star terminates its evolution as a Ne-O WD with a mass of 1.23M_⊙, confirming results by <cit.>. During the evolution of the He-star, mass transfer to the Be star resumes, but the mass transfer is non-conservative, due to the He-star’s convective envelope, which expands on a dynamical timescale, leading to the formation of a CE surrounding the Be star and the degenerate Ne-O core. When the envelope has expanded to the second Lagrangian point, it leads to the formation of a circumbinary disk with a mass of about 0.9M_⊙. This causes the orbital period to decrease by a factor of about 0.8, resulting in a final orbital period of about 17 days (see section 4.3.8 of  for orbital period effects of a circumbinary disk). The disk will finally be expelled from the system. We sketch the proposed evolutionary scenario for   in Figure <ref>. 

This work represents one of the very few cases where a nova outburst in such systems has been thoroughly monitored for most of its duration, from rise to decay. The unprecedented monitoring capabilities of  in the soft X-ray range are particularly well-suited to discovering super-soft outbursts from these sources, offering the opportunity to address many unresolved questions about them. 



This work is based on data obtained with , a space mission supported by Strategic Priority Program on Space Science of Chinese Academy of Sciences, in collaboration with ESA, MPE and CNES (Grant No. XDA15310000), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0550200), and the National Key R&D Program of China (2022YFF0711500). This work is also based on data obtained by: the Neil Gehrels  Observatory (a NASA/UK/ASI mission) supplied by the UK  Science Data Centre at the University of Leicester; , a 0.2–12 keV X-ray telescope operating on the International Space Station, funded by NASA; , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We thank N. Schartel for approving Target of Opportunity observations with  in the Director’s Discretionary Time and the  Science Operation Center for scheduling and carrying out the observations. Similarly, we thank the  team for approving and performing our Target of Opportunity observations.

We acknowledge the support by the National Natural Science Foundation of China (Grant Nos. 12321003, 12103065, 12203071, 12333004, 12373040, 12021003), the China Manned Space Project (Grant Nos. CMS-CSST-2021-A13, CMS-CSST-2021-B11), and the Youth Innovation Promotion Association of the Chinese Academy of Sciences. We acknowledge the data resources and technical support provided by the China National Astronomical Data Center, the Astronomical Science Data Center of the Chinese Academy of Sciences, and the Chinese Virtual Observatory. AM and NR are supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Consolidator Grant "MAGNESIA" No. 817661, PI: NR). FCZ is supported by a Ramón y Cajal fellowship (grant agreement RYC2021-030888-I). AM, FCZ and NR acknowledge support from grant SGR2021-01269 from the Catalan Government (PI: Graber/Rea). HY acknowledges support from China Scholarship Council (No.202310740002). NR acknowledges support from the ESA Science Faculty Visitor program to ESTEC (funding reference ESA-SCI-E-LE-054), where most of this work has been carried out. SG acknowledges the support of the CNES. 


Einstein Probe, , , , 

CIAO v4.16.0 <cit.>, fxtsoftware v1.05, HEASOFT v6.33.2 <cit.>, Matplotlib v3.9 <cit.>, NICERDAS v12, SAS v21.0.0	<cit.>, XSPEC v12.14.0h <cit.>



















aasjournal



 Paper Title: A HD thermal irradiated wind from outer thin accretion disk in LLAGNs.



    1*Aazad Path, New Bengali Tola, Mithapur, Patna-800001;  2 Department of Physics, Indian Institute of Science, Bangalore 560012, India
    
     
    A HD thermal irradiated wind from outer thin accretion disk in LLAGNs
        Nagendra Kumar 12nagendra.bhu@gmail.com; * current address
        
    =====================================================================
    
    § ABSTRACT 
     
     
    
     
      
    
    
    
    
    
    
    
      Evidently, low-luminosity active galactic nuclei  LLAGNs comprise of inner advective disk and outer geometrically thin disk. The wind is inevitable in LLAGNs, mainly interpreted indirect way, also the evidences are growing for the presence of wind in outer thin disk. We present a hydrodynamics HD model for wind  from outer thin disk where the main driver is the inner disk irradiation (which is parameterized by a number x in hydrostatic equilibrium equation with a slightly unbalancing role). The model works for a low intense irradiation or from a height z_s in optically thin medium. We solve the model equations in cylindrical coordinate  along the z-axis for a given radius r with assuming a tiny vertical speed v_z (≪ c_s sound speed).  The sonic point conditions assure an isobaric regime above the sonic height (z^max), in addition from the height z_f (≪ z^max) the radial pressure gradient also supports the fluid rotation, and both conjointly assure a wind ejection from the z^max with fluid speed.
    The z^max increases with x, and beyond a large z^max (say z^max_t corresponded to maximum x) there is no physical solution.
    We start the computation from outer radius r_o^thin to inner r_in^thin with Bondi mass accretion rate Ṁ_Bondi, to explore the r dependency of  mass inflow rate Ṁ and wind properties. We constrain the model by fixing Ṁ at r_in^thin from the observations of NGC 1097, and check the feasibility of model by comparing the energetics with observed bolometric luminosity. The wind is an equatorial with viewing angle i>85degree, and capable to generate red-/blue-shifted lines, it would be a general characteristics for LLAGNs.
    
    
      
    
    
    
    
    
     
     
     
    
     Key words:accretion, accretion discs - hydrodynamics - stars: winds, outflows
    
    
    
    
    
    
    § INTRODUCTION
    
    Active galactic nuclei AGNs  harbour a super massive black hole SMBH 
    which accretes the surrounding materials via an accretion disk.
    The broadband spectral energy distribution (SED) of AGNs 
    indicates that based on the luminosity  AGNs could be classified into low-luminosity AGNs (LLAGNs) and luminous AGNs. In general, L_bol < 0.001 L_Edd is characterized the LLAGNs, where L_bol is the X-ray bolometric luminosity, and L_Edd is the Eddington luminosity.
    The broadband SED of LLAGNs does not exhibit the optical-ultraviolet 'big blue bump' rather it shows a mid- or  near IR 'red bump' with a steep optical-UV slope, commonally, without having a torus
    (; ). It also reflects that underlying accretion process in LLAGNs is differed from the luminous AGNs. 
    The X-ray emission of LLAGNs can not be explained by standard geometrically thin disk, as the temperature of the inner region of disk is comparatively small to generate the X-ray emission even for the mass accretion rate Ṁ_bol
    (= L_bol/η c^2, where c is the speed of light and η is the efficiency) <cit.>.
    An advective type accretion flow <cit.>, especially radiatively inefficient accretion flow RIAF <cit.> is vital for the X-ray generation.
    Many LLAGNs display broad double-peaked (red & blue-shifted) Hα and Hβ emission lines, which affirms the presence of thin accretion disk in outer part of the accretion disk (, ). Recently <cit.> have discovered a cooler ionized gas thin disk (∼ 10^4 K) around Sgr A* within 4 × 10^4 R_g by observing a broad double-peaked H30α recombination emission line, here R_g is the gravitational radius. 
    Therefore, the LLAGNs have both type of accretion flows, a hot flow (or advective flow in the inner region of the accretion disk) and thin disk (cold flow in the outer region) with transition radii R_tr∼ 500 – 2000 R_g (). 
    However, the transition from the cold disk to hot disk is still not well understood.   
    
    
    The high spatial resolution and sensitivity of  Chandra X-ray Observatory
    facilitates to measure the nuclear X-ray emission and diffuse X-ray emission of its surrounding hot interstellar medium (ISM).
    Particularly, the diffuse X-ray emission provides the Bondi accretion radius r_acc (or sphere of influence, the gravitational pull of the SMBH dominates over the internal thermal energy of the gas) and Bondi mass accretion rate Ṁ_Bondi at r_acc by estimating the temperature (T_ism) and electron number density (n_ism) of the gas from the spectrum of diffused (unresolved) X-ray emission and the surface brightness.
    The r_acc and Ṁ_Bondi are written for SMBH of mass M_c as <cit.>
    r_acc≈ 0.05(0.8 keV/kT_ism)(M_c/10^9 M_⊙) kpc
    Ṁ_Bondi≈ 7 × 10^23(n_ism/0.17 cm^-3)(0.8 keV/kT_ism)^3/2(M_c/10^9M_⊙)^2 g s^-1
    In LLAGNs the total bolometric luminosity (L^T_bol) is many orders magnitude lesser than the luminosity generated in accretion disk with Bondi mass accretion rate (L_Bondi = ηṀ_Bondi c^2) <cit.>.
    
    L^T_bol≪ L_Bondi can be understood in RIAF  by (i) the accretion rate is much less than the Ṁ_Bondi or (ii) the accretion rate is of the order of Ṁ_Bondi with a very low radiative efficiency or with an outflow solutions.
    The both arguments are degenerate over the broadband spectrum of LLAGNs, this apparent degeneracy usually can be lifted out by submillimeter polarization and Faraday rotation measurements, which generally predicts a very small mass accretion rate in comparison to the Bondi mass accretion rate 
    near to the SMBH  (r < 100 R_g) 
    <cit.>.
    
    To describe the SED of LLAGNs the more preferable model is RIAF solution with outflow, here the outflow is parameterized in terms of mass accretion rate which decreases with decreasing radius, or Ṁ(r) = Ṁ(r_o) (r/r_o)^s, where r_o is the outer radius of RIAF (e.g., , see also ).
    Numerous advection disk based hydrodynamical (HD) and magnetohydrodynamical (MHD) numerical simulations have been performed and 
    their estimated 
    ranges of s are ∼ (0.5-1)<cit.>. 
    In general, by SED modeling in ADAF/ RIAF for LLAGNs the estimated ranges of s and r_o are ∼ (0.3 - 1) and ∼ (100 - 10^5 R_g) respectively, here the smaller r_o (< 10^3 R_g) is estimated from the observed double peaked Hα line <cit.>. 
    
    The wind outflow is inevitable in LLAGNs. It cannot be limited only in the inner region of the disk (or RIAF/ ADAF) but it would be also  launched from the outer region  (or thin disk).
    
    Usually, the winds are identified through a blueshifted UV and X-ray absorption/emission lines which require a high-resolution  spectroscopy, and  possibly a luminous AGN (atleast moderate one). <cit.> have studied a few moderate luminous AGNs (including one LLAGN) and obtained wind speed ∼ 500 km/s (i.e., for a disk wind the wind would be launched from the outer region; see also,  for a wind detection in radio loud galaxies; note the many LLAGNs are also radio loud ).
    Recently <cit.> claim the detection of an outflow from two nearby LLAGNs (NGC 1052 and Sombrero) in JWST survey by analysing the blue-shifted emission line in mid-IR band.
    A hot wind has been detected in two LLAGNs (M81 and NGC 7213) by <cit.>  by identifying blueshifted (also redshifted) emission line of Fe XXVI Lyα and Fe XXV Kα using  Chandra. The authors claim that these lines are generated in hot accretion flow which also present beyond  10^6 R_g, and disfavour the alternative explanations like AGN photoionization, stellar activities. 
    
    In LLAGNs the evidences for wind outflow are mainly indirect, e.g., the absorption column density in X-ray increases with increasing luminosity <cit.>; a presence of bisymmetric emission features in Hα-EW resolved map (, see also ); Faraday rotation in the jet (, see also for jet driven wind, ).
    
    
    In this work we aim to explore the wind launching mechanism from the outer region of the disk.
    
    
    In thin disk (AGNs), apart from magnetic-driven wind <cit.> the wind can be a line-driven wind <cit.> or Compton heated thermal wind <cit.>. A Compton heated thermal wind is almost impossible in LLAGNs as the requisite Compton temperature (T_C ∼ 10^7 - 10^8 K) can not be generated in thin disk of LLAGNs.
    
    <cit.> have shown that the line-driven wind cannot be possible in LLAGNs even for Ṁ <0.01Ṁ_Edd with M_BH <10^8 M_⊙.
    Recently, <cit.> have studied a thermal irradiation induced  wind outflow from the outer region of thin disk (r>1000R_g), where the inner region of disk irradiates the outer region. The wind is launched from the sonic height with fluid speed for a given radius. The sonic height increases with increasing the magnitude of the irradiation.  The wind is mainly an equatorial wind. They explored the wind characteristics for a range of r for the X-ray binaries with considering a constant mass accretion rate.
    In LLAGNs, the outer disk must be irradiated by the inner region. Thus, the thermal irradiation induced wind mechanism is a plausible in outer region of
    LLAGNs.
    
    
    In this paper, we extend the Paper I studied. We develop the model for the LLAGNs, and due to the low intense irradiation by the inner region, in present model the irradiation effect is accounted from the optically thin region (a region above the scale height of the thin disk). We explore the mass accretion loss due to the wind  outflow as a function of radius, for this we start the computation from the Bondi accretion radius (or outer radius of thin disk) with Bondi mass accretion rate. 
    In general, in LLAGNs the wind is ejected almost along the disk plane (with viewing angle i > 85 degree). 
    In next section, we briefly review the model and solution procedure. 
    In section  3 we describe the general model results at a fixed radius, and examine the assumptions and validity of the solutions.
    In section  4 we study the mass inflow rate (and wind characteristics) as a function of radius started from the outer radius of thin disk (or Bondi accretion radius).
    Finally in section  5 we constrain the model results with observations, followed by summary in section  6. 
    
    
    
    
    
    § MODEL
    
    
    To study the wind outflow in outer region of the disk of LLAGNs, we consider a 2.5 dimensional accretion disk formalism in cylindrical coordinates (r,ϕ,z), and the accretion flow is  
    a steady
    (∂/∂ t≡ 0) and an axisymmetric 
    (∂/∂ϕ≡ 0).
    The hydrodynamics equations are <cit.>1/r∂ (rρ v_r)/∂ r + ∂ (ρ v_z)/∂ z = 0,
    
    v_r ∂ v_r/∂ r + v_z ∂ v_r/∂ z - λ^2/r^3 + 1/ρ∂ p/∂ r + F_r  =1/ρ∂ W_rz/∂ z,
    
    v_r ∂λ/∂ r + v_z ∂λ/∂ z = r/ρ[1/r^2∂ (r^2 W_ϕ r)/∂ r + ∂ W_ϕ z/∂ z],
    
    v_r ∂ v_z /∂ r + v_z ∂ v_z/∂ z + 1/ρ∂ p/∂ z + F_z = 1/rρ∂ rW_zr/∂ r,
    v_r/Γ_3 - 1[∂ p/∂ r- Γ_1p/ρ∂ρ/∂ r] + v_z/Γ_3 -1[∂ p/∂ z - Γ_1 p/ρ∂ρ/∂ z] = 0. 
    
    Here, v_r, v_ϕ and v_z are the radial, azimuthal and vertical veocity components, and λ (=r v_ϕ) is the specific angular momentum. ρ is the mass density, p is the fluid pressure. F_r and F_z are magnitudes of the radial and vertical
    components of Newtonian gravitational force by the compact object
    respectively. Γ_1 and Γ_3 are adiabatic exponents. Equation (<ref>) is the equation of continuity, equations (<ref>-<ref>) are the
    momentum balance equations and equation (<ref>) is the energy equation.
    
    
    
    
     §.§ Model's assumption
    
    
     We adopt a gas pressure dominated regime,  p ≫ p_rad, where p_rad is the radiation pressure. The equation of state is p = kρ T/μ m_p, where k is the Boltzmann constant, m_p is the mass of proton, μ is the mean molecular weight, T is the temperature.
    The sound speed of the medium is c_s∼√(p/ρ).
    
    
    Viscosity :
    We adopt α-prescriptions of <cit.> for tangential
    shear stress W_ϕ r, it is expressed as W_ϕ r(=η r ∂Ω/∂ r)  = α p, where η =α c_s h  ρ is the dynamical viscosity, α is the Shakura-Sunyaev viscosity parameter,
    Ω is the Keplerian angular velocity, h is the scale height of the Keplerian disk at radius r.
    We also account another tangential shear stress  W_ϕ z in calculations, and approximate it in terms of W_ϕ r,  W_ϕ z/W_ϕ r≈∂Ω/∂ z / ∂Ω/∂ r.≈z/r.
    Other shearing stresses W_r z and W_z r due to the motion of v_r and v_z respectively are comparatively negligible, as v_z, v_r≪v_ϕ, and we assume that W_rz = W_zr≈ 0, or  
    
    ∂ v_r/∂ z + ∂ v_z/∂ r = 0.
    
    Here, in notation of the viscous shearing stress W_ij, the first subscript is for the direction of the stress, and the second is for the outward normal to the surface on which it acts. In addition we assume that the α-prescriptions for W_ϕ r is also valid at any height above the scale height, W_ϕ r (z) = α p(z). 
    
    Opacity, and heating & cooling : Like the Keplerian disk, we assume that the medium is optically thick and the total optical depth τ at midplane is very large, i.e., τ^tot≫ 1, where τ at height z is τ(z)= ∫_z^∞κρ dz, κ is the opacity.
    And the annular rings are in the local thermodynamic equilibrium, so the annular rings radiate like a black body. 
    As our interest of solutions is
    focused on the outer region, we consider that the opacity is due to the free free absorption σ_ff,
    is the Rosseland mean opacity.
    The heat is generated  dominantly by viscous process
    
    and disk immediately cools locally in vertical direction by black body emission. Hence, in the right hand side of the energy equation (<ref>) q^+ - q^- = 0, here q^+ is the rate of heat generation per unit volume and q^- is the rate of radiated energy density.
    
    Hydrostatic equilibrium, and external disk irradiation :
    In Keplerian disk <cit.>, it is considered that there is no flow in the vertical z-direction, i.e., the disk is in
    hydrostatic equilibrium vertically. In addition the disk has a concave shape, the inner disk can shine the outer region of the disk. In this work, we consider the irradiation of  outer disk by the inner disk. In principle, the irradiation  can
    introduce a radiation pressure p^irr_rad, 
    and can unbalance the vertical hydrostatic equilibrium.
    
    
    
    
    
    
    
    
    
    
    The irradiated flux ϵ^irr at outer radius r by the inner disk of luminosity L_bol can be  expressed by equation (<ref>). The radiation pressure is defined as  p^irr_rad∼ϵ^irr/c. We find that p^irr_rad≪p for L_bol<10^45erg/s and Ṁ∼ 0.0001Ṁ_Edd, M_c>5 × 10^6 M_⊙, here  Ṁ is the mass accretion rate, M_c is mass of the compact object. Therefore, in outer region the radiation pressure due to the irradiation is negligible in comparison to the gas pressure.
    
      The deviation from the vertical hydrostatic equilibrium of the fluid is expressed as
    
    1/ρ∂ p/∂ z= -(1-x)F_z
    
    Here,  x (< 1) is a number, and for x = 0 the disk is in vertical
    hydrostatic equilibrium. After rearranging the terms, the equation (<ref>) can be expressed for x at a given height z as, x = 1 + .(1/ρ F_zΔ p/Δ z)|_z, here Δ p = [p(z+Δ z) - p(z)] and Δ z is a small increment at height z.
    We can note here that for a given height z, the pressure will increase with x, which indicates an external heating effects as mentioned in above paragraph (see the detail discussions in Paper I).
    
    
    
    
    
    
    
     §.§ Solution procedure and Discussions
    
    
    Combining  the  equation of continuity (<ref>), the momentum balance equations (<ref>-<ref>), the
    energy equation (<ref>) and equations (<ref>)-(<ref>) we obtain
    
    
    
        ∂ v_z/∂ z[v_z^2-v_r^2/v_r(-α r)Γ_1c_s^2/v_r^2-Γ_1c_s^2]    =3 W_rϕ/ρ+ α z1/ρ∂ p/∂ z -v_r ∂λ/∂ r-v_z ∂λ/∂ z
         
        +   α r [v_zxF_z/v_r +f^r_bal -v_r^2/r- 
        1/ρ∂ p/∂ zv_zv_r/Γ_1 c_s^2]Γ_1 c_s^2/ v_r^2-Γ_1 c_s^2 
        ,
    
     where f^r_bal = -λ^2/r^3+F_r.
    Above, ∂ v_z/∂ z is expressed in terms of ∂λ/∂ r, and ∂λ/∂ z. For 
    unique solution, we compute the derivatives
    of λ(z) as a function of height at
    a given radius
    (i.e.,∂λ/∂ r, ∂λ/∂ z) using equation (<ref>) 
    by neglecting the higher order derivatives.
    In addition to obtain the unique solution one has to know one more variable prior to the computation. For this we assume that 1/ρ∂ρ/∂ r (r,z) = 1/ρ∂ρ/∂ r(r). This assumption can be justified as, since the fractional change of density between radiu r and r+Δ r.Δρ/ρ|_r = ρ(r+Δ r,z)-ρ(r,z)/ρ(r,z) is a very small quantity, so one may assume that it does not vary with height, i.e., ρ(r+Δ r,z)-ρ(r,z)/ρ(r,z) = ρ(r+Δ r,z=0)-ρ(r,z=0)/ρ(r,z=0).
    We solve the partial differentials ∂ v_z/∂ z, ∂ v_r/∂ z and ∂ c_s/∂ z  simultaneously.
    
    (a) Initial conditions & Irradiation height: 
    We solve the governing equations along the z-axis 
    at a given launching radius r from the midplane.
    To start, we assume a very small initial vertical speed on the midplane (z=0) in comparison to the sound speed, and define its magnitude in  ratio of radial velocity magnitude as, 
     v_z = f_v |v_r|, 
    here f_v is a number. We notice that the condition v_z(z=0) ≪ c_s
    assures these conditions
    v_z∂ v_r/∂ z, v_r∂ v_r/∂ r≪1/ρ∂ p/∂ r (while, 1/ρ∂ p/∂ r≪F_r already); and v_z∂ v_z/∂ z, v_r∂ v_z/∂ r≪1/ρ∂ p/∂ z.
    Therefore, at least near to the midplane, the governing equations (<ref>)-(<ref>) of the disk  become equivalent to the Keplerian disk <cit.>. 
    With this equivalency of the present formalism to the Keplerian disk, we
    initialize the flow variable to its respective Keplerian value, especially the
    outer-region solution, where the opacity comes mainly from the free-free absorption <cit.>.
    The initial values of flow variables would be a function of Ṁ, M_c and α. 
    In short, we start to solve the governing equations for wind outflow along the
    z-axis from the midplane of the disk at radius r and we initialize the flow variables to its respective  Keplerian value of the outer region. 
    
