The Tidal Disruption Event AT2021ehb: Evidence of Relativistic Disk Reflection, and Rapid Evolution of the Disk–Corona System

We obtained ZTF ²⁶ forced photometry (Masci et al. 2019) in the g and the r bands using the median position of all ZTF alerts up to MJD 59550 (α = 03^h07^m4782, $\delta =+40^\circ {18}^{{\prime} }40\buildrel{\prime\prime}\over{.} 85$). We performed baseline correction following the procedures outlined in Yao et al. (2019).

The peak of the optical light curve probably occurred during Sun occultation and cannot be robustly determined. Therefore, we fitted a five-order polynomial function to the r_ZTF-band observations, which suggested that the optical maximum light was around MJD≈59321. Hereafter we use δ t to denote rest-frame days relative to MJD 59321. The Galactic extinction-corrected ZTF light curves are shown in Figure 1. All ZTF photometry is provided in Appendix A.1 (Table 8).

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We obtained additional ugri photometry using the Spectral Energy Distribution Machine (SEDM; Blagorodnova et al. 2018; Rigault et al. 2019) on the robotic Palomar 60 inch telescope (P60; Cenko et al. 2006), and the optical imager on the Liverpool Telescope (LT; Steele et al. 2004). The SEDM photometry was host-subtracted using the automated pipeline FPipe (Fremling et al. 2016). The LT photometry was host-subtracted using SDSS images.

We found a mismatch between the SEDM/LT gr photometry and the ZTF photometry. This is probably a result of different reference images being used. The ZTF difference photometry is more reliable since the reference images were constructed using P48 observations taken in 2018–2019. The reference images of SEDM/LT come from SDSS images (taken in 2005), and long-term variability of the galactic nucleus will render the difference photometry less robust. Therefore, we present the SEDM and LT photometry in Appendix A.1 (Table 8), but exclude them in the following analysis.

We obtained low-resolution optical spectroscopic observations using the Low Resolution Imaging Spectrograph (LRIS; Oke et al. 1995) on the Keck I telescope, the Double Spectrograph (DBSP; Oke & Gunn 1982) on the 200 inch Hale telescope, the integral field unit (R≈100) spectrograph of SEDM, and the De Veny Spectrograph on the Lowell Discovery Telescope (LDT). We also obtained a medium-resolution spectrum using the Echellette Spectrograph and Imager (ESI; Sheinis et al. 2002) on the Keck II telescope.

Figure 2 shows the low-resolution spectra. The instrumental details and an observing log can be found in Appendix B.

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AT2021ehb was observed by the X-Ray Telescope (XRT; Burrows et al. 2005) and the Ultra-Violet/Optical Telescope (UVOT; Roming et al. 2005) on board Swift under our GO program 1619088 (as ZTF21aanxhjv; target ID 14217; PI: Gezari) and a series of time-of-opportunity (ToO) requests (PI: Yao). All Swift data were processed with heasoft v6.29 c.

2.4.1. XRT

2.4.2. UVOT

AT2021ehb was observed by NICER (Gendreau et al. 2016) under Director's Discretionary Time (DDT) programs on 2021 March 26, 2021 July 2–7, and from 2021 November 13 to 2022 March 29 (PIs: Yao, Gendreau, Pasham). The NICER data were processed using nicerdas v9 (2021-08-31_V008c). We ran nicerl2 to obtain the cleaned and screened event files. Background was computed using the nibackgen3C50 tool (Remillard et al. 2022). Following the screening criteria suggested by Remillard et al. (2022), we removed "good time intervals" (GTIs) with hbgcut = 0.05 and s0cut = 2.0.

We extracted one spectrum for each obsID, excluding obsIDs with 0.3–1 keV background rate>0.2 count s⁻¹ or 4–12 keV background rate>0.1 count s⁻¹. Using observations bracketed by the two NuSTAR observations, we also produced two NICER spectra with exposure times of 8.2 ks and 36.6 ks, which we jointly analyzed with the NuSTAR spectra (see Section 4.4.1 and Section 4.4.2).

All NICER spectra were binned using the optimal binning scheme (Kaastra & Bleeker 2016), requiring at least 20 counts per bin. Following the NICER calibration memo, ²⁹ we added systematic errors of 1.5% with grppha.

We obtained two epochs of follow-up observations with XMM-Newton under our Announcement of Opportunity program (PI: Gezari), on 2021 August 4 (obsID 0882590101) and 2022 January 25 (obsID 0882590901). The observations were taken in full-frame mode with the thin filter using the European Photon Imaging Camera (EPIC; Strüder et al. 2001).

The observation data files were reduced using the XMM-Newton Standard Analysis Software (Gabriel et al. 2004). The raw data files were then processed using the epproc task. Since the pn instrument generally has better sensitivity than MOS1 and MOS2, we only analyze the pn data. Following the XMM-Newton data analysis guide, to check for background activity and generate GTIs, we manually inspected the background light curves in the 10–12 keV band. Using the evselect task, we only retained patterns that correspond to single and double events (PATTERN< = 4).

The source spectra were extracted using a source region of r_src = 35'' around the peak of the emission. The background spectra were extracted from an r_bkg = 108'' region located in the same CCD. The ancillary response files (ARFs) and response matrix files (RMFs) files were created using the arfgen and rmfgen tasks, respectively. We grouped the spectra to have at least 25 counts per bin, and limited the over-sampling of the instrumental resolution to a factor of five.

The location of AT2021ehb was scanned by eROSITA as part of the planned eight all-sky surveys. Hereafter, eRASSn refers to the nth eROSITA all-sky survey. ³⁰ During eRASS4, AT2021ehb was independently identified by SRG as a TDE candidate. A log of SRG observations is given in Table 1. We grouped the eRASS4 and eRASS5 spectra to have at least three counts per bin.

Table 1. Log of SRG Observations of AT2021ehb

eRASS	MJD	δ t	0.3–10 keV flux
		(days)	(10−13 erg s−1 cm−2)
1	58903.59–58904.59	−409.5	<0.25
2	59083.36–59084.70	−232.8	<0.23
3	59253.16–59254.16	−66.1	<0.23
4	59442.45–59443.62	+119.9	${76.8}_{-2.4}^{+2.5}$
5	59624.53–59625.70	+298.7	${30.7}_{-2.3}^{+2.4}$

Note. Upper limits are computed assuming an absorbed power-law (PL) spectrum with Γ = 2.5 and N_H = 9.97 × 10²⁰ cm⁻², and presented at 90% confidence.

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We obtained NuSTAR (Harrison et al. 2013) observations under a preapproved ToO program (PI: Yao; obsID 80701509002) and a DDT program (PI: Yao; obsID 90801501002). The first epoch was conducted from 2021 November 18.8 to 19.9 with an exposure time of 43.2 ks. The second epoch was conducted from 2022 January 10.4 to 12.1 with an exposure time of 77.5 ks.

To generate the first epoch's spectra for the two photon-counting detector modules (FPMA and FPMB), source photons were extracted from a circular region with a radius of r_src = 40'' centered on the apparent position of the source in both FPMA and FPMB. The background was extracted from an r_bkg = 80'' region located on the same detector. For the second epoch, since the source was brighter, we used a larger source radius of r_src = 70'', and a smaller background radius of r_bkg = 65''.

All spectra were binned first with ftgrouppha using the optimal binning scheme developed by Kaastra & Bleeker (2016), and then further binned to have at least 20 counts per bin.

We began a monitoring program of AT2021ehb using the Very Large Array (VLA; Perley et al. 2011) under program 20B-377 (PI Alexander). All of the data were analyzed following standard radio continuum image analysis procedures in the Common Astronomy Software Applications (CASA; McMullin et al. 2007). The first three observations used a custom data reduction pipeline (pwkit; Williams et al. 2017), while the final observation used the standard NRAO pipeline. AT2021ehb was not detected in any of our observations. All data were imaged using the CASA task clean. We computed 3σ upper limits using the stats command within the imtool package of pwkit. The result of the first epoch was reported in Alexander et al. (2021). The full results are presented in Table 2.

