Application of The Wind-Driven Model to A Sample of Tidal Disruption Events
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Application of The Wind-Driven Model to A Sample of Tidal Disruption Events
KOHKI UNO and KEIICHI MAEDA Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan (Received September 8, 2020; Revised October 26, 2020; Accepted November 13, 2020)
Submitted to ApJLABSTRACTAn origin of the Optical/UV radiation from tidal disruption events (TDEs) has recently been dis-cussed for different scenarios, but observational support is generally missing. In this Letter, we testapplicability of the ‘Wind-Driven model’ (Uno & Maeda 2020) to a sample of UV/Optical TDEs. Withthe model, we aim to derive the physical properties of the Optical/UV TDEs, such as mass-loss ratesand characteristic radii. The model assumes optically thick continuous outflows like stellar winds, andone key question is how the wind-launched radius is connected to physical processes in TDEs. Asone possibility, through a comparison between the escape velocities estimated from their black-holemasses and the wind velocities estimated from observed line widths, we propose that the outflow islaunched from the self-interaction radius ( R SI ) where the stellar debris stretched by the tidal force in-tersects; we show that the escape velocities at R SI are roughly consistent with the wind velocities. Byapplying the model to a sample of Optical/UV TDE candidates, we find that explosive mass ejections( (cid:38) M (cid:12) yr − ) from R SI ( ∼ cm) can explain the observed properties of TDEs around peak lu-minosity. We also apply the same framework to a peculiar transient, AT2018cow. The model suggeststhat AT2018cow is likely a TDE induced by an intermediate-mass black hole ( M BH ∼ M (cid:12) ). Keywords:
Transient sources — Tidal disruption — Stellar winds INTRODUCTIONWhen a star approaches a supermassive black hole(SMBH) into its tidal radius, the tidal force of theSMBH destroys the star. These phenomena are knownas Tidal Disruption Events (TDEs). In the TDE,roughly half of the stellar debris is bound and then ac-cretes onto the SMBH, while the rest is unbound andescapes the system (Rees 1988). As a result, the systemis believed to release a large amount of energy to powera bright transient for a relatively brief time ( (cid:46) L ∝ t − / (Phinney 1989). The model predicts that the TDEhas a peak in the radiation energy around the soft X-rays. Indeed, the first TDE candidate was discovered by Corresponding author: KOHKI [email protected]
ROSAT (Donley et al. 2002). However, in recent years,new generation surveys such as Pan-STARRS (Kaiseret al. 2002), PTF (Law et al. 2009), ASAS-SN (Shappeeet al. 2014), and ZTF (Bellm et al. 2019), discover TDEswhich are bright in the Optical/UV wavelengths.Typical features of the optical/UV TDEs include ahigh peak bolometric luminosity ( L peak ∼ erg s − ),a high blackbody temperature at a few × K, a bluecontinuum component, and broad spectral lines of H,He, and N corresponding to the velocity of ∼ km s − (Arcavi et al. 2014; van Velzen et al. 2020). These fea-tures may depend on the physical properties of a dis-rupted star (e.g., MacLeod et al. 2012), a behavior ofa stream stretched by the tidal force (e.g., Strubbe &Quataert 2009), radiative transfer effects (e.g., Roth &Kasen 2018), or other factors.In recent years, it has been suspected that opti-cally thick outflows of the stellar debris form the Op-tical/UV photosphere (Strubbe & Quataert 2009; Met-zger & Stone 2016), and that the direct radiation fromthe disk is observable only in the late phase. As one ori- a r X i v : . [ a s t r o - ph . H E ] N ov UNO & MAEDA gin of the outflows, some models have been proposedsuch as super-Eddington winds (Strubbe & Quataert2009; Lodato & Rossi 2011) and the stream-collision out-flow (Jiang et al. 2016; Lu & Bonnerot 2020). However,the details still remain unclear.In this study, by applying the ‘Wind-Driven model’(Uno & Maeda 2020) to a sample of Optical/UV TDEcandidates, we aim to understand the origin of the Op-tical/UV radiation. The model assumes optically thickcontinuous outflows characterized by the mass-loss rate( ˙ M ) and the wind velocity, which are analogous to stel-lar winds. We note that similar models have been