PS18kh: A New Tidal Disruption Event with a Non-axisymmetric Accretion Disk

We retrieved archival optical ugriz model magnitudes of SDSS J075654.53+341543.6 from SDSS Data Release 14 (DR14; Abolfathi et al. 2018) and infrared W1 and W2 magnitudes from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) AllWISE catalog. The host is not detected in archival data from, or was not previously observed by, the Two Micron All-sky Survey, Spitzer, Herschel, the Hubble Space Telescope, the Chandra X-ray Observatory, the X-ray Multi-mirror Mission, or the Very Large Array Faint Images of the Radio Sky at Twenty-cm survey. It is also not detected in Galaxy Evolution Explorer (GALEX) UV data, but we obtain 3σ 60 upper limits on the UV magnitudes of NUV > 23.65 and FUV > 23.69 using single-epoch data obtained on 2008 January 19. The archival host magnitudes and limits are listed in Table 1.

To place constraints on any X-ray emission prior to the flare that could be indicative of an active galactic nucleus (AGN), we take advantage of data from the ROSAT All-sky Survey (Voges et al. 1999). We do not detect X-ray emission associated with the position of the host galaxy with a 3σ upperlimit on the count rate of 8 × 10⁻³ counts s⁻¹. Assuming an absorbed power law redshifted to the distance of the host galaxy and a photon index similar to that of known AGNs (Γ = 1.75: e.g., Tozzi et al. 2006; Marchesi et al. 2016; Liu et al. 2017b; Ricci et al. 2017), we derive a limit on the absorbed (unabsorbed) flux of 2.3 (2.6) × 10⁻¹³ erg cm⁻² s⁻¹ in the 0.3–10.0 keV energy band. At the distance of PS18kh this flux limit corresponds to an X-ray luminosity of 3.2 × 10⁴² erg s⁻¹. This is lower than the average luminosity of known AGNs (e.g., Ricci et al. 2017), suggesting that the host galaxy of PS18kh does not harbor a strong AGN.

We fit the spectral energy distribution (SED) of the host galaxy to the archival limits and magnitudes from GALEX, SDSS, and WISE using the publicly available Fitting and Assessment of Synthetic Templates (FAST; Kriek et al. 2009). For the fit we assumed a Cardelli et al. (1989) extinction law with R_V = 3.1 and a Galactic extinction of A_V = 0.128 mag (Schlafly & Finkbeiner 2011) and we adopted an exponentially declining star formation history, a Salpeter initial mass function, and the Bruzual & Charlot (2003) stellar population models. In order to make a more robust estimate of the host SED and the uncertainties on its physical parameters, we generated 1000 realizations of the archival fluxes, perturbed by their respective uncertainties assuming Gaussian errors. Each realization was then modeled with FAST. The median and 68% confidence intervals on the host parameters from these 1000 realizations are: ${M}_{\star }={1.4}_{-0.4}^{+0.4}\times {10}^{10}$ M_⊙, age = ${5.0}_{-1.9}^{+2.1}$ Gyr, and a star formation rate ${\rm{SFR}}={6.8}_{-4.9}^{+4.0}\times {10}^{-3}$ M_⊙ yr⁻¹. We scaled the stellar mass of SDSS J075654.53+341543.6 using the average stellar-mass-to-bulge-mass ratio from the hosts of ASASSN-14ae, ASASSN-14li, and ASASSN-15oi (Holoien et al. 2014, 2016a, 2016b), to get a bulge mass estimate of M_B ≃ 10^9.5 M_⊙. Using the M_B–M_BH relation from McConnell & Ma (2013), we obtained a black hole mass of M_BH = 10^6.9 M_⊙, comparable to what has been found for other optical TDE host galaxies (e.g., Holoien et al. 2014, 2016a, 2016b; Brown et al. 2017; Wevers et al. 2017; Mockler et al. 2019).

Our photometric follow-up campaign included ugri photometry, for which the archival SDSS data can be used to subtract the host flux and isolate the transient flux. For the Swift UVOT and Johnson–Cousins BV data, there are no available archival images. To obtain 50 aperture host flux measurements to use for host subtraction in the ugri filters, we measured 50 aperture magnitudes from the archival SDSS images using the IRAF apphot package, with the magnitudes calibrated using several stars in the field with well-defined magnitudes in SDSS DR14. In order to estimate the host flux in the filters without archival data, we used the bootstrapped SED fits for the host galaxy to derive synthetic host magnitudes for each photometric band in our follow-up campaign. For each of the 1000 host SEDs, we computed synthetic 50 aperture magnitudes in each of our follow-up filters. This yields a distribution of synthetic magnitudes for each filter, and we report the median and 68% confidence intervals on the host magnitudes, along with the measured ugri magnitudes, in Table 2. These host magnitudes were used to obtain host-subtracted transient magnitudes for the non-survey data in our analyses.

The Pan-STARRS1 telescope, located at the summit of Haleakalā on Maui, has a 1.8 m diameter primary mirror with a f/4.4 Cassegrain focus. The telescope uses a wide-field 1.4 gigapixel camera mounted at the Cassegrain focus, consisting of 60 Orthogonal Transfer Array devices, each of which has a detector area of 4846 × 4868 pixels. The 10 micron pixels have a plate scale of 026, giving a full field-of-view area of 7.06 deg², with an active region of roughly 5 deg². Pan-STARRS1 uses the grizy_P1 filters, which are similar to those of SDSS (Abazajian et al. 2009), with the redder y filter replacing the bluer SDSS u filter. The Pan-STARRS1 photometric system in discussed in detail in Tonry et al. (2012).

Pan-STARRS1 images are processed with the Image Processing Pipeline (IPP; see details in Magnier et al. 2013). The IPP runs new images through successive stages of processing, including device "de-trending," a flux-conserving warping to a sky-based image plane, masking and artefact location that involves bias and dark correction, flatfielding, and illumination correction obtained by rastering sources across the field of view (Waters et al. 2016). After determining an initial astrometric solution, corrected images are then warped onto the tangent plane of the sky using a flux-conserving algorithm, which involves mapping the camera pixels to a defined set of skycells. For nightly processing, the zero-points of the camera chips are set using a catalog of photometric reference stars from the "ubercal" analysis of the first reprocessing of the PS1 3π data (Schlafly et al. 2012; Magnier et al. 2013). The internal calibration of this catalog has a relative precision of roughly 1%, but the automated zero-point applied in difference imaging is an average full-field zero-point, which can result in variations across skycells of up to ±0.15 mag.

Transient searching is aided by having pre-existing sky images from the Pan-STARRS1 Sky Surveys (Chambers et al. 2016). The IPP creates difference images by subtracting stacked reference images from the PS1 3π from newly observed images, and transient sources are then identified by the IPP through analysis of the difference images (e.g., Huber et al. 2015). Catalog source files from the IPP are transferred from Hawaii to Belfast and ingested into a MySQL database. A series of quality cuts are implemented (McCrum et al. 2015; Smartt et al. 2016) together with a machine-learning algorithm that distinguishes real sources from bogus sources (Wright et al. 2015). Sources are accumulated into unique objects and spatially cross-matched against all large catalogs, therefore providing both a real-bogus value and a classification of variable star, AGN, supernova, cataclysmic variable, or nuclear transient. The grizy_P1 lightcurve presented in this paper was produced from this Pan-STARRS transient processing pipeline as described in McCrum et al. (2014, 2015) and Smartt et al. (2016). The Pan-STARRS1 griz photometry is presented in Table 3 and is shown in Figure 1; we do not present the y photometry as PS18kh was only detected in one y-band epoch.

