# Pre-peak Emission in Tidal Disruption Events

Xiaoshan Huang (黄小珊) California Institute of Technology, TAPIR, Mail Code
350-17, Pasadena, CA 91125, USA  Department of Astronomy, University of
Virginia, Charlottesville, VA 22904, USA  Shane W. Davis Department of
Astronomy, University of Virginia, Charlottesville, VA 22904, USA  Yan-fei
Jiang (姜燕飞) Center for Computational Astrophysics, Flatiron Institute, 162
Fifth Avenue, New York, NY 10010, USA

###### Abstract

The rising part of a tidal disruption event light curve provides unique
insight into early emission and the onset of accretion. Various mechanisms are
proposed to explain the pre-peak emission, including shocks from debris
interaction and reprocessing of disk emission. We study the pre-peak emission
and its influence on the gas circularization by a series of gray radiation
hydrodynamic simulations with varying black hole mass. We find that given a
super-Eddington fallback rate of $10\dot{M}_{\rm Edd}$, the stream-stream
collision can occur multiple times and drive strong outflows of up to
$9\dot{M}_{\rm Edd}$. By dispersing gas to $\gtrsim 100r_{\rm s}$, the outflow
can delay gas circularization and leads to sub-Eddington accretion rates
during the first few stream-stream collisions. The stream-stream collision
shock and circularization shock can sustain a luminosity of $\sim 10^{44}\rm
erg~{}s^{-1}$ for days. The luminosity is generally sub-Eddington and shows a
weak correlation with accretion rate at early time. The outflow is optically
thick, yielding a reprocessing layer with a size of $\sim 10^{14}\rm cm$ and
photospheric temperature of $\sim 4\times 10^{4}$K.

galaxies: ISM — CR-magnetohydrodynamics — ISM: jets and outflows —method:
numerical simulation

##  1 Introduction

A star orbiting around a black hole can be tidally disrupted when its orbital
pericenter is smaller than the tidal radius $R_{\rm T}=R_{\ast}(M_{\rm
BH}/M_{\ast})^{1/3}$, where $R_{\ast}$ and $M_{\ast}$ are the radius and mass
of the star, $M_{\rm BH}$ is the black hole mass. The accretion of the stellar
debris can temporarily light up an inactive black hole, yielding a transient
event that emits across a broad waveband (Rees, 1988; Ulmer, 1999; Evans &
Kochanek, 1989). These tidal disruption events (TDEs) now constitute a class
of well-characterized transients (Gezari, 2021), becoming important probes to
the accretion physics (Shiokawa et al., 2015; Piran et al., 2015; Hayasaki et
al., 2016; Liptai et al., 2019), the properties of black holes such as their
mass function (Rosswog et al., 2009; Metzger & Stone, 2016; Mockler et al.,
2019) and spin distribution (Kasen & Ramirez-Ruiz, 2010; Gafton & Rosswog,
2019; Mummery, 2024), as well as properties of their host galaxies (French et
al., 2020; Hammerstein et al., 2021; Somalwar et al., 2023).

TDEs are mostly discovered in the optical/ultraviolet (UV) band (Van Velzen et
al., 2011; Hung et al., 2017; Kochanek et al., 2017; Van Velzen et al., 2021;
Yao et al., 2023; Hammerstein et al., 2022) or the X-ray band (Grupe et al.,
1999; Lin et al., 2011; Saxton et al., 2012; Sazonov et al., 2021). Early
results suggested TDEs roughly formed two classes, with a single event
preferentially emitting in either the optical/UV or X-ray bands, but rarely in
both simultaneously.

One theoretical interpretation for these two classes involves super-Eddington
accretion leading to a puffed up, optically thick disk and outflow combined
with a corona of hot electrons in the funnel region near the pole (Dai et al.,
2018; Thomsen et al., 2022). The exclusive optical/UV or X-ray emission is
then determined by the observers’ viewing angle: the optical/UV events are
identified with lines of sight through the disk/outflow that blocks the inner
accretion region, and the X-ray events are from angles receiving emission
directly from the corona region. In other models, accretion generates high
energy photons, while the optical/UV emission is produced by other dynamical
processes. For example, the collision shock due to the debris stream self-
intersection (stream-stream collision) (Piran et al., 2015; Jiang et al.,
2016) and the dynamical shocks from the gas around the black hole prior to
accretion disk formation (Ryu et al., 2023; Steinberg & Stone, 2024).
Alternatively, after the accretion flow is established, the lower energy
emission could arise from reprocessing by collision-induced outflow (Lu &
Bonnerot, 2020; Bonnerot et al., 2021; Huang et al., 2023), optical-thick disk
wind (Lodato & Rossi, 2011; Miller, 2015; Metzger & Stone, 2016; Kara et al.,
2018) or a quasi-spherical, weakly bound envelope (Strubbe & Quataert, 2009;
Roth et al., 2016; Metzger, 2022; Price et al., 2024).

Recent observations reveal a more diverse relationship between optical/UV and
X-ray emission, complicating the earlier picture, where the lower energy and
higher energy emission were largely exclusive to each other. Malyali et al.
(2023) report an example of X-ray detection roughly 14 days prior to its
optical peak. Guolo et al. (2023) study the X-ray emission for optically
selected samples from Zwicky Transient Facility. They show that the X-ray
emission can emerge in different stages through out the optical light curve,
and show diverse temporal evolution in addition to power-law decay. The ratio
between black body optical luminosity and X-ray luminosity spans a wide range,
suggesting there might be different types of dynamical evolution in TDEs. TDEs
are also associated with both early-time and late-time radio emission with a
range of different behaviours (Alexander et al., 2020; Goodwin et al., 2022;
Cendes et al., 2023) and potential powering mechanism (Matsumoto & Piran,
2021; Spaulding & Chang, 2022; Bu et al., 2023; Matsumoto & Metzger, 2023). It
is likely that some mechanisms that proposed by different theoretical models
operate simultaneously during TDE evolution, giving rise to the observed
multi-band emission.

The rising part of the light curve is relatively less understood and
particularly interesting for understanding the underlying physical processes
driving the multiband emission. It has been shown that the optical rising time
is weakly correlated to the black hole mass, but more likely to be affected by
photon diffusion timescale (Van Velzen et al., 2021), suggesting reprocessing
might play an important role in pre-peak light curve. Wang et al. (2023);
Huang et al. (2024) find that some TDEs can present a pre-cursor flare in
optical light curve before the main peak, which might be powered by stream-
stream collision. Recent simulation work suggests that after the initial
debris stream dissipates energy via stream-stream collision, gas
circularization might be inefficient or delayed. The dynamical shocks are
instead the leading emission mechanisms in early time (Ryu et al., 2023;
Steinberg & Stone, 2024). Understanding the pre-peak dynamics is essential to
determine early energy dissipation mechanisms, which can be significantly
distinct from standard accretion dominated dynamics.

In Huang et al. (2023), we study the collision between two idealized TDE
debris streams to understand the outcome of stream-stream collision by local
high resolution radiation hydrodynamic (RHD) simulations assuming zero black
hole spin. We find that stream-stream collision can yield a luminosity $\sim
10^{42-44}\rm erg~{}s^{-1}$ and drive strong outflow away from the orbital
plane, favouring the picture of shock powered emission and reprocessing by
optically thick outflow from stream-stream collision. In this paper, we extend
the simulation domain to global region around the black hole and follow the
stream from the first passage of pericenter to tens of orbital periods at the
stream-stream collision radius. We include time-dependent calculation of
radiation field with tabulated opacity, which allows us to better capture
radiation’s role in gas dynamics and energy dissipation. The RHD simulations
adopt gray opacity, but we also show one Monte Carlo post-processing
simulation and one multi-group RHD simulation to explore the potential
spectral information.

In Section 2, we introduce the numerical method, simulation set-up and the
choice of stream orbits. The gray RHD simulations are listed in Table 2 with
key parameters. Section 3 shows the detailed results. We focus on the effect
of black hole mass $M_{\rm BH}$, which strongly affect the debris stream orbit
for given stellar mass, radius, and penetration factor. Our fiducial run
RSI50Edd10 is presented in Section 3.1, where the simulated system corresponds
to the case of a solar type star disrupted by $8\times 10^{6}M_{\odot}$ black
hole. In Section 3.2 we vary the black hole mass and compare the dynamical
processes. We show a separate suite of simulations to understand the potential
effects of resolution in RHD simulations in Section 3.3, focusing on the
pericenter stream structure. In Section 4, we discuss the pre-peak emission
sources (Section 4.1) and energy dissipation mechanisms (Section 4.2). We
discuss the potential high energy emission regarding to the viewing-angle
effect, and potential radio emission from unbound gas produced in the
collision (Section 4.5). We compare the results with previous studies in
Section 4.6.

##  2 Simulation Set-up

###  2.1 Equations and Units

We solve the following equations in Athena++ with the explicit radiation
transfer module (Jiang, 2021)

$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0,$
(1) $\displaystyle\frac{\partial(\rho\mathbf{v})}{\partial
t}+\nabla\cdot(\rho\mathbf{v}\mathbf{v}-\textsf{P}^{*})=-\mathbf{G}+\rho\mathbf{a}_{\rm
grav}$ (2) $\displaystyle\frac{\partial E}{\partial
t}+\nabla\cdot[(E+P)\mathbf{v}]=-cG^{0}+\rho\mathbf{a}_{\rm
grav}\cdot\mathbf{v},$ (3) $\displaystyle\frac{\partial I}{\partial
t}+c\textbf{n}\cdot\nabla I=cS_{I}$ (4) $\displaystyle
S_{I}\equiv\Gamma^{-3}[\rho(\kappa_{\rm s}+\kappa_{\rm a})(J_{0}-I_{0})$
$\displaystyle+\rho(\kappa_{\rm s}+\kappa_{\rm
P})\left(\frac{a_{R}T^{4}}{4\pi}-J_{0}\right)]$ (5) $\displaystyle
cG^{0}\equiv 4\pi c\int S_{I}d\Omega$ (6) $\displaystyle\mathbf{G}\equiv
4\pi\int\mathbf{n}S_{I}d\Omega$ (7)

Equation 1 \- Equation 3 are gas hydrodynamic equations, where $\rho$,
$\mathbf{v}$, $E=E_{g}+(1/2)\rho v^{2}$ are fluid density, velocity and total
energy density. We assume ideal gas law that relates internal energy $E_{g}$
and pressure $P$ by $E_{g}=P/(\gamma-1)$. P is the pressure tensor. In the
right-hand side of Equation 2 and Equation 3, radiation couples with gas
through the components of the radiation four force $-\mathbf{G}$ and
$-cG^{0}$, which are the momentum and energy exchange between radiation and
gas. We adopt the generalized Newtonian gravitational description from Tejeda
& Rosswog (2013). The gravitational force and gravitational potential are
$\rho\mathbf{a}_{\rm grav}$ and $\rho\mathbf{a}_{\rm grav}\cdot\mathbf{v}$, we
list detailed expressions in Huang et al. (2023).

