Continuum-fitting the X-Ray Spectra of Tidal Disruption Events

The accretion rates in disks that form after tidal disruption can vary from highly super-Eddington to quite sub-Eddington. When the accretion rate of the disk is significantly higher than $\sim 0.1{\dot{M}}_{\mathrm{Edd}}$, the gas orbits become significantly non-Keplerian, the aspect ratio of the disk approaches unity, and advective heat losses due to radial gas inflow can no longer be neglected (Abramowicz et al. 1988). As a result, the standard thin disk model (Novikov & Thorne 1973; Shakura & Sunyaev 1973) is not adequate to generally model TDE disks, and a slim disk model is more accurate. Slim disks are optically thick accretion flows, first developed by Abramowicz et al. (1988), which incorporate advective cooling and do not assume Keplerian angular momentum, allowing them to work in both super- and sub-Eddington regimes.

Since the development of the first slim disk models, many analytic and numerical studies have further developed the structure and dynamics of slim disks (Lasota 1994; Abramowicz et al. 1996, 1997; Beloborodov 1998; Gammie & Popham 1998). Based on the work of Abramowicz et al. (1996, 1997), Sądowski (2009; 2011) developed a general relativistic (GR) slim disk code to solve the disk equations numerically. However, this model was only used to study accretion disks around small back holes (Straub et al. 2011). Following the work of Sądowski (2009), we estimate the viscous heating term analytically instead of numerically and rewrite the code by using the shooting method (Press 2002) to determine the sonic point. For convenience, we write the underlying stationary disk equations in the Appendix B. Solving these equations, we find the local flux F(r), the surface density Σ(r), the ratio of disk height to radius H/r, and the four-velocity (u_t, u_r, u_θ, u_ϕ) of the gas at radius r. There are several important assumptions made in this model, such as the imposition of a zero-torque inner boundary condition, the large assumed optical depth, neglect of self-irradiation due to light bending, an absence of magnetic pressure support, and no angular momentum loss due to radiation or outflows.

Early studies (Shimura & Takahara 1993, 1995) demonstrated that the local X-ray spectrum of the disk does not behave as a perfect multicolor blackbody. Due to electron scattering and the temperature gradient in the atmosphere, the real X-ray flux at each annulus will be higher than the corresponding blackbody flux. In principle, one has to solve the full vertical radiation transfer equation to determine the local X-ray flux. The solution is determined by the vertical structure of the disk, which is determined by the effective temperature T_e = (F(r)/2σ)^1/4 (σ is the Stefan–Boltzmann constant), surface density Σ(r), and the strength of vertical gravity Q(r) at each annulus of the disk. Working in the context of thin disks, Shimura & Takahara (1995) found that the X-ray flux can be well approximated by introducing a color correction factor f_c,

$$\begin{eqnarray}&&{I}_{{\rm{d}}}(\nu )=\displaystyle \frac{2h{\nu }^{3}{c}^{-2}{f}_{{\rm{c}}}^{-4}}{\exp (h\nu /{k}_{{\rm{B}}}{f}_{{\rm{c}}}{T}_{{\rm{e}}})-1}.\end{eqnarray} \tag{ 1 }$$

Here, h is the Plank constant and k_B is the Boltzmann constant; f_c is also commonly called the spectral hardening factor. Since both slim disks and thin disks share the same radiation transfer physics, we assume Equation (1) is also true for slim disks, but slim disks will adopt a different f_c value due to different disk parameters when compared with thin disks. For a thin disk, f_c is about 1.7, and is insensitive to disk parameters (Shimura & Takahara 1995). Later studies (Merloni et al. 2000; Gierlinski & Done 2004; Davis et al. 2005) showed that f_c may increase as the accretion rate increases. Recently, Davis & El-Abd (2019, hereafter DE19) gave an approximate formula to estimate f_c for a non-spinning SMBH,

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{\rm{c}}} & = & 1.74+1.06({\mathrm{log}}_{10}{T}_{{\rm{e}}}-7)-0.14[{\mathrm{log}}_{10}Q(r)-7]\\ & & -\ 0.07\{{\mathrm{log}}_{10}[{\rm{\Sigma }}(r)/2]-5\}.\end{array}\end{eqnarray} \tag{ 2 }$$

This equation holds for cases with accretion rates between 0.01 to 1 Eddington accretion. In the super-Eddington regime, f_c would not increase to infinity as T_e increases and would instead saturate (Davis et al. 2006) at

$$\begin{eqnarray}&&{f}_{{\rm{c}}}^{\star }\approx {\left(\displaystyle \frac{72\mathrm{keV}}{{T}_{{\rm{e}}}{k}_{{\rm{B}}}}\right)}^{1/9}.\end{eqnarray} \tag{ 3 }$$

This implicates that ${f}_{{\rm{c}}}^{\star }\approx 2.4$ for an SMBH accreting near Eddington. Above this value, f_c could increase only weakly.

The spectral hardening factor f_c can significantly change the local X-ray flux, and is thus a key input to constrain parameters in X-ray spectral fitting. One may either treat it as a nuisance parameter to be fitted independently or use the prescriptions above to estimate a priori values for f_c. In this work, we take the latter approach as our fiducial model, but explore the impact of letting f_c float as a free parameter in Appendix A. Given the range of T_e, Q(r), and Σ(r) predicted from the slim disk solution, Equation (2) is unlikely to exceed the saturation value ${f}_{{\rm{c}}}^{\star }$, but we also explore this issue more carefully in Appendix A.

We use a general relativistic ray-tracing code (Johannsen & Psaltis 2010; Psaltis & Johannsen 2012) to calculate the trajectories of photons reaching the observer from the surface of the disk. With this code, we trace the null geodesics of photons backwards in time to the disk, and calculate their energy (E_d) and position (r, ϕ), given initial (i.e., observed) energies (E_obs) and positions (x_o, y_o) in the image plane of the observer. This ray-tracing calculation self-consistently accounts for gravitational redshift, the Doppler effect, and strong lensing in Schwarzschild or Kerr spacetimes. The observed flux can then be calculated by integrating over the flux on the image plane. For the reader's convenience, we write the underlying ray-tracing equations in Appendix C.

A monochromatic flux of disk radiation, as seen by a distant observer, can be calculated as (Baubock et al. 2015)

$$\begin{eqnarray}&&{F}_{\mathrm{obs}}({E}_{\mathrm{obs}})=\displaystyle \frac{1}{{d}^{2}}\int {I}_{\mathrm{obs}}({E}_{\mathrm{obs}},{x}_{{\rm{o}}},{y}_{{\rm{o}}}){{dx}}_{{\rm{o}}}{{dy}}_{{\rm{o}}}.\end{eqnarray} \tag{ 4 }$$

Here, d is the distance from the observer to the disk, and I_obs(E_obs, x_o, y_o) is the observed intensity of photons with energy E_obs in the image plane. Since I/E³ is Lorentz invariant, we have

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{{\rm{d}}}}{{E}_{{\rm{d}}}^{3}}=\displaystyle \frac{{I}_{\mathrm{obs}}}{{E}_{\mathrm{obs}}^{3}},\end{eqnarray} \tag{ 5 }$$

where I_d and E_d are the specific intensity and photon energy, respectively, as measured on the surface of the disk. By defining the redshift factor

$$\begin{eqnarray}&&{\rm{\Upsilon }}=\displaystyle \frac{{E}_{\mathrm{obs}}}{{E}_{{\rm{d}}}},\end{eqnarray} \tag{ 6 }$$

Equation (4) can be rewritten as

$$\begin{eqnarray}&&{F}_{\mathrm{obs}}({E}_{\mathrm{obs}})=\displaystyle \frac{1}{{d}^{2}}\int {{\rm{\Upsilon }}}^{3}{I}_{{\rm{d}}}({E}_{\mathrm{obs}}/{\rm{\Upsilon }},r,\phi ){{dx}}_{{\rm{o}}}{{dy}}_{{\rm{o}}},\end{eqnarray} \tag{ 7 }$$

with an emitted intensity I_d(E_d/ϒ, r, ϕ) that is calculated by Equation (1). Therefore, we can calculate the observed flux at a single frequency by integrating Equation (7) numerically.