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ASAS-SN is an ongoing project that monitors the full visible sky on a rapid cadence to find bright, nearby transients (Shappee et al. 2014; Kochanek et al. 2017). ASAS-SN uses units of four 14 cm telescopes on a common mount located at multiple sites in both hemispheres and hosted by the Las Cumbres Observatory global telescope network (Brown et al. 2013). The ASAS-SN network was expanded in 2017 and now comprises five units located in Hawaii, Chile, Texas, and South Africa. With its current capacity, ASAS-SN observes the entire visible sky every ∼20 hr to a depth of g ≃ 18.5 mag, weather permitting. ASAS-SN has proven to be a powerful tool for discovering TDEs, and it has discovered three of the four nearest and brightest TDEs to-date: ASASSN-14ae (Holoien et al. 2014; Brown et al. 2016), ASASSN-14li (Holoien et al. 2016b; Prieto et al. 2016; Romero-Cañizales et al. 2016; Brown et al. 2017), and ASASSN-15oi (Holoien et al. 2016a, 2018). The three ASAS-SN TDEs have since become some of the most well-studied TDEs, with multiwavelength data sets spanning multiple years.

ASAS-SN processes new images using a fully automatic pipeline that incorporates the ISIS image subtraction package (Alard & Lupton 1998; Alard 2000). After the discovery of PS18kh, a host-galaxy reference image was constructed for each ASAS-SN unit that could observe it. As the transient was still brightening, we only used images obtained at least 35 days before the discovery of PS18kh to ensure that no transient flux was present in the references. These reference images were then used to subtract the host galaxy's background emission from all science images. Aperture photometry was computed for each host-template subtracted science image using the IRAF apphot package, with the magnitudes being calibrated using multiple stars in the field of the host galaxy with known magnitudes in the AAVSO Photometric All-sky Survey (Henden et al. 2015). For some of the pre-discovery epochs when PS18kh was still very faint, we stacked multiple science images in order to improve the signal-to-noise ratio (S/N) of our detections. All ASAS-SN photometric measurements (detections and 3σ limits) are presented in Table 3 and shown in Figure 1, with error bars on the X-axis used to denote the date ranges of epochs that were combined.

ATLAS is an ongoing survey project with the primary goal of detecting small (10–140 m) asteroids that are on a collision course with Earth (Tonry et al. 2018). ATLAS uses fully robotic 0.5 m f/2 Wright Schmidt telescopes located on the summit of Haleakalā and at Mauna Loa Observatory to monitor the entire sky visible from Hawaii every few days. During normal operations, each telescope obtains four 30 second exposures of 200–250 target fields per night, allowing the two telescopes to cover roughly a quarter of the visible sky each night. The four observations of a given field are typically obtained within less than an hour of each other. ATLAS uses two broad filters for its survey operations, with the "cyan" filter (c) covering 420–650 nm and the "orange" filter (o) covering 560–820 nm (Tonry et al. 2018).

Every ATLAS image is processed by a fully automated pipeline that performs flat-fielding, astrometric calibration, and photometric calibration. A low-noise reference image of the host field was constructed by stacking multiple images taken under excellent conditions and this reference was then subtracted from each science image of PS18kh in order to isolate transient flux. We performed forced photometry on the subtracted ATLAS images of PS18kh as described in Tonry et al. (2018), and then combined the intra-night photometric observations using a weighted average to get a single flux measurement for each epoch of observation. The ATLAS o-band photometry and 3σ limits are presented in Table 3 and are shown in Figure 1. We do not present the c photometry as there were few c observations during this period due to weather and the design of the ATLAS survey. Because of this, PS18kh was only detected in two c-band epochs.

After PS18kh was classified as a TDE candidate, we were awarded 20 epochs of Swift TOO observations of PS18kh between 2018 March 27 and 2018 May 29, after which it became Sun-constrained. The UVOT observations were obtained in the V (5468 Å), B (4392 Å), U (3465 Å), UVW1 (2600 Å), UVM2 (2246 Å), and UVW2 (1928 Å) filters (Poole et al. 2008) for all epochs. As each epoch contained two observations in each filter, we first combined the two images in each filter using the HEAsoft software task uvotimsum, and then extracted counts from the combined images in a 50 radius region using the software task uvotsource, with a sky region of ∼400 radius used to estimate and subtract the sky background. The UVOT count rates were converted into magnitudes and fluxes based on the most recent UVOT calibration (Poole et al. 2008; Breeveld et al. 2010).

We corrected the UVOT magnitudes for Galactic extinction assuming a Cardelli et al. (1989) extinction law. Using the synthetic 50 host fluxes calculated from the FAST fits, we then subtracted the host flux from each UVOT observation to isolate the transient flux in each band. To enable direct comparison to ASAS-SN magnitudes and other ground-based follow-up photometry, we converted the UVOT B- and V-band data to Johnson B and V magnitudes using publicly available color corrections.²⁴ The host-subtracted Swift UVOT photometry and 3σ limits are presented in Table 3 and are shown in Figure 1.

PS18kh was also observed using the Swift XRT. All observations were taken in photon counting mode, and were reprocessed from level one XRT data using the Swift XRTPIPELINE version 0.13.2. As suggested in the Swift XRT data reduction guide,²⁵ standard filters and screening were applied, along with the most up-to-date calibration files. We used a source region centered on the position of PS18kh with a radius of 30'', and a source-free background region centered at (α, δ) = (07:57:07.71, +34:20:59.97) with a radius of 1500. All extracted count rates were corrected for the encircled energy fraction (a 300 source radius contains only ∼90% of the counts from a source at 1.5 keV; Moretti et al. 2004).

To increase the S/N of our observations, we combined the individual XRT observations using XSELECT version 2.4d. We combined our observations into three time bins spanning the full Swift observing campaign and merged all observations together to extract an X-ray spectrum with the highest S/N possible. From these merged observations, we used the task XRTPRODUCTS to extract both source and background spectra. Ancillary response files were derived using XRTMKARF and merged exposure maps were created from the individual observations using XIMAGE version 4.5.1. We took advantage of the ready-made response matrix files, which are obtained from the most up-to-date Swift CALDB. The XRT fluxes and 3σ upper limits measured from the merged observations are given in Table 4.

The spectral data were analyzed using the X-ray spectral fitting package (XSPEC) version 12.9.1 and χ² statistics. Each spectrum was grouped using FTOOLS command grppha to have a minimum of 10 counts per energy bin. Due to the faintness of the X-ray emission from this source, the S/N of the resulting spectrum was quite low. As such, the spectrum is insufficient to constrain the column density (N_H) and so we fixed it to N_H = 4.42 × 10²⁰ cm⁻², which is the Galactic H i column density in the direction of PS18kh (Kalberla et al. 2005).

