EIC emission from jetted TDEs

External Inverse-Compton Emission from Jetted Tidal Disruption Events

Wenbin Lu, Pawan Kumar,
Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA
wenbinlu@astro.as.utexas.edupk@astro.as.utexas.edu
July 9, 2019
Abstract

The recent discoveries of Sw J1644+57 and Sw J2058+05 show that tidal disruption events (TDEs) can launch relativistic jets. Super-Eddington accretion produces a strong radiation field of order Eddington luminosity. In a jetted TDE, electrons in the jet will inverse-Compton scatter the photons from the accretion disk and wind (external radiation field). Motivated by observations of thermal optical-UV spectra in Sw J2058+05 and several other TDEs, we assume the spectrum of the external radiation field intercepted by the relativistic jet to be blackbody. Hot electrons in the jet scatter this thermal radiation and produce luminosities in the X/-ray band.

This model of thermal plus inverse-Compton radiation is applied to Sw J2058+05. First, we show that the blackbody component in the optical-UV spectrum most likely has its origin in the super-Eddington wind from the disk. Then, using the observed blackbody component as the external radiation field, we show that the X-ray luminosity and spectrum are consistent with the inverse-Compton emission, under the following conditions: (1) the jet Lorentz factor is ; (2) electrons in the jet have a powerlaw distribution with and ; (3) the wind is mildly relativistic (Lorentz factor ) and has isotropic-equivalent mass loss rate . We describe the implications for jet composition and the radius where jet energy is converted to radiation.

keywords:
X-rays: general—Radiation mechanisms: Inverse Compton radiation
pagerange: External Inverse-Compton Emission from Jetted Tidal Disruption EventsReferencespubyear: 2015

1 Introduction

A tidal disruption event (TDE) occurs when a star passes close to a massive black hole (BH). Rees (1988) described the basic physics of tidal disruption, where the star’s self gravity causes the exchange of angular momentum. The outer half of the star gains angular momentum and is ejected, and the inner half is left in bound elliptical orbits. The bound matter circularizes due to shocks and then accretes onto the BH. If the BH mass , the accretion could be highly super-Eddington and is believed to produce optical-UV to soft X-ray flares with luminosities Eddington luminosity lasting for weeks to months (e.g. Strubbe & Quataert, 2009; Lodato & Rossi, 2011). Recently, many TDE candidates were discovered in the optical-UV (e.g. Gezari et al., 2012; Chornock et al., 2014; Holoien et al., 2014; van Velzen & Farrar, 2014; Arcavi et al., 2014) and X-rays (e.g. Komossa et al., 2004; Gezari et al., 2009; Saxton et al., 2012). Usually, blackbody radiation at a temperature of and luminosity is observed.

The recent discoveries of Swift J164449.3+573451 (hereafter Sw J1644+57, e.g. Levan et al., 2011; Bloom et al., 2011; Burrows et al., 2011; Zauderer et al., 2011) and Swift J2058.4+0516 (hereafter Sw J2058+05, Cenko et al., 2012; Pasham et al., 2015) show that the accretion can launch relativistic jets which produce bright multiwavelength emission from radio to X/-ray. Hereafter, we call these events “jetted TDEs”. If the X-ray radiation efficiency is 0.1, the isotropic jet kinetic power reaches for and then decreases roughly as . Modeling of the radio emission from Sw J1644+57 shows that the total kinetic energy of the disk outflow is (e.g. Zauderer et al., 2013; Barniol Duran & Piran, 2013; Wang et al., 2014; Mimica et al., 2015), which means that either the jet beaming factor is (half opening angle about ) or the jet being narrow () but there is another outflow component carrying times more energy.