in-deed proposed for the Optical/UV TDEs (e.g., Piro &Lu 2020), but so far the models have not been applied toa sample of observed TDEs (but see Matsumoto & Pi-ran 2020). For example, in this Letter we show that thewinds are likely to be launched from the self-interactionradii ( R SI , e.g., Piran et al. 2015; Dai et al. 2015),through the comparison between escape velocities ( v esc )and wind velocities ( v wind ). The radius is one of thecharacteristic radii of TDEs, in which the stellar debrisstream stretched by the tidal force intersects. We fur-ther estimate the mass-loss rates of these TDEs at theirpeak luminosity and discuss the physical properties ofthe disrupted stars.The Letter is structured as follows. In Section2, we in-troduce observational properties of the sample of TDEs.In Section3, we apply the Wind-Driven model to theseobserved TDEs and compute their R SI . We also es-timate the peak mass-loss rate ( ˙ M peak ) for each TDEs.By applying the same framework to a peculiar transient,AT2018cow, we discuss its origin in Section4. The Let-ter is closed in Section5 with conclusions. THE PROPERTIES OF 21 TDESWe introduce some observational properties of the Op-tical/UV TDE candidates. We select the candidates dis-covered by surveys such as PTF, ASAS-SN, and ZTF. Toobtain unique solutions using the Wind-Driven model,we need to select TDEs with sufficient information. Therequirements are as follows; (1) the BH mass ( M BH )has been estimated using the M BH − σ relation or the M bulge − M BH relation (McConnell & Ma 2013; Kor-mendy & Ho 2013), (2) the peak luminosity and tem-perature have been obtained, and (3) the spectral linewidths have been derived around the peak luminosity.As the candidates that satisfy the above criteria, weselect 21 Optical/UV TDEs. We summarize their ob-servational properties in Table1.In Table1, we present the classification based on theobserved spectra presented in van Velzen et al. (2020);TDE-H, TDE-Bowen, and TDE-He. We also show the Full-Width Half Maximum (FWHM) of a selected line.We use the FWHM as their v wind . We use H α for objectswhere H α is observed, i.e., TDE-H and TDE-Bowen, andHe II instead for those where H α is not observed, i.e.,TDE-He. PS1-11af has featureless spectral lines andwe cannot identify the spectral lines. We do not alsoidentify the classification of ASASSN-15lh. As v wind ofPS1-11af, we use the spectral line width around 2680˚A,which is presumed to be Mg II (Chornock et al. 2014).We use the width of the main feature at 4200˚A for v wind of ASASSN-15lh (Leloudas et al. 2016). WIND-DRIVEN MODEL FOR A SAMPLE OFTDESIn some models, it is believed that optically thickflares or outflows originated in stellar debris form in theOptical/UV photosphere and emission and/or absorp-tion in their spectral lines (Strubbe & Quataert 2009;Metzger & Stone 2016; Lu & Bonnerot 2020; Piro &Lu 2020). Here, we consider the Wind-Driven modelby Uno & Maeda (2020) to test whether continuousoutflows can explain the properties of the Optical/UVTDEs.We apply the Wind-Driven model to the sample ofthe 21 TDE candidates. By applying the model, we canestimate some physical properties (the mass-loss ratesand some physical scales) from observational proper-ties (luminosity, temperature, and wind velocities). Themodel defines the innermost radius ( R eq ) where the windis launched. In the original formalism, it is assumedthat equipartition is realized between the internal en-ergy (dominated by radiation) and the kinetic energy.However in this Letter, we follow an inversed approach;we first assume R eq = R SI , and then test whether theequipartition is indeed realized there. This way, we willshow below that this assumption is supported by TDEobservations.3.1. The Assumption: R eq = R SI R SI depends on physical properties of the BH and thedisrupted star. It is described by Dai et al. (2015) andWevers et al. (2017) as follows: R SI = R t (1 + e ) β (1 − e cos( δω/ , (1)where R t is the tidal radius given as R t ≈ R ∗ ( M BH /M ∗ ) / , where R ∗ and M ∗ are the radius andmass of the disrupted star. The impact parameter isgiven as β = R t /R p , where R p is the pericenter distance. e is the orbital eccentricity. δω is given by Wevers et al.(2017) as follows: δω = A S − A J cos( i ) , (2) Table 1.