Since the height of the slim disk cannot be neglected, we trace the ray back to the surface of the disk, a finite height above the midplane. We note that when the observer is at a high inclination (observing nearly edge-on), the height of the edge can shield the X-rays that are preferentially emitted from the inner, hottest annuli of the disk.⁹ Even a perfectly edge-on disk will not be completely dim in the X-rays, however, due to strong lensing effects.

The evolution of the mass fallback rate in TDEs at late times can be expressed as

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{t}}}={\dot{M}}_{\mathrm{peak}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{fall}}}\right)}^{n},\end{eqnarray} \tag{ 8 }$$

where ${\dot{M}}_{\mathrm{peak}}$ is the peak fallback rate, t is the time since the most tightly bound debris has returned to pericenter, t_fall is the fallback time of the most tightly bound debris, and n is the power-law decay index, which is equal to −5/3 in simple analytic models of the disruption process (Rees 1988; Phinney 1989). While analytic estimates for these variables exist (Rees 1988; Stone et al. 2013), they can be more precisely calibrated via hydrodynamic simulations of the disruption process (Guillochon & Ramirez-Ruiz 2013; Mainetti et al. 2017; Ryu et al. 2020a). In Newtonian gravity,¹⁰ ${\dot{M}}_{\mathrm{peak}}$, t_fall, and n are determined primarily by the SMBH mass M_•, the victim star's mass m_⋆ and radius r_⋆, and the star's pericenter r_p. Guillochon & Ramirez-Ruiz (2013) provide fitting formulas for the quantities ${\dot{M}}_{\mathrm{peak}}$, t_fall, and n in the aftermath of disruption of a polytropic star:

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{peak}}=46{A}_{\gamma }{\eta }_{-1}{M}_{6}^{-3/2}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{r}_{\star }}{{R}_{\odot }}\right)}^{-3/2}{\dot{M}}_{\mathrm{Edd}},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{fall}}={B}_{\gamma }{M}_{6}^{1/2}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{r}_{\star }}{{R}_{\odot }}\right)}^{3/2}\,\mathrm{yr},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{n}_{\infty }={D}_{\gamma }.\end{eqnarray} \tag{ 11 }$$

Here, γ is the polytropic index, η is the radiative efficiency of accretion, η₋₁ = η/0.1, M₆ = M_•/(10⁶M_⊙), and M_⊙ is the solar mass. We have defined the peak fallback rate in terms of the Eddington-limited accretion rate,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{Edd}}=1.37\times {10}^{21}\,\mathrm{kg}\,{{\rm{s}}}^{-1}{\eta }_{-1}^{-1}{M}_{6}.\end{eqnarray} \tag{ 12 }$$

Hereafter, we refer to accretion and fallback rates in dimensionless Eddington units, i.e., $\dot{m}=\dot{M}/{\dot{M}}_{\mathrm{Edd}}$. A_γ, B_γ, and D_γ are fitting functions that depend only on γ and the penetration parameter β ≡ r_t/r_p (note that tidal disruption radius r_t = r_⋆(M_•/m_⋆)^1/3). The precise form of the A_γ, B_γ, and D_γ functions can be found in the erratum of Guillochon & Ramirez-Ruiz (2013).

We estimate m_⋆ from r_⋆ using the mass–radius relation of Tout et al. (1996), for m_⋆ > 0.1M_⊙. If m_⋆ < 0.1M_⊙, we set r_⋆ = 0.1R_⊙. The numerical fitting formulae above are valid for 0.6 ≤ β ≤ 2.5, and are calibrated for two specific values of γ: 4/3 and 5/3. These adiabatic indices are appropriate for high-mass (m_⋆ ≳ M_⊙) and low-mass (m_⋆ ≲ 0.5M_⊙) stars, respectively (Lodato et al. 2009). For stars in the transition region (0.5M_⊙ < m_⋆ < M_⊙), we use a linear interpolation,

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{t}}}=2{\dot{M}}_{{\rm{t}}}^{\gamma =5/3}({M}_{\odot }-{m}_{\star })+{\dot{M}}_{{\rm{t}}}^{\gamma =4/3}(2{m}_{\star }-{M}_{\odot }),\end{eqnarray} \tag{ 13 }$$

to estimate the fallback rate. Thus, in order to calculate the fallback rate at t, one needs to employ four free parameters, which are M_•, t_peak, m_⋆, and β. Since it is difficult to detect the exact date of peak fallback, t is determined by:

$$\begin{eqnarray}&&t={t}_{\mathrm{obs}}-{t}_{\mathrm{returned}}={t}_{\mathrm{obs}}-{t}_{\mathrm{peak}}+{t}_{\mathrm{fall}}.\end{eqnarray} \tag{ 14 }$$

Here, t_obs, t_returned, and t_peak are the date of observations (which varies from epoch to epoch), the first material returned date, and the peak luminosity date. In order to calculate the accretion rate from Equation (8), one must know the five parameters, M_•, β, m_⋆, t_obs, and t_peak.

Equation (8) is approximate in two significant ways. The first is that the fallback rate is only a power law at late times, and this simple functional form neglects the early rise to peak fallback that characterizes the first days to weeks of a TDE (Lodato et al. 2009; Guillochon & Ramirez-Ruiz 2013). More important, however, is the nontrivial translation between mass fallback rates ${\dot{M}}_{{\rm{t}}}$ and inner disk accretion rates $\dot{M}$. As the most tightly bound debris returns to pericenter, it can begin to form an accretion flow through dissipational processes. While the circularization and disk formation process in TDEs remains an unsolved problem, there is evidence that circularization rates vary enormously across the parameter space of TDEs. Gas circularization may be rapid if r_p ∼ r_g, in which case relativistic apsidal precession will cause debris streams to self-intersect and thermalize their orbital energy in shocks (Rees 1988; Hayasaki et al. 2013; Dai et al. 2015). In this regime, one can expect that the accretion rate of the inner disk, $\dot{M}$, follows the fallback rate of the gas, ${\dot{M}}_{{\rm{t}}}$.

On the other hand, if r_p ≫ r_g, apsidal precession will be very weak, and self-intersection shocks will occur far from the SMBH, leading to inefficient circularization (Dai et al. 2015; Piran et al. 2015; Shiokawa et al. 2015; Bonnerot et al. 2017). In this regime, $\dot{M}$ will probably not track ${\dot{M}}_{{\rm{t}}}$. Even in the r_p ∼ r_g regime, circularization may be delayed if the SMBH is spinning rapidly at an angle misaligned from the debris orbital angular momentum (Guillochon & Ramirez-Ruiz 2015; Hayasaki et al. 2016). In this work, we take a primarily agnostic view on these open theoretical questions. In our fiducial multi-epoch X-ray spectral fitting, we allow $\dot{M}$ to float between epochs as free parameters, but in non-fiducial modeling, we consider how our parameter estimation changes if circularization is rapid, and we can make the strong assumption that $\dot{M}$ has the functional form of Equation (8).

Here we summarize the assumptions we have made in the previous subsections, and give a brief description of our actual fitting procedure.

Throughout this paper, we assume that:

• 1.  
  The dynamic inner accretion flow can be approximated by a series of stationary, circular, and equatorial general relativistic slim disk models. In all cases, we assume that the disk, if initially misaligned from the SMBH equatorial plane, has become aligned by the time of the observations.
• 2.  
  The disk structure is determined assuming a zero-torque inner boundary condition, large optical depth, no self-irradiation,¹¹ no magnetic pressure support, and no angular momentum losses through winds or radiation.
• 3.  
  The local X-ray emissivity of the disk can be described by Equation (1): a multicolor blackbody modified by a (radius-dependent) spectral hardening factor given in Equation (2).
• 4.  
  The X-rays are generated at the finite-height photospheric surface of the disk, which we assume to be equal to its scale height H, and if a photon's trajectory intersects the disk after emission, it will be absorbed completely.
• 5.  
  The inner edge of the disk is set to be the ISCO, and the outer edge of the disk is twice the tidal disruption radius (2r_t).
• 6.  
  While we do not usually assume a prior for the disk accretion rate $\dot{M}(t)$, in the non-fiducial cases when we do, we assume that the circularization process is rapid, so that $\dot{M}(t)={\dot{M}}_{{\rm{t}}}(t)$, as in Equation (8).

