Cooling Envelope Model for Tidal Disruption Events

We present a toy model for the thermal optical/UV/X-ray emission from tidal disruption events (TDE). Motivated by recent hydrodynamical simulations, we assume the debris streams promptly and rapidly circularize (on the orbital period of the most tightly bound debris), generating a hot quasi-spherical pressure-supported envelope of radius R_v ~ 1e14 cm (photosphere radius ~1e15 cm) surrounding the supermassive black hole (SMBH). As the envelope cools radiatively, it undergoes Kelvin-Helmholtz contraction R_v ~ t^(-1), its temperature rising T_eff ~ t^(1/2) while its total luminosity remains roughly constant; the optical luminosity decays as nu L_nu ~ R_v^2 T_eff ~ t^(-3/2). Despite this similarity to the mass fall-back rate Mdot_fb ~ t^(-5/3), envelope heating from fall-back accretion is sub-dominant compared to the envelope cooling luminosity except near optical peak (where they are comparable). Envelope contraction can be delayed by energy injection from accretion from the inner envelope onto the SMBH in a regulated manner, leading to a late-time flattening of the optical/X-ray light curves, similar to those observed in some TDEs. Eventually, as the envelope contracts to near the circularization radius, the SMBH accretion rate rises to its maximum, in tandem with the decreasing optical luminosity. This cooling-induced (rather than circularization-induced) delay of up to several hundred days, may account for the delayed onset of thermal X-rays, late-time radio flares, and high-energy neutrino generation, observed in some TDEs. We compare the model predictions to recent TDE light curve correlation studies, finding agreement as well as points of tension.


INTRODUCTION
A tidal disruption event (TDE) occurs when a star orbiting a supermassive black hole (SMBH) on what is typically a parabolic orbit, comes sufficiently close to the SMBH to be strongly compressed and torn apart by tidal forces (Hills 1975;Luminet & Carter 1986;Rees 1988;Evans & Kochanek 1989;Stone et al. 2013;Guillochon & Ramirez-Ruiz 2013;Coughlin & Nixon 2022).
These issues bear crucially on the energy source powering TDE flares. If circularization is significantly delayed (e.g., by many orbits of the most tightly bound debris), then powering the optical luminosities of TDEs requires tapping directly into the limited amount of energy dissipated by stream-stream collisions (e.g., Piran et al. 2015). On the other hand, if even a modest fraction of the bound debris reaches small scales around the SMBH, the resulting accretion power can be sufficient to power the observed UV/optical emission, e.g. via the reprocessing of disk-emitted X-rays by radially-extended material (e.g., Metzger & Stone 2016;Roth et al. 2016;Dai et al. 2018), such as bound tidal debris (e.g., Guillochon & Ramirez-Ruiz 2013), wide-angle unbound outflows generated during the circularization process (e.g., Metzger & Stone 2016;, or accretion disk winds (e.g., Strubbe & Quataert 2009;Miller 2015;Dai et al. 2018;Wevers et al. 2019;Uno & Maeda 2020). The geometric beaming of thermal X-rays from the inner accretion flow along the low-density polar regions of the reprocessing structure offers a unification scheme for the optical and X-ray properties of TDEs based on the observer viewing angle (e.g, Metzger & Stone 2016;Dai et al. 2018). However, while some TDEs exhibit clear evidence for outflows (e.g., Miller et al. 2015;Alexander et al. 2017;Kara et al. 2018;Blagorodnova et al. 2019;Nicholl et al. 2020;Hung et al. 2021), the mass-loss rates required to sustain the large pho-tosphere radii in outflow reprocessing scenarios may in some events be unphysically large (e.g., Matsumoto & Piran 2021). Some observations hint that the peak SMBH accretion rate can be significantly delayed with respect to the optical peak. While a handful of powerful jetted TDEs exhibit bright non-thermal X-ray and radio emission (e.g., Bloom et al. 2011;Burrows et al. 2011;Levan et al. 2011;Zauderer et al. 2011), most TDEs are radio dim (e.g., Bower et al. 2013;van Velzen et al. 2013;Alexander et al. 2020) excluding powerful off-axis jets (e.g., Generozov et al. 2017). Nevertheless, several TDEs exhibit late-time radio flares, indicating mildly relativistic material ejected from the viscinity of the SMBH, but delayed from the optical peak by several months to years (e.g., Horesh et al. 2021a,b;Perlman et al. 2022;Sfaradi et al. 2022;Cendes et al. 2022). A potentially related occurrence is the coincident detection of high-energy neutrinos from three optical TDEs by IceCube (Stein et al. 2021;van Velzen et al. 2021b;Reusch et al. 2022), each of which also arrived several months after the optical peak. State transitions in the accretion flow offer one potential explanation for the delayed onset of jetted accretion activity (e.g., Tchekhovskoy et al. 2014). Delayed circularization leading to delayed disk formation (as invoked to explain similarly delayed rises in the X-ray emission; e.g., Gezari et al. 2017), offers another.