The thermal optical-UV emission could come from a super-Eddington wind launched from the accretion disk (e.g. Strubbe & Quataert, 2009). Due to the large optical depth, photons are advected by electron scattering in the wind. As a result of adiabatic expansion, the radiation temperature drops to at the radius where photons can escape. Piran et al. (2015) proposed that the energy dissipated by shocks from stream-stream collisions will also produce optical-UV emission consistent with many TDE candidates. Both models show that the thermal emission should be ubiquitous in all TDEs and more or less isotropic. This is supported by comparisons between the TDE rate selected by optical-UV observations and the rate predicted from galactic dynamics (e.g. Donley et al., 2002; Wang & Merritt, 2004). Therefore, in a jetted TDE, we expect a strong external radiation field (ERF) surrounding the jet, and electrons in the jet will inevitably inverse-Compton scatter the ERF.

In this work, we model the ERF simply as a blackbody (motivated by TDEs found in optical-UV and soft X-ray surveys) and calculate the luminosity from inverse-Compton scattering of ERF by electrons in the jet. If the jet has Lorentz factor and electrons have thermal Lorentz factor in the jet comoving frame, external photons’ energy will be boosted by a factor of . For typical seed photon energy and bulk Lorentz factor , the scattered photons have energy . Therefore, the external inverse-Compton (EIC) process produces X/-ray emission that could be seen by observers with line of sight passing inside the relativistic jet cone.

One of the biggest puzzles in the two jetted TDEs Sw J1644+57 and Sw J2058+05 is the radiation mechanism of X-rays (see Crumley et al., 2015, for a thorough discussion of X-ray generation processes in TDE jets) Is it possible that the X-rays are from EIC emission? Thermal emission from Sw J2058+05 is detected in near-IR, optical and UV bands (Cenko et al., 2012; Pasham et al., 2015), thanks to the small dust extinction in the host galaxy (). Therefore, we use the observed thermal component as the ERF and test if the X-ray data is consistent with being produced by the EIC process.

This work is organized as follows. In section 2, we describe the characteristics of the jet. In section 3, we calculate the expected EIC luminosities from above and below the ERF photosphere. In section 4, we apply the model to Sw J2058+05. We discuss uncertainties in our model and suggestions for future observations in section 5. A summary is given in section 6. Throughout the work, the convention and CGS units are used. If not specifically noted, all luminosities and energies are in the isotropic equivalent sense.

2 Jet Characteristics

We assume a baryonic jet with bulk Lorentz factor and half opening angle . By “on-axis observer”, we mean that the angle between the jet axis and the observer’s line of sight is smaller than the relativistic beaming angle . The jet is assumed to be steady111Fluctuations of on a timescale light-crossing time of Schwarzschild radius are inevitable and might be the reason for the fast variability seen in X-ray. Here, by “steady”, we mean the averaged level on timescales . and the (isotropic) kinetic power is denoted as . Electron number density in the lab frame (BH rest frame) is . Throughout the work, we assume inverse-Compton scattering by the electrons in the jet has Thomson cross-section (Klein-Nishina suppression is negligible).

Consider a small radial segment of the jet as a cylinder of height and radius . For external photons traveling across the jet in the transverse direction, the optical depth is equal to the total number of electrons within this cylindrical volume times divided by the area of the side, i.e.

(1)

We call the radius where “self-shielding radius”

(2)

below which external photons cannot penetrate the jet transversely. For external photons moving in the radial direction towards the origin (against the jet flow), the optical depth of the jet is

(3)

The jet becomes transparent in the radial direction () at radius

(4)

which is the radius where the jet has largest scattering cross section. We can see that it is easier for photons to penetrate the jet in the transverse direction than in the radial direction, since the jet is narrow. Note that is different from the “classical” jet photospheric radius (e.g. Mészáros & Rees, 2000), which is based on the optical depth for photons comoving with the jet

(5)

The difference between (Eq. 3) and (Eq. 5) is: the former is for photons moving against the jet flow, so photons can interact with electrons at all radii from to ; the latter is for photons moving along the jet flow, so photons can only interact with electrons in the local casualty connected thickness . From Eq.(5), we can see that once an external photon is scattered by a jet electron at radius , it will escape freely along the jet funnel.