In Figure 3, we compare the radio luminosity of AT2021ehb with other UV- and optically selected TDEs. We note that AT2021ehb looks to be significantly (by more than an order of magnitude) radio-underluminous compared to previously observed non-jetted TDEs at similar times post-peak. It has the deepest limits on any TDE radio emission at>150 days post-discovery.

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The host galaxy absorption lines are prominent in the optical spectra (see Figure 2). Using our medium-resolution (R = 5350) spectrum taken with Keck II/ESI, we measured the line centers of strong absorption lines, and determined the redshift to be z = 0.0180.

Following previous TDE work (Wevers et al. 2017, 2019b; French et al. 2020), we measured the stellar velocity dispersion by fitting the normalized ESI spectrum (see pre-processing procedures in Appendix B) with the penalized pixel-fitting (pPXF) software (Cappellari & Emsellem 2004; Cappellari 2017). pPXF fits the absorption line spectrum by convolving a library of stellar spectra with Gauss-Hermite functions. We adopted the ELODIE v3.1 high-resolution (R = 42,000) template library (Prugniel & Soubiran 2001; Prugniel et al. 2007).

To robustly measure the velocity dispersion and the associated uncertainties, we performed 1000 Monte Carlo (MC) simulations, following the approach adopted by Wevers et al. (2017). In each fitting routine, we masked wavelength ranges of common galaxy emission lines and hydrogen Balmer lines. The derived velocity dispersion is $\sigma ={92.9}_{-5.2}^{+5.3}\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at the 95% confidence interval.

According to the M_BH–σ relation (Kormendy & Ho 2013), the measured σ corresponds to a black hole mass of log (M_BH/M_⊙) = 7.03 ± 0.15(stat) ± 0.29(sys), where 0.29 is the intrinsic scatter of the M_BH–σ relation. If adopting the Ferrarese & Ford (2005) M_BH–σ relation, then log (M_BH/M_⊙) =6.60 ± 0.20(stat) ± 0.34(sys). Hereafter we adopt the result from the Kormendy & Ho (2013) relation because it includes more low-mass galaxies.

We note that although the Kormendy & Ho (2013) relation was originally calibrated mainly at an M_BH regime that is too massive to produce a TDE, recent studies show that the same relation holds in the dwarf galaxy regime (Baldassare et al.2020).

We constructed the pre-TDE host galaxy SED using photometry from SDSS, the Two Micron All-Sky Survey (2MASS; Skrutskie et al. 2006), and the AllWISE catalog (Cutri et al. 2014). The photometry of the host is shown in Table 3.

Our SED fitting approach is similar to that described in van Velzen et al. (2021). We used the flexible stellar population synthesis (FSPS) code (Conroy et al. 2009), and adopted a delayed exponentially declining star formation history (SFH) characterized by the e-folding timescale τ_SFH. The Prospector package (Johnson et al. 2021) was utilized to run a Markov Chain Monte Carlo sampler (Foreman-Mackey et al. 2013). We show the best-fit model prediction of the host galaxy optical spectrum at the bottom of Figure 2.

From the marginalized posterior probability functions, we obtain the total galaxy stellar mass log $({M}_{* }/{M}_{\odot })={10.18}_{-0.02}^{+0.01}$, the metallicity, $\mathrm{log}Z=-0.57\pm 0.04$, ${\tau }_{\mathrm{SFH}}={0.19}_{-0.07}^{+0.18}$ Gyr, the population age, ${t}_{\mathrm{age}}={12.1}_{-0.6}^{+0.3}$ Gyr, and negligible host reddening (E_B−V,host = 0.01 ± 0.01 mag). The best-fit SED model is shown in Figure 5.

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Following Gezari (2021), we use the M_BH–M_* relation from Greene et al. (2020) to obtain a black hole mass of log (M_BH/M_⊙) = 7.14 ± (0.10+0.79), where 0.79 is the intrinsic scatter of the scaling relation. This is consistent with the M_BH inferred from the M_BH–σ relation (Section 3.1).

To summarize, the host galaxy of AT2021ehb has a total stellar mass of M_*≈10^10.18 M_⊙ and a BH mass of M_BH≈10^7.03 M_⊙. The measured black hole mass is on the high end of the population of optically selected TDEs (French et al. 2020; Nicholl et al. 2022), and is too massive to disrupt a white dwarf (Rosswog et al. 2009).

To capture the general trend of AT2021ehb's UV/optical photometric evolution, we fit the data in each filter using a combination of five-order polynomial functions and Gaussian process smoothing, following procedures described in Appendix B.4 of Yao et al. (2020). The model fits in r_ZTF, uvw1, and uvw2 are shown as semitransparent lines in Figure 1.

We then define a set of "good epochs" close in time to actual multiband measurements, and fit a Planck function to each set of fluxes to determine the effective temperature T_bb, photospheric radius R_bb, and blackbody luminosity of the UV/optical emitting component L_bb. We initially assume E_B−V,host = 0 mag, and then repeat the procedure under different assumptions about the host reddening. We find that the fitting residual monotonically increases as E_B−V,host increases from 0 mag to 0.2 mag, suggesting negligible host reddening. Therefore, for the reminder of this discussion, we assume E_B−V,host = 0 mag.

We also define a set of "ok epochs" where we only have photometric observations in the optical (or only in the UV). Due to a lack of wavelength coverage, T_bb and R_bb can not be simultaneously constrained. As such we fix the T_bb values by interpolating the T_bb evolution of "good epochs," and fit for R_bb values of "ok epochs."

The physical parameters derived from the blackbody fits are presented in Table 10 (Appendix A.2) and shown in Figure 6, where they are compared with a sample of recent TDEs with multiple X-ray detections. We have measured the blackbody parameters of other TDEs using the same procedures described above.

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While the temperature of AT2021ehb (T_bb∼2.5 × 10⁴ K) is typical among optical and X-ray bright TDEs, its peak radius (R_bb∼3 × 10¹⁴ cm) and luminosity (L_bb∼3 × 10⁴³ erg s⁻¹) are at the low end of the distributions. We note that in the ZTF-I sample of 30 TDEs (Hammerstein et al. 2022), only two objects (AT2020ocn and AT2019wey) have peak radii smaller than that of AT2021ehb (see the discussion in Section 5.4.1).

Figure 2 shows that no broad line is evident in the optical spectra of AT2021ehb. To search for weak spectral features from the TDE, we fit the Galactic extinction-corrected long-slit spectra in rest-frame 3600–5400 Å using a combination of blackbody emission and host galaxy contribution: f_λ,obs = A₁ f_λ,BB +A₂ f_λ,host. Here ${f}_{\lambda ,{BB}}=\pi {B}_{\lambda }({T}_{\mathrm{bb}})({R}_{\mathrm{bb}}^{2}/{D}_{L}^{2})$, where T_bb and R_bb are obtained by linearly interpolating the blackbody parameters derived in Section 4.1 at the relevant δ t. f_λ,host is the predicted host galaxy spectrum obtained in Section 3.2 convolved with the instrumental broadening σ_inst (see Appendix B). A₁ and A₂ are constants added to account for unknown factors, including the varying amount of host galaxy flux falling within the slit (which depends on the slit width, slit orientation, seeing conditions, and target acquisition), uncertainties in the absolute flux calibration, and the adopted blackbody parameters. We note that f_λ,host is the predicted spectrum for the whole galaxy, and therefore might not be a perfect description of the bulge spectrum.

The fitting results are shown in Figure 7. We mark locations of emission lines commonly seen in TDEs, including Balmer lines, He II, the Bowen fluorescence lines of N III and O III, as well as low-ionization Fe II lines (Blanchard et al. 2017; Wevers et al. 2019a; van Velzen et al. 2021). The observed spectra of AT2021ehb can be well described by a blackbody continuum (dotted lines) plus host galaxy contribution. The spectra at δ t>170 days are mostly from the host, and therefore it is not very surprising that no discernible TDE lines were detected. However, at δ t< 170 days, the blackbody component contributes 25%–80% of the total flux. As such, it is surprising that no prominent lines from the TDE itself can be identified. We further discuss this result in Section 5.4.1.

