Paper Title: Chemical abundance gradients of organic molecules within a protostellar disk.




















































Levi G. Walls
levigwal@umich.edu

0000-0001-6253-1630]Levi G. Walls
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107

0000-0002-2555-9869]Merel L.R. van 't Hoff
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107



0000-0003-4179-6394]Edwin A. Bergin
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107















§ ABSTRACT 
Observations of low-mass protostellar systems show evidence of rich complex organic chemistry. Their low luminosity, however, makes determining abundance distributions of complex organic molecules (COMs) within the water snowline challenging. However, the excitation conditions sampled by  differing molecular distributions may  produce substantive changes in the resulting emission.  Thus, molecular excitation may recover spatial information from spatially unresolved data. By analyzing spatially-unresolved NOrthern Extended Millimeter Array (NOEMA) observations of CH_3OH and CH_3CN, we aim to determine if CH_3OH and CH_3CN are distributed differently in the protostellar disk around HOPS-370, a highly-luminous intermediate mass protostar. Rotational diagram analysis of CH_3OH and CH_3CN yields rotational temperatures of 198 ± 1.2 K and 448 ± 19 K, respectively, suggesting the two molecules have different spatial distributions. Source-specific 3D LTE radiative transfer models are used to constrain the spatial distribution of CH_3OH and CH_3CN within the disk. A uniform distribution with an abundance of 4×10^-8 reproduces the CH_3OH observations. In contrast, the spatial distribution of CH_3CN needs to be either more compact (within ∼120 au versus ∼240 au for CH_3OH) or exhibiting a factor of ≳ 15 increase in abundance in the inner ∼55 au.  A possible explanation for the difference in spatial abundance distributions of CH_3OH and CH_3CN is carbon-grain sublimation.























 



§ INTRODUCTION

Complex organic molecules (COMs) are carbon-bearing molecules with 6 or more atoms <cit.>. Many COMs are formed in ice mantles on dust grains. For example, hydrogenation of CO ice can lead to the formation of simple oxygen-bearing COMs, such as methanol (CH_3OH), up to biologically relevant molecules like glycerol <cit.>. Whether a COM will remain in the ice or transitions into the gas depends on energetics and the local temperature <cit.>. The binding energies of many COMs are similar to that of water, resulting in the transition between gas and ice to occur roughly around the water snowline (∼ 150 K for pressures P ∼ 10^-12 bar).  For example, the binding energy is 5640 K for amorphous solid water <cit.>, 5728 K for methanol <cit.>, and 4954 K for methyl cyanide  <cit.>, as measured for carbonaceous grains.   

Based on this picture, the spatial distribution of COMs in circumstellar material is expected to be uniform post-sublimation within the snowline in the absence of gas phase chemical processing. However, for some species, for example, the nitrogen-bearing COM methyl cyanide, CH_3CN, (additional) gas phase formation routes have been proposed <cit.>. In addition, chemical processing in hot (≳300-500 K) gas may be possible, potentially following the destruction of carbonaceous grains <cit.> or through high-temperature pathways that become active <cit.>. 

For low-mass sources, the snowline lies within a few au from the star in disks up to a few tens of au from the star in protostellar envelopes, making resolving the snowline location technically challenging. As such, most studies of low-mass protostellar sources have focused on the detection of COMs <cit.> or the chemical inventory of individual sources <cit.>, rather than their spatial distribution (see reviews by e.g., ). 

Evidence for chemical changes in hot gas have been found toward high-mass protostars. <cit.> is one of the first studies of the high mass star forming region Orion KL to indicate the observed kinematic and spatial difference in N-COM and O-COM abundances could be due to hot gas phase chemistry. More recent studies on the Orion KL region also show abundance enhancements of N-COMs relative to O-COMs <cit.>.  
<cit.> finds the ratio of column densities in hot gas relative to warm gas for N-COMs is approximately 5 times higher than for O-COMs. While all of these studies toward high-mass systems indicate resolved chemical segregation <cit.> no systematic study of the spatial distribution of COMs within the snowline of low-mass sources has been conducted. 


The disk around the highly-luminous protostar HOPS-370 <cit.> also displays evidence of hot-gas chemistry based on observations with the NOrthern Extended Millimeter Array (NOEMA; van 't Hoff et al. accepted). Being an intermediate mass protostar <cit.> with a luminosity of ∼314 L_⊙ <cit.>, this system is warm, which pushes the water snowline out to ∼225 au. This source has been well-studied and well-modeled: ALMA observations resolved a disk structure in the dust continuum emission (of 62 au) and rotation in multiple molecular lines perpendicular to the outflow direction, suggesting the presence of a near edge-on (inclination of 74) gas disk <cit.> that subtends a radial extent of ∼ 05. As such, HOPS-370 is a good target to study the spatial distribution of COMs within the snowline.

In this paper, we use spatially unresolved NOEMA observations of  and  in the disk around HOPS-370 to probe their spatial abundance distributions within the snowline using molecular excitation and source-specific radiative transfer modeling.
In  <ref>, we show how  can be and is used as a temperature probe. 
In  <ref> and  <ref>, we present the  and  observations toward HOPS-370 and our analysis that indicates differences in the spatial distributions of these species within the NOEMA beam.  These results are discussed in  <ref> and the main conclusions are summarized in  <ref>.




§  AS A TEMPERATURE PROBE
 
Symmetric-top molecules like  are excellent probes for the gas temperature. These molecules break symmetry along the rotational axis in one dimension.  Thus their energy states are characterized by the two quantum numbers J (total angular momentum) and K (the projection of J onto the axis of symmetry).  
Radiative transitions for symmetric-tops have ΔJ = 1 and ΔK = 0, while ΔK≠ 1 transitions are forbidden. 
The selection rules for dipole transitions imply collisions dominate over radiative processes when relative populations are explored across K ladders (same J - different K), and thus probe the gas temperature <cit.>.  To further enhance this capability, the rotational emissions of symmetric top molecules for transitions with the same J but different K are found closely spaced in frequency. Thus a single observation might capture multiple energy states at once.


In general, the excitation temperature, T_exc, defines the relative population of upper level u to lower level l in the Boltzmann distribution:


    N_u/N_l = g_u/g_lexp( -E_u - E_l/k_B T_exc).


If molecular emission is in LTE then the excitation temperature is equal to the kinetic temperature (T_k).  For complex molecules, with multiple emission lines sampling a range of excitation states, it is common to use the rotation diagram method <cit.> to measure the rotational temperature, T_rot.  Here the rotational temperature is a relative measure of the excitation temperature.  Below we explore the relation of T_rot and T_k for the J=14-13 transition of CH_3CN using a series of statistical equilibrium calculations to lay the ground work for our subsequent observational analysis.  

The methodology laid out by <cit.> states

    ln( N_u/g_u) = ln( N_T/Q(T_rot)) - E_u/T_rot,

where N_u is the upper level column density of the molecule, g_u the statistical weight of that level and E_u is the upper energy level in units of K. The total column density of the molecular gas is given by N_T and the partition function is Q(T_rot). 

For optically thin transitions, ln (N_u / g_u) is related to the observed total integrated flux, W in units of K km s^-1, via

    N_u/g_u = 8π k_B/h c^3ν^2 W/A_ul g_u,

where k_B is the Boltzmann constant, G is the gravitational constant, h is Planck's constant, c is the speed of light, A_ul is the Einstein coefficient for spontaneous emission, and ν is the transition frequency.  

With Eq. <ref> presented as such, a linear regression can be fit in semi-log space to the data, ln(N_u/g_u), as a function of upper energy level, E_u. The intercept is then proportional to the total column density and the negative slope yields the rotational temperature. For a fixed abundance, lowering the rotational temperature, T_rot, steepens the slope.

We use   slab model radiative transfer simulations <cit.> of  J=14_K - 13_K, K=0-12 transitions for different kinetic (gas) temperatures, H_2 density, and total CH_3CN column densities.  The spectroscopic data and collisional rate coefficients we adopt in these simulations are provided by <cit.>. For a given set of physical conditions  calculates the line emission, from which upper level column densities and rotational temperatures are derived using the method outlined above.  Fig. <ref> displays the resulting rotational temperatures versus gas density for different kinetic (gas) temperatures, and total   column densities.   Based on this analysis, LTE (in this case T_rot = T_k) is reached near densities of n_H_2∼ 10^6 cm^-3.  Furthermore, LTE conditions are retrieved (within a factor of 2) when n_H_2 < 10^6 cm^-3 provided N_T >10^16 cm^-2 and the emission is optically thin. 

Additionally, a statistical equilibrium argument illustrates that LTE is appropriate for realistic disk temperatures and densities such as those described above (see  <ref> for our disk model). Ignoring radiative processes, a system is in LTE when densities exceed the critical density, i.e. n_H_2≫ n_cr≡ A_ul/C_ul, where C_ul is the collisional rate coefficient from the upper level to the lower level. <cit.> show updated collisional rate coefficients for E- (K = 3n ± 1 for n ≥ 0) which are lower by at most a factor of 5 as compared with the rates from <cit.>; A- (K = 3n for n ≥ 0) is even closer to agreement. We thus approximate the most-up-to-date E- collisional rate cofficients of <cit.> at 140 K (roughly corresponding to the lowest temperature in our disk model). Calculating the critical density in this most conservative case, we find values do not exceed ∼3.4×10^7 cm^-3 (occurring for K≤ 2 transitions). This is approximately an order of magnitude lower than 2.2×10^8 cm^-3 (corresponding to the lowest gas number density in our disk model). Other locations in the disk are increasingly dense and hot and, thus, would only further solidify the fact the disk is in LTE. This effectively validates the use of CH_3CN to probe the gas temperature as demonstrated previously <cit.>. 


In our observational analysis we will explore, and contrast, the rotational temperatures of CH_3OH to CH_3CN.  
CH_3OH is an asymmetric top molecule and its rotational temperature relationship to the gas kinetic temperature is more complex.  However, if its emission is in LTE then a comparison between measured rotational temperatures may be revealing.
For analysis of observations in the PErseus ALMA CHEmistry Survey (PEACHES), <cit.> performs 1D, isothermal non-LTE radiative transfer calculations of . They find that for the ( 5_1,  4) A^- - ( 4_1,  3) A^-  transition (ν_t = 0) at (243915.788 MHz) transition, the excitation temperature approached the kinetic temperature to within 10-30% for gas density estimates appropriate for this type of sources (between ∼4×10^9 and 10^12 cm^-3). A similar analysis is done by <cit.> in their study of IRAS 16293, albeit for a different frequency range, but with a similar conclusion. Furthermore, models of <cit.> show LTE is appropriate for  at densities around ∼10^9 cm^-3, albeit for a low kinetic temperature, T_k = 50 K. We target the same frequency range as the PEACHES sources and the density for the HOPS-370 disk falls within the same range (see  <ref>), suggesting LTE is a valid assumption for our modeling of the optically thin emission spectrum of .    




 

§ OBSERVATIONS
 
HOPS-370 was observed by the Northern Extended Millimeter Array (NOEMA) as a part of the Chemistry And Temperature: Searching with NOEMA for Destruction Of carbon GrainS project (CATS-N-DOGS; HOPS-370 project code: W20AF) four times from December 2020 to March 2021 (van 't Hoff et al. accepted). In this work, we use the first three observations, which were at 1 mm spanning the frequency ranges of 240.9–248.6 GHz and 256.4–264.1 GHz each with 2 MHz (Δ v = 2.3 km s^-1 spectral resolution) and spent 3.0, 1.9, and 2.6 hours, on source in configuration 11C, 11C, and 10A, respectively.  

The data are calibrated using the CLIC package of the  software[<https://www.iram.fr/IRAMFR/GILDAS>], using the most recent  version available at the time of data reduction (July 2021). For each observation the bandpass calibrator was 3C84. The phase calibration was done using J0542-0913 and J0458-020 for each observation. The flux was calibrated using LKHA101 for all observations except for the second, in which MWC349 was used. 

Imaging and cleaning was done using the  package of . Due to the line richness of the spectrum, the data was imaged without continuum subtraction and no self-calibration was performed. The resulting beam size is  3.37× 0.6 (9^∘ ). The gas disk of HOPS-370 is ∼0.5in size (∼200 au in diameter at ∼400 pc; ), and thus unresolved in these observations. The noise level has been measured in a region off source in the image cube and is 16 mK in 2 MHz channels.

After extracting spectra toward the continuum peak position, the continuum was subtracted using a statistical method as, for example, used by <cit.> for the line-rich spectrum of the protostellar binary IRAS16293. This method entails the formation of a density distribution of flux values for the spectrum of each sideband. In the case of only continuum emission, the density distribution would represent a symmetric Gaussian centered at the continuum level. In the case of continuum and line emission, the Gaussian will contain an exponential tail toward higher values. The continuum level is then determined by fitting a skewed Gaussian to the distribution, and subtracted from the spectrum. The continuum-subtracted   J = 14_K - 13_K spectrum is shown in Fig. <ref>. 

We securely detect 12 transitions of the 14_K - 13_K ladder of  shown in Fig. <ref> (see also Table <ref>). The 16 mK noise level results in a weak detection of the K=11  transition. There are multiple other weak lines present near the K=12 frequency, which makes a detection of the K=12 transition very uncertain. We therefore do not report a detection of the K=12 transition. The K=5  transition at 257403.585 MHz coincides with the (18_3, 16) A^+ - (18_2, 17) A^-  transition (ν_t = 0) at 257402.086 MHz. We thus report a detection of the K=5  transition, but do not include it (nor the  transition) in the observed rotation diagram analysis. 






§ ANALYSIS
 
In this section, we model the emission of  and  via full 3D  radiative transfer calculations. The disk around HOPS-370 has a well-characterized physical structure <cit.>. We use this structure for 3D radiative transfer modeling with the LIne Modeling Engine <cit.> which includes both gas and dust emission and their respective optical depth effects. We then convolve the predicted emission to the NOEMA beam (at the source distance) to match the spatially unresolved observations. Using the strategy outlined in Sect. <ref>, we derive the rotational temperature for each molecular species.  A focus of our efforts will be to use the observed molecular excitation to set constraints on the overall spatial distribution of  and . In  <ref> we first describe the physical model in detail, and in  <ref> and  <ref> - <ref> we present the results of our modeling efforts for  and , respectively.


  



 §.§ Physical structure and model setup
  
Following the approach of <cit.>, we adopt the gas disk structure  from their modelling results of  emission, where 
the gas surface density, Σ_g, of the disk is assumed to be a power-law with an exponential cut-off <cit.>

    Σ_g(r) = Σ_0 ( r/r_c)^-γexp[-( r/r_c)^2 - γ],

where Σ_0 = 51.25 g cm^-2 is the gas surface density normalization, r is the cylindrical radius, r_c = 110 au is the disk critical radius, and γ = 0.89 is the surface density power-law index. 

The vertical structure of the disk is assumed to be in hydrostatic equilibrium, such that:

    h_g(r) = (k_B r^3 T_g(r)/G M_* μ_m m_H)^1/2

where k_B is the Boltzmann constant, G is the gravitational constant, M_* = 2.37 M_⊙ is the mass of the central protostar, μ_m = 2.37 is the mean molecular weight <cit.>, and m_H is the mass of the hydrogen atom. A radial power-law is adopted for the temperature profile, T_g(r), 

    T_g(r) = T_0 ( r/au)^-q

where T_0 = 998.7 K is the temperature at 1 au <cit.>. The temperature power-law index q=0.35 was chosen based on results for other protostellar disks, e.g., L1527 <cit.>. 
With this temperature profile, as shown in the right panel of Fig. <ref>, the 150 K water snowline is located at 225 au. 
See  <ref> for more discussion on the effect temperature profile choice has on our modelling work. 
 

The gas (mass) density then is found by integrating Eq. <ref>:

    ρ_g(r) = 1/√(2π)Σ_g(r)/h_g(r)exp[-1/2(z/h_g(r))^2],

where Σ_g(r) is given by Eq. <ref>, h_g(r) is given by Eq. <ref>, and z is the cylindrical height above the midplane. The gas number density, n_H_2, shown in the left panel of Fig. <ref> is found by dividing the gas mass density (Eq. <ref>) by the mean molecular weight.

The dust density is assumed to be 1% of the gas density <cit.>. Furthermore, the dust size distribution is assumed to follow n(a) ∝ a^-3.5 <cit.>, where a is the radius of the dust, which is assumed to be between 0.005 and 1 μm. Dust grains are assumed to be mixtures of water ice (mass fraction: 1.19×10^-3), iron  (mass fraction: 2.53×10^-4), olivine (mass fraction: 2.51×10^-3), refractory and volatile organics (mass fraction: 4.13×10^-3), orthopyroxene (mass fraction: 7.33×10^-4), and troilite (mass fraction: 5.69×10^-4). Following the results of <cit.>, we model the mass-weighted dust opacities presented in <cit.> using  <cit.>.  






The main focus of this work is investigating the effects of varying the abundance distribution (e.g., constant abundance uniformly distributed over the entire disk) of  and . The physical structure outlined above is used to perform LTE radiative transfer calculations for different transitions of  and  with  <cit.>.  solves the radiative transfer equation on a grid of Voronoi cells constructed from the Delauney triangulation of randomly chosen points, whose acceptance can be weighted based on, for example, a provided molecular abundance distribution. The result is an image cube containing the intensity for a prescribed spectral resolution. 

For both  and  we provide as input a Keplerian velocity profile and thermal Doppler broadening parameter in addition to the gas density (e.g. Eq. <ref>) and temperature structure (e.g. Eq. <ref>) as modeled by <cit.>. Additionally we adopt in  the distance <cit.> and inclination angle <cit.> for the disk around HOPS-370. The molecular abundance and its distribution for each molecular species is also provided and is the main quantity we vary.  

The spectroscopic information for both  and  (i.e. ν, g_u, E_u, A_ul) is present in  <cit.> and provided in a file conforming to the  format <cit.>. The spectroscopic data for  is provided 
by <cit.> (see also Table <ref>). 

Since  calculates the partition function internally using the energy levels in the provided molecular data file, we calculate the partition function in the same way for the analysis of the observations using 

    Q(T) = ∑_i g_i exp(-E_i/k_B T),

where g_i and E_i is the statistical weight and energy of level, i, respectively. In calculating the total column density for each species, the partition function is evaluated at the rotational temperature. This is justified given LTE conditions have been established within the disk (see  <ref>).

Given the temperatures considered here, however, higher  vibrational states can contribute to its partition function. We determine the multiplicative vibrational correction factor at the rotational temperature (a factor ≲2) by interpolating the correction factors provided in CDMS. Importantly, the derived rotational temperature does not depend on the partition function.

We include in our models the K=0-12 lines of the J=14_K-13_K transition, each at a velocity resolution of 0.16 km s^-1. While we include the  K=0 and 1 lines in our observational and model rotation diagrams, we do not include them when performing the linear regression as these transitions are blended (see Table <ref> for more information on what transitions are used in constructing the rotation diagrams in Figs. <ref>, <ref>, and <ref>).

The spectroscopic data for  is provided by <cit.> (see also Table <ref>).
From the  entry for , which consists of  torsionally-excited ν_t ≤ 2 states up to J=40 with A- and E-symmetry, we construct a molecular data file containing 38,643 total transitions up to 3 THz. The Einstein A_ul values are calculated as detailed by <cit.>, and we calculate the partition function from the line list as detailed above. Each line is modeled with a velocity resolution of 1 km s^-1.  


Each model yields an image cube containing the intensity in Jy/pixel for each line considered per molecule. For each line, the continuum is determined as the mean intensity in the first three velocity channels devoid of line emission. This is then subtracted from the rest of the velocity channels per line per molecule. This data is then converted to brightness temperature via the Rayleigh-Jeans law <cit.>:

    T_b = 1.222×10^3I/ν^2 θ_min θ_max,

where I is the spectral intensity in mJy, ν is the line frequency in GHz, and θ_min = 06 and θ_max=337 are the beam minor and major axes, respectively. After extracting the spectrum from the data cube, these modeled fluxes are then convolved with a 1D Gaussian assuming the velocity resolution of the NOEMA observations, i.e. Δ v = 2.3 km s^-1 and integrated with respect to velocity. The models, in which abundances and spatial distributions for  and  are varied, are analyzed with rotation diagram analysis and, in this way, compared with observations. 



 §.§ 
 
Fig. <ref> shows the rotation diagram for the  observations of HOPS-370 (see also van 't Hoff et al. accepted). A fit to the data using least-squares minimization yields a rotational temperature of T_rot = 198 ± 1.2 K and a total column density of N_T = (3.42 ± 0.04)×10^16 cm^-2. Given the higher number of  transitions with E_u ≳ 400 K relative to lower energy transitions, it is the higher energy transitions which strongly influence the derived rotational temperature. 

To model the  emission, we adopt a constant abundance of X(CH_3OH) = 4×10^-8 distributed uniformly throughout the 240 au disk, based on the results by <cit.> towards HOPS-370. This model is also shown in Fig. <ref> and results in a rotational temperature of T_rot = 197 K. 
Fitting the ν_t = 0 and 1 torsional states or the A- and E-symmetries separately results in very similar results. This is in agreement with the observed value.







 §.§ : constant abundance model
 
The rotation diagram for the observed K-ladder of J=14_K-13_K   transition is shown in Fig. <ref>. A least squares minimization results in an observed  rotational temperature of T_rot = 448 ± 19 K and a total column density of N_T = (8.28 ± 0.13)×10^14 cm^-2, given the discussion in  <ref>. The temperature, which is unaffected by our treatment of the partition function, is approximately 250 K higher than observed for  and well above the estimated uncertainties. This difference likely indicates different spatial distributions for both COMs. 

To study the spatial distribution of , we first explore a constant abundance distributed uniformly throughout the 240 au disk, as done for . The  abundance is varied to find the abundance that reproduces the observed peak K=4 flux to within 1σ (16 mK). We choose the K=4 line as it has an intermediate upper-level energy (E_u = 207 K) and large signal-to-noise (S/N ∼ 70). The K= 0 - 2 lines show some degree of blending, obfuscating the true fluxes of each line and the K=3 line (and K=6, 9, ...), while having high signal-to-noise, has a different statistical weight  due to the threefold symmetry of the methyl group and nuclear spin statistics. The K=5 transition is blended with a  line. As such, the line K=4 has the highest signal-to-noise of unblended transitions, is unaffected by statistical weights, and has an upper energy level that is readily observable.

The rotation diagram for the full J=14-13 ladder using the resulting constant abundance X(CH_3CN) = 5.2×10^-10 is shown in Fig. <ref>. This model severely underpredicts high K fluxes. This results in a rotational temperature of only T_rot = 225 K, which is ∼220 K cooler than observed. To obtain a higher rotational temperature, a higher  abundance is required in the warmer gas closer to the star. However, simply increasing the  abundance, results in low-K transitions being overpredicted.
Thus, a constant abundance  distribution is unable to reproduce the NOEMA observations, in contrast to . 






 §.§ : compact abundance models
 
The simplest way to increase the calculated rotational temperature while ensuring simulated fluxes agree with observations is by limiting the distribution of  to warmer gas. In these models, we assume a constant  abundance inside a given radius, r_out, and zero outside. Beginning with a radius, r_out∼ 25 au, we first systematically increase r_out in steps of ∼ 50 au. Based on the effect increasing r_out (that is, increasing the relative amount of warm gas we are introducing) has on the T_rot and K=4 line flux, we refine the step size such that a better agreement between model and observations is found. At each step, we determine whether the model and observed rotation diagram agree. This is accomplished when the K=4 line flux agrees with observations to within 1σ (16 mK) and the rotational temperature agrees to within 1σ (19.0 K). We limit the  abundance to be below 10^-4, which corresponds to the canonical CO abundance. Table <ref> provides the results of a set of models that met one or both of the above listed criteria.


cccccccc 

0pt 
Summary of compact model results with varying outer extent of emission

 r_out [au]    T_g(r_out) [K]     X_in(CH_3CN)     Peak T_b,K=4     Fits?^a     T_rot [K]     Fits?^b    N_T×10^14 [cm^-2]  

110     193     6.4× 10^-9     1.727     √     531                 12.15
 
110     193     4.0× 10^-9     1.518                 444     √     8.56 
 
115     190     4.2× 10^-9     1.726     √     436     √     9.39 
 
120     187     5.0× 10^-9     1.985                 452     √     11.17
 
120     187     3.1× 10^-9     1.713     √     386                 8.02 
 
167     167     9.0× 10^-10     1.731     √     257                5.13 
 
The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 
aIf a checkmark is present, this model reproduces the observed peak K=4 flux to within 1σ (16 mK).
bIf a checkmark is present, this model reproduces the observed rotational temperature to within 1σ (19.0 K).


When r_out≲ 25 au, the emitting area is too small, regardless of abundance; even with an abundance of 10^-4, the peak K=4 flux is not reached. Systematically increasing r_out, and thereby the emitting area, the smallest r_out for which the K=4 line flux can be reproduced is ∼ 100 au. However, for these models, the rotational temperature is generally ≳ 1σ larger than observed. 
A model with r_out = 110 au and abundance of X(CH_3CN) = 4.0×10^-9 (see Fig. <ref>, panel a) is able to reproduce the observed rotational temperature, but yields a K=4 line flux that is >1σ lower than observed. Increasing the abundance slightly to X(CH_3CN) = 6.4×10^-9 reproduces the K=4 flux, but results in a too high rotational temperature, as was the case for the model with r_out = 100 au. 
The smallest r_out that matches both criteria is 115 au (Fig. <ref>, panel b). This model has an abundance of X(CH_3CN) = 4.2×10^-9.  
Thus, the inner boundary for a compact model is ∼110 au.

Moving r_out further out quickly produces results comparable to the models with r_out = 110 au. As shown in Fig. <ref> (panel c), when r_out = 120 au, a model with X(CH_3CN) = 3.1×10^-9 is able to reproduce the observed K=4 line flux, but not the rotational temperature. On the other hand, a model with a higher abundance of X(CH_3CN) = 5.0×10^-9 reproduces the rotational temperature (to within 1σ), but does not reproduce the K=4 line flux. The outer boundary for a compact model is thus ∼120 au. In summary,  can be distributed as a compact abundance inside ∼ 110 - 120 au, while the observed rotational temperature and K=4 line flux cannot be simultaneously reproduced (to within 1σ) for models outside this radial range.








 §.§ : jump abundance models
 
The compact models illustrate the sensitivity of  to warm gas or more strictly the need for its emission to have significant contributions from gas closer to the star.  However, these models are somewhat inconsistent with chemical expectations.  That is, nearly the entire disk lies within the  sublimation point.  Thus  should have an abundance that is distributed throughout the disk, albeit with a strong emission component that samples the hot gas closer to the star, but also the warm gas that lies just inside the water snowline. Another way to increase the contribution of hot gas compared to a constant or compact abundance distribution is by including a non-zero abundance outside r_out that is lower than the inner abundance. This is accomplished with a jump abundance model <cit.>. 

We systematically increase r_jump and the relative inner and outer  abundances such that again the K=4 flux and rotational temperature are within 1σ of observed values.  As before, we only consider abundances less than the canonical CO abundance; that is, X_in(CH_3CN) < 10^-4. 
For realistic lower limits on the  abundance in the outer disk, we adopt an abundance ratio X(CH_3CN)/X(CH_3OH) of 0.1% based on the weighted-average column density ratio presented by <cit.> using observations of high-mass protostars in the ALMAGAL sample and literature values toward low- and intermediate-mass protostars. This ratio is ∼10 times lower than found by, for example, <cit.> (CALYPSO survey) and <cit.> (PEACHES survey), so the lowest outer abundance we consider (X_out(CH_3CN) ∼ 10^-11) may be conservative. 



Table <ref> shows the result of varying the jump location and corresponding inner/outer abundances required to reproduce the observations. In each case, there is a  enhancement of ≳ 15 inside the jump radius relative to outside. Fig. <ref> shows rotation diagrams for a range of these models.

ccccccc 

0pt 

Summary of jump-abundance models that can reproduce the observations.

 r_jump [au]     T_g(r_jump) [K]     X_in(CH_3CN)     X_out(CH_3CN)     Peak T_b,K=4 [K]     T_rot [K]     N_T×10^14 [cm^-2]

35      288     9.60× 10^-7     4.90× 10^-10     1.721     444     8.00 

55      246     1.90× 10^-8     4.60× 10^-10     1.727     447     8.64 

90      207     6.10× 10^-9     2.70× 10^-10     1.726     445     9.66 

100     199     5.30× 10^-9     1.70× 10^-10     1.712     448     9.70 

105     196     4.95× 10^-9     1.15× 10^-10     1.715     448     9.72 

The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 





While not shown in Fig. <ref>, when the jump location is located inside 35 au (where the disk temperature is ≳ 290 K), levels with K≤4 are generally well-populated, while higher K-levels are not. For K ≥ 6, peak line fluxes are ∼45% lower than observed on average. This ultimately results in a derived rotational temperature about 200 K lower than observed, even with an inner  abundance in excess of the canonical CO abundance.
This is primarily due to having too small an emitting area as well as optically thick dust obscuring the emission from the inner most region.  However, a jump at 35 au with an abundances of X_in(CH_3CN) = 9.60× 10^-7  and X_out(CH_3CN) = 4.90× 10^-10 provides a good match to the  observations. 

Good agreement between model and observation is found for models with jump radii out to 105 au (Fig. <ref> and Table <ref>). As r_jump is increased we also see that while inner/outer abundances decrease (as expected and also seen in the compact abundance distributions), the inner abundance is consistently ≳15 larger than the outer abundance.  Furthermore, we see that despite retrieving the observed rotational temperature and K=4 peak flux, the total column density increases as the jump radius is moved outward. This is expected since more material is being added as the radius increases. Our most concentrated jump model, with r_jump = 35 au has a total column ∼2σ lower than observed. In contrast, our best-fit compact abundance model with r_out=115 au, results in a total column that is ≳ 11σ larger than observed. 
This is to say, the observed total column necessitates a jump abundance model with  being distributed inside ∼55 au, much more concentrated than the best-fit compact abundance model. 

When the jump location is shifted outward of 105 au (where the disk gas temperature is <196 K), the jump model stops being as effective as, for example, the compact emission model with r_out=110 au at reproducing observed values. Outside 105 au, the contribution from the warm rather than hot gas begins to dominate, and it becomes difficult to reproduce emission from the high K-levels. 

To summarize, models with 35 < r_jump < 105 au are able to simultaneously reproduce the observed rotational temperature as well as the peak K=4 flux. For models with r_jump < 35 au, the emitting area is too small, and, as such, the rotational temperature cannot be reproduced despite using an abundance of 10^-4. For models where r_jump is larger than 105 au, we cannot distinguish whether  is distributed as a jump model or compact abundance. Furthermore, when the total column density is considered, models indicate  needs to have a compact component with the jump radius being r_jump≲ 55 au.   




§ DISCUSSION
 



We observe  to have a rotational temperature of T_rot = 448 ± 19 K, which is approximately 250 K warmer than observed for . Models of their spatial distribution indicate that  must indeed be different from , with  shown to be enhanced in the warm gas relative to . Nominally this requires a  production mechanism in the hot gas or a destruction mechanism preferentially diminishing  in the warm gas. Below we explore how detailed excitation might influence our results and discuss potential mechanisms that could enhance  in the hot gas or deplete it in the warm gas.




 §.§ Effect of temperature profile choice
 
The choice of temperature structure could have a significant impact on the outcome of our modelling work <cit.>. A potentially influential choice in our model is the assumption of a vertically isothermal temperature profile. <cit.> and <cit.> illustrated that the temperature structure for an evolved disk is not simply vertically isothermal. However, for deeply embedded protostars, the addition of an envelope (and its opacity) acts to intercept and reprocess the incident starlight <cit.>. Ultimately, this results in only a weak vertical temperature gradient compared to more evolved disks. This is confirmed for HOPS-370 specifically by 2D  calculations <cit.>, as shown in Fig. <ref>; the temperature does not vary greatly as a function of polar angle (height), especially between 1 and 100 au (demarcated by the gray arrows). In other words, the assumption of a vertically isothermal structure is reasonable for HOPS-370. 


Assuming the temperature is vertically isothermal implies that a 1D temperature profile can be used to describe the entire 2D disk temperature. We adopt the power-law temperature profile from <cit.>, who model the molecular line emission and dust continuum, albeit separately. This is done because calculating the dust temperature self-consistently using the radiative transfer calculations of  is computationally expensive. This quickly becomes intractable when full dust thermal calculations are to be done for each model with a different spatial distribution for multiple molecular species. To assess the validity of this simplified temperature profile, Fig. <ref> shows the difference between the dust temperature calculated self-consistently with  and the power-law temperature profile adopted in  <ref> based on the NOEMA  observations.


A power-law fit to the 2D  data yields:

    T_g+d(r) = 1470K (r/au)^-0.47

whereas in  <ref> we find that the observed  rotation diagram can be described with the following power law as modelled by <cit.>:

    T_g(r) = 998.7K (r/au)^-0.35.

With a larger T_0 and steeper power-law index, q, the fit to the  calculated temperature profile is hotter at 1 au but colder further out in the disk (> 10 au) than the power-law we assume (i.e. Eq. <ref>). This could affect our constraint on the jump location. However, even at 240 au, where there is the largest deviation between the two temperature profiles, we see there is at most a difference of ∼35 K between models. Any difference between the temperature structures where the jumps were placed is less than a factor of 2, which is less than ∼50 K. Even though a slightly different radial temperature profile may influence the exact location of the outer radius in the compact models or the jump radius in the jump-abundance models, it will not affect the conclusion that  is differently distributed than  within the snowline of HOPS-370.


    






 §.§ Pebble drift
  
One potential explanation for the existence of an increase in  in the inner disk could be pebble drift. From the perspective of dynamics, dust of different sizes interacts differently with the gas. The small dust is dynamically well-coupled to the gas, while large dust experiences maximum inward radial drift. The presence of a jump abundance distribution could in principle be explained by the inward drift of large dust, causing an enhanced abundance in the inner parts of the disk where the ice sublimates <cit.>. 
 


However, if radial drift were occurring and giving rise to the jump abundance seen in , we would also expect to a jump abundance distribution for  unless there is some additional ice chemistry serving to keep  enhanced outside the  outer radius or jump location in the outer disk. Since the simpler constant abundance model reproduces the  fluxes observed by NOEMA, a jump model is unnecessary. Thus, radial drift is unlikely to be the reason for the difference in abundance distributions between  and .




 §.§ Destroying CH_3CN in warm gas
 
Models of their spatial distribution indicate inner  abundances are heightened by a factor of ∼ 15 relative to outer within ∼105 au, where warm gas (150 K≲ T_g ≲ 300 K) resides. This enhancement could be due to a chemical mechanism reducing the  in the warm gas. 

Two such destruction mechanisms could be gas-phase reactions between He^+ or H_3^+ for both  and . The reaction between  He^+ and  occurs on a timescale similar to that for He^+ and .  However,  the reaction between  H_3^+ and  is approximately a factor of three faster than for  <cit.>. After ice sublimation this would lead to quicker re-equilibration towards chemical equilibrium for  with a timescale that is a factor of three shorter.   This could lead to a shift in the abundance of  relative to  in the outer disk gas, but any model would have to be fine-tuned in timing to capture this change within the short period of time as timescales are < 10^5 yrs at these high densities <cit.>.
Thus, the elevated  abundance is difficult to explain by a gas-phase destruction mechanism. 