Sample of 21 TDE candidates.Object log M BH log L peak T peak spectral type FWHM L edd /L Ref. R SI [ M (cid:12) ] [erg s − ] [10 K] [km s − ] line(˚A) day [10 cm]TDE2 7 . +0 . − . > . . α - 0.032 a 2.62PTF09ge 6 . +0 . − . . . II (4686) − . +0 . − . . . α +7 0.50 b,c 3.94PTF09djl 5 . +0 . − . . . α +2 0.93 b,c 4.67PS1-10jh 5 . +0 . − . . . II (4686) < . +0 . − . . . II (2680) +24 0.082 f,e 3.14ASASSN-14ae 5 . +0 . − . . . α +3 2.4 g,c,e 2.75ASSASN-14li 6 . +0 . − . . . α +10 0.29 h,c,e 6.17ASASSN-15lh 8 . +0 . − . . . . +0 . − . . . II (4686) +7 0.57 j,e 5.97iPTF15af 6 . +0 . − . . . II (4686) +7 0.15 k,c 3.28iPTF16axa 6 . +0 . − . . . α +6 0.38 l,c,e 6.11iPTF16fnl 5 . +0 . − . . . α . +0 . − . . . α +11 0.43 n,e 3.94PS18hk 6 . +0 . − . . . α +6 0.085 o,e 3.16ASASSN-18jd 7 . +0 . − . . . α average 0.087 p 0.782ASASSN-18pg 6 . +0 . − . . . α - 0.17 q,e 2.67AT2018hyz 6 . +0 . − . . . α peak 1.2 r,e 5.89ASASSN-19bt 6 . +0 . − . . . α +8 0.15 s,e 3.89ASASSN-19dj 7 . +0 . − . . . α - 0.38 t,e 2.17AT2019qiz 5 . +0 . − . . . α - 0.64 u,e 4.30 Note —We show the observed properties ( M BH , L peak , T peak , and FWHM) for a sample of 21 TDEs studied in this Letter. Thespectral type refers to van Velzen et al. (2020). In the 8th column, we list the date when their FWHM was observed since their peakluminosity. We calculate R SI assuming R ∗ = R (cid:12) , M ∗ = M (cid:12) , and β = 1 (see the main text in Section3.1).(References) a: van Velzen et al. (2011), b: Arcavi et al. (2014), c: Wevers et al. (2017), d: Gezari et al. (2012), e: Hinkle et al.(2020a), f: Chornock et al. (2014), g: Holoien et al. (2014), h: Holoien et al. (2016a), i: Leloudas et al. (2016), j: Holoien et al.(2016b), k: Blagorodnova et al. (2019), l: Hung et al. (2017), m: Blagorodnova et al. (2017), n: Nicholl et al. (2019), o: Holoienet al. (2019a), p: Neustadt et al. (2020), q: Holoien et al. (2020), r: Short et al. (2020), s: Holoien et al. (2019b), t: Hinkle et al.(2020b), u: Nicholl et al. (2020) where i is the inclination. A S and A J are given by Mer-ritt et al. (2010) as follows: A S = 6 πc GM BH R p (1 + e ) , and (3) A J = 4 πa BH c (cid:18) GM BH R p (1 + e ) (cid:19) / , (4)where c , G , and a BH are the light speed, the Newtonianconstant of gravitation, and the BH spin, respectively.For TDEs, the condition, β (cid:38)
1, needs to be satisfied.In this study, we assume β = 1. This assumption isappropriate in comparing the model with the observa-tions, since β = 1 means a large collision cross-section,i.e., a high event rate. In addition, a low β , i.e., a lowangular momentum, is likely preferred to produce lowenergy radiation such as the Optical/UV wavelengths (e.g., see Dai et al. 2015). We also assume that theradii and masses of the disrupted stars are R ∗ = R (cid:12) and M ∗ = M (cid:12) . This assumption would be acceptable,as the main-sequence stars like the sun are most likelyto be destroyed since their abundance. Under these as-sumptions, we compute R SI as shown in Table1.To test the assumption, R eq = R SI , we estimate v esc at R SI , using v esc = (cid:112) GM BH /R SI . We regard theFWHM as v wind , and show the comparison between v esc and v wind in Figure1. Figure1 shows that v esc is roughlyconsistent with v wind within a factor of 2. Discussingfurther the possible correlation is however difficult; howto accurately derive v wind from observational data in-volves a large uncertainty. In addition, the present sam-ple for this analysis is still limited. Indeed, the values of v wind of most of the TDE samples here fall into a lim- UNO & MAEDA
Figure 1.