In addition to the survey data and Swift observations, we obtained photometric observations from multiple ground observatories. BVgri observations were obtained from the 2 m LT (Steele et al. 2004) and from the 24 inch PO robotic telescopes located in Mayhill, New Mexico, and Sierra Remote Observatory in California. Additional u-band data were obtained with MegaCam (Boulade et al. 1998) on the CFHT. After flat-field corrections were applied to these follow-up data, we measured 50 aperture magnitudes using the IRAF apphot package, with the magnitudes calibrated using several stars in the field with well-defined magnitudes in SDSS DR14. B and V reference star magnitudes were calculated from the SDSS ugriz magnitudes using the corrections.²⁶

As was done with the Swift UVOT magnitudes, after calculating the 50 aperture fluxes in each image, we corrected for Galactic extinction and subtracted the host flux using the synthetic host magnitudes calculated from the FAST fits. The host-subtracted ground-based follow-up photometry is presented in Table 3 and shown in Figure 1.

After classifying PS18kh as a TDE candidate, we began a program of spectroscopic follow-up to complement our photometric follow-up. The telescopes and instruments used to obtain follow-up spectra as part of this campaign included SNIFS on the University of Hawaii 88 inch telescope, the Inamori-Magellan Areal Camera and Spectrograph (IMACS; Dressler et al. 2011) on the 6.5 m Magellan-Baade telescope, the Gemini Multi-Object Spectrograph (GMOS; Hook et al. 2004) on the 8.2 m Gemini North telescope, the SPectrograph for the Rapid Acquisition of Transients (SPRAT) on the Liverpool Telescope, the Low-Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) on the Keck I 10 m telescope, and the Multi-Object Double Spectrographs (MODS; Pogge et al. 2010) mounted on the dual 8.4 m Large Binocular Telescope (LBT).

We reduced and calibrated the majority of the spectra using IRAF following standard procedures, including bias subtraction, flat-fielding, 1D spectral extraction, and wavelength calibration by comparison to an arc lamp. The MODS spectra were reduced using the MODS spectroscopic pipeline.²⁷ The observations were flux calibrated with spectroscopic standard star spectra obtained on the same nights as the science spectra. In some cases, we also performed telluric corrections using the standard star spectra, and in other cases we masked prominent telluric features. In order to increase the S/N of later observations from SNIFS, spectra taken within two to three days of each other were coadded, with each spectrum weighted by its uncertainty. Details of all spectra obtained for PS18kh are presented in Table 5.

We futher calibrated the spectra using the photometric measurements. We extracted synthetic photometric magnitudes for each filter that was completely contained in the wavelength range covered by the spectrum and for which we could either interpolate the photometric light curves or extrapolate them by onr hour or less. We fit a line to the difference between the observed and synthetic flux as a function of central wavelength and scaled each spectrum by this fit. We corrected the observed spectra for Galactic reddening using a Milky Way extinction curve and assuming R_V = 3.1 and A_V = 0.128 (Schlafly & Finkbeiner 2011).

The spectroscopic evolution of PS18kh is shown in Figure 2. For cases where multiple observations were obtained on a given night, only one spectrum is shown. The SNIFS spectra labeled "2018 May 12" and "2018 May 19" are coadded spectra combining data from 2018 May 11–12 and 2018 May 17–19, respectively. The SNIFS dichroic split falls very close to the Hβ line, and some of the SNIFS spectra (2018 March 7, 18, 30, 31, April 27, May 12 and 19) show residual noise around Hβ as a result.

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After calibrating the spectra, we synthesized photometric magnitudes from each follow-up spectrum for each filter that was completely contained in the wavelength range covered by the spectrum. These magnitudes were corrected for Galactic extinction and host fluxes were subtracted using the synthetic and measured host 50 mag, as was done with the Swift and ground follow-up data. The host-subtracted synthetic photometry is presented in Table 3 and shown in Figure 1.

We used the discovery i-band image obtained by Pan-STARRS1 on 2018 March 2 and the corresponding Pan-STARRS1 i-band reference image to measure an accurate position of the transient. We first measured the centroid position of the transient in the host-subtracted discovery image and the centroid position of the host galaxy nucleus in the reference image using the IRAF task imcentroid, then calculated the offset between the two positions. From this method, we obtain a position of R.A. = 07:56:54.53, decl. = +34:15:43.58 for PS18kh. We calculated an offset of 028 ± 029 from the host nucleus, corresponding to a physical projected distance of 0.45 ± 0.48 kpc at the distance of the host.

We initially obtained a redshift of the transient using a Gaussian fit to the Hα emission line in the Magellan IMACS spectrum obtained on 2018 March 25, as this spectrum had both high S/N and was obtained before the double-peaked feature started to appear in the emission lines. This preliminary redshift was z = 0.074, but this was uncertain due to being measured from such a broad feature. We were later able to refine this measurement using Ca ii H and K lines from the host galaxy that are visible in the LBT MODS spectrum obtained on 2018 May 21. From these narrow features, we obtained a redshift of z = 0.071, corresponding to a distance of d = 322.4 Mpc.

To estimate the time of peak light, we fit a parabolic function to the ASAS-SN g and ATLAS o light curves near peak. In order to estimate the uncertainty on the peak dates, we used a procedure similar to the one used to estimate the uncertainties on the host galaxy parameters: we generated 10,000 realizations of the g and o light curves near peak, with each magnitude perturbed by their respective uncertainties and assuming Gaussian errors. We then fit a parabola to each of these light curves and calculated the 68% confidence interval and median t_peak values. For the g-band, we obtained ${t}_{g,\mathrm{peak}}={58195.1}_{-0.8}^{+0.8}$ and m_g,peak = 17.4, while for the o-band we obtained ${t}_{o,\mathrm{peak}}={58198.5}_{-0.6}^{+0.5}$ and m_o,peak = 17.6. This discrepancy between filters is not unexpected, as PS18kh was becoming redder in optical filters, which will result in later peak dates in redder filters. We adopt the median g-band peak of t_g,peak = 58195.1, corresponding to 2018 March 18.1, when discussing data with respect to peak time throughout the paper.

The ASAS-SN and ATLAS survey data make PS18kh one of the few TDE candidates with a well-sampled rising light curve. PS18kh brightened by roughly 2.1 mag over 40 days in the g-band, reaching a peak of m_g,peak = 17.3. It brightened by a similar amount in the ATLAS o-band over the same time frame, but the rise was less dramatic in redder filters such as i and z. After peak, PS18kh faded gradually in all optical filters redder than U, but was still brighter than the magnitude of first detection in the g-band in the observations obtained 78 days after peak. At the i-band, in contrast, the transient was fainter in later data than it was in the discovery epoch, and in some cases was consistent with the measured host magnitude. In the Swift UV+U bands, the flux plateaus, or begins to re-brighten ∼50 days after peak, with the effect being more pronounced in bluer filters.