Equation 4 \- Equation 7 represent the frequency integrated radiation transfer
equation and the corresponding source terms. In the Athena++ implementation,
the specific intensities $I$ are first transformed to the fluid comoving
frame, where the opacities and emissivities are approximated to be isotropic
(Jiang, 2021). The radiation source terms $S_{I}$, $\mathbf{G}$ and $cG^{0}$
are evaluated and updated along with the comoving frame gas internal energy
equation. The resulting source terms are integrated over frequency and angle
and then transformed back to the lab frame. Here, $c$ is the light speed.
$I_{0}$ and $J_{0}$ are the comoving frame intensity and mean intensity. n is
the unit vector corresponding to rays in discretized angular grids, with the
same angular discretization in Davis et al. (2012). In the simulations, we
used $n_{\mu}=4$, resulting $n_{\rm oct}n_{\mu}(n_{\mu}+1)/2=80$ angles across
$n_{\rm oct}=8$ octants.
$\Gamma=\Gamma(\textbf{n},\mathbf{v})=\gamma(1-\textbf{n}\cdot\mathbf{v}/c)$
is the scaled Lorentz factor, with $\gamma=1/\sqrt{1-(v/c)^{2}}$ is the
Lorentz factor. Equation 6 and 7 represent radiation’s effect on gas energy
and momentum. In the source term, $a_{R}$ is the radiation constant,
$\kappa_{\rm s}$ and $\kappa_{\rm a}$ are scattering opacity and Rosseland
mean absorption opacity, $\kappa_{\rm P}$ is the the Planck mean opacity. We
use OPAL opacity (Iglesias & Rogers, 1996) for $\kappa_{\rm a}$ and TOPS
opacity for $\kappa_{\rm P}$ (Abdallah & Clark, 1985) based on the local gas
density and temperature, and assume $\kappa_{\rm s}=0.34~{}\rm
cm^{2}~{}g^{-1}$. In the code, we solve the unit-less equations with the
scaling of density unit $\rho_{0}=10^{-10}\rm g~{}cm^{-3}$, velocity unit
$v_{0}=0.005c$ and length unit $l_{0}=r_{\rm s}=2GM/c^{2}$, where $M$ is the
black hole mass. In the rest of paper, we report unit-less quantities unless
explicitly specified with c.g.s unit.

###  2.2 Resolution, Initial and Boundary Condition

We set the $\theta$ direction boundary to be outflow, $\phi$ direction
boundary condition is periodic. The $r$ direction boundaries are set to be
single direction outflow for hydrodynamical variables, which copies all the
values from the first active cells but set any velocity that enters the
calculation domain to be zero. The radiation boundaries in the $r$ direction
is “vacuum” radiation boundaries, which copies all the intensities with n
pointing outward, but sets all intensities with n pointing inward to be zero.

The simulations are performed in spherical polar coordinate with logarithmic
grid in r direction. The calculation domain for RSI50Edd10 are $(2.1r_{\rm
s},~{}267.2r_{\rm s})\times(0,\pi)\times(0,2\pi)$. RSI20Edd10 have domain size
$(1.1r_{\rm s},~{}140.0r_{\rm s})\times(0,\pi)\times(0,2\pi)$, RSI100Edd10 run
has domain size $(2.9r_{\rm s},~{}377.2r_{\rm s})\times(0,\pi)\times(0,2\pi)$.
The base resolution in each direction is $64\times 32\times 64$. We add five
levels of adaptive mesh refinement (AMR) to follow the high density regions.
The refinement gives pericenter resolution $\delta R\approx
0.018,0.021,0.028r_{\rm s}$ in RSI20, RSI50 and RSI100 runs, equivalent to
$\delta R\sim 0.7R_{\odot}$ in all three simulations.

We model the debris stream as a thin, cold stream injected near the black hole
with a fixed mass fallback rate and angular momentum. In each simulation, the
stream is injected at $r=r_{\rm inj}$, $\phi=\phi_{\rm inj}$ and $\theta_{\rm
inj}=\pi/2$. The stream temperature is assumed to be $T_{\rm inj}=10^{5}$K.
Following similar argument in Jiang et al. (2016), the kinetic energy density
in the streams is about six orders of magnitude higher than the internal
energy density or radiation energy density before stream-stream collision, so
the exact temperature of stream is unlikely to affect the shock dynamics. We
also compared a simulation similar as RSI20Edd10 but with $T_{\rm
inj}=10^{4}$K, we did not find significant difference in the dynamics or
thermodynamics.

We set density injection rate $\rho_{\rm inj}$ and velocity $\textbf{v}_{\rm
inj}=(v_{\rm r,inj},~{}0,~{}v_{\rm\phi,inj})$ in the neighboring four cells in
$r$ and $\phi$ directions. The density injection rate at these cells is set by
$\rho_{\rm inj}=f_{\rm Edd}\dot{M}_{\rm Edd}/(\mathbf{v}_{\rm
inj}\cdot\mathbf{A}_{\rm inj})$, where $\dot{M}_{\rm Edd}=40\pi GM_{\rm
BH}/(C\kappa_{\rm s})$ is the Eddington accretion rate, $\textbf{A}_{\rm inj}$
is the total injection area, and $f_{\rm Edd}$ is the Eddington ratio. We fix
$f_{\rm Edd}=10$ in this work to study the super-Eddington fallback rate with
different black hole mass. For higher mass, spin-less black hole and solar
type star, $\beta=1.0$ orbits is more likely to yield partial disruptions
(Mockler et al., 2019; Law-Smith et al., 2020), $f_{\rm Edd}=10$ for might be
an idealized assumption. Nevertheless, RSI20Edd10 provides an interesting
comparison in the limit of strong apsidal precession. In simulations,
$\textbf{A}_{\rm inj}$ varies as we change $r_{\rm inj}$ and $\phi_{\rm inj}$,
so we compute $\textbf{v}_{inj}\cdot\textbf{A}_{\rm inj}$ numerically for
precise calculation.

We set injection velocity $(v_{\rm r,inj},~{}0,~{}v_{\rm\phi,inj})$ and
injection cells location $(r_{\rm inj},\pi/2.0,\phi_{\rm inj})$ based on
ballistic trajectories that are integrated according to equations of motion
from Tejeda & Rosswog (2013) for a spin-less black hole. We assume the orbit
eccentricity $e_{\rm orb}$ as the eccentricity for most bound material
estimated as Dai et al. (2015)

$e_{\rm orb}\approx
1-0.02\left(\frac{M_{*}}{M_{\odot}}\right)^{1/3}\left(\frac{M_{\rm
BH}}{10^{6}M_{\odot}}\right)^{-1/3}\beta^{-1},$ (8)

where $\beta=r_{\rm T}/r_{\rm p}$ is the orbit penetration factor. $r_{\rm
T}=R_{*}(M_{\rm BH}/M_{*})^{1/3}$ is the tidal radius, and $r_{\rm p}$ is the
pericenter radius. The orbit integration initial velocity
$(0.0,0.0,v_{\phi,0})$ and location $(r_{0},\pi/2.0,0.0)$ are corresponding
Keplerian values $r_{0}=r_{\rm p}(1+e_{\rm orb})/(1-e_{\rm orb})$ and
$v_{\phi,0}=\sqrt{GM_{\rm BH}/r_{0}}$.

Figure 1: The integrated ballistic trajectories for black hole mass $M_{\rm
BH}=10^{7},~{}8\times 10^{6},~{}6\times 10^{6}M_{\odot}$ (blue, orange and
red), according to (Tejeda & Rosswog, 2013). The solid dots labels stream
injection location in each simulation.

We fix stellar mass and radius to be solar type star $M_{*}=M_{\odot}$ and
$R_{*}=R_{\odot}$ in all simulations. Therefore, with fixed $\beta=1.0$,
varying black hole mass $M_{\rm BH}$ changes stream orbit and stream-stream
collision (stream self-intersection) location $r_{\rm SI}$. Table 2 summarizes
the $M_{\rm BH}$ adopted in each simulation. Figure 1 shows the ballistic
orbits for $M_{\rm BH}=10^{7},~{}8\times 10^{6},~{}6\times 10^{6}M_{\odot}$ in
this work. As we decrease the black hole mass, $r_{\rm SI}$ moves further away
from the black hole.

For RSI50Edd10, $r_{\rm inj}=80.1r_{\rm s},~{}\phi_{\rm inj}=0.53$,
$\textbf{v}_{\rm inj}=(-0.1c,~{}0,~{}0.03c)$, with $\rho_{\rm inj}=2.15\times
10^{-6}\rm g~{}cm^{-3}$ for $f_{\rm Edd}=10.0$. For RSI20Edd10, $r_{\rm
inj}=30.3r_{\rm s},~{}\phi_{\rm inj}=0.83$, $\textbf{v}_{\rm
inj}=(-0.16c,~{}0,~{}0.07c)$, with $\rho_{\rm inj}=3.18\times 10^{-5}\rm
g~{}cm^{-3}$ for $f_{\rm Edd}=10.0$. For RSI100Edd10, $r_{\rm inj}=200.2r_{\rm
s},~{}\phi_{\rm inj}=0.35$, $\textbf{v}_{\rm inj}=(-0.06c,~{}0,~{}0.01c)$,
with $\rho_{\rm inj}=1.83\times 10^{-7}\rm g~{}cm^{-3}$ for $f_{\rm
Edd}=10.0$. The injection points are shown as the solid dots in Figure 1.
Before injecting stream, we set the initial background density and pressure to
be low values of $\rho_{\rm init}=2\times 10^{-8}$ and $P_{\rm init}=2\times
10^{-12}$. We set density and pressure floor for the hydrodynamic Riemann
solver to be $\rho_{\rm floor}=2\times 10^{-8}$ and $P_{\rm floor}=2\times
10^{-12}$. The temperature floor for radiation transfer module is set to be
$T_{\rm floor}=2\times 10^{-4}$.

##  3 Results

We perform three primary simulations RSI50Edd10, RSI20Edd10, RSI100Edd10, with
a stream-stream collision radius (stream self-intersect radius) $r_{\rm
SI}\sim 20,50,100r_{\rm s}$, respectively. We first describe the fiducial
simulation RSI50Edd10 in Section 3.1, and then in Section 3.2, we describe
results from RSI20Edd10 and RSI100Edd10 to study the effect of changing black
hole mass, with other parameters fixed in Equation 8.