Comparison between v wind and v esc . The sym-bols are different for different spectral types of TDEs (vanVelzen et al. 2020). The filled circles, diamonds, squares,and stars show TDE-H, TDE-Bowen, TDE-He, and un-specified spectral type, respectively. The blue, orange, andgreen dashed lines show v esc = 0 . v wind , v esc = v wind , and v esc = 2 v wind , respectively. The magenta dash-dot line showsthe mean value of v esc of the sample, excluding ASASSN-15lh, ASASSN-18jd, and ASASSN-14li (see the main text).The region enclosed by magenta shows the 1-sigma region( µ − σ ≤ v esc ≤ µ + σ ). ited range within a factor of 3 (except for ASASSN-15lh,ASASSN-18jd, and ASASSN-14li, in which the formertwo are outliers). Therefore, the present sample wouldnot allow such a detailed investigation of the correla-tion; the correlation coefficient between v esc and v wind is indeed smaller than 0.2, but this may simply be anoutcome of the currently limited samples. Alternatively,we may simply discuss the mean and standard deviationin the distribution of v esc (but excluding the above men-tioned three objects). The mean and standard deviationare µ = 1 . × cm / s and σ = 1 . × cm / s, re-spectively. This is roughly consistent with the aboveestimate within a factor of 2. In the future, we hope toanalyze the possible correlation further, once the suffi-ciently large sample covering a range of v wind becomesavailable and the relation between v wind and the ob-served line width is better clarified.In the two outliers; ASASSN-15lh and ASASSN-18jd, v wind is significantly lower than v esc . It suggests thatthese objects are beyond the applicability of this model.Indeed, the observations show that ASASSN-15lh maybe induced by an SMBH with mass above the upperlimit to produce TDEs ( ∼ M (cid:12) ). ASASSN-18jd isnot robustly identified as a TDE; we cannot dismiss thepossibility that ASASSN-18jd is an active galactic nu-cleus or an unknown type of transients. We may need to consider different scenarios or emission mechanisms forthese objects, including a possibility of a high BH spin(Mummery & Balbus 2020, see also Section4).In Figure1, different spectral types of TDEs (vanVelzen et al. 2020) are shown by different symbols. Noclear difference is seen in the distribution of v esc and v wind for different spectral types.3.2. Estimate of Physical Properties
In Uno & Maeda (2020), R eq was one of the outputparameters. However, in the present work, we treat R eq as an input parameter under the assumption R eq = R SI .We alternately introduce a new parameter, f , into theequations. The new parameter f is the ratio of the ki-netic energy ( ε kin ) to the thermal energy ( ε th ) per unitof volume at R SI . f is described as f = ε th /ε kin . In Uno& Maeda (2020), it is assumed that ε th = ε kin holds at R eq , but this time we incorporate the ratio as a newunknown parameter. We expect that the derived valueof f should be an order of unity, if the model is self-consistent.Uno & Maeda (2020) defines three typical physicalscales; the wind-launched radius ( R eq ), the photon-trapped radius ( R ad ), and the color radius ( R c ). In thisstudy, we replace R eq by R SI . At R SI , we assume thefollowing relation: aT ( R SI ) = f ρ ( R SI ) v , (5)where the density structure, ρ ( r ), is given as follows: ρ ( r ) = ˙ M πr v . (6)We assume that v wind is constant as a function of radius. R ad is defined by τ s ( R ad ) = c/v , where τ s is the opticaldepth for electron scattering. R c is defined by τ eff ( R c ) =1, where τ eff is the effective optical depth, consideringnot only electron scattering but also absorption pro-cesses. The formation of the photosphere depends on arelative relation between R c and R ad . The photosphericradius ( R ph ) is given as R ph = max( R ad , R c ). We alsodefine the luminosity as follows: L ( r ) = − πr ac κ s ρ ∂∂r T . (7) Figure 2.