 §.§ Carbon-grain sublimation
 
We see  is characterized by hot gas, where there is a relatively concentrated  abundance enhanced relative to . This could indicate the presence of hot gas phase chemistry taking place, which could be the result of carbon grain sublimation. Carbon-grain sublimation is the process by which molecules in refractory carbon grains are released to the gas. This occurs at ≳ 300 K; the so-called “soot line" <cit.>. 


When dust grains are destroyed at the soot line, more carbon and nitrogen will be observed in the hot gas relative to oxygen, since the cometary carbon and nitrogen budget is higher for dust grains than for ices, while the reverse is true for the cometary oxygen budget <cit.>.  After the initial release of carbon- and nitrogen-bearing species in the hot gas,  could then form there via reactions between CH_3^+ and HCN, the product of which, CH_3CNH^+, rapidly equilibrates to  via electronic dissociative recombination <cit.>. The conditions in HOPS-370 could facilitate these gas phase reactions. 


<cit.> provide three potential signatures of carbon grain sublimation and top-down chemistry to occur within young disks: 


  * An increased abundance of  N-COMs e.g.,  inside the soot line,

  * N-COMs being distributed within a smaller spatial area than O-COMs e.g., , and therefore

  * N-COMs exhibiting higher excitation temperatures than O-COMs. 

Potential evidence for carbon-grain sublimation has been observed by <cit.> for high mass protostars in the ALMAGAL survey. However, we also see potential evidence in HOPS-370.


The excitation temperature for the 1 mm NOEMA observations of  (see also van 't Hoff et al. accepted) is T_rot = 448 ± 19 K, approximately 250 K warmer than observed for  (T_rot = 198 ± 1.2 K). 

While we find a compact  abundance out to 115 au is able to reproduce the observations, so do several jump abundance models. All of these jump abundance models have the jump occurring between 35 ≲ r_jump≲ 105 au, corresponding to 196 < T_g < 288 K. Models with smaller jump radii (e.g. r_jump≲ 55 au) are favored based on the total column density calculated being most similar to observations. 
While the 300 K line is expected to correspond to the soot line, our analysis is agnostic of this.  However, we clearly see an enhanced  abundance inside of the expected soot line location. Thus, criteria 1 and 3 are satisfied.


The rotation diagram analysis we perform significantly constrains the location where any hot gas chemistry might be taking place, despite the size of the NOEMA beam. Regardless, future observations with high angular resolution could spatially resolve the location of the water snowline (∼225 au or 095) and maybe even the putative soot line (∼31 au or 013 assuming the soot line occurs at 300 K for HOPS-370).   Regardless, there is clear evidence for chemical shifts in this system that hint at changes that can be resolved.




§ CONCLUSION
 
With spatially-unresolved 1 mm NOEMA observations of  and  in the protostellar disk around HOPS-370, 
we aim to determine if  and  are distributed differently. 
We use the radiative transfer modelling software  in combination with a source-specific physical model to calculate spectra for each of these molecules for different parameterized spatial distributions. Rotation diagrams are used to analyze the observed and modeled spectra. Our main findings are as follows:


  *  has an observed rotational temperature of T_rot = 198 ± 1.2 K, while  is observed to be significantly warmer with T_rot = 448 ± 19 K.

  * The  observations can be reproduced with a constant abundance model out to 240 au.

  * The  observations cannot be reproduced by a uniform distribution similar to . Instead, the  emission is either described by a more compact abundance distribution within ∼115 au or by a jump-abundance distribution with the increase in abundance happening for radii between ∼35 and ∼105 au. In the inner region, the  is enhanced by a factor of ∼15 compared to the outer region.  






These changes suggest chemical gradients are present inside the water snowline within hot gas.
HOPS-370 is a unique laboratory for characterization of chemistry in hot gas which is needed to understand the formation of inner solar/stellar system planets like our own.  

L.G.W. acknowledges support from the Rackham Merit Fellowship, under which a majority of this work was funded. M.L.R.H. acknowledges support from the Michigan Society of Fellows. 




§ MOLECULAR DATA

Tables <ref> and <ref> provide spectroscopic data for observed and modelled transitions of  and , respectively (see also  <ref>). For , we provide in Fig. <ref> the observed transitions (with quantum numbers as they appear in the  database), along with the resultant Gaussian fits calculated with .

cccccccc 

0pt 
Spectroscopic data for  obtained from 
Upper level     Lower level     Frequency     E_u     g_u     A_ul     Observed flux     Flux error 
 
(J_K_a, K_c)      (J_K_a, K_c)     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1] 
 

            v_t = 0                  


(25_3, 22) A^-     (25_2, 23) A^+     241588.758     803.7     204     8.19e-05     7.4304     0.56  
 
( 5_0,  5) A^+     ( 4_0,  4) A^+     241791.352      34.8      44     6.05e-05     25.436     1.17  
 
(24_3, 21) A^-     (24_2, 22) A^+     242490.245     745.7     196     8.19e-05     9.3002     1.33  
 
(23_3, 20) A^-     (23_2, 21) A^+     243412.610     690.1     188     8.20e-05      9.499     0.193 
 
( 5_1,  4) A^-     ( 4_1,  3) A^-     243915.788      49.7      44     5.97e-05     26.813     0.714 
 
(21_3, 18) A^-     (21_2, 19) A^+     245223.019     585.7     172     8.23e-05     13.751     0.977 
 
(20_3, 17) A^-     (20_2, 18) A^+     246074.605     537.0     164     8.25e-05     17.744     0.978 
 
(19_3, 16) A^-     (19_2, 17) A^+     246873.301     490.7     156     8.27e-05     17.737     0.979 
 
( 4_2,  2) A^+     ( 5_1,  5) A^+     247228.587      60.9      36     2.12e-05     21.876     1.44  
 
(18_3, 15) A^-     (18_2, 16) A^+     247610.918     446.6     148     8.29e-05     19.583     1.39  
 
(17_3, 14) A^-     (17_2, 15) A^+     248282.424     404.8     140     8.30e-05     20.338     0.258 
 
(19_3, 17) A^+     (19_2, 18) A^-     258780.248     490.6     156     9.01e-05     13.311     0.451 
 
(20_3, 18) A^+     (20_2, 19) A^-     260381.463     536.9     164     9.14e-05     15.384     1.22  
 
(21_3, 19) A^+     (21_2, 20) A^-     262223.872     585.6     172     9.28e-05     11.785     0.879 
 
 
(14_1, 14) E_2     (13_2, 11) E_2     242446.084     248.9     116     2.29e-05     21.325     1.32  
 
(16_2, 15) E_1     (15_3, 13) E_1     247161.950     338.1     132     2.57e-05     19.189     1.45  
 
(23_1, 22) E_1     (23_0, 23) E_1     247968.119     661.4     188     4.44e-05     8.6042     0.934 
 
(24_1, 23) E_1     (24_0, 24) E_1     259581.398     717.0     196     4.91e-05     6.8224     1.38  
 
(21_4, 18) E_2     (20_5, 16) E_2     261061.320     623.6     172     3.02e-05     7.3807     1.09  
 
( 2_1,  1) E_1     ( 1_0,  1) E_1     261805.675      28.0      20     5.57e-05     19.046     0.315 
 
            v_t = 1                  


( 5_0,  5) A^+     ( 4_0,  4) A^+     241267.862     458.4      44     6.00e-05     10.786     0.621 

( 5_1,  4) A^-     ( 4_1,  3) A^-     241441.270     360.0      44     5.79e-05     13.471     0.643 

(17_2, 15) A^-     (16_1, 15) A^-     259273.686     652.7     140     5.58e-05     8.2852     0.74  

 
( 5_1,  4) E_2     ( 4_1,  3) E_2     241238.144     448.1      44     5.75e-05     11.515     0.654 
 
(12_2, 11) E_2     (13_3, 11) E_2     247840.050     545.1     100     6.28e-05     12.673     0.696 
 
The quantum numbers (QNs) for each level of the transition are listed as (J_K_a, K_c), where J is the total rotational angular momentum, K_a is the projection of J onto the A and C inertial axes, respectively. Following the parentheses in each entry is the symmetry A or E. For transitions with A-symmetry, the parity is denoted with + or -. For transitions with E-symmetry, the subscript 1 indicates K_a ≥ 0, while the subscript 2 indicates K_a < 0 



    


cccrrccc 

0pt 
Spectroscopic data for  obtained from LAMDA
Upper level     Lower level     Frequency     E_u     g_u^a     A_ul      Observed flux      Flux error  

J_K     J_K     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1]

                v_t = 0             

14_12     13_12     256817.163     1118.7     29.0     3.89e-4     —        —   
 
14_11     13_11     256930.140      955.2     14.5     5.62e-4     1.5713     0.266 
 
14_10     13_10     257033.444      805.8     14.5     7.20e-4     4.6887     1.05  
 
14_9     13_9      257127.036      670.5     29.0     8.63e-4     9.5756     0.78  
 
14_8     13_8      257210.878      549.4     14.5     9.91e-4     7.317      1.2   
 
14_7     13_7      257284.935      442.4     14.5     1.11e-3     9.6442     1.2   
 
14_6     13_6      257349.180      349.7     29.0     1.20e-3     21.106     1.2   
 
14_5     13_5      257403.585      271.2     14.5     1.29e-3     —        —   
 
14_4     13_4      257448.128      207.0     14.5     1.36e-3     21.196     0.301 
 
14_3     13_3      257482.792      157.0     29.0     1.41e-3     31.866     0.269 
 
14_2     13_2      257507.562      121.3     14.5     1.45e-3     28.774     0.305 
 
14_1     13_1      257522.428       99.8     14.5     1.47e-3     30.202     0.198 
 
14_0     13_0      257527.384       92.7     14.5     1.48e-3     23.013     0.338 
 
aThe statistical weights presented in the  entry for  have the contribution from the nuclear spin factored out. However, those present in the  file (and used in our calculation of the total column density) have the nuclear spin factor included. See, for example, <cit.> and <cit.> for discussion of the statistical weights for molecules with C_3ν symmetry, like .
While not used in constructing the observed rotation diagram, we include the K=5 and 12 lines in our models and the linear regressions. In both observed and model rotation diagrams, we do not include the K=0 and 1 lines in the rotation diagram analysis as these transitions are blended. 



 
 

 
  
 
 

 
 














NOEMA






 <cit.>,
         <cit.>, 
         <cit.>,
         <cit.>,
        , <cit.>,
        astropy <cit.>, <cit.>, 
        SciPy <cit.>,
        Matplotlib <cit.>
          











aasjournal












 Paper Title: Dynamical friction in rotating ultralight dark matter galactic cores.







Dynamical friction in rotating ultralight dark matter galactic cores
    V.M. Gorkavenko^1, O.V. Barabash^1,
  T.V. Gorkavenko^1,  
O.M. Teslyk^1,

 A.O. Zaporozhchenko^1, Junji Jia^3,  A.I. Yakimenko^1,4,5, E.V. Gorbar^1,2

^1 Faculty of Physics, Taras Shevchenko National University of Kyiv,

64, Volodymyrs'ka str., Kyiv 01601, Ukraine

^2 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,

14-b Metrolohichna str., Kyiv 03143, Ukraine

^3 School of Physics and Technology, Wuhan University, 299 Bayi Road, Wuhan, Hubei Prov., China

^4 Dipartimento di Fisica e Astronomia ’Galileo Galilei’, Universit'a di Padova,
 via Marzolo 8, 35131 Padova, Italy

^5 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy

    
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================




§ ABSTRACT 
Dynamical friction and stellar orbital motion in spiral galaxies with dark matter composed of ultralight bosons in the state of  rotating Bose-Einstein condensate (BEC) are studied. It is found that the dynamical friction force is significantly affected by the topological charge of the vortex structure of the BEC core with the strongest effect at distances near the galactic center. It is also shown that the ultralight dark matter self-interaction plays an important role in studying the dynamical friction. 

Keywords: dark matter, ultralight  dark matter bosons, vortex structure





§ INTRODUCTION


The problem of nature and composition of dark matter (DM) is one of the most pressing problems of modern physics and astrophysics. DM makes up 26.8% of the mass-energy composition of the Universe <cit.> and is assumed to consist of beyond the Standard Model particles which do not interact with Standard Model particles or interact with them very feebly. 
Among popular candidates for DM particles are weakly interacting massive particles (WIMPs) with mass of ∼100 GeV <cit.>, sterile right-handed neutrinos with masses of several keV  <cit.>, and axion-like (pseudoscalar) particles with masses of μeV <cit.>. At the moment, none of these particles has been directly observed and the corresponding models are known as the cold dark matter (CDM) models <cit.>.

In this paper, we will consider another attractive possibility of ultralight DM composed of spinless bosons with masses ∼ 10^-23-10^-21 whose enormous occupation number results in the formation of the Bose-Einstein Condensate (BEC) with superfluid properties  (for a review, see, e.g., <cit.>). Ultralight dark matter (ULDM) models have very interesting phenomenology. Because of the very large de Broglie wavelength of order kpc, the formation of small-scale
structures in ULDM models is suppressed, unlike CDM models which predict too many dwarf galaxies and too much dark matter in the innermost regions of galaxies.

The fascinating physics of superfluid DM leads to a number of interesting ULDM structures whose observational signatures might help to elucidate the basic properties of DM particles.
Recently, it was found that wavelike ULDM may resolve lensing anomalies of Einstein rings in gravitationally lensed images <cit.>. Further, quantum wave interference of CDM without self-interaction (also known as fuzzy DM <cit.>)  produces stochastically fluctuating density granulation which could drive significant disc thickening, providing a natural explanation for galactic thick discs <cit.>. In addition,
it was shown that ULDM could form stable vortex structures <cit.> which survive even after head-on collision <cit.>.
Repulsive self-interaction is critical for the existence of stable vortex solutions.

Given that direct observation of DM remains elusive, we focus on the influence of dynamic friction on baryonic matter. 
We consider how different soliton structures of bosonic ULDM affect the motion of stars in galaxies.
We concentrate on stellar motion because it admits semi-analytic treatment of dynamical friction, unlike the case of motion of supermassive black holes in the ULDM environment near the galactic centers. Indeed, a recent analysis in <cit.> showed that the presence of a moving supermassive black hole significantly affects  ULDM soliton, leading to dynamical friction oscillations, requiring numerical analysis. Further, it was shown that stochastic density fluctuations of ULDM may stall the inspiral of supermassive black holes or globular clusters due to dynamical friction at radii of a few hundred parsecs and can heat and expand the central regions of galaxies <cit.>.

The classical work by Chandrasekhar <cit.> on dynamical friction experienced by a moving star due to the fluctuating gravitational force of neighbouring stars was generalized to the case of the motion of stars in ULDM in the absence of self-interaction  (the case of fuzzy DM <cit.>) in Ref.<cit.> where analytic expression for dynamical friction was derived. Later the obtained results were numerically checked in <cit.>.  The ULDM self-interaction modifies the dynamical friction force and its role for the star's motion in self-interacting  ultralight dark matter was studied in <cit.>.


The ground and vortex states of the ULDM yield different DM density and velocity distributions. Therefore, the doughnut-like density distribution (vanishing at the vortex core) and vortex flows (rapidly increasing at the vortex axis) of the BEC core of vortex soliton should directly affect the dynamical friction and, hence, star motion and their relaxation time.
Since stellar motion is directly observed, the problem of the impact of the vortex structure of ULDM on the dynamical friction provides the main motivation for the study performed here, which to the best of our knowledge was not considered in the literature.

The paper is organized as follows. Dynamical friction acting on moving stars in ULDM with trivial and nontrivial vortex structures is considered in a simplified approach in the absence of the ULDM self-interaction in Sec.<ref>. Taking the ULDM self-interaction into account, the stellar orbital motion is studied in Sec.<ref>. Conclusions are drawn in Sec.<ref>.




§ MODEL AND DYNAMICAL FRICTION IN SIMPLIFIED MODEL




When a star moves in dark matter in the form of the BEC, it produces perturbation in galactic BEC leading to an overdense wake. This wake induces a drag force on a moving star resulting in dynamical friction. Conceptually, the situation is quite similar to the motion of bodies in fluids.
Since the de Broglie wavelength of the BEC is much larger than stellar radii, we could approximate stars as point objects. As stars move through the galactic BEC, a gravitational wake forms behind them and the
associated overdensity causes resistance producing dynamic friction
acting on moving stars.

Since we are interested in the impact of vortex states of ULDM on the motion of stars and the repulsive self-interaction of ULDM is crucial for the existence of stable  vortex structures, we employ in our analysis of dynamical friction the analytic results obtained in <cit.>, which determined the coefficient of dynamical friction in self-interacting ULDM.
 
Let us begin our analysis by reminding the system of the Gross-Pitaevskii-Poisson equations which govern the dynamical evolution of self-gravitating and self-inter­act­ing BEC field ψ and the gravitational potential Φ

    iħ∂ψ/∂ t = (-ħ^2/2m∇^2 + gN|ψ|^2 + mΦ_g)ψ,


    ∇^2Φ_g = 4π GmN|ψ|^2,

where g = 4 πħ^2a_s/m is the coupling strength of the self-interaction of DM particles with mass m, a_s is the s-wave scattering length, N is the number of boson particles, ħ is the Planck constant, and G is the gravitational constant. 
As for the DM density in the BEC state, it was investigated for the case of the Milky Way galaxy in Refs.<cit.>.

Solutions to the GPP equations were found in <cit.> by using the variational ansatz in cylindrical coordinates z,r,ϕ

    ψ(r, ϕ, z) = A (r/R)^se^-r^2/2R^2 - z^2/2(Rη)^2 + isϕ,

where R and η are variational parameters and A is the normalization constant. The quantum number s is the
topological charge of the galactic BEC condensate. While the case s=0 corresponds to the ground BEC state, the s=1 state describes the BEC soliton rotating around the center of the galaxy. States with higher topological charge s>1 are unstable <cit.>. 

It is very important for the study of dynamical friction that vortex states of BEC affect the DM profile differently. Here we use the DM profile ρ_DM∼ |ψ|^2 obtained in the case of the Milky Way galaxy in <cit.> for distances from the galaxy center less than 1 kpc for the trivial ground s=0 and nontrivial s=1 vortex states given by (for r > 1 kpc, the Navarro-Frenk-White profile was used)

    ρ_s=0=0.34· 10^-17 e^-(r_sph/0.61)^2 kg/ m^3,   r<1  kpc


    ρ_s=1=0.88· 10^-17(r/0.4)^2 e^-(r/0.4)^2 - (z/0.58)^2 kg/ m^3,   r< 1  kpc,

where r_sph^2=r^2+z^2 is the radial spherical distance and r, z are coordinates in the cylindrical coordinate system measured in kpc, see Fig.<ref>. This density function was obtained for
the Milky Way halo mass M= 1.3 × 10^12M_⊙,  where M_⊙ is the solar mass, and radius R_halo = 287  kpc.
We use the model described in <cit.> and studied in <cit.> and choose the
DM particle mass m = 2.92 × 10^-22  eV/c^2 = 0.52 × 10^-57  kg  with the scattering length a_s = 8.17 × 10^-77 meter.




For the vortex state s=1, the velocity of DM is given by <cit.>

    u=cr_u/re_ϕ,

where r_u=2.2· 10^-5 kpc. To determine dynamical friction, we use the rotation curve of the Milky Way galaxy <cit.>, according to which

the velocity of stars at small distances from the galactic center (r<1 kpc) is linear with distance and can be approximated as

    v_star=(r/ kpc)·2.6· 10^5  meter/s.

In this study, we quantify the effect of dynamic friction on moving stars.

Since the self-interaction of ULDM particles is crucial for the existence of stable vortex states of BEC, we leave
the analysis of dynamical friction of the s=1 state for the next section. Here we
will consider dynamical friction only for the ground s=0 state which can be realized also in the fuzzy DM.

Although the DM profile of the s=0 state in Eq.(<ref>) was found in <cit.> in a model with ULDM self-interaction, we will assume that this profile is valid also in the case of fuzzy DM to compare the obtained results for dynamical friction with those found in the correct approach in the next section where self-interaction is taken into account.

In the case of fuzzy DM, dynamical friction was studied in <cit.>. For parameter β≪ 1, where

    β=2π G m_star m/ħ v_star,

the friction force equals <cit.>

    F_fr=4πG^2 m_star^2 ρ_DM/v_star^2 C(β,kr).

Here m_star is the star's mass, v_star is the absolute value of the star's velocity, k=mv_star/ħ, and r is the radius of the stellar orbit.
Since m_star is much less than the host system in which it resides, we have the following simple relation for the function C:

    C ={[ 1+Λ/Λ√(Λ^2 + 2Λ)/1 + Λ - √(
     1 + 2/Λ),                                r<50 pc;                 ln(2 Λ) (1 + 1/Λ) - 1,                               r> 50 pc ],
        .

where

    Λ= kr/β=v^2_starr/G m_star

is the gravitational counterpart of the Coulomb logarithm. Equation (<ref>) implies that the friction force increases for stars with small velocities and in the region with  high DM density, i.e., near the galactic center.

Our main quantity of interest is a characteristic time when a moving star changes notably its velocity <cit.>

    T =v_starm_star/F_fr.

Using Eq.(<ref>), we find

    T=v^3_star/G^2 m_starρ_DM1/4π C(β,kr).


Our results for T in ULDM without self-interaction are presented by dash-dotted line in Fig.<ref>, where we used m_star=M_⊙.  As one can see, T monotonously grows with r. The reason is that while the ground (s=0) DM density does not change much in the DM core for r < 1  kpc, the stellar velocity increases linearly with r (see, Eq.(<ref>)). Hence T should grow in view of Eq.(<ref>). In the next subsection, we will compare these results for the s=0 state with those obtained in a more accurate approach where the ULDM self-interaction is taken into account.



§ MOTION OF STARS IN ULDM WITH VORTEX STRUCTURE




To determine dynamical friction of stars moving in ULDM with different vortex structures, we assume that stars move on circular orbits with constant velocity and constant angular velocity Ω in the self-interacting ULDM and use the expression for dynamical friction derived in Ref.<cit.>.

As we mentioned in the Introduction, a stable DM structure for the case of rotating BEC with the quantum number s=1 is possible only for self-interacting ULDM particles  with coupling constant of repulsive self-interaction exceeding some critical value which is fixed by the soliton core's observable radius and the DM particle mass <cit.>. Following Ref.<cit.>, we take a_s=8.17 · 10^-77 meter and m=0.52 · 10^-57 kg. Then we find g/ħ^2=1.97· 10^-18 meter/kg. 

For our purposes, it is sufficient to consider dynamical friction in the direction tangential to a circular orbit. For a star with mass m_star moving at constant velocity v_star on a circular orbit of radius r, the tangential dynamical friction equals <cit.> 


    F_fric=4π G^2 m_star^2 ρ_DM/v_star^2ℱ,

where function ℱ is given in the form of the sum over the magnetic quantum number m_l and the azimuthal quantum number ℓ (note that the derivation of this dynamical friction force is quite similar to that given by Ostriker's formula <cit.> for the motion of a perturber in gaseous medium)

    ℱ= ∑_ℓ=1^ℓ_ max∑_m_l=-ℓ^ℓ-2γ_ℓ m_lIm(S_ℓ,ℓ-1^m_l-S^m_l+1_ℓ,ℓ-1^*),


    γ_ℓ m_l=  (-1)^m_l(ℓ-m_l)!/(ℓ-m_l-2)!
    ×{Γ(1-ℓ-m_l/2)Γ(1+ℓ-m_l/2)Γ(3-ℓ+m_l/2)Γ(1+ℓ+m_l/2)}^-1 ,


    S_ℓ,ℓ-1^m_l   = π i/2 √(1+m^2_l/ℓ_ q^2)[(-1)^1+θ(m_l)j_ℓ(ℓ_ qℳ  f^-_m_l)j_ℓ-1(ℓ_ qℳ  f^-_m_l)
        
                       + i j_ℓ( ℓ_ qℳ  f^-_m_l)y_ℓ-1(ℓ_ qℳ  f^-_m_l) -j_ℓ(  iℓ_ qℳ  f^+_m_l)h_ℓ-1^(1)(  iℓ_ qℳ  f^+_m_l)] ,

j_l(x) and j_l(x) are the  spherical Bessel functions of the first and second kind,  h_l^(1)(x) is the spherical Hankel function of the first kind, θ(x) is the Heaviside step-function, and

    f^±_m_l = √(2)(√(1+(m_l/ℓ_ q)^2)±1)^1/2,

where parameter ℓ_q is defined as

    ℓ_q=rm c_s/ħℳ.


The important characteristic of the problem of dynamical friction is the Mach number

    ℳ=v/c_s,

where v is the absolute value of the relative velocity of the star and ULDM velocity 𝐮 and  c_s is the adiabatic sound speed of DM superfluid. The latter is given by the first derivative of pressure concerning density <cit.>

    c_s=√(gρ_DM)/m.

Note that pressure is proportional to the coupling constant of self-interaction g, therefore,  c_s vanishes in fuzzy dark matter. 

For the ground BEC state s=0, where 𝐮=0, the relative velocity 𝐯 coincides with the star's velocity v=v_star given in Eq.(<ref>). In the vortex state s=1, there are two possiblities of counter-rotating (v=v_star+u) and corotating star motion (v=|v_star-u|). Since the BEC velocity rapidly decreases with distance from the galactic center, the difference between these two cases is significant only at a small distance from the galactic center. The Mach number as a function of distance to the  galactic center r is plotted in Fig.<ref> for the ground (s=0) and vortex (s=1) for the cases where a probe star corotates and counter-rotates with respect to DM superfluid.





Let us discuss now parameter ℓ_q. This parameter as a function of distance r to the center of the galaxy is presented in Fig.<ref> for the ground s=0 BEC state and s=1 vortex BEC state for corotating and counter-rotating stellar orbits. As one can see, parameter ℓ_q is typically much larger than the unity.
Further, the maximum azimuthal quantum number ℓ_max up to which the sum over l is performed is determined by the size R_ probe of probe  body and orbit's radius r

    ℓ_ max =π r/R_ probe .

Since the radius of a star is usually much less than its orbit's radius, we have ℓ_max≫ 1. Since series (<ref>) is convergent, we could introduce instead of ℓ_max a cut-off whose value is chosen by the required accuracy of calculations. 





It is worth mentioning that there is a simple analytic expression which approximates the tangential component of the friction force for ℳ<1 and ℓ_q→∞ <cit.>

    F_fr=4π G^2 m_starρ_DM/v^2[ℳ- arctanhℳ].

According to Fig.<ref>, the Mach number is less the unity, i.e., ℳ<1, for 0.05 kpc ≲ r≲ 0.82 kpc, but parameter ℓ_q, in view of Fig.<ref>, is large in this region but not infinite. This interval defines the region where the above approximate formula for the dynamical friction force could be applied.

In the case of not rotating DM, the result of computation the function ℱ is presented in Fig.<ref> whose left panel shows the force given by Eq.(<ref>). The right panel of this figure presents the friction force defined by the analytical formula (<ref>). Clearly, the results agree rather well for 0.2 kpc < r < 1 kpc and are very different for r < 0.2 kpc. In the latter case, while the dynamical friction given by (<ref>) tends to zero for r → 0, the friction force defined by the approximate expression (<ref>) blows up.



Having defined the dynamical friction force for ULDM with self-interaction, we can easily determine the characteristic time for a star to decrease its velocity using Eq.(<ref>). The corresponding results are plotted by solid, dashed, and dotted lines in Fig.<ref> for distances r> 1 pc in the z=0 plane and give the following results. (i) The characteristic time needed for a star to decrease its velocity is practically the same for the ground s=0 and vortex s=1  states for r≳ 0.3 kpc, where the ULDM velocity 𝐮 is small and can be neglected. (ii) For r≲ 0.05 kpc, T is approximately the same for the counter-rotating and corotating motion in the s=1 state for which the star's velocity can be neglected compared to the ULDM velocity at small r. Clearly, the decrease of ULDM density in the s=1 state compared to the s=0 state, see Fig.<ref>, strongly diminishes the friction force and makes T very large at small distance. (iii) T strongly increases for the counter-rotating case in the s=1 state at r ≈ 0.16 kpc, where the relative velocity of the star's velocity and the ULDM velocity vanishes. This means  that the ULDM rotation makes stellar orbital motion more stable at a certain distance from the galactic center.

Finally, let us compare the results for the ground s=0 state found in the simplified approach using Eq.(<ref>) for the dynamical friction force where the ULDM self-interaction was not taken into account (dash-dotted line in Fig.<ref>) with the results obtained  in the case where the ULDM self-interaction is taken into account (solid line). Clearly, we see that the ULDM self-interaction is very important, otherwise, T is in direct conflict with observations.






§ CONCLUSIONS




For  ULDM in the state of BEC, we considered two different configurations of BEC: ground (not rotating) and vortex states. We studied the dynamical friction force acting on stars moving in ultralight dark matter with different soliton vortex structures and investigated the characteristic time T for a star to decrease its velocity. The main results of our investigation are presented in Fig.<ref> and shown by solid black (s=0 state), dashed green (the counter-rotating case of s=1 vortex DM state) and dotted red (the corotating case of s=1 vortex DM state) lines.

We found that for large distances r≳ 0.3 kpc, the  characteristic time needed to decrease the stellar velocity is practically the same for the ground s=0 and vortex s=1 states because the ULDM velocity is small. At small distances, r≲ 0.05 kpc, T is approximately the same for the counter-rotating and corotating  stellar motion in the s=1 state for which the star's velocity can be neglected compared to the ULDM velocity at small r.  T strongly increases for the counter-rotating case in the s=1 state at r ≈ 0.16 kpc, where the relative velocity of the star's velocity and the ULDM velocity vanishes. This means the ULDM rotation makes the stellar orbital motion more stable at a certain distance from the galactic center. 

Overall, our analysis shows that the orbital motion of stars of the solar mass is stable in the Milky Way galaxy at least for distances from the galactic center  r>1 pc and the dynamic friction induced by BEC is not sufficient to affect strongly star's motion. Still, we found that the BEC structure and ULDM self-interaction are very important for the analysis of dynamical friction on star's motion in galaxies and much work remains to be done in this direction especially for objects much more massive compared to the Sun because the acceleration due to the friction force grows with mass. 

In addition, we confirmed the conclusion made in previous studies that the ULDM self-interaction essentially affects the dynamical friction. A similar conclusion was reached in recent studies <cit.> of the dynamical friction force for inspiraling black holes in the case of cold dark matter, where self-interaction may solve the final parsec problem of supermassive black holes merger.





Acknowledgements




The authors thank K. Korshynska for useful discussions. V.M.G. is grateful to  Wuhan University for its hospitality during his stay there. A.I.Ya. acknowledge support from the BIRD Project ‘Ultracold atoms in curved geometries’ of the
University of Padova and from INFN.


JHEP




 Paper Title: The extremes of AGN variability: outbursts, deep fades, changing looks, exceptional spectral states, and semi-periodicities.



S. Komossa et al.



 



1]S. Komossacor1
[cor1]Corresponding author: 
  skomossa@mpifr.de


2]D. Grupe


3]P. Marziani
4,5]L. Č. Popović
4,5]S. Marčeta-Mandić
5]E. Bon
4,6]D. Ilić
4]A. B. Kovačević
1]A. Kraus
7,8]Z. Haiman
9,10]V. Petrecca
9,10,11]D. De Cicco
5]M. S. Dimitrijević
12]V. A. Srećković
5]J. Kovačević Dojčinović
13]M. Pannikkote
5]N. Bon
14,15]K. K. Gupta
16]F. Iacob

[1]Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany






[2]Department of Physics, Geology, and Engineering Technology, Northern Kentucky University, 1 Nunn Drive, Highland Heights, KY 41099, USA

[3]National Institute for Astrophysics (INAF), Astronomical Observatory of Padua, Vicolo dell’Osservatorio 5, Padua IT, 35122, Italy

[4]University of Belgrade - Faculty of Mathematics, Department of Astronomy, Studentski trg 16, 11000 Belgrade, Serbia





[5]Astronomical Observatory, Volgina 7, 11000 Belgrade, Serbia





[6]Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany





[7]Department of Astronomy, Columbia University, New York, NY 10027, USA

[8]Department of Physics, Columbia University, New York, NY 10027, USA

[9]Dipartimento di Fisica "Ettore Pancini", Università di Napoli Federico II, via Cinthia 9, 80126 Napoli, Italy

[10]INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 Napoli, Italy





[11]Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of 104, Providencia, Santiago, Chile



[12]Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia 



[13]Physics Department, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133 Rome, Italy



[14]STAR Institute, University of Liège, Quartier Agora, Allée du Six Auôt 19c, 4000, Liège, Belgium

[15]Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, 9000, Gent, Belgium

[16]West University of Timisoara, 300232 Parvan Blvd. no. 4, Timisoara, Timis, Romania

11 July 2024
X X 2024
X X 2024
2024
S. Sarkar


§ ABSTRACT 
The extremes of Active Galactic Nuclei (AGN) variability offer valuable new insights into the drivers and physics of AGN.  We discuss some of the most extreme cases of AGN variability; the highest amplitudes, deep minima states, extreme spectral states, Seyfert-type changes, and semi-periodic signals, including new X-ray observations. 
The properties of changing-look (CL) AGN are briefly reviewed and a classification scheme is proposed which encompasses the variety of CL phenomena; distinguishing slow and fast events, repeat events, and frozen-look AGN which do not show any emission-line response. 
Long-term light curves that are densely covered over multiple years, along with follow-up spectroscopy, 
are utilized to gain insight into the
underlying variability mechanisms including accretion disk and broad-line region physics.
Remarkable differences are seen, for instance, in the optical spectral response to
extreme outbursts, implying distinct intrinsic variability mechanisms.
Furthermore, we discuss methods for distinguishing between CL AGN and CL look-alike events (tidal disruption events or supernovae in dense media). Finally, semi-periodic light curve  variability is addressed and the latest multiwavelength (MWL) light curve of
the binary supermassive black hole (SMBH) candidate OJ 287 from the MOMO project is presented. 
Recent results from that project have clearly established the need for new binary SMBH modelling
matching the tight new constraints from observations, including the measurement of a low (primary) SMBH mass of ∼ 10^8 M_⊙ which also implies that OJ 287 is no longer in the regime of near-future pulsar timing arrays. 







active galactic nucleiquasarsSeyfert galaxieschanging-look AGNsupermassive binary black holesaccretion disksMWL variability and spectra 










§ INTRODUCTION









Active Galactic Nuclei (AGN) are thought to be powered by accretion of matter onto the supermassive black holes (SMBHs) at their centers. The central region is surrounded by two line-emitting systems, the broad-line region (BLR) closer to the nucleus and of higher density, and the narrow-line region (NLR) at larger distances up to kiloparsecs and of lower density <cit.>. 
The unified model of AGN has been successful in explaining the major differences between type 1 and type 2 AGN by viewing angle differences into the central engine <cit.>. In addition, other parameters are known to play a role in explaining variations across the Seyfert and quasar populations and within each population, including the influence of the host galaxy, evolutionary effects, accretion rate, and/or metal abundances, for instance. BLR and NLR emission-line properties as well as continuum (spectral energy distribution; SED) shape have been used to evaluate differences across the AGN population using Principal Component Analyses <cit.>, establishing the accretion rate as a main driver. Another approach involves the  investigation of continuum and emission-line variability. This is the topic of this contribution.  