These assumptions, which allow us to compare multi-epoch observations with theoretical predictions for the soft X-ray continuum, represent idealizations to some degree. Assumption (1) above is probably the most questionable, as TDE disks may form with generic misalignments from the SMBH (Stone & Loeb 2012) and have non-axisymmetric initial accretion flows (Shiokawa et al. 2015). Our approach is likely more accurate at later epochs, as initially tilted TDE disks align over time (Franchini et al. 2016; Xiang-Gruess et al. 2016; Ivanov et al. 2018; Zanazzi & Lai 2019), and initially eccentric motions within the accretion flow circularize (Sądowski et al. 2016). The circular disk assumption may also be more valid for the innermost, X-ray producing annuli than for the disk as a whole, as gas must undergo significant dissipation in order to inspiral from scales ≈r_t to the ISCO. A closely related issue is global disk precession, which may modulate TDE X-ray light curves in a quasiperiodic way at early times (Stone & Loeb 2012; Franchini et al. 2016). These modulations could become severe if the disk precesses into an edge-on configuration. Our simplifying assumption that the disk quickly aligns into the equatorial plane excludes the possibility of precession, an issue to address in future work.

The other assumptions listed above may cause issues in specific cases. For example, if circularization is not radiatively efficient, the disk outer edge can exceed 2r_t (Hayasaki et al. 2016; Bonnerot & Lu 2019). However, because most X-rays are generated at small radii, our predictions are generally insensitive to the choice of outer edge, unless both (a) the disk is nearly edge-on (so that the disk edge strongly shields the X-rays from the inner disk) and (b) the largest scale height, H/r, is achieved at the outer edge. We find that condition (b) is rarely satisfied, and only for $\dot{m}\gtrsim 100$ does H/r rise monotonically with increasing r. Since such severely super-Eddington accretion rates are not expected for main-sequence TDEs, our model's X-ray predictions should not be very sensitive to the choice of outer boundary. With these caveats in mind, we now describe our specific fitting procedure.

First, we build a library of slim disk structural solutions. The disk structure is completely determined by M_•, a_•, $\dot{m}$, and α. Throughout our grid, we set α = 0.1; as we will see later, α has no important effect on the fitting. Given a {M_•, a_•} pair, we solve for 70 disk structures, with $\dot{m}$ varying from 600 to 0.1 (for the case of a_• = 0.998, we extend the solution of $\dot{m}$ to 0.06).

Second, we estimate the theoretical spectrum for each disk model in our library. Given $\dot{m}$, M_•, a_•, and the observer's inclination angle¹²  θ, we use the ray-tracing code to calculate the flux in each bin of photon energy. We use linear interpolation between energy bins with different accretion rates to estimate the model flux. We then fit to observations by using the XSPEC package (version 12.10.1; Arnaud 1996).

Finally, for any given epoch of X-ray observations, we search across our model grid and optimize the model parameters using the χ² statistic (or Cash statistic, Cstat) for a given {M_•, a_•} pair. Both SMBH mass and spin are assumed to be constant between epochs. We then calculate the χ² (or Cstat) across the {M_•, a_•} plane, determining the significance of each {M_•, a_•} pair.

The free parameters for the slim disk are M_•, a_•, ${\dot{m}}_{i}$, and θ. Here, the subscript index i denotes the ith observational epoch. If we estimate ${\dot{m}}_{i}$ with Equation (8) (i.e., assume efficient circularization of returning tidal debris), $\dot{m}$ is replaced by β, m_⋆, and t_peak. Given these parameters, as well as f_c (determined by Equation (2)), we can calculate synthetic X-ray spectra for all observational epochs. The real X-ray spectrum is, however, subject to interstellar extinction, so we introduce the extinction N_H. The fitting procedure is shown schematically in Figure 1. The priors on the free parameters are listed in Table 1.

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Table 1.  Allowed Ranges for TDE Parameters

Name	M•a	a•a	βb	m⋆b	${\dot{m}}_{i}$ c	θ	NH,i
	(106M⊙)			(M⊙)	(Edd)	(deg)	(1022 cm−2)
ASASSN-14li	(2, 20)	(−0.9, 0.998)	(0.6, 2.5)	(0.1, 100)	(0.1, 600)	(5, 90)	(0.02, 1)
ASASSN-15oi	(0.8, 10)	(−0.9, 0.9)	(0.6, 2.5)	(0.1, 100)	(0.1, 600)	(5, 90)d	(0.04, 1)

Notes.

^aIn individual-epoch fits, M_• and a_• are discretely sampled across the given ranges. ^bIn our most general fits, we allow $\dot{m}$ to float over time. Otherwise, we assume that $\dot{m}$ follows the gas fallback rate, which requires us to fit two additional variables: stellar mass m_⋆ and penetration parameter β. ^cFor the largest spin we consider, a_• = 0.998, the lower limit of $\dot{m}$ extends to 0.06. The accretion rate $\dot{m}$ here is calculated by assuming η = 0.1 and listed in dimensionless Eddington units. ^dFor ASASSN-15oi, the inclination prior will be handled more carefully to test a range of hypotheses.

Download table as:  ASCIITypeset image

ASASSN-14li was first detected by the All Sky Automated Search for Supernova (ASASSN) on MJD 56983.6 in a post-starburst galaxy with z = 0.0206 (Jose et al. 2014). ASASSN-14li has been observed extensively at different frequencies (Miller et al. 2015; van Velzen et al. 2016a; Pasham et al. 2017; Bright et al. 2018). We extracted 10 XMM-Newton spectra from archival observations. Table 2 lists some details of these observations.

We run the default SAS v18 (20190531) tools under the HEASOFT ftools software version 6.26.1 to extract source spectra and filter the data. We initially focused on data from the EPIC pn detector (Strüder et al. 2001; Turner et al. 2001) as that provides the highest count rate. We filtered the pn data for periods of enhanced background radiation, where we require that the 10–12 keV detection rate of pattern 0 events is lower than 0.4 counts s⁻¹. The effective exposure time for each observation after filtering is given in Table 2. All of these observations are done with the pn detector in the "small window" mode providing a time resolution of 5.7 ms; however, for reasons we explain below, we also extracted the Reflection Grating Spectrometer (RGS; den Herder et al. 2001) data for the first five epochs of observations.

If the source flux is very high, the XMM-Newton (pn) data might be affected by photon pileup. Pileup spuriously influences the observed spectral properties of the source under study. However, owing to the high time resolution afforded by the pn in small window mode pileup can be avoided in many cases. The XMM-Newton handbook states that pileup is important for point sources with count rates above 25 counts per second. As the issue can be especially important for very soft sources such as typically found for TDEs, we investigate the event pattern distribution. The event pattern is the property that the charge cloud released by the incoming photon can be distributed over one, two, or more detector pixels. For the XMM-Newton pn detector, we use only events detected in one or two pixels. If pileup is present, the observed number of one- and two-pixel events deviates from non-piled-up observations. We use the SAS command epatplot to compare the observed and expected number of single and double event patterns as a function of photon energy. We conclude that pileup is important for the first four observations in Table 2 and probably also for the fifth observation. Furthermore, for a source as bright as ASASSN-14li early on in the observing campaign, there is no area of the readout part of the detector that can be used to estimate a reliable background spectrum. This is a limiting factor for the first five pn observations in Table 2, which we therefore do not use further. Instead, for those first five observations, we extracted the RGS data following the standard SAS procedure, where we defined as good times those where the background count rate on CCD number 9 on RGS1 is lower than 1 count s⁻¹.

In the pn observations from epochs 6–10, background photons were extracted from a source-free rectangular region toward the edge of the field of view of the small window mode. In selecting the location of the rectangular background region, we avoid the wings of the point-spread function of the source, including the counts caused by the diffraction spikes, and the area that is affected by so-called out-of-time events. We extracted the source using a circular aperture of 15'' radius centered on the optical position of the source. The extracted spectra are rebinned to yield a minimum of 30 counts per bin.