3 External inverse-Compton Emission

In this section, we construct a simple model for the EIC interaction between the jet and the ERF, and calculate the EIC luminosities. In the jet comoving frame, electrons are assumed to have a single Lorentz factor . For any distribution of Lorentz factors , another convolution is needed (see section 4.2). We assume the ERF is emitted from a spherically symmetric photosphere and has a blackbody spectrum222Other types of ERF could be produced by the accretion disk (multicolor blackbody spectrum), hot corona (disk + Comptonization spectrum), shocks (powerlaw spectrum if some electrons are accelerated to a powerlaw distribution). They can be dealt with by convolving our simple procedure over the ERF spectrum.. The photospheric radius of the ERF emitting material is determined by solving

(6)

where is the total opacity, is the density profile. If the length-scale of the density gradient is on the order of and is dominated by electron scattering , the photospheric radius can be estimated by . As shown in Fig.(1), the EIC emission could come from above and below .

The Rosseland mean absorption opacity (including free-free and bound-free) is (Rybicki & Lightman, 1979). The density at can be estimated by . Observationally, the temperature at is a few . With such a low density and high temperature, the absorption opacity turns out to be . Therefore, the opacity is dominated by Thomson scattering (assuming solar metallicity). Note that the radiation at may not be thermalized, because the “effective” absorption optical depth (Rybicki & Lightman, 1979)

(7)

could be much smaller than 1 at . The “thermalization radius” is defined as where and photons are thermalized only below . The ratio (always ) depends on the density profile. For example, a wind profile gives . Between and , there’s a purely scattering layer where photons escape via diffusion. Note that, if the observed blackbody luminosity and temperature are and , the radius determined by ( being the Stefan-Boltzmann constant) is usually not the photospheric radius.

In typical TDEs, the luminosity of the ERF is close to Eddington luminosity , peaking around optical-UV. With ideal multiwavelength coverage and small dust extinction, the ERF is observable and can be determined by two parameters: the total luminosity and temperature . In the following two subsections, we treat and as knowns.

Figure 1: Geometry of the external inverse-Compton (EIC) process. The material producing the external radiation field (ERF) is in blue and the jet in orange. The observer is on the jet axis. The photospheric radius is where the optical depth of the ERF emitting material (Eq. 6). The EIC scattering could happen above and below . The self-shielding radius (Eq. 2) is where the jet becomes transparent in the transverse direction. Usually we have , so in the region, photons may penetrate the jet transversely multiple times before getting scattered.

3.1 EIC emission from above the photosphere

If the observed blackbody luminosity is , the ERF flux at the photosphere is

(8)

Since , the ERF at is not far from being isotropic. At radii , the ERF flux drops as and photons are moving increasingly parallel with the jet, so most of the EIC scatterings happen at radius and the (isotropic) EIC luminosity is

(9)

where (Eq. 1) and (Eq. 3) are the optical depth of the jet in the transverse and radial direction. In the second line of Eq.(9), we have used , because, for parameter space relavant to this work, the condition is always well satisfied. From Eq.(9), we can see that the EIC process above the photosphere boosts the ERF’s luminosity by a factor of .

3.2 EIC emission from below the photosphere

Below the photosphere, the radiation energy in the ERF emitting material has a gradient in the direction where the optical depth drops, so radiation diffuses outwards at a flux (Castor, 2004)

(10)

where is the radiation energy density in the ERF emitting material at radius and is speed of light. As mentioned above, there is a purely scattering layer between the photosphere and thermalization radius. If the ERF emitting material is expanding, below the radius where photons are advected by electrons (advection radius , see section 4.1) or the thermalization radius , the radiation temperature is controlled by adiabatic expansion (assuming radiation pressure dominates)

(11)

In the radius range , since Comptonization is not efficient enough to change photons’ energy, the diffusive flux follows the inverse square law from energy conservation

(12)

From Eq.(6), (11) and (12), the radial distribution of radiation energy can be solved, once we know the density profile . This is done in 4.1 (Fig. 6) under the assumption that the ERF emitting material is a super-Eddington wind with . A similar discussion is given in the context of a wind from ultra luminous X-ray source M101 X-1 by Shen et al. (2015). Below, we take — the radiation energy density in the ERF emitting material at polar angle — as known and consider the energy density in the jet funnel.