The extremes of AGN variability provide valuable insights into the underlying drivers and physics of the central engine. 
In X-rays, extreme flux and spectral states can reveal the nature of the inner accretion disk, probe the physics of matter under strong gravity, may offer a way of measuring SMBH spin, and can uncover the mechanisms responsible for the expulsion of matter from the SMBH in form of strong outflows. 

Amplitudes of X-ray variability exceeding a factor 30-50 are still rare, and only a few systems are known to exceed a factor of 100 <cit.>.  
The highest amplitudes of X-ray variability to date from the cores of galaxies, exceeding factors of 1000, have nearly exclusively been observed in non-active galaxies in the form of temporary flares from stars tidally disrupted by dormant SMBHs (stellar tidal disruption events; TDEs). These latter will not be discussed much further in this contribution <cit.>. 

In the optical regime, changing-look AGN (CL AGN hereafter) with highly variable emission lines which change the spectroscopic Seyfert type, provide us with tight new constraints on the physics of the outer accretion disk, the BLR and the coronal-line region (CLR). 

The term CL AGN was initially coined to describe systems which changed their X-ray spectrum from mildly or unabsorbed to heavily absorbed or Compton-thick  <cit.>. However, in recent years the term has been widely used to refer to optical spectroscopic Seyfert-type changes, from type 1 to type 2 (or 1.8, 1.9) and vice versa <cit.>. We use this latter definition throughout this publication. 

Single cases of CL AGN were noticed already decades ago. One prominent example is NGC 4151.  
Traditionally known as intermediate-type Seyfert galaxy <cit.>, its broad Balmer lines became very faint in 1984 <cit.>, barely detectable but still
present <cit.>. They were already back to their bright state in January 1985 <cit.>. 
Two recent examples are shown in Fig. <ref>, initially selected either through their high-amplitude continuum flaring and the CL event was then found in triggered follow-up optical spectroscopy (HE 1136–2304;  results in Fig. <ref> taken from <cit.>), 
or targeted in the course of reverberation mapping and long-term spectroscopy over decades 
(NGC 3516; results in Fig. <ref> taken from <cit.>).  While HE 1136–2304  only changed its Seyfert type from 1.5 to 1.8, NGC 3516 changed from type 1 to type 1.9. 


  


The topic of CL AGN  received particular attention in recent years, because larger samples were selected from optical surveys, more extreme cases were identified, events were detected not only in Seyfert galaxies but also in quasars where the emitting regions are much more extended so that longer variability timescales are expected, and higher amplitudes of variability were detected not only in X-rays but also in the optical band <cit.>. 
The rapid and high-amplitude changes pose challenges to accretion-disk models <cit.>, and many new scenarios for rapid changes in the disk emission have been developed in recent years. 

The first part of this review will focus on extreme cases of variability of radio-quiet AGN and the second part will briefly address semi-periodic variability of candidate binary SMBHs.  In Sect. 2 we review the frequency of CL AGN. In Sect. 3 we present a new scheme which sorts the CL AGN into different phenomenological categories based on their continuum and line variability. 
Sect. 4 briefly summarizes the different theoretical approaches suggested to understand CL AGN, and in Sect. 5 representative  cases of extreme flux and spectral variability are discussed, including new X-ray observations. Sect. 6 provides a critical summary of observational effects which can lead to false CL detection and elaborates on how to distinguish true CL events from CL look-alikes. 
Sect. 7 addresses (semi-)periodic variability of radio-loud systems, which has been suggested to be associated with the presence of binary SMBHs. A focus is on the blazar OJ 287. The article concludes with an outlook into future time domain surveys (Sect. 8).

Throughout this publication, we use a cosmology with 
H_ 0=70 km s^-1 Mpc^-1, Ω_ M=0.3 and 
Ω_Λ=0.7
and the cosmology calculator of <cit.>. 



§ THE FREQUENCY OF CL AGN


Large-area spectroscopic surveys offer excellent data bases for selecting samples of CL AGN. Ultimately, these will provide us with the frequency, timescales, and disk and host properties of CL events in Seyfert galaxies and quasars.
A number of selection effects have to be kept in mind: For instance, if the starting sample consists of type 1 AGN, then turn-offs (decreases in line and continuum luminosity) will be preferentially found in the follow-up spectroscopy. If the starting sample consists only of type 2 AGN, then only turn-ons, if any, will be found at the second epoch of spectroscopy. 
Further, the amplitude of the type change (into type 1.8, 1.9, or 2) will depend on telescope sensitivity, with faint Balmer lines more difficult to detect in spectra of lower signal-to-noise ratio (see also Sect. 6). 
Further, AGN with already faint Balmer lines (lower Eddington accretion rates at fixed SMBH mass) will be preferentially detected as CL AGN (as turn-offs), because only a small change in the luminosity is needed to make the lines undetectable, and therefore make the AGN appear as CL. If, instead the broad lines are luminous, a much larger luminosity change is required to identify a CL event. An alternative could be therefore to require a minimum factor (10) of variability, independent of exact AGN type, to search for CL AGN. 

Finally, the selection criteria used to identify CL AGN (e.g., the required amplitude of the type change) so far have varied between different studies which needs to be taken into account as well when directly comparing results. 

Larger-sample searches for CL AGN have been carried out recently with, for instance, LAMOST <cit.>, 6dFGS/ Skymapper <cit.>, and DESI <cit.>. Otherwise, mostly 
the exceptional data base of the Sloan Digital Sky Survey <cit.> has been employed
for subselecting larger samples of AGN which have been observed at least twice with SDSS or once with SDSS and followed-up at other telescopes <cit.>. 
These have started to quantify the frequency and properties of CL events and types in Seyfert galaxies and quasars. 

<cit.> found that the frequency of Seyfert galaxies significantly changing their spectroscopic type was 10% in a sample of nearby type 1 Seyferts initially selected from SDSS and then followed up with Keck. The same frequency in an SDSS selected sample of quasars was lower at ∼1% <cit.>. Approximately equal numbers of events got brighter and changed towards type 1, or got fainter and changed towards type 2. 
The largest sample of CL AGN selected recently from SDSS V <cit.> contained 67% CL AGN which showed  dimming and 33% which showed brightening. The CL AGN have a median accretion rate of L/L_Edd = 0.025, slightly lower than in a control sample with L/L_Edd = 0.043.    

Archival searches on large spectroscopic samples are excellent for statistical inferences on the CL AGN population. An alternative approach has been to carry out rapid MWL follow-up observations within days to weeks, once new CL AGN candidates were identified by the onset of flaring or dipping in the optical or X-ray band <cit.>. That way, dense monitoring of the ongoing, evolving event is possible. Representatives of such AGN are discussed in Sect. 5.



§ DEFINITION OF DIFFERENT CATEGORIES OF CL AGN


In recent years, many different cases of high-amplitude BLR and continuum variability have been reported, making CL AGN quite a mixed bag, with possibly a variety of different underlying physical processes operating in different systems and at different timescales. 

Therefore, here, we first introduce several different phenomenological CL AGN categories, based on the type of continuum variability and/or line response: 

∙ Category I: Slow switch-on AGN. These are CL AGN which slowly turn on, on a timescale of years to decades.  An example for this pattern may be AT2017bcc <cit.>, even though the event is unusual and was initially interpreted as superluminous supernova or a candidate tidal disruption event (see also our Sect. 6). 

∙ Category II: Slow switch-off AGN. 
In this category are CL AGN which slowly turn off, on a timescale of years to decades.
A prime example for this category is Mrk 590. Once known as a classic Seyfert 1 galaxy, its optical continuum emission diminished by a factor of 100 over two decades <cit.>. In low-state, the only broad emission line still visible in the optical spectrum is a weak component of Hα, implying a change from type 1 to type 1.9. 

∙ Category III: Rapid outbursts. In this category are CL AGN which spend most of their time in fainter states but then show a rapid (timescales of weeks to months) strong increase in their optical–X-ray continuum luminosity, followed by a response in the broad emission lines.  Prime examples are IC 3599 <cit.> and NGC 1566 <cit.> which showed dramatic increases of their continuum flux on a timescale of weeks to months.  

∙ Category IV: Rapid drops. In this category are CL AGN which spend most of their time in brighter states but then show a  rapid (timescales of weeks to months) decrease in their optical–X-ray continuum  and broad-line flux. 

∙ Category V: Regular, repeating brightening and dimming events. In this category are CL AGN which show recurrent medium-amplitude increases and decreases in their line and continuum brightness. Many of the reverberation-mapped AGN are in this category, for instance NGC 5548 <cit.>. 

∙ Category VI: Frozen-look AGN.  
In this category are AGN which do not show any emission-line changes in response to strong continuum variability. 
This category is rare. Examples include 
saturation effects in the BLR ionization in case of particularly high continuum luminosity <cit.>.

A much stronger example is the lack of HeII response despite high-amplitude soft X-ray continuum variability of Mrk 335 <cit.>, explained in this case by changes of (dust-free) absorption occurring only along our line-of-sight, not seen in other directions (i.e., not seen by the bulk of the emission-line gas).    



Several CL events are known to repeat. Repetition may occur in all categories, but it depends on the cadence and duration of the observational coverage how easily recurrence can be detected. Single AGN may even cross categories in the long term.   
Some processes are more common than others. 
Tab. <ref> provides a summary of the different categories of CL AGN. 

The strongest cases of CL AGN are in categories I–IV since they exhibit the largest and most unexpected changes and place most tight constraints on variability models. 

Finally, we note that CL AGN were first identified based on observations, and therefore the proposed categories closely match what has been observed so far. They do not yet include any underlying theoretical scenarios, because it is still unknown how to explain CL AGN, and many different scenarios have been proposed in the literature. Not all of them make predictions for the timing and spectral evolution, yet. 
In the next Section, we provide a short overview over the different models for CL AGN  which have been proposed in recent years. 



§ (NEW) MODELS FOR EXTREME AGN VARIABILITY


There are two fundamentally different ways to explain high-amplitude variability. One of them is intrinsic variability where the emission of the accretion disk and BLR itself changes.
The other is apparent variability where the intrinsic emission remains constant, but is then affected on its way to the observer.  Such extrinsic variability can be caused by the effects of gas absorption and dust extinction, or by the effect of gravitational lensing.

We discuss the different mechanisms in turn.



 §.§ Extrinsic variability: gas absorption and dust extinction


Gas absorption affects the X-ray continuum emission through Hydrogen and metal K-shell absorption. For gas column densities exceeding  ∼10^23-24 cm^-2, the material becomes Compton-thick and no X-rays are detected at all below 10 keV where most X-ray satellites operate.

If an absorber contains dust, it also significantly affects the optical continuum through dust extinction, causing a characteristic reddening of the continuum shape and a deviation of the BLR Balmer decrement from its (case B) recombination value.

Dust extinction and/or gas absorption is therefore an efficient mechanism to change the observed luminosity of an AGN, and if at work the mechanism can be relatively easily identified  through its characteristic effects on the optical and X-ray spectra.

However, while partial obscuration from outflowing dusty clouds in the outer BLR can significantly affect line profile shapes <cit.>, and while single CL AGN do show effects of significant extinction <cit.>, this mechanism has generally not been favored for the majority of CL systems, because the expected BLR reddening was not observed <cit.>, the optical-UV continuum shape was inconsistent with extinction <cit.>, the associated X-ray absorption was not observed <cit.>, the timescale of variability was too fast <cit.>, and/or the level of polarization was very low <cit.>. 



 §.§ Extrinsic variability: gravitational lensing


Lensing or microlensing can magnify (or demagnify) the accretion disk and/or the BLR emission.  Depending on the geometry, it can give rise to achromatic light curve variability and/or it can affect different parts of the emitting region, causing complex line profile shapes and changes.

Microlensing modelling of BLR profile variations has not yet been much applied to changing-look AGN, but studied in a number of known lensed quasars <cit.>. It is rare and is therefore not expected to explain the population of nearby CL AGN as a whole.



 §.§ Intrinsic variability: Modified lampposts, disk instabilities, disk precession, influence of magnetic fields, BLR stability, binaries


CL events challenge our ideas of the properties of the accretion disk in AGN. 
Given the rapid and high-amplitude variability of some CL AGN, a large number of new theoretical studies in the last few years have explored new mechanisms for accretion disk structural and emission changes. Some of these addressed 
mechanisms of a more rapid transfer of changes across the disk (e.g., involving magnetic fields), but left open the origin of the high-amplitude outburst in the first place; others addressed the outburst mechanism directly; and yet others explored changes of the BLR structure itself.

First, a modified lamppost model has been suggested <cit.>, assuming that the X-ray emission which arises from the lamppost is not directly reprocessed by the accretion disk, but rather by a system of dense gas clumps above the disk. This explains a rapid MWL response, but leaves open the mechanism(s) of the intrinsic high-amplitude of the X-ray variability itself. 

Second, different types of accretion disk instabilities have been considered to explain high-amplitude continuum changes, including the radiation-pressure instability of the inner disk <cit.>, a Hydrogen ionization instability <cit.>, or instabilities in warped disks at large radii <cit.>. An instability in the circumnuclear gas supply has been discussed as well 
<cit.>. Tidal effects on the mini-disks surrounding both SMBHs of a binary SMBH system <cit.> have seen suggested as well.  

Third, the influence of magnetic fields, or a combination of several effects, has been considered in different ways <cit.>. 

Fourth, effects of two types of shocks developing in highly tilted, highly precessing disk were discussed <cit.>. 

Finally, some models were suggested where the BLR gas 
itself is directly affected, through an instability <cit.>, initially proposed to explain the absence of a BLR in low-accretion-rate systems.  



 §.§ New approaches of understanding BLR and CLR physics


Independent of the exact mechanism which causes the change in the illumination of the emission-line gas, the reprocessing of the radiation on BLR and CLR spatial scales provides us with a wealth of new applications regarding BLR and CLR physics, and the underlying atomic processes. 
We just give a few examples here. For instance,
some of the highest-amplitude outbursts are known to produce a strong response not only in the BLR, but also in the CLR. Strong high-ionization lines in the CLR temporary excited by an outburst provide tight constraints on the intrinsic, unobservable EUV shape of the SED because their ionization potentials are located in the EUV to soft X-ray regime. Further, atomic parameters of many coronal lines are still poorly known with uncertainties in atomic parameters and collision strengths as large as a factor of 10 <cit.>
given the difficulties of producing them in laboratory plasmas. In addition, rare coronal line transitions can be excited that way, and their ratios provide us with valuable new diagnostics of the density and temperature <cit.> 
of the CLR in AGN.  

A second example regards the BLR physics. The nature of the outer BLR is still not well understood. Once gas temperatures drop sufficiently below ∼2000 K  the gas conditions change significantly, Hydrogen molecules can be present <cit.>, and new ionization/recombination processes start to dominate (see <cit.> for a detailed discussion of the relevant physical processes, including chemi-ionization/recombination processes, the importance of H^*(n) + H(1s) collisional ionization, the influence of inverse recombination processes, and 
in <cit.>
the  importance of excitation/de-excitation processes with (n-n') mixing in such collisions).  Turn-off CL AGN may be ideally suited to probe the presence and physics of such an outer BLR component, as the inner parts of the BLR will fade first in response to a deep drop in the ionizing continuum luminosity, systematically leaving the outer parts of the BLR detectable in high-cadence, time-resolved spectroscopy of the BLR response.  



§ EXTREME CASES OF AGN VARIABILITY AND THEIR INTERPRETATION


Given the different types of extreme continuum and spectral variability of AGN and the relatively low number of sources, we discuss these on an individual basis, focussing on systems for which we have obtained long-term observations, and adding new data from the Swift archive as well. The object properties are listed in Tab. <ref>.













 §.§ High-amplitude outbursts and the cases of IC 3599 and NGC 1566




  §.§.§ IC 3599


The Seyfert galaxy IC 3599 underwent a giant outburst detected in the course of the ROSAT all-sky survey. Its X-ray flux increased by more than a factor 100. Optical spectra taken by chance during the ongoing outburst revealed bright broad Balmer lines and high-ionization iron coronal lines <cit.>. These emission lines then faded strongly in subsequent years <cit.>. In low-state, among the Balmer lines, only broad Hα is still significantly detected, leading to a Seyfert 1.9 classification <cit.>. Faint coronal lines are still present in low-state. Photoionization modelling of the optical outburst spectrum revealed that the high-ionization lines are well explained with gas typical of a CLR, with a gas density of ∼10^9 cm^-3
<cit.>. 
The cause of the CL event of IC 3599 was not discussed in more detail at that time, but Narrow-line Seyfert 1 (NLS1) variability, a TDE, or an accretion-disk instability were briefly mentioned as possibilities; the latter favored by <cit.>. 

In ongoing Swift monitoring, a second X-ray outburst of similar amplitude was detected in 2010, 19.5 yr after the first one <cit.>. The outbursts are consistent with a Lightman-Eardley disk instability, using the observed flare repeat time of 19.5 yrs 
and a SMBH mass range of 10^6-7 M_⊙ inferred from observations, 
which then implies a truncation radius between inner and outer disk of r_ trunc = 5–45 r_ g <cit.>.  
MWL monitoring is ongoing in order to search for new high-amplitude outbursts. No third one has been detected so far <cit.>, but significant lower-amplitude X-ray variability is ongoing (Fig. <ref>). During outbursts, the X-ray spectrum of IC 3599 is exceptionally steep (Γ_ x∼ 4). The cause for the steepness remains unknown, but it seems to point towards the absence of a disk corona. Even in low-states, the X-ray spectrum remains very soft (Fig. <ref>). 

Our Swift monitoring of this extreme optical CL AGN is ongoing. Rapid MWL follow-ups including optical spectroscopy at dense cadence will provide us with a unique method of reverberation-mapping the BLR and CLR following the next giant outburst.


  



  




  §.§.§ Recurrent changing-look activity of NGC 1566
 

NGC 1566 is similar in several respects to IC 3599 including the detection of strong iron coronal lines during outburst, but with a lower amplitude in X-rays, particularly hard (not soft) X-ray spectra, and better optical spectroscopic coverage, so that one can tell that it went through repeated CL events. 

NGC 1566 is a very nearby NLS1 galaxy <cit.> at redshift z=0.005. The widths of the broad Balmer lines of NGC 1566 are FWHM(Hβ) = 1950 km/s and FWHM(Hα) = 1970 km/s <cit.>. 

Fe II emission is detected in the type 1 and 1.5 spectral state <cit.>.
NGC 1566 was known as a bright type 1 AGN in the 1960s <cit.>, but became much fainter in subsequent years.
Spectroscopic observations taken between 1980 and 1982 found it to be a type 2 in 1980, then gradually changing back to a type 1. Another CL event was detected in 1985 <cit.>. Most of the changes occurred rapidly, within only 4 months. The BLR Balmer decrement did not change significantly during the CL event <cit.>.

The most recent outburst of NGC 1566 in 2018 was first detected with INTEGRAL in the hard X-ray band <cit.>. MWL follow-up observations were rapidly triggered <cit.> which resulted in the detection of a new CL event. 
During the outburst, the optical spectrum changed from type 1.8 to type 1, accompanied by strong variations in the helium lines and iron coronal lines <cit.>. 
The FWHM of the Balmer lines remains relatively constant during all observations. The emission lines OIλ8446 and CaIIλλ8498,8542,8662 are double-peaked <cit.>. 

The X-ray spectrum of NGC 1566, measured in rapid X-ray follow-up observations carried out with XMM-Newton, NuSTAR and Swift <cit.>, is a typical type 1 X-ray spectrum at high state, extending to high energies 

and modified by ionized absorption and mild reflection. With the reflection grating spectrometer (RGS) an outflow with velocity of 500 km/s, possibly launched during the outburst, was detected. The amount of X-ray cold absorption does not change between low-state and high-state, implying that variable absorption is not the cause of the outburst. <cit.> concluded that a heating front propagating through the disk caused by the Hydrogen instability provides a consistent explanation of the 2018 outburst. 

An earlier hard X-ray flare is present in the Swift BAT light curve <cit.> in 2010, but was not followed up spectroscopically.

The Swift XRT long-term X-ray light curve of NGC 1566, based on our own monitoring observations and additional archival data, is displayed in Fig. <ref>. The X-rays continue to be at low emission levels in 2024. 

NGC 1566 is the nearest of the CL cases detected so far in NLS1 galaxies <cit.>, and among the nearest repeating CL AGN known. 


  




 §.§ Regular intermediate-amplitude dimmings and brightenings of reverberation-mapped AGN and the cases of NGC 5548 and NGC 3516 

  


 §.§ NGC 5548

As a nearby, bright, intermediate-type Seyfert galaxy, NGC 5548 has been the target of several reverberation mapping campaigns and many optical spectroscopic studies covering several decades <cit.>.  It is variable in the optical and X-ray regime, densely monitored with Swift <cit.>. In low-states its BLR becomes faint, but 
Hα and Hβ are always detected. NGC 5548 can therefore be regarded as a CL AGN, changing between Seyfert type 1.5 and type 1.8. 
In the same sense, many reverberation-mapped AGN would represent CL AGN, since pre-selected to show significant continuum variability. 

The  amplitude of variability of NGC 5548 in X-rays is much lower than in the other AGN discussed in this contribution (Fig. <ref>, based on all archival Swift XRT observations of NGC 5548).

Based on evidence for periodicity of the long-term optical light curve between 1972 and 2015 and accompanying changes in the broad lines with a period of 5700 d, it was suggested that NGC 5548 may host a compact binary SMBH <cit.>.


  




 §.§ NGC 3516


The Seyfert galaxy NGC 3516 has long been known to be highly variable in its continuum and broad emission lines. Part of the initial sample of <cit.>, strong broad-line variability was first reported by <cit.>. 
NGC 3516 stands out as highly variable in the X-ray band as well. For instance, it was one of the most highly variable AGN in a dedicated search for such AGN in the ROSAT data base <cit.>.  
Its long-term Swift XRT light curve <cit.>, demonstrating the high-amplitude X-ray variability of this AGN in recent years, based on data retrieved from the Swift archive, is shown in Fig. <ref> in comparison with the other AGN discussed in this contribution.  

NGC 3516 has been the target of many spectroscopic monitoring campaigns spanning decades  including reverberation mapping   <cit.>, showing that NGC 3516 is a CL AGN which varies between type 1 and type 1.8 or even 1.9 at selected epochs.  

 Based on a detailed study of the Hβ flux and profile variability, <cit.> concluded that an intrinsic mechanism, rather than large-scale obscuration in the outer BLR, is required to explain the CL events in NGC 3516, even though a contribution of dust within the BLR may still play a role at low activity phases. 



 §.§ Frozen-look AGN and the case of Mrk 335


The NLS1 galaxy Mrk 335 was once among the X-ray brightest AGN. However, when first observed with Swift, its X-ray flux had strongly declined (Fig. <ref>). When followed up with Swift in dense, long-lasting monitoring campaigns <cit.>, Mrk 335 did exhibit episodes of high-amplitude flaring back into high-states. However, these episodes became less frequent, less long-lasting and less bright, and since 2018, Mrk 335 has remained in deep X-ray low-states most of the time (Fig. <ref>). This is still the case in our most recent observations from 2024.  


Since bright and nearby, Mrk 335 still provides us with good-quality X-ray spectra even in low-state and this is the reason why it was observed multiple times in deep spectroscopic follow-ups, e.g. with XMM-Newton and NuSTAR <cit.>. However, two very different spectral models still provided successful fits to the complex X-ray low-state spectrum, with no way to distinguish based on the quality of the spectral fit alone: The first one consists of an absorption model where the absorber partially covers our line-of-sight to the continuum source (the accretion disk),  the second one is a reflection model where the intrinsic photons are subject to blurred reflection from the inner accretion disk.  

The solution came from combining X-ray with optical–UV data. This approach showed that X-ray and optical–UV emission are uncorrelated most of the time <cit.> and that the high-ionization, broad HeII emission line did not follow the simultaneous high-amplitude X-ray variability even though its ionization potential is in the soft X-ray regime <cit.>.
These results can be understood if the X-ray variability is predominantly caused by a (partial-covering, dust-free) absorber located along our line-of-sight, but not covering a full 4π sphere shielding the BLR. 
The presence of line-of-sight absorption is independently detected with HST <cit.> and with XMM RGS <cit.> spectroscopy of Mrk 335, revealing UV and X-ray absorption lines.  This result does not exclude a reflection contribution to the X-ray spectrum, but requires a strong absorption component, which drives at least a large fraction of the observed variability. 

In summary, Mrk 335 is a good example for (dust-free) absorption-driven X-ray continuum variability. It is a frozen-look AGN as far as its broad HeII emission is concerned <cit.>, but its Balmer lines still do show a response to the mild optical continuum variability during the epoch it was the target of reverberation mapping <cit.>.  Swift monitoring of Mrk 335 is still ongoing in order to trace the duration of the deep low-state, follow the decoupling between X-ray and UV–optical continuum variability, and catch its rise back to high-state. 



§ APPARENT CL VARIABILITY AND IMPOSTORS


Not all observed cases of CL events are real. Some observational setups can mimic false BLR variability, instrumental sensitivity needs to be taken into account when assessing the presence or absence of faint Balmer-line emission, and some phenomena look spectroscopically similar to CL AGN but are due to very different physical processes. We discuss these in turn. 



 §.§ Artificial variability, biases and challenges
  

(1) Slit or fiber not on the nucleus. 
Nearby galaxies appear largely extended on the sky, occasionally making it difficult to position a (narrow) slit or fiber exactly on the nucleus. Since BLR and continuum are pointlike (only broadened by atmospheric seeing) whereas the NLR is extended, a mis-positioned slit will provide a spectrum with apparently faint BLR and continuum. That way, dimming events and turn-off CLs can be mimicked.  

(2) Telescope sensitivity. Faint broad Balmer lines will increasingly merge with the noise, and it then depends on the exposure time and telescope sensitivity, whether faint Hβ and/or Hα can still be detected in a given spectrum. Accordingly, the turn-off state will be classified as type 1.8, type 1.9, or type 2, even if both Balmer lines are still intrinsically present. It is possible that some of the historic CL events reported to reach type 2 states in the 1970s and 1980s are due to this effect. Complete turn-offs into type 2 states have been rarely observed in recent years. 



 §.§ CL look-alikes: TDEs and SNe (type IIn and/or in dense medium)


The optical spectra of both, TDEs and supernovae (SNe) IIn can show temporary broad (and/or narrow) optical emission lines, and will thus mimic a changing-look AGN. In fact, there is a parameter space, where all three outburst types (AGN, SNe, and TDEs)  look alike, and are challenging to distinguish from each other.  Here, we describe three scenarios, and discuss ways how to distinguish them from bona fide CL AGN variability <cit.>.[Further, there is a new class of  long-duration ambiguous transients <cit.>, even though most or all of them are characteristically different from CL AGN.] 

(1) SN of type IIn: These type of SNe are known to show temporary broad and narrow emission lines in their optical spectra which look very similar to AGN <cit.>. The narrow lines arise from strong interactions of the SN ejecta
with dense circum-stellar matter. Strong ionizing radiation may originate from the
shock break-out, Compton-upscattering, as well as in shocks when the SN ejecta collide with the precursor wind and/or the interstellar medium. 


(2) SN in dense medium, in a quiescent galaxy or an AGN: Supernovae in a dense medium would particulary efficiently reprocess continuum into emission lines, and can even produce temporary low-level hard X-ray emission (e.g., SNe in the outskirts of the dense torus environment could be partially obscured or extincted, and their decline light curves and continuum emission would look different from other SNe). The situation is especially complicated, if the SN does not happen in a quiescent galaxy but in a long-lived AGN, and all of the SN features would superpose on the long-lived AGN spectrum, which can itself be (mildly) variable.

(3a) TDE in a quiescent host or in an AGN. TDEs are most reliably identified in quiescent galaxies (i.e., systems with dormant SMBHs without a long-lived accretion disk and therefore without a NLR), because TDEs are then the only known mechanism to produce giant X-ray flares of quasar-like peak luminosity <cit.>.
In such a scenario, broad emission lines can arise from the temporary accretion disk formed from the disrupted star, and the flung-out stellar debris. If the environment is gas-rich, narrow emission lines can be excited as well, first detected from gas closest to the line of sight (little light travel-time delays).
If a TDE happens in an AGN, all the TDE features overlapp with the long-lived AGN features. 

How can we distinguish all of these impostors from true CL AGN ? One crucial parameter is the X-ray luminosity and X-ray light curve. Another key observation  is repeat optical spectroscopic coverage, ideally including a spectrum before the outburst.
The X-rays are important, because longer-duration (weeks to months) X-rays exceeding  10^41-42 erg/s have never been observed from SNe, thus allowing to exclude an SN interpretation of the spectrum.  
Pre-outburst optical spectroscopy is important, because it tells if the host is active or quiescent, based on the presence or absence of a NLR. Similarly, pre-outburst X-ray detections imply a long-lived AGN if L_ x > 10^42 erg/s and exclude a TDE in a quiescent host galaxy. 
Finally, post-outburst spectroscopy will tell, if the narrow lines fade (SN IIn or TDE) or are permanent.
The most challenging situation is a TDE or SN in an AGN, in which case the long-term spectroscopic evolution of multiple emission lines provides the best diagnostics to distinguish between all three scenarios. Nevertheless, individual sources could be challenging to classify. PS16dtm (SN 2016ezh) is a good example. It shows properties of all three, a superluminous SN, a TDE, and a CL NLS1, and all three classifications have been proposed and discussed in the literature <cit.>, (Ilić, these proceedings).

Finally, TDEs are extremely rare, therefore in a statistical sense we know that at most a tiny fraction of CL AGN  could be mimicked by TDEs, because the observed frequency of CL events in nearby galaxies <cit.> is far too high.



§ (SEMI)-PERIODIC VARIABILITY, CANDIDATE BINARY SMBHS, AND THE CASE OF OJ 287



  




 §.§ Candidate binary SMBH identification from light curves


Supermassive binary SMBHs form in the course of galaxy mergers <cit.>. The evolved binary SMBHs with small separations, of much less than a parsec, can no longer be spatially resolved with current techniques, and we rely on indirect methods to detect them. A large variety of different detection methods has been explored in the literature <cit.>. The most common one makes use of (semi-)periodic signals in long-term MWL light curves. These can be induced as a consequence of orbital motion of one or both SMBHs, or precession of the accretion disk or jet. 
At least several periods should be covered in any light curve to be considered a reliable periodicity candidate; otherwise noise can mimic the periodic signal <cit.>.  

Many recent searches for binary SMBH candidates have been conducted on ongoing optical photometric surveys such as the Catalina Real-time Transient Survey <cit.>, the Palomar Transient Factory <cit.>, and Pan-STARRS <cit.>. Recent examples of systems with sinusoidal light curve variations include PG 1302–102 <cit.>, and PKS 2131–021 <cit.> (see Sect. 5.3 for the case of NGC 5548, and <cit.> for a recent discussion how to search for binary SMBHs via BLR line profiles). 

A different semi-periodic light curve pattern, not sinusoidal, is shown by the candidate binary SMBH OJ 287. Its long-term optical light curve exhibits recurrent bright double-peaks repeating every 11±1 yrs, most clearly present in the 1970–1990 light curve of this blazar <cit.>. The most recent double-peak has been identified in 2016–2017 <cit.>.
OJ 287 has the most densely covered MWL lightcurve of all binary SMBH candidates <cit.> and is discussed in more detail in the next Section. 



 §.§ OJ 287 and constraints on binary SMBH models from the MOMO project


Several different binary SMBH scenarios for OJ 287 have been explored in the literature; involving either disk or jet precession, orbital motion of one or two jet-emitting SMBHs, disk impacts of the secondary SMBH during its orbit around the primary, or the presence of mini-disks around two SMBHs fed by a circum-binary disk <cit.>. 
According to other suggestions, a binary is not required, or its absence is preferred <cit.>. All of the binary models have offered explanations for the  double-peaked nature of the outbursts and are based on excellent theoretical frame works, but none has so far been able to predict correctly the timing (and spectra) of the outbursts of the recent light curve of OJ 287 since 2000 <cit.>. 

Tight new model constraints came from ongoing dense MWL monitoring in the course of the MOMO project. It was initiated in 2015 and is designed to distinguish between different binary SMBH scenario predictions, and it independently sheds new light on blazar disk-jet physics
<cit.>. Dense monitoring of OJ 287 at a cadence of a day to weeks at multiple frequencies in the radio, optical, UV, X-ray and gamma-ray bands, along with dedicated deeper MWL follow-up spectroscopy at selected epochs, provides timing, spectra, and broadband SEDs at all activity states of OJ 287 <cit.>. This is the densest and longest MWL monitoring so far carried of OJ 287 at > 12 frequencies, and among the densest of any blazar (Fig. <ref>). 

Recent MOMO results <cit.> have clearly established that (1) no outburst occurred in October 2022 <cit.> 
and instead the last double-peaked outburst was observed in 2016–2017 (our Fig. <ref>); (2) the observed upper limit on the accretion-disk luminosity of OJ 287 (based on 3 different estimates) is a factor >10-100 fainter than required by a ∼10^10 M_⊙ SMBH accreting at 10% Eddington; 
instead 
(3) the (primary) SMBH mass derived from BLR scaling relations is modest and of order 10^8 M_⊙ consistent with the faintness <cit.> of the host galaxy;  and (4) a thermal Bremsstrahlung spectrum with a predicted spectral index of α_ν,opt ≈ -0.2 has never been observed in the densely covered Swift lightcurve between 2016–2024.

The identification of the last double-peaked outburst in 2016 – 2017 implies that these outbursts are relatively regular spaced at 11±1 yr intervals, but are not exactly periodic. The low (primary) SMBH mass implies that OJ 287 is no longer a candidate for detection by near-future pulsar timing arrays. Instead, it would be in the sensitivity regime of LISA-type space-based gravitational-wave interferometers <cit.> upon its final coalescence. 

In summary, successful future binary SMBH models of OJ 287 should reproduce a regular double peak every ∼ 11 yr but allow for ±1 yr variations, should be based on a moderate primary SMBH mass of ∼10^8 M_⊙ (similar to the approach previously followed by <cit.>, but now without assuming strict periodicity), and should reproduce synchrotron outbursts. One  alternative binary modelling avenue to follow will be  the application of recent binary merger simulations which predict SMBH binaries with mini-disks fed by a larger circum-binary disk <cit.>. <cit.> loosely suggested that such a scenario could apply to OJ 287, but modelling of OJ 287 within that framework has not yet been carried out.   