ASASSN-15oi was discovered on 2015 August 14 in the nucleus of a quiescent early-type galaxy at z = 0.0484 (Holoien et al. 2016b). XMM-Newton observed the source ASASSN-15oi twice, namely, on 2015 October 29 and 2016 April 4 with observation IDs 0722160501 and 0722160701, respectively. These ASASSN-15oi observations are done with the pn detector in "full frame" mode, providing a time resolution of 73.4 ms. We run the default SAS v18 (20190531) tools under the HEASOFT ftools software version 6.26.1 to extract source spectra. In particular, we filtered the data for periods of enhanced background radiation, where we require that the 10–12 keV detection rate of pattern 0 events is lower than 0.4 counts s⁻¹. The remaining, effective exposure time after filtering out periods of high background is given in Table 2. We also checked the data for the presence of pileup, but we found that pileup did not affect the observations of ASASSN-15oi. We extracted the source using a circular aperture of a 15'' radius centered on the optical position of the source. The background counts and spectrum were extracted from a source-free, rectangular region on the same detector.

The sensitivity of the synthetic spectrum and light curve to various model parameters is shown in Figures 2–4, where we assume that the mass accretion rate tracks the fallback rate and that the spectral hardening factor f_c is well described by Equation (2). Figure 2 shows the theoretical X-ray spectra. As is generally the case for steady-state α disks, lower SMBH masses imply smaller disk sizes and higher disk temperatures, although the bolometric luminosity from smaller SMBHs is increasingly Eddington-limited.¹³ Consequently, our calculation shows that the highest total soft X-ray luminosities (0.3–10.0 keV) occur for M_• ∼ 4 × 10⁶M_⊙. Increasing the SMBH spin pushes the prograde ISCO of the SMBH inward, increasing the emitting area of high-temperature gas, the disk effective temperature, and thus the X-ray flux. When the inclination is large (i.e., the disk is nearly edge-on), the observed luminosity decreases dramatically, because X-rays from the inner region are mostly shielded by the outer annuli of the disk, in which case the low-temperature edge is the primary contributor to the observed flux. The strength of this effect, which was first predicted for TDE disks by Ulmer (1999), is weakened somewhat by relativistic lensing, in a way that we have quantified here for the first time.

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The evolution of the synthetic spectra over four epochs is shown in Figure 3. The top panel indicates how the fiducial case changes with time. As the accretion rate decreases, the disk spectrum dims and spectrally softens. A higher SMBH spin (upper middle panel) results in a slower decay of the disk luminosity, because higher radiative efficiencies delay the transition to the sub-Eddington regime. The results for a lower viscosity parameter (bottom panel) are similar to the fiducial case.

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For a more inclined, nearly edge-on disk (bottom middle panel), the total luminosity is initially dominated by the disk edge, as X-ray photons from the inner disk are blocked by the edge. As the accretion rate decreases, the height of the disk also decreases, exposing more photons from the inner disk to the observer's line of sight. As a consequence of this "disk slimming," the disk spectrum initially becomes brighter and harder. At later times, the disk luminosity enters the usual regime of exponential decay, and the spectrum fades and softens. Because the effect of disk slimming is not instantaneous, it introduces some degeneracy in single-epoch spectral fitting; the same overall X-ray luminosity can be produced by two different stages of edge-on disk evolution, e.g., t₁ and t₃, with two very different accretion rates. The X-ray spectral shape is somewhat harder at the later epoch, however.

The sensitivity of the light curve to various model parameters is shown in Figure 4. Lower-mass SMBHs have a longer super-Eddington emission plateau, transitioning later to exponential X-ray flux decay (see Lin et al. 2018 for a possible observational example). Increasing the spin a_• increases the normalization of the disk luminosity and extends the super-Eddington plateau phase. In all of these synthetic light curves, the disk accretion rate decays per Equation (8), and the disk temperature and luminosity also decrease. However, the luminosity does not scale linearly with the accretion rate in a slim disk, as the hot, radially inflowing gas will advect away some fraction of the heat that is generated locally. This effect becomes notable when $\dot{m}\gt 0.3$ and comes to dominate over radiative cooling near the Eddington limit (Sądowski 2009).

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In the slim disk model, continuously increasing the accretion rate increases the fraction of disk power advected as heat into the horizon. As a result, at accretion rates well above the Eddington rate, the total luminosity of the disk increases only slowly with accretion rate (Abramowicz et al. 1988). The X-ray luminosity here actually saturates as the accretion rate becomes highly super-Eddington, because f_c decreases due to an increase in the surface density, which more than counters the slowly increasing total luminosity. As a result, slim disk models predict a nearly flat early-time X-ray light curve.

At late times in TDEs, once the accretion rate becomes significantly sub-Eddington, advective cooling is no longer important. The X-ray light curve decays faster than the (power-law) accretion rate at late times, because (1) X-rays are generally emitted from the Wien tail of each annulus, and (2) the spectral hardening factor decreases with the accretion rate. The former is generally more important for setting the rate at which the X-ray light curve declines, as the luminosity decays exponentially, i.e., $\nu {L}_{\nu }\propto \exp (-{{bt}}^{5/12})$, where b is a normalization constant, as the accretion rate falls.

The light curve behaves interestingly for large (edge-on) inclinations, exhibiting a late, steep rise at t ∼ 100 days. This behavior is due to the same "disk slimming" effect seen earlier in Figure 3. At early times, X-ray emission is blocked by the 3D structure of the disk, but once the disk height decreases sufficiently, X-rays from the hot inner disk annuli can escape along the observer's line of sight, increasing the observed luminosity. This result suggests that the disk vertical structure provides a strong constraint on the inclinations of some TDEs. Given that H/r ∼ 0.6 during the super-Eddington phase, the peak of the X-ray light curve can be delayed in nearly half of TDEs with an initially super-Eddington accretion rate (see also the discussion in Jonker et al. 2020 on the delay between the observed optical and X-ray peak luminosities).

Unlike in a standard thin disk model, where the disk luminosity is independent of the viscosity parameter (for fixed accretion rate), the innermost regions of our slim disk have a higher inner temperature (for fixed accretion rate) when the viscosity parameter is larger (Sądowski 2009). The ultimate effect of viscosity on bolometric luminosity is relatively small, so we fix the viscosity parameter to α = 0.1 for the rest of our analysis. There is a possible secondary effect; holding all else constant, different α values produce different surface density profiles, which in turn can alter the spectral hardening factor f_c. We defer a detailed investigation of this effect to future work.

The quasi-thermal slim disk model of the previous subsection forms the basis for our X-ray spectral fitting and parameter estimation. However, X-ray spectra may be affected by absorption local to the source, on larger scales in the host galaxy, and/or from our own Galaxy. Here we fit the multi-epoch spectra of ASASSN-14li by combining our slim disk model with a free-floating absorption parameter, N_H. The general absorption model phabs in XSPEC (Arnaud 1996) is added as a multiplication model to our slim disk model. For simplicity, we combine absorption from material local to the TDE, from the host galaxy (N_H = 0.026 × 10²² cm⁻² at z = 0.0206), and Galactic absorption (N_H = 0.016 × 10²² cm⁻²) into one parameter (Miller et al. 2015), neglecting (small) redshift effects.

For the first five epochs of XMM-Newton RGS spectra, we find no evidence for a spectral component at energies higher than 1.0 keV. As a result, we use only the absorbed slim disk model to fit those epochs. However, as was reported by Miller et al. (2015), there is evidence for a disk wind at early epochs, in the form of strong absorption lines. Ignoring these strong absorption lines in our fits would artificially bring down the continuum level, thus affecting the best-fit parameters. To avoid such a bias, we ignore the data bins that contain the strong absorption lines at ≈0.4911, ≈0.414, ≈0.3997, and ≈0.3776 keV. For the last five epochs of XMM-Newton pn spectra, we fit the spectra over the 0.3–10 keV range.

The fits to the 10 spectral epochs are required to have a constant M_• and a_•, but all other parameters can, in principle, vary. Even the observational inclination angle could change from epoch to epoch, as the disk may be subject to Lense–Thirring precession (Stone & Loeb 2012; Liska et al. 2018) and/or gradual realignment into the SMBH equatorial plane (Franchini et al. 2016; Zanazzi & Lai 2019). However, because the equations for a tilted relativistic slim disk do not yet exist, we assume the same inclination for all epochs, i.e., that the disk was either born with small tilt or has quickly aligned itself into the equatorial plane. We fit the 10 spectra simultaneously, allowing all 10 accretion rates (${\dot{m}}_{i}$), all 10 absorption parameters (N_H,i), and the one inclination (θ) to float. We minimize the χ² for each combination of (M_•, a_•).