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The middle panel of Figure 8 shows the XRT and NICER (all binned by obsID) light curves. The lower panel of Figure 8 shows the evolution of the hardness ratio, defined as HR ≡ (H − S)/(H+S), where H is the number of net counts in the hard band, and S is the number of net counts in 0.3–1 keV. For XRT, we take 1–10 keV as the hard band, while for NICER, we take 1–4 keV.

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X-rays were not detected at δ t< 0. Pre-peak X-ray upper limits are provided by Swift/XRT (<10^40.9 erg s⁻¹; Table 9) and SRG/eROSITA (<10^40.2 erg s⁻¹; Table 1).

X-rays were first detected by XRT at δ t = 73.9 days. The exact time of the X-ray onset cannot be accurately constrained. The count rate initially exhibited strong variability from δ t = 73.9 days to δ t = 82.3 days, and then gradually increased out to δ t = 250 days. At the same time, the HR gradually increased. From δ t = 250 days to δ t = 271 days, both the X-ray flux and the hardness stayed at the maximum values.

From δ t = 271.0 days to δ t = 273.7 days, the NICER net count rate suddenly decreased by a factor of 10 (Yao et al. 2022b). At the same time, the HR significantly decreased. After an X-ray plateau of≈50 days, the XRT net count rate further decreased drastically by a factor of six (from δ t = 320.9 days to δ t = 327.2 days).

In this subsection, we first present a joint spectral analysis of contemporaneous data sets obtained from NICER and NuSTAR, including the first epoch in 2021 November 18–19 (Section 4.4.1) and the second epoch in 2022 January 10–12 (Section 4.4.2). These observations are of high S/N and cover a wide energy range. As such, the fitting results can guide us to choose appropriate spectral models to fit spectra with lower S/Ns. We then perform analysis on data sets obtained by single telescopes, including XMM-Newton (Section 4.4.3), SRG (Section 4.4.4), Swift/XRT (Section 4.4.5), and NICER (Section 4.4.6).

All spectral fitting was performed with xspec (v12.12, Arnaud 1996). We used the vern cross sections (Verner et al. 1996). The wilm abundances (Wilms et al. 2000) were adopted in Sections 4.4.1 and 4.4.2, whereas the Anders & Grevesse (1989) abundances were adopted in Sections 4.4.3, 4.4.4, 4.4.5, and 4.4.6.

4.4.1.  NICER+NuSTAR First Epoch, 2021 November

We chose energy ranges where the source spectrum dominates over the background. For NICER we used 0.3–4 keV. For NuSTAR/FPMA we used 3–23 keV, and for FPMB we used 3–20 keV. ³¹ All data were fitted using χ²-statistics.

For all spectral models described below, we included the Galactic absorption using the tbabs model (Wilms et al. 2000), with the hydrogen-equivalent column density N_H fixed at 9.97 × 10²⁰ cm⁻² (HI4PI Collaboration et al. 2016). We shifted the TDE emission using the convolution model zashift, with the redshift z fixed at 0.018. We included possible absorption intrinsic to the source using the ztbabs model. We also included a calibration coefficient (constant; Madsen et al. 2017) between FPMA, FPMB, and NICER, with C_FPMA ≡ 1. This term also accounts for the differences in the mean flux between NuSTAR and NICER that results from intrinsic source variability.

First, we fitted the spectrum with a power law (PL), and obtained a photon index of Γ≈2.7. The fit is unacceptable, with the reduced χ² being ${\chi }_{{\rm{r}}}^{2}=3.44$ for 144 degrees of freedom (dof). The residuals are most significant between 0.3 and 2 keV, suggesting the existence of a (thermal) soft component. Therefore, we changed the PL to simpl*thermal_model. Here simpl is a Comptonization model that generates the PL component via Compton scattering of a fraction (f_sc) of input seed photons (Steiner et al. 2009). The flag R_up was set to 1 to only include up-scattering. We experimented with three different thermal models: a blackbody (bbody), a multicolor disk (MCD; diskbb; Mitsuda et al. 1984), and a single-temperature thermal plasma (bremss; Kellogg et al. 1975), resulting in ${\chi }_{{\rm{r}}}^{2}=1.33$, 1.15, and 1.35 (for dof = 142), respectively. The fit statistics favors an MCD.

The best-fit result with an MCD, defined as model (1a), is shown in Figure 9. We present the best-fit parameters in Table 4. Here, T_in is the inner disk temperature, and ${R}_{\mathrm{in}}^{* }\equiv {R}_{\mathrm{in}}\sqrt{\cos i}$ is the apparent inner disk radius times the square root of $\cos i$, where i is the system inclination. ${R}_{\mathrm{in}}^{* }$ is inferred from the normalization parameter of diskbb. Model (1a) gives a good fit with ${\chi }_{{\rm{r}}}^{2}=163/142=1.15$.

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Table 4. Modeling of the First Joint NICER and NuSTAR Observations, δ t = 212 Days

Component	Parameter	(1a)
constant	CFPMB	${1.03}_{-0.05}^{+0.06}$
	CNICER	${0.85}_{-0.05}^{+0.06}$
ztbabs	NH (1020 cm−2)	<0.75
simpl	Γ	2.29 ± 0.05
	fsc	${0.35}_{-0.03}^{+0.02}$
diskbb	Tin (eV)	${164}_{-9}^{+6}$
	${R}_{\mathrm{in}}^{* }$ (104 km)	${25.5}_{-2.0}^{+4.4}$
⋯	χ2/dof	163.17/142 = 1.15

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4.4.2.  NICER+NuSTAR Second Epoch, 2022 January

We chose energy ranges where the source spectrum dominates over the background. For NICER, we used 0.3–7.0 keV; for NuSTAR FPMA and FPMB, we used 3–30 keV. All data were fitted using χ²-statistics. Unlike in Section 4.4.1, here we use tbfeo to model the Galactic absorption. Compared with tbabs, tbfeo allows the O and Fe abundances (A_O, A_Fe) to be free.

We adopted a continuum model of simpl*diskbb, defined as (2a). The result, with ${\chi }_{{\rm{r}}}^{2}=2.04$, is shown in Figure 10 and the upper-left panel of Figure 11. The best-fit parameters are given in Table 5. The residual plot clearly indicates the existence of unmodeled spectral features and a significant offset between NuSTAR and NICER at 6–7 keV.

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Table 5. Modeling of the Second Joint NICER and NuSTAR Observations, δ t = 264 Days