Table 1: Summary of Simulation Parameters Name | $M_{\rm BH}/10^{6}M_{\odot}$ | $\dot{M}/\dot{M}_{\rm Edd}$ | $r_{\rm SI}$
---|---|---|---
RSI20Edd10 | 10.0 | 10.0 | 22.5$r_{\rm s}$
RSI50Edd10 | 8.0 | 10.0 | 47.7$r_{\rm s}$
RSI100Edd10 | 6.0 | 10.0 | 105.9$r_{\rm s}$
Table 2: Simulation parameters including the black hole mass $M_{\rm BH}$ (the
second column) normalized to $10^{6}M_{\odot}$, mass fallback rate $\dot{M}$
normalized to Eddington accretion rate of corresponding black hole mass (the
third column) and stream-stream collision radius $r_{\rm SI}$ (the fourth
column). For $M_{\rm BH}=10.0,8.0,6.0\times 10^{6}M_{\odot}$, the collision r
adius $r_{\rm SI}=6.6\times 10^{13}\rm cm,~{}1.1\times 10^{14}\rm
cm,~{}1.9\times 10^{14}\rm cm$

###  3.1 Delayed Circularization due to collisionally induced outflow

In RSI50Edd10, as the returning stream passes pericenter, apsidal precession
leads to stream-stream collision near $50r_{\rm s}$ at the orbital plane.
Figure 2 shows a series of gas density and radiation energy density snapshots
after the stream-stream collision. The first column shows that collision
drives optically thick outflow in both the orbital plane and off the orbital
plane. The post-shock gas is quickly heated and photons are thermalized,
leading to enhanced radiation energy density near $r_{\rm SI}$.

Figure 2: Gas density (the first and second row) and lab-frame radiation
energy density (the third and fourth row) snapshots from RSI50Edd10. The three
columns corresponds to $t=21.9,~{}94.6,~{}240.8$h, the white texts label the
time since stream-stream collision in the c.g.s unit. The first and third row
are quantities volume averaged over $\Delta\theta\sim 5^{\circ}$ near
$\theta=\pi/2$-plane, providing a “face-on view”. The second and fourth plot
shows volume average from $\Delta\phi\sim 5^{\circ}$ near $\phi_{\rm inj}$
plane, providing a “side view”. The white curve in the third row show the
shape of estimated light curve in log scale (Figure 7), the blue dots labels
time of current snapshot.

####  3.1.1 Strong Outflow Stage

The initial stream-stream collision persists $t\approx 18.0-27.1$h, during
which the the returning stream impacts the fallback stream. The thin fallback
stream is perturbed and thickened by the interaction. Around $t\approx 27.1$h,
the returning stream is disrupted and the stream-stream collision is briefly
paused. The fallback stream reforms around $t\approx 35.9$h, leading to
another similar stream-stream collision around $t\approx 52.5$h. After each
collision, small amount of bound, diffuse gas accumulates between the
pericenter radius $r_{\rm p}$ and collision radius $r_{\rm SI}$.

After the third collision, the returning stream expands significantly after
passing the pericenter. The density contrast of the returning stream and the
surrounding gas drops substantially. The stream structure is similar to the
second column of Figure 2: a thin, high density fallback stream extends to
pericenter, but disperses in the accumulated diffuse gas. The high
eccentricity gas delivered by the expanded returning stream still interact
with the thin fallback stream, but the interaction no longer resembles stream-
stream collision at earlier times. A non-negligible fraction of gas starts to
accumulate around the black hole, increasing the optical depth. The flow
motion becomes more regular, circularizing near the black hole. The third
column of Figure 2 shows the next stage where gas accumulates near the black
hole. We refer to this stage as “circularization” since the eccentricity of
the gas is lower than the incoming stream, and decreases with time. The
eccentricity, however, is not zero and the flow deviates from Keplerian until
the end of the simulation run. Gas density and radiation energy density
between $r_{\rm SI}$ and $r_{\rm p}$ increases, the scale height of the
circularizing gas also increases. The enhanced radiation energy density shifts
from a concentration near the point of stream-stream collision to a more
uniform distribution between $r_{\rm p}$ and $r_{\rm SI}$.

We observe multiple interactions between the fallback and the returning stream
prior to flow circularization. Consistent with recent studies (Lu & Bonnerot,
2020; Steinberg & Stone, 2024; Huang et al., 2023; Ryu et al., 2023), they
produce prompt emission by converting kinetic energy to radiation. The first
three encounters are the most violent ones, where the two streams both
maintain sufficient density contrast with surrounding gas during the
collision.

These major collisions drive significant outflow. Figure 3 shows the total
outflow rate measured at $r=150r_{\rm s}$. (We find that the outflow rate is
roughly constant once measured at $r\gtrsim 90r_{\rm s}$, so the specific
choice of measurement radius is not important). Note that in our definition of
outflow rate, gas is not necessarily unbound but could include bound gas that
currently propagating away from black hole at large radius. The first three
collisions happen around $t\approx 21.9,~{}52.5,~{}87.6$h. The first outflow
rate peak includes contributions from the first two collisions, giving peak
outflow rate $\dot{M}_{\rm out}\approx 7.5\dot{M}_{\rm Edd}$ at $t\approx
65.7,96.3$h.

Given that the mass flux carried by the injected stream is
$\dot{M}=10\dot{M}_{\rm Edd}$, the outflow rate $\dot{M}_{\rm out}\approx
5-7.5\dot{M}_{\rm Edd}$, which means that a significant fraction of gas is
ejected as outflow instead of accumulating near the black hole. Therefore,
accretion rate at these early time is suppressed. The second row in Figure 3
shows the accretion rate measured at the innermost stable circular orbit
(ISCO) $r_{\rm ISCO}=3r_{\rm s}$. It is below $\dot{M}_{\rm Edd}$ during the
first three collisions, and only start to increase at $t\gtrsim 127.0$h when
the outflow rate drops.

Figure 3: The outflow rate (the first row), the accretion rate (the second
row), the luminosity (third row) and mass-weighted average outflow speed (the
fourth row) of RSI50Edd10. In the first row, the solid line shows total mass
outflow rate, normalized to $\dot{M}_{\rm Edd}$. The dotted line shows the
unbound outflow rate. In the third row, the dotted line shows the kinetic
luminosity (Equation 10) of carried by the unbound outflow, and the dotted
line in the fourth row shows the mass-weighted average outflow speed for
unbound gas. The horizontal dashed line labels the escape velocity at the
measuring radius.

We show the mass-weighted average speed of outflow gas in the fourth row. The
outflow in the first collision reaches average speed $\sim 0.1c$, and drops to
$\sim 0.04c$ in the following two collisions. We found that in the second and
third collision, the returning stream expands due to the interaction with the
diffuse gas built up near the black hole from previous collisions. This
slightly decreases the returning stream density but enhances the interaction
area between the two streams, leading to comparable or larger outflow rate but
slower outflow speed in the following two major collisions.

The peak outflow speed $\sim 0.1c$ briefly exceeds the local escape velocity,
indicative of unbound gas in the outflow. Similar to Jiang et al. (2014), we
define “unbound gas” as gas with $E_{\rm t}>0$, where $E_{\rm t}$ is:

$E_{\rm t}=E_{\rm G}+\frac{\gamma P}{\gamma-1}+\frac{4E_{\rm rad}}{3}$ (9)

$E_{\rm G}$ is the sum of kinetic energy and gravitational potential. The
second term is fluid enthalpy, and the third term assumes radiation as a fluid
with effective adiabatic index of $\gamma=4/3$. We note that when radiation
diffusion is present, $E_{\rm t}$ is not a conserved quantity. Nevertheless,
it provides an analogue to Bernoulli number in an adiabatic fluid.

Figure 4: The first two column shows gas density snapshots from RSI50Edd10 at
$t=56.1$h. The first row is a “face-on” view at the orbital plane, the plotted
quantity is averaged over $\Delta\theta\sim 5^{\circ}$ near
$\theta=\pi/2$-plane. The second row is a “side-view” that cut through the
gray dashed line in the first row, the plotted quantity is volume average from
$\Delta\phi\sim 5^{\circ}$ near $\phi=2.07$ plane. The third row is flow speed
normalized to speed of light, the black solid lines shows the velocity
streamlines at this projection. The fourth row shows $E_{\rm t}^{\prime}$
(Equation 9), positive value corresponds to unbound gas. The blue cross
markers label selected sample points to extrapolate ballistic trajectories in
Figure 19.

We find that the fastest unbound gas is located primarily off the orbital
plane. For example, Figure 4 shows outflow in side-view slices at
$\phi=0.66\pi$ during the second collision. The last rows shows the quantity
$E_{\rm t}^{\prime}$ defined Equation 9, where positive $E_{\rm t}^{\prime}$
near $x^{\prime}\sim 25-50$ indicated that this gas is unbound. At this time,
they show bulk speed $\sim 0.15c$ around $\sim 30r_{s}$, decelerating as it
propagates away from the black hole.

To estimate kinetic energy carried by the unbound gas, we define kinetic
luminosity as:

$L_{\rm KE}=\int\mathcal{F}_{\rm KE}\cdot\mathbf{dA}$ (10)

where $\mathcal{F}_{\rm KE}$ is kinetic energy flux through the sphere of
radius $150r_{\rm s}$, where we measure $L_{\rm kE}$. In the first, third and
fourth row of Figure 3, we we show outflow rate, $L_{\rm KE}$ and mass-
weighted average speed of unbound gas as dotted lines. The first two
collisions create transient $\sim 1.25\dot{M}_{\rm Edd}$ unbound outflow,
constituting roughly $\sim 14\%$ of the fallback mass flux. Their average
speed is relatively constant $\sim 0.1c$. The unbound outflow rate drops to
nearly zero after the second collision. The high average velocity in the late
time are from small amount of unbound diffuse gas with density initialized at
the numerical floor value.

In the third row, we compare the luminosity carried by radiation flux and the
kinetic luminosity carried by unbound outflow. Around $t\sim 21.9$h, the
initial stream-stream collision leads to the prompt luminosity $L\approx
8\times 10^{44}\rm~{}erg~{}s^{-1}$, just below the Eddington luminosity
$L_{\rm Edd}=1.2\times 10^{45}\rm~{}erg~{}s^{-1}$. The luminosity stays
roughly constant, with only moderate fluctuations during the first three
stream-stream collisions. The kinetic luminosity of unbound gas peaks during
the first two collisions, which is $\sim 10\%$ of luminosity carried by
photons. At later time, when the outflow subsides, the unbound mass and
kinetic energy flux are dominated by diffuse gas with typical densities close
to the floor density of $\rho\lesssim 10^{15}\rm g~{}cm^{-3}$.

The accretion rate increases simultaneously with the luminosity, albeit well
below $\dot{M}_{\rm Edd}$. We found that the collisions produce a small amount
of more strongly-bound gas that is quickly accreted by the black hole,
contributing to the accretion rate before outflow subsides. The lag between
the rise of outflow rate and the rise of luminosity is consistent with the
time that outflow propagates to the measurement radius, estimated as $\Delta
t_{\rm out}\sim(r_{\rm 150r_{\rm s}}-r_{\rm SI})/\overline{v}_{\rm
outflow}\sim(150r_{\rm s}-r_{\rm SI})/0.1c\approx 21.9$h.