4.2.1. SMBH Mass, Spin, and Other Inferred Parameters

The best-fit parameters and their 1σ errors are shown in Table 3, and the spectral fits are plotted in Figure 5. The reduced χ² for each spectrum indicates that our model describes the spectra well. The total reduced χ² is χ²/v = 1.12. The ${\chi }_{\mathrm{dof}}^{2}$ is larger for epochs 7 and 8, in part due to a decrease in flux at ≈0.6 keV. The first three spectra are fitted with high Eddington ratio accretion rates, indicating an early, mildly super-Eddington accretion phase. The slim disk model predicts a nearly constant flux for these three epochs.

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Table 3.  Best-fit Results for ASASSN-14li, with M_• = 10⁷M_⊙ and a_• = 0.998

Epoch		NH		θ		$\dot{m}$ a		χ2/v
		(1020 cm−2)		(◦)		(Edd)
1		5.3 ± 0.3		76 ± 3		3.0 ± 0.2		629.79/562
2		5.0 ± 0.2		⋯		2.8 ± 0.2		2321.45/2057
3		5.1 ± 0.3		⋯		2.9 ± 0.2		591.98/521
4		5.3 ± 0.4		⋯		1.08 ± 0.03		178.92/155
5		4.0 ± 0.3		⋯		0.81 ± 0.02		601.7/572
6b		4.3 ± 0.3		⋯		0.79 ± 0.02		7.31/9
7		4.0 ± 0.4		⋯		0.53 ± 0.02		16.52/5
8		4.9 ± 0.5		⋯		0.48 ± 0.02		11.4/4
9		2.5 ± 0.8		⋯		0.34 ± 0.02		5.02/3
10		3.4 ± 0.8		⋯		0.30 ± 0.01		4.31/7

Notes. In our simultaneous fitting, the SMBH spin and mass are held fixed, inclination is required to be the same for all epochs, and other parameters float. The total ${\chi }_{{dof}}^{2}$ is 4368.42/3895 = 1.12.

^aThe accretion rate (in dimensionless Eddington units) has been corrected for the radiative efficiency η. ^bThe power-law parameters of epoch 6 are: Γ = 1.5 ± 0.7 and A_pl = 8.0 ± 2.7 × 10⁻⁶ photons s⁻¹ cm⁻² keV⁻¹.

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Figure 6 plots the Δχ² on the (M_•, a_•) plane. The best-fit values lie at (10⁷M_⊙, 0.998). We compute and plot different confidence level (CL) contours with two-parameter thresholds of Δχ² = 2.3 (1σ CL, blue) and Δχ² = 6.18 (2σ CL, pink). Within the 1σ CL, we constrain the SMBH mass and spin to lie in the ranges ${10}_{-7}^{+1}\times {10}^{6}{M}_{\odot }$ and >0.3, respectively. While the marginalized, 1D constraints are fairly broad (e.g., the constraint on M_• has a 1σ CL comparable to that derived from galaxy scaling relations), the joint, 2D constraint is tighter. To our knowledge, this is the first time simultaneous multi-epoch X-ray spectral fitting has been used to constrain M_• and a_• in a TDE.

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Our constraint on M_• is broadly consistent with external, independent estimates, including from galaxy scaling relations such as M_•−M_bulge, which predicts ${M}_{\bullet }={6.3}_{-3.1}^{+6.3}\times {10}^{6}{M}_{\odot }$ (Holoien et al. 2016a; van Velzen et al. 2016a), and M_•−σ, which predicts ${M}_{\bullet }={1.6}_{-1.0}^{+2.4}\times {10}^{6}{M}_{\odot }$ (Wevers et al. 2017). Our SMBH mass is also consistent with the MOSFiT estimate from optical and UV light curves fitting: ${M}_{\bullet }={9}_{-3}^{+2}\times {10}^{6}{M}_{\odot }$ (Mockler et al. 2019). Applying the MOSFiT mass constraint to our Figure 6 narrows our a_• constraint to >0.85.

In general, SMBH spin measurements are more challenging to make than SMBH mass estimates, and the only previous constraint for ASASSN-14li comes from detection of a 133 s X-ray quasiperiodic oscillation (QPO) by Pasham et al. (2019). While the physical origin of this QPO remains uncertain, if one assumes that the maximum possible QPO frequency is that of fundamental test particle motion at the ISCO, the maximum mass is M_• ≈ 2 × 10⁶M_⊙. However, because this mass can only be achieved for near-extremal spins, it is inconsistent with our parameter inference at the 2σ CL. This inconsistency is notable, but, given the questions about the QPO's origin, a detailed comparison must be deferred for now.

In our exploration of the (M_•, a_•) plane, χ² increases quickly for large M_• and small a_•, because these parameter choices produce disks with insufficient X-ray luminosity to fit the first three spectra. Thus, observations of super-Eddington spectra offer significant power to constrain parameters of interest and to test accretion astrophysics. For example, the degeneracy of f_c with accretion rate would be greatly weakened in the super-Eddington limit, as the bolometric luminosities of super-Eddington disks are insensitive to the accretion rate. Super-Eddington accretion would also push the radius of the peak effective temperature inwards, providing more spin information than is available in sub-Eddington accretion disks.

In Figure 7, we plot the time evolution of the fitted TDE parameters. The observed X-ray flux (top panel) is integrated from 0.35 to 1.9 keV. The light curve is nearly constant for the first three epochs and then decays quickly, as ∝t^−2.8. This behavior is similar to the theoretical predictions in Figure 4, i.e., an early plateau corresponding to the super-Eddington phase and then a rapid decline in X-ray luminosity at sub-Eddington accretion rates.

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The evolution of the mass accretion rate (upper middle panel) is well constrained, with small error bars for most epochs. The overall uncertainty is dominated by the range of M_• and a_• permitted within the 1σ CL of Figure 6. For the best-fit SMBH mass and spin, the accretion rate through the disk declines from super-Eddington ($\dot{m}\approx 3.0$) to sub-Eddington ($\dot{m}\approx 0.3$) over 1100 days.

The accretion rate decays as t^−1.1, slower than the canonical t^−5/3 decay of the fallback rate.¹⁴ Figure 7 also shows that the specific choice of (M_•, a_•) pair has little effect on the accretion decay rate. However, the decay rate depends on the specific f_c parameterization. Given that f_c can increase or decrease the predicted flux normalization, it is somewhat degenerate with the accretion rate.

To investigate how our f_c assumptions affect the decay of disk accretion rate, we consider a non-fiducial model with constant f_c, i.e., assuming that the last seven (sub-Eddington) epochs have the same f_c as the first three (super-Eddington) epochs. We then refit the data by fixing M_• = 10⁷M_⊙ and a_• = 0.998 and find that the decay of the accretion rate is t^−1.2, a little steeper than t^−1.1. This result shows that the fitted accretion rate decay is insensitive to the f_c parameterization. In agreement with our theoretical analysis, the decay of the light curve is driven by that of the accretion rate and not the modeled time evolution of f_c. We conclude that the accretion rate decays from $\dot{m}$ ≈ 3 to $\dot{m}$ ≈ 0.3 roughly as t^−1.1, which is insensitive to the specific choice of (M_•, a_•) pair and f_c model.

This analysis is based on a specific f_c parameterization (DE19), which was originally derived for thin (i.e., sub-Eddington) disks around non-spinning SMBHs. We have already examined the sensitivity of the temporal evolution of $\dot{m}$ to the DE19 f_c prescription. In Appendix A, we further explore the effect of alternative treatments of f_c in the super-Eddington regime and find that the DE19 f_c prescription can be applied to slim disks. Different treatments of f_c produce essentially the same best-fit (M_•, a_•) pair and have little impact on the broader estimation of M_• and a_•.¹⁵ By comparing with other treatments of f_c, we find that the DE19 f_c prescription gives a reasonable constraint on the accretion rate; our results are generally insensitive to alternative treatments of super-Eddington spectral hardening.