Comparison between ε kin and ε th . The blue,orange, and green dashed lines show f = 0 . f = 0 .
3, and f = 1, respectively. Using above equations, we can estimate f and ˙ M .When R c < R ad , they are given as follows:˙ M ≈ M (cid:12) yr − (cid:18) L . × erg s − (cid:19) (cid:18) T ph . × K (cid:19) − (cid:18) v . × cm s − (cid:19) − , and(8) f ≈ . (cid:18) L . × erg s − (cid:19) (cid:18) T ph . × K (cid:19) (cid:18) v . × cm s − (cid:19) − (cid:18) R SI . × cm (cid:19) − , (9)where T ph is the photospheric temperature; T ph = T ( R ph ). On the other hand, if R c > R ad holds, theparameters are given as follows:˙ M ≈ M (cid:12) / yr (cid:18) L . × erg s − (cid:19) (cid:18) T ph . × K (cid:19) − (cid:18) v . × cm s − (cid:19) , and (10) f ≈ . (cid:18) L . × erg s − (cid:19) (cid:18) T ph . × K (cid:19) (cid:18) v . × cm s − (cid:19) − (cid:18) R SI . × cm (cid:19) − . (11)Figure2 shows the comparison between ε kin and ε th aswe have derived. This shows a roughly positive corre-lation, but there are two objects which are clearly out Figure 3. ˙ M peak estimated by the Wind-Driven model. of the trend; ASASSN-15lh and ASASSN-18jd. Theseare also the outliers in Figure1. This also suggests thatthey are beyond the applicability of the present model.Generally, the outflow may well be launched from a ra-dius where ε kin and ε th become comparable, i.e., f ∼ . R eq = R SI .Figure3 shows the estimated ˙ M peak . We find thatTDEs have strong outflows around the peak luminos-ity, typically with ˙ M peak (cid:38) M (cid:12) / yr. Assuming thatthe disrupted star is M ∗ ∼ M (cid:12) , they cannot releasethe mass exceeding ∼ M (cid:12) . Extremely large mass-lossrates (e.g., ASASSN-15lh) are not feasible. This is an-other important constraint to identify the limits in theapplication of the model (see also Section4).Figure4 shows the comparison between R SI and R ph .Typically, R ph is formed above R SI , between ∼ R SI and ∼ R SI . This result supports the picture that the originof the Optical/UV radiation and some spectral lines isnot the direct radiation from the accretion disk, but isthe optically thick winds (Strubbe & Quataert 2009). DISCUSSIONWe have generally derived high mass-loss rates for asample of TDEs as an outcome of the wind-driven modelfor TDEs. Indeed, TDEs are expected to have high ac-cretion rates at peak. In Stone et al. (2013), the peakfall-back rate, described as ˙ M fb , is roughly given as fol-lows: ˙ M fb ˙ M Edd ≈ (cid:16) η . (cid:17) (cid:18) M BH M (cid:12) (cid:19) − (cid:18) M ∗ M (cid:12) (cid:19) , (12) UNO & MAEDA
Figure 4.