The long-term light curve of OJ 287 at representative MOMO frequencies is shown in Fig. <ref>, including previously unpublished X-ray, optical, UV, and radio measurements from 2023–2024. It shows that OJ 287 is at a deep low-state in June 2024, particularly obvious in the optical, UV, and radio bands.[A previous deep optical  low-state of OJ 287 was once speculated to possibly represent an epoch in the binary SMBH orbit evolution where the jet of the primary SMBH is briefly misaligned due to the perturbation of a secondary SMBH when it passes near the jet <cit.>, but the deep low-states have happened too often in recent years <cit.> to be consistent with such an interpretation.]  
The next double-peak is expected in 2026–2028 <cit.>, and dense monitoring in the course of the MOMO project is planned.




§ UPCOMING TIME DOMAIN SURVEYS IN THE OPTICAL AND X-RAY REGIME


Upcoming time domain surveys in the optical and X-ray bands will not only detect large numbers of extremely variable AGN and CL AGN, but will also facilitate a search for the characteristic periodic signals of binary SMBHs based on the high-cadence light curves coverage every one to few days. In the optical bands, the Vera C. Rubin Observatory Legacy Survey of Space and Time  <cit.> is scheduled to start operation in the year 2025.
LSST will monitor the entire Southern sky for 10 years in 6 filters (ugrizy) with a 9.6 deg^2 field-of-view and a 5 σ g-band magnitude depth of ∼24.5 in a single 30s exposure [<https://www.lsst.org/scientists/keynumbers>]. 
LSST will detect millions of variability events per night, issuing an alert within 60s from the detection 
to enable prompt follow-ups with other instruments <cit.>.
The combination of joint photometry data sets from LSST and spectroscopy from other facilities will enable accurate classification for a plethora of events, including outbursts, flares and changing states in AGN. The extent of the sample will further increase the possibility to detect rare events, while the implementation of the Difference Image Analysis <cit.> on the entire dataset will allow us to identify variability in obscured or host-dominated AGN.
In the X-ray regime, two new transient missions will carry out high-cadence all-sky surveys. The Einstein Probe <cit.> was launched in January 2024 and the Space Variable Objects Monitor <cit.>) was launched in June 2024. 
Several studies have already highlighted how LSST is expected to produce important results in relation to accretion disc studies, reverberation mapping campaigns, and binary SMBH candidate detections in AGN <cit.>. 
In particular, binary SMBH candidates with periods of months to years, similar to candidates selected from ongoing optical surveys (Sect. 7), are expected to be detected in large numbers in radio-loud and radio-quiet AGN with LSST (longer periods) and in X-rays. 
Further, an advantage of LSST is its high cadence[the exact cadence is still being discussed and may change during its 10 year mission; the higher it is, the more short-period systems will be well covered] so that it can find shorter periods, and it is very deep, therefore it covers AGN with lower-mass SMBHs, of which there are many more, and also they are longer-lived at periods of weeks compared to more massive SMBHs.    This implies that the LSST catalog will contain large numbers of fainter short-period binaries.

Evolved, ultra-compact SMBH binaries of much shorter periods of only days which then coalesce years later in the LISA time window
can be detected in the optical band with LSST through the periodic light-curve modulation from the circum-binary disk feeding the two SMBH's mini-disks <cit.>,
while the orbital modulation from the two mini-disks themselves is best searched for in the X-ray regime, given the compactness of the mini-disks at very advanced stages of merging. 

Other future surveys with a strong time-domain component and expected to start by the late 2020s are the Cosmological Advanced Survey Telescope for Optical and UV Research (CASTOR), and the Nancy Grace Roman Space Telescope, in the NIR range. These two spacecrafts will add further MWL coverage to the variable Universe, and will have a strong synergy with contemporary surveys such as LSST and the EUCLID satellite (even though the latter does not have an explicit time-domain focus).































§ ACKNOWLEDGMENTS

 We would like to thank the Swift team for carrying out the observations we proposed.
 In addition to our own data, we have also used the Swift archive at <https://swift.gsfc.nasa.gov/archive/>. 
 We would like to thank all participants of the “VI. Conference on Active Galactic Nuclei & Gravitational Lensing” (Serbia, June 2024) for enlightening discussions. S.K. and Z.H. would also like to thank the participants of the conference on “Multi-messenger observations of supermassive black hole binaries” which was supported by the Lorentz Center (Leiden, June 2024) for very stimulating discussions. 
 S.K. would like to thank R. Antonucci, D. Hutsemékers, F. Marin, and M. Ochmann for very useful discussions. 
 E.B., N.B., D.I., A.B.K, J.K.D., S.M.M., and L.Č.P.  
 acknowledge funding provided by the University of Belgrade—Faculty of Mathematics (contract 451-03-66/2024-03/200104) and/or Astronomical Observatory Belgrade (contract 451-03-66/2024-03/200002), through grants by the Ministry of Education, Science, and Technological Development of the Republic of Serbia. D.I. acknowledges the support of the Alexander von Humboldt Foundation. A.B.K. and L.Č.P. thank the support by the Chinese Academy of Sciences President’s International Fellowship Initiative (PIFI) for visiting scientists. 
 V.P. and D.D. acknowledge the financial contribution from PRIN-MIUR 2022 and from the Timedomes grant within the “INAF 2023 Finanziamento della Ricerca Fondamentale”. D.D. also acknowledges PON R&I 2021, CUP E65F21002880003. 
 Z.H. acknowledges support by NSF grant AST-2006176 and NASA grants 80NSSC22K0822 and 80NSSC24K0440.
 K.K.G. acknowledges the Belgian Federal Science Policy Office (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the
 European Space Agency (ESA).
 This work is partly based on data obtained with the 100 m telescope of the Max-Planck-Institut für Radioastronomie at Effelsberg.   
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester <cit.>. This work has made use of public Fermi-LAT data <cit.>.  
This research has made use of the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy.
This work has also made use of the NASA Astrophysics Data System Abstract Service (ADS), and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.







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 Paper Title: Recent Update of Gas-Phase Chemical Reactions and Molecular Lines of TiO in Cloudy.



 























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1Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India
2Physics & Astronomy, University of Kentucky, Lexington, Kentucky, USA
3Physics & Astronomy, University of Georgia, Georgia, USA
4Stellar Science, 6565 Americas Pkwy 925, Albuquerque, NM 87110, USA






  * G. Shaw, Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research,  
Mumbai 400005, India (gargishaw@gmail.com).

 
  * G. Ferland, Physics & Astronomy, University of Kentucky, Lexington KY 40506, USA  (gary@uky.edu).

 
  * P. Stancil, Department of Physics and Astronomy and Center for
Simulational Physics, University of Georgia, USA.


  * R. Porter, Stellar Science, 6565 Americas Pkwy 925, Albuquerque, NM 87110, USA
 


Shaw et al.

Gas-phase TiO


G. Shaw et al



Ferland, G.J.
Phillip Stancil
Ryan Porter




We present our current update on the gas-phase chemical reactions and spectral 
lines of TiO in the spectral synthesis code Cloudy. 
For this purpose, we have added 229 Ti-related reactions in the chemical network. 
In addition, we consider 230 fine-structure energy levels, the corresponding 223 
radiative transitions, and 444 
collisional transitions with ortho and para H_2
and predict 66 TiO lines.
We perform spectroscopic simulations of TiO emission from the 
circumstellar region of the oxygen-rich red supergiant VY Canis Majoris to validate 
our update. Our model reproduces
the observed TiO column density. 

This update is helpful in modeling dust-free astrophysical environments where Ti is in the gas phase and TiO can form.




Presentamos nuestra actualizacion actual sobre las reacciones quimicas 
en fase gaseosa y espectrales lineas de TiO en el codigo de sintesis 
espectral Cloudy. 
Para ello, hemos anadido 229 reacciones relacionadas con 
el Ti en la red quimica. 
Ademas, consideramos 230 niveles 
de energia de estructura fina, los correspondientes 223 
transiciones radiativas, y 444 transiciones colisionales 
con orto y para H_2 y predecir 66 lineas de TiO. 
Realizamos simulaciones espectroscopicas de la emision de 
TiO desde el region circunestelar de la supergigante roja 
rica en oxigeno VY Canis Majoris para validar nuestra 
actualizacion. Nuestro modelo reproduce la densidad de la 
columna de TiO observada. Esta actualizacion es util para 
modelar entornos astrofisicos libres de polvo donde el 
Ti esta en fase gaseosa y se puede formar TiO.




Astronomy software
Computational methods
Galaxy
Molecular data




Recent update of gas-phase chemical reactions and molecular lines of TiO in Cloudy
    Gargi Shaw1,
Gary J. Ferland2,
Phillip Stancil 3,
Ryan Porter 4

    
==================================================================================





§ INTRODUCTION

The spectroscopic simulation
code Cloudy simulates the conditions in
a non-equilibrium astrophysical plasma
and predicts the resulting spectrum and column densities of various ions, neutrals, 
and molecules<cit.>. 
Our aim is to predict more molecular lines with better precision. Hence, 
we regularly update our chemical network and incorporate improved chemical reactions
and internal structures for our existing chemistry whenever new or better data are available <cit.>
and add new molecules as the astrophysical need arises.
One such molecule is TiO, a possible precursor of inorganic dust <cit.>. 

TiO is the dominant source of opacity in the atmosphere of cool stars <cit.>
and is observed in the atmosphere of giant M stars <cit.>. 
<cit.> observed pure rotational lines 
of TiO in the circumstellar envelope of the red supergiant VY Canis Majoris (VY CMa). 
The observed column density of TiO in VY CMa is (6.7 ± 0.8) 
× 10^15 cm^-2. They also observed TiO_2. 
In an oxygen-rich circumstellar envelope, nearly all carbon is
locked into CO, and hence, TiO forms and can act as an important seed for 
inorganic dust formation <cit.>. 
<cit.> showed that for VY CMa, the dust and TiO emission come 
from different regions.
Their formation regions are separated by about 0”.15 corresponding to 2.7×10^13cm assuming the 
distance of VY CMa is 1.20 kpc <cit.>.
This confirms that TiO is present in the dust-free region of VY CMa.

 The solar abundance of Ti is 8.91×10^-8 <cit.>. However, TiO is not observed in the ISM due to the high depletion of Ti <cit.>.




In dust-free environments, Ti is in the gas phase and TiO can form.

Here, we aim to include the gas-phase chemical reactions and spectral lines of TiO 
in the spectral synthesis code Cloudy <cit.> which will be useful in modeling dust-free astrophysical environments where TiO is in the gas phase. These updates will be part of the upcoming 2024 release of Cloudy.



§ CALCULATIONS AND RESULTS



 §.§ Updated Ti-chemistry in 

There is a scarcity of reaction rates for Ti-chemistry due to its very low abundance in the ISM. 
It is not included in the UMIST Database for Astrochemistry (UDfA) <cit.> or in Kinetic Database for Astrochemistry (KIDA) <cit.>. We incorporate available 
reactions<cit.>, but these are small in number. 

We adopted Si-based chemistry as a proxy for the missing Ti chemistry.
Both Ti and Si are important refractory elements <cit.>. 
In addition, 
Ti-oxides and Si-oxides are precursors of 
inorganic dust, and Si-chemistry is well-studied and documented.
Hence, for 
the rest of the Ti-related reactions, we copied analogous reactions involving Si from 
UDfA. All the reactions in our new network are given in the appendix. 







A large number of models covering diverse astrophysical environments are publicly 
available with the cloudy download under the directory .
We compare tsuite model predictions every day to monitor the changes in species
abundances and line intensities as a result of changes in the source
code and atomic and molecular data. The Cloudy test suite is extensive and 
includes simulations
at very low and very high temperatures, CMB to 10^10 K. 
The chemistry must work for all conditions.
Often we encounter problems with the extrapolation of simple
fits to the temperature range we need for chemical rate coefficients. 
To overcome this problem, we apply a 
temperature cap, T_cap, for β > 0 (See appendix). For T > T_cap, 
the rate coefficients
retain the same values as at T_cap. Though ad-hoc, we choose 
T_cap=2500 K <cit.>. 
The chemical network is very sensitive to small 
changes in rate coefficients. Hence, for any chemical update to pass through all the 
tsuite models simultaneously is hard and time-consuming. 
We critically examine each update if it results in a relative change of greater than  ± 30% in column density or line intensity. 


In this work, some updates produce a relative change of greater than
± 90% in column density. 
 
We find that the two reactions, H_3O^+ + SiO → SiOH^+ + H2O and
H_3+ SiO → SiOH^+ + H_2, produced significant  changes 
in SiOH^+ column densities with the updated UDfA2012 rate coefficients. 
The UDfA2012 rate coefficients for the reactions, 
HSiO_2^+ + e → SiO_2 + H and 
OH + SiO → SiO_2 + H result in convergence problem for high density 
(> 10^17 cm^-3) and low radiation model. Hence, we retain the UDfA2006 
rate coefficients for these two reactions.   

We calculate the line luminosities/intensities from 
energy levels and
radiative and collisional rate coefficients in the Leiden Atomic and Molecular Database
(LAMDA) format <cit.>. 
TiO energy levels and radiative rate coefficients are from the Cologne Database for
Molecular Spectroscopy (CDMS) (https://cdms.astro.uni-koeln.de/). The ground state 
of TiO is 
triplet-delta with lambda splitting. We consider 230 energy levels, and the
corresponding 223 radiative transitions. In addition, we consider 444 
collisional transitions with ortho and para H_2. The collisional transitions of CO 
with ortho and para H_2 are modified from <cit.> as a proxy 
for ortho and para H_2 collisions with TiO. In cases for J_low=3 there are four possible transitions, so the CO rates were divided by 4. For J_low=2 and J_low=1, CO rates were divided by 3 and 2, respectively. Otherwise, each CO rate was divided by 5. The temperature range for collisional 
rates vary from 2 to 3000 K. 
These were compiled and converted to LAMDA format. This will be part of upcoming 2024 release of Cloudy.
In the future, we will replace  
these collisional rates with actual data when they become available. We will also add collisional 
rate coefficients for other colliders, H and He, when data becomes available. 



 §.§ Modelling TiO from VY Canis Majoris

We present a model for the circumstellar 
envelope of the red supergiant VY CMa to validate our Ti-chemical network. 
All  calculations  
are done using the development version of Cloudy <cit.>. VY CMa has a rich inventory 
of various molecules
<cit.>. However, here we concentrate mainly on TiO. 

We consider a spherical model with the ionizing source, star, at the centre. Cloudy requires basic input parameters 
such as the radiation field, density at the inner radius which is illuminated by the starlight, chemical abundances, etc. Observations reveal
the effective temperature of the 
central star is 3650 K <cit.> with a luminosity of 
(3±0.5) × 10^5 L⊙ <cit.>. We use this observed value 
to set the SED of the radiation field.

VY CMa is a large star with 
R⋆ = 1420 ± 120 R⊙ <cit.>. We consider the inner radius of the circumstellar envelope to be at R⋆.
TiO is expected to form nearer the star, where the temperature is high, above the
sublimation temperature of dust. <cit.> have observed that NaCl forms 
at R < 250 R⋆ from the central star. <cit.> also  
noticed that 
the main component of the TiO emission covers the same velocity range as the NaCl profile, but 
is confined within R = 28 × R⋆. The Dissociation energy of TiO is much higher than NaCl, so in the gas phase it is expected to peak closer to the central star than NaCl. 
Since our main focus is TiO, we further confine our model 
to R = 28 × R⋆ from the central star. 

For an oxygen-rich environment, nearly all carbon is locked up in CO. Thus, formation of silicate 
dusts is more favourable than the formation of graphite dust, when physical conditions permit. 
The sublimation temperature of 
silicate dusts is ≈ 1400 K. Furthermore, <cit.> estimated 
T_rot = 1010 ± 870 K. So, we further constrain our model to have a 
temperature above > 1400 K.

The density, in general, depends on the distance from the central source of ionization. 
We consider the density to be proportional to r^-2, consistent with a v∝ r wind <cit.>. In addition, it is known that the 
density of the circumstellar envelope can vary over a wide range. Hence, 
in our grid models, 
we vary the density n_H(r_in) (the density at the inner radius (r_in) of the circumstellar envelope) from 10^9 to 10^12.5 cm^-3
in steps of 0.25 dex. We consider solar 
abundances of elements as reported in <cit.>, and we do not 
include dust in our model. Finally, we include a background cosmic ray 
ionization rate (important for the ion balance in molecular regions) of 2 × 10^-16 s^-1  s^-1 in our grid models <cit.>.

Figure <ref> shows our model's predicted TiO column density
as a function of hydrogen number density, n_H(r_in). The solid horizontal lines represent
the observed column density range <cit.>. From the plot, we conclude
that the observed TiO column density comes from a region with n_H(r_in)=
10^12.06 to 10^12.08 cm^-3. 








Next, we consider a single model with n_H(r_in) = 10^12.07 cm^-3, 
holding other 
parameters constant. Figure <ref> shows the variation of TiO, TiO^+, 
TiO_2, Ti, Ti^+ as a function of the depth of the cloud from the illuminated surface. The 
temperature varies from 3431 K to 1400 K across the region considered.
We find that Ti is mostly ionic, and the dominant oxide form of Ti is TiO. 

Our model's predicted neutral hydrogen column density is ≈ 8×10^25 cm^-2. Although our model does not
include dust, it predicts a high H_2 column density ≈ 10^18 cm^-2. 
This is due to the presence of a 
significant H^- abundance which helps to form H_2 in dust-free 
environments <cit.>. 
However, this single model predicts a 
TiO_2 column density, 7.16×10^12 cm^-2, much below the observed value. 

Figure <ref> shows the line intensity of TiO lines for this single model assuming the 
distance of VY CMa is 1.20 kpc <cit.>.

Earlier, 
<cit.> have showed that TiO_2 forms at a much lower temperature than 
TiO. So, we recomputed the same single model but removing the 1400 K temperature limit. The 
predicted TiO and TiO_2 column densities are 7.5× 10^15 and 8.7×10^15 
cm^-2, respectively. The observed TiO_2 column density is (5.65 ±1.33) × 
10^15 cm^-2 <cit.>. Both of these molecules are confined within R = 28 × R⋆.
The corresponding ionization structure is shown in Figure <ref>. 
We notice that as the temperature decreases below 1400 K, more TiO_2 is formed. We conclude that
Ti is locked in TiO above 1400 K and in TiO_2 at a lower temperature. Since silicate dust
can form below 1400K, we conclude that TiO_2 remains in the gas phase outside the dust-formation zone, and might play only a minor role in the dust-condensation process around VY CMa. The same conclusion was made by <cit.>. 

















 §.§ Effects of input parameters

There is a debate regarding the luminosity and radius of the star VY CMa. <cit.> have estimated the 
luminosity and radius to be 6×10^4 L⊙ and 600 R⊙, respectively. However, <cit.> have
estimated these to be 5×10^5 L⊙ and R⋆ = 1420 ± 120 R⊙, respectively. Here, we show the effects of
various input parameters on the predicted hydrogen number density, column densities of TiO and 
TiO_2 for the above-mentioned model of VY CMa. 

Firstly, we consider 
a value, 1.0 × 10^5 L⊙, instead of 2.5× 10^5 L⊙ considered in section 2.2.  
We run a similar grid of models varying n_H(r_in) from 10^9 to 10^12.5 cm^-3
in steps of 0.25 dex, keeping all the other parameters the same. For this case, the observed TiO column density comes from a 
region with n_H(r_in) =
10^11.79 to 10^11.80 cm^-3. This is lower by ≈ 0.3 dex from the value derived in section 2.2. 

Next, we consider a lower value of R⋆ =770 R⊙ compared to the earlier value of R⋆=1540 R⊙ for VY CMa, 
and run a similar grid model keeping other parameters fixed. For this case, the observed TiO column density comes from a 
region with n_H(r_in) = 10^12.66 to 10^12.67 cm^-3. This is higher by ≈ 0.6 dex from the value derived in section 2.2.

Further, we change the density to be proportional to r^-3, and run a similar grid model keeping other parameters fixed as 
described in section 2.2. We find that with the new density law,
the observed TiO column density comes from a 
region with n_H(r_in) = 
10^12.28 to 10^12.31 cm^-3. 
This is higher by ≈ 0.2 dex from the value derived in section 2.2. 

We notice that the value of n_H(r_in) is more dependent on the inner radius r_in of the envelope. However, for all these cases, as the temperature decreases below 1400 K, more TiO_2 is formed. Similarly, both TiO and TiO_2 are confined within R = 28 × R⋆.





 §.§ Model limitations

Like every astrophysical model, our model also has some limitations. 
The main limitations are listed below.

Firstly, most of our Ti-related gas-phase chemical reaction rate coefficients involving Ti are copied from analogous Si-related reactions.
Secondly, we have considered only ortho and para–H_2 collisional excitation and
deexcitation. 
Thirdly, we have assumed a spherical geometry. However, observations suggest that 
the geometry is not perfectly spherical.
Despite these limitations, our model predicts the observed column density of TiO and TiO_2.



§ DISCUSSIONS AND CONCLUSIONS

In this work, we updated gas-phase chemical reaction rates and molecular lines arising
from 
TiO in the spectral synthesis code cloudy. We have added 229 reactions in the chemical 
network. 
In addition, we consider 230 energy levels, corresponding radiative transitions, and 444 
collisional transitions with ortho and para H_2. These molecular lines can be observed using ALMA and JWST. 
These updates will be part of the upcoming 2024 release of Cloudy.

Our main conclusions from this work are:



  * In the gas-phase, Ti is mainly in TiO for temperatures above 1400 K. Whereas, TiO_2 dominates at a lower temperature.


  * Our model predicts the observed column densities of TiO and TiO_2 around the oxygen rich red super giant VY CMa. Both of these molecules are confined within R = 28 × R⋆. However, TiO forms closer to the central star when temperatures are above 1400K.


  * Few ISM-appropriate Ti molecular reaction networks exist due to its low gas-phase abundance in the ISM.
Most of our Ti-related gas-phase chemical reaction rate-coefficients  are copied from analogous Si-related reactions. Our model does predict observed TiO and TiO_2 column density around VY CMa. However, we look forward to actual rate coefficients for these reactions. 



In future, we hope to incorporate 
another important precursor of inorganic dust, AlO, into Cloudy's chemical network.
 


§ ACKNOWLEDGMENTS

We thank Peter van-hoof, Kyle Walker, and Krishal Patel for their valuable help. GS acknowledges support from DST/WOS-A/PM-2/2021. 



 





§ APPENDIX

Here we list the rate coefficients used for various Ti-related reactions in
Table <ref>. Ti-related reactions are mainly copied from the analogous 
Si-related reactions from UDfA2006 and UDfA2012. 
For example, in the reference column Si5209 refers a copied reaction from Si 
reaction of UDfA2012 with the reaction number 5209. Ref 1, 2, and 3 refer to <cit.>, <cit.>, and <cit.>, respectively. The rate coefficient for the reaction, Ti^+ + H^- → Ti + H, is taken from <cit.>.

The format for the two-body rate coefficients are given in an Arrhenius-type equation, 

    k=α(T/300)^β exp(-γ /T).

where T is the temperature of the gas.
Whereas the photoreaction rate coefficients are given by

    k= α exp(-γ A_ V)   .

where A_ V is the extinction at visible wavelengths. Details can be found 
in <cit.>. For the two-body reactions and photoreactions, γ 
depends on the activation energy of the reaction and the increased dust extinction at 
ultraviolet wavelengths, respectively. 
It is to be noted, we do not use UDfA cosmic ray ionization rates. Our cosmic-ray 
ionization rates are normalised
to a total rate for electron production
from cosmic-ray ionization. We adopt a mean cosmic-ray ionization
rate of 2×10^-16 s^-1 for atomic hydrogen as the default
background value <cit.>. 
<cit.> showed that this rate is dependent of the presence of PAHs as well. 
For detail, see <cit.>. 



 |c|c|c|c|c|
 List of Ti related chemical reactions 


 5| c |Begin of Table

 Reactions     α     β     γ     Ref 

 

 5|c|Continuation of Table <ref>

 Reactions     α     β     γ     Ref 

 

5|c|Continuation of Table <ref>

 Reactions     α     β     γ     Ref 

 

 

 5| c |End of Table

 