In all of our fits, the disk height H/r steadily decreases with time (middle panel). The total absorption N_H does the same (lower middle panel). Because both of these effects will increase the X-ray flux (when all else is equal), the decline of the light curve is driven by that of the accretion rate. N_H decreases to that of the host galaxy plus Milky Way after ∼300 days, suggesting an early additional contribution from obscuring material near the disk. The fitted (constant) inclination θ depends strongly on the (M_•, a_•) pair (bottom panel).

While we were completing this paper, Mummery & Balbus (2020, hereafter MB20) published an independent analysis of the ASASSN-14li X-ray and UV light curves, using time-dependent models of viscously spreading, general relativistic thin disks. MB20 find 1.45 × 10⁶ < M_•/M_⊙ < 2.05 × 10⁶, with a broad range of permitted SMBH spins. This mass constraint is inconsistent with ours at the 1σ CL, and marginally consistent at the 2σ CL. The origin of this tension could arise from differences between the theoretical models, including our choice to model the local annular emission of the disk as a spectrally hardened blackbody, in contrast to MB20's assumption of a purely thermal spectrum.

Our model allows the accretion rate at each epoch to float, while MB20 compute $\dot{m}(t)$ deterministically by assuming several different initial conditions. Given this difference, it is notable that our accretion rate evolves as an $\dot{m}\propto {t}^{-1.1}$ power law; this decay rate is similar to the classic t^−1.2 power law for a viscously spreading disk with zero stress at the ISCO (Cannizzo et al. 1990) and somewhat steeper than the t^−0.8 power law predicted for finite stresses at the disk inner edge (Mummery & Balbus 2019). However, while spreading disk solutions are likely valid at late times in TDE disk evolution (van Velzen et al. 2019), it is not clear if they apply at early times, when $\dot{m}$ is influenced not just by internal stresses, but also by deposition of tidal debris falling back to pericenter.

The two models have other strengths and weaknesses. Our slim disks can fit the detailed shape of the X-ray spectrum, while their thin disk models are applied to the Swift X-ray light curve and to the late-time UV light curves. MB20 explore a wider range of inner boundary conditions and also account for the spreading of the outer disk edge, which we neglect. It is not clear whether this latter effect should change the X-ray spectrum significantly, given the physical origin of the X-rays in the innermost annuli.

4.2.2. No Evidence for Missing Energy Problem

4.2.3. Implications for Circularization Efficiency and Reprocessing Layers

ASASSN-15oi is notable for its unusual X-ray light curve. The total X-ray flux (0.3–10 keV) stays roughly constant during the first 100 days, and then increases by a factor of 10 (Gezari et al. 2017) about 250 days after discovery. So far, there are two hypotheses to explain this phenomenon, both described in Gezari et al. (2017). The first is the "delayed accretion" theory, in which inefficient circularization of bound debris (e.g., Guillochon & Ramirez-Ruiz 2015; Piran et al. 2015; Shiokawa et al. 2015; Hayasaki et al. 2016) prevents the disk mass accretion rate from tracking the mass fallback rate, with little circularization at early times and more at later times. The second is the "variable obscuration" proposal, which posits a time-dependent absorption column between our line of sight and the hot inner disk of the TDE. If the absorbing column density N_H diminishes over time, the X-ray flare will appear to brighten, even if the intrinsic luminosity is steady or falling. "Variable obscuration" is disfavored by Gezari et al. (2017), who find no evidence for declining absorption.

We suggest a third possible explanation. As we saw in Figure 4, late-time X-ray brightening can also occur due to evolution in the structure of a highly inclined disk. In this "slimming disk" scenario, a nearly edge-on disk with initial $\dot{m}\gtrsim 1$ experiences a decline in its accretion rate, which in turn reduces the aspect ratio H/r and exposes hot, X-ray bright inner annuli to the observer's line of sight. In this section, we initially fit the ASASSN-15oi spectra with few priors (our "general case" fit), as we did for ASASSN-14li. We then add additional priors to systematically study and compare the "slimming disk," "delayed accretion," and "variable obscuration" hypotheses.

For ASASSN-15oi, there are only two epochs with enough X-ray counts for detailed spectrum fitting; both are from XMM-Newton and were reported to be well fit with an absorbed blackbody plus power-law model (Holoien et al. 2016b). However, after careful filtering of background flaring events and background subtraction, we find no evidence for a power-law component related to the source spectrum in either epoch. Because there are only 223 counts for epoch 1 in the 0.3–10 keV band, we employ Cash statistics (Cash 1979) in fitting the data. We require that there be at least one count per bin.

The background spectra themselves cannot be well fit by a single power-law model in band 0.3–10 keV. A broken power-law model (bknpow) in the 0.3–5 keV range works better; for epoch 1, Cstat/ν = 117.66/91, and, for epoch 2, Cstat/ν = 132.98/97. We use the broken power-law plus our slim disk model multiplied by the phabs model in XSPEC to account for absorption and to fit the source and background spectra together. The parameters of the broken power-law component are held fixed to the best-fit values from the background-only fit, except for the normalization, which is permitted to float to allow for differences between the background and source extraction region.

4.3.1. Minimal-prior, "General Case" Fit

We initially fit the two ASASSN-15oi spectra following the same procedure as for ASASSN-14li, with the absorption parameter N_H,i, accretion rate ${\dot{m}}_{i}$, and inclination θ (as well as the broken power-law normalization) allowed to float freely. In this simultaneous, minimal-prior fit, we require M_•, a_•, and θ to be constant between epochs. We fit the spectra across the (M_•, a_•) parameter space: 8 × 10⁵M_⊙ ≤ M_• ≤ 10⁷M_⊙ and −0.9 ≤ a_• ≤ 0.9. Figure 8 shows the best-fit spectra. The corresponding fitting parameters are in the "general case" column of Table 4.

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Table 4.  Fitting Results for ASASSN-15oi for Four Scenarios

	General	Slimming	Delayed	Variable
	Case	Disk	Accretion	Obscuration
NH,1[1020 cm−2]a	10.5 ± 2.3	14.3 ± 5.7	10.1 ± 2.5	26.1 ± 7.9
${\dot{m}}_{1}[\mathrm{Edd}]$ b	0.5 ± 0.1	4.7	1.1 ± 0.3	92
Apl,1[10−6]	2.8 ± 0.5	2.8 ± 0.5	3.5 ± 0.6	2.8 ± 0.5
NH,2[1020 cm−2]	8 ± 3	3.4 ± 1.6	$={{\boldsymbol{N}}}_{{\bf{H}},{\bf{1}}}$	10.9 ± 1.5
${\dot{m}}_{2}[\mathrm{Edd}]$	${84}_{-58}^{+867}$	1.0	2.8 ± 1.2	28
Apl,2[10−6]	2.3 ± 0.4	2.2 ± 0.4	2.4 ± 0.4	2.6 ± 0.4
θ1[◦]c	80 ± 2	${89}_{-1.5}^{+1.0}$	${{\bf{60}}}_{-{\bf{55}}}^{+{\bf{0}}}$	${\bf{58}}.{{\bf{5}}}_{-{\bf{4.7}}}^{+{\bf{1.5}}}$
β	⋯	1.7 ± 0.2	⋯	2.0 ± 0.5
m⋆[M⊙]	⋯	${1.7}_{-0.5}^{+5.1}$	⋯	${100}_{-50}^{+0}$
M•[106M⊙]	3	4	6	8
a•	0.9	0.7	−0.1	−0.9
Cstat/ν	62.83/63	67.1/63	77.02/64	75.33/63
ΔAIC	0	4.27	12.19	12.5

Notes. The four scenarios considered here are a minimum-prior ("general case") scenario and three other idealized scenarios, each detailed in a separate column. Priors, where applied, are marked in bold.