To better quantify the physical parameters of the system, we modeled the UV and optical SED of PS18kh for epochs where Swift data were available as a blackbody using Markov chain Monte Carlo methods, as was done for the previous ASAS-SN TDEs (e.g., Holoien et al. 2014, 2016a, 2016b, 2018; Brown et al. 2016, 2017). So as not to overly influence the fits, we performed the blackbody fits using a flat prior of 10,000 K ≤ T ≤ 55,000 K in all epochs. As can be seen in Figure 3, which shows the best-fit blackbody SED at various epochs compared to the Swift photometry, the blackbody fits provide good fits to the data. The resulting temperature evolution in rest-frame days relative to peak is shown in Figure 4, with time corrected to rest-frame days relative to peak.

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The blackbody fits indicate that for the first ∼45 days after peak, the temperature of PS18kh held relatively constant around T ≃ 14,000 K. This temperature and flat evolution is not uncommon for TDEs (e.g., Holoien et al. 2014, 2016b, 2018; Brown et al. 2016, 2017). However, after the UV flux began to rise, the transient became hotter, with the temperature increasing to T ≃ 25,000 K over the following three weeks. This temperature is similar to that of other TDEs, but the rising behavior seen ∼50 days after peak is unusual. Unfortunately, it is unclear whether the temperature continued to increase further, as PS18kh became Sun-constrained for Swift not long after the source began to rebrighten in the UV.

For those epochs with Swift data, we also estimated the bolometric luminosity of PS18kh from the blackbody fits. In order to better take advantage of the high-cadence light curve, we used the epochs with Swift blackbody fits to calculate bolometric corrections to the g-band data taken within one day of the Swift observations, or to g-band magnitudes interpolated between the previous and next g-band observations if there was no observation within one day of the Swift observation. We then used these bolometric corrections to estimate the bolometric luminosity of PS18kh from the g-band data for epochs when we did not have Swift data, linearly interpolating the bolometric corrections for each g-band epoch. For epochs prior to our first Swift observation, we used the bolometric correction from the first Swift SED fit. We did not correct the data taken after the last Swift observation, as the g-band continued to decline while the UV was re-brightening, and we did not want to extrapolate a rising or falling behavior beyond what our SED fits could tell us. The luminosity evolution calculated from the Swift SED fits and estimated from the g-band light curve is shown in Figure 5.

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As suggested by the Swift light curves, while the luminosity initially drops after peak, it begins to rise again ∼50 (rest-frame) days after peak. As we did with previous TDEs, we fit the initial fading light curve (0 < t < 50 days) with an exponential profile $L={L}_{0}{e}^{-(t-{t}_{0})/\tau }$, a $L={L}_{0}{(t-{t}_{0})}^{-5/3}$ power-law profile, and a power law where the power-law index is fit freely, $L\propto {(t-{t}_{0})}^{-\alpha }$. Our best-fit parameters for each model are as follows: for the exponential profile we obtain L₀ = 10^44.0 erg s⁻¹, t₀ = 58163.3, and τ = 49.8 days; for the t^−5/3 power law we obtain L₀ = 10^46.7 erg s⁻¹ and t₀ = 58142.0; and for the free power law we obtain L₀ = 10^44.4 erg s⁻¹, t₀ = 58190.9, and α = 0.60. We find that both power laws provide better fits than the exponential profile, with χ² = 31.0, χ² = 43.4, and χ² = 57.2, for the free power law, the t^−5/3 power law, and the exponential fit, respectively. All three fits are shown in Figure 5.

As can be seen in the figure and from the χ² value of the fit, the t^−5/3 profile is not a particularly good fit to the data, as the luminosity initially declines at a steeper rate, and then levels off sooner than such a profile would predict. However, it is expected that there should be some deviation from this profile near peak, as the luminosity is not expected to track the fallback rate until later in the flare, and the initial steeper decline after peak could be due to inefficient circularization of the stellar debris (e.g., Dai et al. 2015; Guillochon & Ramirez-Ruiz 2015). The best-fit t^−0.60 power-law profile is closest to the t^−5/12 power law expected for disk-dominated emission (e.g., Lodato & Rossi 2011; Auchettl et al. 2017), though the flare is not expected to exhibit this decline rate until later times after peak. It is clear that the luminosity evolution of PS18kh is more complicated than the simple t^−5/3 rate that would be observed if the luminosity tracked the mass fallback rate, as predicted in Rees (1988) and Phinney (1989), and it is not a good match to any individual theory, implying that multiple physical processes may be contributing to the observed luminosity.

Figure 5 also shows the X-ray luminosity calculated from the binned Swift XRT observations. While there is weak X-ray emission detected in the two earlier time bins, we do not detect any X-ray emission at later times, and the detected X-ray luminosity is 2 or more orders of magnitude weaker than the UV/optical emission in all epochs. The X-ray detections are below the archival limit from ROSAT, and we cannot definitively determine whether it is associated with the host or the transient based on the measured flux. Similar to what was seen with ASASSN-15oi and ASASSN-14li at early times (Holoien et al. 2016a, 2016b), the X-ray emission does not show strong evolution during the period of observation.

Modeling the X-ray spectrum obtained by combining all the XRT data, we find that the X-ray emission favors an absorbed power-law spectrum with a photon index of Γ = 3 ± 1. We also tested an absorbed blackbody model, but find that this produces a significantly worse fit (reduced ${\chi }_{r}^{2}$ ∼ 2) compared to the simple power law (${\chi }_{r}^{2}$ ∼ 1). Auchettl et al. (2017, 2018) showed that the X-ray emission of a non-jetted TDE can be well described by photon indices larger than ∼3, which is consistent with that obtained for PS18kh. These values are much softer than seen for AGNs, which have photon indexes ∼1.75 (e.g., Auchettl et al. 2017), suggesting that the emission we see arises from the TDE, rather than an underlying AGN.

Integrating over the entire rest-frame bolometric light curve calculated from the g-band data and the Swift blackbody fits gives a total radiated energy of E = (3.46 ± 0.22) × 10⁵⁰ erg, with (1.42 ± 0.20) × 10⁵⁰ erg being released during the rise to peak. This shows that a significant fraction of energy radiated from TDEs can be emitting during the rise to peak, and highlights the need for early detection. The total radiated energy corresponds to an accreted mass of ${M}_{\mathrm{Acc}}\,\simeq 0.002{\eta }_{0.1}^{-1}$ M_⊙, where the accretion efficiency is η = 0.1η_0.1. As with other TDEs, a negligible fraction of the bound stellar material appears to actually accrete onto the black hole, or the material is accreting with a very low radiative efficiency.

The dominant spectral features of PS18kh are a strong blue continuum and broad hydrogen emission lines, similar to the features that have been seen in most TDEs discovered at optical wavelengths (e.g., Arcavi et al. 2014). PS18kh falls into the "hydrogen-rich" group of TDEs, with strong Balmer lines, particularly Hα and Hβ, visible in most epochs, but with weak or absent helium emission features. There is some suggestion of emission that is consistent with He i 5875 Å at the redshift of PS18kh, but the He ii 4686 Å line seen in many TDEs is notably absent.