####  3.1.2 Gas Circularization Stage

Around $t\gtrsim 127.0$h, the thin, dense returning stream disperses shortly
after the pericenter passage, well before it could collide with the incoming
stream. The outflow rate drops and gas starts to accumulate near the black
hole at a more rapid rate, the flow motion becomes less eccentric. The
accretion rates grows from $\sim 0.5\dot{M}_{\rm Edd}$ to $\sim
2.8\dot{M}_{\rm Edd}$, the mass accumulated between $r_{\rm p}$ and $r_{\rm
SI}$ increases by $\sim 2.7$ times. The circularization yields relatively
stable luminosity $\sim 10^{45}\rm~{}erg~{}s^{-1}$, with slow increment over
time. The scale height of circularizing flow also increases, as shown in the
side-view density slice in third column of Figure 2.

Although the flow eccentricity drops, we emphasize that it never becomes
circular. It is not well-approximated by an axisymmetric, Keplerian disk. The
fallback stream runs into the accumulating gas with lower angular momentum and
orbital energy, forming localized shocks in the flow. A pair of weak spiral
shock develops near the black hole, where radiation energy density is enhanced
(see, for example, the second row of the third column in Figure 2). The spiral
shock truncates where the dense fallback stream collides with the disk near
$r_{\rm p}$ and does not extends to outer region of circularizing flow.

Figure 5: The average density (upper panel) and average gas temperature (lower
panel) of RSI50Edd10 at $t=240$h. In the upper panel, the solid black line
shows average gas density through spherical shells. The purple solid line
shows the average gas density through a finite range of $\theta$ but excluding
the orbital plane, $\pi/32<|\theta-\pi/2|<\pi/8$ near $\pi/2$. The three
vertical dashed lines are tidal radius (also pericenter radius given
$\beta=1.0$) $r_{\rm p}$, stream-stream collision radius $r_{\rm SI}$ and the
injection radius $r_{\rm inj}$.

Figure 5 shows radial profile of average gas density and temperature. The
black solid lines are the averaged through spherical shells at each radius.
The majority of gas accumulates between $r_{\rm p}$ and $r_{\rm SI}$ (the
first and second vertical dashed lines), where the average gas density roughly
follows power law. We can also describe the total density profile by two power
law indexes, with a shallower inner region and steeper outer region separated
around $\sim 20r_{\rm s}$. The best fitted indexes are $\rho\propto r^{-1.1}$
and $\rho\propto r^{-2.5}$. The inner region is more circularized while the
outer region is affected by the stream-disk interaction and residual outflow.
The black line in lower panel shows the average gas temperature. Between
$r_{\rm p}$ and $r_{\rm SI}$, the average gas temperature $T_{\rm gas}\sim
10^{5}$K, and drops to $T_{\rm gas}\sim 10^{4}$K towards the outer region.

We also show the average density of a region with finite $\theta$ range but
excluding the orbital plane, where the high density stream affects the average
density. The off-orbital plane average density show more clear transition from
the circularizing inner region $\rho\propto r^{-0.77}$ to the outer
accumulating region $\rho\propto r^{-3}$. The average gas temperature profiles
are similar to the spherically-averaged gas temperature and we do not plot
them separately.

To quantify gas circularization, we define a scaled eccentricity following
Oyang et al. (2021) as:

$\mathbf{e}=\frac{1}{GM_{\rm
bh}}\mathbf{v}\times(\mathbf{r}\times\mathbf{v})-\mathbf{\hat{r}}$ (11)

In a circular Keplerian orbit, the magnitude of $|\mathbf{e}|=0$. With the
gravity potential we adopt to approximate general relativistic precession,
$|\mathbf{e}|$ is, however, non-zero for circular orbits at small radius. For
example, $|\mathbf{e}|\approx 4.47\times 10^{-2},1.6\times 10^{-2},1.2\times
10^{-3}$ for $r=10r_{\rm s},30r_{\rm s},400r_{\rm s}$. For the highly
eccentric ballistic trajectory that we use to initialize the stream in
RSI50Edd10 (Figure 1), $|\mathbf{e}|$ drops from 1.0 at $r\gtrsim 100r_{\rm
s}$ to its minimum $|\mathbf{e}|\approx 0.6$ at $r_{\rm p}$.

Figure 6: The magnitude of flow eccentricity (Equation 8) in the orbital-plane
at $t=240$h (RSI50Edd10). The dark gray lines are flow velocity streamlines.
The gas with high eccentricity $|\mathbf{e}|\approx 0.6-0.8$ roughly tracks
the fallback stream. The circularizing gas $r<20r_{\rm s}$ show lower
eccentricity $|\mathbf{e}|\approx 0.2$. The time in the label is time since
the first stream-stream collision, equivalent to the last column in Figure 2.

In Figure 6, the inner circularizing region appears as the low eccentricity
region near the black hole. The flow velocity stream lines are also more
regular. The fallback stream’s eccentricity is consistent with ballistic
trajectory calculation. The kinks in velocity streamlines near $x\sim
20-40,~{}y\sim 20-40$ corresponds to the “stream-disk” shock between the high
eccentricity fallback stream and the low eccentricity circularizing gas, where
the local energy dissipation yields enhanced radiation energy density in
Figure 2.

###  3.2 Varying the Black Hole Mass at Fixed $\beta$

In RSI20Edd10 and RSI100Edd10, we fix the orbital penetration parameter
$\beta$ (Equation 8) and alter the ballistic trajectories by changing central
black hole mass $M_{\rm BH}$. Figure 1 shows the corresponding orbits. When
fixing the $\beta$, $M_{*},R_{*}$, for spin-less black hole, increasing
$M_{\rm BH}$ leads to more apsidal precession and smaller $r_{\rm SI}$, so the
stream-stream collision happens closer to the black hole (Dai et al., 2015; Lu
& Bonnerot, 2020). Therefore, the velocity of streams before collision becomes
larger, increasing the available kinetic energy to dissipate during the
collision. Table 2 lists $r_{\rm SI}$ in these runs, note that $r_{\rm SI}$ is
in unit of $r_{\rm s}$ of corresponding $M_{\rm BH}$.

Figure 7: Upper panel: the outflow rates $\dot{M}_{\rm out}$ (dashed lines)
and accretion rates (solid lines) in RSI50Edd10 (orange lines), RSI20Edd10
(blue lines) and RSI100Edd10 (red lines). The accretion rate is measured at
$3r_{\rm s}$ in all three runs. The outflow rate is measured over spherical
shells at radius of $100r_{\rm SI}$, $150r_{\rm SI}$ and $300r_{\rm SI}$ in
RSI20Edd10, RSI50Edd10, RSI100Edd10. These are relatively outer radius in each
simulation where the mass flux and luminosity is not sensitive to measuring
radius. Lower panel: total luminosity $L$ carried by radiation flux measured
over the same spherical shell as $\dot{M}_{\rm out}$, normalized to Eddington
luminosity assuming $10\%$ efficiency. The prompt rise on $L/L_{\rm Edd}$
correspond to the emission from stream-stream collision. We note that such
short rising timescale is related to the fixed $\dot{M}$ when injecting
stream, the rising time in the simulations should not be interpreted as
physical rising time for TDEs.

First, in RSI100Edd10, we consider the case where the stream-stream collision
moves further away from black hole to $r_{\rm SI}=102r_{\rm s}$. The overall
dynamics is similar to RSI50Edd10. The first stream-stream collision is the
most prominent one, followed by weaker collisions between the expanded
returning stream and thin fallback stream. Around $t\approx 200.3$h, the
fallback stream disperses near the pericenter, and gas begins to more rapidly
accumulate near the black hole.

Figure 7 shows the outflow rate measured at $300r_{\rm s}$ and accretion rate
measured at $3r_{\rm s}$. Compared with RSI50Edd10, the larger collision
radius in RSI100Edd10 reduces the stream velocities before collision, leading
to weaker interaction. The outflow rate peaks at $\dot{M}_{\rm out}\approx
5\dot{M}_{\rm Edd}$ around $t\approx 137.0$h. Similar to RSI50Edd10, the first
outflow peak covers the first two collisions. The outflow subsides around
$t\approx 6.1$ due to the significant expansion of returning stream, the
average density near the black hole increases, and the flow eccentricity
drops. Gas circularization near the black hole enhances the subsequent
accretion rate. The lower panel of Figure 7 shows that at late time, the
accretion rate in RSI100Edd10 is slightly larger than RSI50Edd10 with an
increasing trend.

The stream-stream collision again yields a prompt rise in the luminosity. The
luminosity is slightly less variable and remains at a slightly lower quasi-
steady-state value of $L\approx 6\times 10^{44}\rm erg~{}s^{-1}$, which is
consistent with weaker collision. Overall, the luminosity is only weakly
dependent on the location of the collision radius in these runs, and shows
similar low fraction of Eddington luminosity despite the super-Eddington
fallback rate.

In the strong outflow stage, the dynamics is modulated stream-stream collision
at $r_{\rm SI}$. In the circularization stage, the gas accumulation usually
extends between pericenter radius $r_{\rm p}$ and $r_{\rm SI}$. So the orbital
period $t_{\rm SI}\equiv\pi/\sqrt{2GM_{\rm BH}/r_{\rm SI}^{3}}$ at the $r_{\rm
SI}$ is a characteristic dynamical timescale. For example, $t_{\rm
SI}=1.71\approx 56$h in RSI100Edd10 and $t_{\rm SI}=0.51\approx 22$h in
RSI50Edd10. We ran RSI50Edd10 and RSI100Edd10 for about $12t_{\rm SI}$ and
$6t_{\rm SI}$. Hence, despite the longer physical time, RSI100Edd10
experiences fewer $t_{\rm SI}$ than RSI50Edd10. The circularization in
RSI100Edd10 begins more sooner due to the weaker stream-stream collisions and
outflow, allowing gas to accumulate near the black hole within less $t_{\rm
SI}$.

Figure 8: The first row: the unbound outflow rate in RSI50Edd10 (orange
lines), RSI20Edd10 (blue lines) and RSI100Edd10 (red lines). The second and
third row: kinetic luminosity $L_{\rm KE}$ (Equation 10) and mass weighted
outflow flow velocity for unbound gas. The outflow rate, kinetic luminosity
and average velocity is measured over spherical shells at radius of $100r_{\rm
SI}$, $150r_{\rm SI}$ and $300r_{\rm SI}$ in RSI20Edd10, RSI50Edd10,
RSI100Edd10. The arrows in the third row indicates the first stream-stream
collision in the corresponding simulation. The horizontal dashed line in the
third row shows the escape velocity at the measuring radius.

Weaker collision in RSI100Edd10 leads to less unbound gas in the outflow.
Figure 8 shows the unbound outflow rate, kinetic luminosity and average
velocity measured over the sphere shell at $300r_{\rm s}$ for RSI100Edd10. The
first three collisions are the primary source for the two unbound outflow rate
peaks at $t=120.6,~{}170.7$h, yielding a peak unbound mass flux $\dot{M}_{\rm
out,ub}\approx 0.6\dot{M}_{\rm Edd},~{}0.45\dot{M}_{\rm Edd}$. The mass
weighted average velocity indicates that the outflow is marginally unbound
with velocity close to escape velocity. The arrows in Figure 8 label the time
of the first stream-stream collision. Due to the slower outflow and larger
measurement radius at $300r_{\rm s}$, the delay between the first stream-
stream collision and the rise in the unbound outflow rate is longer than in
run RSI50Edd100. The kinetic luminosity $L_{\rm KE}$ in run RSI100Edd10 is
lower than in run RSI50Edd10.