Even rapid and efficient circularization may not however create a compact disk, at least initially. Loeb & Ulmer (1997) assume the TDE debris forms a spherical radiation-dominated hydrostatic envelope encasing the SMBH. Coughlin & Begelman (2014) emphasize that the low angular momenta of TDE debris relative to their binding energy (i.e., "circularization" radii "virial" radii) endow the circularized structure with properties quite unlike thin Keplerian disks, due to their much larger radial extent and propensity to launch outflows/jets along an extremely narrow polar funnel. While this structure may briefly manifest as a higheccentricity disk (e.g., Cao et al. 2018;Liu et al. 2021;Wevers et al. 2022), dissipation within this geometrically thick (e.g., Steinberg & Stone 2022) confluence of differentially-precessing annuli will likely be strong (e.g., Bonnerot et al. 2017;Ryu et al. 2021). And once thermal pressure provides the support against gravity, the bound debris may arguably be modeled most simply as a quasi-spherical "envelope" (Loeb & Ulmer 1997). A quasi-spherical emission surface is supported by spectropolarimetry observations of some TDEs (e.g., Patra et al. 2022).
Recently, Steinberg & Stone (2022) presented threedimensional radiation hydrodynamical simulations of the tidal disruption of a solar-mass star by a 10 6 M black hole, for the most common (but most computationally challenging) case of a β = 1 orbit penetration factor. Unlike the findings or assumptions of most previous works, they find rapid circularization of the debris streams within a short time ( 70 days), comparable to the fall-back time of the most tightly bound debris. A possible explanation for their result is stronger tidal compression and heating of the streams as they pass through pericenter compared to that found in previous work (e.g., Guillochon & Ramirez-Ruiz 2013;Bonnerot & Lu 2022) due to the inclusion of recombination energy in the assumed equation of state. Steinberg & Stone (2022) further show that radiative diffusion from the extended circularized envelope generates a rising optical light curve with an effective temperature consistent with those of optically-selected TDE flares. Rapid circularization was also found in general-relativistic hydrodynamical simulations of a similar 1:10 6 mass-ratio system by Andalman et al. (2022), in the case of a higher β = 7 encounter for which rapid stream-stream collisions driven by general relativistic precession play a decisive role in circularization.
Motivated in part by these recent findings of rapid circularization even across the most commonly sampled regions of TDE parameter space, here we present a model for the long-term evolution and emission from TDE envelopes following their formation. Though following in spirit Loeb & Ulmer (1997), we make different assumptions and track the time-evolution of the envelope size in light of various cooling and heating processes, including SMBH feedback. The proposed model of TDE emission as being driven by the thermal evolution of a spherical envelope, while clearly oversimplified in many respects, may nevertheless provide a new view on open questions, such as the timescale and shape of the light curve decay, and the origin of the observed delay between the optical peak and those physical processes (soft X-ray emission, radio flares, fast outflows, neutrino production) instead driven by the innermost SMBH accretion flow. This paper is organized as followed. In Sec. 2 we present the model for the envelope evolution. In Sec. 3 we present our results, first focusing on a single fiducial model and dissecting the impact of different physical processes (Sec. 3.1) and then comparing the light curve predictions across a range of star and SMBH properties to TDE observations (Sec. 3.2). In Sec. 4 we summarize our findings, expand on some implications, and comment on directions for future work. Schematic illustration of the model. Rapid circularization of the TDE debris forms a quasi-circular pressuresupported envelope around the SMBH of characteristic radius Rv and photosphere radius R ph ∼ 10Rv which powers the early optical emission. The envelope luminosity L rad primarily derives from the gravitational energy released from its cooling-driven contraction at close to the Eddington limit, though fall-back accretion (which deposits its energy at a radius Racc Rv, where the stream dissolves inside the envelope after passing through pericenter) may contribute significantly at early times. As the envelope cools and contracts, roughly as Rv ∝ R ph ∝ t −1 initially, the effective temperature rises T eff ∝ t 1/2 while the optical luminosity νLν ∝ R 2 ph T eff ∝ t −3/2 drops. The SMBH accretion rate rises in tandem, as controlled by the envelope density and viscous time near the circularization radius, potentially powering thermal X-ray and/or jetted activity along the initially narrow polar funnel. The inner accretion flow also acts as a source of energy to the envelope, which can delay the envelope's contraction in a regulated manner, flattening the late-time optical and X-ray light curve decay.