    
    We  adopt 
    a positive sign convention, i.e., the radially inward direction is negative,
    and vertical outward direction is positive. In this sign convention the radial inflow velocity v_r is negative and vertical outflow velocity v_z is positive. We consider a negative α to  ensure the angular momentum conservation
    in this sign convention as prescribed by <cit.>.  Irradiation height: In paper I, for simplicity we obtained the model solutions  with considering x (or irradiation effect) from the midplane.
    In general we found that for a given parameters set and given x
    the fluid temperature increases in such a way that the pressure and density both decrease.
    As the required energy to increase the medium temperature depends on the opacity (or mainly optically thick vs thin) of the medium, and it is measured by using flux formula of black body (∝ T^4) for optically thick medium while for optically thin medium it is measured in terms of enhancement in internal energy density (∝ T).
    However, the present formalism works only for p^irr_rad≪ p, in addition for LLAGNs (even for L_bol< 0.01L_Edd) we find that p^irr_rad≪ p_rad(=σ T_c^4/c) for r<2× 10^6R_g, Ṁ >0.0005Ṁ_Edd and M_c =10^8 M_⊙. Or another words the irradiated flux from the inner region is significantly smaller than the radiated flux of disk (σ T_c^4) at that r.
    Hence in above circumstance the model  will work only in optically thin medium.
    Therefore, we have to account the irradiation effect from the appropriate height z_s above the midplane rather than midplane (which is an optically thick and used in paper I). 
    In literature, the authors have a different choices for the height z_s,  where heat due to irradiation is deposited, e.g. disk scale height (z_s=h) or disk photosphere (τ(z=z_s) =2/3) or disk surface (Σ(z>z_s) = 0, where Σ is the surface density at height z, Σ(z)=∫_z^∞ρ dz) <cit.>.
    For convenient we choose z_s =h.
    However, in general, an effective equilibrium z_s must be established, where an irradiated energy is almost deposited into the medium, depending on the irradiated intensity and other factors of diffusing nature of irradiated photon. And below this height z<z_s there is no any irradiation effect.
    As the irradiated photon moves from optically thin to thick medium, the diffusing mean free path of this photon (=1/ρκ) will decrease and finally after travelling appropriate distance from top, the photon will deposit owns energy to the medium.    
    On the other hand, the disk radiated photons do not loose energy comparatively as it moves from optically thick medium to thin.
    Due to the uncertainty over z_s height we also  check the results with  z_s =1.5h.
    In summary, due to the low irradiation intensity the present formalism is only applicable in optically thin medium 
    with base of the wind at height z_s (from the midplane).
    We solve the model equations for z<z_s with x=0 and for z>z_s with a given x. 
    
    
    
    (b) Critical point
    For v_z(z)  = v_r(z), the equation (<ref>) has a singular point. At that z, a smooth velocity field would be obtained when the RHS of equation (<ref>) be zero, i.e.,
    
    
        3 W_rϕ/ρ+ α z1/ρ∂ p/∂ z   -v_r ∂λ/∂ r-v_z ∂λ/∂ z = -α r [v_zxF_z/v_r +f^r_bal. 
               .   -v_r^2/r-  1/ρ∂ p/∂ zv_zv_r/Γ_1 c_s^2]Γ_1 c_s^2/ v_r^2-Γ_1 c_s^2
    
     For v_r≪Γ_1 c_s^2, x ≪ 1 and with equation (<ref>)  the above condition is 
    always satisfied. Hence, ∂ v_z/∂ z is smooth at that height, where v_z(z)  = v_r(z) and v_r(z)^2 ≪Γ_1 c_s^2. (c) Solution behavior at height where v_r^2 →Γ_1 c_s^2 : 
    For v_r^2 →Γ_1 c_s^2, the corresponding height is termed as a sonic point, the equation (<ref>) reduces
    as,
    
     
        ∂ v_z/∂ z[v_z^2-v_r^2/v_r] =  -v_zxF_z/v_r -f^r_bal  +v_r^2/r + 1/ρ∂ p/∂ zv_zv_r/Γ_1 c_s^2,
    
     which has a singular point for v_z = v_r. Thus for a smooth velocity field at singular point, we have
    
     
        f^r_bal≈ -v_zxF_z/v_r  +v_r^2/r + 1/ρ∂ p/∂ z      or      1/ρ∂ p/∂ r + 1/ρ∂ p/∂ z≈ -v_r^2/r.
     
    Here, f^r_bal = -1/ρ∂ p/∂ r. For x≪ 1 and using equation (<ref>) one can have magnitude wise
    |1/ρ∂ p/∂ z| = |F_z|. Since at outer region,
    v_r^2/r<1/ρ∂ p/∂ z, we can write
    
    
        |1/ρ∂ p/∂ r| ≈|1/ρ∂ p/∂ z| = |F_z|.
    
    1/ρ∂ p/∂ r can be expressed  by using equations (<ref>), (<ref>) and (<ref>) as
    
    
        1/ρ∂ p/∂ r(v_r^2/Γ_1 c_s^2-1)=v_z ∂ v_r/∂ z+f^r_bal-v_r^2/r-v_r ∂ v_z/∂ z-1/ρ∂ p/∂ zv_r v_z/Γ_1 c_s^2.
    
     Using equations (<ref>) and (<ref>), we note v_z ∂ v_r/∂ z≈ v_r ∂ v_z/∂ z.
    Finally, we find the  relations |1/ρ∂ p/∂ z| ≈|v_z ∂ v_z/∂ z| and  |1/ρ∂ p/∂ r| ≈|v_r ∂ v_r/∂ r| by using equations (<ref>) and (<ref>) magnitudewise respectively.
    In summary, at a height where v_r or v_z is comparable to the sound speed (or sonic height/ point), we find mainly two results (i)  |1/ρ∂ p/∂ z| ≈|v_z ∂ v_z/∂ z| and  |1/ρ∂ p/∂ r| ≈|v_r ∂ v_r/∂ r|, (ii) 1/ρ∂ p/∂ r + 1/ρ∂ p/∂ z≈ -v_r^2/r.
    
    
    (d) Sign flip of ∂ p/∂ r :
    In the Keplerian disk, 1/ρ∂ p/∂ r(≪F_r (z=0))  acts in radially outward direction, so in the present sign convention its sign is positive. 
    The quantity (-F_r(z)+.λ^2/r^3|_z=0+.1/ρ∂ p/∂ r|_z=0)  flips the sign  around z = 0.92 h, which is
    equivalent to the quantity 1/ρ∂ p/∂ r (z) for a constant λ within scale height (here, it should note that in Keplerian disk   . -F_r (z=0) +.λ^2/r^3|_z=0→ 0 ).
    To compare this, we evaluate 1/ρ∂ p/∂ r  as a function of height at a fixed radius for x = 0. We notice sign flip of ∂ p/∂ r around 0.83, 0.87 and 0.90 h for f_v = 2.0, 1.02 and 0.1 respectively. Hence it is consistent with the result of Keplerian disk.
    In general, the sign flip of ∂ p/∂ r (or ∂ p/∂ r = 0) at height z = z_f can be determined, by using equation (<ref>), as
    
     
    3α c_s^2  = - (α z_f 1/ρ∂ p/∂ z-v_r ∂λ/∂ r-v_z ∂λ/∂ z). 
     
     (e) Force term in radial direction:
     In equation (<ref>), for v_r(z), v_z(z) ≪Γ c_s^2, the terms (v_r ∂ v_r/∂ r-v_z ∂ v_r/∂ z) are negligible in comparison to the λ^2/r^3 (or even, 1/ρ∂ p/∂ r). Using equation (<ref>), this terms can be expressed as ( ∂ (v_r^2 - v_z^2)/∂ r), thus for v_r^2 →Γ_1 c_s^2, where v_r = v_z, this will tend to zero. Hence, effectively the force term in radial direction can be expressed as
     
    
     F_r + 1/ρ∂ p/∂ r = λ^2/r^3.
    
    Here, we highlight with discussion point (d) that for z < z_f, ∂ p/∂ r acts opposite to the direction of gravity by compact object, while for z >z_f in the direction of the gravity. 
    
    (f) Isobaric regime and wind launching :
    As mention earlier, the external heating raises the temperature (or enhances the internal energy of fluid), and in interested region the pressure is the gas
    dominated. The increment in fluid velocity is happened due to the expense of the internal energy, and the acceleration is driven by the pressure gradient.
    At sonic point, see discussion point (c), we obtain a condition (i) which
    asserts that the kinetic energy of fluid is now comparable to the internal energy. Therefore, fluid meets to the equipartition of energy states, and there is no pressure gradient (no acceleration), i.e., above the sonic point an isobaric regime exists.  
    We term sonic height as the maximum reachable  height due to an acceleration and denote it by z^max (see paper I for the detail discussions).
    
    Later, we will find that the height z_f is far below to the height z^max, i.e., z_f ≪ z^max. Near sonic height, the 1/ρ∂ p/∂ r acts in direction of F_r, the both F_r and 1/ρ∂ p/∂ r are supporting the fluid rotations. And above the sonic height the term 1/ρ∂ p/∂ r  is absent and only F_z and F_r act on the fluid particle. Hence at z^max if 1/ρ∂ p/∂ r≪ F_r then F_r is alone able to
    support the rotation and the fluid is rotationally bound.
    
    In other case where the magnitude of 1/ρ∂ p/∂ r is significant in comparison to the |F_r|, then F_r is not able to support the rotation alone, and the fluid materials get ejected from the disk at height z^max with speed √(v_r^2+v_ϕ^2+v_z^2). Hence in last case (1/ρ∂ p/∂ r≮≮ F_r ) the wind outflow is launched at radius r from the sonic height z^max with speed v_wind = √(v_r^2+v_ϕ^2+v_z^2), otherwise the fluid is rotationally bound.  
    Above the height z^max, in present work there is a no point of interest,
     we perform calculations upto the height near to z^max. (g) Mass loss by wind :
    In paper I, we explore the general
    characteristics of wind outflow for a wide range of the launching radius, simply assuming a fixed accretion rate.
    
    
    Expectantly, the mass accretion rate will decrease with decreasing r due to a wind outflow.
    The mass accretion rate at radius r-Δ r, Ṁ (r-Δ r), can be written as (with Δ r/r≪ 1),
    
    Ṁ (r-Δ r) =  Ṁ (r) - Ṁ_out (r)
    
    where Ṁ_out = 2(2π r Δ r v_windρ_w) is mass outflow rate at radius r, ρ_w is the density at wind ejection height z^max, and Ṁ (r) is the mass accretion rate at radius r.
    (h) Energetic for wind  :
    
    
    
    
    The present model works in optically thin regime, and the driver of wind is the pressure gradient. 
    The fluid acceleration occurs on expense of the internal energy where the internal energy rises due to the irradiation.
    Hence, by comparing the enhancement in the internal energy for a given x (i.e., modeled value) with irradiated energy one can constrain the range of free parameters, like z^max and f_v, or in general, the required range of mass accretion rate at transition radius R_tr to produce the X-ray emission in RIAF for a known Bondi mass accretion rate and Bondi accretion radius.
    
    
    The vertically averaged internal energy density  for a given x, u^x, at launching radius r can be determined as
    
    u^x = 1/z^max∫3/2(c_s(z)^2)ρ(z) dz
    
    
    
    
    
    
    
    
    
    
    The irradiated flux ϵ^irr at radius r and height z_s =h by the inner region of temperature T_in at radius r_in is expressed  as
    ϵ^irr
    ≈∫ 2π r_inσ T_in^4 dr_in/4 π r^2   (1-β) C_sph = L_bol/4π r^2 (1-β) C_sph
    here, we approximate the distance between inner and outer region r-r_in to
     r for r_in≪ r. β is
     the albedo, C_sph (≈h/8r, see <cit.>)
     measures the normally irradiated energy on the surface 2π r dr at height
     h, and L_bol=∫ 2π r_inσ T_in^4 dr_in is the bolometric luminosity of the inner region.
     The total irradiated energy in annular area 2π r Δ r in time t_w  is ϵ^irrt_w (2π r Δ r), here t_w is the time interval  in which the fluid raises from height z_s to the sonic point z^max.
     By dividing the volume  (2π r Δ r(z^max-z_s)) in total energy, the total irradiated energy density is ϵ^irr/⟨ v_z ⟩, where ⟨ v_z ⟩ = (z^max-z_s)/t_w= 1/z^max∫ v_z dz is the vertically averaged v_z.
    
     For a comparison of irradiated one with modeled value, in place of using energy density we consider a flux. 
     For that we define the vertically averaged internal energy flux ϵ^x for a given x as 
    ϵ^x = ⟨ v_z ⟩ u^x
    here,⟨ v_z ⟩ is the denominator term of total irradiated energy density.
    The vertically averaged  enhancement in the internal energy flux ϵ_exess^x for a given x at launching radius r can be determined as
    
     ϵ_exess^x =   ( . 2π/z^max⟨ v_z ⟩∫3/2(c_s(z)^2)ρ(z) dz |_arbitrary x - .2π/z^max⟨ v_z ⟩ u^x|_x=0), 
      
    Here, the z^max and   ⟨ v_z ⟩ for a given x, are different from the respective value  for x = 0. 
     
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    § GENERAL RESULTS
    
    The solutions  are mainly characterized by model free parameters, the initial vertical speed (which is parameterized by f_v) and the index of external heating x, and by disk parameters Ṁ, M_c. We examine the general behavior of solutions at fixed launching radius first like paper I, for a supermassive black hole. Next we extend the paper I work by studying the decrement of mass accretion rate due to the mass loss by a wind outflow as a function of radius.
    As we noted earlier that the wind ejection/ sonic height z^max depends on x and we see later in this section that the z^max increases with x. Thus to explore the wind properties we consider z^max as a paremeters in place of x, and for a fixed viewing angle i we parameterize the z^max in ratio of r as
    
    z^max = f_z r,
    here f_z is a number, and i = tan^-1(f_z).
    To explore it, without loss of generality, we consider a SMBH of mass M_c = 10^8 M_⊙, coefficient of viscosity α = 0.1, and for fix launching radius we take r = 2000 R_g.
    
    
    
    
     §.§ Vertical disk structure for x= 0 
    
    
    As stated, the present formalism is equivalent to the Keplerian disk at least near the midplane in limit of v_z ≪ c_s. 
    Here, we examine the validation of these equivalency in vertical direction 
    first for x= 0.
    
    In the Keplerian disk, for an isothermal disk the pressure, density profile are expressed as <cit.>,  ρ(z,r) /ρ_c(r) = p(z,r)/p_c(r) = exp(-z^2/2h^2),
    here p_c (r), ρ_c (r) are the pressure and density on the midplane respectively. The density and pressure scale height both are same as h.
    In present case the disk is slightly deviated from an isothermal profile (see c_s profile in Figure <ref>) within the scale height, as an acceleration occurs on expense of internal energy. As a consequence, we obtain a different isothermal pressure and density profiles, which are expressed as (see the middle panel of Figure <ref>)
     p(r,z) = p_c (r) exp(-z^2/2 (0.93h)^2);   ρ(r,z) = ρ_c (r) exp(-z^2/2 (1.2h)^2).
    Here, we observe a different pressure (h_p) and density scale height and these are ∼ 0.93h and 1.2h respectively.
    
    In addition we notice that the  height h_p and  z_f both are related  by z_f = h_p^2/h as from the 1/ρ∂ p/∂ r curve in right panel of figure <ref>, z_f ∼ 0.87h.
    
    In general, we find that the z_f decreases with increasing f_v.  Since this analysis is independent of the launching radius r, the above result is similar to the paper I.
    
    In the right panel of figure <ref>, we show the variation of different quantities, which have a force dimension, v_z∂ v_z/∂ z, v_r∂ v_r/∂ r, 1/ρ∂ p/∂ r, F_r and F_z as a function of height z. We observe v_r∂ v_r/∂ r, v_z∂ v_r/∂ z≪1/ρ∂ p/∂ r≪F_r; also   v_z∂ v_r/∂ z≪F_z for z < h. 
    Hence within the h_p, the Keplerian limit is valid in the present formalism without external heating, and in general also for f_v < 10.
    
    
    The velocities profile v_r, v_z and c_s are shown in the left panel of
    figure <ref>. The sonic point (v_r^2 →Γ c_s^2) occurs at height around 2.2 h, or z^max = 2.2 h.
    We notice that the sonic point condition (i) of discussion point (c) 
    ( i.e., |1/ρ∂ p/∂ z| ≈|v_z ∂ v_z/∂ z|; |1/ρ∂ p/∂ r| ≈|v_r ∂ v_r/∂ r| ) is satisfied.
    For z > 0.87 h(=z_f), 1/ρ∂ p/∂ r acts in  direction of the gravity. Since, here 1/ρ∂ p/∂ r≪ F_r at z ∼ z^max, hence the disk material is rotationally bound.
    
     Disk Photosphere (r_phot): We find here that the  z^max is ∼ 2.2h, which slightly decreases with increasing f_v (even for f_v =0.001, z^max∼ 2.2h) and slightly increases with increasing r. 
    Hence, Keplerian disk photosphere is ∼ 2.2h.
    Since the Keplerian value (or here initial value) of flow variables at given r are tuned with vertical hydrostatic equilibrium (see equation (<ref>) with x=0) as h = c_s/v_ϕr<cit.> with approximating an isothermal disk or 
    
    
    p ∝exp(-z^2/2 h^2), and see <cit.> for another approximations ∂ p/∂ z≈p/h & z ∼ h.
    This is the reason the disk photosphere is almost same in unit of h over r, and the disk shape is concave as h ∝ r^9/8. 
    
    However, for this calculations in principle, we should coupled the model equation with the radiative transfer equations <cit.>, but to avoid the complexity we left this exercise for a future work.
    
    
    As, we can take any sets of initial value of fluid variables, to understand the impact of equation  (<ref>) on model solutions we take
    arbitrary value of p and ρ. 
    We find that for a given r if we change p, ρ in such a way that c_s is constant then the photosphere remains same, and if c_s increases than photosphere also increases and vice varsa. For a tuned value of p, ρ and r with equation (<ref>), if we increase r only then after sufficient large r the photosphere increases with increasing r, and it decreases with decreasing r and shrinks to zero for appropriate small r and further there is no solution.   
    
    
    
    
     §.§ Vertical disk structure for fixed x and f_v at given r
    
    As stated, for irradiation case we first solve the model equations upto  height z_s from midplane with x = 0, then for z>z_s we solve it for a given x.
    The results are shown in Figure <ref> for  x = 7.8 × 10^-9 (a limiting maximum value of x, see the curve 5 of Figure <ref> or curve 1 of Figure <ref>), z_s = h and f_v = 1. We obtain the sonic height z^max∼ 60h and z_f∼ 12.2h. 
    At z = h, we find a sharp change in fluid variables, this can be understood by horizontal shifting of the curve. If we shift these curves by z/h-1, then the sharp change is vanished and it appears similar to the respective curves of paper I. And If one does a reverse exercise on the curves which have z_s =0, then a sharp change will be appeared at z ∼ h, it is shown in the inset of Figure <ref>b  by the dashed curve which is shifted by z/h+1 (also for a comparison it is also lowered by factor 0.72) and computed with z_s =0, x=7.0× 10^-9 (z^max=55h) and rest parameters are same as the curve 1.
    
    The numerical results for density and pressure profile are shown in  Figure <ref>b.
    To measure the change due to the irradiation from the height z_s, we define the pressure scale height h_P^irr as
    
    p(z=z_s+h_p^irr)=e^-1/2 p(z=z_s),
    
    which is equivalent to disk pressure scale height for z_s = 0, i.e., h_p=h_p^irr, since for x=0 case by definition z_s = 0.
    We find that for a given x (or z^max) the h_p^irr decreases with increasing z_s which is mainly due to the x is started to act in large F_z region with increasing z_s also the initial value of c_s decreases with z_s.
    Here, h_p^irr∼ 2.5h and similarly the density scale height 
    is 0.038h (above the z_s).
    At height z = 2h, the density decreases to the ∼0.034ρ_c,  the pressure decreases slightly ∼ 0.42 p_c, and the sound speed is increased by ∼3.5 times from its midplane value, which reveals the external heating  interpretation for x (but in optically thin medium).
    In  Figure <ref>d, for z < √(h z_f) we find that v_r∂ v_r/∂ r, v_z∂ v_r/∂ z≪1/ρ∂ p/∂ r≪F_r; also   v_z∂ v_r/∂ z≪F_z. Hence, the equivalency of present formalism to Keplerian disk is still valid within the height √(h z_f), 
    and one can initialize the flow variables to its  Keplerian values.  
    
    The velocities profile v_r, v_z and c_s are shown in Figure <ref>a.
    In comparison to x=0 case, here v_r and v_z are accelerated more and at z = z^max their magnitudes are v_z(r, z=z^max) ∼ 2.5c_s(r, z=0).
    The variation of v_ϕ and escape velocity v_esc(=√(2 G M_c/√(r^2+z^2))) have been shown in Figure <ref>c, and at z=z^max, v_ϕ≫ v_r,  v_z.
    Like x=0, the sonic point condition (i) (see the discussion point (c) of section <ref>) is fulfilled, which assures a smooth solution around z^max (see the inset figure of Figure <ref>a), and also the existence of an isobaric regime beyond the z^max.
    Since, z_f ≪ z^max and at z = z^max, 1/ρ∂ p/∂ r∼ 0.035F_r (see Figure <ref>d), hence the matter will be ejected (as a wind) tangentially with speed √(v_ϕ^2+v_r^2+v_z^2). The wind is an equatorial wind with v_wind∼ v_ϕ, and it will not escape the system as v_ϕ < v_esc.
    
    
    
    
    Next we study the solution behaviours by varying x.
    We obtain the solutions for seven different x = 0, 0.5, 1.4, 3.6, 7.8, 8.36 and 8.69  × 10^-9, and notice a different sonic point height for each x which are  z^max = 2.2, 4, 10, 25, 60, 70 and 100h respectively (see, also the curve 1 of Figure  <ref>).
    We present the results in Figure <ref>, where we only study the
    vertical structure of ρ, p and ∂ p/∂ z which are shown in left, middle and right panels respectively.
    We note that for z^max< 60h, the profile of ρ and p change significantly by varying x, however for z^max> 60h the ρ and p profiles effectively do not change as their respective scale height is same to the scale height at z^max∼ 60h, e.g., h_p^irr∼ 2.5h for z^max> 60h.
    Consequently, the ∂ p/∂ z profile and the internal energy density (3/2ρ c_s^2) of fluid are effectively remained same for z^max> 60h.
    We reexamine above by studying the variation of power-law index around z=2h which is shown in the inset of respective figure. 
    The power-law index changes, from -3.0 to -1.8 for ρ, from -1.9 to -0.4 for p, and from -2.0 to -0.8 for ∂ p/∂ z, when the z^max varies from 10h to 60h. 
    Since the p and ρ profile do not change after z^max = 60h, we term this z^max as a maximum physically possible z^max and denote as a z^max_t and corresponding x as a x^max.
    