When we increase black hole mass to $M_{\rm BH}=10^{7}M_{\odot}$ in run
RSI20Edd10, the apsidal precession is stronger and yields $r_{\rm SI}\sim
20r_{\rm s}$. This implies larger encounter velocities and a stronger
collision. Given the $\dot{M}=10\dot{M}_{\rm Edd}$ that could be higher than
physical fallback rate of solar type star, RSI20Edd10 provides a comparison
study in the strong-collision limit. It shows a dynamical evolution that is
different from the RSI50Edd10 and RSI100Edd10.

Figure 9: Gas density (the first and second row) and lab-frame radiation
energy density (the third and fourth row) snapshots from Rsi20Edd10. The three
columns corresponds to $t=7.6,10.4,14.7$h, the white texts label the time
since stream-stream collision in the c.g.s unit. The first and third row are
quantities volume averaged over $\Delta\theta\sim 5^{\circ}$ near
$\theta=\pi/2$-plane, the second and fourth plot shows volume average from
$\Delta\phi\sim 5^{\circ}$ near $\phi_{\rm inj}$ plane. The white curve in the
third row show the shape of estimated light curve in log scale, the blue dots
labels corresponding time of current snapshot.

Figure 9 shows process of the first stream-stream collision in RSI20Edd10.
Similar to RSI50Edd10, the returning stream expands and collides with fallback
stream, driving gas outflow and producing prompt emission. The radiation
energy density is enhanced near $r_{\rm SI}$. A prominent outflow is launched
away from the orbital plane, an a larger fraction of the outflowing gas gains
sufficient angular momentum to flow over the black hole instead of directly
falling into the black hole from the polar region. When this gas returns
orbital plane from above and below, they further collide and dissipate energy
via shocks. As a result, a bright emitting column forms near the pericenter
around $t=10.4-14.7$h, about $180^{\circ}$ from stream-stream collision site
in the orbital plane.

The disrupted fallback stream reforms and leads to another stream-stream
collision. Due to the close collision radius, the $t_{\rm SI}=9$h is rather
small. We ran the simulation for about $12t_{\rm SI}$ to obtain a physical
duration comparable to previous runs. Unlike RSI50Edd10 and RSI100Edd10,
however, the system repeats similar streams-stream collision for about eight
times and show little evidence of transition to a gas circularization stage
where matter accumulates near the black hole.

Figure 10: Gas density snapshots from Rsi20Edd10. The three columns
corresponds to $t=23.5,35.6,95.8$h, the white texts label the time since
stream-stream collision in the c.g.s unit. The first row are quantities volume
averaged over $\Delta\theta\sim 5^{\circ}$ near $\theta=\pi/2$-plane, the
second row show volume average from $\Delta\phi\sim 5^{\circ}$ near $\phi_{\rm
inj}$ plane. The white curve in the third row show the shape of estimated
light curve in log scale, the blue dots labels corresponding time of current
snapshot.

Figure 10 compares the gas density at a similar dynamical stage during the
second, third and eighth collision. At this moment in each collision, the
returning stream already encounters the fallback stream and about to break the
fallback stream. The average gas density around the black hole is increased
compare to the first collision in the second column of Figure 9, but barely
changes between these following collisions.

Figure 7 show the significant increase in the outflow rate during these
repeating collisions. The $\dot{M}_{\rm out}$ of the first two collisions is
comparable to the peak outflow rates in the other two runs with lower $M_{\rm
BH}$ and larger $r_{\rm SI}$, but the following collisions shows persistently
higher outflow rates. The majority of the mass flux carried by the injected
fallback stream is ejected as outflow. Consequently the accretion rate through
$r_{\rm ISCO}$ is low due to lack of gas accumulation near the black hole.

Despite the persistence of episodic collisions, the luminosity shows only
moderate variations. The periodicity of emission from stream-stream collision
is smoothed over the time it takes photons to slowly diffuse out from
optically thick region surrounding the collision point. The shocks produced by
interaction of outflowing gas in between collisions also contributes to the
emission. For example, some of the initially outflowing gas falls back to
orbital plane and collides, yielding emissions by shock dissipation. The
periodically-launched outflow also form multiple shells, the collision between
the shells can also contributes to emission. These outflow interaction further
reducing the luminosity difference between the active collision and stream
reformation stages.

The strong collisions in RSI20Edd10 creates more unbound outflow, even though
these collision occur deeper in the potential. In Figure 8, the unbound
$\dot{M}_{\rm out}$ peaks during the first two collisions around
$3\dot{M}_{\rm Edd}$, with average velocity $v_{\rm out,ub}\sim 0.12c$. The
unbound outflow rate drops below $\dot{M}_{\rm Edd}$ in the following
collisions but remains higher than in the other runs. The average velocity
approaches to local escape velocity $\sim 0.1c$. The persistent kinematic
luminosity in the following collisions provides a potential source of kinetic
energy from these marginally unbound gas. If this unbound gas can travel to
large radius without significant dissipation, it may play an important role in
early radio emission when shocking ambient gas, a possibility we discuss in
Section 4.5.

We note that the periodic dynamics in RSI20Edd10 might be related to the
assumption of a constant angular momentum and a rather higher fallback rate
fixed at $\dot{M}=10\dot{M}_{\rm Edd}$ in our injected stream. The effect of
changing $\dot{M}$ and angular momentum distribution within debris stream will
be a focus of future work. Nevertheless, the fallback rate for $M_{\rm
BH}=10^{7}M_{\odot}$ is likely to be within similar order of magnitude for a
few $t_{\rm SI}=8.8$h. The lack of gas accumulation in RSI20Edd10 suggests the
potential scenario that the prominent mass loss from outflow pauses the
formation of accretion flow. The three runs with different $r_{\rm SI}$ show
consistent trend that a stronger collision leads to longer delay of gas
circularization. Increasing $\dot{M}$ of the fallback stream could also
enhance the outflow because of larger kinetic energy flux carried by the
streams. It is likely that a rising $\dot{M}$ at pre-peak times could produce
non-linear mass accumulation near the black hole.

###  3.3 Resolution and Pericenter Stream Width

Near the pericenter, the gas ballistic trajectories with slightly different
inclinations intersect near the pericenter. This effect results maximum
vertical compression at the pericenter (perpendicular to the orbital plane),
forming the “nozzle shock”. The nozzle shock compression is commonly
identified in previous simulations (Shiokawa et al., 2015; Hayasaki et al.,
2016; Liptai et al., 2019; Curd, 2021; Steinberg & Stone, 2024; Ryu et al.,
2023; Price et al., 2024). Depending on the black hole mass, pericenter radius
and prescribed physical processes, it dissipates stream kinetic energy and
leads to different levels of orbital-plane and vertical stream expansion.

Recent studies suggest that the stream can be compressed to be orders of
magnitudes thinner near the pericenter compared to its original width
(Bonnerot et al., 2022; Bonnerot & Lu, 2022; Coughlin, 2023), making it
numerically challenging to resolve the stream structure in proximity to the
black hole. Other physical processes such as recombination might also impact
stream width during the pericenter passage (Coughlin, 2023; Steinberg & Stone,
2024). In our simulations, the pericenter resolution for RSI20Edd10,
RSI50Edd10 and RSI100Edd10 are $R\Delta\theta\approx 0.018,0.021,0.028r_{\rm
s}\approx 0.7R_{\odot}$. For the simulations reported in other sections, we
set the maximum refinement level to five and ensure the pericenter is on
highest refinement level. In this section, we use the orbit in RSI50Edd10 and
compare five runs with maximum refinement level from three to seven. We ran
the level three to six simulations to $t=32.8$h (except for RSI500Edd10 at
level five), roughly right before the first stream-stream collision, the level
seven run only ran to $t=13.1$h, when the stream just arrives pericenter.

We find that in the orbital plane, the stream expands after the first
pericenter passage regardless of the resolution. Interestingly, the orbital-
plane spreading is rather moderate in level three and four runs, where the
returning stream roughly maintains its width before the pericenter
compression. Increasing resolution to level five and above, however, leads to
enhanced orbital-plane expansion. The returning stream expansion in level six
($R\Delta\theta\approx 0.01r_{\rm s}$) is similar to level five
($R\Delta\theta\approx 0.018r_{\rm s}$), with slightly reduced width. In other
words, thinner streams are produced at lower resolution, but the orbital-plane
stream width shows relatively weak dependence on resolution from five
refinement level and above. We note that we do not reach stream width
convergence even given the highest resolution, stream orbital-plane expansion
can still be related to numerical diffusion. The convergence study in Price et
al. (2024) also suggests orbital-plane expansion can be overestimated with
insufficient resolution.

As the stream is compressed, energy is dissipated and much of it turns into
radiation. If the stream remains optically thick, and the diffusion time is
long compared to the pericenter passage, significant radiation pressure is
present to re-expand the stream as it leaves pericenter, and the evolution is
approximately adiabatic. Alternatively, if the stream is sufficiently
optically thin so that cooling is significant, there will be less radiation
pressure available to re-expand the stream after pericenter passage. For
physically likely parameters, the stream should remain sufficiently optically
thick so that the diffusion time is much longer than pericenter passage and
little radiation escapes (Bonnerot et al., 2022; Coughlin, 2023).

When injecting the stream and before the stream experiences strong tidal
compression, an important length scale to resolve is the vertical optical
depth of the stream. The transition of horizontal expansion from level four to
level five is reasoned as follow: similar to (Huang et al., 2023), we keep
$\dot{M}$ fixed when changing our resolution and inject the stream over a
fixed width $H$ in both the $\theta$ and $\phi$ directions. The density
$\rho\propto H^{-2}$, where $H$ is our injected stream width, which scales
proportional to the resolution. The optical depth across the stream is roughly
given by $\tau\sim\rho\kappa_{s}H\propto H^{-1}$. Insufficient resolution
leads to artificially larger $H$ and the density $\rho$ is underestimated by
factor of $H^{2}$ consequently, the optical depth $\tau$ is underestimated by
factor of $H$. Before the stream arrives pericenter, energy might be lost via
radiation diffusion due to the artificially reduced optical depth.
Equivalently, radiative loss is too efficient in low resolution cases, the
thin width of returning steam is closer to the isothermal limit (for example,
radiative efficient models in Hayasaki et al. (2016). When increase resolution
to resolve the optical depth, the tidal compression on the stream is more
adiabatic instead of isothermal.

Figure 11: Snapshots of vertical stream density (upper panel) and lab-frame
radiation energy density (lower panel) near pericenter. From left to right we
show the stream vertical cross section for simulations with maximal refinement
level from four to seven, the simulations reported in other sections are with
level five refinement (the second column). In the highest refinement level,
the simulation domain is the same as others, i.e. $\theta$ ranges from
$0-2\pi$, we only read out and plot the data near orbital plane. The snapshot
is cut at $r\approx 4.75r_{\rm s}$, $\phi=0.1\pi$ and $t=13.1$h. The orbital
plane is at $z=0$. The solid gray curves label radius of $r=4.75$.