MODEL
We model the long-term evolution of a quasi-spherical TDE envelope under the assumption of prompt circularization (see Fig. 1 for a schematic illustration). We first describe the initial conditions imparted by the tidal disruption process (Sec. 2.1) and then the details of the envelope evolution (Sec. 2.2).

Tidal Disruption and Envelope Formation
A star of mass M = m M and radius R is tidally disrupted if the pericenter radius of its orbit, R p , becomes less than the tidal radius (e.g., Hills 1975) and we assume here and in analytic estimates to follow a massradius relationship R ≈ m 4/5 R appropriate to lower main sequence stars (Kippenhahn & Weigert 1990). The orbital penetration factor is defined as β ≡ R t /R p > 1. Disruption binds roughly half the star to the SMBH by a specific energy |E t | = kGM • R /R 2 t corresponding roughly to the work done by tidal forces over a distance ∼ R t . The most tightly bound matter falls back to the SMBH on the characteristic fall-back timescale set by the period of an orbit with energy E t , where the factor k has a weak dependence on the penetration factor β Guillochon & Ramirez-Ruiz 2013) and hereafter we shall take k 0.8 corresponding to a β = 1 disruption for a γ = 5/3 polytropic star. The resulting rate of mass fall-back at time t t fb is given by 1 (e.g., Phinney 1989;Evans & Kochanek 1989), where M acc 0.4M is the fraction of the star accreted at t > t fb (the remainder ∼ 0.1M being accreted during the rise phase at t < t fb ; this fraction in general depends on the outer density structure of the star).
Motivated by recent simulations (e.g., Steinberg & Stone 2022;Andalman et al. 2022) we assume rapid circularization of the initial tidal debris into a quasispherical envelope, on a timescale t circ ∼ t fb . We further assume the envelope possesses a power-law radial density profile with a characteristic radius R v and a sharp outer edge: where ξ 3. In what follows we take ξ = 1, i.e. ρ ∝ r −1 for r < R v ; 2 however, the qualitative features of the model should be preserved for other choices 1 ξ 3 (the "ZEBRA" models of Coughlin & Begelman 2014 predict 1/2 < ξ < 3 for an adiabatic index γ = 4/3). Neglecting wind mass-loss or SMBH accretion, the envelope mass grows with time t > t fb as We estimate the characteristic initial radius of the envelope R v ("virial radius") by equating the energy of the bound stellar debris, |E t | = kGM • M R /R 2 t , to half of its gravitational binding energy where in the second line and hereafter we take k = 0.8. The envelope is notably much larger than the circularization radius, R circ = 2R t , Rotational support is thus sub-dominant initially compared to thermal pressure, rendering the envelope quasispherical. The envelope energy we assume is smaller (equivalently, R v is larger) than Loeb & Ulmer (1997), who in adopting a steeper density profile ρ ∝ r −3 effectively take R v ∼ R circ (their Eq. 14), a more tightly bound envelope than imparted by the TDE. The characteristic density of the envelope at r = R v is given by From the virial theorem, the envelope's internal energy E int = 3 P rad 4πr 2 dr ≡ 4πR 3 vPrad equals |E t |, leading to an estimate of the interior temperature: Thus, as the envelope contracts and R v decreases, both ρ c and T c will rise. Radiation pressure P rad = aT 4 /3 dominates over gas pressure P gas = ρkT /µm p at all times: For ρ ∝ r −ξ , hydrostatic balance dP rad /dr ∝ −GM • ρ/r 2 implies T ∝ r −(1+ξ)/4 and hence our assumed density profile (ξ = 1; Eq. 5) implies T ∝ r −1/2 . The envelope entropy profile s ∝ T 3 /ρ ∝ r (ξ−3)/4 is therefore unstable to convection (ds/dr < 0). While efficient convection may try to drive s const (ρ ∝ r −3 ), for ξ = 3 such a configuration would be more tightly gravitationally-bound than permitted by its initial energy. The envelope structure may therefore try to evolve towards ξ = 3 as it cools and contracts, but for simplicity we neglect this possibility and assume ξ = 1 at all times.