C + TiH → TiC + H   6.59e-11   0   0    Si5209

C + TiH_2 → HCTi + H   1.e-10   0   0    Si5207

N + TiH → TiN + H   1.66e-10   -0.09   0    Si5513

N + TiH_2 → HNTi + H   8.0e-11   0.17   0    Si5511

N + TiC → Ti + CN   5.0e-11   0   0    Si5509

N  + TiC → TiN + C   5.e-11   0   0    Si5510

CH_2 + Ti → HCTi + H   1.e-10   0   0    Si5241

O + TiH → TiO + H   1.e-10   0   0    Si5665

O + TiH_2 → TiO + H_2   8.e-11   0   0    Si5661

O + TiH_2 → TiO + H + H   1.2e-10   0   0    Si5662

O + TiC → TiO + C   5.e-11   0   0    Si5660

O + TiC → Ti + CO   5.e-11   0   0    Si5659

O + HCTi → TiO + CH   2.e-11   0   0    Si5626

O + TiN → NO + Ti   5.e-11   0   0    Si5666

O + TiN → TiO + N   5.75e-11   0.1   200    Si5667

O + TiNC → TiN + CO   1.e-11   0   0    Si5668

O + TiC_2 → TiC + CO   4.e-11   0   0    Si5656

OH + Ti → TiO + H   1.e-10   0   0    Si5698 

OH + TiO → TiO_2 + H   2.e-12   0   0    Si5699 

C_2H_2 + Ti → TiC_2 + H_2   1.30e-10   -0.71   29     Si5100

Ti + O_2 → TiO + O   1.72e-10   -0.53   17    Si5710

H^+ + TiH → Ti^+ + H_2   1.70e-9   0   0     Si2586

H + TiH^+ → Ti^+ + H_2   1.9e-9   0   0    Si3069

H^+ + TiH_2 → TiH^+ + H_2   1.5e-9   0   0     Si2583

H^+ + HCTi → TiC^+ + H_2   1.5e-9   0   0     Si2568

H^+ + HNTi → TiN^+ + H_2   1.5e-9   0   0     Si2571

H + TiS^+ → HS + Ti^+   1.90e-9   0   0     Si3070

H_2 + TiC^+ → HCTi^+ + H   1.5e-9   0   0     Si2693

H_2 + TiO^+ → TiOH^+ + H   3.2e-10   0   0     Si2696

H_3^+ + Ti → TiH^+ + H_2   3.7e-9   0   0     Si2964

H_3^+ + TiH → TiH_2^+ + H_2   2.e-9   0   0     Si2977

H_3^+ + TiC → HCTi^+ + H_2   3.e-9   0   0     Si2971

H_3^+ + TiN → HNTi^+ + H_2   2.e-9   0   0     Si2978

H_3^+ + TiO → TiOH^+ + H_2   2.e-9   0   0     S2981

H_3^+ + TiS → HTiS^+ + H_2   8.7e-9   -0.5   0     Si2982

He^+ + TiH → Ti^+ + He + H   1.80e-9   0   0     Si3549

He^+ + TiH2 → Ti^+ + He + H_2   1.e-9   0   0     Si3543

He^+ + TiH2 → TiH^+ + He + H   1.e-9   0   0     Si3544

He^+ + TiC → TiC^+ + He   2.e-9   -0.5   0     Si3538

He^+ + TiC → Ti^+ + C + He   2.e-9   -0.5   0     Si3537

He^+ + HCTi → Ti^+ + CH + He   1.e-9   0   0     Si3478

He^+ + HCTi → TiC^+ + He + H   1.e-9   0   0     Si3479

He^+ + TiN → Ti^+ + N + He   2.e-9   -0.5   0     Si3550

He^+ + HNTi → TiN^+ + He + H   2.e-9   0   0     Si3488

He^+ + TiO → Ti^+ + O + He   1.e-9   0   0    1 
             
He^+ + TiO → Ti + O^+ + He   8.6e-10   -0.5   0     Si3554

He^+ + TiC_2 → Ti^+ + C_2 + He   2.e-9   -0.5   0     Si3528

He^+ + TiNC → Ti + CN^+ + He   2.e-9   -0.5   0     Si3551

He^+ + TiO_2 → O_2 + Ti^+ + He   2.e-9   0   0     Si3552

He^+ + TiS → S + Ti^+ + He   3.8e-9   -0.5   0     Si3556

He^+ + TiS → S^+ + Ti + He   3.8e-9   -0.5   0     Si13555

C^+ + TiH → TiC^+ + H   1.1e-9   0   0     Si1666

C + TiH^+ → TiC^+ + H   2.e-10   0   0     Si2146

C^+ + TiH_2 → TiC^+ + H2   1.e-9   0   0     Si1661

C^+ + TiH_2 → HCTi^+ + H   1.e-9   0   0     Si1660

C + TiH_2^+ → HCTi^+ + H   1.1e-9   0   0     Si2147

C^+ + TiC → Ti^+ + C_2   2.50e-9   -0.5   0     Si1655

C^+ + HCTi → TiC_2^+ + H   2.e-9   0   0     Si1631

C^+ + TiN → TiC^+ + N   1.e-9   -0.5   0     Si1667
 
C^+ + HNTi → TiNC^+ + H   2.e-9   0   0     Si1635

C^+ + TiO → Ti^+ + CO   1.e-9   0   0     1

C + TiO^+ → Ti^+ + CO   1.e-9   0   0     Si2153

C^+ + TiS → TiC^+ + S   2.3e-9   -0.5   0     Si1669

CH + Ti^+ → TiC^+ + H   6.3e-10   -0.5   0     Si2487

CH + TiH^+ → Ti + CH_2^+   6.e-10   -0.5   0     Si2488

CH + TiO^+ → HCO^+ + Ti   5.9e-10   -0.5   0     Si2489

N + TiH^+ → TiN^+ + H   2.e-10   0   0     Si3692

N + TiH_2^+ → HNTi^+ + H   1.e-10   0   0     Si3693

N + TiC^+ → Ti^+ + CN   7.7e-10   0   0     Si3690

N + TiO^+ → NO^+ + Ti   9.e-11   0   0     Si3696

N + TiO^+ → NO + Ti^+   2.1e-10   0   0     Si3697

CH_2 + Ti^+ → HCTi^+ + H   8.7e-10   0   0     Si2239

CH_2 + TiO^+ → H_2CO + Ti^+   7.88e-10   0   0     Si1589

NH + Ti^+ → TiN^+ + H   1.e-9   -0.5   0     Si3846

CH_3^+ + Ti → HCTi^+ + H_2   2.e-10   0   0     Si2318

CH_3 + Ti^+ → HCTi^+ + H_2   1.e-9   0   0     Si2323

O + TiH^+ → TiO^+ + H   4.e-10   0   0     Si3977

O + TiH_2+ → TiOH^+ + H   6.3e-10   0   0     Si3978

O + TiC^+ → TiO^+,C   6.e-10   0   0     Si3974

O + TiO^+ → O_2 + Ti^+   2.e-10   0   0     Si3985

O + TiNC^+ → TiN^+ + CO   1.e-10   0   0     Si3983

OH + Ti^+ → TiO^+ + H   6.3e-10   -0.5   0     Si4030

OH^+ + Ti → TiH^+ + O   1.9e-9   0   0     Si4012

OH^+ + TiH → TiH_2^+ + O   1.e-9   0   0     Si4014

OH^+ + TiC → HCTi^+ + O   4.5e-9   -0.5   0     Si4013

OH^+ + TiO → TiOH^+ + O   9.e-10   -0.5   0     Si4015

NH_3 + TiH^+ → Ti + NH_4^+   1.e-9   -0.5   0     Si3824

NH_3 + TiOH^+ → NH_4^+ + TiO   2.5e-9   -0.5   0     Si3825

NH_3 + HTiS^+ → NH_4^+ + TiS   9.7e-10   -0.5   0     Si3812

H_2O + Ti^+ → TiOH^+ + H   2.30e-10   -0.5   0     Si2790

H_2O + TiH^+ → Ti + H_3O^+   8.e-10   -0.5   0     Si2792

H_2O + HNTi^+ → TiN + H_3O^+   2.e-9   -0.5   0     Si2772

H_2O + HTiS^+ → TiOH^+ + H_2S   1.1e-9   -0.5   0     Si2777

H_3O^+ + Ti → TiH^+ + H_2O   1.8e-9   0   0     Si3039

H_3O^+ + TiH → TiH_2^+ + H_2O   9.7e-10   0   0     Si3045

H_3O^+ + TiC → HCTi^+ + H_2O   3.e-9   -0.5   0     Si3042

H_3O^+ + TiO → TiOH^+ + H_2O   2.0e-9   -0.5   0     Si3047

HF + Ti^+ → TiF^+ + H   5.7e-9   -0.5   0     Si3236

C_2 + TiO^+ → TiC^+ + CO   7.6e-10   0   0     Si1686

C_2H + Ti^+ → TiC_2^+ + H   1.e-9   0   0     Si1972

C_2H_2^+ + Ti → TiC_2^+ + H_2   2.e-10   0   0     Si1750

HCN + Ti^+ → TiNC^+ + H   1.4e-12   -0.5   0     Si3106

HNC + Ti^+ → TiNC^+,H   1.e-9   -0.5   0     Si3267

HCN + HTiS^+ → HCNH^+ + TiS   6.1e-10   -0.5   0     Si3101

CO + HNTi^+ → TiN + HCO^+   2.e-9   0   0     Si2508

CO + TiO^+ → CO_2 + Ti^+   7.9e-10   0   0     Si2513

HCNH^+ + Ti → TiNC^+ + H_2   5.e-10   0   0     Si3117

H_2NC^+ + Ti → TiNC^+ + H_2   5.e-10   0   0     Si2723

Ti + HCO^+ → TiH^+ + CO   3.3e-10   0   0     S3210

Ti^+ + C_2H_5OH → TiOH^+ + C_2H_5   2.2e-9   -0.5   0     Si4080

Ti^+ + HC_4H → C_4H^+ + TiH   1.6e-9   0   0     SiRATE12

Ti^+ + NCCN → TiNC^+ + CN   1.5e-10   0   0     Si4091

Ti^+ + OCS → TiS^+ + CO   9.e-10   0   0     Si409
 
HCO^+ + TiH → TiH_2^+ + CO   8.7e-10   0   0     Si3223

HCO^+ + TiC → HCTi^+ + CO   2.e-9   -0.5   0     Si3218

HCO^+ + TiO → TiOH^+ + CO   7.9e-10   -0.5   0     Si3226

HCO^+ + TiS → HTiS^+ + CO   3.3e-9   -0.5   0     Si3227

TiH + S^+ → TiS^+ + H   4.2e-10   0   0     Si4106

TiH_2^+ + O_2 → TiOH^+ + OH   2.4e-11   0   0     Si4103

TiH_2^+ + S → HTiS^+ + H   1.1e-9   0   0     Si4104

O_2 + TiS^+ → SO^+ + TiO   6.23e-11   0   0     Si3896

O_2 + TiS^+ → TiO^+ + SO   2.67e-11   0   0     Si3897

S + TiO^+ → SO + Ti^+   1.e-9   0   0     Si4063

H_2S + HTiS^+ → H_3S^+ + TiS   2.9e-10   -0.5   0     Si2801

H^+ + Ti → Ti^+ + H   9.9e-10   0   0     Si418 

H^+ + TiH → TiH^+ + H   1.7e-9   0   0     Si431

H^+ + TiH_2 → TiH_2^+ + H   1.5e-9   0   0     Si428

H^+ + TiC → TiC^+ + H   3.e-9   -0.5   0     Si425

H^+ + HCTi → HCTi^+ + H   1.5e-9   0   0     Si392

H^+ + TiN → TiN^+ + H   3.e-9   -0.5   0     Si432

H^+ + HNTi → HNTi^+ + H   1.5e-9   0   0     Si394

H^+ + TiO → TiO^+ + H   1.e-9   0   0     1

H^+ + TiC_2 → TiC_2^+ + H   3.e-9   -0.5   0     Si419

H^+ + TiNC → TiNC^+ + H   3.e-9   -0.5   0     Si433

H^+ + TiS → TiS^+ + H   1.50e-9   -0.5   0     Si435

He^+ + Ti → Ti^+ + He   3.3e-9   0   0     Si528

C^+ + Ti → Ti^+ + C   2.1-9   0   0     Si194

C^+ + TiH_2 → TiH_2^+ + C   1.e-9   0   0     Si201

C^+ + TiC → TiC^+  + C   2.50e-9   -0.5   0     Si198

C^+ + TiN → TiN^+ + C   1.e-9   -0.5   0     Si203

C^+ + TiC_2 → TiC_2^+ + C   2.e-9   -0.5   0     Si195

C^+ + TiS → TiS^+ + C   2.3e-9   -0.5   0     Si204

CH^+ + Ti → Ti^+ + CH   2.e-10   0   0     Si257

NH_3^+ + Ti → Ti^+ + NH_3   1.9e-9   0   0     Si597

H_2O^+ + Ti → Ti^+ + H_2O   3.e-9   0   0     Si477

Na + Ti^+ → Ti + Na^+   2.7e-9   0   0     Si645


Mg + TiO^+  → TiO + Mg^+   1.e-9   0   0     Si541

Ti + NO^+ → NO + Ti^+   1.6e-9   0   0     Si720

Ti + P^+ → P + Ti^+   1.e-9   0   0    Si722

Ti + O_2^+ → O_2 + Ti^+   1.6e-9   0   0     Si721

Ti + S^+ → S + Ti^+   1.6e-9   0   0     Si723

Ti + HS^+ → HS + Ti^+   1.4e-9   0   0     Si719

Ti + H_2S^+ → H_2S + Ti^+   1.60e-9   0   0     Si718

Ti + CS^+ → CS + Ti^+   1.50e-10   0   0     Si716


HCO + TiO^+ → TiO + HCO^+   6.6e-10   -0.5   0     Si504

TiH + S^+ → S + TiH^+   4.2e-10   0   0    Si724

NO + TiO^+ → TiO + NO^+   7.2e-10   0   0     Si631

S^+ + TiC → TiC^+ + S   3.7e-9   -0.5   0     Si705

S^+ + TiS → TiS^+ + S   3.2e-9   -0.5   0     Si706

TiO^+ + Fe → Fe^+ + TiO   1.e-9   0   0     Si725


C^- + Ti^+ → Ti + C   7.51e-8   -0.5   0     Si4138

Ti^+ + S^- → S + Ti   7.51e-8   -0.5   0     Si5084

TiH^+ + e^- → Ti + H   2.e-7   -0.5   0     Si1498

TiH_2^+ + e^- → Ti + H + H   2.e-7   -0.5   0     Si1500

TiH_2^+ + e^- → Ti + H_2   1.5e-7   -0.5   0     Si1499

TiH_2^+ + e^- → TiH + H   1.5e-7   -0.5   0     Si1501

TiC^+ + e^- → Ti + C   2.e-7   -0.5   0     Si1471

HCTi^+ + e^- → Ti + CH   1.5e-7   -0.5   0     Si1362

HCTi^+ + e^- → TiC + H   1.5e-7   -0.5   0     Si1363

TiN^+ + e^- → Ti + N   2.0e-7   -0.5   0     Si1508 
 
HNTi^+ + e^- → Ti + NH   1.5e-7   -0.5   0     Si1376

HNTi^+ + e^- → TiN + H   1.5e-7   -0.5   0     Si1377

TiO^+ + e^- → Ti + O   1.e-7   0   0     1
       
TiOH^+ + e^- → Ti + OH   1.5e-7   -0.5   0     Si1515

TiOH^+ + e^- → TiO + H   1.5e-7   -0.5   0     Si1516

TiF^+ + e^- → Ti + F   2.0e-7   -0.5   0     Si1497  
 
TiC_2^+ + e^- → Ti + C_2   1.5e-7   -0.5   0     Si1472

TiC_2+ + e^- → TiC + C   1.5e-7   -0.5   0     Si1473

TiNC^+ + e^- → Ti + CN   3.0e-7   -0.5   0     Si1509

TiS^+ + e^- → S + Ti   2.0e-7   -0.5   0     Si1517

HTiS^+ + e^- → HS + Ti   1.5e-7   -0.5   0    Si1400

HTiS^+ + e^- → TiS + H   1.5e-7   -0.5   0     Si1401

Ti^+ + e^- → Ti + PHOTON   4.26e-12   -0.62   0     Si6178

H + Ti^+ → TiH^+ + PHOTON   1.17e-17   -0.14   0     Si6127

H_2 + Ti^+ → TiH_2^+ + PHOTON   3.e-18   0   0     Si6114

O + Ti^+ → TiO^+ + PHOTON   9.22e-19   -0.08   -21.2     Si6136

O + Ti → TiO + PHOTON   5.52e-18   0.31        Si6137

Ti + PHOTON → Ti^+ + e-   3.1e-9   0   2.3     Si6017

TiH + PHOTON → Ti + H   2.8e-9   0   1.6     Si6038

TiH^+ + PHOTON → Ti^+ + H   2.7e-9   0   1.2     Si6029

TiC + PHOTON → Ti + C   1.0e-10   0   2.3     Si6025

TiO + PHOTON → Ti + O   1.6e-9   0   2.3     Si6044

TiO + PHOTON → TiO^+ + e^-   2.4e-10   0   2     Si6045

TiO^+ + PHOTON → Ti^+ + O   1.0e-10   0   2     Si6042

TiS + PHOTON → S + Ti   1.0e-10   0   2.3     Si6046

Ti + CRPHOT → Ti^+ + e^-   2115   0   0     Si4424

TiH + CRPHOT → Ti + H   250   0   0    Si980

TiC + CRPHOT → Ti + C   250   0   0     Si974

TiO + CRPHOT → Ti + O   250   0   0     Si984

TiS + CRPHOT → S + Ti   250   0   0     Si985

Ti + O → TiO^+ + e^-   1.0e-11   0   1973     chu80,opp77

TiO^+ + e^- → TiO + PHOTON   5.0e-10   0   0     chu80

Ti^+ + O_2 → TiO^+ + O   1.0e-9   0   0     1

Ti^+ + H_2O → TiO^+ + H_2   1.0e-9   0   0     1

TiO^+ + Ca → TiO + Ca^+   1.0e-9   0   0     chu80

TiO^+ + Na → TiO + Na^+   1.0e-9   0   0     chu80

TiO^+ + Al → TiO + Al^+   1.0e-9   0   0     chu80

TiO^+ + K → TiO + K^+   1.0e-9   0   0     chu80

TiO + C → Ti + CO   1.0e-10   0   0     chu80

Ti^+ + CH_3OH → TiOH^+ + CH_3   1.65e-9   -0.5   0     Si4088 

TiH + H → Ti + H_2   8.3e-11   0   0     2

TiH + O → Ti + OH   1.66e-10   0   0     2

TiH + CH_3 → Ti + CH_4   1.0e-10   0   0     2

TiC + H → Ti + CH   1.0e-10   0   20109    2

Ti + CO → TiC + O   8.0e-10   0   68842     2

Ti + CN → TiC + N   5.0e-10   0   29543     2
     
Ti + NO → TiN + O   5.0e-11   0   19706.     2
 
TiN + H → Ti + NH   1.e-10   0   15655     2

Ti + NO → TiO + N   9.e-11   -0.96   0      2 

TiO_2 + O → TiO + O_2   1.e-10   0   11877     2

TiO_2 + CO → TiO + CO_2   1.e-10   0   9160     2

TiO + O → Ti + O_2   5.42e-13   0   20794     2

TiO + N → Ti + NO   4.76e-12   0   4772     2

TiO + N_2 → Ti + N_2O   5.9e-12   0   62193     2

Ti + CO_2 → TiO + CO   7.01e-11   0   1790     2 
   
TiO + NO → TiO_2 + N   7.8e-12   0   0     3
  
TiO + O_2 → TiO_2 + O   1.5e-12   0   0     3

TiO + N_2O → TiO_2 + N_2     4.e-13   0   0     3









 Paper Title: Reverberation mapping of high-mass and high-redshift quasars using gravitational time delays.























































Miriam Golubchik
golubmir@post.bgu.ac.il


0000-0001-9411-3484]Miriam Golubchik
Department of Physics,
Ben-Gurion University of the Negev, P.O. Box 653,
Be'er-Sheva 84105, Israel
0000-0003-3780-6801]Charles L. Steinhardt
Cosmic Dawn Center (DAWN), Denmark
Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, Copenhagen Ø 2100, Denmark
0000-0002-0350-4488]Adi Zitrin
Department of Physics,
Ben-Gurion University of the Negev, P.O. Box 653,
Be'er-Sheva 84105, Israel
0000-0002-7876-4321]Ashish K. Meena
Department of Physics,
Ben-Gurion University of the Negev, P.O. Box 653,
Be'er-Sheva 84105, Israel
0000-0001-6278-032X]Lukas J. Furtak
Department of Physics,
Ben-Gurion University of the Negev, P.O. Box 653,
Be'er-Sheva 84105, Israel
0000-0002-4830-7787]Doron Chelouche
Department of Physics, Faculty of Natural Sciences, University of Haifa, Haifa 3498838, Israel
Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Haifa 3498838, Israel
0000-0002-9925-534X]Shai Kaspi
School of Physics & Astronomy and the Wise Observatory, The Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Israel





















§ ABSTRACT 
Mass estimates of black holes (BHs) in the centers of Active Galactic Nuclei (AGN) often rely on the radius-luminosity relation. However, this relation, usually probed by reverberation mapping (RM), is poorly constrained in the high-luminosity and high-redshift ends due to the very long expected lag times. Multiply imaged AGN may thus offer a unique opportunity to explore the radius-luminosity relation at these ends. In addition to comprising several magnified images which enable a more efficient light-curve sampling, the time delay between multiple images of strongly lensed quasars can also aid in making such RM measurements feasible on reasonable timescales: If the time delay is, for example, of the order of the expected time lag, changes in the emission lines in the leading image can be observed around the same time as the changes in the continuum in the trailing image. In this work we probe the typical time-delay distribution in galaxy-cluster lenses and estimate the number of both high-mass (∼10^9-10^10 M_⊙), and high-redshift (z≳4-12) quasars that are expected to be strongly lensed by clusters. We find that less than one, very massive (and luminous, L_UV>10^46.5 erg s^-1) multiply imaged quasar is expected across the sky down to 30 AB mag. Nonetheless, up to several dozen thousand M_BH∼10^6-10^8 M_⊙ broad-line AGN at z>4 should be multiply imaged by galaxy clusters and detectable with JWST, hundreds with Euclid and several thousands with the Roman Space Telescope, across the whole sky. These could supply an important calibration for the BH mass scaling in the early Universe.
















§ INTRODUCTION




Super massive black holes (SMBHs) are found in the centers of most galaxies. The masses of those SMBHs are of much interest as their evolution is not well understood, and since they are believed to play a major role in their host galaxy evolution. In fact, tight relations seem to hold between the mass of an SMBH and the properties of its host galaxy such as the luminosity or the velocity dispersion of the bulge <cit.>. Measuring SMBH masses as a function of redshift can also shed light onto black hole (BH) formation and the physics of accretion processes <cit.>, and their mass evolution through cosmological time might help constrain different cosmological models <cit.>.

SMBHs are believed to be the central engines of Active Galactic Nuclei (AGNs). The rapid accretion onto the BH results in a high luminosity across a broad range of wavelengths and unique spectral features from which one can estimate the virial mass of the BH. Using a single epoch spectrum, one can resolve the velocity of the material in the broad line region (BLR) surrounding the BH and, through further assumptions of virial motion, obtain an estimate for the BH mass <cit.>. However, this technique strongly relies on the ability to determine the distance R from the central engine to the BLR (often referred to as the size of the BLR) which is usually obtained through empirical scaling relations with the AGN's continuum luminosity L (often referred to as an R-L relation).

R-L relations can be empirically derived using Reverberation Mapping (RM). The RM technique aims to measure the distance to the BLR in AGNs by measuring the time lag between variations in the continuum flux from the accretion disk, and the response of the emission lines in the BLR to those variations, assuming that this response propagates with the speed of light <cit.>.
Almost a 100 AGNs were studied using RM to determine the sizes of their BLRs, resulting in an empirical relation connecting the continuum luminosity to the BLR size for different emission lines <cit.>. Indeed, those R-L relations are widely used for virial mass estimates of SMBHs in the centers of AGN, varying in luminosity and redshift. However, because those relations were mostly obtained using (relatively) low redshift and lower-luminosity AGN, RM studies of high-luminosity and high-redshift quasars are needed in order to extend and calibrate these relations to those extreme ends where the relation is not well established and where the physical picture may be different (for example, the BLR might not be necessarily virialized <cit.>). 

Unfortunately, RM of high-luminosity and high-redshift quasars faces several difficulties. High-luminosity quasars are presumed to have large BLR sizes, resulting in long time lags between continuum and emission-line variations. Similarly, high-redshift quasars and AGN <cit.> will also show longer observed time lags compared to their lower-redshift counterparts, due to cosmological time dilation. As a result, observations for several decades are needed for those RM campaigns <cit.>. Nevertheless, several attempts have been recently made to conduct RM for those types of AGN. In one study, <cit.> observed photometrically 11 quasars at redshifts ∼ 2<z<3.4 and luminosities ∼ 10^46.9<λ L_λ(1350 Å)<10^48.0 erg  s^-1 for almost twenty years while six of them were monitored spectrophotometrically for 13 years. Their study manages to significantly measure time lags of the C iv λλ1548,1551Å doublet in three objects and a C iii] λλ1907,1909Å time lag in one object. Scaling the BLR size with the UV continuum luminosity they conclude that their objects lie on the previously measured relation. <cit.> preformed RM using 11-year photometric and a 7-year spectroscopic data for 849 broad-line quasars. They report 339 significantly measured time lag detections for the different emission lines. Although they claim that on average the R-L relations follow existing ones, they report an intrinsic scatter of ∼ 0.3 dex in the lags for the Hβ line at 4861 Å and the Mg ii] λλ2796,2803Å doublet, whereas the C iv R-L relation has an even more substantial scatter of ∼ 0.5 dex. It is thus becoming important to understand whether these large scatters stem from intrinsic physical processes at high luminosities or due to the fact that there is often a difficulty to significantly measure time lags for such objects given the monitoring times needed. 

In this work, we discuss the possibility to probe the R-L relation for the high-luminosity or high-redshift regimes using multiply imaged quasars. If the strong-lensing (SL) time delays between different multiple images of a lensed quasar are of the same order as the time lags, the expected changes in the emission lines can be observed in a leading multiple image around the same time as the changes in the continuum from a trailing multiple image, so that RM of these types of objects can be done with a reasonable and shorter monitoring time-span than for similar, un-lensed quasars (nominally shorter by one lag-time, but as we also discuss here in practice the gain will be larger, as a lensed quasar will be lensed into several magnified images, supplying a more efficient sampling of the light curve). 

About a handful of quasars and AGN, mostly around cosmic noon, are known to be multiply imaged by galaxy clusters <cit.>, whereas some have also been used for cosmography <cit.>. Notably, a successful RM of a strongly lensed quasar has been recently done by <cit.> where they exploit the time delays between the multiple images of the strongly lensed quasar SDSS J2222+2745 to construct a detailed light curve spending over 6 years in the observed frame with 4.5 years of observations with high cadence. They significantly detect a time lag of τ_cen = 36.5^+2.9_-3.9 rest-frame days for the C iv emission line, whereas the time delays from the two other images they use are of ∼40 and ∼700 days. The quasar SDSS J2222+2745 ranged in continuum luminosity with the average value of log_10[λ L_λ(1350 Å)/erg  s^-1] = 44.66 ± 0.18. In this illustrative case the time lag was short enough so that the gain from image multiplicity was mainly manifested in additional data points on the light curve, thus importantly allowing to shorten the overall monitoring time. Here, we wish to focus on objects at the high luminosity end, namely continuum UV luminosities L_UV > 10^46.5 erg s^-1 (BH masses of log_10M_BH≳9), or at high redshifts (i.e., z≳4), for which we expect time lags of ∼300 - ∼1000 days, depending on redshift. For example, a redshift z∼7, M_BH∼ 10^9 - 10^9.5 M_⊙ quasar <cit.> would be expected to show an observed time lag of about 3 or 6 years respectively, requiring decades of monitoring to measure it without lensing multiplicity. In this paper we suggest that SL time delays of lensed quasars may be a useful route to constraining the R-L relation at those regimes. This is becoming especially timely now that JWST has been uncovering a new abundant population of Type I AGN at high redshifts <cit.>, including the first example of a cluster-lensed multiply imaged one at z∼7 <cit.>. 

The work is constructed as follows. In section <ref> we start by estimating the expected time lags for high-luminosity and high-redshift quasars (<ref>), and then estimate the distribution of time delays between the multiple images of lensed systems in typical lensing clusters (<ref>). In section <ref> we estimate the expected number of lensed quasars and AGN that could be found with e.g., the JWST, Euclid, or with the Roman Space Telescope, in the near future. In section <ref> we demonstrate the suggested idea using simulated light curves, and in section <ref> we conclude the work.
Throughout we assume a standard flat ΛCDM cosmology with H_0=70 km/s Mpc, Ω_Λ=0.7, and Ω_m=0.3. Magnitudes are quoted in the AB system <cit.> and all quoted uncertainties represent 1σ ranges unless otherwise stated. 

In this work we use the terms “quasars" and “AGN" synonymously. In addition, whenever we discuss lensed AGN in this work, we explicitly refer to multiply imaged examples.




§ THE DISTRIBUTIONS OF TIME LAGS AND LENSING TIME DELAYS



 §.§ RM of quasars: the distributions of time lags





As was noted above, RM of quasars relies on the ability to detect temporal variations in the continuum emission from a quasar that lead to, after a time lag which depends on the BLR size, similar variations in the emission lines from the BLR. In this section, we estimate the BLR sizes – and thus time lags – of a representative sample of quasars, namely the SDSS quasar catalog, which is based on Data Release 7 (DR7) <cit.> and is available online. This catalog supplies continuum luminosities for either λ L_λ(1350 Å) or λ L_λ(5100 Å), depending on redshift. 

The distributions of BLR sizes are shown in Fig.<ref>. The expected Hβ time-lag distribution is obtained assuming the R-L relation from <cit.>, for those quasars that have a measurement of the continuum luminosity at 5100 Å.
The expected C iv time-lag distribution is obtained assuming the R-L relation from <cit.> (see their eq. 1), using the continuum UV luminosity at 1350 Å.  Note that the C iv subsample consists of 52,427 objects, which have a higher redshift (median redshift of 1.96) and are typically brighter (median λ L_λ(1350Å)=10^46.1 erg s^-1) than the Hβ subsample (23,093 objects, median redshift of 0.59, median λ L_λ(5100Å)=10^44.6 erg s^-1). As can be seen in Fig. <ref>, the lag-time distributions, plotted as log_10(R) where R is in light days, are approximately Gaussian, with a mean and width of [μ,σ]=[1.86,0.21] and [μ,σ]=[1.78,0.16] for Hβ and C iv, respectively. About 9%, roughly, of the quasars show C iv time lags of 100 days or longer in the rest-frame, and about 26% of the quasars show Hβ time lags of 100 days or longer in the rest-frame. This translates to 500-1300 days for redshifts of z∼4-12, respectively, which we define here as the high redshift regime. 




 §.§ The distribution of strong-lensing time delays


In this subsection we aim to obtain a general distribution of the time delays caused by strong lensing in clusters. Galaxy clusters act as gravitational lenses, causing the bending of light. When the projected mass density is larger than some critical density for lensing, strong lensing will occur comprising multiple images of background sources. Using the thin lens approximation, light that travels from a source to the observer will be delayed by some time t <cit.>:

    t(β⃗,θ⃗) = D_lD_s/D_ls[1/2(θ⃗ - β⃗)^2 - ψ(θ⃗)](1+z_l)/c ,

with D_l, D_s and D_ls being the angular diameter distances to the lens, source and between the lens and the source, respectively; z_l is the redshift of the cluster; θ⃗ and β⃗ represent the position of the image and source, respectively; and ψ(θ⃗) is the lensing potential in that position in the image plane (see <cit.> for more details).
The relevant lensing quantities can be obtained for different clusters using SL lens modeling <cit.>. We use mass models made previously by our group <cit.> for three typical high Einstein mass galaxy clusters: RXC0018, MACS0035 and Abell 2744, where the Einstein mass is the projected mass enclosed within the Einstein radius, and three typical lower Einstein mass clusters: Abell 383, Abell 611 and RX J2129. While all these clusters have typical masses of a few times 10^14 M_⊙ to ∼10^15 M_⊙, the high Einstein mass clusters have a corresponding Einstein radii of ∼20-30 (and thus higher enclosed mass) and the low Einstein mass clusters have Einstein radii of about ∼10-15 typically. Two of the three high mass clusters also happen to show a multiply lensed AGN or quasar-like system with three multiple images each: MACS0035 contains a Type I AGN at z=2.06 <cit.>, and a redshift z=7.05 “red-dot" AGN was found behind Abell 2744 <cit.>.

In order to create the time delay (TD) distribution, we randomly plant sources in the source plane and re-lens them to the image plane using the deflection fields from the model. Typically we use a 100 multiply imaged sources in each case. We calculate the time delays between all available pairs using eq. <ref> and repeat this process for different source redshifts, in the ranges of 1<z<2, 2<z<3, 3<z<5, 5<z<8, 8<z<10 and 8<z<12. The SL time-delay distributions are plotted in Fig.<ref>.

We parameterize the TD distributions and note that they can be well fit (R-square of ≃0.98) by the following parametrization, inspired by a log-normal distribution function:,  P(x)= 1/a·(d-x)·exp{-[ln(d-x)-b]^2/c^2}, with a∼10-16 typically (depending on the redshift bin), b∼0.1, c∼0.8 and d∼4.6 for most redshift bins. Various previous works <cit.> have also looked into the TD statistics. Another useful way to parameterize the TD distribution is in log-log space, where we find a slope of ∼0.8, similar to what was found for, e.g., Abell 1689 <cit.>.






In Fig. <ref> we also compare the lensing TD probability distributions to the time lags expected from the BLR sizes, recalculated to match each redshift bin. We see that there is an overlap in the distributions such that time delays by both types of galaxy clusters cover the entire range of expected time lags, with a higher probability of having the relevant time delays, i.e., of order the time lag, for higher-luminosity and higher-redshift quasars. About 20% - 40% of the TD pairs in all clusters overlap with the range of time lags, depending on the redshift of the lensed source. We see that the peak of the TD distribution caused by lower mass clusters, which should be more abundant, agrees better with the peak of the time lag distributions, but those clusters would have also less multiple images per cluster in proportion to their smaller critical area size (which is roughly proportional to the square root of the mass), such that both populations should be useful.
Also overplotted is a line marking a typical time delay expected by a massive galaxy lens – where we nominally adopt for this calculation a SIS galaxy of σ_vel∼250 km s^-1 placed at redshift 0.5 <cit.>, demonstrating that time delays in the case of lens galaxies are too short for a significant gain in the wait time needed for RM of high-redshift, high-mass quasars, and that galaxy clusters are needed for that purpose.  







§ EXPECTED NUMBERS OF LENSED QUASARS

To examine whether the proposed idea for RM of multiply imaged quasars at the high luminosity and high redshift ends might become realistic in the near future, we estimate the expected number of quasars that would be multiply imaged by galaxy clusters. Given the many uncertainties involved (see for example section <ref>), our goal is merely to obtain an order-of-magnitude estimate. We focus on the observational properties of three observatories: Euclid, the Roman Space Telescope, and JWST. The basic idea is to calculate for each observatory, i.e., a filter set and depth limit, how many quasars in the relevant BH mass and redshift regimes fall within the caustics of lensing clusters so that they would be multiply imaged, and would be bright enough, folding in the magnification, to be detected. We adopt relevant quasar luminosity functions (QLFs) and generally follow known prescriptions for the lensing optical depth, where we assume an explicit cluster mass function – from which an Einstein radius function is calculated – and then calculate the magnification bias and expected fraction (and total number) of multiply imaged quasars in the relevant cluster mass and redshift ranges. 



 §.§ The Quasar Luminosity Function

For the calculations below we adopt two complementary QLFs: For UV selected quasars we use the one from <cit.>, and to account for recent populations of AGN uncovered with JWST for example, we also use the QLF from <cit.>. Both QLFs are given in terms of the UV absolute magnitude at 1450 Å, M_1450. The former also has a term for evolution with redshift, which we assume applies for the range of redshifts spanned here (4<z<12), while the latter is divided into four discrete QLFs, one for each of the following redshift bins - z∼4, z∼5, z∼6, and z∼7, where we assume that it does not change significantly from z∼7-12 – but as seen below, even as such the redshift window from z∼8-12 only negligibly contributes to the result.
Note that, given that we are interested in an order of magnitude and since the results heavily rely on the various assumptions made throughout, for the QLFs we adopt only the best-fit parameters, without propagating their uncertainties (which would affect the uncertainty on our estimates, too). Also, while the QLF from <cit.> is less constrained at the faint end, the QLF from <cit.> is better constrained at that regime.




 §.§ The total expected number of quasars

For each observatory we then do the following. We first divide the relevant redshift range into bins. For each bin, we redshift a quasar's SED and define the selection according to the band in which the quasar would be brightest. 

Then, using the QLF, which itself is divided into magnitude bins for summation, we calculate how many quasars are expected in that redshift bin down to the observational depth. The total number of quasars expected to be observable across the sky is then obtained by summing this number up over the different redshift bins. 




For the Euclid survey we focus on the detection of typical "blue" quasars and adopt the spectrum from <cit.>. Following the Euclid overview paper <cit.>, we primarily use the QLF from <cit.>, but also for comparison employ the <cit.> one. For each redshift bin we choose as the selection band the band in which the rest frame 1450Å falls. If the redshifted wavelength is lower (or higher) than the survey's available bands, we choose the nearest band to it. Since in this work we are particularly interested in lensed high-mass and high-redshift quasars, we perform the calculation for those two regimes specifically. 



We follow the same procedure for the Nancy Grace Roman Space Telescope. Both observatories cover roughly similar wavelength ranges into the near-infrared. While the nominal FOV for Roman is about half of that of Euclid, Roman has a two times in diameter larger mirror and the observational limits are deeper than the ones for Euclid, hence we may expect higher numbers of quasars to be detectable with it. Again our focus remains on the high-redshift and high-luminosity quasars and as above we adopt the typical SED from <cit.>.





Last we turn to perform the same calculation for JWST. In addition to typical "blue" quasars that are detectable with JWST at redshifts higher than ∼ 4, given the new population of "red-dot" AGN at high redshift that has been discovered in JWST observations in increasing numbers <cit.>, we also consider here an additional quasar SED model. The SED of this type of object is heavily reddened and thus is brighter towards the red, with some UV light from star formation or scattered AGN radiation. As the representative SED for red-dot objects we use the one from <cit.>, and we only consider here the QLF from <cit.> which was based on or commensurate with this new population of quasars <cit.>. Given the red SED, for red-dot AGNs we now choose the selection band to be the one in which the rest frame 5500Å falls, shifting all selection to the red-optical end of the spectrum, which indeed, is currently only observable with JWST for these redshifts. We again focus on the high-redshift population for our calculations and estimate the expected numbers of quasars to be detected with JWST for both blue quasars and red quasars. To account for JWSTs extreme depth capabilities we assume a depth of 30 magnitudes, reachable in several hours with JWST. 







 §.§ The lensed fraction

Various works have calculated the effect of lensing on the luminosity function and the fraction of some population of lensed sources of interest such as galaxies, quasars or supernovae, and the prescriptions for doing so are well established. Here we largely follow the outlined procedure in <cit.>, where they obtained an estimate for the fraction of lensed quasars mostly by galaxies. i.e. in a different mass regime that is our interest here, and expand it to massive galaxy clusters. Various components of the formalism are also based on e.g., <cit.>. One other important difference is that in this formalism, given it was usually applied to galaxies, a singular isothermal spheres (SIS) mass density profile is typically assumed, where here we consider the Navarro–Frenk–White (NFW) mass density profile <cit.>, which is a more realistic approximation for the mass distribution of clusters. 


One can define the optical depth for multiple images, which describes how likely it is for a source at z_s to lie within the caustics of a lens (the Einstein radius mapped back to the source plane), and thus be multiply imaged.  The optical depth for multiple images is then given by:

    τ_m=f_sp∫_0^z_sdz_l ∫ dM ϕ(M, z_l) d^2V_c/dΩ dz_lπθ_E(M, z_l, z_s)^2 D_l^2 ,

where ϕ(M,zl) is the lens density function and taken here to simply be the halo mass function (see eq. <ref>), z_l is the redshift of the lens, D_l is the angular diameter distance to it, and f_sp is the fraction of the critical area (πθ_E^2) in which sources would be multiply imaged (for example, for a SIS lens, f_sp=1).
The expression d^2V_c/dΩ dz_d=(1+z_d)^3cdt/dz_d is the differential co-moving volume[Note that in some work the differential co-moving volume is defined such that it contains the D_l^2 factor seen in eq. <ref>.], with dt/dz_d = 1/H(z)(1+z) where H(z) = H_0(Ω_m(1+z)^3+Ω_Λ)^1/2. 

For our calculations we adopt the mass function from <cit.>, namely:

    dn/dM = f(σ)ρ_m/Mdlnσ^-1/dM

with ρ_m = ρ_critΩ_m being the mean matter density, σ is the RMS variance of a spherical top hat containing a mass M and f(σ) is the halo multiplicity function given also in <cit.>. 
In practice, to calculate the mass function (and the power spectrum required therein) we use  and . We refer the reader to <https://cluster-toolkit.readthedocs.io/en/latest/source/massfunction.html> for more information.
The Einstein radius of a NFW lens depends on the mass, concentration and the redshifts of lens and the source. To obtain the Einstein radius for the NFW lens for each combination of parameters we follow the procedure from <cit.> and <cit.>, while we assume the concentration-mass (c-M) relation from <cit.>:

    c(M,z) = A (1.34/1+z)^B (M/8×10^14h^-1M_⊙)^C


with A=3.757±0.054, B=0.288±0.077 and C=-0.058±0.017 (although note, we only use here the best fit values to translate the mass to Einstein radius; we do not propagate the scatter in this relation).

Analytically, for a NFW lens multiple images will occur if the source lies within the radius of the radial critical curve <cit.>. We however adopt here the more empirical assumption of f_sp=0.1, which we obtained from an ensemble of cluster models for known lensing clusters <cit.>: We find that in practice, the angular source-plane area enclosed by the caustics is about ∼10% of the angular area enclosed by the critical curves in the image plane (see also <cit.>).


Next we need to consider that in flux-limited surveys, objects that lie intrinsically below the detection threshold can be detected if they are sufficiently magnified. However, there exists a trade-off with the smaller area probed in the source plane and the actual gain in number count would depend also on the shape of the luminosity function <cit.>. The magnification bias can be written as:

    B=∫_μ_min^+∞dμ p(μ)N(>L_lim/μ)/N(>L_lim),

where p(μ) is the magnification distribution of lensed sources, L_lim is the survey flux limit, and N(>L_lim) is the cumulative luminosity function, i.e., the number of background sources brighter than L_lim.


We use μ_min=2 for the magnification
bias calculation, and, as the magnification probability function we adopt p(μ)∝ 1/(μ)^3 (truncated at μ=100). This distribution seems to approximately describe the magnification distribution of lensed images, also in NFW lenses (see for example Figure 9 in <cit.>). We also verify this by simulating a NFW lens and calculating the magnification distribution of randomly planted sources. 

As is also indicated in <cit.>,
out of all sources with apparent luminosity brighter than L_lim, the lensed fraction is given by <cit.>:

    F_multi = B τ_m/B τ_m + B'(1-τ_m)

where B' is the magnification bias of non lensed sources, and we assume here it is B'≈1 as in <cit.>. 












 




















                            
























                            








After calculating the expected number of quasars to be detected as explained in the previous section, the number is multiplied by the lensed fraction F_multi for each redshift bin. Eventually, a total lensed fraction is calculated. We perform the calculation for three different regimes - quasars at all redshifts and luminosity ranges, high luminosity quasars (i.e. M_1450 < -26.5 which corresponds to λ L_UV,1450 > 10^46.5 erg s^-1) and high redshift quasars, namely 4<z<12. Last, out of the high redshift candidates we aim to obtain the number of quasars for which the notions we highlight here might be particularly useful, i.e, those with long lag times. We follow the same procedure, only this time for each redshift bin we find the lower limit for the luminosity such that the expected time lag of the C iv line is at least, e.g., a 100 days. As can be seen in Fig.<ref>, the expected time lags for the H_β emission line will be even longer.

 
 




§ RESULTS

We summarize the results in Table <ref>.
For all observatories, we see that the chances to detect a multiply imaged, high-luminosity quasar are very small - with <1 that are expected to be detected across the sky, with both QLFs. Nonetheless, for the high redshift regime, we obtain that many such lensed AGN should be found. Explicitly, we find that ∼ 3-400 multiply imaged AGN may be detected with Euclid over the whole sky, with the exact number depending on the choice of QLF. For the Roman Space telescope, we similarly estimate ∼ 13-9,800 detectable, multiply imaged quasars, and for JWST we find that ∼ 3 × 10^4 - 5 × 10^4 quasars multiply imaged by galaxy clusters might be detected all over the sky. In general, then, the numbers of multiply imaged AGN above correspond to about ∼10^-4 of the total (unlensed) population.

It is interesting to note that JWST has imaged to date about ∼10 clusters and in one of them, Abell 2744, a conspicuous multiply imaged AGN at z≃7 has already been discovered <cit.>. If we take this 1 in 10 ratio as representative, then many more such multiply imaged AGN should be detected soon with JWST, as it expands its observations to a larger cluster sample. Moreover, with ∼100,000 clusters in the sky one could expect about ∼10,000 multiply-imaged AGN detectable with JWST, which is of the same order of magnitude as the result obtained in our more detailed calculation, implying that the expected number is representative. 