To elaborate the transition from level five and above, we show the “side view”
density and radiation energy density in Figure 11 through a cross section at
$\phi=0.1\pi$, near pericenter passage. In the set of comparison, we fix the
fallback rate among all different resolutions.

At this time, the stream is significantly compressed compare to the injected
width. In the level four run (the first column), the stream is only a couple
cells thick due to strong cooling before arriving pericenter. Insufficient
resolution leads to an artificially large radiation flux. As a result, the
radiation energy outside the dense stream is noticeably larger than in the
other runs. After pericenter passage, there is little radiation pressure
remaining to re-expand the stream and it remains thin. (Note that the level
three run shows similar vertically collapsed stream as level four so we do not
show it in Figure 11.)

We discuss the impact of resolution on the vertical structure of the streams
further in Appendix A, but the upshot is that for the runs with maximum
refinement level five and above, we start to resolve the optical depth and
capture the vertical structure of the streams. Increasing resolution from
level five to seven, the stream vertical height continues to decrease and the
radiation energy outside the stream also decreases.

In other words, we find that before the stream arrives pericenter and
experiences strong compression, the radiative cooling decreases as the
resolution increases, the stream becomes more optically thick. The scale
height of the stream in the vertical direction is still set by our resolution
threshold and estimates suggest that we need at least an order-of-magnitude
higher resolution to resolve the vertical height of the stream, and the
corresponding horizontal expansion dependence on resolution from level seven
and above is unclear. We find, however, that five levels of refinement reduces
the flux to the point where the cooling is low enough that the radiation
energy density within the stream remains high and the re-expansion of the
stream is nearly independent of resolution.

##  4 Discussion

###  4.1 Pre-peak Emission from shocks: from stream-stream collision to
circularization shock

In the conventional perspective, TDE emission originates from the accretion of
stellar debris. For a significant fraction of TDEs, the late time light curve
roughly follows the theoretical mass fallback rate, which sets the accretion
rate. While this may still be a reasonable picture of the later decay phase,
we find that the rising time emission is more likely powered by various shocks
instead of accretion. Figure 7 shows in all three simulations, the measured
luminosity is about a few $\sim 10^{44}\rm erg~{}s^{-1}$ and rarely exceed
Eddington luminosity. The accretion rate near $r_{\rm ISCO}$ is usually well
below the mass fallback rate $10\dot{M}_{\rm Edd}$. To quantify the relation
between accretion and luminosity, we adopt the definition of radiation
efficiency $\eta_{\rm out}=L/\dot{M}c^{2}$, where $\dot{M}$ is the accretion
rate measured at ISCO.

Figure 12: The upper panel shows outflow rates (dashed lines) and accretion
rate (solid lines), the lower panel shows radiation efficiency $\eta$ for
RSI20Edd10, RSI50Edd10, RSI100Edd10 (blue, orange and red lines). The time is
normalized to the free-fall time at $r_{\rm SI}$ to show the correlation with
dynamics. The upper panel is equivalent to the upper panel of Figure 7 except
for different time normalization. The gray dashed line in the lower panel
labels $10\%$ efficiency.

We show the evolution of radiation efficiency $\eta$ in the lower panel of
Figure 12. We normalize time to the orbital period at stream-stream collision
radius $t_{\rm SI}$, and in the upper panel, we also plot the outflow and
accretion rate for comparison. The radiation efficiency evolution in the pre-
peak time significantly deviates from the picture of a classic steady-state
accretion disk. The efficiency $\eta$ fluctuates with time, and shows a
complex relation to the accretion rate.

In RSI50Edd10 and RSI100Edd10, when the $\dot{M}_{\rm out}>5\dot{M}_{\rm Edd}$
and the accretion rate is well below $\dot{M}_{\rm Edd}$, the efficiency
fluctuates around $\approx 10\%$. When the outflow rate drops and accretion
rate grows (roughly $6t_{\rm SI}$ for RSI50Edd10 and $3t_{\rm SI}$ for
RSI100Edd10), the radiation efficiency tends to be lower and more stable at
around $4\%$. The efficiency drops because the increased accretion rate but
doesn’t commensurately increase the luminosity, which is dominated by stream
interactions. RSI20Edd10 provides a limit where the outflow depletes the gas
near black hole and halts circularization. Dominated by stream collisions, it
shows a consistently low the accretion rate in $12t_{\rm SI}$, but the highest
$\eta$ among the three runs at around $10\%$.

The lack of correlation between $\eta$ and accretion rate suggests that the
pre-peak emission is not directly powered by accretion at small radius. We
find that various shocks in the dynamical process are important sources for
pre-peak emission. For RSI20Edd10, shown in Figure 9, the radiation energy
density is enhanced by the stream-stream collision shock and the interaction
between subsequent outflow. The luminosity from RSI20Edd10 is also roughly
consistent with the comparable run in Huang et al. (2023) and Jiang et al.
(2016), which are local studies focusing on the stream-stream collision itself
without accretion.

In RSI50Edd10 and RSI100Edd10, prior to the circularization stage, the stream-
stream collision is the primary source for emission. After gas accumulates
near the black hole, the high radiation energy density regions shift from
stream-stream collision point to nearer the black hole, as seen in Figure 2.
Small spiral patterns develop near the ISCO, where the radiation energy
density increases locally at the shock front. Another prominent high radiation
energy density region traces the shock where the fallback stream is running
into the circularizing gas. If such “stream-disk” interaction persists, the
density contrast between the stream and circularizing gas might be an
important factor determining the emission. These shocks can be modulated by
the time-dependent mass fallback rate, and understanding their effect will be
important for future work.

###  4.2 Energy Dissipation in Post-shock Flow

In RSI50Edd10 and RSI100Edd10, gas accumulates near the black hole after the
outflow subsides, increasing the average radiation energy density between the
tidal radius $r_{\rm T}=r_{\rm p}$ and the stream-stream collision radius
$r_{\rm SI}$. Figure 6 shows a snapshot of eccentricity magnitude (Equation 8)
in the orbital plane, where $|\mathbf{e}|$ in regions between $r_{\rm p}$ and
$r_{\rm SI}$ is reduced compare to original ballistic eccentricity.

Figure 13: Eccentricity magnitude $|\mathbf{e}|$ (Equation 8) averaged in
$\theta$ and $\phi$ direction in RSI50Edd10. The gray dashed curve shows
$|\mathbf{e}|$ for the initial stream ballistic trajectory. For the general
Newtonian potential we adopt, $|\mathbf{e}|$ drops to $\approx 0.6$ towards to
the pericenter. The different colors of solid curves shows average
$|\mathbf{e}|$ in different times (since the first collision). The vertical
dotted lines labels three radius: the tidal radius $r_{\rm T}=r_{\rm p}$, the
stream-stream collision radius $r_{\rm SI}$ and the radius where we inject the
stream $r_{\rm inj}$.

We show $|\mathbf{e}|$ averaged in $\theta$ and $\phi$ direction in Figure 13.
The gray dashed line is the $\theta$ and $\phi$ direction averaged
eccentricity for the ballistic trajectory of injected stream. The eccentricity
is close to one at the injection radius. Due to the non-Newtonian potential,
it drops to roughly $0.6$ at the pericenter. The non-Newtonian potential also
slightly shifts the pericenter radius from the tidal radius even though we
assumed $\beta=1.0$. At $t=6$h, just after the stream-stream collision begins,
the stream is only perturbed at near $r_{\rm SI}$ so the majority of radius
still follows the eccentricity of the ballistic trajectory. The eccentricity
drops slightly in the strong outflow stage ($t=53$h), and only later in the
gas circularization stage ($t=129,~{}202$h) does the eccentricity
significantly decreases between $r_{\rm T}$ and $r_{\rm SI}$.

The eccentricity decrement suggest energy dissipation is associated with the
outflow launching and gas circularization. The radiation production in the
system provides a perspective to quantify the dissipation of total gas energy,
including kinetic energy, gravitational potential and the relatively small
internal energy. Recent work has investigated the “circularization efficiency”
in terms of the ratio between radiative loss rate and the orbital energy
required to circularize all the supplied gas around $2r_{\rm p}$ (Steinberg &
Stone, 2024; Ryu et al., 2023). The circularization efficiency presented in
early time ranges from $\lesssim 5\%$ to $\sim 30\%$ from previous works
depending on dynamical system configuration.

We compare the radiative loss in all the simulations and found contrasting
results for RSI50Edd10 and RSI100Edd10, which indicate a circularizing flow,
and RSI20Edd10, which lacks evidence for significant gas circularization. We
estimate the radiation loss $e_{\rm loss}(r)$ by normalizing the total
radiation energy $E_{\rm rad}$ to the total circularization energy $E_{\rm
circ}=GM_{\rm BH}M_{r}/2r$ within radius $r$:

$e_{\rm loss}(r)=\frac{E_{\rm rad}(<r)}{M(<r)(GM_{\rm BH}/2r)}$ (12)

Figure 14: The time evolution of normalized radiative loss $e_{\rm loss}$
(Equation 12) in RSI20Edd10 (upper panel) and RSI50Edd10 (lower panel). In
each panel, the time is in code units. The gray lines are $e_{\rm loss}$
evaluated at the tidal radius $r_{\rm T}=r_{\rm p}$, blue lines are at the
stream-stream collision radius $r_{\rm SI}$ and the red line is the average
value between $r_{\rm p}$ and $r_{\rm SI}$.

Figure 14 compares the radiative loss in RSI20Edd10 and RSI50Edd10. In
RSI20Edd10, each stream-stream collision enhances average radiative loss
between $r_{\rm p}=r_{\rm T}$ and $r_{\rm SI}$. The peaks at $r_{\rm SI}$
(gray line) are followed by peaks at $r_{\rm p}$ (blue line), corresponding to
the interacting outflow that collides at the orbital plane in between stream-
stream collisions. In the absence of stream-stream collision shock and outflow
interaction, $e_{\rm loss}$ is low, indicating that shocks are the primary
energy dissipation mechanisms in RSI20Edd10. The average $e_{\rm loss}$
between $r_{\rm p}$ and $r_{\rm SI}$ (red line) stays around $\sim 0.1$ during
the repeated collisions.

In RSI50Edd10, the first three stream-stream collisions lead to the first few
peaks of $e_{\rm loss}$ (gray and blue lines). Due to the larger $r_{\rm SI}$
than RSI20Edd10, the collision strength is reduced, yielding slightly lower
radiative loss at both $r_{\rm p}$ and $r_{\rm SI}$. After the the outflow
subsides and more gas is circularizing, the average $e_{\rm loss}$ between
$r_{\rm p}$ and $r_{\rm SI}$ (red lines) rises to $\sim 0.4$ by the time we
stop the simulation. The shocks in the circularizing flow, including the
spiral pattern and the “stream-disk” shock are the primary sources for
radiative loss.