Envelope Evolution
After forming at time t circ ≈ t fb , the mass and energy, (and hence characteristic radius R v ) of the envelope evolve at later times according to respectively. HereṀ • /Ė • accounts for the effects of accretion onto the SMBH, a specific treatment of which is given in Secs. 2.2.1. The termsṀ w /Ė w allow for massand energy-loss in a wind from the envelope given an appropriate prescription relating them to other properties of the system. Though included for completeness, we neglect outflows hereafter (i.e. we takeṀ w =Ė w = 0); we speculate on when this assumption may be violated in Sec. 4. The luminosity radiated by the envelope can be written where T eff and R ph are the effective temperature and photosphere radius, respectively. 3 The latter is defined by For our assumed density profile at r > R v (Eq. 5), where the characteristic optical depth, and we hereafter take κ = κ es 0.35 cm 2 g −1 (scattering generally dominates other opacity sources given the envelope's high entropy). The envelope is in hydrostatic equilibrium and supported by radiation pressure, so its inner layers must radiate close to the SMBH Eddington luminosity, L Edd ≈ 1.4 × 10 44 erg s −1 M •,6 (Loeb & Ulmer 1997), which therefore enters as a loss-term in Eq. (14). The second term in Eq. (15), accounts for heating of the outer envelope layers by the fall-back stream (specific kinetic energy v 2 ff /2 ≈ GM • /r at radius r), where r ∼ R acc is the characteristic ra-dius at which the the stream material decelerates and becomes incorporated into the envelope. 4 Steinberg & Stone (2022) show that the densest portion of the fall-back stream−that which remains gravitationally self-bound−thickens during compression at pericenter and then dissolves into the envelope on a radial scale ∼ R v,0 (as a part of the same process giving rise to efficient circularization in the first place). Motivated thus, we assume the accretion radius scales with the apocenter distance of the fall-back stream, where the constant ζ accounts for various uncertainties (e.g., in the radial distribution of the envelope heating and the fraction of the stream able to penetrate the envelope). We adopt a fiducial value ζ = 2, because only the fraction (e.g., ∼ 1/2, depending on β; Steinberg et al. 2019) of the stream mass confined in the transverse directions by self-gravity (e.g., Coughlin et al. 2016;Bonnerot et al. 2022) is likely to survive penetration through the envelope to radii ∼ R v .

Black Hole Accretion
Insofar as the envelope retains the same specific angular momentum from the time of disruption, rotation will become important on small radial scale R circ = 2R p = 2R t /β R v , thus limiting accretion onto the SMBH based on the viscous time at this radius. We estimate the accretion rate through the rotationally-supported inner disk region (second term in Eq. 13) aṡ where Σ ρ(R circ )R circ M e /(10πR 2 v ), ν = αc 2 s /Ω = r 2 Ω(H/r) 2 , α is the Shakura & Sunyaev (1973) viscosity parameter, H/r = c s /(rΩ) is the vertical disk aspect ratio, c s = (P/ρ) 1/2 the sound speed and Ω = (GM • /r 3 ) 1/2 the Keplerian orbital frequency. An "accretion" timescale for the envelope can thus be defined, 4 By including L fb in L rad (Eq. 15) but not Eq. (14), we have implicitly assumed that L fb is "instantaneously" radiated by the envelope. This is generally a good assumption because the timescale over which L fb evolves ∼ t is typically long compared to the Kelvin-Helmholtz time over which the envelope can radiate any deposited energy (see Eq. 29).