Due to the removal of photons by jet scattering, the energy density in the funnel will be smaller than in the surrounding material far from the funnel. However, since the jet is narrow, when the optical depth of the jet in the transverse direction is small enough, the radiation field in the funnel will not feel the existence of the jet, i.e. it will isotropize and reach energy density . We define an “isotropization radius” where the removal of photons by the jet is balanced by the flux entering the jet funnel , i.e.

(13)

In the range , the radiation energy density in the funnel is smaller than and is roughly given by

(14)

In the range , the radiation energy density in the funnel equals to . Physically, photons cross the funnel back and forth in the transverse direction times before getting scattered by electrons in the jet, and when , the radiation field can no longer distinguish between the funnel and the region far from the funnel and hence will isotropize. Fig. (2) roughly shows the changing of radiation energy density with polar angle at different radii .

Figure 2: A sketch for the radiation energy density in the jet funnel as a function of polar angle at different radii R. Above the “isotropization radius” (defined by Eq. 13), the removal of photons by the jet is balanced by the supplying diffusive flux from the surrounding material, so the radiation energy density at the funnel reaches the same as in the surrounding material . Below the “self-shielding radius” (defined by Eq. 2), the jet is optically thick in the transverse direction, so the radiation energy density at the funnel center is approximately zero. In between the two characteristic radii , the radiation energy density in the funnel can be estimated by Eq.(14) as .

The order of , and depends on the density profile , jet Lorentz factor and jet kinetic power . In the case of a wind density profile in the TDE context, we typically have (see section 4.1). The EIC luminosity below the photosphere is mostly produced at radius and we have

(15)

where we have normalized the diffusive flux at to the total luminosity by . The EIC scattered photons’ peak energy is . Eq.(15) means that the EIC process below the photosphere boosts the ERF’s luminosity by a factor of .

3.3 Corrections for mildly relativistic wind

If the ERF comes from a super-Eddington wind launched from the disk, the wind velocity could be mildly relativistic. In this subsection, we show that relativistic effects make the EIC scattered photons’ energy and EIC luminosities (Eq. 9 and 15) smaller. Depending on , the corrections could be significant. Quantities in the wind comoving frame are denoted by a prime () and those in the lab frame are unprimed.

If the wind Lorentz factor is , the relative Lorentz factor between the jet and wind is . For example, if , gives . After EIC scattering, external photons’ energy is only boosted by a factor of , which could be much smaller than .

If the observed blackbody luminosity and temperature are and , the radiation energy density at the wind photosphere in the wind comoving frame is

(16)

The wind photospheric radius is different from the non-relativistic case of Eq.(6) by a factor and is given by

(17)

where the rest mass density is related to the (rest) mass loss rate by mass conservation

(18)

Therefore, the EIC luminosity from above the photosphere is

(19)

The EIC luminosity from below the photosphere is mostly produced at the isotropization radius and can be estimated as

(20)

Here, the normalization from the diffusive flux to is different from the non-relativistic case used in Eq. (15) by a factor of

(21)

which will be derived in section 4.1. The EIC scattered photons’ peak energy is

(22)

4 Applications to Sw J2058+05

Similar to the more widely studied event Sw J1644+57, Sw J2058+05 has a rich set of data, in terms of multiwavelength (radio, near-IR, optical, UV, X-ray, -ray) and time coverage (a few to days, in the host galaxy rest frame). In this section, we use the data published by Cenko et al. (2012); Pasham et al. (2015) and test if the X-rays from Sw J2058+05 are consistent with the EIC emission from the jet. We focus on Sw J2058+05 because it suffers from a small amount of host galaxy dust extinction and reddening (, while Sw J1644+57 has ). All quantities (time, frequencies and luminosities) are measured in the host galaxy rest frame at redshift (Cenko et al., 2012).