The $e_{\rm loss}$ evolution in RSI100Edd10 is similar to RSI50Edd10 and also
increases during the gas circularization stage. Averaging over time, the
stream-stream collision is less efficient in dissipating energy compare to the
circularization shocks. When the outflow is too strong, there is little gas
accumulation near the black hole to interact with the high eccentricity
stream, so the radiative loss is low in between collisions. Once a moderate
amount of gas accumulates near the black hole, the circularization shocks can
increase $e_{\rm loss}$ rather rapidly. One potentially important caveat is is
that our purely hydrodynamics simulations neglect the potential role of
magnetic fields in the circularizing flow, which could enhance energy
dissipation and angular momentum transport.

###  4.3 Angular Momentum Redistribution

Figure 15: Histograms of Keplerian orbital period $t_{\rm orb}$ and
circularization radius $r_{\rm circ}$ for bound gas in RSI50Edd10, the color
shows probability for each bin weighted by mass. The upper panel is
$t=101.4$h, roughly at the end of strong outflow phase and corresponds to the
middle column of Figure 2. The lower panel is $t=225.7$h, when we stopped the
simulation and more gas is circularizing. The two gray dashed vertical lines
are the circularization orbit of original stellar at $2r_{\rm p}$ and the
stream-stream collision radius $r_{\rm SI}$. The horizontal dashed line is the
$t_{\rm orb}$ for a circular orbit at the stellar circularization radius.

For a given time snapshot, we can calculate the gas Keplerian orbital period
$t_{\rm orb}$ based on their specific angular momentum $h=r^{3}\psi/(r-1)$ and
orbital energy $e=(r^{2}\dot{r}^{2}/(r-1)^{2}+r^{3}\psi^{2}/(r-1))/2.0-GM_{\rm
BH}/r$ (Tejeda & Rosswog, 2013). We approximate gas circularization radius by
$r_{\rm circ}=h^{2}/GM_{\rm BH}$. The original stellar orbits has $t_{\rm
orb}\approx 250$ days and $r_{\rm circ}\approx 11r_{\rm s}$.

Figure 15 shows two snapshots of orbital period and circularization radius
distribution for all bound gas in strong outflow stage and gas circularization
stage. The concentration near $\log t_{\rm orb}\sim 2.4$ days, $\log r_{\rm
circ}\sim 2r_{\rm p}$ corresponds to gas in dense stream with original stellar
orbital periods and specific angular momentum.

The upper panel of Figure 15 corresponds to the time near the end of strong
outflow phase. The stream-stream collision redistributes gas specific angular
momentum, so that $r_{\rm circ}$ spans between $\approx 6r_{\rm s}-r_{\rm
SI}$. A majority of gas has $t_{\rm orb}\approx 10-100$ days, which is
significantly less than the original stellar $t_{\rm orb}\approx 250$ days,
but still much longer than the free fall time. The lower panel is when the
circularization dominates dissipation. Angular momentum is still redistributed
broadly, but as eccentricity drops, more gas acquires orbital properties that
are close to circular orbits.

In a simplified re-processing picture, the stream-stream collision unbinds
part of the marginally-bound stellar debris. The unbound gas propagates
outward and assembles the optically-thick reprocessing layer. The more-bound
gas will quickly circularize around the black hole near the circularization
radius on the free-fall timescale, and eventually be accreted on the viscous
timescale.

In contrast, the simulations indicate that the stream-stream collision
disperse gas angular momentum without efficiently damping the orbital
eccentricity. So that even if the gas is still bound, they can stay on orbits
with relatively high eccentricity and will not return to the black hole after
$t_{\rm orb}\sim 10^{1-2}$ days. We conclude that adopting a circularization
radius of $2r_{\rm p}$ and free fall timescale may underestimate the initial
disk size and the time it takes to circularize bound gas. Furthermore, this
bound gas can extend to large radius and will contribute to the reprocessing
layer. Hence estimates that relate the mass outflow rate derived from the
inferred optical depth of the unbound layer (e.g. $\tau\sim\kappa
R\dot{M}_{\rm out}/(4\pi R^{2}v_{\rm out})$) may be skewed by the presence of
bound gas on eccentric orbits.

###  4.4 Photosphere Properties

After the first stream-stream collision, photons are produced via shocks at
smaller radius $r\lesssim r_{\rm SI}$ and diffuse through the optically thick
surrounding medium. The relatively constant luminosity (Figure 7) implies that
the diffusion time is longer than the relevant dynamic timescales. The
photosphere appearance is determined by both the driving dynamics and the
radiation transfer processes.

Figure 16: Evolution of thermalization radius $R_{\rm th}$ (Equation 13, upper
panel) and average gas temperature (lower panel) for RSI20Edd10 (blue lines),
RSI50Edd10 (orange lines) and RSI100Edd10 (red lines). The prompt rise of
$R_{\rm th}$ and $\overline{T}_{\rm gas}$ is driven by stream-stream
collision.

We define a thermalization radius $R_{\rm th}$:

$\int_{R_{\rm th}}^{R_{\rm out}}\sqrt{\kappa_{\rm a}(\kappa_{\rm
s}+\kappa_{\rm a})}\rho dr=1$ (13)

where $\sqrt{\kappa_{\rm a}(\kappa_{\rm s}+\kappa_{\rm a})}$ is mean opacity
from scattering and absorption, $R_{\rm out}$ is the outer boundary radius,
where gas density is low and close to floor value. We assume all the lines-of-
sight are along radial direction. When $\kappa_{\rm s}\ll\kappa_{\rm a}$ this
corresponds to the photosphere. Figure 16 shows the evolution of average
$R_{\rm th}$ and the average gas temperature at $R_{\rm th}$. With this
definition of $R_{\rm th}$, before steam-stream collision, it traces the dense
injected stream and roughly follows the average radius of stream orbit. As the
initial stream-stream collision launches outflow, $R_{\rm th}$ promptly
increases by about an order of magnitude. Afterward, $R_{\rm th}$ continues
increasing, but with a slower rate and reaches $\sim 10^{14}$cm by the time we
stop the simulation. In RSI50Edd10 and RSI100Edd10, $R_{\rm th}$ is comparable
to $r_{\rm SI}$ because gas is primarily circularizing between $r_{\rm ISCO}$
and $r_{\rm SI}$. In RSI20Edd10, however, $R_{\rm th}$ primarily traces the
outflow moving around the black hole. The thermalization radius expands to
$R_{\rm th}\sim 2r_{\rm SI}$ with $R_{\rm th}\propto t^{0.4}$.

We also compute the time evolution of the average temperature at $R_{\rm th}$,
which peaks $\sim 10^{5}-10^{6}\rm K$ when the gas is heated by the first
stream-stream collision and there is very little gas surrounding the stream.
Once an outflow develops, the average temperature drops to around a few
$10^{4}$K as the average $R_{\rm th}$ moves outward, forming an optically-
thick region extends to $\sim 10^{2}r_{\rm s}$.

In RSI20Edd10, where the optically thick gas flows around the black hole, the
distribution of $R_{\rm th}$ and gas temrpature at $R_{\rm th}$ are relatively
isotropic with view angles. In RSI50Edd10 and RSI100Edd10, as the
circularizing flow forms, $R_{\rm th}$ and $\overline{T}_{\rm gas}$ show
greater variation along different lines-of-sight.

In Figure 17 and Figure 18, we show the radiation temperature $T_{\rm rad}$
and the $r_{\rm SI}$ distribution in for RSI20Edd10 and RSI50Edd10 together
with $R_{\rm th}$. At this time, RSI20Edd10 corresponds to the middle panel in
Figure 10, the stream is reforming and optical-thick gas fills the surrounding
region. RSI50Edd10 corresponds to the third column in Figure 2, gas are
circularizing near the black hole and flow eccentricity drops.

Figure 17: The “face-on” radiation temperature snapshot of RSI20Edd10 at
$t=36.1$h (the upper panel) and RSI50Edd10 at $t=240$h (the lower panel). The
temperature is volume averaged over $\Delta\theta\sim 5^{\circ}$ near
$\theta=\pi/2$-plane. The green solid line shows $R_{\rm th}$ along different
radial line-of-sight. Figure 18: The “side-view” radiation temperature
snapshot of RSI20Edd10 at $t=36.1$h (the upper panel) and RSI50Edd10 at
$t=240$h (the lower panel). The temperature is volume average from
$\Delta\phi\sim 5^{\circ}$ near $\phi_{\rm inj}$ plane. The green solid line
shows $R_{\rm th}$ along different radial line-of-sight.

The gas circularization in RSI50Edd10 increases the overall density and
radiation energy density near the black hole, yielding $T_{\rm rad}\sim
10^{6}K$, which is higher than RSI20Edd10 in the central region. Near $R_{\rm
th}$, $T_{\rm rad}$ is roughly $\sim 10^{4}$K in both runs in the orbital
plane. Perpendicular to the orbital plane, circularizing gas in RSI50Edd10
yields a disk-like geometry and the temperature at $R_{\rm th}$ varies with
viewing angle. In the polar region, gas density is lower and more optically
thin, $R_{\rm th}$ traces gas $\lesssim 50r_{\rm s}$ with temperature of $\sim
10^{5}$K. Near the orbital plane, the higher density circularizing gas extends
$R_{\rm th}$ to larger radius, giving $T_{\rm rad}\sim 10^{4}$K.

In RSI100Edd10 and RSI50Edd10, we find a similar anisotropy from circularizing
gas. The orbital plane is more optically thick than the polar region and
$R_{\rm th}$ is larger. The temperature variation along different radial
directions can range one to two orders of magnitude from $\sim 10^{4}$K to
$\sim 10^{6}$K. In comparison, the strong outflow in RSI20Edd10 disperses
optically thick gas more uniformly across latitudes and creates a more
isotropic photosphere. The anisotropic photosphere may imply potential viewing
angle effect on TDE appearance (Dai et al., 2018). But $R_{\rm th}$ primarily
considers the radially integrated directions, to understand the viewing angle
effect on observed spectrum, exploring photosphere properties along more
lines-of-sight is of interest for future work.

###  4.5 Implication to Pre-peak Multiband Emission

The estimated average temperature at $R_{\rm th}$ (Figure 16) will produce
optical to UV emission broadly consistent with observations if radiation is
nearly thermal. The temperature variation with lattitude, however, suggests
there might be different spectral energy distribution (SEDs) when varying
lines-of-sight. For example, in Figure 18, the residual outflow is optically
thick and $R_{\rm th}\gtrsim 100r_{\rm s}$ on the right hand side of the
snapshot. On the left hand side of Figure 18 (also the last column in Figure
2), gas density is particularly low in the mid-to-high altitude region, and
radiation temperature at $R_{\rm th}$ is $\sim 10^{5-6}$K, potentially
yielding UV to soft X-ray photons. These regions are likely to be optically
thin to scattering, and high energy photons will likely escape more easily.
The outflows can also advect out higher energy photons emitted where the flow
is hotter.