    
    In general, we find a one-one mapping between x and z^max, where z^max increases with increasing x. In another words, with the interpretation of external heating of x, the sonic point/ height raises with intensity of the external heating. 
    The x vs z^max curve is shown in  Figure <ref>, in which the left, middle and right panels are obtained by varying f_v, Ṁ and r respectively. In all panels, we note that first x increases with z^max (with having power law index ∼ 0.93 for z^max> 60h for curve 1) but after some z^max a very little increment in x leads to a large increment in z^max (see the horizontal region). 
    We find that the starting z^max of the horizontal region is same to the z^max_t (e.g., see z^max_t of Figure <ref> and here curve 1).
    The z^max_t increases with increasing either f_v, or Ṁ, or r, while x^max decreases with increasing r. Here, the curves 1, 2, 3 and 4 are for f_v = 1, 2.5, 5 and 10; the curves 1b, 1a and 1 are for Ṁ = 0.1, 0.01 and 0.001 Ṁ_Edd; and the curves 5, 1, 6 and 7 are for r = 500, 2000, 10^4, 10^5R_g respectively, and the rest common parameters are same as curve 1, or f_v =1, Ṁ = 0.001Ṁ_Edd, r=2000 R_g and z_s =h.
    
    In left panel of Figure <ref>,  z^max_t is ∼60, 85, 130 and 200h for curves 1, 2, 3 and 4 respectively.
    In addition, we find that the non-horizontal region of all curves is overlapped each other. It means that for any f_v the same amount of external heat is needed to launch the wind from a particular height. 
    To check it we compute the internal energy density of fluid for each x(∫ρ c_s^2 dz /h) and we find a similar overlapping region for all f_v cases.
    Since here initial value of all fluid variables except v_z is same, so the role of higher f_v is only to raise the wind launching height.
    
    
    
    
    
    
    
    
    
    
    
    
     §.§ Limiting value  of x and physical accessible regime of the solutions
    
    To understand the solution behaviour around x = x^max (or z^max=z^max_t), we study the fluid variables profiles (mainly p, ρ and ∂ p/∂ z) at x= x^max.
    In Figure <ref>, the upper, middle and bottom panels are for
    p, ρ and ∂ p/∂ z respectively.
    These variables are studied by varying f_v (left panel) and r (middle panel). In addition due to the uncertainty over z_s, we also study it by varying the z_s, the results are shown in the right panel, where the curves 1, 2 and 3 are for z_s  = 1.0, 1.5 and 1.95 respectively. Also mentioned earlier, here the initial value of c_s decreases with increasing z_s.
    We find that by increasing f_v or r the h_p^irr and z^max_t increase, while by increasing z_s the  h_p^irr decreases and z^max_t increases.
    Since for z^max=z^max_t the pressure scale height is maximum, hence the increment in z^max_t can be attributed by respective decrement in  ∂ p/∂ z, as for the lowest  ∂ p/∂ z the pressure scale height is highest, and consequently a smaller density scale height (the same is noted here).
    
    
    These can be also interpreted by using the equations (<ref>) and (<ref>). The equation (<ref>) asserts that for larger initial value of v_z (or f_v) or for smaller c_s the 
    ∂ p/∂ z magnitude would be smaller, see the left and right panels respectively.
    Equation (<ref>) reveals that for a given parameters set the 
    |∂ p/∂ z| would be smaller for a small F_z.
    However, this situation can not be achieved by simply varying F_z for a given parameters set, therefore we consider   
    
    three different r (see the middle panel), where the initial value of flow variables are different with c_s^2 ∝ r^-3/4 also F_z ∝z/r^3 for r ≫ z, and we obtain a similar trends as predicted.
    The curve 1 of left, curve 2 of middle and curve 3 of right panel are same to the respective curve 5 of Figure <ref>. 
    Finally, it appears that the x^max or z^max_t is associated to that pressure and density profiles where ∂ p/∂ z∝ z^∼ -0.8 and consequently ρ∝ z^∼ -1.8 for 2h<z<6h and z_s>h, and these dependencies are almost similar with variation of either f_v or r or z_s.
    
    
    
     Physically accessible regime:  For z^max>z^max_t, we noted earlier that the pressure and density profile do not change effectively, also the internal energy density does not enhanced (see Figure <ref>).  Therefore these results indicate that the model solution after z^max=z^max_t is not physical accessible.
    In addition, the later results indicate that the z^max_t is seemed to be an inflection point on the curve of flow variables as a function of z^max (see Figure <ref>, or <ref>).
    For clarity, we investigate the ρ-T_wind curve at z^max height, here T_wind is the temperature of fluid at z=z^max. 
    The results are shown in Figure <ref>, in which it is obtained, by varying f_v for fixed r (in left panel), and for a different r at fixed f_v=1 (in right panel). The curve symbols (and respective parameters) are same as the Figure <ref>.
    Here, the x (or z^max) is increasing in right to left direction, i.e., the right bottom corner corresponds to the x=0 (or z^max∼ 2.2h).
    In both panels, around z^max = z^max_t the behaviour of variations of ρ-T_wind curve with increasing x is deviated from the general trend (which is, by increasing x the p and ρ at z^max decrease in such a way that T_wind increases). And for z^max≫ z^max_t, where x∼ x^max, it again shows the general trend with smaller slope, which is an unphysical situation. Hence these results again emphasize that the model solution after z^max=z^max_t is not of a physical interest.
    
    
    Interestingly, as noted earlier, for a given parameters set at z^max = z^max_t the minimum wind density is  independent of f_v also independent of the magnitude of z^max_t, here its magnitude is 1.6 × 10^-16 g cm^-3. 
    Hence, the minimum wind density only depends on F_z for a given z_s. 
    Therefore, for a different r it will change in same ratio of ρ_c (initial value of ρ) and F_z, or the minimum wind density ∝r^-15/8r^-3, where the initial value of ρ∝ r^-15/8 for a given Ṁ, M_c, α, and F_z ∝ r^-3.
    We find the minimum wind density in same ratio (shown in  right panel), and it is ∼1.3 × 10^-13 and 0.6 × 10^-19 g cm^-3 for r= 500 and 10^4 R_g respectively.
    
    
     Disk photosphere &z^max_t : Since the z^max_t mainly depends on f_v, c_s and F_z; and the disk photosphere is associated  to the equation (<ref>) or, in general, it depends on c_s, F_z (see section <ref>). Next we explore the connection existed  between z^max_t and r_phot.
    For this we consider  a fixed f_v (=1), and r (=2000R_g, so F_z is also). And for simplicity we take z_s = 0, however we note that the results are qualitatively same for z_s > h also.
    For c_s, we take arbitrary values of ρ and p of this form, ρ_c  × a^15/8 and  p_c × a^21/8 respectively, where a is the number.
    The results are shown for 9 different a in Figure <ref>, where the curves 1, 2, 3, 4, 5, 6, 7, 8 and 9 are for a = 0.01, 0.1, 0.5, 1, 2, 10, 100, 1000 and 5000 respectively.
    Here, the curve 4 is for the Keplerian disk. 
    Since the r_phot is the height z^max for x =0, and in the x vs z^max curve it corresponds to the z^max of vertical region (see Figure <ref>). For curves 1, 2, 3, 4, 5, 6, 7, 8 and 9 the r_phot are ∼ 0.4, 0.9, 1.7, 2.2, 2.8, 5.2, 12.3, 28.9 and 52.1h respectively.
    For all curves the z^max_t is ∼ 55h, however the corresponding x^max slightly decreases with increasing c_s.
    Particularly for curve 9 the disk photosphere is almost equal to the z^max_t, it means that without any irradiation/ external heating the disk fluid could rise up to the height z^max_t, only one has to increase the sound speed of the fluid. If one further increases the c_s from the maximum c_s (where r_phot= z^max_t), the obtained x vs  z^max curve is similar to curve 9 with large  r_phot. 
    Since, the horizontal region of  x vs  z^max curve is not a physical accessible regime.
    In the present model the z^max_t is the maximum height  where the disk can hold the fluid with maximum c_s in hydrostatic equilibrium. 
    
     §.§ 1/ρ∂ p/∂ r& shear stress at large height
    
    As the 1/ρ∂ p/∂ r flips the sign at height z_f and above this height it starts to support the fluid rotation along with F_r.
    We first examine the variation in v_ϕ with height 
    in view of the equation ( <ref>). To examine at large height (z≫ h), we consider r=10^4 R_g, x = 1.5× 10^-9 and the rest parameters are same as the curve 6 of Figure <ref>. 
    The results are shown in upper left panel of Figure <ref>, here z_f∼ 55h and z^max∼ 240h. In which the solid curve is for a calculated one, and dotted-dashed and dashed curves are an analytic one which is computed using the relations v_ϕ^2 =rF_r and v_ϕ^2= rF_r(1+1/ρ∂ p/∂ r/F_r) respectively, however for later case the 1/ρ∂ p/∂ r is taken from the calculations. At large height the v_ϕ differs significantly from the analytic one when the gravity is supported alone the rotation.
    
    Next we examine the sign flip of 1/ρ∂ p/∂ r at given r. For this we compute the p at two adjacent annular radii  around r=10^4R_g, r ±Δ r. Therefore this analysis also checks 
    
    the consistency of the numerical set up of the model with radial (+ vertical) grid points.
    The results are shown in the right panel of Figure  <ref>, in which the red, blue and green solid curves are for Δ r = +r/800, 0 and -r/800 respectively, and the rest parameters are same as upper left panel. The left and right inset figures are for z<z_f(4<z/h<6) and z>z_f (76<z/h<78) respectively.
    The pressure gradient acts in radially outward direction in left inset (or p(r+Δ r,z)<p(r-Δ r,z)) while in right inset in radially inward (or p(r+Δ r,z)>p(r-Δ r,z)), i.e., it flips the sign at some height.
    Hence, although we 
    solve the equation along the z-axis for a given r, but the results are approximately consistent with one which would have obtained it with having also 
    radial grid points.
    
     Shear stress: 
    Since the shear stress generates due to the differential rotations, to check the consistency of the solution, we examine the presence of differential rotation at large height by computing the v_ϕ as a function of height for three adjacent annular radii. The results are shown in left bottom panel of Figure  <ref>, where the curves symbol and parameters set are same as the right panel. We find that the differential rotations are present at large height, and ∂ v_ϕ/∂ r(z) (or ∂Ω/∂ r(z)) decreases  with increasing z.
    In present model, we assume that the  α-prescription for viscosity within scale height is also valid for any higher height.
    For consistency, we examine a turbulent eddy of size h'(z) (≈c_s(z)/v_ϕ(z)r ) around a higher height, say z=100h, we find that the pressure decreases very slowly inside the eddy, so within this eddy one can approximate  p ≈ c_s h' ρ r ∂Ω/∂ r|_eddy. Hence throughout the height W_ϕ r = α p assumption is consistent in the solution. And, W_ϕ r will decrease with height as p, and it becomes almost negligible at wind injection height z^max, hence the wind is non-viscous. 
    
    
    
     §.§ General Wind Characteristics
    
    We now study the flow variables only at the sonic height (or explore the wind characteristics) by varying either f_v or Ṁ or r. 
    The results are shown in Figures <ref> and <ref>, here the curves symbol and parameters are same as the Figure <ref>.
    Figure <ref>a shows v_z (solid curves) and v_r (dotted curves).
    The ρ/ρ_c (solid curves) and p/p_c (dotted curves) are shown in Figure <ref>b, and the Figure <ref>c and <ref>d present
    
    T_wind and ϵ^x_excess respectively.
    In Figure  <ref>, the panel [a] shows v_z; the panel [b] is for v_wind (solid curves), v_ϕ (double dotted-dashed curves) and v_esc (single dotted-dashed line); The panels [c], [d], [e] and [f] are for T_wind, ρ/ρ_c, |1/ρ∂ p/∂ r|/F_r and ϵ^x_excess respectively. 
    
    We note that around z^max=z^max_t almost all flow variables are started to change the variation pattern from the previous one (i.e., for z^max < z^max_t), e.g., for curves 1, 2, 4, 1a and 1b at height z^max_t the v_z, v_r and T_wind change from increasing behaviour to almost constant. Hence, the z^max_t is seemed to be an inflection point on these curves. 
    We find that in physically accessible regime (or z^max<z^max_t) for a given z^max, in general, the v_z, v_r and T_wind increase with either increasing f_v, or increasing Ṁ, or decreasing r.
    However, ρ/ρ_c increases with increasing either f_v or Ṁ or r. Consequently, the required external flux or corresponding ϵ^x_excess increases with either increasing f_v, or increasing Ṁ, or decreasing r. In addition, ϵ^x does not follow 1/r^2 (like ϵ^irr) dependency but it decreases faster than this, e.g., here ϵ^x(r=500R_g)/ϵ^x(r=10^4R_g):ϵ^x(r=2000R_g)/ϵ^x(r=10^4R_g):ϵ^x(r=10^5R_g)/ϵ^x(r=10^4R_g)∼ 6000:100:1/900.
    In another words, to raise the fluid at similar height in unit of h one needs larger external flux (bolometric flux) for smaller r in comparison to the large r.
    
    
    
    
    
    
    
    
    
    
    We find that v_wind∼ v_ϕ, and v_wind increases with decreasing r. The v_wind exceeds from v_esc at z^max∼ 550h and 395h for curves 6 and 7 respectively. Since for curve 6 the z^max_t is ∼ 650h, hence
    
    for considered parameter sets
    for r<10^4 R_g the wind will not escape the system. In addition for larger r (r>10^4 R_g), the wind will escape the system from the smaller height.
    Within the physical accessible regime, at given  z^max the ratio |1/ρ∂ p/∂ r|/F_r increases with increasing either f_v or Ṁ or r. It means that the wind will be started to launch from smaller z^max either for larger f_v or Ṁ or r.
    As noted earlier (see left panel of Figure <ref>), the density for z^max = z^max_t is constant for curves 1, 2 and 4, however the pressure and temperature have larger value for the large f_v. It is mainly because of that at z = z^max firstly  one reaches an equipartition of energy state and secondly v_z and v_r increase with increasing f_v which will increase the pressure so also temperature.
    
    For a given parameters set, at some z^max the T_wind is same to the T_c, and we term this  z^max as a z^max_temp, and for z^max<z^max_temp, T_wind < T_c.
    We find the range of z^max_temp for considered parameter sets is ∼(8-15 h). In  general z^max_temp  decreases with increasing either f_v or r or Ṁ. Although in this range of z^max, T_wind < T_c, the value of x is significant or to reach the z^max_temp  a significant  amount of irradiated flux is required (see panel f).
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
     Irradiation equilibrium height z_s:
    Due to the uncertainty over z_s, in Figure <ref> we study the wind characteristics for four different z_s. Here,  curves 10, 1, 8 and 9 are for z_s = 0, 1, 1.5 and 1.95h respectively. 
    We find that z^max_t increases with increasing z_s.
    The z^max_t is ∼ 55, 60, 70 and 110 h for curves 10, 1, 8 and 9 respectively. 
    In physically accessible regime, for  a given z^max  the T_wind increases with increasing z_s while the density decreases.
    Like density,  for  a given z^max the ϵ^x_excess decreases with increasing z_s, or in another words for larger z_s one needs a comparatively small irradiated/ external flux to launch the wind from the almost same height.
    Particularly for curve 10 we still use an optically thin approximation while in this case one should also include optically thick  approximation. Hence the curve 10 does not show actual things, and here it is presented only for a comparison purpose. 
    
    
    For the region 2.2h< z^max <3.5h we have  (1/ρ∂ p/∂ r)/F_r <0.001, here the fluid is either rotationally bound or just in the position to launch the wind. One can exclude this region for the wind analysis. Therefore,  the  ϵ^x_excess =ϵ^irr line will  intersect only
    some range of curves among these series of curves (which is obtained by varying  z_s). One may approximately argue in reverse way that for a given ϵ^irr these range of z_s is viable, or in another words for a given ϵ^irr an equilibrium height (from the midplane) 
    would be established at this z_s.
    For example, for L_bol =8.5 × 10^41ergs^-1 from equation (<ref>),  ϵ^irr∼1.2× 10^4ergcm^-2 s^-1 at r=2000R_g and Ṁ =0.001Ṁ_Edd, and for this ϵ^irr the probable range of z_s is ≈ (1.5±0.3)h.
    However, as noted earlier the ϵ^x_excess decreases faster than the  1/r^2 for a fixed z^max in unit of h, for a given L_bol the z_s would be different for  different r. In general,  the z_s will decrease with increasing  r for a fixed z^max in unit of h and for a given L_bol. 
    
    
    
    
    
    
    
    
    § MASS INFLOW RATE AND WIND  OUTFLOW
    
    
    We compute the mass accretion rate as a function of r in presence of wind outflow for a wide range of the model free parameters. 
    In next section we  compare the model results with observations for a LLAGN source NGC 1097, which has the estimated SMBH mass 1.2 × 10^8 M_⊙ (, ).
    To explore the general results, here also we  consider  M_c = 1.2 × 10^8 M_⊙. We take  two different outer radius for thin disk, r_o^thin (∼ r_acc) = 5 × 10^5 and 2 × 10^6 R_g (which corresponds to the hot ISM temperature 1.6 and 0.4 keV respectively, see equation <ref>), as for many LLAGNs (including this source) there is no estimation for r_acc in the literature.
    We consider three different Bondi mass accretion rate at radius r=r_o^thin, Ṁ_Bondi = 0.01, 0.05 and 0.1 Ṁ_Edd, in which (0.01, 0.05Ṁ_Edd) and (0.01, 0.1 Ṁ_Edd) are for r_o^thin = 5 × 10^5 and 2 × 10^6 R_g respectively. And the hot ISM electron number densities are n = 4.1 × 10^2 and ∼ 10^2cm^-3 for Ṁ_Bondi = 0.05 and 0.1 Ṁ_Edd respectively (see equation <ref>).
    We obtain the model solutions from the outer radius to inner radius r_in^thin∼ 10^3 R_g ().  
    
    
    
    
    
    
    The general results are shown in left (panel I) and middle (panel II) columns of Figure
    <ref> which are for r_o^thin = 5× 10^5 and 2× 10^6R_g respectively. Here, the curves 1, 2 and 3 are for f_v = 1, 4 and 8, and the suffix a and b are for f_z = 0.1 and 0.2 respectively, and the panels IA, IB, IIA and IIB are for Ṁ_Bondi = 0.01, 0.05, 0.01 and 0.1 Ṁ_Edd respectively.
    The rows (i), (ii)u, (ii)l, (iii)u, (iii)l and (iv) are for Ṁ, n_e(wind), n_e(dis k), T_wind, T_c and ϵ^x_excess respectively, here suffix u and l stand for upper and lower panels, and n_e=ρ/μ m_p is the hydrogen number density. And respective x and v_wind are shown in the rows named (v) and (vi) of Figure <ref>.
    In Figure  <ref>, we show the respective z^max in unit of h.
    In rows (iv) the dotted dashed black curves i1 and i2 are for ϵ^irr, in which the curve i1 is computed for L_bol = 8.5× 10^41ergs^-1 (the bolometric luminosity of NGC 1097 estimated by <cit.>) using equation (<ref>). And the curve i2 is obtained by raising the curve i1 with factor 10 to incorporate the uncertainties due to the factor C_sph<cit.> and due to the unspherical region of inner disk emitting region. 
    
    In present model, the wind characteristics mainly depend on z^max in unit of h. However, for the observation point of view, we consider f_z = 0.1 and 0.2 which reflects the viewing angle i∼ 84.2 and 78.6 degree respectively.
    Since, in the Keplerian disk  r/h(r)∝ r^-1/8Ṁ^-3/20, the z^max (in unit of h) will increase with decreasing r, particularly z^max(r=r^thin_in)/z^max(r=r^thin_o)|_in unit of h∼ 2.1 and 2.5 for panels I and II respectively for constant Ṁ (which is nearly true for curves 1a of all panels, see Figure  <ref>).
    This is the reason, where we note that,
    in general, the Ṁ decreases faster around radius r_o^thin and  comparatively very slower around radius r_in^thin.
    For  example, in panel IA the Ṁ∝r^2.3 and  r^0.13 (at ∼ 2.5× 10^5 R_g), ∝r^0.3 and  r^0.054 (at ∼ 10^4 R_g) for curves 3b and 1b respectively.
    As noted in Figure <ref> and <ref>, for a given z^max, the wind density is larger either for larger f_v  or for larger Ṁ or smaller r with rest parameters are fixed, and also it decreases with increasing z^max (or f_z). These trends would be also true for Ṁ_out (see equation <ref>). Also, for considered parameter sets we have v_wind∼ v_ϕ, or v_wind=√(c^2/r) (r is in R_g). 
    Consequently, 
    Ṁ decreases more  either by increasing f_v or by decreasing f_z or for larger Ṁ at fixed r (e.g., see at r_in^thin).
    In addition, we find that for a given parameters  Ṁ decreases more  by increasing r_o^thin (see panels I and II).
    
    
    We find that the n_e(disk) and T_c both are deviated from the r dependency of the Keplerian disk, 
    as here for all curves the Ṁ is not a constant. However, at given r both will have to take
    the Keplerian values, therefore the n_e(disk) and T_c of curve 1a are larger (due to the higher Ṁ) than the respective values of curve 1b, the same is true for the curves 2 and 3 of all panels.
    Interestingly, we note that for curves 3b (also curves 2b) the n_e (wind) initially decreases with decreasing r then after some r it starts to increase like other curves. It is mainly due to a sharp decrement of Ṁ in this range of r, which leads a comparatively lower value of n_e (disk) and which further makes a smaller h (or in another words, the fixed z^max corresponds to a more times of h).
    
    However, in those ranges the respective T_wind is deviated slightly from the
    general trend while ϵ^x_excess more, see by comparing the curves with suffixes a and b.
    The T_wind and  ϵ^x_excess of curves with suffix a are comparatively higher than the magnitude of respective curves with suffix b. 
    For a considered parameter sets, on averaged the n_e(wind) is almost 10^10  smaller than the respective n_e(disk), T_wind∼10^4K (a completely ionization temperature for the hydrogen) occurs around the ranges of r (1-2× 10^4 R_g) and (6-10 × 10^4R_g) for curves with suffixes a and b respectively.
    For few curves (e.g., 1a, 1b, 2b) near to r_in^thin the wind parameters (like n_e(wind), T_wind) start to decrease with decreasing r, which corresponds to the situation z^max > z^max_t, so that range of r of corresponding curves is not a physical interested situations.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    § COMPARISON WITH OBSERVATION
    
    NGC 1097 is a LLAGN in LINER (Low-Ionization Nuclear Emission-Line Region), 
    it exhibits broad double-peaked Balmer (Hα) lines which was monitored more than two decades.  
    These broad lines are believed to generate in thin accretion disk at r ≈ 550 R_g, hence the outer accretion disk 
    is a thin disk <cit.>.
    <cit.> had explained the broadband spectral energy distribution (SED) in an inner RIAF plus an outer thin disk with transition radius R_tr = 450 R_g, and having SMBH mass M_c =1.2 × 10^8 M_⊙. Their estimated mass accretion rate at R_tr is ∼ 0.0064 Ṁ_Edd, and computed bolometric luminosity L_bol = 8.5 × 10^41 erg s^-1<cit.>.
    However, in NGC 1097 an indirect evidence for wind is growing, e.g., from ALMA observations <cit.> have claimed that at  r = 40pc (or ∼ 7× 10^6 R_g) the molecular gas inflow rate is 0.033Ṁ_Edd and molecular and ionized gas inflow rate is 0.073Ṁ_Edd.
    On other hand one has from SED (optical to X-ray band) modeling Ṁ = 0.0064Ṁ_Edd at r ∼ 450 R_g, which indicates a wind outflow from the thin disk regime of NGC 1097.
    As there is no direct evidence / measurements for wind characteristics in this source, we constrain the model results indirectly. We start the calculations from the Bondi accretion radius r_acc with  Ṁ_Bondi, and consider f_z in such a way that Ṁ(r=r^thin_in) ∼ 0.006Ṁ_Edd. Next we examine the energetics with observed L_bol.
    