Component	Parameter	(2a)	(2b)	(2c)	(2d)
constant	CFPMB	1.03	1.03 ± 0.01	1.03 ± 0.01	1.03 ± 0.01
	CNICER	0.98	1.02 ± 0.01	1.02 ± 0.01	1.03 ± 0.01
tbfeo	AO	1.40	${1.48}_{-0.08}^{+0.11}$	1.26 ± 0.13	${1.45}_{-0.07}^{+0.10}$
	AFe	1.80	${2.07}_{-0.64}^{+0.63}$	${1.99}_{-0.62}^{+0.61}$	${2.37}_{-0.39}^{+0.50}$
ztbabs	NH (1020 cm−2)	0.00	<0.12	0.70 ± 0.28	<0.01
diskbb	Tin (eV)	187	${198}_{-6}^{+8}$	257 ± 8	${180}_{-2}^{+7}$
	${R}_{\mathrm{in}}^{* }$ (104 km)	31.7	${28.4}_{-1.7}^{+1.2}$	${10.5}_{-0.9}^{+1.0}$	47.3 ± 2.8
simpl	Γ	2.09	2.11 ± 0.01	⋯	2.26 ± 0.01
	fsc	0.52	${0.49}_{-0.03}^{+0.02}$	⋯	0.61 ± 0.01
Gaussian	Eline (keV)	⋯	${4.92}_{-0.71}^{+0.36}$	⋯	⋯
	σline (keV)	⋯	${2.18}_{-0.32}^{+0.50}$	⋯	⋯
	Norm (10−4 ph cm−2 s−1)	⋯	${2.52}_{-0.51}^{+1.01}$	⋯	⋯
relxill	q1 = q2	⋯	⋯	3 (frozen)	⋯
	a	⋯	⋯	0 (frozen)	⋯
	i (◦)	⋯	⋯	${43.4}_{-9.6}^{+8.5}$	⋯
	Rin (RISCO)	⋯	⋯	1 (frozen)	⋯
	Rout (Rg)	⋯	⋯	400 (frozen)	⋯
	Γ	⋯	⋯	1.86 ± 0.02	⋯
	logξ (erg cm s−1)	⋯	⋯	${4.09}_{-0.12}^{+0.20}$	⋯
	AFe	⋯	⋯	${1.86}_{-0.63}^{+1.46}$	⋯
	Ecut (keV)	⋯	⋯	${54.0}_{-9.5}^{+13.4}$	⋯
	RF	⋯	⋯	1 (frozen)	⋯
	Norm (10−5)	⋯	⋯	${6.1}_{-0.40}^{+0.40}$	⋯
xstar	NH (1023 cm−2)	⋯	⋯	⋯	${2.22}_{-0.86}^{+0.49}$
	$\mathrm{log}\xi$ (erg cm s−1)	⋯	⋯	⋯	${1.51}_{-0.32}^{+0.34}$
	fcover	⋯	⋯	⋯	0.31 ± 0.02
	Redshift	⋯	⋯	⋯	0 (frozen)
⋯	χ2/dof	609.43/299 = 2.04	330.72/296 = 1.12	306.64/295 = 1.04	318.23/296 = 1.08

Note. Parameter uncertainties of model (2a) cannot be calculated since ${\chi }_{{\rm{r}}}^{2}\gt 2$.

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First, we study whether this offset is brought about by a cross-calibration difference between NICER and NuSTAR. To this end, we replaced constant with crabcorr (Ludlam et al. 2022), which multiplies the spectrum by a PL of C · E^−ΔΓ. When ΔΓ = 0, crabcorr is equivalent to constant. We fixed ΔΓ_FPMA = ΔΓ_FPMB = 0, and allow ΔΓ_NICER to be free. The best-fit model gives ${\rm{\Delta }}{{\rm{\Gamma }}}_{\mathrm{NICER}}=-{0.128}_{-0.023}^{+0.014}$, which is too large compared with the value of ΔΓ_NICER≈− 0.06 found by Ludlam et al. (2022). Therefore, we conclude that a difference in the cross-calibration slope is likely not the primary reason for the 6–7 keV offset.

Next, we investigate whether this offset is caused by imperfect NICER calibration at 2–3 keV. NICER effective area changes rapidly in the 2–3 keV band. Calibration issues in that range may cause the model to overestimate the data at 2–3 keV, and to badly underestimate it above 3 keV. As a test, we performed the fit omitting the 2–3 keV region in the NICER data. However, the best-fit result still leaves a significant offset between NICER and NuSTAR at 6–7 keV, similar to that shown in Figure 10.

We are left to conclude that the offset is likely caused by either an underestimate of NICER background at the high-energy end or systematic uncertainties in NICER calibration that is not well characterized. This conjecture is based on the fact that NICER uses X-ray concentrators optics, and its 3–7 keV background is>10 times brighter than that of NuSTAR (Figure 10). On the other hand, NuSTAR adopts X-ray focusing optics, which enables more robust background estimation using regions close to the object of interest.

In the following, we attempted to improve the fit by three approaches: adding a Gaussian line, adding reflection emission features, and adding absorption features.

Modeling with a Gaussian Line Profile—The result with adding a Gaussian line component is shown in the upper-right panel of Figure 11. This model, defined as (2b), provides a much better fit compared with (2a). The best-fit parameters (Table 5) give a very broad emission profile with a central energy at E_line∼5 keV and a line width of σ_line∼2 keV. If the 3–7 keV spectral feature indeed comes from an emission line, its central energy is different from the emission line at E_line∼8 keV that has been found in the jetted TDE Sw J1644+57, which has been interpreted as highly ionized iron Kα emission blueshifted by∼0.15c (Kara et al. 2016; Thomsen et al. 2019). Instead, what is shown here indicates the possible existence of a relativistically broadened iron line (either redshifted or with a more distorted red wing).

Modeling with Disk Reflection—In this method, we fit the data using a combination of MCD and relativistic reflection from an accretion disk.

We utilize the self-consistent relxill model to describe the direct PL component and the reflection part (García et al. 2014; Dauser et al. 2014). In relxill, we fixed the outer disk radius (R_out) at a fiducial value of 400 R_g, since it has little effect on the X-ray spectrum. The redshift parameter in relxill was fixed at 0 since the host redshift was already included by the zashift model. To reduce the complexity of this model, we froze the reflection fraction R_F (ratio of the reflected to primary emission; Dauser et al. 2016) at 1. The inner and outer emissivity index q were fixed at 3 throughout the accretion disk, making R_break obsolete. We assume the inner disk radius is at the innermost stable circular orbit (ISCO), i.e., R_in = R_ISCO. Other parameters in relxill include the PL index of the incident spectrum Γ, the cutoff energy of the PL E_cut, the black hole spin a, the inclination i, the ionization of the accretion disk ξ, the iron abundance of the accretion disk A_Fe, and the normalization parameter Norm_rel. We first fit the data allowing a to be free, finding that the fit is not sensitive to a. Therefore, we performed the fit with a fixed at zero.

The best-fit model, hereafter (2c), gives χ²/dof =306.65/296 = 1.04 and is shown in the lower-left panel of Figure 11. The best-fit model parameters are given in Table 5. In Figure 12, the solid black line shows the best-fit model; Modifications of the best-fit model are shown as dotted (if a is changed from 0 to 0.998), dashed (if i is changed from 434 to 70°), and dashed–dotted (if logξ is changed from 4.09 to 3.50) lines. The shape and width of the extremely broad iron emission are mainly determined by the high disk ionization state and the moderate inclination. We note that the best-fit ionization of ξ∼10⁴ erg cm s⁻¹ is greater than the typical values observed in Seyfert 1 active galactic nuclei (AGNs; Walton et al. 2013; Ezhikode et al. 2020).

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Modeling with Absorbers—In this method, we attempt to improve the fit by adding absorption features. First, we added partial covering of a neutral absorber using the pcfabs model. In pcfabs, a fraction f_cover of the X-ray source is seen through a neutral absorber with hydrogen-equivalent column density N_H, while the rest is assumed to be observed directly. The best-fit model gives χ²/dof = 386/297 = 1.30. If we add a new free parameter (redshift of the neutral absorber) by replacing pcfabs with zpcfabs, χ²/dof becomes 370/296 = 1.25. However, both models leave 5–8 keV flux excess in the residual.

Therefore, we next allow the absorber to be partially ionized by replacing zpcfabs with a photoionized absorber. This is also motivated by the fact that a good fit to the Chandra LETG observation conducted on 2021 November 29 was found with such a model by Miller et al. (2022a). This fit utilized the zxipcf model (Reeves et al. 2008), which is a grid of photoionization models computed by the XSTAR code (Kallman & Bautista 2001). However, Reynolds et al. (2012) noted that zxipcf only has a very coarse sampling in ionization space, and so in this work, we use an updated XSTAR grid that is suitable for use with AGNs (computed in Walton et al. 2020). This grid assumes an ionizing continuum of Γ = 2 and a velocity broadening of 100 km s⁻¹, and allows the ionization parameter, column density, absorber redshift, and both the oxygen and iron abundances to be varied as free parameters (although for simplicity we assume these abundances are solar). Fitting the data with the redshift of the absorber fixed at zero yields χ²/dof = 318.2/296 = 1.08. If the redshift is allowed to be free, we have χ²/dof =317.7/295 = 1.08. Since χ² only reduces by 0.5 for 1 dof, the redshift parameter cannot be well constrained by our data. Therefore, we name the model with the absorber redshift fixed at zero as (2d), and show it in the lower-right panel of Figure 11. The model parameters are given in Table 5. In Figure 12, the solid crimson line shows a high-resolution version of model (2d).