Our earliest spectroscopic follow-up was obtained prior to or within a few days of the g-band peak, and some interesting trends can be seen in the spectra. In particular, the spectral slope becomes steeper near peak before beginning to slowly flatten again over the course of our observations, which is unsurprising given that the TDE was optically brightest at peak. The emission lines become stronger as time progresses, and only become clearly visible shortly after peak light. Unfortunately our first spectrum, the classification spectrum obtained on 2018 March 7, was taken through clouds, making it difficult to determine whether there were emission lines prior to peak. As was seen with the optical photometry, there is little evidence of the UV re-brightening in the optical spectra—the continuum level remains relatively flat, and the lines show no significant evolution.

The spectra of PS18kh differ from the majority of other TDEs in one respect: the Hα, and in some cases Hβ, lines show evidence of an evolving, boxy shape that becomes more prominent over time, and in some later epochs there is a suggestion of double peaks in the Hα profile. A similar double-peaked Hα profile was seen in the TDE PTF09djl, though in that case the peaks showed a much larger separation (Arcavi et al. 2014; Liu et al. 2017a).

The possibility that TDEs could lead to the formation of line-emitting (elliptical) disks was discussed by Eracleous et al. (1995) and Guillochon et al. (2014). In the cases of two recent TDEs, PTF09djl and ASASSN-14li, an elliptical disk model has been used to fit the emission line profiles and model the properties of the accretion disk (Liu et al. 2017a; Cao et al. 2018). Here we use similar models to infer the properties of the accretion disk, and potentially the stellar debris, of PS18kh.

We consider models for the profiles of the broad Hα emission lines that attribute the emission to gas in a relativistic Keplerian disk. We were motivated by the success of such models in describing the Balmer line profiles of active galaxies and quasars in general (e.g., Popović et al. 2004; Bon et al. 2009; La Mura et al. 2009; Storchi-Bergmann et al. 2017) and recent theoretical scenarios that associate the broad-line region with the accretion disk in quasars and active galaxies (e.g., Elitzur et al. 2014), as well as the studies of PTF09djl and ASASSN-14li mentioned above. An alternative family of models, which we do not consider here, attribute the emission lines to spherically expanding outflow (see Roth et al. 2016; Roth & Kasen 2018). Those models employ more rigorous radiative transfer calculations than ours and incorporate electron scattering (our models adopt the Sobolev apprximation for radiative transfer in an accelerating medium). They can also produce asymmetric line profiles in the early stages of the evolution of the event with asymmetries resulting from radiative transfer effects. In contrast, in our models the asymmetries result from relativistic effects. In fact, an interpretation of the Balmer line profiles of PS18kh in terms of an outflow model is discussed in a recent paper by Hung et al. (2019). The blueshifted broad absorption lines found by Hung et al. (2019) in the UV spectra of PS18kh can be explained by both their model and ours, since the models are qualitatively similar: they invoke accretion-powered outflows (our models are based on the accretion-disk wind calculations of Murray et al. 1995). The models do differ, however, in the exact geometry and velocity field of the outflow, the layers taken to emit the Balmer lines (we attribute the Balmer lines to the base of the outflow, i.e., the accretion disk atmosphere), and the methods they used to treat radiative transfer.

The model line profiles are obtained in the observer's frame by adopting the formalism detailed in Chen et al. (1989), Chen & Halpern (1989), Eracleous et al. (1995), and Flohic et al. (2012) by computing the integral

$$\begin{eqnarray}&&{f}_{\nu }\propto \int d\varphi \int \xi \,d\xi \,{I}_{\nu }(\xi ,\varphi ,{\nu }_{{\rm{e}}}){D}^{3}(\xi ,\varphi ){\rm{\Psi }}(\xi ,\varphi )\,\end{eqnarray} \tag{ 1 }$$

over the surface of the disk. The functions in the integrand are expressed in polar coordinates in the frame of the disk where φ is the azimuthal angle in the plane of the disk, ξ ≡ r/r_g is the dimensionless radial coordinate, ${r}_{{\rm{g}}}\equiv {{GM}}_{\bullet }/{c}^{2}$ is the gravitational radius, and M_• is the mass of the black hole. The axis of the disk makes an angle i with the line of sight to the observer (the "inclination" angle) and the line-emitting portion of the disk is enclosed between radii ${\xi }_{\mathrm{disk}}^{\mathrm{in}}$ and ${\xi }_{\mathrm{disk}}^{\mathrm{out}}$ (see Figure 1 of Chen et al. 1989).

The functions D and Ψ describe the gravitational and transverse redshifts and light bending, respectively, in the weak-field approximation. The function I_ν represents the apparent emissivity of the disk and includes terms that account for the intrinsic brightness distribution of the disk, the (potentially anisotropic) escape probability of line photons in the direction of the observer, and local line broadening (see Equation (2) of Flohic et al. 2012, and the associated discussion). The local profile of the line is assumed to be a Gaussian of standard deviation σ that includes contributions from local turbulence, electron scattering, and blurring resulting from the finite cells used in the numerical integration. The intrinsic brightness profile of the disk is parameterized by a power law of the form ξ^−q where q takes values between 1 and 3, inspired by the results of photoionization calculations by Dumont & Collin-Souffrin (1990a, 1990b). This axisymmetric emissivity pattern can be perturbed either by making the disk elliptical or by superposing a logarithmic spiral, as we explain below.

At early times, the observed profile of the Hα line in PS18kh appears bell-shaped and somewhat asymmetric with an extended red wing. At late times, the profile evolves to a flat-topped or, sometimes, double-peaked shape. It maintains its red wing and it sometimes shows a blue shoulder. We interpret this sequence of line profiles as indicating a progressive decline in the optical depth of the line-emitting region of the disk. This interpretation is based on the behavior of the theoretical line profiles with optical depth and on the expected evolution of the accretion rate through the disk and onto the black hole. At early times the high accretion rate is likely to lead to the emission of a wind from the surface of the accretion disk, consisting of stellar debris from the disruption, whose dense base layers will provide a substantial optical depth to the line photons. As the accretion rate drops and the debris moves outward from the black hole, the density of the wind and the optical depth of the surface layers of the disk decline accordingly. We also note that the blue shoulder in the observed late-time Hα profiles cannot be reproduced by a model of an axisymmetric disk. Therefore, we postulate that a non-axisymmetric perturbation is present and we explore whether an elliptical disk or a disk with a spiral arm can describe this perturbation successfully.