    
    
    In literature, we do not find the estimated r_acc and Ṁ_Bondi, we
    perform the calculations with same sets of free parameters, r_acc, Ṁ_Bondi, r_in^thin and M_c of section <ref> with restriction on Ṁ at r= r_in^thin (or at r=r_in^thin, Ṁ∼ 0.006Ṁ_Edd).
    In addition, to account the smaller irradiation intensity we consider also z_s in more optically thin regime, z_s=1.5h.
    The results are shown in right column (or panel III) of Figure <ref>. The panels IIIA and IIIB are for r^thin_o = 5 × 10^5 and 2 × 10^6 R_g respectively.
    The curves 1, 2 and 3 are for (f_v, z_s) = (1, h), (1, 1.5h) and (4, h) respectively. The suffix a and b are for Ṁ_Bondi = 0.01 and 0.05 Ṁ_Edd in panel IIIA, and for panel IIIB Ṁ_Bondi = 0.01 and 0.1 Ṁ_Edd respectively.
    In panel IIIA the f_z for curves 1a, 2a, 3a, 1b, 2b and 3b are 0.1, 0.085, 0.2,  0.0675, 0.055 and 0.125 and for panel IIIb  are 0.11, 0.09, 0.2, 0.073, 0.059 and 0.136 respectively. In present notations, the tan^-1(f_z) will reflect the viewing angle  i.
    The different rows are same as the corresponding row of panel I or II.
    
    
    
    
    
    
    
    
    We find that for a given Ṁ_Bondi the Ṁ vs r curves are almost independent of f_v and z_s (see the respective curves a and b of both panels), mainly due to the fixed value of Ṁ at r_in^thin.
    Like, panels I and II, here also Ṁ decreases faster around r_o^thin and comparatively slower at inner thin disk radius (see the right panel of Figure <ref> for the corresponding z^max in unit of h).
    In panel IIIA, the Ṁ∝r^0.13 and r^0.85 (at r=2.5× 10^5R_g), and  Ṁ∝r^0.06 and r^0.18 (at r=10^4R_g) for curves 1a and 1b respectively.
    In panel IIIB, the Ṁ∝r^0.13 and r^1.1 (at r=10^6R_g), and  Ṁ∝r^0.06 and r^0.16 (at r=10^4R_g) for curves 1a and 1b respectively.
    There are two branches in n_e(disk) and T_c curves around r_o^thin, in which lower branch corresponds to the lower value of Ṁ (or curves with suffix a) and upper branch is for curves with suffix b.
    
    
    
    
    
    
    We find that for r ≲ 0.1r_o^thin the n_e(wind) ∝1/r in all curves with suffix b, and in case of curves with suffix a the n_e(wind) ∝1/r^1.2. 
    For a given Ṁ profile, T_wind does not depend on z_s but it increases with increasing f_v.
    For curve 1b of panel IIIA, the T_wind is equal to the T_c around radius r_o^thin. Hence, in general, for f_v =1 one obtains T_wind < T_c for Ṁ > 0.05Ṁ_Bondi. 
    In case of curve 1b (or curve 2b) of panel IIIB, T_wind < T_c for r ≳ 1.5× 10^6R_g, and around r_o^thin the T_wind drops to ∼76K (or ∼0.8T_c). Therefore,  for larger Ṁ_Bondi, one will have a comparatively larger n_e(wind) and smaller T_wind (<T_c) near to r_o^thin, these may  provide a favourable condition to form a molecular gas in this wind medium (here, we simply speculate about this, in future we will study this in details).
    In NGC 1097, a molecular gas has been observed on parsec-scale <cit.>.
    However, the estimated range of hydrogen number density and temperature  from the emission molecular line ratio are comparatively higher than the wind density and temperature of this work <cit.>.
    Finally, we compare the modeled values of ϵ^x_excess with ϵ^irr of observed L_bol, and all considered parameter sets can launch the wind in some extent. Particularly, for curves 2 the wind can launch upto the radius 4× 10^3 and ∼4× 10^5 R_g for ϵ^irr curves i2 and i1 respectively.
    The wind is an equatorial wind with v_wind∼ v_ϕ. The wind medium is capable to generate blue-/red-shifted emission or absorption lines. Here the i>85degree, the wind is almost along the disk plane. Since, in present analysis we consider the arbitrary values for r_o^thin and Ṁ_Bondi, hence in general the predictions are feasible for any LLAGNs with Ṁ < 0.005Ṁ_Edd at r=r_in^thin.
    
    In present work we consider a constant f_z (for z^max) over r, in general f_z can vary with r (e.g., a smaller f_z near to the r_o^thin and larger f_z near to the r_in^thin, or reverse) and it may provide a refined constrain on model by comparison with observations.
    In summary, the winds are present in outer thin disk of NGC 1097 with speed Keplerian v_ϕ (< v_esc, so it does not escape the system), which can be visible almost along the disk (i>85degree). Therefore, the red-/blue-shifted emission lines generated in wind medium are visible mainly along the disk plane. In general, it would be true for a wide range of LLAGNs.  The wind medium around r_o^thin may provide a favourable conditions for molecular gas formation at higher value of  Ṁ_Bondi, which are consistent with observation of molecular gas at pc-scale in NGC 1097.
    
    
    
    
    
    § SUMMARY
    
    
    We have extended the paper I work, mainly, by studying the mass inflow rate as a function of radius in the presence of wind outflow,  and applied the model for  outer thin disk of LLAGNs with taking into account of inner disk irradiation,  where the calculations are started from the Bondi accretion radius r_acc (or outer radius of thin disk r_o^thin) with Bondi mass accretion rate Ṁ_Bondi.
    We have also clarified the few assumptions, especially due to the low-intense external heating at outer region from inner disk of LLAGNs the present formalism is only applicable for the optically thin medium, that is now the base of wind launching is not a midplane (like paper I) but at some height from the midplane z_s (also termed as an effective irradiation equilibrium height where the irradiated energy is almost deposited into the medium, see point 'a' of section <ref>, 
    also see the Figure <ref> for a schematic diagram of the model).
    In particular, in LLAGNs the high energy emission and double-peaked broad Hα emission lines reveal that it comprises of both types of accretion flow, an inner hot accretion flow (mainly RIAF) and an outer thin disk <cit.>. The wind outflow is ubiquitous in LLAGNs, mainly inferred from indirect way (like, two independent methods estimate mass inflow rate at two radii inner and outer, in which Ṁ at inner radius is smaller in comparison to the outer, e.g., in NGC 1097 Ṁ (r)∼ 0.006Ṁ_Edd (at 450R_g by SED modeling, ) and ∼ 0.07 Ṁ_Edd (at 7× 10^6 R_g by the presence of molecular clouds, ), however for few cases it is interpreted indirectly by blue-shifted absorption/ emission lines <cit.>.
    
    
    Similar to the paper I, we have considered a steady, axisymmetric disk in cylindrical coordinates and set up a formalism for a wind outflow 
    along the z-axis at a given launching radius r from the height z_s.
    We have assumed a very small vertical speed v_z in comparison to the sound speed c_s, and taken its magnitude in ratio of the radial speed v_r; v_z =f_v v_r ≪ c_s at the midplane for a given r.
    We have taken account of both tangential shearing stresses W_ϕ r and W_ϕ z and assumed other shearing stress is negligible in comparison to the 
    W_ϕ r, W_ϕ z; or W_rz∼ 0.
    Like Keplerian disk, we adopted α-prescriptions for W_ϕ  r, W_ϕ r = α p and assumed that at any height  this is also valid, i.e., W_ϕ r(z)=α p(z) (however we find a consistency in solution under this assumption, see section <ref>). 
    Our interest for solution is the outer region of the disk (where the gas pressure dominates over radiation pressure, p ≫ p_rad), we incorporate the irradiation effects of inner region onto the outer, which can only unbalance the hydrostatic equilibrium as the pressure due to the irradiation p_rad^irr<p_rad≪ p. And it is parameterized by x, where x=0 reflects that flows are in vertical mechanical equilibrium. 
    As Keplerian disk we have assumed that within scale height the medium is optically thick (where opacity is mainly due to the free-free absorption) and the viscous generated heat radiates out immediately in vertical direction by blackbody emission.
    
    As in present case, the irradiated energy get almost absorbed in optically thin medium at height z_s (≳ h), so it does not contribute to increase the blackbody temperature. 
    Since, we have v_z ≪ c_s, and in this limit the present formulation behaves like a Keplerian disk at least around the midplane. Therefore, for the initial value of flow variable we take the corresponding  Keplerian value at given r. 
    We have solved the model equations for z< z_s with x=0, and for z≳ z_s with given x (see the Figure <ref> for a schematic diagram for the model).
    
    
    
    The present framework is equivalent to the Keplerian disk for  x=0 and f_v ≪ 1. For Keplerian disk, if one computes either the quantity [-F_r(z)+λ^2/r^3|_z=0 +1/ρ∂ p/∂ r|_z=0](≡1/ρ∂ p/∂ r(z) for the constant λ within the scale height) or 1/ρ∂ p/∂ r(z) in present model at given r then they flip the sign at height z_f ∼ 0.92h.
    In another words, for z<z_f the radial component of pressure gradient acts radially outward direction like Keplerian disk, while for z>z_f it supports the rotation of fluid.
    In general, the sign flip height z_f increases with increasing x for a given parameters set. 
    The sonic point (v_z^2 ∼ v_r^2 →Γ c_s^2) provides two conditions for smooth solution, in which the condition (i) |1/ρ∂ p/∂ z| ≈ |v_z ∂ v_z/∂ z| and  |1/ρ∂ p/∂ r| ≈ |v_r ∂ v_r/∂ r| states that fluids reach to the equipartition of energy state and there is no pressure gradient above the sonic point. The sonic height is the maximum attainable height by fluids and termed as z^max, and z^max increases with increasing x.
    We find z_f ≪ z^max. Since for z>z_f the 1/ρ∂ p/∂ r and F_r both support the fluid rotation and at z^max the fluid reaches in isobaric regime (or there is no pressure gradient), therefore if  
    1/ρ∂ p/∂ r≪ F_r at z^max then F_r can alone support the rotations and the fluid would be rotationaly bound.
    In other case (or sufficiently large  1/ρ∂ p/∂ r in comparison to the F_r), the F_r cannot support the rotations alone and the fluid would be ejected from the z^max as a wind outflow with velocity (v_r,v_ϕ,v_z) and speed v_wind= √((v_r^2+v_ϕ^2+v_z^2)).
    For considered parameter sets, here we note that 
    v_wind∼ v_ϕ, that is the wind is an equatorial wind. 
    
    For x=0 case, we obtain the disk photosphere r_phot∼ 2.2h, which increases very slightly with  decreasing f_v, and is also independent of r. 
    The Keplerian disk photosphere is 2.2h, and the disk shape is concave as h ∝ r^9/8.
    Here, the vertical profiles of pressure p and density  ρ are isothermal (like Keplerian) but both have different scale height (unlike Keplerian disk).
    For non-zero x, and for given parameters set the p scale height increases with increasing x (or z^max) while ρ scale height decreases (see left and middle panels of Figure <ref>), here the scale height is defined by equation (<ref>) for given z_s. 
    And. interestingly. after some large x
    the scale height of p and ρ do not change (or effectively profile also) we term this x as a x^max and corresponding z^max as  z^max_t.
    The x^max can also be depicted by a horizontal region in x vs z^max curves (see Figure <ref>), where a very little increment in x leads to a large increment in z^max. Therefore,  we argue that the model solution with z^max>z^max_t is not a physical (see also Figure <ref>),
    In addition, we note that z^max_t is related to the disk photosphere, in which z^max_t at given r is a maximum possible r_phot where the gravity can hold  the gas with highest sound speed in hydrostatic equilibrium (see Figure <ref>). 
    
    We have explored the wind characteristics (or another words, the fluid variables as a function of z^max) by varying the free parameter either initial v_z (or f_v), or mass accretion rate Ṁ, or r while keeping rest parameters are constant.
    The z^max_t increases with increasing f_v, however the x vs z^max curves for different f_v overlap each other (see left panel of Figure <ref>), also the wind density at z^max = z^max_t is constant for all f_v. It signify that the role of f_v is only to raise the z^max_t.
    The z^max_t increases with increasing r. Interestingly for considered parameters set,  at z^max∼ z^max_t the v_wind becomes equal to the escape velocity v_wind∼ v_esc for r=10^4R_g. For r<10^4 R_g the wind can not escape the system. And for r>10^4R_g the z^max, where wind starts to escape the system, decreases with increasing r. Since, for higher z^max where v_wind>v_esc there is still v_wind∼ v_ϕ (see Figure <ref>) therefore the escaped wind (which is an equatorial wind, and ejected in all directions, see inset of Figure <ref>) material may contaminate the rotation curve of its galaxy on kpc scale.
    
    In general, v_z, v_r, T_wind, |1/ρ∂ p/∂ r|/F_r increase with increasing z^max, while p and ρ decrease with increasing z^max.
    Within physically accessible regime, for a given z^max the v_z, v_r, T_wind and ϵ^x_excess increase with either increasing f_v, or increasing Ṁ, or decreasing r. And ρ/ρ_c, and |1/ρ∂ p/∂ r|/F_r increase with increasing either f_v or Ṁ, or r.
    Interestingly, for considered parameter sets the T_wind becomes smaller than the midplane disk temperature T_c for (on averaged) z^max < 11h.
    We have also studied the wind characteristics by varying z_s. Particularly at a given r within physically accessible regime for a given z^max the ϵ^x_excess decreases with increasing z_s. For a given L_bol, one can find irradiation flux ϵ^irr for a given r using equation (<ref>). By comparing ϵ^irr with computed ϵ^x_excess for different z_s one can estimate the desired range of z_s  where the irradiated energy completely deposited into the medium.
    In addition we found that  ϵ^x_excess decreases faster than 1/r^2, so for a given L_bol the z_s would be different for different r.
    
    We have studied the general trend of mass inflow rate in presence of wind outflow and corresponding wind characteristics as a function of r at z_s=h for two different z^max with f_z = 0.1 and 0.2 (where z^max = f_z r). We have started the computation with two different outer radii of thin disk r_o^thin (∼ r_acc) = 5× 10^5 and 2× 10^6R_g and for each r_o^thin we have two different Ṁ_Bondi namely (0.01, 0.05 Ṁ_Edd) and (0.01, 0.1 Ṁ_Edd) respectively with having inner radius of thin disk r_in^thin =10^3R_g.  
    Since, in present model the wind parameters mainly depend on z^max in unit of h. And in the Keplerian disk r/h(r)∝ r^-1/8Ṁ^-3/20, 
    for a given f_z, the z^max (in unit of h) for higher r is smaller in comparison to the lower r (see Figure <ref>). As a results Ṁ decreases faster around r_o^thin and comparatively very slower around r_in^thin (see first row of Figure <ref>).  
    In general the Ṁ decreases with faster rate by either increasing f_v or decreasing f_z.
    The wind is an equatorial wind with speed v_wind∼ v_ϕ, blowing out in all direction, hence blue- /red-shifted emission/absorption lines are expected  in wind medium.
    
    We have constraint the model parameters with observations of LLAGNs, NGC 1097, where one has Ṁ (r=450R_g) = 0.0064 Ṁ_Edd (from SED modeling) and L_bol = 8.5 × 10^41 erg s^-1<cit.>.
    As in literature there is no estimation of r_acc and Ṁ_Bondi, we considered same sets of r_o^thin, Ṁ_Bondi and r_in^thin of the general case and took three sets for (f_v,z_s) = (1,h), (1,1.5h) and (4,h).
    We constraint f_z in such a way that Ṁ(r=r_in^thin)∼ 0.006 Ṁ_Edd.  
    For higher Ṁ_Bondi, the approximately the hydrogen number density for wind n_e(wind) ∝ r^-1 for r< 0.1 r_o^thin and for lower Ṁ_Bondi the n_e(wind) ∝ r^-1.2 for all r (see the 2nd row of right column of Figure <ref>).
    Importantly, for Ṁ_Bondi∼ 0.05Ṁ_Edd, the temperature of wind medium T_wind∼ T_c  at r=r_o^thin. Therefore for Ṁ_Bondi > 0.05Ṁ_Edd the T_wind would be smaller than T_c near r_o^thin, and here T_wind = 0.8T_c at r_o^thin=2× 10^6R_g (or ∼ 11pc) for Ṁ_Bondi = 0.1Ṁ_Edd, which may provide a favourable conditions to form molecular gas. In NGC 1097 the ALMA observation had traced the HCN molecule around 40 pc radius <cit.>, the model results are consistent with molecular observations and predict a higher Ṁ_Bondi (> 0.05Ṁ_Edd) for NGC 1097.
    We compared the computed internal energy flux enhancement of fluid (ϵ^x_excess) for a given x
    with irradiated flux ϵ^irr (of observed L_bol), all parameter sets can launch the wind upto some extent of radius.
    The wind is an  equatorial wind with speed v_wind∼ v_ϕ and viewing angle i>85 degree. That is, the wind will be visible almost along the disk plane and it would be a general characteristics for the LLAGNs provided  Ṁ (r=R_tr) < 0.005 Ṁ_Edd.
      
    
    
    
    
    
    
    
    
    § ACKNOWLEDGEMENTS
     Author is partly supported by the Dr. D.S. Kothari Post-Doctoral Fellowship (201718-PH/17-18/0013) of University Grant Commission (UGC), New Delhi.
    Author thanks Banibrata Mukhopadhyay and his group members (2019-2020) for discussions and critical comments.
    
    
    
    
    § DATA AVAILABILITY STATEMENT
    No datasets are analysed.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    § FEW PLOTS OF FIGURE <REF>
    A&A
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 Paper Title: A study in scarlet -- II. Spectroscopic properties of a sample of Intermediate Luminosity Red Transients.

 


   

   

   Valerin et al.
   

   Affiliations are listed after the acknowledgements
             

   



 
  
  
   
  
   We investigate the spectroscopic characteristics of Intermediate Luminosity Red Transients (ILRTs), a class of elusive objects with peak luminosity between that of classical novae and standard supernovae. Our goal is to provide a stepping stone in the path to unveil the physical origin of such events, thanks to the analysis of the datasets collected.
  
   We present the extensive optical and near-infrared (NIR) spectroscopic monitoring of four ILRTs, namely NGC 300 2008OT-1, AT 2019abn, AT 2019ahd and AT 2019udc. First we focus on the evolution of the most prominent spectral features observed in the low resolution spectra, then we discuss more in detail the high resolution spectrum collected for NGC 300 2008OT-1 with the Very Large Telescope equipped with UVES. Finally we analyse late time spectra of NGC 300 2008OT-1 and AT 2019ahd through comparisons with both synthetic and observed spectra.
  
   Balmer and Ca lines dominate the optical spectra, revealing the presence of slowly moving circumstellar medium (CSM) around the objects. The line luminosity of Hα, Hβ and Ca II NIR triplet presents a double peaked evolution with time, possibly indicative of interaction between fast ejecta and the slow CSM. The high resolution spectrum of NGC 300 2008OT-1 reveals a complex circumstellar environment, with the transient being surrounded by a slow (∼30 km s^-1) progenitor wind. At late epochs, optical spectra of NGC 300 2008OT-1 and AT 2019ahd show broad (∼2500 km s^-1) emission features at ∼6170Å and ∼7000Å which are unprecedented for ILRTs. We find that these lines originate most likely from the blending of several narrow lines, possibly of iron-peak elements.
  
   

   

   A study in scarlet - II. Spectroscopic properties of a sample of Intermediate Luminosity Red Transients
    G. Valerin1E–mail: giorgio.valerin@inaf.it,
          A. Pastorello1,
          E. Mason2,
          A. Reguitti1,3,
          S. Benetti1,
          
          Y.-Z. Cai4,5,6,             
          T.-W. Chen7,         
          D. Eappachen8,9, 
          N. Elias-Rosa1,10,
M. Fraser11, 
A. Gangopadhyay12,13,  
E. Y. Hsiao14,           
D. A. Howell15,16,
C. Inserra17,             
L. Izzo18,19,
J. Jencson20,         
E. Kankare21,
R. Kotak21, 
P. Lundqvist12,22,
P. A. Mazzali23,24,         
K. Misra25, 
G. Pignata26,
S. J. Prentice27,           
D. J. Sand28,        
S. J. Smartt29,30,
M. D. Stritzinger31, 
L. Tartaglia32,
S. Valenti33,
J. P. Anderson34,35,
J. E. Andrews28,
R. C. Amaro28,              
C. Barbarino12,
S. Brennan12,
F. Bufano36,          
E. Callis12,
E. Cappellaro1,
R. Dastidar37,35,
M. Della Valle18,38,               
A. Fiore1,39,40,          
M. D. Fulton30,
L. Galbany10,41,
M. Gromadzki42,
T. Heikkilä21,            
D. Hiramatsu14,15,43,44,             
E. Karamehmetoglu12,31,    
H. Kuncarayakti21,45,
G. Leloudas46,  
M. Limongi47,48,49
M. Lundquist28,                
C. McCully15,
T. E. Müller-Bravo10,41, 
M. Nicholl30,
P. Ochner1,50, 
E. Padilla Gonzalez15,16,    
E. Paraskeva51,
C. Pellegrino52,     
A. Rau53,
D. E. Reichart54,
T. M. Reynolds20,55,56,      
R. Roy57,                                    
I. Salmaso1,    
M. Shahbandeh58,59,
M. Singh60,
J. Sollerman12,
M. Turatto1,               
L. Tomasella1,                                 
S. Wyatt28,
D. R. Young30
          
    Received —-; accepted —-
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION


The transient sky is brimming with a wide variety of phenomena, and spectroscopic observations are a crucial tool to navigate through such a complex landscape. Indeed, remarkable efforts have been made in order to promptly classify newly discovered transients, culminating in the formation of dedicated spectroscopic surveys such as the “Public European Southern Observatory Spectroscopic Survey of Transient Objects" (PESSTO) consortium <cit.>.
The swift collection of spectra is rewarded with key insights on the ongoing physical processes which could not be otherwise probed. In the past years there has been a growing number of studies on objects displaying narrow and intense emission lines coupled with a slow spectral evolution: these are the hallmarks of optically thick circumstellar medium (CSM) surrounding the transient (e.g. ). Such a scenario has been investigated primarily in Supernovae IIn (SNe IIn): terminal stellar explosions within an opaque CSM formed during the years prior to the event <cit.>. 