While it is encouraging to see that many AGN multiply imaged by galaxy clusters could be detected across the sky, it is also insightful to examine which fraction of this lensed population has a large expected time lag, such that the gain from lensing would be most significant. Columns 5 and 6 in Table <ref> list the expected number of multiply imaged AGN that are expected to have lag times of, as an example, >50 and >100 days. For Euclid, we expect ∼3 - 380  multiply imaged AGN with >50 days observed time lags and ∼2 - 35 with >100 days; ∼4-2150 and ∼2-207 with the Roman space telescope; and ∼2750 - 3500 and ∼140 - 190 with JWST, respectively, across the sky. This is about 1-2 orders of magnitude lower than the total number of high-redshift AGN that are multiply imaged by galaxy clusters, but shows that there exists a lensed population in the sky which can gain substantially from RM campaigns to probe the R-L relation at high-redshift, as we also discuss further below (section <ref>). 

In addition to the above, it is seen in Table <ref> that for both Euclid and the Roman space telescope higher numbers of quasars are generally expected assuming the QLF from <cit.> compared to the QLF from <cit.>. The order of magnitude difference is not surprising given the high number density of broad-line AGN recently detected with JWST to fainter magnitudes <cit.>.

In Table <ref>, for completeness, the first column lists the numbers of quasars across all redshifts that are expected to be multiply imaged by galaxy clusters. It should be noted that both the QLFs we adopt for this calculation are, however, limited in redshift. For the <cit.> QLF, which is defined between redshifts z∼4 and z∼8, we assume no additional evolution: We use the z∼8 QLF for all redshifts above z∼8, which contributes only negligibly to the final numbers, and the z∼4 QLF to all redshift below z∼4. Since the QLF is expected to rise down to z∼2 and then turn over in resemblance to the cosmic star formation density <cit.>, we note that the exact effect of this assumption is not clear, in particular since the evolution of “ little red-dot" and other JWST selected AGN is not well established yet to lower redshifts. Regarding the QLF from  <cit.>, we note that it has a term for a monotonic evolution with redshift, although it was defined in practice over the redshift range ∼ 5.5-6.5 and the redshift evolution term assumes a monotonically increasing number density towards lower redshifts. In that case, assuming this QLF below z ∼ 4 might cause an over estimate of the number of quasars at the low redshift regime. A detailed examination of the effect of the z∼1-4 range on the total numbers is out of scope for this work, in particular, since here we focus on the high-redshift regime z ≳4.








§ DISCUSSION


To examine the feasibility of the idea proposed in this work, in the first part of this work we have shown the distribution of time delays by typical lensing clusters and compared it to the time lag distribution of quasars across their mass range and at various redshifts. After showing that indeed galaxy clusters supply useful time delays that can easily be as large as the the time lags expected for highest luminosity and highest redshift AGN, in the second part of this work we then estimated the number of AGN that are expected to be multiply imaged by galaxy clusters and observable with different observatories of interest. 









Before concluding the work and while perhaps quite intuitive, we now aim to provide a more visual demonstration of the idea, using an idealized case.

In particular, we consider an illustrative scenario in which the expected time lag is equal to the time delay between two images of a quasar. 

We end by referring to a couple of recent lensed examples for which such RM observational campaigns may be feasible using existing facilities.

To simulate a typical continuum light curve of a quasar, we use mock light curves of the quasar HE0435-1223 generated by H0LiCOW collaboration and available online <cit.> . We simulate the emission-line light curve by replicating the continuum light curve shifted by a certain time lag and add for illustration purposes, a Gaussian noise with a standard deviation = 0.05 in flux. 


In Fig. <ref> we show the mock light curves for two lensed images A and B for a SL time delay of 1000 days, where it is in this case equal to the observed time lag. An observed BLR size of a 1000 days corresponds to a physical BLR size of 1000/(1+z) light days. At a redshift of z=2, for example, this corresponds to a SMBH luminosity of λ L_λ∼4×10^47 erg s^-1, according to the local scaling relations. For each image we show the same time-frame of 1200 days in the observer frame, while the duration was  arbitrarily chosen for illustration purposes. For each image separately, similar to the case in which the quasar were unlensed, the pattern seen in the first 1000 days in continuum is seen repeating itself in emission lines a 1000 days later. In practice, to securely identify such an offset it would require at least several times this timeframe, hence order a decade or more of continuous monitoring. However, due to the TD being similar to the time lag, the same changes in the continuum seen in the trailing image are seen at the same time in emission lines in the leading image, essentially saving the years-long wait time to observe the lagging emission line changes. Suppose that in order to detect the ∼3 year time lag a monitoring campaign spanning 15 years were needed for an unlensed source to accumulate enough data points and timespan (the true value depends on a variety of factors that we do not go into here). Investing 15 years of observing time would be looked upon as an uneasy and perhaps risky investment given that the output is uncertain. In contrast, imagine a lensed quasar such that the source is multiply imaged with at least three images, which - more often than not - would be the case for wide separation lensed quasars as we consider here (e.g., ). In such a case the investment becomes much more reasonable: each observation triples the number of data-points compared to the unlensed case, and more importantly, assuming that t_TD∼ t_lag then the corresponding emission line changes can be observed (even if not immediately identified) already in the first few batches of monitoring. Furthermore, each image would then be magnified, which would also ease the observational follow-up. 







As another example, in the case of a smaller BLR size, causing a shorter time lag than the SL time delay, the pattern seen in the first days of observations in continuum would already be seen repeated in the emission lines after a short time lag. This is similar to the way it would appear for an unlensed quasar, where as for the lensed case we mainly gain magnification and additional sampling from the different multiple images, making the reverberation mapping more efficient overall. Also in the opposite scenario where the SL time delay is much shorter than the expected time lag, while we do not gain much in terms of overall wait time, we gain in several other aspects as above. First, we obtain additional points in the light curve with the same observation, compared to the unlensed case (an advantage that increases even further when more than two quasars images are seen, as is often the case in cluster lensing). Second, the images are magnified, which facilitates their follow-up. Third, while of singly imaged (or unlensed) objects can only be observed for several months a year, when they are away from the Sun's direction, a multiply-imaged quasar may allow us to fill gaps in the light curves also in those otherwise down times.

It should however be noted that even if TDs are used to facilitate RM, the monitoring time should still be sufficiently significant so as not to bias the measurements towards the available TDs.



















Recently, a successful reverberation mapping was performed on the multiply imaged quasar, SDSS J2222+2745, at z=2.805 <cit.>, demonstrating the usefulness of multiply imaged quasars in that respect. A time lag of τ_cen = 36.5^+2.9_-3.9 restframe days was significantly detected for the C iv emission line, whereas the time delays to the two other images of the quasar are about ∼40 and ∼700 days (see also ). Hence, one time delay is of the order of the time lag, similar to first case discussed above (but with a shorter time lag than what we consider here, which is aimed for the high-redshift and high luminosity regimes), and the second time delay is much longer than the time lag, similar to second case discussed above.


Note that throughout we assume that any object detected by a certain observatory could also be followed up spectroscopically (by the same or other observatories). In addition, given that we only wish to demonstrate the basic idea, we only show an idealized case. We also neglect here other sources of noise such as local microlensing for some of the images, which is known to often be present and cause its own variations to the light curve for some of the components <cit.> – depending on the mass of the point mass and size of the emitting source – although even in such cases the effects can be remedied (see e.g. <cit.>). Similarly, we do not take into account so-called “changing-look" AGN <cit.>, for which some of the images may not show, for example, broad-line emission, while the other images might (in fact, lensing TDs may be a route for detecting lensed AGN of this type). 

One should also bear in mind that in reality the TD is not necessarily known from the lens model to the required accuracy to remove it from the fit altogether. However, it might be first determined from the temporal shift between the continua of the two images before determining the time lag. This may improve the error on the TDs compared to the errors from the lens model's TD, which would otherwise propagate into the R-L relation as well.


It is also worth highlighting a couple of interesting recent objects which may become useful targets for RM of lensed AGN (other examples naturally exist, too). First we note the triply-imaged Little Red Dot AGN reported by <cit.> at =7.045. This object is currently the highest-redshift multiply imaged AGN. It has according to their lens model a TD of ∼3 years, roughly, between the two closest images, and a TD 
of about 19 years between the first of these two and the third, more distant image, which is first to arrive. The NIRSpec spectrum of the source was published in <cit.>. Based on its luminosity, with a few hours per visit we could detect both the continuum and the Hβ line (no C iv lines were detected in the object, despite the very deep integration time). One can thus envision that a NIRSpec/JWST program, for example, targeting the three images simultaneously, could be used to measure both the continuum and emission lines in each epoch. From the width of Hβ, <cit.> derived a BH mass of 4×10^7 M_⊙, which, according to the scaling relations by <cit.>, corresponds to an expected or observed time lag of a couple dozen days. This time lag is relatively short (corresponding to the second case discussed above), and while its measurement could in principle also be done were it not lensed, its measurement would be facilitated by measuring the three images simultaneously. While the expected time lag for this object does not appear to be very long (so that the gain in wait time is not maximized), it could supply a unique example for probing the RM relation at high-redshift.

A second, recent example is the large separation z=3.27 quasar COOL J0335-1927 <cit.>, for which time delays of ∼500 days and ∼130 days between two pairs of its images were predicted from the lens model, although with quite substantial uncertainty. To estimate the expected time lag, we adopt the typical g-band luminosity seen in Fig. 4 in their work, and translate it to an absolute UV magnitude of ∼25.5 AB, which, based on existing scaling relations <cit.>, corresponds to a an observed time lag of ∼300 days. The object therefore might benefit significantly from such a campaign. For more discussion on the latter object, see also <cit.>.


We thus conclude that designated observations of multiply imaged quasars can open the door for RM studies in a much more efficient manner, as was already demonstrated at lower-redshifts <cit.>. For the most massive and/or high-redshift quasars for which the time lags may be otherwise too long, this might possibly be the only, feasible way.




§ SUMMARY

In this work we discuss the possibility of using lensing time delays to facilitate the RM of high-luminosity and high-redshift quasars, for which the radius-luminosity relation is currently, poorly constrained. The expected time lags for very-massive or high-redshift quasars can reach several years and would likely require much longer monitoring times than can be comfortably accommodated by typical observing campaigns. Multiply imaged AGN may, however, offer a unique opportunity to explore the radius-luminosity relation at these ends. Not only do strongly lensed quasars comprise several magnified images which boost the efficiency of the light curve sampling, but also, the time delay between the quasar's multiple images can save the long wait times that are typically required. For example, if the time delay is roughly of the order of the expected time lag, changes in the emission lines in a leading image can be observed around the same time as the changes in the continuum in a trailing image, thus allowing a much reduced wait times and overall monitoring times-spans. 

We start by showing the typical distribution of BH time lags compared to the lensing TD distribution from cluster lenses and show that a strong overlap exists. We then estimate the number of multiply imaged quasars that may be detectable in the near future with Euclid, Roman, or JWST. We find that indeed, a fair amount of multiply imaged high-redshift quasars, out of which some would experience a significantly long observed time lag, should be detectable. Using the time delays between the multiple images, RM of these possible future candidates could be obtained in a reasonable few-year monitoring timescale, probing the poorly known R-L relation in the early Universe. 




§ ACKNOWLEDGEMENTS

M.G would like to thank Yuichi Harikane for a useful discussion. M.G is thankful for the hospitality of the SURF program and cosmic DAWN center where much of this work has been carried out.
The BGU group acknowledges support by grant No. 2020750 from the United States-Israel Binational Science Foundation (BSF) and grant No. 2109066 from the United States National Science Foundation (NSF); by the Israel Science Foundation Grant No. 864/23; and by the Ministry of Science & Technology, Israel. 
D.C. acknowledges support by grants from the German Research Foundation (HA3555-14/1, CH71-34-3) and the Israeli Science Foundation (2398/19, 1650/23).
This research made use of ,[<http://www.astropy.org>] a community-developed core  package for Astronomy <cit.> as well as the packages ,  <cit.>,  <cit.>,  <cit.> and the astronomy library for  <cit.>. The  package <cit.> was used to create some of the figures in this work.



aasjournal





 Paper Title: Analysis of Polarized Dust Emission from the First Flight of the SPIDER Balloon-Borne Telescope.




pider_authors.tex


§ ABSTRACT 

Using data from the first flight of and from HFI, we
probe the properties of polarized emission from interstellar dust in
the observing region.  Component separation algorithms
operating in both the spatial and harmonic domains are applied to
probe their consistency and to quantify modeling errors associated
with their assumptions.  Analyses spanning the full region
demonstrate that i) the spectral energy distribution of diffuse
Galactic dust emission is broadly consistent with a modified-blackbody
(MBB) model with a spectral index of β_d=1.45±0.05
(1.47±0.06) for E (B)-mode polarization, slightly lower than
that reported by for the full sky; ii) its angular
power spectrum is broadly consistent with a power law; and iii)
there is no significant detection of line-of-sight decorrelation of
the astrophysical polarization.  The size of the region
further allows for a statistically meaningful analysis of the
variation in foreground properties within it.  Assuming a fixed dust
temperature T_d=19.6 K, an analysis of two independent
sub-regions of that field results in inferred values of
β_d=1.52±0.06 and β_d=1.09±0.09,
which are inconsistent at the 3.9 σ level.
Furthermore, a joint analysis of and 217 and 353 GHz
data within a subset of the region is inconsistent with a
simple MBB at more than 3 σ, assuming a common morphology of
polarized dust emission over the full range of frequencies.  These
modeling uncertainties have a small—but non-negligible—impact on
limits on the cosmological tensor-to-scalar ratio derived from
the dataset.  The fidelity of the component separation
approaches of future CMB polarization experiments may thus have a
significant impact on their constraining power.









§ INTRODUCTION


The polarization of the cosmic microwave background (CMB) contains a wealth of information about the contents, history, and origins of the Universe <cit.>. 
Observations of the E-mode (“gradient”, scalar, even-parity) polarization pattern reveal details of acoustic oscillations in the primordial plasma.
A measurement of a cosmological B-mode (“curl”, pseudoscalar, odd-parity) pattern on large angular scales would imply the presence of a background of primordial gravitational waves, and thus provide remarkable new insights into early Universe models <cit.>.

Precision studies of CMB polarization are complicated by polarized emission at similar frequencies from foreground sources within our own Galaxy, notably synchrotron radiation and thermal emission from dust particles <cit.>.
Even at the highest Galactic latitudes, foreground emission is known to be far brighter than any primordial B-mode signal <cit.>.
Observations at multiple frequencies can be used to differentiate CMB from foreground emission based upon their differing spectral energy distributions (SEDs) <cit.>.
Several such component separation methods have been developed: see, e.g., <cit.> and references therein.
In order to stitch together such multi-frequency data, these methods must necessarily make assumptions about the nature of foregrounds: their spatial morphology, SED, or some combination thereof.
Component separation is thus an imperfect process, and there are always some uncertainties due to these assumptions <cit.>.





At frequencies above 100 GHz, Galactic dust emission is the dominant polarized foreground on large angular scales <cit.>.
Dust grains, heated by stellar radiation to ∼20 K, thermally re-emit that energy with an efficiency that depends on the physical characteristics of the grains.
Because emission is inefficient at
wavelengths much larger than the grain itself, dust emissivity is
expected to drop in the mm-wave regime where CMB measurements occur <cit.>.
Aspherical grains that are ferro- or para-magnetic will preferentially align relative to the local magnetic field, leading to linearly-polarized emission with large-scale correlations <cit.>.
Motivated by this picture, it is common to model the dust SED[Note that, while Equation <ref> may be used to model both the intensity and polarization of dust emission, the same model parameters need not apply to both; indeed, has found a slightly steeper power law for polarization than for intensity in an all-sky analysis <cit.>.]
as a thermal blackbody at temperature T_d, modified by a power-law emissivity with exponent β_d:

    I(ν) = τ_0 ( ν/ν_0)^β_d B(ν, T_d).

Here B is the Planck spectrum for a blackbody emitter, ν_0 is some reference frequency, and τ_0 is the optical depth at the reference frequency.
Equation <ref> is referred to as a modified blackbody (MBB).



Though widely-used, this simple model is unlikely to reflect the full range of dust emission across the sky.
The detailed nature of dust grains remains a subject of ongoing research <cit.>.
Their composition, size distribution, and density are all expected to vary from place to place, as are their radiative and magnetic environments.
Thus, we may naturally expect different dust populations in our Galaxy to exhibit different SEDs.
Such variation of the spatial morphology of a foreground component with observation frequency—or, equivalently, of the component SED across the sky—is often called frequency decorrelation.
Furthermore, observations of the CMB constitute a collection of line-of-sight (LOS) integrals through the materials surrounding us, each of which may traverse heterogeneous populations.
In addition to complicating the dust SED, such LOS decorrelation can yield variation in polarization angle with frequency even within a single map pixel <cit.>.

Decorrelation may significantly complicate efforts at component separation.
For example, <cit.> have shown that naive extrapolation of dust templates made from 353 GHz data to lower frequencies can bias the recovery of the tensor-to-scalar ratio r ≤ 1.5× 10^-3.
While an analysis by the BICEP team <cit.> showed no evidence of dust decorrelation within their observing region, there is evidence for its presence in larger regions.
By identifying regions of the sky likely to contain magnetically misaligned clouds, <cit.> demonstrate that LOS frequency decorrelation is detectable within the dataset, largely in the Northern Galactic cap.
<cit.> has further demonstrated LOS frequency decorrelation in large-scale regions close to the Galactic plane.
It is thus critically important to test the assumptions behind common foreground separation techniques through detailed studies of dust polarization.
Such studies will inform both the field of Galactic astrophysics as well as the prospects for foreground removal by future CMB experiments <cit.>.






This publication is part of a series describing results from the first flight of , a balloon-borne CMB telescope optimized to search for B-modes on degree angular scales.
These data from provide a measurement of the angular power spectra of 4.8% of the polarized sky at two frequencies: 95 and 150 GHz <cit.>.
Dust emission is the dominant foreground at these at these frequencies.
Since the first instrument did not include a channel optimized for dust, the analysis of these data relies heavily on 217 and 353 GHz maps for component separation.
The published B-mode results from these data relied primarily on template subtraction using the XFaster power spectrum estimator (Section <ref>) alongside comparisons with alternate estimators and foreground-cleaning techniques (e.g., Section <ref>).

This paper exploits the sensitivity and sky coverage of the data to explore the properties of the Galactic emission and to quantify the impact of the methods and models employed on component separation.
Section <ref> reviews both component separation techniques, and in Section <ref> they are applied to two distinct subsets of the region to probe for evidence of spatial variation in the nature of the diffuse polarized dust emission.
The fidelity of the MBB model is tested in a comparison of the results from the two sub-regions using two -derived dust templates (353-100 GHz and 217-100 GHz).
Section <ref> examines different choices for modeling the dust SED within the full region: the validity of the MBB model, the constancy of MBB parameters with angular scale, and the use of power-law models for the dust angular power spectrum.
Section <ref> searches for LOS decorrelation by developing a custom estimator to look for evidence of variation of dust polarization angle with frequency in the observing region.
Finally, Section <ref> compares the measured dust power spectrum for this sky region to templates generated by the PySM foreground modeling package.













§ COMPONENT SEPARATION TECHNIQUES



As in <cit.>, our analyses in this paper rely on two main
component separation techniques: foreground-template subtraction, and
Spectral Matching Independent Component Analysis (SMICA).
In this section, we present baseline versions of these methods.
In Sections <ref>
and <ref> we explore variations in the
assumptions presented below in order to test our understanding of the
polarized emission from interstellar dust.




 §.§ Template Subtraction



In the baseline cosmological analysis of the first flight
data <cit.>, we remove polarized dust emission from
the 95 and 150 GHz maps by subtracting from them a scaled
template for this emission derived from data.[In both
the analyses presented in <cit.> and in this
publication, we use release 3.01 of the data <cit.> unless otherwise noted.]
The implementation of foreground-template
subtraction follows the approach pioneered
by <cit.> in the analysis of the polarization
data.
Within any given map pixel observed at frequency ν, the measured
Stokes parameters, S_ν, are each modeled as the sum of three
components: CMB, dust, and noise:

    S_ν = S^ CMB + A_ν,ν_0 S_ν_0^ dust + n_ν .

The uniform (pixel-independent) scaling factor A_ν,ν_0 connects the dust map
at the observing frequency ν to that at the reference frequency
ν_0=353 GHz.


We construct our dust template by subtracting the 100 GHz map from the 353 GHz map:
S_ν_0^ t≡ S_ν_0-S_100.
Since the CMB component is independent of frequency in these units,
this removes the CMB signal at the cost of a modest increase in noise.
To produce a cleaned CMB map, we scale this dust template by a fitting parameter α
and subtract it from S_ν, giving

    S_ν - α S_ν_0^ t    =     S^ CMB +
    (A_ν,ν_0 - α [1-A_100,ν_0]) S_ν_0^ dust
        +     n_ν - α n_ν_0^ t.

Here n_ν is the noise component at the frequency ν, and
n_ν_0^ t≡ n_ν_0-n_100.


Following <cit.>, we use the XFaster
algorithm <cit.> to fit for values of α=A_ν,ν_0/(1-A_100,ν_0)
and the tensor-to-scalar ratio, r, in a simultaneous fit to the
EE and BB spectra.
Note that, before
subtraction, the dust template is “reobserved,” i.e., injected as
an input sky through the full simulation pipeline; it is thus
subjected to the same scan strategy, beam smoothing, time-domain
filtering, and map-making as the sky observed by .
The reobserved template, like the data, is corrected in map space
for the temperature-to-polarization leakage induced by the pipeline.

The template method implements a single scalar-valued factor α
for each frequency multiplying the entire map.
Thus, this method assumes that the morphology of the polarized dust emission
is independent of frequency (at least between 353 GHz  and 100 GHz) or, equivalently,
that the dust SED is constant across the relevant sky area.
Note, however, that this method does not impose assumptions on the dust SED itself.
We will revisit this assumption in our analysis below, notably in
Sections <ref> and <ref>.







 §.§ SMICA



Spectral Matching Independent Component Analysis <cit.>
is a method for constructing linear combinations among a collection of maps in order
to isolate desired sky components (e.g., dust or CMB).
In contrast to the map-space template analysis described above, SMICA operates in
multipole space, acting on the spherical harmonic coefficients, a_ℓ m, for each map.
Given a collection of
n input frequency maps,[Throughout this paper, n=6: for each polarization mode we use maps at
all four polarized  HFI frequencies (100, 143, 217, and 353 GHz) and
the two frequencies (95 and 150 GHz).] we construct
Y = (y_1, ⋯, y_n), where each y_i is a row vector of
a_ℓ m coefficients for a single map i.
We use the SMICA algorithm to construct the column vector
of map weights, w, such that the linear combination
s≡ w^T Y is an estimate of the spherical harmonic coefficients for our desired sky component.
Throughout this section, lower case letters denote vectors, and upper case bold letters denote matrices.

The weights are optimized with respect to some model of the assumed sky components.
Given a column vector, a, containing the frequency scaling of a given sky component across
the various maps (uniform for CMB, MBB for dust), the multipole-binned weights, w_b, for this sky component are given by

    w_b^T =
            (a^T R_b^-1 a )^-1 a^T R_b^-1,

where R_b is the binned modeled covariance matrix containing CMB, dust, and noise components. These weights minimize the variance of the reconstructed harmonic map, var(s) = w_b^T R_b w_b under the
signal-preserving constraint w_b^T a = 1  <cit.>.

When reconstructing component maps, each harmonic mode a_ℓ m is scaled by the corresponding w_b for its appropriate bin. The reconstructed maps, S, are the inverse spherical harmonic transform of these weighted collections of harmonic modes.

SMICA's power and flexibility lie in the freedom to impose (or relax) different assumptions about the components in this model, and thus about the relationships among the weights.
As discussed in detail in <cit.>, the SMICA implementation for
the B-mode analysis assumed a single dust component with a MBB SED.
In this analysis we make two improvements to SMICA dust modeling: we do not impose an SED on
the dust component, and we improve the power-spectrum estimation for the dust through a more accurate transfer function. Both changes are described further below.


In this analysis we do
not assume a MBB dust SED, but instead fit for the dust a as for any other parameter of the
model defining R_b (which includes the CMB, dust
emission, and noise) using the same maximum-likelihood fitting
procedure as in <cit.>.
We choose the arbitrary
overall normalization of a such that its 353 GHz element is set to 1.
Fitting for the dust a in this manner effectively trades precision for accuracy:
the frequency-dependence of the reconstructed dust component will not be biased by any
assumptions about the dust SED, but the uncertainty on the
other fitted parameters (e.g., the amplitude of the emission) will
likely be larger; it could be smaller, but only if the SED we would otherwise
assume were an especially poor description of the data.

In previous analyses <cit.>, the sky components of R_b are corrected for effects from filtering, beam smoothing, and mode-coupling kernels through a J_b b' transfer matrix, as per <cit.>.
While this approach is accurate for CMB spectra, the different angular power spectrum of dust make it inaccurate for an unbiased recovery of the full-sky dust spectrum, especially at larger scales.
In this paper we modify the model to apply J_b b' only to the CMB component, and instead apply a diagonal F_b to the foreground component.
F_b is the ratio of the output to input power for each bin and is constructed from an ensemble of reobserved Gaussian dust realizations.
These realizations follow a power-law scaling of amplitude with angular scale, with an exponent γ = -2.30, pivoting around ℓ = 80.







Figure <ref> shows
the weights by which each and  HFI frequency map contribute
to the component maps computed by SMICA. For the dust component, the
relative contribution of the map at frequency ν is given by its
SMICA (a_ℓ m) weight multiplied by
f_ν≡ a_ν^dust, the ν-component of the vector
a^dust, the dust frequency scaling vector fitted to the
data by SMICA. The main contributor to both the E-mode and B-mode
dust signals in Figure <ref> is the
353 GHz map, which has the highest ratio of
dust signal to noise. In the B-mode dust reconstruction, all
contributions are positive, as expected in the absence of CMB B-mode
power.

For the CMB, the SMICA weight for a given frequency map directly represents its relative contribution, since
a^CMB is the identity vector. The maps provide the dominant
contribution to both the E and B-modes, reflecting the higher ratios of CMB signal to noise in
these maps than in any of the HFI maps.

The Q and U maps
derived by SMICA, along with their estimated uncertainties,
are shown in Figures <ref> and <ref> for the CMB and dust components, respectively.
The covariance σ_w of the weights and the noise covariance σ_Y contribute in quadrature to the total per-pixel uncertainty on the estimates of the component maps S.
The former represents the uncertainty due to signal reconstruction, while the latter (the dominant term) represents the per-pixel noise after a weighted average.
Both are presented in separate panels of Figures <ref> and <ref>.


In Figure <ref>, the morphology of σ_Y^dust is similar to noise estimates for 353 GHz, as that map dominates the weight.
The uncertainty on the dust reconstruction, σ_w^dust, is morphologically distinct from the noise.
In particular, the larger uncertainty in the upper third of the σ_w^dust maps does not trace the reconstructed dust signal strength or the noise.
This unexpected structure could come from
unidentified systematic errors in the input or maps,
although evidence for the latter was not found in the expansive
suite of null tests published in <cit.>. A physical
origin is also possible: frequency decorrelation, in which dust
emission at a given frequency is not well correlated with that at
other frequencies, would prevent an accurate reconstruction of this
emission by SMICA, as the formalism here
assumes that a single frequency scaling factor applies to all
pixels. Given a specific frequency decorrelation model, e.g., a
mixture of two dust populations, each characterized by its own
spectral index, simulations could be run through SMICA in an attempt
to reproduce the structure observed in the dust uncertainty maps. Such
a study is beyond the scope of our analysis.

Figure <ref> also compares the SMICA dust component map S_353^dust to the dust template maps S_ν_0^t constructed with ν_0 = 353 GHz and ν_0 = 217 GHz.
The template maps have been band-limited to ℓ∈ [8, 258], to match the scales for which SMICA weights are fit.
The S_353^dust and S_353^t maps look very similar, which is not surprising since both are dominated by 353 GHz data.
S_217^t has much lower dust amplitude than S_353^t, and lower signal-to-noise ratio.
The visible structures in these two template maps are similar but not identical.







§ SPATIAL DEPENDENCE OF POLARIZED DUST EMISSION PROPERTIES



In our baseline cosmological analysis <cit.>, we use
foreground-template subtraction (Section <ref>) to
subtract the polarized emission from interstellar dust from the
maps. As reflected in the scalar nature of A_ν,ν_0
(and therefore of α) in Equation <ref>, this
procedure assumes that the morphology of this emission
is independent of frequency. This assumption could
easily be inadequate, e.g., if the temperature of dust grains
changes across the region, or if two or more dust
populations—even if they are well mixed—permeate it. In
Section <ref>, we perform
foreground-template subtraction independently in two non-overlapping,
physically motivated subregions of the field
(described in Section <ref>), and assess the consistency
of the α values derived in the two subregions. We discuss the physical interpretation of these results in Section <ref>.








 §.§ Delineating Subregions






We define two physically motivated subregions within the field based upon a SMICA-generated map of 
the total polarized dust emission power, P^2=Q^2+U^2 (Figure <ref>, top).
This map is constructed from the Q_353^dust and U_353^dust Stokes
parameter maps described in Section <ref> and shown in Figure <ref>.
After smoothing this map with a Gaussian of 8^∘ FWHM, we separate the map into two complementary 
subregions along the P^2 = 100 μK_CMB^2 contour   (Figure <ref>, bottom).
We denote by () the subregion in which P^2 is greater
(less) than this threshold. 
This procedure produces two simply-connected subregions (prior to masking point sources) that are separated
by a smooth boundary, features that facilitate power spectrum estimation <cit.>.
After point-source masking according to
the procedure described in <cit.>, and cover roughly equal
sky fractions of 2.4% and 2.3%, respectively.
We note in passing that overlaps with the “Southern Hole,” a region of the
Southern sky that has long been known to exhibit low total dust
emission, and that has more recently been mapped with
high polarization sensitivity
<cit.>.



 §.§ Template Subtraction by Subregion: Evidence for Spatial Variation






Following the procedure described in
Section <ref>, we perform foreground template
subtraction independently on each of the two subregions
defined in Section <ref>.
The left panel of Figure <ref> shows, for each subregion
and for the entire field, the resulting 1-D and 2-D likelihoods for the
tensor-to-scalar ratio, r, and for the foreground-template fitting
parameters, α_95 and α_150, where the added indices
refer to the frequencies to which we fit the 353 GHz foreground template.
Following the method in Section 4.3 of <cit.>, the results in this section are validated with a suite of 300 simulations with input
r, α_95, and α_150 fixed to the maximum likelihood values in Figure <ref>.
Each simulated map is constructed from ensembles of CMB plus noise, along with either the 217 or 353 GHz dust template.
Fitted templates add noise from FFP10 realizations.
To produce robust estimates of the α parameters recovered from the data, we correct the data likelihoods for a small (≪ 1 σ) bias in the XFaster estimator, as measured from this validation suite.

With the 353 GHz template, we find the maximum-likelihood values for r and α_95 to be
consistent to better than 0.7 σ and 2.0 σ, respectively,
between all three regions. However, the maximum-likelihood values for
α_150 differ by 3.6 σ between the two subregions, with a 
significantly higher value of α_150 in than in .
In other words, the template-correlated dust signal in the more diffuse 
region, , is brighter than would be inferred by scaling the template 
based upon observations of the region with more polarized dust emission, (or of the combined region, 
where the fit is naturally driven by the brighter areas).
We would thus expect residual dust signal after subtraction of a template that 
did not account for such variability.

Note that these data are not sufficient to establish any general causal relationship 
between variation in α_150 and the intensity of the dust P map. 
This would require comparisons
between a larger number of subregions, which is beyond the scope of
this work.

We are able to reject a number of compact structures as the
origin of this discrepancy. Two regions of particular interest are the
Orion-Eridanus Superbubble <cit.> and the
Magellanic Stream <cit.>.
The former, which features
prominently in Hα maps, occupies about 20% of the full region, mostly located in .
Although the Magellanic Stream falls mostly outside
the region, some high-velocity clouds have been mapped on the
outskirts of its bottom-right corner <cit.>. Even though
we do not necessarily expect the dust emission from these
regions to dominate the polarized emission in our maps, high levels of
polarization along the edge of the Orion-Eridanus Superbubble and
hints of variation in dust properties between the Galaxy and the
Magellanic Clouds <cit.> warrant caution. As documented
in Appendix <ref>, we find the measured spatial
variation in α_150 to be robust to masking aimed at probing
the impact of these structures.




We further explore the discrepancy in α between subregions by repeating this template subtraction analysis with an alternate 
dust template, constructed by substituting the 217 GHz map for
the 353 GHz map.
The 217  GHz data also contain substantial polarized dust emission, as evidenced 
by the fact that they provide the second-largest contribution to the
SMICA dust reconstruction (Figure <ref>).
The right
panel of Figure <ref> shows the 1-D
and 2-D likelihoods for r, α_95, and α_150 in this
configuration.  We again find r to be consistent (within
0.2 σ) between the two subregions, a moderate discrepancy in
α_95 (2.7 σ), and a larger discrepancy in
α_150 (3.5 σ).  However, while the latter discrepancy
has similar significance to that found with the 353 GHz template, it
is in the opposite direction: α_150 is found to
be lower in than in . 



 §.§ Implications for a Modified-Blackbody Model






In the context of the simple MBB model of Equation <ref>, 
the template scaling parameter α is a function of the dust model parameters—the dust temperature, T_d, and the spectral index of polarized emission, β_d—as well as the frequencies of the maps used to build the dust template and the
frequency of the map to which the template is fitted.
Specifically,

    A_ν, ν_0 =
      (e^ x - 1)^2/x^2 e^ xx_0^2 e^ x_0/(e^ x_0 - 1)^2(ν_0/ν)^2(ν^d/ν_0^d)^β_dB(ν^d,T_d)/B(ν^d_0,T_d),

where the superscript “d” added to the frequency of a map indicates
that its effective band center is computed for a dust SED instead of
the CMB SED, as discussed in
Appendix <ref>, and we have used the
shorthand x≡ hν/(k T_CMB). 

Using this relationship, we can translate the 1-D likelihoods for α_95 and α_150
shown in Figure <ref> into contour plots in 
(T_d,β_d).
The results are shown in
Figure <ref>.
These parameter constraints allow us to perform several internal comparisons 
within the MBB model context, which are discussed in the paragraphs below.




  
Comparing frequencies
The β_d values inferred from the fitted α_95 and 
α_150 show excellent consistency (<1 σ) with one another for 
either choice of dust template, except perhaps in at the very lowest dust 
temperatures (T_d≲ 7 K).
In other words, if we confine ourselves to a single -derived dust template, 
the data provide no significant evidence of deviation from an MBB SED in either subregion.



  
Comparing subregions
The values of α_150 obtained using the ν_0 = 353 GHz
dust template translate into lower values of β_d
in compared to for any value of
T_d. Specifically, if we fix
T_d=19.6 K <cit.> throughout, we
find β_d = 1.52 ± 0.06 in and 1.09 ± 0.09
in . Importantly, the value of β_d inferred
independently from α_95 is consistent with that of
α_150, with comparable significance.
Taken at face value, this 3.9 σ inconsistency in β_d would indicate a
spatial variation in dust composition between the and subregions.  