The spectra obtained prior to 2018 March 25 show a weak Hα emission line, suggesting that the optical depth of the material surrounding the accretion disk is too large to obtain a model fit to the data. To represent the observed early-time Hα profiles (those between 2018 March 25 and 2018 April 1) we adopt the wind model discussed by Murray et al. (1995; see also Chiang & Murray 1996; Murray & Chiang 1997; Flohic et al. 2012; Chajet & Hall 2013). In these models, the apparent brightness profile of the disk is non-axisymmetric, as shown, for example, in Figure 4 of Flohic et al. (2012), because of the large optical depth and anisotropic escape probability of photons through the emission layer. The resulting line profiles have round or somewhat flat tops and an extended red wing because of relativistic effects (see examples in Figure 5 of Flohic et al. 2012). The free parameters of the model are the inner and outer radii of the line-emitting portion of the disk, ${\xi }_{\mathrm{disk}}^{\mathrm{in}}$ and ${\xi }_{\mathrm{disk}}^{\mathrm{out}}$, the local line width, σ (in km s⁻¹), the emissivity power-law index, q, the disk inclination angle, i, the angle of the wind streamlines relative to the plane of the disk, λ (see Figure 1 of Murray & Chiang 1997), and the normalization of the position-dependent optical depth pattern, given in terms of τ, the optical depth in the direction of the observer at a fiducial position in the disk of $(\xi ,\varphi )=({\xi }_{\mathrm{disk}}^{\mathrm{in}},0)$.^{28 ,29}

For a circular disk with an axisymmetric emissivity pattern, the line profile has a net redshift and a red wing that is more pronounced than the blue wing because of a combination of gravitational and transverse redshifts. If ${\xi }_{\mathrm{disk}}^{\mathrm{out}}/{\xi }_{\mathrm{disk}}^{\mathrm{in}}\lesssim 10$, there are two well-separated peaks and the blue peak is stronger than the red peak because of Doppler boosting. As this ratio increases, the two peaks get closer together, eventually blending to form a profile with a flat or round top. If the emitting layer of the disk is accelerating to form a wind, radiative transfer effects make the apparent emissivity non-axisymmetric, as illustrated in the left panel of Figure 6. The emissivity is enhanced at low projected velocities and depressed at high projected velocities, which enhances the core and depresses the two peaks of the profile, making it flat- or round-topped.

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To fit the profiles at late times, we tried two different models, an elliptical disk (see Eracleous et al. 1995), and a circular disk with a single spiral arm (see Gilbert et al. 1999; Storchi-Bergmann et al. 2003). Figure 6 shows illustrations of these models, which are described in more detail below.

• 1.  
  The axisymmetric emissivity of a circular disk is perturbed by a logarithmic spiral, as described in Equation (2) of Storchi-Bergmann et al. (2003). In addition to the five free parameters that describe a circular disk, there are five free parameters that describe the spiral pattern: the pitch angle, width, and azimuth of the spiral arm at the inner disk, p, w, and φ_in, respectively, its brightness contrast relative to the underlying axisymmetric disk, A, and its outer radius, ${\xi }_{\mathrm{spiral}}^{\mathrm{out}}$. We take its inner radius to be the same as that of the line emitting portion of the disk, i.e., ${\xi }_{\mathrm{spiral}}^{\mathrm{in}}={\xi }_{\mathrm{disk}}^{\mathrm{in}}$. The disk may also emit a wind of modest optical depth that modifies the line profiles because of radiative transfer effects, as discussed earlier in this section.
• 2.  
  The disk streamlines are nested ellipses with aligned semimajor axes whose eccentricity increases linearly with distance from the center (from 0 to a maximum value of e). The emitting gas is optically thin to the line photons. There are two more free parameters in addition to those of an axxisymmetric circular disk (without a wind), the outer eccentricity and orientation of the semimajor axis relative to the observer, e and φ₀. A model of this type was considered by Guillochon et al. (2014) in their discussion of the evolution of the TDE PS1-10jh and applied to PTF09djl and ASASSN-14li by Liu et al. (2017a) and Cao et al. (2018). Moreover, a structure resembling an elliptical disk is discernible in the simulations of Shiokawa et al. (2015) that follow the evolution of the post-disruption debris to late times.

Introducing non-axisymmetric perturbations, such as a spiral arm or eccentric orbits, enhances the disk emissivity at specific projected velocities. Thus the asymmetries present in axisymmetric disk models can be changed (e.g., reversed or eliminated) or the two peaks can become less pronounced because the valley between them is filled in. In the case of a spiral arm, the profile modifications are determined largely by the shape, orientation, and contrast of the spiral arm. In the case of an eccentric disk of the type employed in this work, the modifications of the line profile are controlled largely by the combination of eccentricity and orientation of the major axis.

While a complete exploration of the model parameter space is beyond the scope of this work, we carried out a limited, qualitative exploration where the goodness of all fits was assessed by eye. We focused our attention on disks with low inclination (i.e., closer to face-on) so as to obtain fits with models that have small disk radii. We took this approach in order to reduce the angular momentum of the debris in the disk so that it did not exceed the initial angular momentum of the approaching star; we discuss this issue in detail in Section 3.4. We note that in models of line profiles from a non-relativistic disk, the line profiles are symmetric and their widths depend on the combination ${({M}_{\bullet }/R)}^{1/2}\sin i\propto {\xi }^{-1/2}\sin i$, making the inclination angle and disk radius degenerate. Once special and general relativistic effects are included, the line profiles become progressively more asymmetric and redshifted as ξ decrases, and the degeneracy between i and ξ is broken. In the models that we explore here, we look for the minimum inclination angle that can reproduce the shape/asymmetry of the line profiles.

In practice, we first fitted the 2018 March 25 spectrum with a wind model (including a spiral arm) and then adjusted the optical depth and disk radii to reproduce the March 30, 31, and April 1 spectra. We found that the minimum inclination angle for which the model can match the red wing of the Hα profile well and the blue wing approximately is 26°. At smaller inclination angles the model profiles are too asymmetric to fit the data well. We then fitted the 2018 April 25 spectrum with a disk and spiral model and an elliptical disk model of the same inclinaiton and compared that model with the other observed spectra obtained after 2018 April 1 to check whether they could adequately describe those spectra as well. We estimated the uncertainties in the model parameters by perturbing them about their best-fit values and adjusting the other parameters to get a good fit until no good fit was possible. Thus, we found that the inner disk radius can be determined to ±20 r_g, the outer radius to ±200 r_g, the emissivity power-law index to ±0.2, and the broadening parameter to ±300 km s⁻¹. The wind optical depth could be determined to a factor of 3 while the wind opening angle was held fixed at 15° based on the physical considerations discussed in Murray & Chiang (1997). The orientation of the spiral pattern could be determined to ±5°, its pitch angle to ±10°, its angular width to ±20°, its outer radius to ±200 r_g, and its contrast to ±2. The eccentricity of the elliptical disk could be determined to ±0.2 and the orientation of its major axis to ±10°. The best-fit parameters for the disk+wind+spiral arm models are given in Tables 6 and 7. The former table gives the values of the model parameters that were held fixed while the latter gives the values of the parameters that were allowed vary with time in order to reproduce the evolution of the line profiles.

Table 7.  Variable Parameters for Circular Disk Models

Date	$\tau ({\xi }_{\mathrm{disk}}^{\mathrm{in}},0)$	ξdisk	ξspiral	q
Mar 25	>3.0	50–1600	⋯	1.8
Mar 30	>3.0	50–1600	⋯	1.5
Mar 31	0.9	50–1600	⋯	1.3
Apr 1	0.3	50–1600	⋯	1.2
Apr 11 and later	0.1	80–1500	80–350	1.0

Note. Parameters for the disk+wind+spiral model used to model the Hα emission line profile that change with time. All optical depths are in the inner disk the direction of the observer, i.e., at $\xi ={\xi }_{\mathrm{disk}}^{\mathrm{in}}$ and φ = 0. Radii are the outer radius of the disk and the outer radius of the spiral arm in the disk (the inner radius of the disk and the spiral arm are fixed). The parameter q is the index of the power law that describes the axisymmetric portion of the disk emissivity.