The X-ray lightcurve and spectrum of Sw J2058+05 are similar to Sw J1644+57. The main X-ray properties are as follows: (1) The isotropic luminosity stays for and then decline as until a sudden drop (by a factor ) at . (2) Rapid variability () is detected before the drop off, suggesting the X-ray emitting region is at radius . (3) The spectra could be fit by an absorbed powerlaw, with early time (, from Swift/XRT) spectral index () and late time () . We note that the early time index comes from combining333Similar to Sw J1644+57, Sw 2058+05 could have different spectral indexes at different flux levels (Saxton et al., 2012). However, single Swift/XRT observations do not have enough statistics to constrain the spectral parameters in Sw 2058+05. all the XRT PC-mode data within , and hence should be taken with caution. We use as a typical spectral index in the following.

Figure 3: The early time optical-UV spectra of Sw J2058+05. They are consistent with blackbody. The blue dashed line represents a blackbody spectrum of temperature , normalized by . Since the data points only cover the Rayleigh-Jeans tail, the adopted temperature is a lower limit of the true blackbody temperature. All quantities (time, frequencies and luminosities) are measured in the host galaxy rest frame.

The reported optical-UV magnitudes are not corrected for dust extinction. We correct the reddening from the Milky Way (in the direction of this event), using (Cenko et al., 2012, and refs therein). The extinction in any band is calculated by using the tabulated value (at ) from Schlafly & Finkbeiner (2011). The host galaxy is at redshift , so the luminosity distance is , if a standard CDM cosmology is assumed with , , and . We refer to the time of discovery as 00:00:00 on MJD = 55698, following Cenko et al. (2012). The rest-frame time is estimated as . We use the effective wavelengths of different filters and the rest-frame frequencies are calculated by .

Figure 4: The optical-UV spectra of Sw J2058+05 from to . Apart from the blackbody spectrum, a powerlaw component shows up on the low frequency end. The dashed lines are blackbodies with parameters {, } (red) and {, } (green). Since the data points only cover the Rayleigh-Jeans tail, the adopted temperatures are lower limits. The black dotted line is a representative powerlaw (not a fit to the data). Considering the large errorbars and uncertainties from host galaxy reddening, a powerlaw of index to () could fit the data. We note that the two data points at (from HST/F160W, see Table 2 of Pasham et al. (2015)) seem not consistent with the powerlaw, which could be due to the under estimation of errorbars. However, the blackbody component is not affected. All quantities (time, frequencies and luminosities) are measured in the host galaxy rest frame.

The optical-UV spectra at different time are shown in Fig.(3, 4, 5). The spectrum is purely a blackbody at early time , then a powerlaw component shows up on the low frequency end at , and when , the powerlaw component dominates and the blackbody component becomes invisible. For our purpose, we focus on the blackbody component hereafter (see section 5 for a discussion about the powerlaw component).

Figure 5: The optical-UV spectra of Sw J2058+05 from to . There is no visible blackbody component, so we can get an upper limit . Considering the large errorbars and uncertainties from host galaxy reddening, a powerlaw of index to () could fit the data. The black dotted line is a representative powerlaw (not a fit to the data). All quantities (time, frequencies and luminosities) are measured in the host galaxy rest frame.

A blackbody spectrum can be described by two parameters, the bolometric luminosity and the temperature , as follows

(23)

where is the Planck constant and is the Boltzmann constant. Unfortunately, optical-UV observations only cover the Rayleigh-Jeans tail, which is insufficient to fully describe a blackbody spectrum. From Fig.(3) and (4), we can get two pieces of information: (i) a lower limit on the temperature

(24)

where , , when , , , respectively; (ii) the normalization

(25)

where , , when , , , respectively. Making use of Eq.(23), we can rewrite Eq.(25) as

(26)

where is the blackbody temperature (constrained by Eq. 24). Hereafter, we use the approximation , which is accurate to when . Eq.(24) and (26) are all the information we can get from the observed spectra.

In Fig.(5), there’s no visible blackbody component from to , so we get an upper limit . The jet might have been turned off at this time, because the X-ray sharp drop occurs at .

We note that the host galaxy may contribute a small amount of reddening444Pasham et al. (2015) fit the XMM-Newton X-ray () spectra with a single powerlaw and obtain an absorbing column density. After subtracting the Galactic column density (Kalberla et al., 2005), we get . similar to the Milky Way, which will make the spectra slightly steeper, but the conclusions on the blackbody component (Eq. 24 and 26) are only mildly affected. These uncertainties could be taken into account by the two dimensionless parameters and .