Comparison between R SI and R ph . The blue,orange, and green dashed lines show R ph = R SI , R ph = 5 R SI ,and R ph = 10 R SI , respectively. where ˙ M Edd = 4 πGM BH / ( κ s ηc ), or,˙ M fb ≈ . M (cid:12) / yr (cid:18) M BH M (cid:12) (cid:19) − (cid:18) M ∗ M (cid:12) (cid:19) . (13)Since this is the super-Eddington accretion, it is likelythat most of the accretion may indeed be ejected (i.e.,outflows) and therefore the accretion rate here may rep-resent the mass-loss rate. Therefore, the high mass-lossrate we derived is in line with this expectation from theTDE physics. While it is true that the theoreticallyexpected rate here is a few times smaller than our esti-mate, we note that both estimates involve uncertaintieswhich would not allow detailed comparison.Since TDEs are highly time-dependent transients, fur-ther discussion requires the effect of time evolution (see,e.g., Uno & Maeda 2020). One way to take this intoaccount is to check the total mass ejected by the out-flow. The timescale in the TDE light curve may differfrom one object to another; some TDEs rapidly fade in ∼
20 days, while others stay bright for about a year(van Velzen et al. 2020). With this caveat in mind, weadopt one month as the typical time scale, since mostTDEs typically fade substantially in this timescale. Bymultiplying this time scale to the peak luminosities, thetotal ejected masses are expected to be a few M (cid:12) (seealso Figure3).Since we assume M ∗ = M (cid:12) , the system cannot ejectmass larger than M (cid:12) . However, R SI , which affects themass-loss rates, depends on M ∗ as follows: R SI ∝ R t ∝ R ∗ M − / ∗ . For larger disrupted stars, we thus expectthat R SI roughly stays the same because both R ∗ and M ∗ become larger. Therefore, considering more massivestars would not alter the present results substantially, and then the mass ejection of a few M (cid:12) can be easilyaccommodated.Indeed, Matsumoto & Piran (2020) has recently esti-mated the mass ejection of some TDEs using the modelsimilar to the present work. They have applied themodel to a few TDEs, partially taking time evolutioninto account. They thereby derived ∼ M (cid:12) for the to-tal ejected mass. This is roughly consistent with ourestimate. With this value, they indeed have arguedagainst the Wind-Driven model for TDEs. However,given that both models still lack detailed treatment oftime evolution and also use some simplified assumption(e.g., treatment of opacity), and that we find a few otherindependent supports for the wind-driven model, we dotake this rough agreement in the ejected mass betweenour estimate and the TDE expectation as another sup-port for the applicability of the Wind-Driven model toTDEs.As mentioned above, further detailed analysis will re-quire the full time-dependent treatment. Before beingable to sophisticate the model at this level, we indeedneed to overcome several limitations in our current un-derstanding of the nature of TDEs in their observationaldata. For example, our current understanding of theevolution of v wind is not sufficient, which requires deeperunderstanding of the line formation processes (see, e.g.,Uno & Maeda 2020). Also, a sample of TDEs with well-sampled time evolution data is still limited. Therefore,in this work, we focus on the properties of TDEs at theirpeaks, aiming at understanding the general/statisticalproperties of TDEs using as large an observed sample asis currently available. In the future, we hope to addressthe time evolution effects and apply such a model to asample of TDEs with the time-evolution data available,but this is beyond a scope of the present work.Mummery & Balbus (2020) presented that ASASSN-15lh is a peculiar TDE which has a high BH spin closeto a BH ≈ .
99. Adopting the high BH spin, we have ap-plied our model to ASASSN-15lh. The main change isseen in the value of f , but this is only about 2%. Takinginto account the BH spin thus does not have a signifi-cant effect on the derived properties based on the windmodel, e.g., mass-loss rate. ASASSN-15lh thus remainsan outlier, which seems to be beyond the applicabilityof our present model.Using the Wind-Driven model and the results, we canobtain insights for TDEs or other astronomical tran-sients driven by explosive mass ejection. Namely, wecan constrain some physical properties for transientswith insufficient observational data. For example, wecan roughly estimate M BH or v wind .As one such application, we estimate M BH for apeculiar transient, AT2018cow (Prentice et al. 2018).AT2018cow is a luminous blue transient ( L peak ≈ × erg s − , T peak ≈ v ≈ . c ) discoveredby ATLAS on MJD 58285 (Perley et al. 2019). In Per-ley et al. (2019), they argued that AT2018cow is a TDEinduced by an intermediate-mass black hole (IMBH).However, AT2018cow occurred far from the galactic cen-ter, which makes it difficult to estimate M BH from the M bulge − M BH relation. They estimated M BH using Mos-fit TDE model (Guillochon et al. 2018). In our model,we can constrain M BH independently from Mosfit. As-suming R ∗ = R (cid:12) , M ∗ = M (cid:12) , β = 1, and f = 0 .