Critically, SNe IIn are not the only events showing evidence of optically thick CSM. Violent eruptions of Luminous Blue Variables (LBVs) mimic the characteristics of a genuine SN, despite being non-terminal events: for this reason they earned the designation of “SN impostors"  (e.g. ). 

The picture was further complicated by the discovery of other classes of transients that display clear signs of CSM in their spectra. Luminous Red Novae (LRNe) have been argued to arise from mergers of non-degenerate stars after a common envelope phase where the final product of the merger event is expected to survive <cit.>. Finally, perhaps the most poorly understood among the objects discussed so far, there are the subjects of the present study: the Intermediate Luminosity Red Transients (ILRTs), whose nature is still debated. Together, LBVs, LRNe and ILRTs make up a wider class of objects with luminosity intermediate between Classical Novae and standard SNe, called “Gap Transients" (see  and  for a review).
While their physical origin may be remarkably different, all these transients have one thing in common: the presence of optically thick CSM surrounding the system.
The resulting similarity among the spectra of the Gap Transients and SNe IIn makes a proper classification of a newly discovered transient quite difficult: it can be challenging to unmask an impostor <cit.>.




Photometric follow-up can provide some important clues for this type of investigation. The light curve shape can be a useful indicator for classification: LBVs alternate periods of quiescence with erratic and powerful outbursts <cit.>, LRNe often show distinctive double peaked light curves, while ILRTs are characterized by SN-like light curves <cit.>. 
It is still essential to complement the photometric follow-up with a spectroscopic study to characterize each class, despite the partial degeneracy in the properties of these transients.
From a spectroscopic point of view, ILRTs can be summarily described as a black body continuum with narrow emission lines superimposed. Strong Balmer emission lines dominate their spectra together with several Ca features: Ca II H&K in absorption, strong Ca II NIR triplet in emission as well as the [Ca II] forbidden doublet. Opposed to LRNe, ILRTs show no sign of molecular bands in the late phases of their evolution <cit.>.



A key difference between ILRTs and SNe IIn is that both the peak observed temperature (T_ILRT∼7500 K) and peak luminosity (L_ILRT ≲ 5x10^41 erg s^-1) of ILRTs are significantly lower compared to SNe IIn, which instead tend to be hot (T_IIn∼10000 K, , but up to ∼15000 K, e.g. ) and even brighter than average SNe II (e.g. ). The photospheric temperature and luminosity observed for SNe IIn can be sustained even for years (e.g. ), thanks to the interaction between fast ejecta and slow moving CSM, a mechanism that can efficiently convert kinetic energy into radiation <cit.>. When the interaction region between ejecta and CSM is visible, specific spectral features can be identified, most notably a blue shoulder on the Hα line <cit.>. ILRTs, however, do not show clear signs of strong dynamical interaction between ejecta and CSM in their spectra: their low temperature and luminosity could imply a rather weak ejecta-CSM interaction, or at least that such interaction is hidden underneath layers of unshocked CSM.




As previously mentioned, the physical mechanism behind ILRTs is still debated. Since the photosphere lies within an optically thick gas shell, it is not possible to probe the material underneath the CSM, which could reveal key information on the nature of the transient itself. This uncertainty has led to a variety of interpretations regarding their true nature.
Some researchers have advocated outbursts of post-main sequence stars <cit.>, while others have suggested that ILRTs are the result of jets originating from interacting binary stars <cit.>. Finally, ILRTs were also associated with Electron Capture Supernovae (ECSNe, e.g. ) originating from Super AGB stars <cit.>.
ECSNe are expected to have some distinctive features that separates them from classic Fe core collapse events. As was discussed for SN 2018zd, nebular spectra of ECSN are expected to show Ni features stronger than Fe lines, due to the different nucleosynthesis yields in the two scenarios <cit.>. Indeed, <cit.> provide synthetic nebular spectra of a 9 M_⊙ Fe core collapse SN, and approximate the behaviour of an ECSN by enhancing the Ni content in their Fe core collapse model. The Ni enriched spectra display particularly strong [Ni II] emission lines.
Until now, such an analysis has not been performed for ILRTs due to the lack of high quality nebular spectra, as well as the lack of detected features beside the ones originating in the CSM.

In this paper we present and analyse spectra of four ILRTs, following the photometric analysis of the same targets performed in “A study in scarlet I. Photometric properties of a sample of Intermediate Luminosity Red Transients" (Valerin et al. 2024, hereafter "Paper I"). The work is structured in the following manner: in Section 2 the data reduction techniques are described. In Section 3  we present the optical and near infrared (NIR) data collected. In Section 4 the line luminosity and velocity are measured and compared within the sample. Section 5 is dedicated to the study of the high resolution spectroscopy obtained for NGC 300 2008OT-1, while in Section 6 we discuss the broad features appearing in the late time spectra of NGC 300 2008OT-1 and AT 2019ahd. Finally, in Section 7 we summarize our work and draw the conclusions.




§ DATA REDUCTION

The analysis presented in the following sections is based on the spectroscopic observations of four different ILRTs over the course of several months. Our follow-up campaigns are focused on the optical and Near-Infrared (NIR) domains, employing several telescopes equipped with different instruments. The data reduction procedure is based on the Image Reduction and Analysis Facility (, ), a software specifically designed for image processing developed within the astronomical community.
Every image undergoes the same preliminary reduction procedure: trimming, bias and flat-field correction. NIR spectra go through an additional sky subtraction step when combining the different nodding positions.
A one dimensional spectrum of the target is extracted from the two dimensional image, while simultaneously performing a background subtraction in order to avoid contamination from the sky emission lines and the host galaxy signal.
The one dimensional spectrum is then calibrated in wavelength on standard spectral lamps, typically showing He, Ne, Ar, Hg and Kr lines.


These procedures are repeated on the spectrum of a standard star (belonging to the list of ESO spectrophotometric standards[https://www.eso.org/sci/observing/tools/standards/spectra/stanlis.html]), which is routinely obtained during every observation run. Thanks to the comparison between the observed and tabulated flux values of a standard star, it is possible to accurately flux calibrate the spectrum of the target. Standard star observations are also useful to remove the telluric absorption features which affect ground-based observations. An additional, wavelength independent flux calibration is performed, ensuring that the fluxing of the spectrum is compatible with the values measured from coeval photometric observations. Finally, each spectrum is corrected for redshift and reddening, in particular assuming a reddening law R_V = 3.1 both for Galactic and internal extinction <cit.>. The distance, reddening and redshift adopted for each transient are summarized in Table <ref>: for a discussion on the choice of the values used, see Paper I.









Many facilities and instruments were involved in the follow-up campaigns of our targets. Indubitably, the backbone of our spectroscopic monitoring was obtained through the Nordic Optical Telescope (NOT) within the “NOT Unbiased Transient Survey 2" collaboration (NUTS2, ) and the instruments available within the “Global Supernova Project" (GSP, ). Other telescopes used during this project include the Gran Telescopio Canaria equipped with OSIRIS, the William Herschel Telescope equipped with ACAM and LIRIS, the South African Large Telescope equipped with RSS, and the New Technology Telescope plus EFOSC2, thanks to the ePESSTO+ consortium <cit.>. Additionally, multiple spectra of NGC 300 OT were taken with the 6.5 m Magellan Telescopes, and the 4.0-m R-C Spectrograph at the Cerro Tololo Inter-American Observatory, and with the Very Large Telescope, most notably a high resolution spectrum with UVES <cit.> and a late spectrum with X-Shooter <cit.>. We also integrated our dataset with some publicly available spectra to have a more complete sampling of each event ( for AT 2019ahd and  for AT 2019abn).
The detailed logs listing all the spectroscopic observations for each target are reported in Tables  <ref>, <ref>, <ref> and <ref>.





















§ OVERVIEW OF THE DATA




 §.§ Optical Spectra
 
 Spectra obtained for NGC 300 OT are displayed in Figures <ref> and <ref>, while Figures <ref> and <ref> show the spectral sequence of AT 2019abn. Figures <ref> and <ref> present the spectral evolution of AT 2019ahd and AT 2019udc, respectively. All spectra were corrected for redshift and total extinction along the line of sight: values adopted for these corrections are reported in Table <ref>. The most prominent features noticeable in these objects are the emission lines of the Balmer series. Their narrow profiles (≲1000 km s^-1) indicate that the spectral lines are forming in a region where the material is moving slowly, compatible with an opaque CSM. We do not detect clear P-Cygni profiles in the emission lines, which would be caused by absorbing gas approaching the observer along the line of sight. The absence of the P-Cygni features may imply that the photosphere of the transient is located at the outer edge of the CSM, that the transient is intrinsically asymmetric, or that the covering fraction of the absorbing material is low.
The Ca II H & K absorption lines (λλ 3934,3968) are found in all objects, especially at early phases. The photons absorbed through the Ca II H & K resonance lines excite the electrons of the Ca II atoms from the ground state 3s^2 3p^6 4s ^2S_1/2 to 3s^2 3p^6 4p ^2P^o_1/2,3/2. From this energy level the electrons can radiatively decay to the level 3s^2 3p^6 3d ^2D_3/2,5/2, giving rise to a triplet of lines (λλλ 8498,8542,8662). Indeed this Ca II NIR triplet is detectable at most epochs in ILRTs, and it becomes dominant at later times, as displayed in Figures <ref> and <ref>. The compatible shape of the Ca II NIR and Ca II H& K lines in velocity space corroborates this interpretation (e.g. see Figure <ref>).

















 Finally, the transition that closes this “loop" envisions the de-excitation of the electrons from the level 3s^2 3p^6 3d ^2D_3/2,5/2 back to the ground state 3s^2 3p^6 4s ^2S_1/2. Such transition, which produces the [Ca II] doublet (λλ7291,7323), is forbidden.
 Even if the metastable energy level 3s^2 3p^6 3d ^2D_3/2,5/2 is populated, the formation of [Ca II] requires a large amount of gas at moderate excitation and most of all at low density (e.g. ). In fact, in a dense environment, collisions with other atoms or ions de-excitate the electrons back to the ground state before the radiative decay process can take place, consequently suppressing the formation of these lines (e.g. ). Therefore, this spectral feature is only found in regions where the gas density is below the critical density N_crit ∼ 10^7 cm^-3 (for temperatures typically found in ILRTs, adopting the collision strength reported by ).
 Based on these premises, the ubiquitous presence of the [Ca II] doublet is one of the most characteristic features in ILRTs spectra. While not unique to ILRTs, the persistent presence of the [Ca II] forbidden doublet in all the objects analysed so far has been proposed as a key feature to distinguish ILRTs from other transients that present similar spectra <cit.>. 

 
 
 

 Another noteworthy feature in the low resolution optical spectra of our sample of ILRTs is the presence of O I lines. In the spectra of all the considered objects the O I triplet λλλ 7771.9,7774.2,7775.4 is found in absorption. This triplet is detected at multiple epochs in NGC 300 OT (Figure <ref>), AT 2019abn (Figures <ref> and <ref>) and AT 2019ahd (Figure <ref>), while in the case of AT 2019udc it is confidently detected only in the GTC spectrum obtained on 2019-11-22, 8 days after maximum. The O I triplet in absorption is a sign that the metastable level 2s^22p^3 (^4S^o)3s^5S^o_2 is populated. We do not detect any emission lines from transitions that may feed this energy level: first of all the λλλ 7771.9,7774.2,7775.4 triplet is never clearly detected in emission at any epoch, but we also do not detect transitions from higher energy levels (e.g. O I λ3947). This suggests that the electrons populated the metastable energy level s^22p^3 (^4S^o)3s^5S^o_2 by recombination after an initial excitation. With time, the O I absorption triplet becomes less pronounced, until its detection becomes uncertain in the latest phases. The absence of a clear emission component of the O I triplet could be due to efficient absorption by electrons in the metastable energy level.
 Few months after maximum luminosity there is also the appearance of the O I λ8446 emission line, which becomes progressively more prominent with time. In AT 2019abn this feature appears around +155 d, in AT 2019ahd around +93 d and in NGC 300 OT around +123 d (but this emission line is detected already in the high S/N spectrum at +47 d discussed in Section <ref>). This feature is not found in the spectra of AT 2019udc, but its spectroscopic monitoring had to be stopped at +67 d, therefore this emission line could have developed after the end of our observational campaign. The appearance of the O I λ8446 line is particularly interesting, since it cannot be the result of the de-excitation of O I atoms, which would result in the coeval formation of the λλλ 7771.9,7774.2,7775.4 triplet in emission. Instead, as discussed above, the O I triplet has only been detected in absorption.
 We conclude that the O I λ8446 emission line found in ILRTs is the consequence of UV pumping. 
 This process is also known as Bowen fluorescence, and it is caused by the resonance between the Lyβ λ1025.72 Å and the O I λ1025.76 line, which excites the electrons from the ground level 2p^4 ^3P_2 to 2p^33d ^3D^0_3 <cit.>. The electrons subsequently decay to the levels 2p^33p ^3P_2 (emitting λ11287 Å), 3p^33s ^3S^0_1 (emitting λ8446 
 Å) and finally back to the ground state (through the UV triplet λλλ1302.2,1304.9,1306.0). Indeed, in the NIR spectra of NGC 300 OT presented in the following section we detect the O I λ11287 Å emission line, corroborating the fact that in these ILRTs we are observing the fluorescence process described above, rather than a different pumping mechanism (e.g. see the different processes described by ). Since the excitation of the O I electrons to the 2p^33d ^3D^0_3 energy level requires Lyβ photons, the appearance of prominent O I λ8446 Å at relatively late phases (and not at early ones) needs to be addressed. In the context of faster ejecta expanding underneath a slow moving CSM, the Lyβ could be shifted off resonance due to Doppler effect: in fact, to have a perfect wavelength match, the Lyβ photons should be redshifted by 11.7 km s^-1 with respect to the oxygen atoms. 
 In the early phases, when the resonant lines are opaque, the Lyβ escapes the ejecta from a compact pseudophotosphere which is moving faster than the CSM above, greatly reducing the coupling between the Lyβ and the O I λ1025.76 line. As time progresses and the ejecta expand, the opacity of the lines drops and CSM is finally reached by the redshifted Lyβ photons that can trigger the Bowen fluorescence, hence explaining the delayed appearance of the O I λ8446 line. 

 
 
 

 
 
 
 It is worth noting that all spectra of ILRTs in our sample become bluer during the rise to maximum: such behaviour is observed in other interacting transients (e.g. SN 2023ixf, ) and predicted by models of interacting SNe <cit.>, and therefore it could be a result of increasing interaction between the ejected material and the CSM. This interpretation is also supported by the increasing flux of the emission lines during the rise to maximum luminosity in all observed ILRTs (see Section <ref>). Finally, the late time spectra (≳100 days) collected for NGC 300 OT, AT 2019abn and AT 2019ahd do not show clear blueshift nor change in the shape of the emission lines. These are telltale signs of dust along the line of sight, and dust formation was inferred for these objects in Paper I from the NIR excess in their SED. A possible explanation for the lack of these spectroscopic signatures may be that the dust is not located along the line of sight: in this scenario, ILRTs would be intrinsically asymmetric events.

In Figure <ref> we display spectra of six ILRTs taken in the first weeks after maximum luminosity. Beside the four targets in our sample, SN 2008S <cit.> and AT 2017be <cit.> are also shown. SN 2008S is considered to be a prototype of the ILRT class thanks to its prominent Ca lines, with a peak luminosity well in line with our sample of objects. On the other hand, AT 2017be lies in the lower end of the luminosity distribution of ILRTs (see Figure 5 of Paper I).
On the right panels of Figure <ref> we zoom in on the most characterising features of this class, in particular the [Ca II], Ca H&K and Hα lines. The strength of each feature varies among the sample: for instance, AT 2019udc is characterised by a strong Ca H&K absorption, while the [Ca II] doublet is most prominent in NGC 300 OT and SN 2008S. Albeit present, the Ca II NIR triplet becomes more prominent in all ILRTs at later phases.










 §.§ Near Infrared Spectra




Beside the optical spectra, we gathered a total of five NIR spectra of our targets. NIR spectra of AT 2019abn were collected during the early rise, immediately after discovery. The single spectrum available for AT 2019udc was taken in a more advanced phase, three weeks after peak luminosity. Finally, the two spectra of NGC 300 OT monitor the object at late phases, hundreds of days after maximum light. As displayed in Figure <ref>, AT 2019abn and AT 2019udc appear to be quite similar: a nearly featureless continuum, without strong emission or absorption lines superimposed. 
Conversely, in the spectra collected for NGC 300 OT it is possible to identify several emission features. Most notably, these include hydrogen lines of the Paschen (Pa β,γ,δ,ϵ) and Brackett series (Br γ), He I (λ10830), O I (λ11287) and Fe I (λλλ15831,16679,16693). As mentioned in the previous section, the detection of O I λ11287 Å has been key to confirm that the optical line O I λ8446 Å is produced by the Bowen fluorescence.






§ LINE PARAMETERS




 §.§ Line luminosity
 





In this section we estimate the parameters of the main lines observed in our spectra. In order to avoid an additional source of systematic uncertainty, we do not assume a specific line profile (e.g. Gaussian or Lorentzian): instead we estimate the luminosity contribution of a given feature by integrating the flux over the continuum in the considered spectral region. This procedure is repeated multiple times in order to estimate the error associated with the measurement. The measured line flux is then converted into the luminosity of the line by accounting for the distance of the transient (the spectra were already corrected for extinction during the calibration procedure).
The luminosity evolution of Hα, Hβ, Ca II NIR triplet and [Ca II] doublet is shown in Figure <ref>. In each panel we also represent the continuum level in the proximity of each spectral feature as a solid line. For clarity, the continuum level is scaled to match the line luminosity at the first measured epoch. The continuum was estimated in the range 4725-4825 Å for Hβ, 6700-6850 Å for Hα and [Ca II] and 8200-8350 Å for Ca II NIR triplet. In the same Figure, the bottom panels report the equivalent widths (EW) for Na ID and Ca II H&K in absorption. We remark that the spectral features examined can present multiple components in high resolution (see in particular Section <ref>), therefore the values reported here track the behaviour of the sum of all components of each line. An additional complication is given by the fact that in the case of optically thick lines we are not sampling the whole emitting volume.


Considering the emission features, we note that the luminosity evolution of Hα and Ca II NIR triplet is not monotonic: after an early peak, the luminosity declines sharply over the course of a few weeks. A minimum is then reached, and subsequently the lines start to strengthen again, forming a sort of broad, long lasting second peak. This trend is even more clear when compared to the evolution of the spectral continuum at the same phases, which is monotonically declining after maximum brightness. The same evolution can be detected also in Hβ, although less pronounced. For NGC 300 OT, only the second broad peak is visible, because the object was behind the Sun during the first phases of its evolution. This line rebrightening has already been noticed in other ILRTs and it was tentatively attributed to CSM interaction <cit.>. In this scenario, there is a collision between rapidly expanding ejecta and the slower CSM lying above. The energy released from this interaction could lead to the re-ionization of the gas above, and its subsequent recombination. The rebrightening of the hydrogen emission lines being coeval with the Ca II NIR strengthening (see Figure <ref>) suggests that these features originate in a similar environment and are produced by material undergoing similar physical processes. 
At the same time, we note that during this rebrightening of the lines there are no relevant changes in the ionisation state nor in the profile of the lines in the low resolution spectra, which are typical telltale signs of ongoing interaction. This means that either the CSM along the line of sight is optically thick enough to hide the interaction region, or that the secondary peak in line luminosity is caused by a mechanism other than CSM interaction. An alternative scenario to CSM interaction could be presence of material that is reached by the photons produced by the transient only after some time after its onset (in the order of one or two months). This material would become excited as it is overflown by photons, producing Balmer and Ca II NIR emission lines as it de-excites. Based on the onset time of the second peak (30 days for AT 2019udc, up to 55 days for AT 2019abn) and the speed of light, in this scenario the material providing the line luminosity for the secondary peak would be located at least at 8 x 10^16 cm (1.4 x 10^17 cm) from the transient. In any case, the secondary luminosity peak observed for the emission lines favours the scenario of a complex environment surrounding ILRTs, which could have been formed by a stellar wind or eruptive episodes of the progenitor star.

 Whatever the mechanism driving its evolution, the marked rebrightening shown by the Ca II NIR and Hα lines is a characteristic commonly found in ILRTs.
 In contrast, the luminosity of the [Ca II] forbidden doublet displays an overall monotonic decline. We point out that the [Ca II] evolution appears to mimic the shape of the continuum luminosity in all four objects. This behaviour is consistent with the fact that the [Ca II] doublet originates from low density gas, farther from the dense CSM. Its evolution is therefore dictated by the incoming radiation from the transient, in particular by the radiation absorbed by the Ca H& K lines. As previously described, the emission of the [Ca II] doublet can only happen if the medium is below the critical density. In this configuration, the hypothetical interaction between ejecta and CSM can cause the collisional excitation of Ca and H atoms in the dense CSM, which is over the critical density for the production of the [Ca II] doublet, therefore resulting in a rebrightening only for Balmer and Ca II NIR triplet lines. At the same time the slow material producing [Ca II] lines is too distant to be affected by the collision, and keeps following the evolution of the spectral continuum.












The Na ID EW depends on the properties of the CSM, the interstellar gas along the line of sight, possible ejecta and even the viewing angle in case of non-spherically symmetric ejecta: for these reasons, its behaviour presents a large variance within the sample. AT 2019udc shows a strong Na ID feature in the first two observed epochs (EW∼5 Å), which become rapidly weaker during in the following days: a similar behaviour was observed for the ILRTs AT 2010dn and AT 2012jc <cit.>. On the contrary, AT 2019abn initially displays a weak Na ID absorption feature (EW∼1 Å), which becomes more significant in the following weeks (up to EW∼3 Å). AT 2019ahd and NGC 300 OT present a similar behaviour, although scaled down compared to AT 2019abn. The increase and following decrease in EW of the Na ID over time can be qualitatively reproduced by considering the column density of Na I laying in front of a shock travelling through a thick CSM <cit.>: in this simple scenario the recombination of Na II is responsible for the initial increase in Na ID EW, while the shock travelling outward and sweeping the gas eventually causes its decline.
Finally, the EW evolution of Ca II H&K appears to be quite similar in AT 2019ahd and AT 2019udc: an initial rapid decline followed by a somewhat broad "peak". NGC 300 OT, missing the first phases of evolution, displays a slow decline in EW that is resembling the evolution of AT 2019udc in particular. In stark contrast with the behaviour of the other three ILRTs, AT 2019abn displays Ca II H&K absorption features that become significantly more pronounced after the first month of evolution. In principle, high resolution and multi-epoch observations of Na ID and Ca II H&K components in ILRTs would allow us to discriminate if the evolution of these absorption features is driven by a change in the ionization state of the CSM or by the presence of clumps of matter, as prescribed by <cit.>. However, Na ID and Ca II H&K absorption features often saturate in ILRTs, even despite the broadening introduced by the velocity field: this inevitably hinders the physical interpretation of these measurements.