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Figure 7 shows the evolution of the Hα emission line and the model fits for each epoch after the line emerges. As described above, the profiles between 2018 March 25 and 2018 April 1 are well-fit by a disk+wind model, shown by a cyan line in the figure. As the optical depth of the wind drops, the emission from the underlying disk becomes apparent. We show three fits to the 2018 April 1 spectrum, a disk+wind model, an elliptical disk model (shown in magenta), and a disk with spiral arm model (shown in red). The parameters of the elliptical disk model are i = 26°, ξ_disk = 60–1400 (pericenter distances), e = 0.25, φ₀ = 15°, q = 1.4, and σ = 800 km s⁻¹ (see the illustration in Figure 6).

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All models can provide a reasonable fit to the top of the 2018 April 1 line profile and its red wing but none can reproduce the blue wing very well. The same is true for the models in most of the later epochs, although the blue wing of the line becomes weaker in some epochs in late April and early May and the models can approximate it better. The model parameters that do not change with time are summarized in Table 6 while the parameters that vary with time are in Table 7. Each of the later epochs has been fit with the same models, with the only difference between epochs being a flux scaling factor ranging from 1.15 to 1.8. The flux scaling factor increases with time until 15 May 2018 after which it begins to fall again. The changing scaling factor and and varying difference between the models and the blue wing of the line profile, as well as small changes in the peak and wings of the emission line from epoch-to-epoch, are likely a result of short-term variability in the disk structure on spatial scales too small to properly capture with the models used here.

Finally, we note that we also experimented with disk models with higher inclination angles, up to i = 60°. These models have correspondingly larger radii and the other parameters have to be adjusted somewhat to obtain a good fit. For example, the models with i = 60° have ξ_disk ∼ 500–15,000 and $\tau ({\xi }_{\mathrm{disk}}^{\mathrm{in}},0)$ of order a few. We do not favor such models because they imply that the debris carries too much angular momentum, as we discuss in detail in Section 3.4.

Taken as a whole, the evolution of the Hα profile in PS18kh shows that as the TDE is brightening toward its peak, the disk is obscured by optically thick material, likely debris from the disruption. Between 2018 March 25 and 2018 April 1, this material becomes progressively more optically thin while the line-emitting portion of the disk grows in size, and the emission lines are well-fit by a disk+wind model. After 2018 April 1, the wind becomes feeble, the emission from the disk is clearly seen, and the double-peaked/boxy profile is well-fit by a disk with a spiral arm model. The scale of the disk is similar to that seen in PTF09ge and ASASSN-14li (Liu et al. 2017a; Cao et al. 2018), indicating that this is likely a common feature of TDEs. The disk has non-axisymmetric perturbations that are approximated by the spiral arm. A slight variation in the scaling of the models to the line profiles after 2018 April 1 suggests that the perturbations are changing with time.

We also measured the luminosities of the Hα and Hβ emission lines from our follow-up spectra for all epochs where the lines were pronounced enough to measure the flux. In Figure 8 we show the luminosity evolution of these two emission features from the spectra of PS18kh. As estimating the true error on the line fluxes is difficult given their complex shape, we assume 30% errors on the emission fluxes calculated in each epoch. From the spectra taken at and shortly after peak, we can see the Hα line becoming more luminous and more pronounced, peaking at a luminosity of L_Hα ∼ 6 × 10⁴¹ erg s⁻¹ roughly 20 rest-frame days after the continuum peaks. After peaking, the Hα luminosity remains relatively constant for the rest of the period of observation. Similarly, though it is not measurable prior to peak, the Hβ luminosity remains at roughly L_Hβ ∼ 1–2 × 10⁴¹ erg s⁻¹ in all epochs where it is measurable. This roughly constant evolution of the line luminosities differs from that of ASASSN-14li, which showed declining line luminosities following discovery (Holoien et al. 2016b; Brown et al. 2017).

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Figure 8 also shows the Hα emission that would be expected given the measured Hβ emission, assuming the emission is driven by case B recombination. The Hα/Hβ ratio is largely consistent with what would be expected from recombination, within noise, similar to what was seen in ASASSN-14li (Holoien et al. 2016b). This also indicates there is little additional extinction from the host galaxy. The measured luminosities for both lines are given in Table 8.

The excellent spectroscopic coverage of PS18kh before, during, and after peak light may also allow us to trace the evolution of the accretion state in the TDE. Few TDEs prior to PS18kh have exhibited such a strong evolution in emission line profiles as we see here with H_α, and this evolution may be due to changes in the accretion state. For example, some theories predict that, shortly after disruption, the accretion in a TDE is expected to be super-Eddington, launching a wind where strong optical reprocessing can occur (e.g., Roth et al. 2016; Dai et al. 2018). The accretion rate is expected to drop to roughly Eddington shortly after peak for a 10^6.9 M_⊙ black hole, when the Hα line begins to emerge, and later as the accretion becomes sub-Eddington the wind is expected to become optically thin, allowing the disk to be observed. While there does not seem to be significant evolution in the Hα profile of PS18kh after it emerges, indicating this theoretical picture may not be exactly correct, there have been few TDEs observed at such early times and even fewer that are observed before the lines appear, indicating more objects with similarly early spectra are needed to refine these models.

To further evaluate the plausibility of the models for the Hα line profiles, we compare the angular momentum of the line-emitting gas to the angular momentum of the star prior to disruption. We assume that the star is on a parabolic orbit around the black hole with a pericenter distance equal to the tidal disruption radius, r_t, and a pericenter speed ${\upsilon }_{{\rm{p}}}{({r}_{{\rm{t}}})=(2{{GM}}_{\bullet }/{r}_{{\rm{t}}})}^{1/2}$. The specific angular momentum of the star is, then, ${j}_{\star }={r}_{{\rm{t}}}\,{\upsilon }_{{\rm{p}}}{({r}_{t})=(2{{GM}}_{\bullet }{r}_{{\rm{t}}})}^{1/2}$. If the post-disruption debris conserves specific angular momentum and settles down into a circular disk (or ring), the radius of that disk should be r_d = 2r_t, which is two orders of magnitude smaller than the disk radius inferred from fitting the line profiles (see the illustration below). Thus, in order to reconcile the size of the line emitting disk with the dynamics of the debris we must adopt a picture in which the line-emitting gas represents a small fraction of the mass of the debris and as much larger specific angular momentum than the initial star. This implies that angular momentum is transported very quickly, perhaps with the help of shocks, (e.g., Shiokawa et al. 2015; Bonnerot et al. 2016; Hayasaki et al. 2016), and redistributed so that a small fraction of the mass of the debris ends up at a large distance from the center and emits lines. We assess this scenario below by comparing the total angular momentum of the star to that of the line-emitting disk.