where α −2 ≡ α/(10 −2 ) and we are motivated to consider a geometrically thick disk H/r 0.3, consistent with the near-Eddington accretion rates of interest (Abramowicz et al. 1988;Shen & Matzner 2014). Though initially much longer than other timescales in the problem, t acc will shorten ∝ R 2 v as the envelope cools and contracts. Finally, the third term in Eq. (14) accounts for energy released by accretion onto the SMBH which is transferred outwards to the envelope through radiation or convection. We assume the feedback luminosity scales with the accretion rate at R circ , where the dimensonless efficiency η 0.1 encapsulates a number of uncertain factors related to efficiency of reprocessing by the envelope of radiation/outflows/jets from the inner disk, including the potential for diskwind mass-loss from the (potentially super-Eddington) accretion flow between R circ and R isco (e.g., Blandford & Begelman 1999). A canonical minimum value, set by the conditionĖ • = GM •Ṁ• /R circ , is

Summary of the Model
As summarized in Table 1, a given model is defined primarily by the masses of the star M and SMBH M • . Secondary variables, whose values we shall typically fix within a fiducial value or range, include: orbital penetration factor β = 1; density radial profile power-law ξ = 1; fall-back heating efficiency ζ = 2; viscosity α = 0.01, aspect ratio H/r = 0.3, and feedback efficiency η ∼ η min ∼ 10 −2 − 10 −1 of the inner accretion disk. Starting the calculation at time t = t fb , when the envelope mass M e (t 0 ) = 0.1M (Eq. 6) and radius R v = R v,0 (Eq. 7), we solve Eqs. (13), (14) for the evolution of M e , radius R v , photosphere radius R ph , and effective temperature T eff of the envelope, as well as the SMBH accretion rateṀ • . We evolve the system until the envelope mass reaches zero, though the assumptions of the model may break down before this, once the radius contracts to R v R circ = 2R t /β violating the assumption of negligible rotational support. Given the bolometric luminosity L rad and T eff we calculate the luminosity νL ν at a given optical waveband ν assuming blackbody emission (e.g., we neglect the distinction between the scattering photosphere and frequency-dependent thermalization surface, which can lead to an underestimate of the luminosity on the Rayleigh-Jeans tail; e.g., ).

RESULTS
We begin in Sec. 3.1 by showing results for a fiducial model with M = M , M • = 2 × 10 6 M , β = 1. Rather than including all of the physics into the model at once, we begin (Sec. 3.1.1) by artificially neglecting the effects arising from accretion onto the SMBH (i.e., we assumė M • =Ė • = 0) and walking through some analytic arguments which reproduce the results. Then we move on to models which include SMBH accretion, first just a mass-loss term for the envelope (Sec. 3.1.2) and then finally including also SMBH energy feedback (Sec. 3.1.3). Finally, Sec. 3.2 presents the full-model optical/X-ray light curves for a range of star and SMBH properties.

Pure Cooling (Kelvin-Helmholtz Contraction)
Solid lines in Fig. 2 show results for the envelope evolution, neglecting accretion or feedback onto the SMBH. The envelope radius R v and photosphere radius R ph begin large, but gradually decay with time, with R v reaching R circ by around day 130 measured with respect to the envelope assembly (time t circ ∼ t fb after the disruption). Likewise, while the bolometric luminosity is at or slightly above the Eddington luminosity of the SMBH at all times, the optical decays from its initial value νL ν 10 43 erg s −1 roughly ∝ t −3/2 , as the effective temperature rises. These results can largely be understood analytically.
Neglecting fall-back heating, Eq. (14) becomes, Further approximating the envelope mass M e as a constant (in reality, M e grows gradually from 0.1M to 0.5M ), we obtain The radius contracts as, is the "Kelvin-Helmholtz" time and t KH,0 = t KH (R v = R v,0 ) defines its initial value. 5 Making use of Eq.
(2), 5 The thermal timescale t KH also equals the photon diffusion time through the envelope, t diff ∼ Λ(Rv/c), where Λ is the characteristic optical depth (Eq. 18). Me/M , and photosphere radius R ph , while the bottom panel shows the envelope luminosity L rad , optical luminosity νLν at frequency ν = 6 × 10 14 Hz (g-band), effective temperature T eff , and LX ≡ 0.01Ṁ•c 2 , taken as a proxy for the X-ray luminosity from the inner disk (potentially observable only through a narrow polar region). The envelope evolution concludes roughly once Rv decreases to Rcirc (horizontal dashed line), as occurs roughly at the times t disk (Eq. 33) and t acc (Eq. 34), respectively, in the two models. A dashed black line illustrates ∝ t −3/2 decay.
we see that The fact that t KH /t 1 at times t t fb ∼ t circ implies that (1) thermal equilibrium can be established on the timescale the envelope is being assembled and will remain so at later times; (2) if the assembly pro-cess itself is not rapid (taking place over a timescale 0.1 − 1t fb , depending on the SMBH mass), then the light curve properties near peak light will depend on the assembly history and hence may not be captured by our model, which assumes instantaneous assembly (we return to this point in Sec. 3.2). Fig. 2 shows that the envelope luminosity (Eq. 15) roughly obeys L rad L Edd with the fall-back luminosity L fb (Eq. 19) boosting this value only moderately at early times. Indeed, from Eqs. (3), (20), (27) we obtain, a result which is notably independent of m and M • . Fallback accretion thus contributes at an order-unity level to the envelope luminosity at early times t ∼ t fb , but becomes comparatively less important with time relative to the passive envelope cooling.