In the following two subsections, we first show that the blackbody component can be produced by a super-Eddington wind. Next, we use the observed blackbody component as the ERF and test if the X-ray lightcurve and spectrum are consistent with the EIC emission from above or below the photosphere. Constraints on the jet parameters from the two cases are summarized in Table (1). Note that, since the EIC model in section 3 is under the assumption of the jet being ultra-relativistic (), if the constraints lead to , the model is inconsistent with the data.

4.1 Wind Model

The high X-ray luminosity of Sw J2058+05 implies that the accretion stays super-Eddington for a few months. Super-Eddington disks are known to be accompanied by strong winds. For instance, Poutanen et al. (2007) show that strong winds combined with the X-rays from the disk around super-Eddington accreting stellar-mass BHs are in good agreement of the observational data from ultra luminous X-ray sources. The super-Eddington wind could be launched by radiation pressure (e.g. Shakura & Sunyaev, 1973). Rencent radiation-magnetohydrodynamic (rMHD) simulations by Ohsuga & Mineshige (2011, 2D) and Jiang et al. (2014, 3D) show that the kinetic power of (continuum) radiation driven wind can be much higher than . However, the 3D general relativistic rMHD simulations by McKinney et al. (2014) show that the kinetic power of wind from super-Eddington disks around rapidly spinning BHs remains at the order of . Laor & Davis (2014) proposed that the strength of line driven winds sharply rises when the local temperature of the accretion disks around supper massive BHs reaches . It is also likely that magnetic fields (MFs) play an important role in the wind launching process, since angular momentum is removed from an accretion disk through MFs. For example, Blandford & Payne (1982) proposed that the wind could be driven centrifugally along open MF lines.

Up to now, a systematic study of the role of MFs and (line- and continuum-) opacity is still lacking and the detailed wind launching physics is still not well understood. In the context of TDEs, the fact that the fall-back material is very weakly bound is very different from the initial conditions used in the aforementioned numerical simulations. Since the fall-back material evolves nearly adiabatically, the energy released from the accretion of a fraction of the material on bound orbit could push the rest outwards as a wind.

Hereafter, we use upper case to denote the true radii (in ) and lower case for the dimensionless radii normalized by the Schwarzschild radius . Also, the true accretion, outflowing (subscript “w”), and fallback (subscript “fb”) rates (in ) are denoted as upper case and the dimensionless rates are normalized by the Eddington accretion rate as . The Eddington accretion rate is defined as , and , where is BH mass in and we have assumed solar metallicity with Thomson scattering opacity .

For a star with mass and radius , the (dimensionless) tidal disruption radius is

(27)

The star’s original orbit has pericenter distance . When the star passes for the first time, the tidal force from the BH causes a spread of specific orbital energy across the star (Stone et al., 2013)

(28)

Bound materials have specific orbital energies and the corresponding Keplerian orbital periods are given by

(29)

Therefore, if circularization is efficient enough (within a few orbital periods), the fall back rate is

(30)

which means that a flat distribution of mass per orbital energy gives the mass fall-back rate . The leading edge of the fall-back material has the shortest period

(31)

Therefore, the normalized fall-back rate profile is

(32)

Following Strubbe & Quataert (2009), we assume a fraction of the fall-back gas is gone with the wind, and hence the wind mass loss rate is at early time () and later on (if stays constant). Note that, in the absence of the wind, the jet might be draged to a halt due to the IC scattering of radiation from the disk as follows. From the conservation of angular momentum, the disk size is , which is larger than the self-shielding radius (Eq. 2). Therefore, at , disk photons penetrate the jet funnel in the transverse direction and hence the inverse-Compton power of each electron in the jet is . The ratio of EIC drag timescale, (assuming electrons and protons are coupled), and the dynamical timescale, , is

(33)

As we show in this paper, an optically thick mildly relativistic wind alleviates this IC drag problem and links the observed optical-UV to the X-ray emission in a self-consistent way.