Model Comparison and Comments—Between (2b) and (2c), we consider (2c) to be superior since (i) its χ² is smaller by 24 for only 1 dof, and (ii) it adopts a physically motivated model instead of a mathematical function.

To compare (2c) and (2d), we use the Bayesian information criterion (BIC) to assess the goodness of fit. Here

$$\begin{eqnarray}&&\mathrm{BIC}=k\cdot \mathrm{ln}(N)-2\mathrm{ln}{L}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&=\,k\cdot \mathrm{ln}(N)+{\chi }^{2}+\mathrm{constant}\end{eqnarray} \tag{ 2 }$$

where k is the number of free parameters, N is the number of spectral bins, and ${ \mathcal L }$ is the maximum of the likelihood function. Models with lower BIC values are favored. According to Raftery (1995), a BIC difference between 2 and 6 is positive, a difference between 6 and 10 is strong, and a difference greater than 10 is very strong. Since BIC(2c) − BIC(2d) = − 5.9, model (2c) is slightly favored over (2d). The energy range over which model (2c) performs better than (2d) is ∼8–12 keV. This is because absorption by ionized iron adds a relatively sharp flux decrease at ∼7 keV, while the blue wing of the iron emission in relxill is smoother (see the lower-right panel of Figure 11).

We note that the residual below 0.7 keV is strong in all model fits, and is likely caused by underestimated NICER calibration uncertainties at the lowest energies.

4.4.3. XMM-Newton Analysis

We chose energy ranges where the source spectrum dominates over the background. For XMM E1, this is 0.2–2.6 keV, while for XMM E2 this is 0.2–7.0 keV. All data were fitted using χ²-statistics. Following Sections 4.4.1 and 4.4.2, all models described below have been multiplied by tbabs*ztbabs*zashift to include Galactic absorption, host absorption, and host redshift.

Although the XMM E1 spectrum is very soft, a single MCD results in a poor fit and leaves a large residual above 1 keV, suggesting the existence of a nonthermal component. A continuum model of simpl*diskbb gives a much better fit with ${\chi }_{{\rm{r}}}^{2}=1.35$. The best-fit model is shown in the upper panel of Figure 13. The XMM E2 spectrum is much harder than that from XMM E1. Fitting with simpl*diskbb gives a good fit with ${\chi }_{{\rm{r}}}^{2}=1.19$ (see Figure 13, lower panel).

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We note that although the ${\chi }_{{\rm{r}}}^{2}$ of our best-fit XMM-Newton models are acceptable, there seems to be some systematic residuals. For example, eight consecutive bins of positive residuals are seen in the 1.7–2.6 keV XMM E1 data. A possible explanation is that there exist spectral features created by absorbing materials in the TDE system, such as the blueshifted absorption lines reported in the TDE ASASSN-14li (Miller 2015; Kara et al. 2018). Seven consecutive bins of positive residuals are seen in the 4.0–7.0 keV XMM E2 data. This might indicate the existence of disk reflection features, such as an iron emission line. We note that 2–4 days after our second XMM-Newton epoch, XMM-Newton/EPIC observations obtained under another program also reveals the existence of interesting features in the iron K band (Miller et al. 2022b). More detailed modeling of the XMM-Newton spectra is beyond the scope of this paper, and is encouraged in future work.

4.4.4. SRG/eROSITA Analysis

We chose energy ranges where the source spectrum dominates over the background. For eRASS4 this range is 0.2–3 keV, while for eRASS5 this range is 0.2–2 keV. All data were fitted with the C-statistic (Cash 1979).

Following Section 4.4.1 and Section 4.4.3, we fitted the SRG/eROSITA spectra with tbabs*ztbabs*zashift* simpl*diskbb. For the eRASS5 spectrum, since the source is only above background at 0.2–2 keV, the PL index cannot be constrained from the SRG/eROSITA spectrum alone. Therefore, we fixed Γ at the best-fit value of the XMM E2 spectrum (Table 6, Γ = 2.92), and allowed other parameters to be free. This choice is based on the fact that the XMM E2 and eRASS5 observations appear to show the same properties on the light curve and hardness evolution diagrams (Figure 8).

Table 6. Modeling of Two XMM-Newton Observations

Component	Parameter	XMM E1	XMM E2
ztbabs	NH (1020 cm−2)	${1.09}_{-0.45}^{+0.99}$	<1.22
diskbb	Tin (eV)	${68}_{-4}^{+1}$	125 ± 8
	${R}_{\mathrm{in}}^{* }$ (104 km)	${511}_{-75}^{+144}$	${39}_{-6}^{+10}$
simpl	Γ	>4.57 a	2.92 ± 0.15
	fsc	${0.13}_{-0.01}^{+0.03}$	0.16 ± 0.03
⋯	χ2/dof	70.26/52 = 1.35	97.49/82 = 1.19

^a Upper limit of Γ is at 5.

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The fitting results are shown in Figure 14. The best-fit parameters are shown in Table 7.

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Table 7. Modeling of Two SRG/eROSITA Observations

Component	Parameter	eRASS4	eRASS5
ztbabs	NH (1020 cm−2)	${0.21}_{-0.20}^{+2.19}$	<3.41
diskbb	Tin (eV)	${89}_{-13}^{+7}$	${96}_{-22}^{+32}$
	${R}_{\mathrm{in}}^{* }$ (104 km)	${210}_{-38}^{+179}$	${73}_{-21}^{+243}$
simpl	Γ	${4.15}_{-0.73}^{+0.82}$	2.92 (frozen)
	fsc	${0.14}_{-0.08}^{+0.12}$	${0.21}_{-0.09}^{+0.06}$
⋯	cstat/dof	126.43/140	76.45/85

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4.4.5. XRT Analysis

The temporal coverage of each time-averaged XRT spectrum (generated in Section 2.4.1) is shown as "s1," "s2," ..., "s9" in the lower panel of Figure 8. We fitted the 0.3–10 keV spectra using a simple model of tbabs*zashift*(diskbb+powerlaw). We did not include the ztbabs component, as host galaxy absorption was found to be negligible or much smaller than the Galactic absorption in all previous spectral analyses (see Tables 4, 5, 6, and 1). The adopted continuum model does not give realistic model parameters. For example, the disk radii will be underestimated when the source spectrum is hard (see a detailed discussion in Steiner et al. 2009). The main goal of this fitting is to compute the multiplicative factor to convert the 0.3–10 keV XRT net count rate to X-ray fluxes, including (i) the observed 0.3–10 keV flux f_X(0.3–10 keV), (ii) the Galactic absorption corrected 0.3–10 keV flux f_{X, 0} (0.3–10 keV), (iii) the Galactic absorption corrected 0.5–10 keV flux f_{X, 0} (0.5–10 keV), and (iv) the Galactic absorption corrected flux density at the rest-frame energies of 0.5 keV and 2 keV (i.e., f_ν (0.5 keV) and f_ν (2 keV)). All data were fitted using C-statistics.

The best-fit models are shown in Figure 15. Scaling factors to convert 0.3–10 keV net count rate to X-ray fluxes can be computed using values provided in Table 11 (Appendix A.2). The observed isotropic equivalent 0.3–10 keV X-ray luminosity, L_X, is shown in the upper panel of Figure 16. Note that for the initial four XRT nondetections, we assume a spectral shape similar to "s1."