The total angular momentum of the star at the tidal disruption radius, r_t, is ${J}_{\star }={m}_{\star }{r}_{{\rm{t}}}\,{\upsilon }_{{\rm{p}}}({r}_{t})$, which we re-write as ${J}_{\star }={m}_{\star }{({{GM}}_{\bullet }/c)(2{\xi }_{{\rm{t}}})}^{1/2}$, where ξ_t ≡ r_t/r_g. We use the expression for r_t from Phinney (1989) and re-cast it in terms of the average density of the star. We then take advantage of the mass–radius relation for zero-age main-sequence stars (r_⋆ ∝ ${m}_{\star }$^0.57 for ${m}_{\star }$ > 1.66 M_⊙; e.g., Demircan & Kahraman 1991) to relate the average density of the star to its mass and obtain ${\xi }_{{\rm{t}}}\,=7.1{(K/0.7){M}_{7}^{-2/3}({m}_{\star }/{M}_{\odot })}^{0.237}$, where ${M}_{7}={M}_{\bullet }/{10}^{7}\,{M}_{\odot }$. The parameter K is equivalent to the combination (k/f)^1/6 used by Phinney (1989) to parameterize the structure of the star and takes values between 0.52 and 0.82. Substituting this expression for ξ_t into the equation for J_⋆ and carrying out some algebra we obtain ${J}_{\star }=3.3\times {10}^{56}{\left(K/0.7\right)}^{1/2}{\left({m}_{\star }/{M}_{\odot }\right)}^{1.12}{M}_{7}^{1/3}\,{\rm{g}}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$.

To compute the angular momentum of the line-emitting debris, J_d, we assume that the debris is concentrated at the outer boundary of the disk adopted in the models for the line profiles. Using the same approach as for the angular momentum of the star, and assuming that the disk is generally elliptical, we write ${J}_{{\rm{d}}}={m}_{{\rm{d}}}\,({{GM}}_{\bullet }/c){\left[{\xi }_{\mathrm{disk}}^{\mathrm{out}}(1+e)\right]}^{1/2}$, where m_d is the mass of the debris and ${\xi }_{\mathrm{disk}}^{\mathrm{out}}$ and e are, respectively, the outer pericenter distance and eccentricity of the disk (inferred from the fits to the line profiles). To get the mass of the debris, we use the Hα luminosity, assuming that it is produced by case B recombination in an ionized, thermal plasma of uniform density. The luminosity per unit volume is, then, ${L}_{{\rm{H}}\alpha }/V\,={n}_{{\rm{e}}}\,{n}_{{\rm{p}}}\,{\epsilon }_{{\rm{H}}\alpha }^{\mathrm{eff}}(T,{n}_{{\rm{e}}})$, where n_e is the electron density, n_H and n_p are the electron and proton densities, and ${\epsilon }_{{\rm{H}}\alpha }^{\mathrm{eff}}(T,{n}_{{\rm{e}}})$ is the case B effective recombination coefficient (a function of the temperature, T, and n_e). Expressing n_p in terms of m_d and V and re-arranging, we obtain the following expression for the mass of the debris: m_d = 0.084 (L₄₂/n_{e,11 −25}) M_⊙, where L₄₂ = L_Hα/10⁴² erg s⁻¹, n_e,11 = n_e/10¹¹ cm⁻³, and ${\epsilon }_{-25}\,={\epsilon }_{{\rm{H}}\alpha }^{\mathrm{eff}}(T,{n}_{{\rm{e}}})/{10}^{-25}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-3}$. For the range of temperatures and densities of interest here, ₋₂₅ has values of order a few (see Storey & Hummer 1995). The electron density is unknown, but we can draw an analogy with the broad-line region of Seyfert galaxies and quasars where recent work modeling the Fe ii UV line complex (Bruhweiler & Verner 2008; Hryniewicz et al. 2014), the optical and UV intermediate-width lines (Adhikari et al. 2016), and the far-UV resonance lines (Moloney & Shull 2014) suggests densities in excess of 10¹¹ cm⁻³. Thus, we scale n_e by 10¹¹ cm⁻³ in the above expression. The angular momentum of the line-emitting debris is, then, ${J}_{{\rm{d}}}=2.3\,\times {10}^{56}\,{L}_{42}\,{M}_{7}\,{n}_{{\rm{e}},11}^{-1}$ ${\epsilon }_{-25}^{-1}{\left[{\xi }_{\mathrm{disk}}^{\mathrm{out}}(1+e)/1000\right]}^{1/2}$ ${\rm{g}}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$.

Combining the above estimates, the ratio of the angular momentum of the debris to that of the star is

$$\begin{eqnarray}&&\displaystyle \frac{{J}_{{\rm{d}}}}{{J}_{\star }}=0.7\,\displaystyle \frac{{L}_{42}\,{M}_{7}^{2/3}}{{n}_{{\rm{e}},11}\,{\epsilon }_{-25}}{\left[\displaystyle \frac{{\xi }_{\mathrm{disk}}^{\mathrm{out}}(1+e)}{1000}\right]}^{1/2}{\left(\displaystyle \frac{K}{0.7}\right)}^{-1/2}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-1.12}.\end{eqnarray} \tag{ 2 }$$

Inserting values for the quantities that we can constrain observationally, ${\left[{\xi }_{\mathrm{disk}}^{\mathrm{out}}(1+e)/1000\right]}^{1/2}\lesssim 1.3$, L₄₂ ≈ 0.5, ${M}_{7}^{2/3}\,=0.86$, and ₋₂₅ ≈ 2 (from Storey & Hummer 1995 for T = 1–3 × 10⁴ K and ${n}_{{\rm{e}}}={10}^{9}\mbox{--}{10}^{13}\,{\mathrm{cm}}^{-3}$), we obtain J_d/J_⋆ ≲ 0.2. The value of K, although unknown, does not change the result by more than 10%. The distance of closest approach of the star to the black hole is unknown but it cannot be less than 0.14 of ξ_t, otherwise the star would enter the event horizon even for a maximally spinning black hole; since J_⋆ ∝ ξ_t^1/2 a closer encounter could increase J_d/J_⋆ to ≲0.5, at most. The mass of the star is also unknown but it is clear from the above analysis that a more massive star could be disrupted more easily because of its lower average density and that would also lead to a smaller value of the angular momentum ratio. For example, assuming ${m}_{\star }$ = 4 M_⊙ would lead to J_d/J_⋆ ∼ 0.06. Moreover, if the mass of the black hole is lower than the value we have estimated, the fraction of the angular momentum carried by the line-emitting gas is correspondingly lower. Taken at face value, these estimates suggest that the angular momentum of the line-emitting debris can be considerably smaller than the angular momentum of the star, reinforcing the plausibility of the model fits to the Hα line profiles. However, two significant uncertainties must be borne in mind: (i) the density of the debris is unknown, and (ii) there may be a substantial amount of neutral gas associated with the line-emitting gas that would contribute to the mass, hence the angular momentum, of the debris.