Approximating L rad L Edd and using Eqs. (15), (17) The predicted gradual rise in T eff ∝ t 1/2 is consistent with that shown in Fig. 2 and similar to that of observed TDE UV/optical flares (e.g., van Velzen et al. 2021a;their Fig. 5).
Envelope contraction as we have modeled it will continue until rotational support becomes important, as occurs once R v decreases to R circ = 2R t . Again neglecting accretion onto the SMBH, this timescale for the envelope to transform into a disk, can be estimated using Eq. (27) (in the t t KH,0 limit): i.e. typically several months to a year, in agreement with where R v crosses R circ in Fig. 2.

Mass-Loss from SMBH Accretion
A dotted line Fig. 2 shows an otherwise identical model to that presented in the previous, but which now includes envelope mass-loss due to SMBH accretion (Eq. 21, assuming α = 10 −2 and H/r = 0.3), yet still neglects any feedback heating from the accretion. At early times the solution is similar to that neglecting accretion, until around day 100 when the envelope mass reaches a maximum and begins to decrease. This in turn drives R v and R ph to decrease, and thus T eff to increase and νL ν to drop, at a faster rate than they would otherwise without accretion.
We can estimate the time required for the envelope to be fully accreted by setting t = t acc (Eq. 22) with in rough agreement with where M e begins to fall rapidly in Fig. 2. Depending on the value of α(H/r) 2 , t acc can be larger or smaller than t disk , the maximum disk formation time absent accretion (Eq. 33). Accretion onto the central SMBH can in principle power X-ray emission, which may begin to escape along what may be a narrow accretion funnel (e.g., Kara et al. 2018;Dai et al. 2018) to a greater and greater fraction of external observers as the envelope becomes more disklike (R v → R circ ). A brown dotted line in Fig. 2 shows an estimate of this proxy X-ray luminosity, where the prefactor 10 −2 is an estimate of the radiative efficiency of super-Eddington accretion disks (e.g., Sadowski & Narayan 2016). Although the normalization of the X-ray power is clearly uncertain, and the observed luminosity be highly inclination-dependent (e.g., Dai et al. 2018), the key feature of note is the delayed rise of the X-ray light curve relative to the optical peak. Though such delayed X-ray rises are observed in some TDEs, they have frequently been attributed to inefficient or delayed circularization (e.g., Shiokawa et al. 2015;Gezari et al. 2017). The physics here is instead delayed cooling and envelope contraction, which leads to accelerating growth in the SMBH accretion ratė M • ∝ t −1 acc ∝ R −2 v (Eq. 22).

Feedback from SMBH Heating
Finally, in Fig. 3 we show the effects of adding SMBH accretion heating to the envelope evolution (Eq. 23), by comparing a model with a low feedback efficiency η = 10 −3 (dotted line) to one with higher efficiency η = 1.5 × 10 −2 (solid line). The η = 10 −3 model follows a similar evolution to models excluding feedback altogether (Fig. 2). However, the η = 1.5 × 10 −2 model differs markedly, exhibiting a much more gradual decline in the rate of envelope contraction and optical luminosity. In effect, the energy provided by SMBH accretion keeps the envelope "puffed up" for longer, which in turn slows the SMBH accretion rate.
This regulated state, in which SMBH feedback reaches a balance with the radiated luminosity L rad L Edd , can be expressed as a condition on the SMBH accretion rate (see also Loeb & Ulmer 1997) Equating this with Eq. (21), the corresponding envelope radius in the SMBH-regulated state is given by The timescale for the envelope to be completely accreted at the regulated rate (Eq. 36) is thus given by, (38) From when R v decreases from its initial value to R • v , until the time the envelope is accreted t ∼ t • acc , the envelope radius and optical luminosity will exhibit a flat plateau-like time-evolution (suggestive of the late-time behavior of some TDE light curves; e.g., Leloudas et al. 2016;van Velzen et al. 2019;Wevers et al. 2019).