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4.4.6. NICER Analysis

To assist comparison with TDEs from the literature, we computed the UV to X-ray spectral index α_OX (Tananbaum et al. 1979; Ruan et al. 2019; Wevers et al. 2021) and α_OSX (Gezari 2021), which are commonly used in AGN and TDE literature to characterize the ratio of UV to X-ray fluxes. ³² Here

$$\begin{eqnarray}&&{\alpha }_{\mathrm{OX}}\equiv \displaystyle \frac{\mathrm{log}[{L}_{\nu }(2500\,\mathring{\rm{A}} )/{L}_{\nu }(2\,\mathrm{keV})]}{\mathrm{log}[\nu (2500\,\mathring{\rm{A}} )/\nu (2\,\mathrm{keV})]},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\alpha }_{\mathrm{OSX}}\equiv \displaystyle \frac{\mathrm{log}[{L}_{\nu }(2500\,\mathring{\rm{A}} )/{L}_{\nu }(0.5\,\mathrm{keV})]}{\mathrm{log}[\nu (2500\,\mathring{\rm{A}} )/\nu (0.5\,\mathrm{keV})]},\end{eqnarray} \tag{ 4 }$$

where L_ν is the luminosity at a certain frequency (corrected for N_H and E_B−V,MW). We use the Swiftuvw1 host-subtracted luminosities (rest-frame effective wavelength at 2459 Å for T_eff = 3 × 10⁴ K) as a proxy for L_ν (2500 Å). We measure f_ν (0.5 keV) and f_ν (2 keV) by converting the XRT net count rates to flux densities using the scaling factors derived in Section 4.4.5. We note that f_ν (2 keV) mainly traces the evolution of the nonthermal X-ray component, while f_ν (0.5 keV) traces both the thermal and nonthermal components. The results are shown in the lower panel of Figure 16.

Based on Figure 16, we divide the evolution of AT2021ehb into five phases. In phase A (δ t ≲ 0 days), the UV/optical luminosity brightens, while X-rays are not detected (<10^40.9 erg s⁻¹). In phase B (0 ≲ δ t ≲ 100 days), the UV/optical luminosity declines, and X-rays emerge. Entering into phase C (100 ≲ δ t ≲ 225 days), the X-ray spectrum gradually hardens, while the UV/optical luminosity stays relatively flat. In phase D (225 ≲ δ t ≲ 270 days), the X-ray further brightens two times (indicated by D1 and D2), and the UV/optical plateau persists. In phase E, the X-ray luminosity drops two times (indicated by E1 and E2), while the UV/optical luminosity only slightly declines. Interestingly, the dramatic X-ray evolution in phase D+E does not have much effect on the UV/optical luminosity. Typical SEDs in each phase are shown in Figure 17.

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To calculate the bolometric luminosity L_bol at the epochs of the Swift observations, we assume that the bulk of the radiation is between 10,000 Å and 10 keV. We estimate that when the X-ray spectrum is the hardest (i.e., model 2c), the 0.3–10 keV flux still constitutes 72% of the 0.3–100 keV flux. Therefore, this assumption at most underestimates logL_bol by 0.14 dex.

We compute the 10,000 Å to 10 keV luminosity by adding the luminosities in three energy ranges (see a demonstration in Figure 18).

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From 10,000 Å to 2500 Å, we integrate below the blackbody model fitted to the UV/optical photometry (Section 4.1).

From 2500 Å to 0.5 keV, we assume that the TDE spectrum is continuous and can be approximated by a PL of ${f}_{\nu }\propto {\nu }^{{\alpha }_{\mathrm{OSX}}}$. Hence, the luminosity is

$$\begin{eqnarray}&&L={\int }_{{\nu }_{1}}^{{\nu }_{2}}{L}_{\nu }d\nu \approx {\int }_{{\nu }_{1}}^{{\nu }_{2}}\displaystyle \frac{{L}_{\nu }({\nu }_{1})}{{\nu }_{1}^{{\alpha }_{\mathrm{OSX}}}}{\nu }^{{\alpha }_{\mathrm{OSX}}}d\nu \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}=\displaystyle \frac{{L}_{\nu }({\nu }_{1})}{{\nu }_{1}^{{\alpha }_{\mathrm{OSX}}}}\times \left\{\begin{array}{ll}\displaystyle \frac{{\nu }_{2}^{{\alpha }_{\mathrm{OSX}}+1}-{\nu }_{1}^{{\alpha }_{\mathrm{OSX}}+1}}{{\alpha }_{\mathrm{OSX}}+1} & \mathrm{if}\ {\alpha }_{\mathrm{OSX}}\ne -1\\ \mathrm{ln}({\nu }_{2}/{\nu }_{1}) & \mathrm{if}\ {\alpha }_{\mathrm{OSX}}=-1\end{array}\right.\end{eqnarray} \tag{ 6 }$$

where ν₁ = 10^15.08 Hz, ν₂ = 10^17.08 Hz. In this range, we assume that the uncertainty of L is 0.3L.

From 0.5 keV to 10 keV, we calculate the luminosity by converting the 0.3–10 keV XRT net count rate to Galactic absorption corrected 0.5–10 keV luminosity using the scaling factors derived in Section 4.4.5.

Note that for the first four Swift epochs, since X-rays were not detected, we use the UV/optical blackbody luminosity L_bb as an approximation of L_bol.

The evolution of logL_bol as a function of α_OX is shown in Figure 19. The data points are color coded by their phases (from A to E; see Figure 16). The right y-axis converts L_bol to the Eddington ratio λ_Edd ≡ L_bol/L_Edd. For pure hydrogen, an M_BH of 10^7.03 M_⊙ (Section 3.1) implies an Eddington luminosity of L_Edd≈10^45.13 erg s⁻¹. We further discuss this figure in Section 5.5. The maximum luminosity was reached at δ t = 253 days, with L_bol = (7.94 ± 0.66) × 10⁴³ erg s⁻¹ and ${\lambda }_{\mathrm{Edd}}={6.0}_{-3.8}^{+10.4}$%. As a cautionary note, the relatively low value of λ_Edd (<16%) does not necessarily imply that the accretion is in the sub-Eddington regime, as the TDE broadband SED may peak in the extreme-UV (EUV) band (Dai et al. 2018; Mummery & Balbus 2020).

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The soft X-ray emission of many TDEs has been attributed to the inner regions of an accretion disk (Saxton et al. 2020). Assuming R_in≈6R_g≈10¹³ cm, the maximum effective temperature of an optically thick, geometrically thin accretion disk is ${T}_{\mathrm{eff}}\approx 20{\left(\tfrac{\dot{m}}{{M}_{7}{\eta }_{-1}}\right)}^{1/4}$ eV (Shakura & Sunyaev 1973). With a maximum black hole spin of a → 1, R_in → R_g, and ${T}_{\mathrm{eff}}\approx 78{\left(\tfrac{\dot{m}}{{M}_{7}{\eta }_{-1}}\right)}^{1/4}$ eV. The top panel of Figure 20 shows that in phase D, when the X-ray spectrum is the hardest, the measured T_in is ∼2.5 times greater than the maximum allowed T_eff. This high color temperature causes the inferred disk radius ${R}_{{\rm{d}},\mathrm{in}}={R}_{\mathrm{in}}^{* }/\sqrt{\cos i}$ to be much less than R_g throughout the evolution. The projection factor $\sqrt{\cos i}$ should not be much less than unity, since for a nearly edge-on viewing angle, the X-rays from the inner disk will be obscured by the gas at larger radii. The relativistic disk reflection model (2c) also suggests $\sqrt{\cos i}={0.85}_{-0.07}^{+0.06}$. We note that disk radii much less than R_g have also been inferred in a few other X-ray bright TDEs (see, e.g., Figure 8 of Gezari 2021).

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This discrepancy may be due to Compton scattering (Shimura & Takahara 1995), which makes the measured temperature greater than the effective inner disk temperature by a factor of f_c (Davis & El-Abd 2019), i.e., T_in = f_c T_eff. The physical reason is that, as the X-ray photons propagate in the vertical direction away from the disk mid-plane, the color temperature is determined by the thermalization depth (corresponding to the last absorption), which could be located at a high scattering optical depth τ ≫ 1—this causes T_in to be higher than T_eff by a factor of∼τ^1/4. As a result of ongoing fallback, the vertical structure of the TDE disk (see Bonnerot et al. 2021) is likely substantially different from the standard thin disk as studied by Davis & El-Abd (2019), who concluded f_c ≲ 2, so the color correction factor may be different. More detailed radiative transport calculations in the TDE context are needed to provide a reliable f_c based on first principles.