AT 2019abn, the brightest and longest-lasting ILRT observed to date, also displays the brightest Hα and [Ca II] features among the sample, reaching 1.6 × 10^39 erg s^-1 and 10^38 erg s^-1 at peak luminosity, respectively. Conversely, NGC 300 OT is characterised by the least luminous Balmer lines within the sample. However, beside the simple luminosity of these emission features, it is useful to investigate their evolution relative to other lines. 
For this purpose, in Figure <ref> we track the change in value of the Balmer decrement (Hα / Hβ) over time for our sample of ILRTs.
In their first spectra, both AT 2019ahd and AT 2019udc display large values for the Balmer decrement (>4 and >6, respectively), which quickly decrease down to 2.1 and 2.4 in the following two weeks. AT 2019abn and NGC 300 OT do not show this early decline, but this can be caused by the absence of very early spectra, especially for NGC 300 OT. 
With time the Balmer decrement of the whole sample grows significantly, reaching values of 7 and 23 for AT 2019abn and NGC 300 OT, respectively. 
The Balmer decrement evolution just described matches the one observed in SN 2008S. Based on the measured values of Balmer decrement and temperature at early phases, <cit.> constrain the electron density in the CSM to be 10^10 cm^-3 < N_e < 10^12 cm^-3, according to the work performed by <cit.>.
Such high electron densities are not unheard of in the context of SNe IIn: for example SN 2006gy presented an electron density in the CSM above 6×10^10 cm^-3 ().

However, these considerations are based on the assumption of case B recombination, meaning that the gas is optically thick for lines of the Lyman series and optically thin for lines of the Balmer series. Under this conditions, it has been shown that the Balmer decrement is close to ∼3 for a wide range of temperature and density values of astrophysical interest <cit.>. As reported in Figure <ref>, ILRTs display a ratio Hα / Hβ which varies with time, reaching values well above the expected one of ∼3.
There are several possible explanations for this departure from case B recombination, such as the collisional excitation of H atoms, which would boost the Hα luminosity, and dust formation, which would extinguish the Hβ more than the Hα <cit.>. Another mechanism that can cause an increase in the Balmer decrement is an increased opacity of the lines of the Balmer series, as explained by <cit.> (Chapter 4.5). 















 §.§ Width of the spectral features


Measuring the full width at half maximum (FWHM) of the emission lines of ILRTs can be challenging: narrow features such as the [Ca II] doublet are typically unresolved in our spectra. As previously done for the line flux, we do not assume a specific fitting function, but we manually measure the FWHM of the lines. The resulting FWHM is corrected for the resolution of the instrument (typical values of R∼500) and then converted into a FWHM velocity, which can be taken as an order of magnitude estimate of the speed at which gas is moving in the emitting region. This value, however, should not be treated as a precise velocity measurement: the broadening of the line profiles can be caused by the combination of Doppler shift due to bulk motion of the emitting material as well as by the electron scattering of the photons within a hot medium <cit.>. Electron scattering is more marked at higher electron temperatures as well as higher optical depths, resulting in the appearance of extended wings <cit.>.
In Figure <ref> we display the FWHM velocities inferred from the Hα line for our sample of ILRTs in the first 150 days of evolution. Analysing the Hα line has the significant advantage that this feature is bright at all epochs, therefore allowing for an accurate FWHM measurement. Furthermore, especially at early times, the Hα line is resolved in most spectra, and it is not blended with other features (as happens for the Ca II NIR triplet), making it the ideal feature for comparing the evolution of different objects with our dataset. 


During the earliest phases, when the luminosity is still rising, the FWHM velocity of Hα quickly increases in AT 2019abn and AT 2019ahd and AT 2019udc. AT 2019udc, on the other hand, displays the broadest FWHM in the first spectrum; it is impossible to confirm if this is the case also for NGC 300 OT, since we are missing the pre-maximum part of its evolution. The ILRTs in our sample reach a peak FWHM velocity between ∼700 and 1100 km s^-1, with AT 2019abn as a clear outlier and the other three objects clustering around v_FWHM∼800 km s^-1 at peak. 
After the maximum FWHM is reached, ILRTs display a rapid decline over the course of 3-5 weeks. Afterwards, NGC 300 OT settles on an almost constant v_FWHM∼550 km s^-1 for over 100 days, while the other three transients are characterised by a secondary peak in v_FWHM after a local minimum, increasing  by ∼200 km s^-1 in about 5 weeks. During this phase, AT 2019ahd displays the broadest lines, with a v_FWHM up to 750 km s^-1. Eventually the Hα narrows once again, down to 320 km s^-1 at 135 days for AT 2019abn (and unresolved thereafter), 470 km s^-1 at 302 days for AT 2019ahd and 170 km s^-1 at 536 days for NGC 300 OT.
The behaviour of the FWHM velocity just described shows remarkable differences from the photometric features of the transients. For example, NGC 300 OT is two magnitudes fainter than AT 2019abn (see Paper I), yet its FWHM velocity is not only compatible, but oftentimes larger than the one observed for AT 2019abn.
It is important to stress that these FWHM velocities are linked to the properties of the CSM where the line is produced: as the line formation region moves within the CSM, different values of gas speed, electron temperature and optical depth mold the profile of the lines, which is reflected in the evolution of the FWHM velocity shown in Figure <ref>. However, as happens in SN IIn, the low FWHM velocities measured do not exclude the presence of fast ejecta lying below the CSM (e.g. ).














§ HIGH RESOLUTION SPECTROSCOPY OF NGC 300 OT
 

NGC 300 OT is the closest ILRT observed (see  and  for a review of known ILRTs). Thanks to the proximity of its host galaxy (1.92±0.14 Mpc), it was possible to obtain a high resolution optical spectrum of the transient on 2008-07-02 with the Very Large Telescope (VLT) equipped with UVES, yielding a resolution between 0.12 and 0.36 Å. In Figure <ref> the identification of the main spectral lines on this high resolution spectrum is shown. The measurements pertaining to the most prominent features are reported in Table <ref>, in particular their observed central wavelength, their FWHM and their estimated flux.
Apart from the H, Ca, O and Na lines discussed in the previous sections, easily recognizable also in the low resolution spectra, in this high resolution and high S/N we find a number of metal lines, in particular Fe I (λλλλ 5535,8688,8757,8824), Fe II (multiplets 40,42,46,49,73,74 and forbidden line λ 7155), Mg I (λ8806) and Sc II lines (Moore multiplet 29). Additionally, we identify [O I] (λλ 6300,6364), faint Ca II absorption doublet (λλ8912,8927) and He I (λλλ 5876,6678,7065). All these features are narrow (FWHM ≲ 10 Å) but resolved, therefore originating within the opaque CSM or above it. 
Thanks to the high resolution spectrum, it is possible to accurately compare the FWHM velocities associated with different emission features. The broadest feature observed at this epoch is the Hα emission line, with a FWHM velocity of 520 km s^-1. The Ca II NIR line λ 8498 is marginally narrower, with a FWHM velocity of 440 km s^-1: the fact that Hα and Ca II NIR have compatible shapes in velocity space corroborates the scenario outlined in the previous section, where both these features are generated in neighbouring environments. The [Ca II] forbidden doublet, instead, is characterised by a FWHM of 80 km s^-1, pinpointing its origin in a separated region, likely characterized by lower gas velocity, electron temperature and optical depth compared to the region where permitted lines are formed. The [O I] forbidden lines are comparable in FWHM and shape with the [Ca II] doublet.
In general, the range of FWHM velocity displayed by the lines (Table <ref>) suggests a stratification in the emission lines of different ions, and even within the emission lines of the same ions. 









Spectral features that appear simple in low resolution may actually hide a more complex structure, which can only be unveiled through high resolution observations. The shape of such lines carries relevant information on the geometry of the emitting region. In Figure <ref> we show the profiles of the lines of the Balmer series, of some prominent Fe II lines, of the Ca features (NIR λ8498, [Ca II] λ 7291 and Ca II K absorption) and the Na ID absorption doublet. The Hα line in the UVES spectrum shows a composite profile: superimposed to the emission line are two narrow absorption components. The most prominent absorption component has a FWHM of 30 km s^-1 and is only 15 km s^-1 redward of the rest wavelength of Hα. The second absorption component is even narrower, with a FWHM close to the spectral resolution, and it is situated 140 km s^-1 blueward of the emission peak. In the top left panel of  Figure <ref>, the most prominent and redward of the two absorption features can be seen in other coeval lines of the Balmer series, although with lower S/N. The second, more shallow and blueward absorption does not have a counterpart in the other H lines, and therefore it could result from a transition of a different element. The Hγ line presents a unique shape, which deviates from the trend of the other Balmer lines, showing in particular a marked absorption in the blue wing which also results in a narrower FWHM. The same behaviour for the Hγ line has been observed in the yellow supergiant IRC+10420, where it has been explained as the result of absorption components found in stellar atmosphere models (Figure 6 in ). 

In the top right panel of Figure <ref> it is possible to study the different profiles of the Fe II lines. Despite originating from the same atomic species, some lines display a dominant emission component, with only traces of absorption at around the rest wavelength (e.g. λ6517), while other lines display a more complicated profile, with marked absorption features and a redshifted emission component, compatible with a P-Cygni profile (e.g. λλ5018,5169). The Fe II lines that display a stronger absorption, such as the multiplet 42, are characterized by a larger oscillation strength, which favours self absorption. Additionally, the spatial stratification of the emission lines may further enhance this effect.






















The analysis of the Ca II lines (bottom left panel of Figure <ref>) offers valuable insight on the geometry of the line forming region.
The Ca II NIR line λ8498 displays, as said, a prominent absorption component with a FWHM of 30 km s^-1 situated just 10 km s^-1 blueward of the rest wavelength position of this line. 
On the other hand, the [Ca II] doublet presents a very different line profile: the line is markedly asymmetric, with the blue wing almost completely missing. <cit.> tentatively attribute this behaviour to self absorption, in a scenario where the [Ca II] doublet originates in an low density, inflowing gas. However, <cit.> state that self absorption is quite rare for a forbidden transition such as [Ca II] (although not impossible, as shown for example by ): the peculiar line profile is postulated to be formed through electron scattering in an expanding envelope, which gives rise to the extended red wing observed.
We expand on this interpretation, based on the similarity between the shape of the absorption feature superimposed to the Ca II (λ8498) emission line and the [Ca II] doublet: a steep blue edge and an extended red wing (lower left panel of Figure <ref>). We deduce that both features are produced in the same region, in which the gas density must be low to allow the production of the [Ca II] doublet. 
The velocity field of the gas in this region molds the shape of both spectral features. We speculate that the observed asymmetry reflects the dynamics of a stellar wind which reached the terminal velocity. The blueward sharp cutoff would indicate the terminal wind velocity.
By extension, all the observed narrow absorptions superimposed to the emission lines (Balmer lines, Ca II NIR triplet, He I, Fe II and O I) can be traced back to this low density, pre-existing wind produced by the progenitor star. With the redshift we adopted for NGC 300 OT, the blue cutoff for the [Ca II] is located at ∼-30 km s^-1, which provides an estimate of the terminal wind velocity in the scenario just described. Such value is well in line with the wind velocities observed for AGB stars (e.g. ).
On the other hand, the red wing of the [Ca II] extends over 200 km s^-1 redward: this could be the result of photons scattered into the line of sight by interaction with free electrons. Indeed, a similar line profile has been obtained by simulating the scattering of an emission line within a stellar wind (Figure 10 in  and Figure 4 in ). 

In the Na ID absorption profile (bottom right of Figure <ref>) it is possible to differentiate multiple components: a narrow, barely resolved (v_FWHM<10 km s^-1) component associated with the Milky Way gas (which is unchanging with time, according to ), a second narrow component associated with the host galaxy, NGC 300, and a broader absorption (FWHM velocity ∼200 km s^-1 for each component of the doublet) originating from the gas expanding around the transient (for a more detailed discussion of Na ID in ILRTs, see ).
Lastly, the Ca II H line presents a FWHM velocity of 420 km, ruling out an interstellar origin in favour of production within the more turbulent CSM surrounding the transient. The other line of the doublet, Ca II K, is contaminated by the Hϵ emission line, resulting in a misleading asymmetric profile with a short blue wing and an extended red wing.






















§ BROAD FEATURES IN LATE TIME SPECTRA


Late time spectra of ILRTs display some intriguing features. Both in NGC 300 OT and AT 2019ahd, two broad features start appearing at 6168 Å and 7002 Å a few months after peak luminosity (the position of the peak was estimated through a Gaussian fit). Notably, so far such broad lines were never detected in any spectrum of ILRTs. In Figure <ref> we show the late time spectra of NGC 300 OT and AT 2019ahd superimposed, in order to highlight the similarities between the two, in particular regarding these broad features. The red wing of the bluer feature (at 6168 Å) partially overlaps with the narrow [O I] λ6300 line, while another emission feature can be detected at λ6247. On the redder feature (at 7002 Å) there is a minor contamination from He I λ7065. It is challenging to exclude the presence of absorption components, similar to the ones detected in the H and Ca II NIR lines, given the rather low S/N. Both broad features present a FWHM velocity of ∼2500 km s^-1.
The lower panels of Figure <ref> show their evolution over time in NGC 300 OT, from their first detection at +123 d up to +243 d. Their development is remarkably slow, as the features appear barely different after 120 days. As for AT 2019ahd, we only detect the broad emission components in a single spectrum, therefore it is not possible to study their evolution in time.
Given the rather symmetrical distribution of these broad features around the Hα line, in principle they could be explained through Hα emission from two fast jets of matter moving at ∼20000 km s^-1 (based on the displacement of their peak with respect to the Hα rest wavelength), one approaching and one receding along the line of sight. However, since the features appear to be almost identical in both AT 2019ahd and NGC 300 OT, it is extremely unlikely to have observed two jets oriented exactly in the same direction relative to the Earth, and therefore we disfavour this scenario.









In the context of a terminal, explosive event, it is tempting to associate these features with the products of nucleosynthesis emerging from the slow moving CSM that was obscuring our line of sight during the photospheric phase. However, these broad features are challenging to identify, as they do not match the common nebular lines found in SNe (e.g. ).
Few atomic species can account for both the emission lines: according to the tabulated values reported by <cit.>, a candidate could be Ni I λλ 6177,7002, which would corroborate the scenario in which explosive nucleosynthesis took place during the ILRT events.
In order to test this hypothesis, in Figure <ref> we compare the late time spectra of NGC 300 OT and AT 2019ahd with the models of nebular spectra presented by <cit.>. The synthetic spectra have been scaled down for clarity. These models aim to reproduce the late time spectra of SNe arising from the core collapse of a 9 M_⊙ star (at zero age main sequence). The first synthetic nebular spectrum considered is relative to 300 days after explosion and has a solar abundance of elements. A high Ni content is expected to be an indicator of an ECSN event <cit.>: for this reason we consider a second synthetic spectrum (again provided by ), at the same evolutionary stage, in which the ratio Ni/Fe is 30 times the solar value. In the lower panels of Figure <ref> we display a zoomed in view of the broad lines found in NGC 300 OT, together with the same spectral region of the synthetic nebular spectra. In the region of the feature peaking at ∼7002 Å, there is a noticeable contribution in flux that is present in the Ni rich spectrum. 
However, the broad emission component at 6168 Å is not well reproduced by either synthetic spectrum, and an enhanced abundance of Ni does not seem to significantly influence this spectral region. Furthermore, a notable shortcoming of this interpretation is the lack of prominent [Ni II] lines (λλλ6667,7378,7412, marked with dashed lines in the upper panel of Figure <ref>) in our ILRTs spectra, which would instead be expected in the nebular spectra of a Ni-rich explosion <cit.>. Having considered these elements, we deem unlikely that the broad features observed originate from the nebular emission of fast, Ni-rich ejecta.








We now perform comparisons with empirical data, starting with the well studied SN 2005cs, a low luminosity SN IIP <cit.>. In the upper panel of Figure <ref> we plot the nebular spectrum of SN 2005cs, taken 280 days after explosion, together with the late time spectra of NGC 300 OT and AT 2019ahd. We observe a promising match for the broad feature at 6168 Å: interestingly, during the photospheric phase of SN 2005cs there are two absorption components compatible with P-Cygni profiles in that region (bottom left panel of Figure <ref>). The broad emission components could therefore originate from the blend of multiple Fe II (and possibly Fe I) lines, which form a P-Cygni profile during the photospheric phase of SN 2005cs (+64 d), but then turn to pure emission when the continuum fades (+280 d). Subsequently, almost one year after explosion (+336 d), the broad emission component appears weakened. On the other hand, SN 2005cs does not present strong emission around the region around 7002 Å, as shown in the bottom right panel of Figure <ref>, although there could arguably be a feeble P-Cygni profile in this region during the photospheric phase.

As a final comparison, we present the red nova V1309 Scorpii (hereby V1309 Sco, ), which is interpreted as the result of a merger of a contact binary system <cit.>. The spectrum taken 48 days after the start of the outburst shows remarkable similarities with the late time spectra of our ILRTs (Figure <ref>). Thanks to the high resolution provided by the VLT equipped with UVES, it is possible to distinguish multiple narrow features in the spectrum of V1309 Sco which corroborate the scenario in which the broad emission components originate from the blend of multiple narrow lines. Fe I and II lines are likely candidates, although pinpointing the exact transitions involved is quite challenging. As shown in the bottom panels of Figure <ref>, the insurgence of this forest of emission lines is sudden, since there is no hint of emission over the continuum in the regions around 6168 Å and 7002 Å in the spectrum of V1309 Sco taken 26 days after the outburst. 
Based on this comparison, we favour the interpretation that the broad emission lines observed in late time spectra of ILRTs are blends of multiple narrow (possibly Fe) lines, as observed for V1309 Sco.




This newly discovered resemblance between the spectra of V1309 Sco, AT 2019ahd and NGC 300 OT shows that similar chemistry and physical conditions of the emitting regions can be found in red novae and ILRTs. However, while there are some objects showing hybrid properties (e.g. AT 2022fnm ), this does not mean that ILRTs and LRNe can be easily blended into one class.
First of all, the timescales over which V1309 Sco and the ILRTs develop these broad emission features are quite different. This could be linked to the different scale of the phenomena: the peak optical luminosity of V1309 Sco is about ∼150 times lower than the peak luminosity of NGC 300 OT and AT 2019ahd, and it is characterised by a faster decline rate, dimming by 3 magnitudes in 50 days. 
Furthermore, late time spectra of red novae such as V1309 Sco are characterised by a red continuum eroded by the absorption of countless molecular bands, most notably TiO, VO, CO and H_2O (e.g. ).
 On the contrary, ILRTs never display the absorption features typical of molecular bands. This difference is made evident by comparing the spectrum of NGC 300 OT taken 514 days after maximum with X-Shooter with the spectrum of a red nova that already developed molecular band absorptions. Even at 514 days after maximum luminosity, the surviving continuum of NGC 300 OT does not show signs of molecular bands, which are instead prominent in the case of V1309 Sco, 244 days after the outburst (see Figure <ref> in Appendix).



§ ON THE NATURE OF ILRTS


The physical mechanism driving ILRTs has been a puzzle since the identification of this class of objects, 15 years ago. We quickly outline the main observed features of these transients before reviewing some scenarios that could explain their origin:


  
  ∙ ILRTs present a single peaked, monotonically declining light curve. The total radiated energy spans from few 10^47 to few 10^48 erg (Paper I).
  

  
  ∙ Whenever detected, their progenitors were heavily obscured in the optical and NIR domain, but bright sources in the mid-infrared. These objects were associated with dust-enshrouded, extreme AGB stars in the range of 8-12 M_⊙ <cit.>.
  

  
  ∙ When monitored for several years, ILRTs decline well below the progenitor level before disappearing, even in the MIR domain (, Paper I). This favours the interpretation of ILRTs as terminal events.
  

  
  ∙ Their spectra evolve slowly and are characterized by low excitation, narrow lines, with no signs of broad features even 17 months after the discovery.


Eruption of an intermediate-mass star – A first scenario proposed for the formation of ILRTs was the non-terminal eruption of a ∼9-15 M_⊙ star, which reached its Eddington limit and became dynamically unstable after shedding a considerable amount of mass <cit.>. The similarity between NGC 300 OT and the yellow hypergiant IRC+10420 () corroborated the idea of a stellar outburst producing an optically thick expanding envelope. However, this scenario struggles to explain the disappearance of the star observed years later in the mid infrared domain, since in the original view the progenitor would survive the eruption.

Stellar mergers – Thanks to the careful analysis of V1309 Sco (e.g. ) the class of transients known as red novae was soundly associated with non-degenerate stellar mergers. 
Depending on the masses of the progenitor stars, the peak magnitude of red novae can span from being significantly fainter than a classical nova (M_r∼–4.5 mag, e.g. OGLE-2002-BLG-360, ) up to competing with genuine core collapse SNe (M_r∼–15.5 mag, e.g. AT 2017jfs, ). On the other hand, ILRTs known to date peak consistently in the range –15 mag ≲ M_r≲–11 mag (Paper I, ).
Thus, there is some overlap in the energy radiated by ILRTs and the brightest red novae: this degeneracy is worsened by the similar spectral features shown by ILRTs and red novae during their photospheric phase. This can give rise to ambiguous objects, for which a clear classification is uncertain (e.g. M85 OT2006-1, ). At the same time, there are some peculiarities in these two classes of transients that help with their identification: several (although not all) red novae present a double-peaked light curve, typically weaker or absent [Ca II] emission lines, and most importantly red novae always display molecular absorption bands in their late time spectra <cit.>, while ILRTs never show such bands. Furthermore, the progenitor systems of different red novae were observed in the optical domain (e.g. ), excluding the presence of large dust quantities along the line of sight, which is instead a key feature of ILRTs progenitors. In principle, these differences could be tentatively explained with a different viewing angle on the same type of progenitor system (such as binary stars surrounded by a dust torus): it was indeed suggested that the same physical mechanism could power both ILRTs and red novae (e.g. ). However, the expected result of a stellar merger struggles to be reconciled with the severe mid infrared dimming observed for ILRTs over the course of a few years. Prolonged monitoring is therefore crucial to confirm the complete disappearance of the object, since there are known cases of transients settling on a lower luminosity compared to their progenitor (e.g. ).