The regulated plateau state is only be achieved if t • acc > t acc , as occurs for sufficiently high accretion feed- Figure 3. Models calculated for the same parameters as in Fig. 3, but now including the effects of SMBH feedback on the envelope structure for two different values of η = 1.5×10 −2 (solid lines) and η = 10 −3 (dotted lines). Powerful feedback (large η) acts to slow the rate of envelope cooling and accretion, flattening the late-time νLν optical light curve decay. A dashed black line shows ∝ t −3/2 decay, as roughly expected absent efficient feedback. back efficiency, consistent with the large difference in the light curve duration between the η = 1.5 × 10 −2 and η = 10 −3 models in Fig. 3. For fiducial values of α and H/r, η crit is comparable to η min (Eq. 24); this suggests that events both with and without a self-regulated plateau phase could occur amongst the TDE population depending on the precise system parameters. Figure 4 shows the optical νL ν and proxy X-ray L X = 10 −2Ṁ

Dependence on SMBH/Star Properties
• c 2 light curves for a series of models which adopt fiducial parameters but varying the star and SMBH mass as marked and fixing η = η min (M , M • ) (Eq. 24). The optical luminosity is higher for more massive stars or SMBHs, consistent with Eq. (32). TDEs of lower-mass stars and/or by higher-mass SMBH also tend to produce faster decaying optical light curves (and correspondingly faster rising proxy X-ray light curves), as expected because of their shorter envelope cool times (Eq. 28). Since η min η crit for the assumed values of {β, α, H/r}, the total light curve durations (400, 150, 250 d, respectively) Velzen et al. (2021a, hereafter V21) analyze the optical light curve properties of a sample of 17 TDEs detected by the Zwicky Transient Facility, exploring internal correlations between the light curve properties (e.g., blackbody luminosity L rad , blackbody[our photosphere] radius R ph , effective temperature T eff , rise time t rise , decay time t decay ) and with the host galaxy stellar mass M gal (a rough proxy for the SMBH mass given M gal -M • and related correlations; e.g. Magorrian et al. 1998). We briefly describe comment on our model's predictions in terms of their findings.
Albeit with large scatter, V21 find evidence for a positive correlation between the blackbody luminosity and host galaxy mass. This supports more luminous TDEs arising from higher mass SMBH, consistent with the Eddington-limited luminosity L rad ∝ L Edd of a hydrostatic envelope (as predicted in our model given the likely sub-dominant role played by fall-back accretion luminosity throughout the bulk of the light curve; Eq. 30). V21 also find that the flare rise-time is correlated with peak luminosity and anti-correlated with the photosphere radius. Insofar as t rise is set by the timescale of envelope assembly ∼ t circ (e.g., Steinberg & Stone 2022), it could be expected to scale with the fall-back time (Eq. 2), i.e.
leading a correlation between t rise ∝ M 1/2 • ∝ L 1/2 rad . The situation regarding the light curve decay-time is more complicated. At face value our model predicts that the initial decay-time should scale with the initial envelope cooling time (Eq. 28), i.e.
thus predicting a negative correlation between t decay and M • and hence between t decay and M gal , contradicting the positive correlation found by V21. However, as already mentioned, because t KH,0 /t fb < 1 (Eq. 29) the light curve shape near peak may be influenced by the envelope assembly processes if the process is not sufficiently abrupt (assembly duration t fb ). A significant contribution from fall-back heating at t ∼ t fb (Eq. 30) could also imprint some t fb −dependence into the early light curve decay. Finally, feedback heating from the SMBH also acts to flatten the light curve and increase t decay , and may become more efficient for higher SMBH masses because the circularization radius is typically deeper within the gravitational potential well.
The strongest correlation found by V21 is between L pk /R ph and t rise . Taking L pk ∝ M • and R ph ∝ R v ∝ m 2/15 M 2/3 • , and eliminating the SMBH mass, our model would predict While the weak dependence on stellar-mass is encouraging for generating a tight correlation, the scaling with t rise is somewhat too shallow compared to the data (V21; their Fig. 9).

CONCLUSIONS
We have presented a model for TDE light curves, which starts from the assumption that circularization of the most tightly bound stellar debris is prompt (i.e., occurs on the fall-back time of the most tightly bound debris), resulting in the creation of a quasispherical pressure-supported envelope surrounding the SMBH (Loeb & Ulmer 1997) with a characteristic size much larger than the circularization radius defined by the angular momentum of the original orbit (Coughlin & Begelman 2014). This assumption is motivated by recent hydrodynamical simulations which find prompt circularization and rising optical emission consistent with observed early phases of TDE flares, even for the most common "garden-variety" β = 1 disruptions (Steinberg & Stone 2022). Our model builds on earlier works starting with Loeb & Ulmer (1997), but focuses on predicting the long-term evolution of the envelope, and its accretion rate onto the SMBH, under the influence of different sources/sinks of mass and energy, in a flexible and simple to implement and interpret format.