5, weestimate M BH as ∼ . × M (cid:12) . This is consistentwith the estimate by Perley et al. (2019). Our modelthus supports that AT2018cow may be a TDE inducedby an IMBH. In addition, we note that R SI estimatedby Perley et al. (2019) is a factor of 10 smaller thanthe observed photosphere. This is also consistent withour result (see Figure4), supporting by the Wind-Drivenmodel. CONCLUSIONSUsing the Wind-Driven model presented by Uno &Maeda (2020), we have aimed to constrain the origin ofthe optical/UV radiation in TDEs. The comparison be-tween the escape velocities and the wind velocities sup-ports that the wind is launched from the self-interactionradius. Generally, the wind is expected to be launchedfrom a position where the ratio of kinetic to thermalenergy per unit volume is roughly equal (i.e., equiparti-tion). We also estimate the ratio at the self-interaction radius through the Wind-Driven model, and it turns outto be an order of unity. This result supports the assump-tion that the stream collision induces the wind.We find TDEs have strong outflow around the peak.The mass-loss rates are typically over 10 M (cid:12) yr − . Wealso show that the photospheric radii are 1-10 timeslarger than the self-interaction radii. This result sup-ports the picture that the Optical/UV radiation is emit-ted not from the accretion disk directly, but from anoptically thick wind.Applying the framework to TDEs or other astronomi-cal transients driven by explosive mass ejections, we canobtain constraints on the physical properties that cannot be obtained from observations. We apply the frame-work to a peculiar transient, AT2018cow. The modelsuggests that AT2018cow is likely a TDE induced by anintermediate-mass black hole ( ∼ M (cid:12) ).The Wind-Driven model still has significant room forimprovement. The present model assumes a steady-state, and in this Letter we estimate physical quantitiesonly at the peak. To obtain detailed constraints, it isnecessary to create a non-steady-state model that takesinto account the time evolution and radial dependenceof the velocity. We postpone such work to the future.ACKNOWLEDGMENTSWe thank Girogs Leloudas and Tatsuya Matsumotofor helpful comments and discussions. K.M. acknowl-edges support provided by Japan Society for the Pro-motion of Science (JSPS) through KAKENHI grant(18H05223, 20H00174, and 20H04737).REFERENCES Arcavi, I., Gal-Yam, A., Sullivan, M., et al. 2014, ApJ, 793,38, doi: 10.1088/0004-637X/793/1/38Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019,PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbeBlagorodnova, N., Gezari, S., Hung, T., et al. 2017, ApJ,844, 46, doi: 10.3847/1538-4357/aa7579Blagorodnova, N., Cenko, S. B., Kulkarni, S. R., et al. 2019,ApJ, 873, 92, doi: 10.3847/1538-4357/ab04b0Chornock, R., Berger, E., Gezari, S., et al. 2014, ApJ, 780,44, doi: 10.1088/0004-637X/780/1/44Dai, L., McKinney, J. C., & Miller, M. C. 2015, ApJL, 812,L39, doi: 10.1088/2041-8205/812/2/L39Donley, J. L., Brandt, W. N., Eracleous, M., & Boller, T.2002, AJ, 124, 1308, doi: 10.1086/342280Gezari, S., Chornock, R., Rest, A., et al. 2012, Nature, 485,217, doi: 10.1038/nature10990 Guillochon, J., Nicholl, M., Villar, V. A., et al. 2018, ApJS,236, 6, doi: 10.3847/1538-4365/aab761Hinkle, J. T., Holoien, T. W. S., Shappee, B. J., et al.2020a, ApJL, 894, L10, doi: 10.3847/2041-8213/ab89a2Hinkle, J. T., Holoien, T. W. S., Auchettl, K., et al. 2020b,MNRAS, doi: 10.1093/mnras/staa3170Holoien, T. W. S., Prieto, J. L., Bersier, D., et al. 2014,MNRAS, 445, 3263, doi: 10.1093/mnras/stu1922Holoien, T. W. S., Kochanek, C. S., Prieto, J. L., et al.2016a, MNRAS, 455, 2918, doi: 10.1093/mnras/stv2486—. 2016b, MNRAS, 463, 3813, doi: 10.1093/mnras/stw2272Holoien, T. W. S., Huber, M. E., Shappee, B. J., et al.2019a, ApJ, 880, 120, doi: 10.3847/1538-4357/ab2ae1Holoien, T. W. S., Vallely, P. J., Auchettl, K., et al. 2019b,ApJ, 883, 111, doi: 10.3847/1538-4357/ab3c66