Another possible reason for the seemingly small disk radii is that a scattering dominated, Compton-thick gas layer can suppress the X-ray flux without causing any significant change to the spectral shape. For a spatially uniform layer, the transmitted flux is exponentially suppressed for a large scattering optical depth ≫ 1. A more likely configuration is that the layer is like an obscuring wall with small holes where a fraction of the source X-rays can get through, and the rest of the area contributes negligibly to the observed flux. In this scenario, the inferred disk radius is reduced by a factor of the square root of the transmitted over emitted fluxes.

Hard X-rays can be generated by Compton up-scattering of soft X-rays from the accretion disk by the hot electrons in the (magnetically dominated) coronal regions above the disk, as is the case in AGNs and XRBs. The physical situation in TDEs is more complicated than in AGNs in that the hard X-rays must make their way out of the complex hydrodynamic structures. An X-ray photon undergoes∼τ² electron scatterings as it propagates through a gas slab of Thomson optical depth τ. In each scattering, the photon loses a fraction E_γ /m_e c² of its energy (where E_γ is the photon energy) as a result of Compton recoil, and hence the cumulative fractional energy loss is∼τ² E_γ /m_e c². This means that photons above an energy threshold of∼1 keV(τ/20)⁻² will be Compton down-scattered by the gas.

Our NuSTAR observations clearly detected hard X-ray photons up to 30 keV, which requires that the optical depth along the pathways of these photons from the inner disk ( ≳ R_g∼10^12.2 cm) to the observer is less than about 4. On the other hand, the UV/optical emission indicates that the reprocessing layer is optically thick up to a radius of the order of R_bb∼10¹⁴ cm. Therefore, our observations favor a highly nonspherical system—there are viewing angles that have very large optical depths such that most X-ray photons are absorbed (and reprocessed into the UV/optical bands), and there are other viewing angles with scattering optical depth τ ≲ 4 so that hard X-ray photons can escape.

5.3.1. Soft to Hard Transition: Corona Formation

5.3.2. Hard to Soft Transition: Thermal–Viscous Instability?

The rapid X-ray flux drop (D → E) is likely due to a state transition in the innermost regions of the accretion disk. Under the standard α-viscosity prescription where the viscous stress is proportional to the total (radiation + gas) pressure (Shakura & Sunyaev 1973), the disk undergoes a thermal–viscous instability as the accretion rate drops from super- to sub-Eddington regimes (Lightman & Eardley 1974; Shakura & Sunyaev 1976). This instability causes the disk material to suddenly transition from a radiation pressure-dominated to gas pressure-dominated state on a sound-crossing timescale, and the consequence is that the disk becomes much thinner and hence the accretion rate drops. Shen & Matzner (2014) considered the thermal–viscous instability in the TDE context but concluded that the instability should occur within a few months since the disruption and the accretion rate drops by several orders of magnitude—these, taken at face value, are inconsistent with our observations. More detailed work on the disk evolution is needed to draw a firm conclusion. Here, we provide two arguments for the disk state transition explanation.

First, TDEs with relativistic jets (e.g., Bloom et al. 2011; Burrows et al. 2011; Cenko et al. 2012; Pasham et al. 2015) also show a sharp drop in X-ray luminosity 200 to 300 days (in the rest frame) after the discovery and that has been interpreted as the thick-to-thin transition of the inner disk (Tchekhovskoy et al. 2014). Second, from the mass fallback rate ${\dot{M}}_{\mathrm{fb}}\simeq {M}_{* }/3{P}_{\min }{(t/{P}_{\min })}^{-5/3}$ (M_* being the stellar mass and ${P}_{\min }$ being the minimum period of the fallback material), one can estimate the time t_Edd at which the fallback rate drops below the Eddington accretion rate of∼10L_Edd/c², and the result is (Lu & Kumar 2018)

$$\begin{eqnarray}&&{t}_{\mathrm{Edd}}\simeq 309\,{\rm{d}}\,{M}_{{\rm{h}},7}^{-2/5}{\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{(1+3q)/5},\end{eqnarray} \tag{ 7 }$$

where we have taken the normal mass–radius relation of main-sequence stars ${R}_{* }\simeq {R}_{\odot }{({M}_{* }/{M}_{\odot })}^{q}$ (q = 0.8 below one solar mass stars and q = 0.57 above one solar mass stars). We expect the inner disk to collapse into a thin state on the timescale t_Edd, under the condition that an order-unity fraction of the fallback rate directly reaches near the innermost regions of the disk. We note that the condition is satisfied for a≈10⁷ M_⊙ MBH since the tidal radius is only≈10R_g for a solar-type star.

If the rapid X-ray flux drop (D → E) is indeed caused by a disk state transition, then after the transition, the disk mass will accumulate over time due to ongoing fallback. This causes the accretion rate to increase, and eventually the disk briefly goes back to a thick state (with a very short viscous time) followed by another transition to the thin state. This is qualitatively consistent with the second rapid X-ray flux decline at δ t≈325 days (E1 → E2 in Figure 16).

5.4.1. Featureless Optical Spectrum

As shown in Section 4.2, AT2021ehb's optical spectroscopic properties are dissimilar to the majority of previously known TDEs (i.e., H-rich, He-rich, N-rich, Fe-rich; Leloudas et al. 2019; van Velzen et al. 2021; Wevers et al. 2019a). It is most similar to a few recently reported TDEs with blue and featureless spectra (Brightman et al. 2021; Hammerstein et al. 2022). Hammerstein et al. (2022) found that compared with TDEs that develop broad emission lines, the UV/optical emission of four featureless events have larger peak L_bb, peak T_bb, and peak R_bb.

Figure 21 compares the rest-frame g-band light curve of AT2021ehb with 30 TDEs from phase I of ZTF (Hammerstein et al. 2022). Solid lines are the results of fitting the multiband light curves (δ t<100 days) with a Gaussian rise + exponential decay model (see Section 5.1 of van Velzen et al. 2021). We highlight the TDEs lacking line features by plotting the data as colored symbols. Here we have chosen an observing band with good temporal sampling, and converted the observations in this band into ν₀ = 6.3 × 10¹⁴ Hz by performing a color correction.

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Our study suggests that not all featureless TDEs are overluminous. In fact, the peak g-band magnitude (M_g,peak) and peak L_bb of AT2021ehb are faint compared with other optically selected TDEs (Figure 6, Figure 21). It is unclear whether M_g,peak of the TDE-featureless class forms a continuous or bimodal distribution between −17 and −22. This question will be addressed in a forthcoming publication (Y. Yao et al. 2022, in preparation). A detailed analysis of Hubble Space Telescope UV spectroscopy (E. Hammerstein et al. 2022, in preparation) will be essential to reveal if AT2021ehb exhibits any spectral lines in the UV.

5.4.2. Origin of the NUV/optical Emission

In stellar-mass black hole XRB outbursts, some objects are observed to transition between a soft disk-dominated state (SDS) and a hard Comptonized state (HCS). In the SDS of XRBs, the inner radius (R_{in, d}) of an optically thick, geometrically thin disk stays around the ISCO of R_ISCO∼a few × R_g. When the outbursts transition to the HCS, ${\dot{M}}_{\mathrm{acc}}$ decreases, and R_{in, d} progressively moves outwards to∼few × 100R_g, leaving a radiatively inefficient, advection-dominated accretion flow (Yuan et al. 2005; Yuan & Narayan 2014). At the same time, a region of hot corona is formed close to the BH (Done et al. 2007). For MBH accretors, Seyferts have been proposed as the high-M_BH analogs of XRBs in the SDS, whereas low-luminosity AGNs and low-ionization nuclear emission-line regions are considered similar to XRBs in the HCS (Falcke et al. 2004).