^aWe set N_H,1 = N_H,2 for the delayed accretion scenario, but allow it to float for the slimming disk and variable obscuration scenarios. ^bFor the slimming disk and variable obscuration scenarios, we assume that $\dot{m}$ tracks the theoretical fallback rate ${\dot{M}}_{{\rm{t}}}$ with a peak date of MJD 57248.2; the fitting determines the values of TDE parameters β and m_⋆. In contrast, we allow the accretion rates for the general case and delayed accretion scenarios to float. ^cWe set the range of priors for θ to be (5°, 60°) and (5°, 60°) for the delayed accretion and variable obscuration scenarios, respectively, to prevent disk slimming effects from contributing to their explanatory power. A_pl is the normalization of the broken power-law model. The asymmetric errors are calculated using the XSPEC error command. The general case denotes the best fit under minimal priors from the ΔCstat minima in the (M_•–a_•) plane in Figure 9. The ΔAIC values are calculated with respect to the general case, which fits the data best overall. The slimming disk scenario also fits the data well. The delayed accretion and variable obscuration scenarios fit the data poorly and are disfavored on the basis of their ΔAIC values. Variable obscuration is further disfavored by the exceptionally large required m_⋆.

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Epoch 1 is fit well with a mildly sub-Eddington accretion rate ($\dot{m}=0.5\pm 0.1$), while epoch 2 prefers a large super-Eddington accretion rate ($\dot{m}={84}_{-58}^{+867}$). The epoch 2 accretion rate implies that ≈2M_⊙ is accreted during the ≈200 day high-luminosity plateau in the X-ray light curve (Gezari et al. 2017), which requires the disruption of a massive star (m_⋆ ≳ 4M_⊙). The best-fit θ is nearly edge-on, which suggests that the disk height H/r impacts the fitting result when the accretion rate is high.

Does the accretion rate evolution inferred from the general case fit make sense? The large inferred progenitor star mass already provides reason to question this fit, as disruptions of high-mass stars that could lead to highly super-Eddington accretion rates are relatively rare. Also troubling is that our theoretical calculations show that, for the best-fit (nearly edge-on) θ, the peak X-ray flux in a color-corrected SMBH slim disk is achieved near $\dot{m}\approx 4.5$ (at lower Eddington ratios, the decreasing bolometric luminosity suppresses the X-ray flux, while at higher Eddington ratios, X-rays decline due to the increasing H/r). If $\dot{m}=84$ really is the correct description of epoch 2, then at later times, as the accretion rate decreases to $\dot{m}=4.5$, the X-ray flux should increase to ≈1.7 × 10⁻¹² erg cm⁻² s⁻¹, i.e., four times higher than for epoch 2. The same peak in X-ray flux should also have been observed between epochs 1 and 2, assuming a continuous increase from ${\dot{m}}_{1}\approx 0.5$ to ${\dot{m}}_{2}\approx 84$. Yet this prediction of a double-humped ("Bactrian") light curve was not seen.

The absence of two peaks in the X-ray light curve, before and after epoch 2, does not definitively disprove our general case fit, because of inconvenient gaps in the Swift record, one of ≈100 days between epochs 1 and 2, and another of ≈200 days, starting after the ≈150 day X-ray flux plateau that follows epoch 2 (Gezari et al. 2017). It is possible that both of our predicted X-ray flux peaks occurred during these two data gaps. In summary, both the large m_⋆ required by our general case fit and the lack of an observed double peaked light curve are reasons for doubt, but a more rigorous test would have been possible with higher-cadence X-ray observations.

Figure 9 shows the joint constraints on M_• and a_• for our minimal-prior, general case fits to ASASSN-15oi. The best fit lies at (3 × 10⁶M_⊙, 0.9). The best-fit parameters are all listed in Table 4. Past estimates of the SMBH mass for ASASSN-15oi vary. Using a combination of galaxy scaling relations, Holoien et al. (2016b) estimated M_• ∼ 1.3 × 10⁷M_⊙. Subsequent estimates generally have been lower; Gezari et al. (2017) found ${M}_{\bullet }={2.5}_{-1.8}^{+6.4}\times {10}^{6}{M}_{\odot }$, and Wevers et al. (2019) got ${M}_{\bullet }={0.5}_{-0.4}^{+1.4}\times {10}^{6}{M}_{\odot }$. The range estimated from MOSFiT is ${4}_{-1}^{+1}\times {10}^{6}{M}_{\odot }$. Our marginalized 1σ CL on M_• is consistent with these independent, external estimates, except for Holoien et al. (2016b), with whom we agree to within 2σ.

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Figure 10 shows the time evolution of the best-fit parameters, including the X-ray light curve derived from the spectral fits to both epochs. Because the background dominates the modeled disk flux above 1 keV, we only integrate the flux from 0.3 to 1.0 keV.¹⁶ The flux increases by a factor of six from epochs 1 to 2, consistent with the result from Gezari et al. (2017).

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Our fit to the X-ray spectrum shows that the observed flux at epoch 1 is 6.2 × 10⁻¹⁴ erg cm⁻² s⁻¹, while the flux calculated from the best-fit accretion rate and inclination is higher, 3.2 × 10⁻¹³ erg cm⁻² s⁻¹. From ray-tracing and running our model through XSPEC to account for absorption, we find that 1.0 × 10⁻¹⁴ erg cm⁻² s⁻¹ is blocked by the disk edge, and 2.4 × 10⁻¹³ erg cm⁻² s⁻¹ is absorbed by gas. Because both epochs 1 and 2 are affected similarly by absorption, i.e., there is no significant evolution in N_H in the best fit, the ∼6 times lower flux at epoch 1 relative to epoch 2 is driven primarily by the much lower accretion rate then (${\dot{m}}_{1}\ll {\dot{m}}_{2}$).

We also explore the effects of allowing epoch 2 to have a lower inclination than epoch 1. We obtain a good fit, with a lower Cstat (62.1) than for the general case, and a best-fit pair of SMBH parameters (2 × 10⁶M_⊙, 0.1). This result hints that precession or realignment of the inner accretion disk in ASASSN-15oi may occur. Future TDE discoveries will offer ample opportunity to explore disk precession and realignment. In a somewhat edge-on disk, with π/2 − θ ≈ H/r, a small variation in inclination or aspect ratio can be easily detected; the X-ray flux will exhibit large fluctuations in response to small changes in self-shielding. Studying such reorientation effects is beyond this paper, as our model, which does not yet account for a tilted disk nor ray-trace from a non-equatorial disk, cannot make self-consistent predictions.

Another important caveat is our simplistic treatment of the disk outer edge. As described previously, we model the edge of the disk as a single-temperature blackbody and assume that photons are absorbed completely when they intersect with the 3D disk structure. In reality, an accretion disk is not a solid body with infinite optical depth, but instead has a tenuous, optically thin atmosphere some distance above its local height. An accurate treatment of photon propagation through this atmosphere could alter our fits for highly edge-on models. For now, we assume that the complicated physics of disk atmosphere self-absorption can be encapsulated in the floating absorption parameter N_H.

4.3.2. Testing Physically Motivated Hypotheses

The minimal-prior fit described above suggests a unreasonable accretion rate for epoch 2. Therefore, we consider alternative scenarios, with additional priors, to explain the late-time brightening of ASASSN-15oi's unusual light curve:

• 1.  
  Slimming disk: a disk with a declining accretion rate that increases its observed flux through slimming in an edge-on configuration. In this model, the mass accretion rate does not float freely, but is forced to follow the mass fallback rate ${\dot{M}}_{{\rm{t}}}$, which requires additional parameters such as the penetration parameter β and the progenitor star mass m_⋆. To account for the uncertain physics of X-rays propagating through an optically thin, edge-on disk atmosphere, we allow N_H to float. We do not place a prior on θ, under the expectation that the best fit will involve a high-inclination disk.
• 2.  
  Delayed accretion: a disk that increases its luminosity between epochs by increasing its accretion rate, possibly due to delayed circularization of tidal debris (Gezari et al. 2017). Here the mass accretion rate and N_H are allowed to float freely as long as N_H,1 = N_H,2. We enforce 5 ≤ θ ≤ 60° to avoid self-shielding effects associated with an edge-on disk.
• 3.  
  Variable obscuration: a disk that increases its luminosity due to a large decrease in photoelectric absorption, possibly due to the dilution of a reprocessing layer on larger scales (Gezari et al. 2017). Here we assume that the mass accretion rate follows the mass fallback rate, N_H is allowed to float freely, and 5 ≤ θ ≤ 60° is enforced to eliminate self-shielding.