The proposed "cooling envelope" model accounts for a variety of TDE observations, including (1) large photosphere radii and correspondingly high optical luminosities; (2) optical light curve decay, driven largely by passive cooling of the envelope, which roughly follows a power-law νL ν ∝ t −3/2 , coincidentally similar to the canonical ∝ t −5/3 rate of fall-back decline; (3) potential at late times for a shallower or plateau-shaped light curve decay, due to self-regulated energy input from SMBH accretion; (4) gradually rising effective temperature T eff ∝ t 1/2 as the envelope contracts; (5) delay in the peak of the SMBH accretion rate, and hence of thermal X-ray (for opportunely oriented viewers) or jetted emissions, relative to the time of optical peak by up to several hundred days. This delay is notably driven by envelope cooling (either acting in isolation, or temporarily offset by SMBH accretion heating), rather than requiring a delay in the circularization process.
An ∼Eddington-limited hydrostatic envelope scenario appears broadly consistent with correlations between TDE light curve properties and host galaxy (proxy SMBH) mass (van Velzen et al. 2021a). On the other hand, the model is challenged to explain the observed positive correlation between proxy SMBH mass and optical decay-time assuming the latter tracks the Kelvin-Helmholtz time t KH,0 ∝ M −7/6 • (Eq. 28); however, the shortness of t KH relative to the fall-back time t fb ∝ M 1/2 • (Eq. 29) suggests the light curve shape near peak will be sensitive to the envelope assembly process and earlytime fall-back heating (Eq. 30), possibly mixing some t fb −dependence into the decay time.
The end of our calculation, and thus of the most optically-luminous phase, is defined by when the envelope contracts to the circularization radius, after which point rotational effects dominate and the disk structure should better resemble the pure α-disk models originally envisioned (e.g., Rees 1988;Ulmer 1999;Lodato & Rossi 2011). The properties of the remaining envelope at this transition may then define the initial conditions for a viscous disk evolution phase (e.g., Cannizzo et al. 1990;Shen & Matzner 2014), which can power longer lasting UV/X-ray emission (e.g., Auchettl et al. 2017;van Velzen et al. 2019;Jonker et al. 2020). The timescale of this transition depends on whether the envelope contraction is limited by radiative cooling or accretion, and whether SMBH feedback slows the latter, but in general can roughly be written as: where t disk , t acc , t • acc are given in Eqs. (33), (34), (38), respectively.
The prediction of a cooling-induced time delay of several months or longer between the peak of the optical light curve and the SMBH accretion rate, may also bear on other puzzling TDE observations. One of these is the discovery of late-time radio flares or rebrightenings (e.g., Horesh et al. 2021a,b;Perlman et al. 2022;Sfaradi et al. 2022;Cendes et al. 2022), which may indicate the delayed ejection of mildly relativistic material from the immediate viscinity of the SMBH several months to years after the optical peak. We speculate these could arise from jets or winds from the inner accretion disk that suddenly become more powerful as the SMBH accretion rate rises rapidly near the termination of the envelope cooling-contraction phase. Shocks driven by such outflows into the surrounding wind/envelope material could in principle accelerate relativistic ions, generating a source of high-energy gamma-rays and neutrinos (Senno et al. 2017;Lunardini & Winter 2017;Guépin et al. 2018;Fang et al. 2020;Murase et al. 2020), perhaps explaining the significant observed delay between the high-energy neutrino detections from a growing sample of TDE and the optical light curve maximum (Stein et al. 2021;van Velzen et al. 2021b;Reusch et al. 2022).
Although our model is constructed to allow for the effects of winds or outflows from the envelope on its evolution (the sink termsṀ w ,Ė w in Eqs. 13, 14), we have neglected this possibility for simplicity in this work. Strong outflows could occur from the envelope if energy is deposited below its surface at a highly super-Eddington rate (Quataert et al. 2016). We speculate this may occur at two phases in the TDE: (1) at early times, when the envelope is being assembled and R acc R v is small and henceĖ fb L Edd is possible (Eq. 30); (2) at late times as R v → R acc and the SMBH accretion rate is quickly rising to high values, on a timescale faster than the envelope can radiate the received energy.

ACKNOWLEDGMENTS
I am grateful to Elad Steinberg and Nicholas Stone for helpful comments and for sharing an early draft of their manuscript, which imparted momentum to this work.