We assume that the wind is launched from radius at a speed . Due to inadequate understanding of the wind launching physics, the radius is uncertain and hence taken as a free parameter in this work. The rMHD simulations mentioned at the beginning of this subsection show that a few.

At the wind launching radius , we assume that radiation energy and kinetic energy are in equipartition:

(34)

The radiation temperature at the base of the wind is related to the radiation energy density by ( being the radiation density constant), so from Eq.(34), we have

(35)

Combining Eq. (17) and (18), we obtain the photospheric radius of the wind

(36)

Below , photons escape by diffusion or advection, and the radius where diffusion time equals to the dynamical time (i.e. ) is called the “advection radius”

(37)

At smaller radii , the wind evolves adiabatically, so the radiation pressure, which dominates over gas pressure (), decreases with density as . Under the assumption of a steady wind with constant velocity and spherical symmetry, the density profile is , so the radiation temperature (in the comoving frame) evolves as

(38)

Here, at a temperature , the thermalization radius (defined by according to Eq. 7) is related to the photospheric radius by . Since , we usually have . In the range , photons only interact with baryons by electron scattering (or Comptonization), which is not efficient enough to change photons’ energy significantly. Therefore, the radiation temperature stays constant as

(39)

Combining Eq.(35), (37) and (39), we find the radiation temperature at the advection radius (in the wind comoving frame). The blackbody temperature to be observed555Strictly speaking, the spectrum integrated over the whole photosphere is not Plankian, because the temperature is a function of latitude angle (see Eq. 43 below). The blackbody approximation makes the equations explicitly solvable and hence greatly simplifies the model. We have verified that the error in the integrated spectrum resulting from the blackbody approximation is less than , if . is and is given by

(40)

In the range , photons escape by diffusion and the diffusive flux follows the inverse square law (since radiation energy is conserved), so we have

(41)

The evolution of radiation energy density and temperature with radius in the wind model is shown in Fig.(6).

Figure 6: A sketch for the evolution of radiation energy density and temperature with radius . Below the “advection radius” (defined by Eq. 37), the energy density is controlled by adiabatic expansion. Above , the energy density decreases with because of diffusion. Since Comptonization is not efficient enough to change photons’ energy, the radiation temperature stays constant at .

Next, we Doppler-boost the radiation field from the wind comoving frame to the lab frame to calculate the luminosity seen by the observer. The specific intensity at in the wind comoving frame is

(42)

After Lorentz transformation , the specific intensity in the lab frame is still a blackbody and the only difference is that the temperature is a function of the emission latitude angle , i.e.

(43)

where . Note that the difference between relativistic and non-relativistic solutions is the latitude dependence of , and the flux ratio is a function of wind Lorentz factor

(44)

where has been used. Note that in the ultra-relativistic limit and in the non-relativistic limit. The isotropic equivalent luminosity for an observer at infinity is

(45)

from which, we can see that the wind luminosity can mildly exceed the Eddington luminosity (when ). Putting the optical-UV constraints from Sw J2058+05 (Eq. 24 and 26) into the wind model (Eq. 40 and 45), we find

(46)

where , , and , , when , , , respectively. We note that, due to the strong dependence on the temperature (through ) and wind velocity , the upper limit of mass loss rate has large uncertainties and so does the lower limit of BH mass . However, the product only depends on , decreasing from to 2 when . Therefore, the true wind mass loss rate can be estimated by

(47)

Note that the derived mass loss rate is in the isotropic equivalent sense. The wind is expected to be somewhat beamed along the jet axis (towards the observer), so Eq.(47) is consistent with a typical TDE and the optical-UV blackbody component is consistent with being produced by a super-Eddington wind. Note that the advection radius only depends on the product and is hence not affected by the uncertainties on the temperature:

(48)

And the photospheric radius is a factor larger.

In section 3.2, we defined the “isotropization radius” by balancing the radiation flux entering the jet funnel through the interface with the wind and the flux removed due to EIC scattering. In the relativistic case, is given by

(49)

where