Two of these idealized models use a theoretical prior for the evolution of the accretion rate, Equation (8). There are three free parameters in this equation, but only two observing epochs, so we fix one parameter, the peak date. We set the time of peak to the discovery date, MJD-57248.2, and allow the other two parameters, β and m_⋆, to float. Adopting this prescription for $\dot{M}$ can allow us to test whether the disk accretion rate follows the mass fallback rate, to constrain β and m_⋆, and to tighten constraints on M_• and a_•. However, in our exploration of these idealized scenarios, we focus on a simpler question: can the X-ray spectra of ASASSN-15oi be fit by assuming efficient circularization and relying on evolution in the absorption and/or in the 3D disk structure to produce the observed late-time flux increase?

The slimming disk fit answers this question affirmatively. In Figure 11, we plot ΔCstat values on the (M_•, a_•) plane. The best-fit (M_•, a_•) pair is (4 × 10⁶M_⊙, 0.7), with Cstat/ν = 67.1/63. The best-fit results are listed in the slimming disk column of Table 4. The marginalized, 1σ CL constraint on M_• is consistent with the range obtained from the Reines & Volonteri (2015) relation between M_• and host galaxy luminosity, the M_•−σ relationship (Wevers et al. 2019), and MOSFiT (Mockler et al. 2019), but is inconsistent with the M_•−M_bulge estimate of Holoien et al. (2016b).

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This slimming disk has more constraining power than our general case fit and can rule out extreme M_• values; e.g., M_• ≳ 6.5 × 10⁶M_⊙ is excluded at the 1σ CL. Unlike for ASASSN-14li, however, a_• is unconstrained. For a larger SMBH mass, the peak fallback rate (in Eddington units) decreases as ${M}_{\bullet }^{-3/2}$, and the radial extent of the disk (in gravitational radii) decreases as ${M}_{\bullet }^{-2/3}$. Both scalings decrease the maximum H/r of the disk. Consequently, a higher M_• makes it easier for relativistic lensing to propagate X-ray photons to observational sightlines in nearly edge-on disks, thus increasing the X-ray flux from the inner disk.

The slimming disk fit also constrains β = 1.7 ± 0.2 and ${m}_{\star }={1.7}_{-0.5}^{+5.1}{M}_{\odot }$. Both of these values differ from those inferred from MOSFiT, where $\beta ={0.91}_{-0.02}^{+0.06}$ and ${m}_{\star }={0.11}_{-0.01}^{+0.04}{M}_{\odot }$. This tension is hard to reconcile, as a smaller β and m_⋆ would drive the accretion rate too low to fit epoch 2. Part of this tension may arise from our assumption that the mass accretion rate follows the mass fallback rate here, in contrast to the greater number of free parameters in the MOSFiT accretion rates. Alternatively, the MOSFiT assumptions about the relationship between the accretion rate and the properties of the optical/NUV photosphere may be incorrect. A joint analysis of the X-ray and optical/NUV observations will prove valuable in future work.

In Figure 12, we plot the evolution of the parameters inferred for the slimming disk scenario. Except for the (prior-constrained) behavior of $\dot{m}$ and H/r, the evolution of the other parameters is similar to that of the general case. Our calculations show that the observed flux in epoch 1 is 6.1 × 10⁻¹⁴ erg cm⁻² s⁻¹, while the total 0.3–1 keV flux generated by the best-fit accretion rate and inclination is 3.9 × 10⁻¹² erg cm⁻² s⁻¹. The bulk of the missing flux, 3.3 × 10⁻¹² erg cm⁻² s⁻¹, has been blocked by the edge-on disk, while another 5.3 × 10⁻¹³ erg cm⁻² s⁻¹ has been absorbed by gas (which could be interpreted as the optically thin atmosphere of a nearly edge-on disk). The low flux of epoch 1 is mainly due to the disk edge blocking part of the generated X-rays.

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If we assume that the inferred value of ${\dot{m}}_{2}=1.0$ is constant over the ≈200 day plateau in the Swift X-ray light curve, we find a lower limit of ΔM ≈ 0.05M_⊙ for the best-fit values of M_• = 4 × 10⁶M_⊙ and a_• = 0.7. Although the lack of temporal coverage makes this estimate quite uncertain, it is compatible with the disruption and efficient accretion of a lower main-sequence star and appears once again to circumvent the missing energy problem.

Figure 12 also shows that the fitted absorption parameter N_H decreases in time. The fitted observational inclination angle θ is nearly edge-on. The two results, taken together, agree with the predictions of some accretion reprocessing theories (Lu & Bonnerot 2020); for an edge-on configuration, there is more debris in the line of sight at early times. Here, as for ASASSN-14li, N_H declines to roughly the host galaxy plus Milky Way value after several hundred days, when the accretion becomes sub-Eddington.

We now compare the results of the general case, slimming disk, delayed accretion, and variable obscuration scenarios. For each scenario, we fit the two spectra through the parameter space 0.8 × 10⁵M_⊙ ≤ M_• ≤ 10⁷M_⊙ and −0.9 ≤ a_• ≤ 0.9. We then use the Akaike information criteria (AIC; Akaike 1974) to test which scenario is favored by the data.¹⁷ Table 4 gives the fitting results and recapitulates the priors. The slimming disk hypothesis fits the data well, with a Cstat = 67.1 and a ΔAIC of only 4.27 relative to the general case fit. On the other hand, the variable obscuration and delayed accretion hypotheses, at least in the simple forms tested here, are disfavored. We now discuss these latter two scenarios in greater depth.

For delayed accretion, the spectra are best fit with an inclination on the high end of its prior range, suggesting that the fit minimizes Cstat (77.02) and ΔAIC (12.19) only by ingesting some of the underlying physics of the slimming disk. The best-fit Cstat here is still 7.92 larger than for the slimming disk. Unless additional X-ray spectral components are invoked, such as a power-law component due to a hot corona, a low accretion rate cannot adequately fit the epoch 1 spectrum at high energies. The challenge comes from the need to simultaneously fit two spectra with comparable levels of hard (≈0.7 keV) X-ray emission, but radically different soft X-ray fluxes. The general case fit accomplishes this task by using (1) a high N_H value to absorb most of the soft X-rays emitted in a low-$\dot{m}$ epoch 1 and (2) an edge-on disk to preferentially obscure hard X-rays in a high-$\dot{m}$ epoch 2. The slimming disk fit succeeds, albeit with a lower quality of fit, by relying on high early-time N_H, and obscuration by an edge-on disk, to absorb X-rays (mostly at softer energies) from a medium-$\dot{m}$ epoch 1 solution. Later on, a greatly diminished N_H and a reduced H/r allow a comparable hard X-ray flux and a much greater soft X-ray flux to emerge from a low-$\dot{m}$ solution in epoch 2. Without these extra degrees of freedom, the delayed accretion scenario fails to fit both epochs satisfactorily.

This failure does not, however, imply that the accretion rate must track the mass fallback rate precisely. Assuming the delayed accretion best-fit (M_•, a_•) pair, we refit the spectra, allowing N_H,1 to float freely. This modification yields a better fit (Cstat = 67.2). However, the accretion rate of epoch 1, $\dot{m}=1.6$, is closer to that of epoch 2, $\dot{m}=2.7$, and the low flux of epoch 1 is mainly due to absorption. The failure of the delayed accretion scenario to find a good fit when θ ≤ 60° is enforced indicates that its goodness of fit is being driven mostly by geometric shielding effects.

The variable obscuration scenario is likewise excluded by its large ΔAIC value of 12.5. Another problem is that the best fit selects an extreme progenitor star mass of ${m}_{\star }={100}_{-50}^{+0}{M}_{\odot }$, the high bound of the permitted prior range. The heavily absorbed slim disk fits neither the soft spectrum nor the hard spectrum well.

In summary, our minimum-prior, general case fits the data best, but with a potentially problematic late-time accretion rate, followed by our proposed idealized slimming disk scenario. The two other simplified scenarios, delayed accretion and variable obscuration, are disfavored. Some degree of geometrical change, such as the diminishing H/r that reduces the self-shielding of an edge-on disk in the slimming disk scenario, is required for these other scenarios to achieve better fits.