In Section 5.3.2, we propose that in phases B–D, the accretion flow of AT2021ehb is in a radiation-trapped, super-Eddington regime (see, e.g., Figure 2 of Narayan & Quataert 2005), which is different from the typical sub-Eddington X-ray states observed in AGNs and XRBs. Such a difference is further corroborated by two properties. First, on the hardness–intensity diagram (HID; see Figure 22), the evolution of AT2021ehb is neither similar to the "turtlehead" pattern observed in XRBs (Fender et al. 2004; Munoz-Darias et al. 2013; Tetarenko et al. 2016), nor similar to the "brighter when softer" trend observed in X-ray bright AGNs (Auchettl et al. 2018). On the L_bol–α_OX diagram (Figure 19), its evolution is also different from that observed in CLAGNs (Ruan et al. 2019). Second, in the canonical hard state (i.e., sub-Eddington accretion rates), there is a correlation among radio luminosity, X-ray luminosity, and M_BH, which applies to both XRBs and hard-state AGNs (Merloni et al. 2003; Falcke et al. 2004). A recent fit to this "fundamental plane of black hole activity" was provided by Gültekin et al. (2019):

$$\begin{eqnarray}\begin{array}{rcl}(1.09\pm 0.10)R & = & (\mathrm{log}{M}_{7}-1)-(0.55\pm 0.22)\\ & & -(-{0.59}_{-0.15}^{+0.16})X,\end{array}\end{eqnarray} \tag{ 8 }$$

where $R\equiv \mathrm{log}[{L}_{5\,\mathrm{GHz}}/({10}^{38}\,\mathrm{erg}\,{{\rm{s}}}^{-1})]$ and $X\equiv \mathrm{log}[{L}_{2-10\,\mathrm{keV}}/({10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1})]$. In the hard state of AT2021ehb, the 2–10 keV luminosity in the Swift/XRT "s6" spectrum gives X =2.96 ± 0.02. Using Equation (8), the expected 5 GHz radio luminosity on the fundamental plane is $\mathrm{log}[{L}_{5\,\mathrm{GHz}}/(\mathrm{erg}\,{{\rm{s}}}^{-1})]=R+38\,=\,38.21\pm 0.59$. The uncertainty of R is calculated from the distribution of 10⁵ MC trials. Using our radio limit at δ t = 228 days (Table 2) and assuming a flat radio spectrum of f_ν ∝ ν⁰, we find a 5 GHz equivalent radio upper limit of $\mathrm{log}[{L}_{5\,\mathrm{GHz}}/(\mathrm{erg}\,{{\rm{s}}}^{-1})]\lt 35.76$. According to Gültekin et al. (2019), the scatter on the fundamental plane correlation is a factor of ∼7.6 in the L_{5 GHz} direction (or ∼10 in the M_BH direction), which is not enough to solve the discrepancy. This again argues that AT2021ehb is not in a canonical hard state where we would expect a radio jet to be launched.

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The radio behavior of BH binaries at the Eddington accretion rates might be more relevant to AT2021ehb (at least in phases B–D). A few XRBs get above the Eddington limit for brief periods, when they are seen to undergo a sequence of bright radio flares for a short period of time. Examples include the 2015 outburst of V404 Cygni (Tetarenko et al. 2017, 2019) and the ultraluminous X-ray source in M31 (Middleton et al. 2013). In AT2021ehb, the slow cadence of our radio follow-up observations does not allow us to rule out the existence of such radio flares, which last for hours to weeks, not months.

Finally, we note that the evolution of the X-ray properties of AT2021ehb are different from a few other TDEs. For example, Wevers et al. (2021) constructed the logλ_Edd–α_OX diagram for AT2018fyk, finding that the 2 keV (corona) emission is stronger when λ_Edd is lower. Figure 19 shows that this is clearly not the case for AT2021ehb. Separately, Hinkle et al. (2021) studied the evolution of AT2019azh on the canonical HID, showing that when the X-ray luminosity is higher, the X-ray spectrum is softer. Figure 22 shows that AT2021ehb does not follow this trend either. It remains to be seen whether AT2021ehb is peculiar among the sample of optically selected TDEs with significant X-ray spectral evolution. To this end, constructing a systematically selected sample and analyzing the multiwavelength data in a homogeneous fashion is the key.

UV and optical photometry are presented in Table 8. Swift/XRT observations are summarized in Table 9.

Table 8. UV and Optical Photometry of AT2021ehb

MJD	Instrument	Filter	fν (μJy)	${\sigma }_{{f}_{\nu }}$ (μJy)
59250.1643	ZTF	r	−3.31	12.48
59250.2031	ZTF	g	2.07	10.91
59299.0783	UVOT	uvw1	567.68	27.76
59299.0798	UVOT	U	551.85	41.60
59299.0808	UVOT	B	487.52	76.95
59299.0831	UVOT	uvw2	587.34	22.36
59299.0855	UVOT	V	260.66	146.91
59299.0875	UVOT	uvm2	528.68	19.49

Note. f_ν is observed flux density before extinction correction.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 10 presents the photospheric parameters derived from fitting a blackbody function to the UV/optical data. Note that the uncertainty of T_bb is only computed for "good" epochs (see details in Section 4.1).

The XRT spectral parameters are presented in Table 11.

Table 11. X-Ray Fluxes from Modeling of XRT Spectra

Observation	Net 0.3–10 keV Rate	fν (0.5 keV)	fν (2 keV)	fX (0.3–10 keV)	fX, 0 (0.3–10 keV)	fX, 0 (0.5–10 keV)
	(count s−1)	(μJy)	(μJy)	(10−13 erg s−1 cm−2)	(10−13 erg s−1 cm−2)	(10−13 erg s−1 cm−2)
s1	0.0276 ± 0.0019	${1.133}_{-0.185}^{+0.103}$	${0.030}_{-0.023}^{+0.002}$	${9.76}_{-1.08}^{+1.39}$	${18.47}_{-2.05}^{+2.63}$	${11.34}_{-2.72}^{+0.37}$
s2	0.1476 ± 0.0041	${9.639}_{-0.595}^{+0.178}$	${0.106}_{-0.012}^{+0.008}$	${48.67}_{-1.03}^{+1.79}$	${129.40}_{-2.74}^{+4.76}$	${48.96}_{-2.53}^{+1.34}$
s3	0.2116 ± 0.0047	${10.930}_{-0.694}^{+0.287}$	${0.312}_{-0.018}^{+0.017}$	${78.12}_{-1.95}^{+2.87}$	${166.66}_{-4.15}^{+6.13}$	${86.04}_{-3.95}^{+1.75}$
s4	0.2584 ± 0.0062	${7.565}_{-0.683}^{+0.224}$	${0.544}_{-0.035}^{+0.026}$	${97.63}_{-3.31}^{+3.67}$	${158.72}_{-5.37}^{+5.97}$	${109.87}_{-5.73}^{+2.56}$
s5	0.2382 ± 0.0058	${5.485}_{-0.583}^{+0.211}$	${0.678}_{-0.054}^{+0.029}$	${113.80}_{-4.00}^{+5.90}$	${159.82}_{-5.62}^{+8.29}$	${127.47}_{-7.81}^{+3.50}$
s6	0.4776 ± 0.0083	${8.675}_{-1.257}^{+0.247}$	${1.702}_{-0.063}^{+0.057}$	${233.38}_{-4.48}^{+8.72}$	${310.81}_{-5.97}^{+11.61}$	${256.24}_{-11.39}^{+7.73}$
s7	0.9314 ± 0.0129	${14.647}_{-1.061}^{+0.724}$	${3.575}_{-0.164}^{+0.082}$	${528.81}_{-13.21}^{+20.29}$	${662.69}_{-16.55}^{+25.43}$	${579.24}_{-27.10}^{+11.66}$
s8	0.0780 ± 0.0029	${2.843}_{-0.351}^{+0.121}$	${0.132}_{-0.014}^{+0.010}$	${27.30}_{-0.96}^{+1.56}$	${50.03}_{-1.77}^{+2.85}$	${30.99}_{-2.74}^{+1.06}$
s9	0.0185 ± 0.0015	${1.218}_{-0.741}^{+0.006}$	${0.027}_{-0.005}^{+0.006}$	${7.59}_{-0.52}^{+1.11}$	${21.62}_{-1.48}^{+3.15}$	${6.78}_{-1.38}^{+0.48}$

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