Failed supernovae – A hypothetical link was suggested between ILRTs and fallback supernovae, events in which a large mass fraction of the exploding star in unable to escape the gravitational potential of the compact object formed during the core collapse (e.g. ). <cit.> found that the low energy radiated and the low photospheric temperature reached by ILRTs is compatible with radiation hydrodynamical simulations of matter ejected by a failed supernova which is interacting with pre-existing CSM. This interpretation has the merit of linking together a terminal event with the low luminosity and the narrow lines spectra of these transients. However, as the authors remarked, their simulations involve Wolf-Rayet and blue supergiants, which are not compatible with the progenitors of ILRTs discovered so far. Additionally, the monitoring of NGC 300 OT and SN 2008S spanning several years has not shown signs of fallback accretion on the black hole, which would be expected in this scenario. 


Electron capture supernovae – Multiple works have proposed the association between ILRTs and ECSNe <cit.>. The 8-12 M_⊙ extreme AGB progenitors identified by <cit.> are the ideal candidates for giving rise to an ECSN explosion, and the intrinsic weakness of the event expected from the theory <cit.> is encouraging when considering these underluminous objects. ILRTs, despite showing some degree of individuality, appear to be a relatively homogeneous family of objects, especially compared with the profound variability of the observable parameters within the red novae class. This can be interpreted as transients arising from similar progenitors (8-12 M_⊙ extreme AGB stars) whose differences are caused by the interaction with different environments.
The declining of the transient below the progenitor level is definitely expected in this scenario, given the terminal nature of ECSNe. In this context, the late time spectra characterized by narrow emission lines which were shown in this work deserve additional commentary. Type IIn SNe have shown that large quantities of CSM can bottle up the ejecta of violent stellar explosions for several years (at least 4 years in the case of SN 2015da; ). Actually, among SNe IIn it is rare to witness the emergence of the ejecta from the CSM, as observed for example for SN 1995N years after its explosion <cit.>. In analogy, the absence of ejecta signatures in our latest ILRTs spectra does not exclude their presence underneath the CSM. The ECSN origin of these transients therefore remains a valid interpretation of the observed data, also corroborated by indirect indicators such as event rates (, ) and dust chemical composition ().



§ SUMMARY AND CONCLUSIONS


We analysed a rich collection of optical and NIR spectra of four ILRTs. Their slow evolution coupled with the presence of narrow (few 10^2 km s^-1) emission lines is indicative of the presence of optically thick CSM surrounding the transients. Balmer lines as well as Ca features (Ca H&K absorption doublet, [Ca II] doublet and Ca II NIR triplet) dominate the optical spectra and are recognizable at most epochs.
The O I λ 8446 line appearing at late time is a consequence of the Bowen fluorescence, and could be the result of coupling between the O I λ1025.76 line in the CSM and the Lyβ line emitted by underlying ejecta.
The luminosity evolution of the H emission lines and Ca II NIR triplet display a characteristic double peaked behaviour, an opposite trend compared with the neighbouring continuum flux: this is possibly linked to the interaction between fast ejecta and the slow moving CSM. 

The high resolution spectrum of NGC 300 OT obtained with VLT+UVES shows the presence of a complex environment, possibly multiple shells of gas, causing narrow absorption features to appear on the most prominent emission lines. The characteristic shape of the [Ca II] lines is likely a result of electron scattering in an expanding medium, and allows us to postulate the presence of an extended, slow (∼30 km s^-1) wind surrounding the progenitor star.
Late time spectra of NGC 300 OT and AT 2019ahd display broad features never observed in any ILRT so far. We find a striking similarity with the behaviour of the red nova V1309 Sco, and we tentatively explain these broad emission components with a blend of several narrow lines, possibly Fe transitions. 
In Paper I we discussed how the low energy released by ILRTs coupled with their terminal nature (at the very least for NGC 300 OT) make them appealing ECSN candidates. The spectroscopic analysis performed in this work provides a characterisation of the CSM surrounding these transients: while we do not find a smoking gun evidence for the explosion scenario, our findings are overall consistent with a weak ECSN event within a thick CSM.





We thank Steve Shore and Valentina D'Orazi for the useful and lively discussions.
We thank the staff of the various observatories where data were obtained for their assistance.
Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
Observations from the Nordic Optical Telescope were obtained through the NUTS2 collaboration which are supported in part by the Instrument Centre for Danish Astrophysics (IDA) and the Finnish Centre for Astronomy with ESO (FINCA) via Academy of Finland grant nr 306531. The data presented here were obtained in part with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA).
Time-domain research by the University of Arizona team, and D.J.S. is supported by NSF grants AST-1821987, 1813466, 1908972, 2108032, and 2308181, and by the Heising-Simons Foundation under grant #2020-1864.
H.K. was funded by the Research Council of Finland projects 324504, 328898, and 353019.
This work makes use of data from the Las Cumbres Observatory network. The LCO team is supported by NSF grants AST–1911225 and AST–1911151, and NASA SWIFT grant 80NSSC19K1639. 
Data were also obtained at the Liverpool Telescope, which is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos with financial support from the UK Science and Technology Facilities Council. 
Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma.
Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, as part of ePESSTO+ (the advanced Public ESO Spectroscopic Survey for Transient Objects Survey).
ePESSTO+ observations were obtained under ESO program IDs 1103.D-0328 and 106.216C (PI: Inserra).
LCO data have been obtained via OPTICON. The OPTICON project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant no. 730890.
Time domain research by the University of Arizona team and D.J.S.  is supported by NSF grants AST-1821987, 1813466, 1908972, & 2108032, and by the Heising-Simons Foundation under grant #20201864. 
G.V., N.E.R. and I.S. acknowledge financial support from PRIN-INAF 2022 “Shedding light on the nature of gap transients: from the observations to the models".
A.R. acknowledges financial support from the GRAWITA Large Program Grant (PI P. D’Avanzo) and the PRIN-INAF 2022 "Shedding light on the nature of gap transients: from the observations to the models.
Y.-Z. Cai is supported by the National Natural Science Foundation of China (NSFC, Grant No. 12303054), the Yunnan Fundamental Research Projects (Grant No. 202401AU070063) and the International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001).
L.G. acknowledges financial support from the Spanish Ministerio de 
Ciencia e Innovación (MCIN), the Agencia Estatal de Investigación 
(AEI) 10.13039/501100011033, and the European Social Fund (ESF).
This work was funded by ANID, Millennium Science Initiative, ICN12_009
M.N. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948381). Part of the funding for GROND (both hardware as well as personnel) was generously granted from the Leibniz-Prize to Prof. G. Hasinger (DFG grant HA 1850/28-1).
H.K. was funded by the Academy of Finland projects 324504 and 328898. 
R.K. acknowledges support from the Research Council of Finland (340613).



















aa




§ AFFILIATIONS


^1INAF – Osservatorio Astronomico di Padova, Vicolo 
             dell’Osservatorio 5, I-35122 Padova, Italy

^2INAF – OATS, Via G.B. Tiepolo 11, 34143 Trieste, Italy

^3INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807, Merate (LC), Italy 

^4Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^5International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216, P.R. China 

^6Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, P.R. China 

^7Graduate Institute of Astronomy, National Central University, 300 Jhongda Road, 32001 Jhongli, Taiwan 

^8SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA, Leiden, The Netherlands 

^9Department of Astrophysics/IMAPP, Radboud University Nĳmegen, P.O. Box 9010, 6500 GL, Nĳmegen, The Netherlands 

^10Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, E-08193 Barcelona, Spain 

^11School of Physics, O’Brien Centre for Science North, University College Dublin, Belfield, Dublin 4, Ireland 

^12The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden 

^13Hiroshima Astrophysical Science Center, Hiroshima University, Japan 

^14Department of Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL 32306, USA 

^15Las Cumbres Observatory, 6740 Cortona Dr. Suite 102, Goleta, CA, 93117, USA

^16Department of Physics, University of California, Santa Barbara, CA, 93106, USA

^17Cardiff Hub for Astrophysics Research and Technology, School of Physics & Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK 

^18INAF, Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy 

^19DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark 

^20Caltech/IPAC, Mailcode 100-22, Pasadena, CA 91125, USA 

^21Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 20014, Turku, Finland 

^22Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden 

^23Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK 

^24Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, D-85748 Garching, Germany 

^25Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital 263001 India 

^26Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

^27School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland 

^28Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

^29Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH 

^30Astrophysics Research Centre, School of Mathematics and Physics, Queens University Belfast, Belfast BT7 1NN, UK 

^31Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 

^32INAF – Osservatorio Astronomico d’Abruzzo, via M. Maggini snc, Teramo, I-64100, Italy 

^33Department of Physics, University of California, Davis, CA 95616, USA 

^34 European Southern Observatory, Alonso de Córdova 3107, Casilla 19, Santiago, Chile 

^35Millennium Institute of Astrophysics, Nuncio Monsenor Sótero Sanz 100, Providencia, 8320000 Santiago, Chile 

^36INAF-Osservatorio Astrofisico di Catania, Via Santa Sofia 78, I-95123 Catania, Italy 

^37Instituto de Astrofísica, Universidad Andres Bello, Fernandez Concha 700, Las Condes, Santiago RM, Chile 

^38ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy 

^39Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 

^40INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, I-38123 Trento, Italy 

^41Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain. 

^42Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland 

^43Center for Astrophysics, Harvard & Smithsonian, Cambridge, Massachusetts, MA 02138, US 

^44The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA 

^45Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500 Piikkiö, Finland 

^46DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 

^47Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monteporzio Catone, Italy 

^48Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 

^49INFN. Sezione di Perugia, via A. Pascoli s/n, I-06125 Perugia, Italy 

^50Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli studi di Padova Vicolo dell’Osservatorio 3, I-35122 Padova, Italy 

^51IAASARS, National Observatory of Athens, Metaxa & Vas. Pavlou St., 15236, Penteli, Athens, Greece 

^52Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA 

^53Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße 1, 85748, Garching, Germany 

^54Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA 

^55Cosmic Dawn Center (DAWN), Denmark 

^56Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 København N, Denmark 

^57Manipal Centre for Natural Sciences, Manipal Academy of Higher Education, Manipal - 576104, Karnataka, India 

^58Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA 

^59Space Telescope Science Institute: Baltimore, MD, US 

^60Indian Institute Of Astrophysics, 100 Feet Rd, Santhosapuram, 2nd Block, Koramangala, Bengaluru, Karnataka 560034, India 









§ APPENDIX: ADDITIONAL TABLES AND FIGURES
 




































 Paper Title: Discovery of Polarized X-Ray Emission from the Accreting Millisecond Pulsar SRGA J144459.2-604207.





Alessandro Papitto
alessandro.papitto@inaf.it

0000-0001-6289-7413]Alessandro Papitto
INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy

0000-0003-0331-3259]Alessandro Di Marco
INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy

0000-0002-0983-0049]Juri Poutanen
Department of Physics and Astronomy, 20014 University of
Turku, Finland
Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia

0000-0001-6356-125X]Tuomo Salmi
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098XH Amsterdam, the Netherlands
Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy
Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133 Roma, Italy
Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy


0000-0001-8916-4156]Fabio La Monaca
INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy

0009-0009-3183-9742]Anna Bobrikova
Department of Physics and Astronomy, 20014 University of
Turku, Finland


INAF Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italia


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy
Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy



INAF/IASF Palermo, Via Ugo La Malfa 153, 90146 Palermo, Italy
Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu, KM 0.7, Monserrato 09042, Italy


INAF Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italia


Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain
Institut d’Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels (Barcelona), Spain


Dipartimento di Fisica e Chimica – Emilio Segrè, Università di Palermo, Via Archirafi 36, 90123 Palermo, Italy


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy


Department of Physics and Astronomy, 20014 University of
Turku, Finland


Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy


ASI - Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Rome (RM), Italy


0000-0002-0940-6563]Mason Ng
MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

0000-0001-7397-8091]Maura Pilia
INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy


Dipartimento di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu, KM 0.7, Monserrato 09042, Italy


INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy


Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Joint Space-Science Institute, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA


Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
 
§ ABSTRACT 
We report on the discovery of polarized X-ray emission from an accreting millisecond pulsar. During a 10 day-long coverage of the February 2024 outburst of , the Imaging X-ray Polarimetry Explorer () detected an average polarization degree of the 2–8 keV emission of 
2.3%± 0.4% at an angle of  59±6 (East of North; uncertainties quoted at the 1σ confidence level). The polarized signal shows a significant energy dependence with a degree of 4.0%±0.5% between 3 and 6 keV and <2%  (90% c.l.) in the 2–3 keV range.
We used , , and  observations to obtain an accurate pulse timing solution and perform a phase-resolved polarimetric analysis of  data. We did not detect any significant variability of the Stokes parameters Q and U with the spin and the orbital phases. We used the relativistic rotating vector model to show that a moderately fan-beam emission from two point-like spots at a small magnetic obliquity (≃ 10) is compatible with the observed pulse profile and polarization properties.  also detected 52 type-I X-ray bursts, with a recurrence time Δ t_ rec increasing from 2 to 8 h as a function of the observed count rate C as Δ t_ rec∝ C^-0.8. We stacked the emission observed during all the bursts and obtained an upper limit on the polarization degree of 8.5% (90% c.l.).







§ INTRODUCTION
 



The simultaneous measurement of the mass M and the equatorial radius R_ eq of a few neutron stars (NSs) is of paramount importance for constraining the equation of state of ultra-dense matter because of its one-to-one mapping to the NS mass–radius dependence <cit.>.
Relativistic effects bend the light trajectory and modify energy of the photons emitted by hot spots on the surface of millisecond pulsars  <cit.>. As a result, modeling the X-ray pulse profiles of MSPs is one of the most powerful techniques to obtain the desired information on M and R_ eq <cit.>.  The pulse shape also depends on several geometrical parameters (e.g., the binary inclination i, the spot co-latitude θ, and its angular size  ρ) as well as spectral parameters (e.g., the angular pattern of the emitted radiation; see, e.g., Fig. 1 of  for the geometry assumed). Breaking this degeneracy with X-ray spectral-timing data requires a very large number of counts <cit.>. A few megasecond-long NICER observations of rotation-powered MSPs met this requirement and led to constraints on M and R_ eq with a relative uncertainty of ∼ 10% for three of them <cit.>.

Accreting millisecond pulsars <cit.> represent an intriguing alternative.  During accretion outbursts, they attain an X-ray luminosity of L_ X≃ 10^36–10^37 , i.e. 5–6 orders of magnitude brighter than rotation-powered MSPs. 
The degeneracy between M and R_ eq and the spot geometrical and spectral parameters <cit.> prevented to get meaningful constraints from the data taken with, e.g., the  Rossi X-ray Timing Explorer   (M=1.2–1.6 M_⊙, R_ eq=6–13 km by ; M=0.8–1.7M_⊙, R_ eq=5–13 km by ). 
However, <cit.> showed that if i and θ were known a priori with an accuracy of a few degrees, those data would have allowed measurements with a relative uncertainty 
comparable to or slightly smaller than that for rotation-powered MSPs.  

X-ray polarimetry might provide key information on the geometrical parameters of the spots. Soft photons from the NS surface are up-scattered by hot electrons in the accretion shock <cit.> achieving polarization degree (PD) from a few per cent up to 10–20%  <cit.>. 
The magnetic field strength of AMSPs (10^8–10^9 G) is too weak to affect X-ray photon polarization. As a result, the polarization vector of the X-rays emitted from the accretion shock is expected to lie either in the meridional plane, formed by the photon momentum and the normal to the surface, or perpendicular to it. 
As the NS rotates, the polarization angle (PA) should thus swing as a function of the pulsar spin phase in accordance with the rotating vector model <cit.>. 
In  AMSPs, the relativistic motion of the emission regions causes additional rotation of the polarization plane <cit.>; also the oblate shape of a quickly rotating NS has to be accounted for <cit.>.


Here, we report on the discovery of polarized X-ray emission from the AMSP  (SRGA J1444 in the following)  by the Imaging X-ray Polarimetry Explorer (; ) in the context of a campaign which included also , , and  observations. Discovered in outburst on 2024 February 21 by the Mikhail Pavlinsky ART-XC telescope on-board the Spectrum-Roentgen-Gamma observatory <cit.> at a relatively bright 4–12 keV X-ray flux of 2×10^-9  (see Figure <ref>), SRGA J1444 showed coherent pulsations at 447.9 Hz first detected by , coming from a NS in a 5.2 h orbit around a donor with a mass in the range 0.2–0.7 M_⊙ <cit.>. SRGA J1444 also showed type-I X-ray bursts <cit.> quite regularly, with a recurrence time increasing from 1.6 to 2.2 h  as the X-ray flux was observed to decrease <cit.>. No optical counterpart could be identified <cit.> at the most accurate X-ray position obtained with High Resolution Camera observations on board the  observatory <cit.>, most likely due to the extinction of A_ V≃ 7 mag <cit.>  in that direction. <cit.> reported a radio counterpart with a flux density of ∼ 200 μJy at 5.5 GHz and a flat spectral shape, consistent with a compact jet or discrete ejecta.







§ OBSERVATIONS


lcccccc
Observations analyzed in this paper.
0pt

Observatory     Obs. ID     Start date (MJD)     End Date (MJD)     Instrument     Exposure (s)      Energy band (keV)
    03250101      60367.556     60377.676     DU1     552930     2–8 

                          DU2     553541     

                            DU3     554066     

    620419      60361.831     60363.908     XTI     9536     0.3–10 

     663908      60364.219     60392.971         24805     
 
     80901307002      	60366.459     60369.883     FPMA     157675     3–79 

            	          FPMB     157440     

     0923171501     60368.102     60369.376     EPIC-pn     110077     0.3–10 



Table <ref> lists the observations analyzed in this paper, using HEASoft package version 6.33.2.  started observing SRGA J1444 less than a week after the first detection of the source, in response to the trigger of the GO program 03250101 (PI: A. Papitto). 
Observations lasted ∼10 days 
with visibility gaps giving a duty cycle exceeding 60 per cent.
We used the  algorithm in the  tool of the ixpeobssim package version 30.6.4 <cit.>, along with the  CALDB released on 2024 February 28, to extract polarimetric information using the formalism from <cit.>. We selected source photons from a circular region with a radius of 100 centered at the source position. We use the unweighted analysis implemented in ixpeobssim <cit.> to measure the Stokes parameters (I,Q,U). Figure <ref> shows the 2–8 keV IXPE count rate, the polarization degree PD=√(q^2+u^2) and angle PA=1/2arctan (u/q) measured in the 3–6 keV band, where q=Q/I and u=U/I.
As the source count rate is relatively high (∼2–8 count s^-1), the background contributes at most 5% of the total rate and we did not subtract it following the recommendation of <cit.>. For the  spectral analysis, we used the response matrices  present in the CALDB, computed ancillary response files using  weighting the Stokes parameters following the analysis presented by <cit.> and binned the I spectra to have at least 30 counts per bin. To ensure statistics high enough for a spectro-polarimetric analysis, we binned data to a constant resolution of 200 eV.




<cit.> reported the analysis of NICER observations during the first three days of the outburst using publicly available data. Here, we also include data taken afterwards in the GO program ID 663908 (PI: Papitto). Visibility constraints and high background prevented  to collect data simultaneously with the  (see Figure <ref>). The monitoring briefly resumed on 2024 March 14 and more regularly after March 21, when the source count rate had already decreased close to the background value. We reduced the observations using calibration version xti20240206 and the same filtering criteria adopted by <cit.>, retaining photons with energy ranging between 0.3 and 10 keV for a pulse timing analysis.

The EPIC-pn telescope on-board  observed SRGA J1444 uninterruptedly for ∼110 ks.
We retained 0.3–10 keV source photons extracted from a 21 pixel-wide stripe (equivalent to 861) around the position of the pulsar to perform a pulse timing analysis.

 provided the longest overlap with  data starting on 2024 February 26.
We reduced the data using the calibration version indx20240325, which includes the fine clock correction file ver. 179.
A circular region with a wide radius of 140 was adopted to extract source photons in the 3–79 keV band.


The times of arrival of the photons recorded by all the instruments were transformed into the inertial reference frame centred at the Solar System barycenter using the Chandra coordinates reported by <cit.>, R.A.= 14^ h 44^ m 5894, Dec = -6041553 (J2000) and the DE 405 ephemeris.  We detected 52, 5, 13 and 23 type-I X-ray bursts during , ,  and  exposures, respectively. When analysing the persistent emission, we discarded intervals between 5-s before and 60-s after the onset of each burst. 




§ RESULTS




 §.§ X-ray pulse timing


lcccc
Pulse phase timing solutions of .

0pt

                     

Interval (MJD)     60361.837–60383.544     60368.106–60369.374     60366.465–60369.881     60367.615–60377.619 

ϕ(T_0)     0.47(1)     0.500(7)     0.396(2)     0.37(1)

ν(T_0) (Hz)     447.871561115(24)     447.87156196(24)     447.87156122(17)     447.87156138(8)  

ν̇ (Hz s^-1)     <1.2×10^-13     <7.0×10^-12     <7.4×10^-13     <3.0×10^-13 

a_1 sin i /c (lt-s)     0.65052(2)     0.650491(5)     0.650494(4)     0.650486(13) 

P_ orb (s)     18803.65(1)     18803.65(1)     18803.690(4)      18803.665(4) 

T_ asc (MJD)     60361.641306(3)     60361.641305(6)     60361.641291(2)     60361.641310(5) 

e     <9×10^-5     <9×10^-5     <3×10^-5     <1.0× 10^-4 

χ^2 (d.o.f.)      93.0 (53)     368.5 (212)     713.0 (378)     121.3 (53) 

All the solutions presented in the table were obtained by folding data around ν_ fold=447.87156116029 Hz with a reference epoch MJD 60361.8. To facilitate the comparison, the frequency estimates were evaluated at T_0=MJD 60368.0. All the epochs reported here are given in the Barycentric Dynamical Time (TDB) system. Uncertainties at the 1σ confidence level are reported in parentheses. Upper limits are given at a 90% confidence level.


We used the pulsar timing solutions based on the first few days of  observations <cit.> to correct the arrival time of X-ray photons for the light travel time delays due to the pulsar orbit.
We used two harmonic components to model the pulse profiles obtained by folding data taken in ∼500 s-long intervals and sampled by 16 phase bins. Table <ref> shows the timing solutions based on the observed evolution of the phase computed over the fundamental harmonic for the various instruments considered here, plotted in Fig. <ref> with the best-fitting linear trend). We attribute the residual random phase variability by ∼0.05–0.1 cycles to pulse timing noise, as observed from other AMSPs with an even larger magnitude <cit.>. The two values of the pulse phase measured by  at MJD 60384,  after an interruption of the monitoring due to visibility constraints, line up with the trend observed earlier and  allow us to put a tight upper limit on the spin frequency derivative (ν̇<1.2×10^-13 Hz s^-1). However, we caution that such an alignment could be fortuitous, as the pulse phase of AMSPs often shows erratic variability, especially in the late stages of an outburst <cit.>. 