If we instead fix the dust spectral index to
be β_d = 1.53 <cit.> in both
subregions, we find a large difference in implied dust temperatures:
T_d=19.1^+3.7_-2.5 K in , 
T_d=9.2^+1.1_-0.8 K in .  Although we certainly
expect some lack of uniformity in dust temperature—stemming
from spatial variation in the dust's radiative environment—such a
large variation in the temperature of the diffuse interstellar medium
at high Galactic latitude may be unphysical given our current
understanding of dust physics <cit.>.

A similar discrepancy between subregions (though of opposite sign) is visible with the 217 GHz dust 
template, as shown in the middle panel of Figure <ref>.
Fixing 
T_d=19.6 K with
the α_150 constraint results in β_d = 1.37 ± 0.08 in and 
2.1^+0.3_-0.2 in , a 3.2 σ discrepancy. Alternatively, fixing
β_d = 1.53 gives
T_d=12^+3_-2 K for , but 
no value of T_d accommodates β_d = 1.53 within .



  
Comparing dust templates within 
The values of β_d
derived from α_150 within the subregion using the ν_0 = 217 GHz dust template
are inconsistent with those derived using the ν_0 = 353 GHz
template. The bottom panel of Figure <ref> shows the significance of this
discrepancy as a function of dust temperature: it exceeds 3 σ
(4 σ) for any T_d≳ 5 K
(T_d≳ 15 K). Under the assumption that the
morphology of the polarized emission from interstellar dust is
identical in both templates, this result corresponds to a
statistically significant break in β_d for any physical
T_d, with the index having a lower value at the higher
frequency.

Furthermore, we find that the value of α_150 increases
from to with the ν_0=353 GHz template,
but decreases from to with the ν_0=217 GHz
template.
This observation is difficult to interpret under the assumption that the
morphology of the polarized emission from interstellar dust is
identical in both templates.  This can be reconciled by assuming the
existence of more than one dust population (which may be inferred
from the spatial variation in β_d detected with the
ν_0=353 GHz template) combined with a break in
β_d between 217 and 353 GHz. 





  
Discussion
The physical interpretation in this section of the results derived in
Section <ref> points to a more complex
picture of the polarized emission from interstellar dust in
the region than that provided by a single-component MBB
model. 

Taking the observations above together, we conclude: i)
Unless T_d exhibits large variations at
high Galactic latitudes, the spatial variation in α
between and may be evidence of multiple dust
populations (i.e., differing compositions, and thus emissivities);
ii) Unless the ν_0=217 GHz dust template is
morphologically inconsistent with the ν_0=353 GHz template, the
inconsistency between the value of α derived in with one template and that found in the same subregion with the other
template would indicate a break in the value of
β_d between 217 and 353 GHz;
iii) The break in β_d may itself be composition dependent in order to simultaneously explain the
increase in α between and observed with the
ν_0=353 GHz template and its decrease obtained with the
ν_0=217 GHz template.
If this rather complex picture holds as more data (such as the 280 GHz maps) become available, it may provide new insights into the
properties of the interstellar medium, and associated modeling
uncertainties would need to be taken into account in upcoming searches
for cosmological B-modes.












§ TESTING STANDARD DUST MODELING









The SMICA framework offers a high degree of flexibility in modeling the
characteristics of dust.  The SMICA pipeline implemented for the analysis in
<cit.> incorporates a MBB model for the dust foreground component
with the following characteristics:



  * A single β_d parameter that is common across all multipoles and polarizations,

  * A dust amplitude A_d that is free to vary across multipoles and polarizations,

  * A δ-function prior of T_d = 19.6 K to fix the dust temperature at the all-sky value, and

  * A Gaussian prior on β_d centered at the all-sky value
  β_d = 1.53, with a generous width of σ_β_d = 1.5.

In this section, we quantify how these foreground modeling choices impact the
best-fit CMB and foreground parameters recovered with the combined and
data.  Throughout this section, we leave the dust temperature fixed,
due to a lack of sufficient constraining power and frequency coverage
to simultaneously estimate both β_d and T_d,
while varying the constraints on A_d and/or
β_d.  Here, unlike the discussion in
Section <ref>, we apply the analysis to the full region.  The results in this section are therefore driven by the higher signal-to-noise component, corresponding approximately to the contours of Figure <ref>.



 §.§ SED Modeling of Polarized Dust Emission







We begin by testing the SED model of dust emission by fitting an independent
amplitude A_d at each frequency, operating under the assumption that this
amplitude remains consistent across multipoles.  As discussed in
<cit.>, dust and noise exhibit similar scaling behaviors with frequency,
making it challenging for an ICA-based
algorithm to distinguish between the two components.
By relaxing the prior constraint on β_d and fitting for the
dust amplitudes directly, we are able to circumvent this noise/dust scaling
degeneracy.  The resulting amplitudes can then be fit to a MBB model to recover
the best-fit β_d value.

The dust amplitudes recovered with this algorithm are listed in
Table <ref> for each of the and frequency
bands, and the resulting MBB model fit is illustrated in
Figure <ref>.  Fitting a spectral index to
each set of six frequency maps results in a best-fit value of β_d of 1.45±0.05
(1.47±0.06) for the EE (BB) components, with χ^2 values of 0.7 (3.0),
respectively.[For comparison, we note that the template fit in the combined region from Section <ref> corresponds to a value β_d=1.50± 0.05 for the combined EE & BB spectra.] With approximately four degrees of freedom, these χ^2 values
indicate that a single-component MBB model is an adequate description of the
+ dataset, thus justifying the use of the MBB model for
constraining foreground power in the CMB B-mode analysis.
Additionally, Figure <ref> illustrates the improved
constraining power of the data at 95 and 150 GHz in this region of the sky
relative to that of the nearby bands at 100 and 143 GHz.



 §.§ Angular Scale Dependence of Dust Spectral Index




Next we test the assumption of a scale-independent SED by fitting the MBB
model independently for each polarization as a function of ℓ.  The results
of this fit for β_d are shown in Figure <ref>.
The best-fit spectral index values agree to within ∼ 0.5 σ between the
E and B polarization components across all angular scales, providing no
evidence for including a polarization-dependence in the model.
Comparing the best-fit β_d values averaged over larger (ℓ < 140) and smaller
(ℓ > 140) angular scales shows agreement to within 2 σ.

When analyzed over the full region, the data do not reveal any strong evidence of variations in spectral
index with polarization mode or angular scale, further justifying the adoption of the
simple MBB dust model.



 §.§ Angular Scale Dependence of Dust Amplitude






Dust emission is often (e.g., <cit.>) modeled as a power law in C_ℓ:

    C_ℓ = A_d(ν) ( ℓ/80)^γ.

We can fit this model to the SMICA-derived amplitudes as a function of ℓ-bin
independently for the E and B polarization components. We assume a
single-component MBB scaling of the amplitude with frequency, common to both
polarizations. The results of this fit are shown in
Figures <ref> and <ref>, with reduced
χ_ν^2 = 1.1 (0.9) for the E (B) polarization components, assuming ν = 7.
This analysis finds a dust spectrum that falls steeply, γ^EE = -2.81±0.23, with
no significant difference in best-fit values between the two polarizations.  We note that has reported values of -2.42 and -2.54 ± 0.02 in the largest area of sky for EE and BB, respectively.  In the most restrictive mask, finds more shallow indices of -2.28 ± 0.08 and -2.16 ± 0.11 for EE and BB <cit.>.

The dataset also provides an additional observational constraint on the asymmetry in
dust power between EE and BB polarization modes, visible in Figure <ref>.
This analysis finds an EE/BB dust power ratio of 2.1 ± 0.3 at large scales (ℓ < 140).
This result aligns with prior findings from <cit.>. The origin of this dust power asymmetry is not
conclusively understood.  Some studies investigate the role of
magnetohydrodynamic (MHD) turbulence in the interstellar medium
(ISM) <cit.>, while others suggest that
this power asymmetry can be a probe of the dust filament aspect
ratio <cit.> and magnetic field alignment <cit.>.



 §.§ Impact on Cosmological B-mode Measurements



At the level of sensitivity of contemporary observational programs,
the contribution of modeling uncertainty to limits on the cosmological
CMB B-mode signal is non negligible even in those regions of the sky
with low levels of Galactic emission.  In this section we attempt to
quantify the impact of these complexities on the uncertainty in the
tensor-to-scalar ratio, σ_r, within the region.

Following the analysis in <cit.>, for each dust model we
can compute the σ_r derived from an ensemble of sky and noise
simulations constructed to match the model under consideration.  In
the configuration where there is no SED modeling and SMICA fits a dust
amplitude per frequency, we find σ_r = 0.130.  If instead a
spectral index β_d is fit per multipole, allowing for
dust behavior to change with scale, we find σ_r = 0.128.
Constraining the spectral index to a single value over all scales, as
in <cit.>, we find σ_r = 0.116.  Finally, the
assumption that dust follows a power-law in multipole space does
not improve σ_r further.  We find instead that this assumption
re-distributes the uncertainties: the uncertainty near the pivot scale
(ℓ∼ 80) is reduced, while that at larger scales increases.
Thus, we find that the complexities in foreground modeling described in this section can contribute at the level of σ_r∼ 0.012, corresponding to a ∼10% impact on  σ_r in the case of .





 §.§ Discussion




When applied to the full area of sky observed by the experiment, the variations of the nominal SMICA algorithm described
above, each employing different assumptions about foreground modeling, all point
to a consistent description of polarized Galactic dust emission. These data are consistent with dust characterized by a single-component, modified black-body model across the and frequencies and for both polarization parity modes.  The dependence on angular scale is consistent with an
approximate power-law dependence on multipole, ℓ.  The combined and data over this restricted region prefer a dust emissivity spectral index β_d = 1.45±0.07, somewhat lower than the all-sky 2018
value <cit.> of β_d = 1.53 ± 0.02 with T_d = 19.6K.
The foreground complexities considered in this section are found to contribute to the uncertainty on the tensor-to-scalar ratio at the level of σ_r∼ 0.012.











§ LINE-OF-SIGHT DECORRELATION OF DUST






In addition to the search for spatial variation in Section <ref>, we use data to perform a search for variation in the polarization angle of dust emission with frequency.
Such a rotation with frequency may indicate the presence of multiple dust clouds along a line of sight with different SEDs and magnetic alignments.
For example, <cit.> used HI line emission maps to identify regions of the sky likely to contain magnetically-misaligned clouds; they compute the difference in polarized dust angle for these different populations to demonstrate that LOS frequency decorrelation is detectable within the Planck dataset, largely in the Northern Galactic cap. An analysis by the BICEP team <cit.> showed no evidence of dust decorrelation within their observation region.

We perform this search by constructing a set of dust templates as in Section <ref>, each derived from a single or map.
In this case, we opted to subtract CMB signal using the SMICA-derived CMB template rather than the  100 GHz map, due to its lower overall noise level. 
From each template, we compute a map of polarized dust angle using the unbiased estimator <cit.>

    ψ = 1/2arctan(U,Q).

We construct a likelihood estimator to evaluate the consistency of the measured dust angle ψ at different frequencies.

In order to construct our likelihood, we approximate the uncertainty on ψ as Gaussian.
We estimate this uncertainty with standard error propagation:

    σ_ψ^2 =  1/4Q^2 σ_U^2 + U^2 σ_Q^2/(Q^2 + U^2)^2
     + Q^2U^2/2(Q^2 + U^2)^4[ σ_Q^4 + σ_U^4 + 1/2(Q^2 - U^2)^2/Q^2U^2σ_Q^2 σ_U^2 ]

where the two grouped terms represent the first- and second-order errors respectively.
We expect these uncertainties to be underestimated when the signal-to-noise is low (σ_U^2 + σ_Q^2 > U^2 + Q^2). However, the Gaussian approximation breaks down as higher-order terms become more significant. To address this, we apply a mask to reject map pixels for which the second-order errors on polarized dust angle exceed the first. This criterion rejects 20-30% of the sky area covered by the instrument.
This thresholding serves two main purposes: (1) to exclude pixels whose uncertainties are challenging to model accurately, and (2) to remove pixels with especially large uncertainties, for which the width of the distribution would wrap around the range of the arctangent function, yielding non-Gaussian behavior. While the statistical properties of ψ are complex and cannot be strictly Gaussian, the Gaussian distribution serves as a reasonable approximation after this thresholding.

From a given pair of dust maps X and Y we calculate a pair of dust angles (ψ^X_i,ψ^Y_i) for each map pixel i.
To this we fit a Gaussian mixture model <cit.> in which each pixel can belong to one of two populations: a base population, for which the dust angle is consistent between our pair of maps up to some constant angular offset Δψ; and a second population of outlier pixels, for which the polarization angle is inconsistent between the two frequencies. 
The first population probes a possible systematic offset in polarization angle between maps at different frequencies; this is expected to be consistent with zero, and unlikely to arise from dust decorrelation given the relatively large sky area covered by .
The second population is our probe of decorrelated dust, in the form of isolated, dense cloud structures with arbitrary polarization angles relative to our line of sight.





Our likelihood takes the form

    ℒ({q_i}, Δψ, p_d, V_d) = 
    ∏_i [ 1/√(2 πΣ_i)exp( -Δ_i^2/2 Σ_i) ]^1 -q_i×
    [1/√(2 π (Σ_i + V_d))exp( -(ψ^X_i)^2 + (ψ^Y_i)^2/2 (Σ_i + V_d)) ]^q_i,

where

    Δ_i    = min_δ_π{1/√(2)( (ψ^Y_i + δ_π) - (ψ^X_i + Δψ) )  }
    Σ_i    = 1/2(σ_X^2 +σ_Y^2) 
    
      q_i    ∼BetaBinomial(α = p_d, β = 1 - p_d)

The boolean q_i assigns each pixel i to one of the two populations.
The generative model of the base population (q_i=0) is taken to be a two-dimensional Gaussian, with an orthogonal distance Δ_i between the two map angles for each pixel (Equation <ref>) and variance Σ_i (the projected variance of the uncertainty of the pixel onto a line of slope of unity). Note that, because the domain of angles lies in the range [-π/2, π/2], differences in ψ by any multiple of π are inconsequential.
We account for this by minimizing over a nuisance parameter δ_π∈{ -π, 0, π} in Equation <ref>.

The second population represents outlier, or rejected, pixels, which show local differences in angle between the maps. We do not a priori have an expectation for the statistics of this population. We choose to model it as a two-dimensional Gaussian with zero mean and an additional variance V_d (in excess of Σ_i).

Finally, we treat the pixel assignments q_i probabilistically, such that each pixel has a probability p_i of being assigned to the base population.
The probability p_i is itself drawn from a beta distribution with α  = p_d, β = 1- p_d, such that the expectation value E(p_i) = p_d. In this way, p_d can be thought of as a global parameter for the fraction of outlier pixels in the total population. All together, our model has N_pix+3 variables: N_pix population assignments q_i, a systematic offset parameter Δψ, and two parameters describing the variance (V_d) and population fraction (p_d) of the outliers.

We conduct two iterations of this analysis using pairs of maps with the highest signal-to-noise ratios for dust: one comparing the 353 GHz and 150 GHz maps, the other comparing 353 GHz to 217 GHz.
Figure <ref> shows the resulting uncertainties and covariances between the fit parameters (Δψ, p_d, V_d).
Both comparisons find angular offsets Δψ consistent with zero to within 1 σ, with p-values of 0.21 (0.47) for Δψ_150 (Δψ_217).
This suggests that the mean polarized dust angle at 353 GHz is consistent with those at lower frequencies.

Only about 5% of the pixels included in our analysis region are rejected and assigned to the outlier population. These pixels display discrepancies in dust angle between 353 GHz and the corresponding lower frequency.
While these outliers could signify spatially-dependent decorrelation, a map of the locations of these pixels (Figure <ref>) reveals two key observations: (1) the regions lack the connectivity we might expect from a real astrophysical population, and (2) the outliers identified by the two analyses share little overlap. Indeed, only about 0.2% of pixels are identified as deviant in both analyses. This argues against interpreting these pixels as a detection of LOS decorrelation.

We can further perform a hypothesis test to estimate the probability of this analysis yielding 5% decorrelated pixels under the null hypothesis that there is no decorrelated dust at any frequency for the entire observed region.
We construct a dust map from 353-100 GHz difference maps, then scale this to both 150 GHz and 217 GHz using a standard modified black-body dust model.
To this we add realistic noise by drawing from an ensemble of simulations, specifically  noise simulations at 150 GHz and FFP10 realizations at 217 GHz and 353 GHz. From these simulated dust + noise maps, we compute the dust angles and perform an identical analysis.
Comparing our measurement of the fraction of outlier pixels, p_d, to this ensemble of simulations, we obtain p-values of 0.18 and 0.10, respectively, for the 150 GHz and 217 GHz analysis.
We thus conclude that the inferred fraction of decorrelated pixels can be readily attributed to noisy measurements of Stokes Q and U.
Consequently, our analysis finds no evidence for LOS decorrelation in the observing region. The dust angles at 353 GHz largely agree with those at the lower frequencies, and those pixels that show outsized deviations are plausibly caused by noise.





§ COMPARISON TO PYSM DUST MODELS






The SMICA dust maps derived from the combination of and
data (shown in Figure <ref>) can also be
used to test full-sky dust models. The Python Sky Model, commonly
known as PySM, is a Python package designed for generating full-sky
simulations of Galactic microwave foregrounds <cit.>.  It
is often used by the CMB community to test component separation
methods and perform sensitivity forecasts. The current release, PySM3,
contains thirteen distinct dust models that vary the assumptions and data
processing of the underlying data, allowing the user to
generate full-sky dust emission maps at any frequency bandpass of
interest.

To test the agreement between each of these models and our SMICA dust
polarization maps, we first use PySM to generate maps in the observing region using the measured 150 GHz frequency bandpass.
We then compute binned EE and BB spectra for each PySM model
using PolSPICE <cit.> and compare these to analogous spectra generated
from the -SMICA dust maps described in
Section <ref>. 



Figure <ref> shows these spectra for a representative
subset of dust models, which were chosen to span the range of
predicted dust amplitudes in the observing region and
highlight several that have been used in recent experiment forecasts
(e.g. <cit.>).  Briefly, each of
these are described as follows:
 
  
 
  * d0: A single-component MBB with fixed (β_d = 1.54, T_d =20 K) using the 2015 353 GHz polarization map <cit.> as a template.
 
  * d4: A generalization of the MBB model with spatially varying temperature and spectral index based on the 2015 results, scaled using the two-component model from <cit.>.
 
  * d9: A single-component MBB model with fixed (β_d = 1.48,  T_d = 19.6 K) based on the 2018 results <cit.>, which uses GNILC component-separated maps as a template <cit.> with additional small-scale fluctuations.
 
  * d10: A variant of d9 that also includes small-scale fluctuations in β_d and T_d.
 
  * d12: A 3-D model of polarized dust emission with six layers as further described in <cit.>.
 
More information about each of these models can be found in
 the PySM documentation.[
   <https://pysm3.readthedocs.io/en/latest/models.html#dust>]
The remaining PySM3 models are compared in Appendix <ref> and Figure <ref>.

From these figures we can make some qualitative observations about how these models behave
in this region of the sky.
For instance, the similarity in predicted power between
models d0 and d9 suggests that differences in
the model parameters and templates between the 2015 and 2018 releases do not
greatly affect the degree-scale dust predictions in this region of sky.
The scale of variation between d10 and d11 reflects
the impact of stochastic modeling of the small scale features.
On the other
hand, despite their common underlying data, several of these models
predict substantially different dust levels from the others in this region.
In particular, d4 predicts significantly more
degree-scale polarized dust emission than the other models while d12
predicts significantly less, though a detailed exploration of the
origin of these differences is beyond the scope of this
work.

Comparing the PySM spectra with those from our SMICA analysis
is complicated by the fact that both SMICA and the various PySM models draw heavily upon the same underlying maps and noise ensembles.
In our own SMICA analysis, for example, the 353 GHz map dominates the relative weights used to construct the dust component and thus drives the
morphology of the dust map and the shapes of the dust
power spectra.
The amplitudes of those power spectra at
150 GHz, however, are driven primarily by the data.
Since different PySM models employ the maps in different ways and combinations,
we expect complex covariances among all of these spectra that are difficult to quantify reliably.
We can nonetheless qualitatively observe closer correspondence of our on-sky results with some
models (e.g., d0, d9, d10) than others (e.g., d4, d12).








§ CONCLUSION





This paper addresses the impact of choices made in component
separation for the cosmological B-mode analysis, and explores
the complexity of the polarized dust emission in the region.
A quantitative comparison of a template-based and
harmonic-independent-component analysis shows that, at the level
of the statistical noise in the dataset, the modeling errors resulting from
the approximations made in each result in a small, but non-negligible
change in σ_r.

Template-based and harmonic domain methods make
entirely different assumptions about the behavior of
foregrounds. The value of employing multiple component
separation pipelines lies in the capacity to explore the properties of
foregrounds while evaluating the sensitivity of the result to
different analysis choices.  Future B-mode observational programs
will need to carefully consider the impact of these modeling choices
on the likelihood of the tensor-to-scalar ratio.

In Section <ref>, we analyze the spectral energy
distribution of polarized diffuse Galactic dust emission using a
template-based approach, showing variations between two selected
half-regions. We find that the statistical significance is robust to
the choice of the dust template, 217 or 353 GHz, with confidence
levels of 3.2 σ and 3.9 σ for β_d, respectively.
Diffuse dust emission in the region is not accurately modeled
by a single scaling of a polarized dust template, α, over the
entire region.  Future CMB component separation analyses will need to
accommodate the possibility of spatial variation in the SED of diffuse
polarized dust, or quantify the impact of any simplifying
approximations.  If confirmed and interpreted as a spatial variation
in the emissivity index, β_d, this would challenge our
understanding of the polarized interstellar medium.

Within a subset of the region, the joint 217 and
353 GHz template analysis provides evidence of a departure from a
simple modified blackbody spectral energy
distribution at more
than 3 σ, under the assumption that the morphology of dust
emission is consistent at all frequencies within each region. In the
context of a single temperature modified black body considered here, a
break in the spectral index would be required to accommodate this
result.

In Section <ref> it is shown that, averaged over
the full region, the spectral energy distribution of polarized dust is
consistent with that of a modified blackbody over the range of
frequencies probed by and HFI. The data provide no
significant evidence for variation of the emissivity index with
angular scale. The spatial distribution of the emission is found to be
consistent with a power law in angular multipole. Given that the
highest signal-to-noise dust emission drives the result in the full region,
this finding is consistent with the analysis of the sub-regions.

Component separation may be complicated by line-of-sight frequency decorrelation, such as may result from
multiple dust clouds with differing SEDs and orientations in the same
line of sight. In Section <ref>,
we look for evidence of this effect in our region
in locally isolated structures and a global rotation of dust angles
with frequency. We do not find any evidence for this effect.



Looking forward, the addition of 280 GHz data from the second
flight of in 2022-23 will provide constraints on dust foregrounds in this
region of sky that are both complementary to and independent
of , enabling further exploration of the topics considered
here.


















is supported in the U.S. by the National Aeronautics and Space Administration under grants NNX07AL64G, NNX12AE95G, NNX17AC55G, and 80NSSC21K1986 issued through the Science Mission Directorate, and by the National Science Foundation through PLR-1043515.
Logistical support for the Antarctic deployment and operations was provided by the NSF through the U.S. Antarctic Program.
Support in Canada is provided by the Natural Sciences and Engineering Research Council and the Canadian Space Agency.
Support in Norway is provided by the Research Council of Norway.
The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto.
The Flatiron Institute is supported by the Simons Foundation.
JEG acknowledges support from the Swedish Research Council (Reg. no. 2019-03959) and the Swedish National Space Agency (SNSA/Rymdstyrelsen). This work is in part funded by the European Union (ERC, CMBeam, 101040169).
J.M.N. acknowledges support from the Research Corporation for Science Advancement.
K.F. acknowledges support from DOE grant DE-SC0007859 at the University of Michigan.
We also wish to acknowledge the generous support of the David and Lucile Packard Foundation, which has been crucial to the success of the project.

The collaboration is grateful to the British Antarctic Survey, particularly Sam Burrell, for invaluable assistance with data and payload recovery after the 2015 flight.
We thank Brendan Crill and Tom Montroy for significant contributions to 's development.
Some of the results in this paper have been derived using the HEALPix package <cit.>.
The computations described in this paper were performed on the GPC supercomputer at the SciNet HPC Consortium <cit.>.
SciNet is funded by the Canada Foundation for Innovation under the auspices of Compute Canada, the Government of Ontario, Ontario Research Fund - Research Excellence, and the University of Toronto.







§ IMPACT OF COMPACT STRUCTURES ON Α_150 SPATIAL VARIATION





In this Appendix, we assess the impact of the compact structures discussed in Section <ref>—the Orion-Eridanus Superbubble and the Magellanic Stream—on the detection of spatial variation in the value of α_150.

Following <cit.>, we use the full-sky Hα map synthesized by <cit.> as a morphological tracer of the Orion-Eridanus Superbubble in the region.
In the background map of the left panel of Figure <ref>, the presence of the Orion-Eridanus Superbubble is evident in the top left corner of our region.
To quantify its contribution to the determination of α_150, we mask the region it occupies with successively more aggressive thresholds.
Specifically, for the reasons discussed in Section <ref>, we start by smoothing the <cit.> map with a 5^∘-FWHM Gaussian.
We then build three masks rejecting all pixels in which the Hα emission is brighter than 5 R(ayleighs), 2 R, and 1 R, respectively.
The threshold-derived borders of these masks, which leave the region with, respectively, 4.39%,  3.81%, and 3.06% of the sky, are shown in the left panel of Figure <ref>, along with the border of the and subregions (f_sky≈ 2.4%) used in the spatial variation analysis of Section <ref>.
Finally, we compute α_95 and α_150 as in Section <ref> using the 353 GHz template, but for the full region masked by each of the three Hα masks. 
We chose the 353 GHz template for this analysis because it provides the dominant contribution to the SMICA reconstructed dust map, as shown in Figure <ref>.
Results from these subregions are shown in the right panel of Figure <ref>.

The value of α_95 is quite robust to the choice of a Superbubble mask.
In fact, the least aggressive mask results in the same value of α_95 as the most aggressive mask, a value which remains consistent with that determined in the region.
Similarly, although the value of α_150 trends higher as we mask more aggressively—i.e., it trends towards its value in the region—it remains
largely consistent with its value in the region, with the and  I < 1R results in agreement at 1.4 σ.

Therefore, the presence of the Orion-Eridanus Superbubble in the region does not appear to drive the detected variation in the value of α_150.

The Magellanic Stream, which falls mostly outside of the region aside from a few high-velocity clouds (HVCs) along the edge of its bottom right corner, also does not appear to be driving the detected spatial variation in the value of α_150.
Indeed, none of the Hα masks shown in Figure <ref> exclude the HVCs.
Nevertheless, the values of α_150 derived in the full region after application of any of these masks are consistent with each other, but inconsistent with those found in the region, in which the HVCs also remain.
If HVCs were the cause of the detected spatial variation in α_150, we would expect these four results to be in agreement.



§ CONVERSION BETWEEN TEMPLATE Α AND SPECTRAL INDEX Β


Galactic dust emission is modeled as a single MBB with a power-law emissivity,

    I(ν) = τ_0 ( ν/ν_0)^β B(ν, T),

where τ_0 is the optical depth at the reference frequency ν_0.  Throughout the text, we use β_d and T_d to refer to the spectral index and temperature of the dust component as defined in Equation <ref>.  We
calculate an effective observing frequency using the dust SED that takes into
account the bandpass of the observing instrument.  In general,

    ν̅ = ∫ d ν ν I(ν) t(ν) /∫ d ν  I(ν) t(ν) ,

where I(ν) is the SED of the observed source and t(ν) is the spectral
transmission of the detector, properly normalized to its optical efficiency
ϵ. This equation is a generalization of Equation 4 of
<cit.>. There, they express I(ν) for a
power-law like SED: I(ν) ∼(ν/ν_0
)^α_Pl.  Note that we identify the spectral index for
this simple SED as α_Pl to differentiate it from the template
coefficients α_95 and α_150 used throughout the main text.



Table <ref> lists effective frequency band centers for a
variety of SED models. The results presented in
<cit.> assume the α_Pl = 4 SED. However, this
assumption holds true only in the Rayleigh-Jeans limit, and only when β≈ 2.  A more accurate effective frequency can be calculated by
considering the true dust SED.

CMB maps are typically shown in so-called thermodynamic units (μ K_CMB),
which encode the amplitude of fluctuations about the CMB temperature,
T_CMB = 2.725 K. The amplitude of dust emission can be expressed in these units
by normalizing the dust emissivity with a first-order Taylor expansion of the CMB blackbody emission about T = T_CMB:

    Δ T (ν̅^d) = ( . ∂ B/∂ T|_T = T_CMB)^-1 I(ν̅^d)
    = (e^x - 1)^2/x^2 e^x1/2k(c/ν̅^CMB)^2  I(ν̅^d).

Here, for convenience, we have defined
x ≡ hν̅^CMB/(kT_CMB). This equation describes
the anisotropy in the mm-map caused by dust emission. It is important to note
that the effective frequency used for conversion to thermodynamic units
(ν̅^CMB) differs from the effective frequency used for the dust
scaling (ν̅^d). The map-domain scaling relationship from a reference
frequency ν_0 to a lower frequency ν is then expressed as

    A(ν̅^d, ν̅_0^d; β_d, T_d)
      = Δ T(ν̅^d)/Δ T(ν̅_0^d)
      = (e^x - 1)^2/x^2 e^xx_0^2 e^x_0/(e^x_0 - 1)^2(ν̅_0^CMB/ν̅^CMB)^2
      (ν̅^d/ν̅_0^d)^β_dB(ν̅^d, T_d)/B(ν̅_0^d, T_d).


Because we construct dust templates from difference maps, specifically
353-100 GHz or 217-100 GHz, there is a minor subtraction of dust at
100 GHz. Consequently, the scaling of the difference template to a frequency
ν is thus given by

    α(ν̅^d, ν̅_0^d; β_d, T_d)
      = Δ T(ν̅^d)/Δ T(ν̅_0^d) - Δ T (ν̅_100^d)
      = A(ν̅^d, ν̅_0^d; β_d, T_d)/1 - A(ν̅_100^d, ν̅_0^d; β_d, T_d),

where ν̅_100^d = 104.5 GHz according to Table <ref>.
To recover a value of β_d given α,
Equation <ref> must be inverted using numerical methods.



§ COMPARISON TO ALL PYSM3 DUST MODELS



Section <ref> and Figure <ref> present a comparison between PySM3 dust models and the dust component estimated by SMICA.
For simplicity, that comparison used only a representative subset of the available models.
The remaining ones are shown in Figure <ref>.

Due to the fact that both the SMICA results and the PySM3
models incorporate—to varying extents—both the data and
noise simulations, the covariance between them is
non-trivial.  While this makes a quantitative measure of the goodness
of fit between the data and the models difficult to assess, we
reiterate that the overall amplitude of the SMICA result is
driven by the 95 and 150 GHz data, and is therefore largely
independent from the amplitude of the models shown.
The amplitudes of the d9–d11 family of models are in closest agreement with the SMICA amplitudes at 150 GHz.
Most other models predict more power than , except d12, which predicts significantly less.

For convenience,
we provide a brief description of the PySM3 dust models shown
in Figure <ref>.  For a definitive
description we refer the reader to the official documentation[
   <https://pysm3.readthedocs.io/en/latest/models.html#dust>].





  Models d0–d4 This family of models treats thermal dust emission as a single-component modified black body (MBB).  Polarized dust templates are based on the 353 GHz channel, and are scaled to other frequencies using an MBB spectral energy distribution (SED). The polarization templates are smoothed with a Gaussian kernel of 2.6^∘ FWHM, with power at smaller scales added according to the model described in the reference.  For the SED, the models use:[Here we define families of models by their class designation within PySM3.]
  
  
  d0 A spatially fixed spectral index β_d=1.54 and a black body temperature, T_d=20K.
  
  d1 A spatially varying spectral index and temperature based upon those provided in the Commander maps <cit.>, which assume the same spectral index for both intensity and polarization.
  
  d2 (d3) An emissivity index that varies spatially on degree scales, drawn from a Gaussian with β=1.59 ± 0.2 (0.3).
  
  d4 The two component (one hot, one cold) dust model of <cit.>.
  

  Models d5, d7, d8 Implementations of the model described in <cit.>, using:
  
  
  d5 The baseline model of that reference.
  
  d7 A model including iron inclusions in the grain composition.
  
  d8 A simplified version of d7 in which the interstellar radiation field strength is uniform.
  

  Model d6 A model that averages over spatially varying dust spectral indices, both unresolved and along the line of sight, using the frequency covariance of <cit.> to calculate the frequency dependence.

  Models d9–d11 A family of models using a single component MBB with templates based on the GNILC analysis <cit.>, using the 353 GHz color correction described in <cit.>. The frequency modeling uses:
  
  
  d9 A fixed spectral index β_d=1.48 and temperature T_d=19.6K.
  
  d10 Additional small scale fluctuations in the template, as well as the dust spectral index and temperature.
  
  d11 Identical to d10, but using a different set of seeds and maximum multipole to generate the small scale features.
  

  Model d12 A three-dimensional model of Galactic dust emission based on <cit.>.







apj




 Paper Title: Apocalypse When? No Certainty of a Milky Way -- Andromeda Collision.




Apocalypse When? No Certainty of a Milky Way - Andromeda Collision]Apocalypse When? No Certainty of a Milky Way - Andromeda Collision

[1,2]Till Sawalatill.sawala@helsinki.fi
1,3]Jehanne Delhomelle
2]Alis J. Deason
2]Carlos S. Frenk
1]Peter H. Johansson
1]Atte Keitaanranta
1]Alexander Rawlings
1]Ruby Wright

[1]Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2, Helsinki, FI-00014, Finland

[2]Institute for Computational Cosmology, Durham University, South Road, Durham, DH1 3LE, United Kingdom

[3]Université Toulouse III - Paul Sabatier, 118 Route de Narbonne, Toulouse, F-31062, France

It is commonly believed that our own Milky Way is on a collision course with the neighbouring Andromeda galaxy. As a result of their merger, predicted in around five billion years, the two large spiral galaxies that define the present Local Group would form a new elliptical galaxy. Here we consider the latest and most accurate observations by the Gaia and Hubble space telescopes, along with recent consensus mass estimates to derive possible future scenarios and identify the major sources of uncertainty in the evolution of the Local Group over the next 10 billion years. We find that the next most massive Local Group member galaxies – namely, M33 and the Large Magellanic Cloud – distinctly and radically affect the Milky Way - Andromeda orbit. While including M33 increases the merger probability, the orbit of the Large Magellanic Cloud runs perpendicular to the Milky Way - Andromeda orbit and makes their merger less likely. In the full system, we find that uncertainties in the present positions, motions, and masses of all galaxies leave room for drastically different outcomes, and a probability of close to 50% that there is no Milky Way - Andromeda merger during the next 10 billion years.

[
    [
    Received July 31, 2024; accepted XX XX, XXXX
================================================

The Local Group (LG) contains two large spiral galaxies, our own Milky Way (MW) and the Andromeda galaxy (hereafter M31), along with approximately 100 known smaller galaxies <cit.>. In addition, it likely hosts additional galaxies yet to be discovered <cit.> and, according to the standard cosmological model, a vast number of completely dark substructures <cit.>.

The negative radial velocity of M31 towards the MW has been known for over a century <cit.>, even before its distance was first accurately measured  <cit.>. However, while indirect methods had since been used to constrain the transverse components of M31's velocity vector <cit.>, direct measurements of the minute proper motions were only achieved much more recently, with the Hubble Space Telescope (HST) <cit.>.

The first numerical studies <cit.> of a possible MW-M31 merger predate even the early estimates of the transverse velocity. The finding that the MW-M31 motion is close to radial immediately led to the prediction of a likely future collision and merger <cit.>. This scenario has since become the prevalent narrative <cit.> and textbook knowledge <cit.>. 



 §.§ Predicting the future of the Local Group

The MW and M31 both contain remnants of past mergers and interactions with other galaxies <cit.>. Predicting future mergers requires knowledge about the present coordinates, velocities, and masses of the systems partaking in the interaction. In addition to the gravitational force between galaxies, dynamical friction is the dominant process in the lead-up to galactic mergers. It describes a transfer of orbital kinetic energy to internal energy of the objects involved, and consequently leads to the decay of galactic orbits.

In this study, we parameterise the density profile of each halo as isotropic Navarro-Frenk-White profiles <cit.>, use standard assumptions about their concentrations and velocity dispersion profiles <cit.> and calculate dynamical friction using an analytic formalism (see Methods for a detailed description). While non-gravitational processes, such as gas drag and star formation leading to increased central densities, etc., also shape the final stages of a merger, the phase of the orbit that defines the occurrence of mergers is largely determined by gravity, which in turn, is dominated by the dark matter component in the standard cosmological model <cit.>. 

The simplest model for the MW-M31 orbit, as considered by <cit.>, contains only the two main galaxies. Due to the planar symmetry, only five parameters are required: the two masses, along with the initial separation and the two velocity components. In general, an orbit with N>2 galaxies requires specifying 3 × (N-1) coordinates and 3 × (N-1) velocity components, along with the N masses. More recently,  <cit.> and <cit.> have considered three-body orbits including M33, the third most massive LG galaxy; and <cit.> also considered the Large Magellanic Cloud (LMC), the fourth most massive LG galaxy. They still conclude that a merger is certain. We will consider two-body, three-body, and four-body systems to study the future evolution of the LG and reveal the distinct effects of the M33 and LMC on the MW-M31 orbit.



 §.§ A fiducial model based on the most accurate values available

In the context of the Local Group, it is important to note that apart from the sky positions, all parameters including distance moduli, line-of-sight velocities, proper motions, masses, and concentrations carry non-negligible uncertainties. We will use Monte Carlo (NC) sampling of all values to investigate how these observational errors propagate to uncertainties on the future evolution, and in particular, the probability of a merger between the MW and M31.

Our fiducial LG model contains the four most massive LG members: the MW, M31, M33, and the LMC, based on the latest and most accurate available data. The mass of the MW has recently been extensively studied using Gaia data, with a consensus emerging of a total mass close to 10^12. We adopt a value of M_200 = 1 ± 0.2 × 10^12 (excluding the mass of the LMC, which we treat separately). For all other galaxies, there is significantly more uncertainty. For M31, we adopt M_200 = 1.3 ± 0.4 × 10^12. For M33, we assume M_200 = 3 ± 1 × 10^11 and for the LMC we assume M_200 = 1.5 ± 0.5 × 10^11. See Methods for a review of mass estimates.

The line-of-sight velocities are well-known and we adopt the values given by <cit.>. For the distance moduli, μ, we choose the most accurate and precise recent values in the literature: <cit.> for the LMC and <cit.> for M33. For M31, we use recent HST Cepheid results <cit.>.

For the proper motions, where we use the notation,  μ_δ for the proper motion in declination and μ_α^* = μ_αδ for the proper motion in right ascension, we use the Hubble Space Telescope (HST) proper motions of <cit.> for the LMC, and the combined HST and Gaia DR2 proper motions of <cit.> for M33. Our fiducial M31 proper motions are the most precise values in the literature based on Gaia DR3 astrometry <cit.>. However, we find very similar results using the combined Gaia DR2 and HST proper motions of <cit.>, as shown in Extended Data Figure <ref>. All assumed values are listed in Extended Table <ref>. We assume Galactocentric coordinates of RA_GC= 266.41^∘, Dec_GC=-28.94^∘ <cit.>, d_GC=8.122 kpc <cit.>, and a velocity of (12.9, 245.6, 7.78) kms^-1 with respect to the Sun <cit.>.

To account for the fact that the true probability distributions may not be Gaussian and to exclude possible effects caused by unrealistic or even unphysical outliers, we truncate all distributions at ± 2σ. This truncation increases the probability of a MW-M31 merger by only ∼ 10% (see Extended Data Figure <ref>). For the fiducial model, we use 50000 MC samples while for all other variants, we use 2500 samples, ensuring that all statistical errors on the merger rate are below 1%.



 §.§ The evolution of the MW-M31-M33-LMC system

The Monte Carlo initial conditions are integrated numerically using a symplectic direct scheme (see Methods for details). Figure <ref> shows 100 realisations each of the MW-M31 orbit, in the two-body MW-M31, and the four-body MW-M31-M33-LMC systems. Figure <ref> shows the evolution of the distances between the MW and M31 for the same sets of orbits, and additionally, for the MW-M31-M33 and MW-M31-LMC three-body systems.

The MW-M31 two-body orbit evolves in a plane and leads to a merger in slightly less than half of cases, the majority of which occur during the second pericentre. The addition of M33 increases the merger probability to ∼ 2/3, with a similar median merger time. However, the addition of the LMC has the opposite effect: the pure MW-M31-LMC system experiences a merger in only slightly more than 1/3 of cases and the merger probability of the full M31-MW-M33-LMC system is just over 50%.

For each system, we also show the single orbit obtained by adopting the most likely value of each observable, either in the fiducial model or assuming HST+Gaia DR2 proper motions for M31 <cit.>. In the case of the MW-M31-M33 system, which is the one considered by <cit.> and <cit.>,
we reproduce similar orbits and times of first pericentre and merger (the differences are likely due to differences in the other model parameters). However, despite using newer and more precise measurements, when we perform a Monte Carlo analysis, we find considerable uncertainty in the outcome that was not previously reported. In particular, in the full MW-M31-M33-LMC system, a merger between the MW and M31 occurs within the next 10 Gyr in only approximately half of the cases. This is in stark contrast to all previous results that had only considered the most likely values without accounting for the numerous and significant uncertainties.

In Figure <ref>, we show the probability distributions of the merger time and of the minimum distance between the MW and M31, as well as the “survival" rate of the MW over time, i.e. the probability that no merger with M31 has occurred. In the fiducial model, we consider a merger to occur when the distance between any two galaxies is below 20 kpc, but we find that our results are not sensitive to this particular choice (see Methods for details). Due to the effect of dynamical friction, we find that there are two distinct possibilities for the eventual fate of the MW and M31: orbits that come within less than ∼ 200 kpc eventually merge, which would likely lead to the formation of an intermediate-mass elliptical galaxy <cit.>. For systems that merge, we find a median time of 7.6 Gyr in the fiducial model, or 8.0 Gyr adopting a 10 kpc threshold. By contrast, orbits with larger pericentres do not decay due to dynamical friction. In this case, the MW and M31 continue to evolve in isolation. Based on the best current data, both outcomes are almost equally likely.



 §.§ The roles of M33 and the LMC

The distinct effects of each satellite on the MW-M31 orbit are illustrated in Figures <ref> and <ref>, where we compare the trajectories of the MW and M31 in two-body systems and in systems that also include either M33 or the LMC. Both satellites provide some additional acceleration in the radial direction of the MW-M31 orbit. However, importantly, both satellites also affect the motion of their respective hosts. Including M33 in the calculation decreases the transverse velocity of M31 with respect to the MW. By contrast, as already pointed out in <cit.>, at its current orbital phase, the recoil due to the LMC results in a lower transverse velocity measured between the MW and M31. During its short orbital period of ∼ 1.5 Gyr, the LMC will accelerate the MW to a higher transverse velocity. In addition, the inclusion of M33 largely provides momentum in the original MW-M31 plane, while the inclusion of the LMC also provides significant momentum perpendicular to the MW-M31 plane. In our analysis, the LMC is certain to merge with the MW, and M33 is highly likely (∼ 86%) to merge with M31 before any possible MW-M31 merger. The net effect of adding M33 to the two-body system is to increase the merger probability, while the net effect of adding the LMC is to decrease it.



 §.§ Sources of uncertainty

In Figure <ref>, we show how the merger probability depends on the different observables. In each panel, we show the dependence on two variables in the ± 2 σ ranges, with the remaining variables Monte Carlo sampled. The effects of the concentration parameters are shown in Extended Data Figure <ref>. We find that all masses, the proper motions of M31 and M33, and the distance moduli of M31 and M33, significantly impact the probability of a merger. The merger probability is positively correlated with the masses of the MW, M31, and M33 and negatively correlated with the mass of the LMC. The impact of the satellite masses is more pronounced for lower combined masses of the MW and M31. The ± 2 σ ranges of the M31 proper motions include values that imply a merger probability above 90%, but also values that imply a merger probability close to zero. The most likely proper motions (assuming no errors) only lead to a merger in ∼ 2/3 of cases.

Future more precise proper motion measurements may significantly change the expected outcome, although they could make the merger either more or less likely. However, if they fall within ± 1 σ of the current most likely values, even precise M31 proper motion measurements alone will not suffice to determine the outcome.

Even the comparatively high precision of the line-of-sight velocity and distance moduli for M33 and M31 contribute significant uncertainty, with the probability of a merger varying between 40% and over 60% for different combinations in the ± 2 σ ranges around the most likely values.

Given the considerable measurement errors, it is worth noting that cosmological simulations result in a present-day MW-M31 transverse velocity prior of v_t = 75_-40^+65 kms^-1 <cit.>. There is thus no reason to assume that the transverse velocity measured using Gaia DR3 (most likely value of v_t=76 kms^-1) is overestimated or to expect that more precise measurements will result in a lower value. It is also worth noting that a perfectly radial present-day M31-MW motion (μ_α^* = 38.03 mas yr^-1, μ_δ = -21.37 mas yr^-1) is neither compatible with current observations nor does it result in the highest merger probability in the four-body system.



 §.§ Summary

Even using the latest and most precise available observational data, the future evolution of the Local Group is uncertain. Intriguingly, we find an almost equal probability for the widely publicised merger scenario (albeit with a later median time to merger) and one where the MW and M31 survive unscathed. We reach this conclusion by including the LMC and importantly, for the first time, considering the relevant uncertainties in the observables.

Our results are not sensitive to the necessary choices of gravitational softening (see Extended Data Figure <ref>), merger threshold (Extended Data Figure <ref>), or dynamical friction scheme (Extended Data Figure <ref>)).

While we have shown that considerable uncertainty results from the proper motion measurements of M31, we also find that a more accurate prediction requires more precise measurements of the positions, motions, and masses of all participating galaxies. The dependence of the evolution of the MW-M31 system on the treatment of other LG galaxies points to further uncertainties. Cosmological simulations suggest that ∼ 25% of the bound mass of the LG is outside the two main haloes <cit.>. The next most massive individual Local Group galaxies that could impact the MW-M31 orbit are M32 (an M31 satellite and possible merger remnant <cit.>) and the Small Magellanic Cloud (SMC), a satellite of the MW. Both are at least a factor of five less massive than the LMC, and we find that including the SMC, for which proper motion measurements are available, does not significantly change the MW-M31 merger probability (see Extended Data Figure <ref>). However, the unaccounted cumulative effects of additional substructures as well as that of the cosmic environment introduce further uncertainty, particularly towards the far future. An accurate prediction of the evolution even from perfectly precise observations may require cosmological constrained simulations <cit.> that can account for these effects. Meanwhile, the assumptions and simplifications we have made here are likely conservative regarding our central claim, that there is considerable uncertainty about the MW-M31 merger.

Upcoming Gaia data releases will improve the proper motion constraints and mass models are continuously refined. However, it is clear that Galactic eschatology is still in its infancy and significant work is required before the eventual fate of the Local Group can be predicted with any certainty. As it stands, proclamations of the impending demise of our Galaxy appear greatly exaggerated.


 




































 



§ METHODS

Our results are based on numerically integrated initial conditions, which are in turn based on Monte Carlo samples of the observational data. We describe here the generation of the Monte Carlo samples, the numerical integration, and our treatment of dynamical friction and galaxy mergers. We also demonstrate the robustness of our findings to the particular choices made and show that our fiducial model and the results are conservative in predicting an uncertain future and relatively low merger probability. We also present results when the SMC is included in the analysis, in addition to the four main galaxies. To facilitate the reproduction of our results and allow future work to easily incorporate new observational data, our analysis code is flexible and public.



 §.§ Monte Carlo Samples

Our fiducial model consists of the Milky Way and three additional galaxies, M31, M33, and the LMC. We assume that the sky coordinates (RA & Dec) for the centres of M31, M33, and the LMC are known. We furthermore assume that the position of the Galactic centre with respect to the sun is fixed at (Ra, Dec) = (266.4051, -28.936175) <cit.>, d=8.122 kpc <cit.>, and that the galactocentric velocity of the Sun is (12.9, 245.6, 7.78) km^-1 <cit.>.

We create Monte Carlo samples for the remaining 20 variables, the four halo masses, M_MW, M_M31, M_M33, M_LMC, the four halo concentration parameters, the three distance moduli, μ, the three sets of proper motions, μ_δ and μ_α^*, and three line-of-sight velocities, v_los. Sampling directly in the space of the observables rather than sampling in a space of derived variables such as Cartesian coordinates or velocities minimises the effect of possible correlations.

To allow reproducibility and identification of individual orbits across our figures when changing parameters of the model such as the set of galaxies included and numerical parameters such as the gravitational softening length and merger threshold, we re-initialise the pseudo-random number generator with the same values for every new set of Monte Carlo samples. While we generally only show the first 100 orbits on each plot, all quoted probabilities are computed from at least 2500 samples each, so that statistical errors are less than 1%.

The most likely values of each variable together with the ± 1 σ uncertainty are either taken directly from single sources in the literature or in the case of masses and concentrations (see below), estimated by us based on multiple sources. In creating our Monte Carlo samples, we assume that each variable follows a Gaussian probability distribution. However, in our fiducial model, we truncate all distributions at ± 2σ, corresponding to the central ∼95% of values. While Gaussian distributions are a natural assumption for measurement errors, this is not always explicitly stated, and it may not reflect the true probability distribution, especially at some distance from the most likely values. Indeed, for some variables, untruncated distributions extend to unphysical values with finite probabilities. A truncation at 2 σ also ensures that all variables in the samples are physical.

Given that our central claim is uncertainty in the future evolution of the LG, truncating the probability distributions of the observables is a conservative assumption that leads to lower uncertainty about the outcome. However, as shown in Extended Data Figure <ref>, our results are not sensitive to the truncation at ± 2 σ, and the merger probability is only slightly lower when the distributions are not truncated. On the other hand, even a truncation at only ± 1 σ leaves approximately 1/4 of orbits that do not merge within 10 Gyr.



 §.§ Numerical Integration

The orbits are integrated using a symplectic leapfrog algorithm in the centre-of-mass frame, with a step size of 1 Myr, approximately the time it takes for a galaxy moving at 100 kms^-1 to travel 0.1 kpc — 1/200^ th of our merger threshold (see below). Our results are not affected by the finite time step.

To account for the fact that the haloes are extended objects, the gravitational force between the haloes is softened with a constant softening length of 20 kpc, similar to the scale radius of an NFW halo in the mass range we consider here. We also consider different choices for the softening and show in Extended Data Figure <ref> results with softening lengths of 10, 20, or 30 kpc, respectively. A softening length that is too small can lead to unphysical hard scattering events during close encounters while a softening length that is too large artificially reduces the gravitational force. In the context of the Local Group, both of these effects could reduce the merger probability. However, we find no strong dependence of the merger probability on the softening length, with our adopted fiducial value of 20 kpc resulting in the highest merger probability.



 §.§ Dynamical Friction

To estimate the effect of dynamical friction, we use a modified Chandrasekhar formula, similar to that used in <cit.>. The classic Chandrasekhar formula assumes a point mass orbiting in the potential of a much more massive host halo, which is in turn composed of much less massive particles. This approach has been expanded to account for extended satellites White-1976, and we use the following expression to calculate the acceleration of a satellite due to dynamical friction <cit.> once inside r_200 of the host halo:


    d v/ dt = -  4 π G^2 M ρlnΛ/v^2[  erf(X) - 2X/√(π) e^-X^2] v/v,


where G is the gravitational constant, M is the mass of the satellite, v is the velocity of the satellite relative to the host, ρ is the density of the host at the position of the satellite, X = v / (2 σ) is the ratio between the orbital speed of the satellite and the 1D-velocity dispersion, σ, of the host at the location of the satellite, and Λ is the Coulomb factor expressed as r / ϵ, with ϵ a scale length that depends on the density of the satellite. To determine ϵ, we adopt an empirical expression derived from N-body simulations Jethwa-2016:


    ϵ =
    
    2.2  r_s - 14  kpc   if  r_s ≥ 8 kpc
    
    0.45  r_s     if  r_s < 8 kpc

where r_s is the scale radius of the satellite's NFW halo. Finally, to approximate the velocity dispersion of the host at the location of the satellite, we use the expression derived in <cit.> for NFW haloes.

Standard dynamical friction schemes assume a clear hierarchy between the (much more massive) host and the (much less massive) satellite. According to the assumptions underlying Equation <ref>, the satellite and host enter the calculation in clearly defined and distinct roles, with the dynamical friction force applied only on the satellite, while the host remains unaffected.

However, in the Local Group context where galaxies and haloes of similar mass are interacting, this introduces an inconsistency. In particular, in the (relatively likely) scenario that M31 has a similar mass to the MW, the roles of the satellite and host are unclear, but their assignment changes the result of the calculation. For example, if the MW is considered the satellite, it would be accelerated by its interaction with M31 while M31 would remain unaffected, and only the motion of the MW with respect to the other galaxies would be affected while that of M31 would remain unchanged. If M31 is considered the satellite, the roles would be reversed. Accelerating only one galaxy also violates momentum conservation.

To make the calculation more symmetrical, in our dynamical friction calculation we distribute the dynamical friction force proportionally between the satellite (s) and host (h), conserving the total momentum:


    d v_s/ dt   =  a_DF M_h/M_s + M_h, 
     d v_h/ dt   = -  a_DF M_s/M_s + M_h,

where v_s and M_s are the velocity and mass of the satellite, v_h and M_h are the mass and velocity of the host, and a_DF is the acceleration computed using Equation <ref>. In the limit that the satellite is much less massive than the host, the standard hierarchical scheme is recovered and only the satellite is accelerated, while in the limit that both galaxies have equal mass, both receive equal and opposite accelerations.

A small inconsistency remains in that even when the differences in mass are small, we still assign the more massive galaxy as the host and the less massive galaxy as the satellite when calculating the magnitude of the dynamical friction, where the velocity dispersion of the host, σ, but not that of the satellite, and through the Coulomb factor, the scale radius of the satellite, but not that of the host, are considered. In our spherical halo models, both the velocity dispersion and Coulomb factor depend only on the assumed masses and concentrations, and as we discuss below, the concentration of the satellite has a greater impact on the dynamical friction force. For two halos of significantly different masses, the assignment of host and satellite is clear. For an individual case of two haloes of nearly identical masses but different (randomly assigned) concentrations, the choice of calculating the dynamical friction force by treating either halo as the satellite seems arbitrary. However, for a large number of samples of nearly equal-mass interactions with randomly assigned concentrations, the dynamical friction calculations are not biased. We also assume identical distributions of concentration parameters for all galaxies.

In Extended Data Figure <ref>, we compare the orbits of the fiducial system using no dynamical friction, “hierarchical” dynamical friction (only from the more massive host to the less massive satellite), and our default “proportional” dynamical friction. It is clear that without dynamical friction a MW-M31 merger is highly unlikely. In fact, the finite merger rate without dynamical friction depends strongly on our default impact parameter threshold of 20 kpc, with a lower threshold, the merger rate can become arbitrarily small. On the other hand, when dynamical friction is included, the evolution of each orbit is quite similar in the “hierarchical” and “proportional” schemes, and the merger rate is not significantly affected by the exact choice of scheme.

It is worth noting that our semi-analytical approach to dynamical friction is still quite simplistic, and while the average behaviour of N-body simulations has been used to calibrate parameters, numerical simulations also show that individual systems with non-zero internal angular momenta and substructures can have different merger times than predicted by these simple equations Boylan-Kolchin-2008. A precise prediction of the MW-M31 orbit will likely require full N-body simulations. On the other hand, we show that even a simple dynamical model that assumes no spin, no triaxiality, and no substructure results in considerable uncertainty in the future evolution of the MW-M31 system.



 §.§ Mergers

Below a certain distance, the interactions of the gas and stellar components become significant, and our simple approach is no longer appropriate for predicting the remaining orbital evolution. Our aim here is not to predict the precise time of the merger (in fact, we argue that such a prediction is futile based on the current data), so we simply assume that any system that passes below a threshold distance will eventually merge, and identify this time as a lower limit for the time of the merger.

In our fiducial model, we have adopted a value of 20 kpc as the merger threshold for all galaxy interactions. When such a merger occurs between two galaxies, we combine the masses and momenta of the two galaxies at their common centre of mass and continue the integration. The concentration parameter is set to that of the most massive galaxy.

In Extended Data Figure <ref>, we compare the sensitivity of our results to adopting merger thresholds of 10, 20, and 30 kpc. With a threshold of 10 kpc, we find a slightly reduced MW-M31 merger probability. However, raising the threshold from 20 to 30 kpc has no significant impact on our results. This confirms our assertion in the main section: due to the effect of dynamical friction, orbits either inspiral and eventually merge, or do not come close enough for dynamical friction to become effective and hence do not merge.

The small fraction of MW-M31 orbits that merge with a threshold of 20 kpc, but do not when the threshold is set to 10 kpc, approach with a small impact parameter and high velocity. This reduces the effect of dynamical friction and also allows them to escape to a large apocentre. While this may be a possible scenario for the Local Group, our methods are not adequate for studying such close interactions between galaxies. To be conservative in our prediction of a low merger probability, by setting the merger threshold at 20 kpc we assume that these orbits also merge, and with an even larger threshold of 30 kpc, we would still predict a similar merger rate.

For comparison, the best-studied merger of two galaxies with similar stellar masses to the MW and M31 are the Antenna galaxies. Their past evolution is reproduced with an orbit with pericentre ∼ 10 kpc Karl-2010, predicted to lead to coalescence ∼ 1.3 Gyr later Lahen-2018.



 §.§ Fate of the LMC and M33

While our main focus here is on the future evolution of the MW-M31 orbit, we naturally also make predictions for the evolution of the LMC and M33.

In the fiducial model, we find that the LMC is certain to merge with the MW before any eventual MW-M31 merger. With a merger threshold of 20 kpc, we find a median time of the LMC-MW merger of 1.3 Gyr, while with a threshold of 10 kpc, we find a median time of 1.9 Gyr.

For M33, with a merger threshold of 20 kpc, we find an ∼ 86% chance of a merger with M31 and a median time of 3.3 Gyr, while with a merger threshold of 10 kpc, we find an ∼ 83 % probability of a merger with M31 and a median time of 3.9 Gyr. In both cases, we also find a small probability of ∼ 1-2 % for a merger of M33 with the MW-M31 remnant after a MW-M31 merger within the next 10 Gyr.

It must be noted that our simulations are not designed to study these mergers in detail and ignore, for example, the impact of the disk of the Milky Way. Nevertheless, they broadly agree with the results of <cit.>, who have previously studied the LMC-MW encounter in the presence of M31.



 §.§ Additional galaxies

The next most massive LG member galaxy for which proper motion data <cit.> is available is the Small Magellanic Cloud (SMC), a satellite galaxy of the MW that has likely been accreted together with the LMC. We repeat our analysis including the SMC as a fifth system, and show results in Extended Data Figure <ref>. Adding the SMC, whose mass is approximately 10% of that of the LMC, has no significant effect on the merger rate. The SMC properties used are listed along with those of the other galaxies in Extended Data Table <ref>.



 §.§ DR2 + HST proper motions

Due to the fact that the M31 proper motions have the largest impact on the probability of a MW-M31 orbit, and for easier comparison with earlier works, particularly <cit.>, we also repeat our analysis adopting the HST+Gaia DR2 M31 proper motions of <cit.>. In Extended Data Figure <ref>, we show the corresponding evolution of the MW-M31 distance in the same two-body MW-M31, three-body MW-M31-M33 and MW-M31-LMC, and four-body MW-M31-M33-LMC systems. The results can be directly compared to Figure <ref> which shows the same quantities in our fiducial model that used Gaia DR3 proper motions. In each case, the merger probability is slightly lower using the HST+Gaia DR2 proper motions: 40% instead of 44% for the MW-M31 system,  56% instead of 63% for the MW-M31-M33 system,  34% instead of 37% for the MW-M31-LMC system, and 48% instead of 54% for the full MW-M31-M33-LMC system. However, both sources of proper motions predict similar distributions of outcomes and a similar uncertainty about the MW-M31 merger. This difference can be attributed to the slightly lower precision of the HST+Gaia DR2 proper motions.



 §.§ Sources for the masses

As discussed in Figure <ref>, the assumed masses of all four galaxies and their associated uncertainties have a strong impact on the likely evolution of the MW-M31 orbit in our fiducial model. Here, we review the mass measurements and show the implications for some alternative scenarios.

MW   The total mass of the MW has been extensively studied with different methods and tracers, and the accurate astrometry of the Gaia space telescope has brought a flurry of recent measurements. Estimates for the total MW mass based on Gaia DR2 or DR3 satellite dynamics include 1.17_-0.15^+0.21× 10^12 Callingham-2019, 1.51^0.45_-0.40× 10^12 Fritz-2020, 1.23^+0.21_-0.18× 10^12 Li-2020, and 1.1_-0.1^+0.1× 10^12 Rodriguez-Wimberly-2022 (where the latter two works also use a simulation based prior). 

Estimates using rotation curves include 1.08^+0.20_-0.14× 10^12 based on Gaia DR2 Cautun-2020, 0.89^+0.1_-0.08× 10^12 using stars in the galkin catalogue Karukes-2020, 0.822 ± 0.052 × 10^12 using classical Cepheids Ablimit-2020, and 1.08^+0.12_-0.11× 10^12 from the H3 survey and Gaia DR3 Shen-2022.  Other recent measurements include  1.54_-0.44^+0.75× 10^12 using combined Gaia DR2 and HST kinematics of globular clusters Watkins-2019, 1.16 ± 0.24 × 10^12 (including the mass of the LMC) using the kinematics of halo stars, 1.26^+0.40_-0.22× 10^12 from high velocity RR-Lyrae stars Prudil-2022, and 1.19_-0.32^+0.49× 10^12 from a Bayesian estimate using dwarf galaxy kinematics from multiple sources Slizewski-2022. 

<cit.> contains an overview of recent measurements and Wang-2020 includes a comprehensive review of earlier results.

Analysis from several different tracers, and particularly from the latest studies using the latest Gaia observations consistently point towards a Milky Way total mass that is close to 10^12. Both the simple mean and median of the above measurements are 1.16 × 10^12, but it is worth noting that most methods of measuring the total mass of the MW include that of the LMC (at a galactocentric distance of ∼ 50 kpc, well within r_ vir or r_200), which is not always made explicit. When the LMC is excluded, the mass of the MW is reduced by ∼ 0.15 × 10^12 (see our discussion of the LMC mass below). We adopt M_MW = 1.0 ± 0.2 × 10^12 excluding the LMC in our fiducial model, reflecting the consensus of recent observations.

M31 While M31 does not enjoy the benefit of accurate Gaia proper motions, there nevertheless exist a number of studies estimating its mass. Most recently, Zhang-2024 measured a total mass of 1.14^+0.51_-0.35× 10^12 using rotation curves based on LAMOST data release 9 and DESI. Watkins-2010 measured  1.4 ± 0.4 × 10^12 derived from satellite kinematics, Tollerud-2012 derived 1.2_-0.7^+0.9× 10^12 from kinematics of M31 dwarf spheroidals, and Tamm-2012 measured a total mass of 1.05^+0.15_-0.15× 10^12 from SED fitting together with the rotation curve and the kinematics of outer globular clusters and satellite galaxies. Chemin-2009 measured 1.0 × 10^12 from the HI rotation curve, Fardal-2013 measured 2.0^+0.4_-0.3× 10^12 from kinematics of the Giant southern stream. Veljanoski-2013 measured 1.35_-0.15^+0.15× 10^12 and Veljanoski-2014 found 1.4_-0.2^+0.2× 10^12 both using outer halo globular clusters, while Kafle-2018 found 0.8 ± 0.1 × 10^12 from high-velocity planetary nebulae, and Sofue-2015 found 1.39 ± 0.26 × 10^12 for the total mass combining disk rotation velocities and radial velocities of satellite galaxies and globular clusters. Both the simple mean and median of the above values are 1.27 × 10^12. Our adopted mass of M_M31 = 1.3± 0.4 × 10^12 in the fiducial model reflects the broad consensus of M31 mass estimates using different methods, but also the considerable remaining uncertainty.

M33   Mass estimates of M33 are much more sparse. More than twenty years ago, Corbelli-2000, Corbelli-2003 measured a dark matter mass of 5 × 10^10, extrapolated out to a virial mass of 5× 10^11 from the measured HI rotation curve, but noted that this results in a very low baryon fraction. More recently, Kam-2017 obtained a similar result using the Hα rotation curve, but noted that because the measurements only extend to a few percent of the virial radius, there are no strong constraints on the total dark matter halo. On the other hand, abundance matching based on the observed stellar mass results in a significantly lower total mass of 1.7 ± 0.55 × 10^11 Guo-2010, Patel-2017. Citing both the direct measurements and abundance matching, Patel-2017 adopt a mass range of 0.8 - 3.2 × 10^11 in their dynamical models of the M33-M31 interaction, while Patel-2018 and <cit.> assume a total mass of 2.5 × 10^11.

We adopt M_M33 = 3 ± 1 × 10^11 in the fiducial model, marginally compatible with both the extrapolated masses from rotation curve measurements and the results of abundance matching and in line with previous studies. However, the results from the two methods are certainly in tension.

LMC   For the mass of the LMC, recent mass estimates based on its effect on Galactic stellar streams include 1.38^+0.37_-0.24× 10^11 Erkal-2019, 1.30 ± 0.3 × 10^11 Vasiliev-2021, 1.88^+0.4_-0.35× 10^11 Shipp-2021 and 1.29^+0.28_-0.23× 10^11. Erkal-2021 obtain good agreement between the perturbations of Milky Way halo stars with an LMC mass of 1.5 × 10^11. Using the abundance of likely satellites of the LMC, Erkal-2020 obtain a lower limit of 1.24× 10^11, and using kinematics of satellites associated with the LMC, Correa-Magnus-2022 find 1.65^+0.47_-0.49× 10^11. Most recently, Watkins-2024 used 30 LMC globular clusters to infer a total mass of 1.80^+1.05_-0.54× 10^11 while <cit.> find an even higher value of 2.5^+0.9_-0.8× 10^11 using a timing argument and to be most consistent with the Hubble Flow around the Local Group. The simple mean of the above values is 1.6 × 10^11 while the median is 1.5 × 10^11. A comprehensive recent review on the effect of the LMC on the Milky Way, including a discussion of mass measurements, is given by Vasiliev-2023, who conclude that the LMC mass is likely in the range 1-2 × 10^11, which matches the choice of M_LMC = 1.5 ± 0.5 × 10^11 in our fiducial model.



 §.§ Halo Concentrations

As explained above, we assume that for the purpose of integrating their orbits, all galaxies are represented by NFW haloes <cit.> with the total masses M_200 defined above. We assume concentration parameters of 10 ± 2 for all galaxies, consistent with results of cosmological simulations Neto-2007, Ludlow-2014, Wang-2015, Correa-2015, Diemer-2015 in this mass range. While cosmological simulations show the average concentration parameter to be mass-dependent, the halo-to-halo scatter is significantly larger than the change in the mean concentration over this narrow mass range Neto-2007. 

The concentration parameter does not affect the orbital calculation, except for the dynamical friction, where (for a given mass), it sets the scale radius, r_s = r_200 / c, of the “satellite" that enters in Equation <ref>.

In Extended Data Figure <ref>, we show the dependence of the merger probability on the concentration. Except for a very low M31 mass, there is only a weak dependence of the merger probability on the concentration of M31, which is more likely to be the more massive “host" galaxy in the MW-M31 encounter. For the concentration of the MW, which, in the MW-M31 interaction is more likely to be the satellite, we find that the merger probability is reduced if the concentration is below ∼ 8, i.e. below - 1 σ of our fiducial value. We find no significant dependence on the merger probability on the concentration assumed for M33 or the LMC. 



Acknowledgments
We thank Marius Cautun for his generous help, and Joonas Nättilä for helpful suggestions. TS and JD are supported by the Research Council of Finland grant 354905, and TS and PHJ are also supported by the Research Council of Finland grant 339127. JD is supported by an Erasmus+ grant. TS and CSF are supported by the European Research Council (ERC) Advanced Investigator grant DMIDAS (GA 786910) and the STFC Consolidated Grant ST/T000244/1. PHJ, AK, AR and RW also acknowledge support from the European Research Council (ERC) Consolidator Grant KETJU (no. 818930). This work used facilities hosted by the CSC—IT Centre for Science, Finland, and the DiRAC@Durham facility, managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk) and funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the UK National e-Infrastructure. We thank the authors of the open source software listed below.

Authors' contributions
TS planned the project and performed the analysis together with JD. TS, AJD, CSF and PHJ planned the paper, and TS wrote the analysis code, created the figures and wrote the first draft. AK and AR contributed to the dynamical friction calculations. TS, JD, AJD, CSF, PHJ, AK, AR and RW jointly discussed and edited the manuscript.




§ DATA AVAILABILITY

All data used in this work is publicly available and provided as part of the analysis code listed below.



§ CODE AVAILABILITY

The analysis in this paper was performed using Python 3.10, and makes extensive use of the following open-source libraries: Astropy 6.0.1 <cit.>, Matplotlib 3.8.3, NumPy 1.26.4 <cit.>, SciPy 1.13.0 <cit.> and Colossus 1.3.5 <cit.>. A documented Juypter notebook containing the code to produce all figures in this paper is available at: <https://github.com/TillSawala/MW-M31>.


Conflict of interest
The authors declare that they have no conflict of interest.























































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