We used these ephemerides to fold 5 ks-long intervals of  data and recover the pulsed signal notwithstanding the lower counting statistics. Figure <ref> shows the fractional amplitude and phase of the fundamental harmonic observed by . An increase of the amplitude from ∼4 to 15%  is observed after MJD 60376.5.




The frequencies measured by  and   are compatible within the errors (δν/ν≃2×10^-10). On the other hand,  and  show a frequency difference of δν/ν≃ 0.6×10^-9 and 2×10^-9 with respect to the NICER measurement, respectively. The latter is compatible with the value quoted in the latest calibration technical note for the  clock relative time accuracy[<https://xmmweb.esac.esa.int/docs/documents/CAL-TN-0018.pdf>] (δν/ν≃ 10^-8).
The pulse phase recorded by  and  at epoch T_0 precedes those observed by  by 0.07–0.1 cycles. The NuSTAR shift can be understood in terms of the dependence of the pulse phase on energy already noticed by SRG/ART-XC <cit.>; the reason for the  mismatch is less clear and will be addressed in a future paper. For consistency, the phase-resolved analysis presented in Sect. <ref> is based on the  timing solution, even if slightly less accurate. 










 §.§ Polarized X-ray emission




Figure <ref> shows the average normalized Stokes parameters observed during the persistent emission by the three IXPE detector units. The average Stokes parameters are q=-1.1±0.4% and u=2.0±0.4%, which translate into an average PD of  2.3 ± 0.4%, and an angle of PA 59±6  (measured East of North, errors are quoted at the 1 σ c.l. unless differently specified; see Fig. <ref>). The probability of detecting such values of the normalized Stokes parameters from noise fluctuations is 1.1×10^-7 (corresponding to a 5.3 σ c.l.).
Figure <ref> show the results of an energy-resolved analysis. Polarized emission is detected at the highest statistical significance in the 3–6 keV energy range with an average  PD_3-6=4.0%±0.5% and PA_3-6=57±3. Between 2 and 3 keV, the PD is not significant (0.7%±0.6%) with an upper limit at 90% confidence level of PD_2-3<2.0%.
Above 6 keV, the 5% upper limit on the PD
becomes larger than both PD_3-6 and the minimum detectable polarization at 99% confidence level in the 3–6 keV band (MDP_99=1.4%). 
We did not detect any significant variation or rotation of the 3–6 keV polarization vector over 2 d-long intervals, even though the PD slightly increases halfway through the observation  (see bottom panels of Figure <ref>). 

Similar to <cit.>, we used the sum of a black-body ( in xspec) and thermal Comptonization  () to satisfactorily model (χ^2=483.9 for 441 d.o.f.) the 2–8 keV  energy spectrum. We modelled absorption with the Tuebingen-Boulder model <cit.> with  the absorption column fixed to N_ H=2.9×10^22 cm^-2
and quote uncertainties on the spectral parameters at 90% c.l.
The thermal component has a temperature of kT_ bb=0.52±0.01 keV and an apparent size of R_ bb=(14.0±0.5) d_8 km, where d_8 is the distance to the source in units of 8 kpc. Comptonization is described by an asymptotic power law with index Γ=1.70±0.03 and a soft photon input with temperature <0.15 keV. The electron temperature was fixed at a value beyond the  band-pass kT_ e=15 keV. The Comptonization component accounts for ∼95% of the 2–8 keV observed flux of (5.6±0.1)×10^-10 .
Spectral analysis of the  observation (Obs. ID 6639080103) performed ∼12 h before the start of the  coverage indicates a smaller black-body size  (kT_ bb=0.68±0.05 keV; R_ bb=(5.7±1.0) d_8 km) and a  steeper slope of the Comptonized emission (Γ=1.90±0.01). 

The decrease of the PD below 3 keV, i.e. where the soft thermal component gives its highest contribution to the 2–8 keV spectrum, motivated us to perform a spectro-polarimetric analysis. 
By fitting the spectra of the three Stokes parameters with a 
constant polarization component to multiply the model presented above,  , gave  PD=2.6%±0.5% and PA=59±6 with a χ^2/d.o.f. of 340/257, compatible with the  results.
By using separate constant polarization for the two spectral components gave a slight improvement of the fit χ^2/d.o.f.=328/255
with a  component polarized with the PD=8%±5% at PA=-43 ± 19 and the  with the PD=4.6%± 1.1% and PA=55± 7. 











 §.§ Phase-resolved variability of the polarized X-rays



To perform a phase-resolved analysis of the polarimetric properties of SRGA J1444, we considered  data corrected with the orbital parameters listed in Table <ref>.  We used the  algorithm to evaluate in n=16 phase bins the Stokes parameters Q and U, which are expected to be normally distributed <cit.>. We restricted the analysis to the 3–6 keV energy band in order to maximize the strength of the polarized signal.  
We used the  algorithm without acceptance corrections to measure the Stokes parameters in counts per second. 

From top to bottom, Figure <ref> shows the phase-resolved  Stokes parameters, the PD and the PA observed before the pulse amplitude increase which occurred at MJD 60376.5.

The Stokes parameters Q and U did not show significant variability. A fit with a constant function gave a χ^2 of 16.6 and 15.6 for 15 d.o.f., respectively, with  average values of q=Q/I=-1.6%±0.5% and  u=U/I=3.5%±0.4%.

We also analyzed separately data taken after the pulse amplitude increase which occurred at MJD 60376.5 (see Figure <ref>). 
Fitting with a constant gives q'=0.5%±1.4% and u'=2.4%±2.7% for χ^2=10.2 and 18.7 for 15 d.o.f.. Both Stokes parameters are compatible with zero within the uncertainties, even though the value of u' measured at phase 0.4 significantly deviated from the trend, leading to an increase of the PD up to ∼30%.

We also measured the Stokes parameters over 15 orbital phase bins to search for orbital variability of the polarization properties but found none significant. Constant values of Q and U modeled well the dependence of these values on the orbital phase (χ^2=9.1 and 12.5 for 14 d.o.f., respectively). 






§ TYPE-I X-RAY BURSTS


During the observations reported here,  detected 52 type-I X-ray bursts which shared similar properties (see the inset of Figure <ref> for an example). The sample of peak count rates had an average of 38.7 cnt s^-1 and a standard deviation of 2.9 cnt s^-1. After subtracting the persistent emission, the distribution of the fluence of the  observed bursts had an average of 649 cnt with a standard deviation of 40; the fluence of the different bursts was very similar and only slightly increased for bursts occurring when the persistent count rate had decreased below 3 cnt s^-1. The relatively low number of counts observed by  prevented us from performing a time-resolved spectral analysis of the single bursts. By stacking all the observed bursts, and fitting the observed spectra with the sum of a thermal component and the two-component model used to fit the persistent spectrum (see Sec. <ref>), we get ⟨ kT_ burst⟩=1.83±0.05 keV and R_  burst=(6.2±0.2) d_8 km. Defining the burst duration as τ=ℱ/F_ peak, and neglecting variations of the spectrum throughout the burst,  gives τ = 16.8±1.6 s. Figure <ref> shows the recurrence times Δ t_ rec between the observed bursts[Consecutive bursts without data gaps could not be observed in IXPE data, but the uninterrupted XMM coverage allowed us to define the recurrence time unambiguously.] as a function of the average persistent count rate C measured during the longest possible contiguous interval before the burst. For 14 bursts, we scaled the time elapsed since the previous burst by a factor of 2 as the immediately preceding burst was likely missed due to interruption in the  coverage (green points in Figure <ref>). For one burst (red point), we used a scaling factor of 3. 






To search for a polarized signal during the X-ray bursts, we stacked the emission observed during all the events, retaining 32 s after the onset of each of the bursts for a total exposure of 1664 s. We measured q_ b=Q_ b/I_ b=-0.8%±3.3% and u_ b=U_ b/I_ b=2.8%±3.3%, resulting in an upper limit on the PD of the sum of the burst and the persistent emission of PD_UL=7.2% (90% c.l.).
We then subtracted the absolute Stokes vectors observed during the persistent emission after normalizing them to the exposure time of 1664 s over which the burst emission was integrated, summing in quadrature the uncertainties, and obtained normalized and background subtracted Stokes parameters q̃_ b=-0.7%± 3.9% and ũ_ b = 3.0%±3.9%, corresponding to an upper limit on the PD of 8.5% (90% c.l.). Such an upper limit is  larger than PD_UL by a factor 1+r≃1.2, where r is the ratio between the burst (only) and the persistent flux.



§ DISCUSSION




 §.§ Average polarization properties


For the first time,  observations of  allowed us to significantly detect polarization from an AMSP with an average 2–8 keV degree of 2.3%±0.4%. Such a value increases to 4.0%±0.5% if the analysis is restricted to the 3–6 keV energy band.   Comptonization by hot (kT_ e∼20–60 keV) electrons in slabs of moderate optical depth (τ≈1–2) located above hot spots (kT_ bb∼ 1 keV)  on the NS surface accounts for more than 90% of the emission in the  bandpass of AMSPs such as SRGA J1444 <cit.>  and represents the most natural explanation of the polarization we detected.  Values of the PD as high as 10–20% were predicted by modelling the scattering in Thomson approximation <cit.>. However, the actual degree of polarization is a function of the physical properties of the scattering medium <cit.>.
Recently, <cit.> applied the formalism for Compton scattering in a hot slab <cit.> to describe the Stokes parameters as a function of energy and angle in terms of the electron temperature and optical depth of the slab and the seed photon temperature. They found values of the PD of a few per cent at the most, decreasing to values undetectable to  at high (>50 keV) electron temperatures. They concluded that chances for  to detect polarization from an AMSP significantly increased for systems with a larger optical depth (τ>2) and lower electron temperature (kT_ e < 30 keV).  <cit.> fitted the 4–30 keV ART-XC spectrum with Comptonization from a medium with kT_ e∼ 25 keV and τ∼ 3.2 (evaluated from the photon index of Γ=1.8 as in, e.g., ). Preliminary fitting of the 3–79 keV  spectrum (Malacaria et al., in prep.) suggests an even lower temperature (kT_ e=10.9^+0.2_-0.1 keV) and higher optical depth (Γ=2.2±0.1, τ=4.0± 0.3).  In the framework developed by <cit.>, SRGA J1444 may thus present favorable spectral conditions to produce a relatively high degree of X-ray polarization compared to other AMSPs which might show even lower values. In addition, <cit.> showed that the observed polarization degree increased when an observer mostly sees the spot under high angles (i.e., |i-θ| close to π). Such a configuration is supported by the results presented in Sect. <ref>.

We detected a significant decrease in the PD  below 3 keV. In the IXPE bandpass, that energy band is where the non-upscattered blackbody emission contributes the most to the total observed flux. However, even though these photons are not expected to be polarized, they only account for 5-10% of the total emission. As a result, the influence of such an unpolarized component at low energies does not readily explain the observed decrease of the PD. The analysis presented in Sec. <ref> shows that, to explain the decrease of the total polarization, the polarization angle of the soft blackbody component  (contributing to 7.7%±1.4% of the 2–3 keV emission, according to the  spectral analysis reported in Sect. <ref>) should differ by ∼ 90 from the  polarization angle compared of the Comptonization component . Yet, the relatively modest improvement of the fit description compared to a model with the same polarization properties for the two spectral components suggests that an intrinsic evolution of the polarization properties of the Compton scattered photons is a more likely explanation of the observed energy dependence. <cit.> predicted a swing by ∼90  of the polarization vector of the Comptonized emission at an energy of ≃5 keV,  only slightly higher than the turn-over energy of ≃3 keV we observed. A comprehensive analysis of the spectral properties of the source, including also  data, is in preparation. 



 §.§ Phase-resolved analysis



Assuming the accretion columns as the source of the polarized X-ray photons we observed, it is necessary to ask what spot geometrical configuration can explain at the same time the observed pulse profile and the non-significant phase-resolved variability of the Stokes parameters Q and U.

Unlike the case of highly magnetized, slowly rotating X-ray pulsars <cit.>, the polarization properties of AMSPs are defined by the geometry of the spots, relativistic effects <cit.>, and possibly eclipses by the inner parts of the accretion disks <cit.>. This does not allow us to use a simple rotating vector model to fit the observed phase-resolved PD and PA, but requires a fit of the total Stokes vector (I,Q,U) which is the sum of the Stokes vectors of the two spots. 

The spots' Stokes parameters depend on how the total and polarized radiation vary with the spin phase, photon energy and zenith angle <cit.>. A modeling that includes an accurate description of all these effects is beyond the scope of this letter and will be presented elsewhere.
Here, we draw considerations on the system's geometry by using the approximate formulae given by <cit.> and <cit.> for the phase-resolved Stokes parameters. We assume two point-like spots on the surface of a rapidly rotating NS which emit X-rays with a power-law spectrum with I(α',E)=I_0 (1+hcosα') E^-(Γ-1). Here, Γ=1.9 is the spectral photon index, α' is the zenith angle (i.e., the angle between the normal to the spot and emission direction) in the co-rotating frame and h is an anisotropy parameter (h=0 for black-body emission, -1<h<0 for a fan-beam pattern such that expected from Comptonization in a thin slab; ). We used Eq. (20–22) of <cit.> to determine the phase intervals in which the photons are visible to the observer and Eqs. (28–34) of the same paper to express the flux observed from each spot, F_i(ϕ_i+ϕ_0,i,θ_i,h) (i=1,2) at a given binary inclination i and colatitude θ_i of the spot, and the resulting total flux F_ tot(ϕ)=N(F_1+F_2). Here, N is a normalization constant, ϕ_0 is the spin phase of the primary spot at the folding reference epoch. We assumed the phase of the secondary spot to be ϕ_2=ϕ_1+π and used the relativistic rotating vector model of <cit.> to express the phase-resolved 
Stokes parameters of each of the two spots (Eqs. 58–59), assuming that N F_i gives the observed flux and that their polarization degree p_1=p_2 does not depend on energy and zenith angle. We then used (Eqs. 21–23) of the same paper to compute the total expected Stokes vector, Q=Q_1+Q_2 and U=U_1+U_2.

We fitted the phase-resolved Stokes parameters observed before the amplitude increase  (see Figure <ref>) with our simple model fixing the NS mass and radius to M=1.4 M_⊙ and R_ eq=12.5 km (see Figure <ref>). The fit suggests a configuration with two visible spots as the fit χ^2 (38.8 for 41 d.o.f) improves compared to a fit with only one spot (χ^2=63.0 for 42 d.o.f.; the probability of chance improvement is ≃ 10^-5 according to an F-test). A reasonably good fit can be obtained for any value of h ranging between 0 and -1; here we show the results obtained for a randomly chosen intermediate value of  h=-0.3. We find a best-fit for i=(74.1_-6.3^+5.8), θ_1=(11.8_-3.5^+2.5), θ_2=(172.6_-1.0^+2.0), ϕ_0=0.57(4), N=6.90(6)×10^4 cnt, p_1=4.0%±2.0% and χ_ p=57.2±0.5, where the latter is the position angle of
the pulsar angular momentum. We evaluated uncertainties at 1σ confidence level from the variation of the fit χ^2 for the number of interesting parameters measured <cit.>. Such uncertainties have to be taken with caution, as they were obtained for fixed values of the NS mass, radius and emission anisotropy parameter. In addition, the large correlation between the parameters warrants a study of the posterior distribution of the parameters fitted using the likelihood function for I, Q and U, which will be presented in a forthcoming paper. The red line in Figure <ref> marks the predicted model of the normalized Stokes parameters obtained for these parameters.  Bottom panels of the same figure show the total PD=√(q^2+u^2) and PA=χ_ p+tan^-1(u/q), respectively. The absence of significant variability with the pulse phase of the phase-resolved observed normalized Stokes parameters Q and U can be explained by two almost antipodal spots if their magnetic inclination is small. Yet, the observed roughly constant phase-resolved trend gives little information on the other parameters which are essentially set by the total pulse profile alone. <cit.> used a modeling similar to ours (even though they assumed extended spots) and found qualitatively similar results (i=58 and θ_1=14, θ_2=π-θ_1). They modeled a pulse with a higher amplitude (9–12%) than ours (A_1=5.2%±0.1%) and a plateau close to the pulse minimum, which they interpreted in terms of occultation by the inner accretion disk extending down to ∼25 km. The flattening of the pulse profile observed by  is less pronounced and partial occultation is not required to obtain qualitative modeling of the profile. Because the pulse profile <cit.> modeled was observed when the X-ray flux was roughly double that observed by , this could indicate a recession of the inner accretion disk at a decreasing mass accretion rate. 

Interestingly, the increase of the pulsed fraction observed by  after MJD  60376.5 to A_1=18.7%±0.8% (see Fig. <ref> and Fig. <ref>) can be qualitatively explained assuming a fixed value of the binary inclination if the magnetic inclination of the two spots varies by δθ_1≃ -δθ_2 ≃ 10, the longitude shifts by ∼20and the polarized degree of each spot decreases to 2.0%± 1.5%. The profile predicted using our model assumption and keeping the rest of the parameters to the values quoted above is plotted as a red line in Figure <ref>. 






 §.§ Type-I X-ray bursts


To our knowledge, the observations presented here are the first in which  caught thermonuclear X-ray bursts from an accreting NS. 
The ∼10 d coverage of the outburst of  allowed us to detect 52 bursts and stack them when searching for a polarized signal. The upper limit we put on the PD of the  burst emission (8.5%) is higher than the maximum value estimated by <cit.> assuming reflection of the burst emission by the accretion disk (3.7% for an inclination of 72). With a recurrence time variable between ≃ 1.6 hr at the beginning of the outburst <cit.> and ≃ 8 hr at the end of the IXPE observation presented here,  was immediately recognized as a very prolific burster. The 11 Hz pulsar IGR J17480–2446  (which showed a recurrence time decreasing from ≃ 30 to ≃ 3 minutes; ) and 4U 1636–536 (with a recurrence time of ≃ 30 minutes and even less energetic short recurrence time bursts; ) are similar cases. Among accreting MSPs, SAX J17498–2021 showed bursts between 1 and 2 hr <cit.>, IGR J17511–3057 between 7 and 21 hr <cit.> and IGR J17498–2921  every 16–18 hr <cit.>, in all cases with a recurrence time increasing as the X-ray flux declined. The ≈ 15 s duration of the bursts seen from SRGA J1444 suggests that they ignite in a mixed H/He environment. Hot CNO cycle is expected to deplete the accreted hydrogen in 9.8 (X/0.7) (Z/0.02)^-1 hr, where H and Z are the hydrogen and metals abundances <cit.>.  The range of observed recurrence times (<8 hr) suggests that complete depletion of hydrogen could only be achieved if the H abundance is sub-solar. For bursts igniting in a mixed H/He environment, the burst ignition depth is not expected to depend much on the mass accretion rate Ṁ and the recurrence time scale should increase as Δ t ∝Ṁ^-1. Assuming that the IXPE count rate is a good tracer of the mass accretion rate, the observed dependence (see Fig. <ref>) is broadly compatible with such an expectation. Malacaria et al. (in prep.) will present a detailed analysis of the burst ignition process on this fast rotating pulsar allowed by the long and high duty cycle allowed by IXPE.








§ CONCLUSIONS


We presented the first X-ray polarimetric IXPE observation of an accreting millisecond pulsar in outburst, SRGA J1444, in the context of an observational campaign involving also NICER, NuSTAR and XMM-Newton observations. The main results are the following:




  * The 2-8 keV emission is significantly polarized, with an average degree of 2.3%± 0.4% at an angle of  59±6 (East of North). 


  * We observed a significant change of the polarization properties with energies. The PD is maximum between 3 and 6 keV (4.0%±0.5%) and decreases to <2%  (90% c.l.) in the 2–3 keV range.



  * The pulse phase of SRGA J1444 is rather stable throughout the observations presented here, with an upper limit on the spin frequency derivative of ν̇<1.2×10^-13 Hz s^-1. During IXPE observations the pulse has an amplitude of ∼ 5%, before abruptly increasing to almost 20%, roughly one day before the end of the coverage.



  * The observed phase-resolved Stokes parameters are compatible with a constant function. A simple approximate model of the Stokes vector expected from the sum of two almost antipodal spots on the NS surface reproduces the observed values for a binary inclination of i=(74.1^+5.8_-6.3) and relatively small magnetic inclination of the spots θ_1=(11.8_-3.5^+2.5), θ_2=(172.6_-1.0^+2.0). The little phase-resolved variability shown by the Stokes parameters, if any, prevented us from obtaining constraints on the NS mass and radius based on our simple modeling. 


  * IXPE observed 52 type-I X-ray bursts sharing similar properties (e.g. burst peak fluence, duration, fluence) with a recurrence time increasing as a function of the observed count rate, Δ t_ rec∝ C^-0.8. After subtracting the persistent emission, the upper limit on the PD observed in the stacked burst emission is  8.5% (90% c.l.).



The first significant detection of polarization from an accreting millisecond pulsar confirms theoretical expectations and strengthens the prospects of using this technique to measure the geometrical parameters of accreting MSPs and obtain constraints on the mass and radius of the NS from pulse profile modeling. Future IXPE observations of accreting millisecond pulsars outbursts will hopefully detect a larger phase-resolved variability and enable more accurate determination than those allowed by the present dataset.



§ ACKNOWLEDGMENTS


We warmly thank the Directors and the Science Operations Team of IXPE, NICER, NuSTAR and XMM for promptly scheduling the observations reported here, as well as Anna Watts and Bas Dorsman for useful discussions.

NICER is a 0.2–12 keV X-ray telescope operating on the International Space Station, funded by NASA. The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian mission.  The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C).  The Italian contribution is supported by the Italian Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2022-13-I.0, agreements ASI-INAF-2022-19-HH.0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in Italy.  This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC).  The NuSTAR mission is a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. Data analysis was performed using the NuSTAR Data Analysis Software (NuSTARDAS), jointly developed by the ASI Science Data Center (SSDC, Italy) and the California Institute of Technology (USA). XMM-Newton is an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.

AP, GI, FA, RLP, CM and LS are supported by INAF (Research Grant ”Uncovering the optical beat of the fastest magnetised neutron stars (FANS)”) and the Italian Ministry of University and Research (MUR) (PRIN 2020, Grant 2020BRP57Z, ”Gravitational and Electromagnetic-wave Sources in the Universe with current and next-generation detectors (GEMS)”).
JP and VL thank the Academy of Finland grant 333112 and Minobrnauki grant 075-15-2024-647 for support. 
AB acknowledges support from the Finnish Cultural Foundation grant 00240328. MCB acknowledges support from the INAF-Astrofit fellowship. FCZ is supported by a Ramón y Cajal fellowship (grant agreement RYC2021-030888-I).



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