The flow geometry is reminiscent of the viewing angle model (Dai et al., 2018;
Thomsen et al., 2022), with the important difference that at this time, the
accretion flow has not formed a traditional “hot coronae” or a super-Eddington
disk. Instead, the high energy photons are primarily from shocks in the inner
circularization region. In such a flow geometry and velocity field, X-ray
emission may emerge in TDE systems before a classical geometrically and
optically thick super-Eddington disk forms.

Figure 8 suggests that $\sim 10-20\%$ of outflow launched during the first few
collisions is unbound. If we integrate unbound $L_{\rm KE}$ in Figure 8 for
times when $\dot{M}_{\rm out,ub}>0.1\dot{M}_{\rm Edd}$ and $L_{\rm
KE}>10^{42}\rm erg~{}s^{-1}$, the total unbound kinetic energy through the
spherical shell at $150r_{\rm s}$ is $E_{\rm KE}\approx 1.2\times 10^{49}\rm
erg$ in RSI50Edd10, and the total unbound mass is $M_{\rm ej,ub}\approx
1.1\times 10^{-3}M_{\odot}$. With similar estimation, we have total unbound
kinetic energy $E_{\rm KE}\approx 6.8\times 10^{48}\rm erg,~{}1.9\times
10^{49}\rm erg$, total unbound mass $M_{\rm ej,ub}\approx 7\times
10^{-4}M_{\odot},~{}1.7\times 10^{-3}M_{\odot}$ in RSI100Edd10 and RSI20Edd10.
If the unbound outflow can escape without significant energy dissipation,
their collision with circum-nuclear medium may be relevant to producing the
synchrotron emission at larger radius.

Goodwin et al. (2023) found that the early radio emission in AT2020vwl is
likely related to sub-relativistic outflow launched before the optical peak.
For AT2020vwl, the estimated black hole mass is smaller than the adopted
$M_{\rm BH}$ in this work. Considering for the emission radius $\sim
10^{16}\rm cm$ (unbound outflow travel distance), reducing $M_{\rm BH}$ but
keepin a similar $\beta=1$ will lead to a larger collision radius $r_{\rm
SI}\sim 10^{3}r_{\rm s}$ and weaker collision. Higher $\beta\approx 4$ can
give $r_{\rm SI}\sim 50r_{\rm s}$, which is similar to RSI50Edd10. The
different black hole mass and unknown $\beta$ introduce uncertainties whhen
extrapolating our simulation results. Nevertheless, estimating the unbound gas
properties from simulation near $\sim 10^{16}\rm cm$ provides an interesting
comparison.

Figure 19: Inferred unbound outflow travel distance from the black hole (the
upper row) and total velocity magnitude (the lower panel) in RSI50Edd10. We
randomly sample velocity and position from five points in Figure 4 to
represent gas parcels in the unbound outflow when they are just launched. Then
we integrate the ballistic trajectories to extrapolate their travel distances
and velocities, and infer the sub-relativistic outflow associated with
potential radio emission. The blue, orange, green, red and purple lines
corresponds to sample points with initial
$(r,~{}\theta)=(35.08,0.78),~{}(17.91,1.05),~{}(40.05,1.48),~{}(49.80,1.48),~{}(24.95,2.26)$
on the $\phi=1.66\pi$ plane in Figure 4.

Here we use RSI50Edd10 as an example to extrapolate unbound outflow properties
at $\sim 10^{16}\rm cm$, where the radio emission is estimated to be produced.
The total unbound kinetic energy is $\approx 10^{49}\rm erg$ at $150r_{\rm
s}$. To reach the emission radius $\sim 10^{16}\rm cm$, the outflow need to
travel to $\approx 4000r_{\rm s}$. The kinetic energy change between
$150r_{\rm s}$ and $\sim 4000r_{\rm s}$ is roughly within order of magnitude,
giving approximately consistent total kinetic energy.

To estimate the outflow velocity and travel distance beyond the simulation
domain, we randomly sample five points from unbound outflow in Figure 4, about
$12$ hours after they are launched by the second stream-stream collision in
RSI50Edd10. By integrating the ballistic trajectories using the sampled points
as initial condition, we show their distance to the black hole and velocities
in Figure 19. The sampled unbound outflow travels to $\sim 10^{16-17}\rm cm$
after 100-200 days since launched, with velocity $\sim 0.1c$. Comparing with
Goodwin et al. (2022), the selected unbound outflow in RSI50Edd10 seem to be
best-fitted by the assumption of electron energy fraction $\epsilon_{e}=0.1$,
with slight preference of conical geometry. The exact comparison requires a
lower black hole mass and time-dependent mass fallback rate. Unbound outflow
along different lines-of-sight might show distinct properties too. These
factors will be considered in the future work.

We can also apply a similar extrapolation based on ballistic trajectories to
bound outflow. The majority of outflow in RSI50Edd10 is bound and may return
to the black hole on the timescale of their orbital period. We also randomly
sample five points from bound gas on the same plane as the above unbound gas
snapshot, which is not shown in Figure 4. We found they will return to the
black hole on timescales of $10-100$ days, and can travel as far as $\sim
10^{15}\rm cm$. The timescale and distance is generally order-of-magnitude
consistent with Figure 15. If some of the earliest-launched outflow gas
survives traveling through the surrounding medium, it will return to the black
hole with velocity $\sim 0.1c-0.2c$ after $10-100$ day, and a super-Eddington
disk may already form from the circularizing gas. The interaction between the
returning outflow and the accretion disk or disk wind may also produce
multiband emission.

###  4.6 Comparison with Previous Works

The repeating interactions in RSI20Edd10 provide a limit of strong stream-
stream collision when $r_{\rm SI}$ is small and $\dot{M}$ is high. Dropping
$M_{\rm BH}$ in RSI50Edd10 and RSI100Edd10, $r_{\rm SI}$ moves away from the
black hole. The debris stream in both of simulations show multiple collisions
before settling to circularization. However, only the first few collisions
drive strong outflow before the returning stream disperses near pericenter. In
the limit of low black hole mass, $r_{\rm SI}$ is likely to be near the
apocenter instead of near the pericenter. In this scenario, the returning
stream might be more expanded compare to small $r_{\rm SI}$, and the apocenter
shock between the streams might be the primary shock dissipation mechanism
(Shiokawa et al., 2015; Piran et al., 2015; Krolik et al., 2016). With non-
zero spin and large penetration factor $\beta$, Andalman et al. (2022) also
finds multiple stream-stream collisions before accretion disk formation.
Understanding how different $r_{\rm SI}$ and debris stream orbits lead to
different stream-stream collision and their effects on outflow rate and
emission source will be interesting future works.

In the simulations, we found that various shocks are the leading pre-peak
emission mechanism instead of accretion, consistent with recent studies (Lu &
Bonnerot, 2020; Steinberg & Stone, 2024; Ryu et al., 2023). The simulations
presented in this paper does not include magnetic fields, which are essential
to excite magnetic rotational instability (MRI) (Balbus & Hawley, 1991). MRI
may dominate angular momentum transport and facilitate more efficient
accretion, and possibly lead to jet formation (Shiokawa et al., 2015; Andalman
et al., 2022; Curd et al., 2023). Our results are also not likely to be a good
approximation if the black hole spin is large. The Lense-Thirring precession
induced by spinning black hole can introduce vertical offsets between the
fallback and returning stream (Hayasaki et al., 2016; Jiang et al., 2016;
Batra et al., 2023; Jankovič et al., 2023) and delay stream self interaction
(Guillochon & Ramirez-Ruiz, 2015; Hayasaki et al., 2016). General relativistic
effects may also affect the post-collision circularization process and
accretion disk formation (Sadowski et al., 2016; Liptai et al., 2019; Curd,
2021, see e.g.).

The circularizing flow formed in RSI50Edd10 and RSI100Edd10 is moderately
eccentric. Lacking magnetic field, the truncated spiral shock near the black
hole ISCO is likely to be the primary angular momentum transport mechanism.
Similar spiral patterns are identified in previous work (Bonnerot & Stone,
2021; Curd, 2021; Ryu et al., 2023, see e.g.). The spiral shock is truncated
by the dense fallback stream in our simulations. If the circularizing flow
eventually reaches an average density that is comparable to the fallback
stream, the spiral shock may be able to extend to larger radius and further
facilitate accretion. Curd et al. (2023) studies the scenario where a debris
stream running into a magnetically dominated accretion disk. They find that
for a range of density contrast between the stream and disk, hydrodynamical
shocks is more important to the angular momentum transport than magnetic
stress. However, it seems no significant spiral structure forms globally over
the disk. Understanding the relative importance between the hydrodynamical
shock, magnetic stress and radiation viscosity will help understand the
angular momentum transport mechanism in TDEs accretion.

The circularizing flow shows noticeable eccentricity in the vertical
direction. Due to the interaction between thickened returning stream and
fallback stream, the side that is closer to stream-stream collision has larger
scale height ($H/R\sim\rm tan(\pi/4)$) than the side closer to pericenter
($H/R\sim\rm tan(\pi/8)$). Eccentric TDE disks are discussed by previous work
assuming inefficient circularization. The models qualitatively agrees on the
eccentric geometry of small (large) scale height at pericenter (apocenter),
with different primary dissipation mechanisms (Piran et al., 2015; Liu et al.,
2017; Zanazzi & Ogilvie, 2020). In our simulations, the different scale height
near the pericenter side and stream-stream collision side is more related to
the outflow instead of compression by nozzle shock. Understanding the thermal
emission from eccentric disk and their difference to near-Keplerian disk can
bring new insights to TDE emission. When including magnetic field,
understanding the competing effect of angular momentum transport and orbital
energy dissipation by MRI and stress tensors in eccentric TDE disk may also be
relevant to shape the long-term accretion physics (Chan et al., 2018, 2022)
and jet launching processes.

##  5 Conclusions

We find that stream-stream collision can delay gas circularization by
launching strong outflows and depleting gas near the black hole. For fallback
rates of $10\dot{M}_{\rm Edd}$, the instantaneous outflow rate can reach
$\dot{M}_{\rm out}\approx 5\dot{M}_{\rm Edd}-10\dot{M}_{\rm Edd}$ depending on
the black hole mass. The accretion rate onto the black hole rarely exceeds
Eddington value. The stream-stream collision is likely to happen multiple
times until the returning stream expands significantly after the pericenter
passage. The outflow subsides when the returning stream disperses near the
pericenter, and gas accumulation and circularization only starts after the
outflow rate drops.

For black hole masses $6-10M_{6}$ and mass fallback rate $10\dot{M}_{\rm
Edd}$, the pre-peak emission in TDEs is primarily powered by various shocks
instead of direct accretion. The initial stream-stream collision can produce
prompt emission $\sim 10^{44}\rm erg~{}s^{-1}$ (Figure 7). When the system in
in the strong outflow stage, the collision shock and shocks produced by
outflow interaction are the major emission sources. After the outflow rate
decreases and gas start to circularize, the accretion flow develops multiple
circularization shocks to power the emission, including the “stream-disk”
shock between the dense stream and circularizing gas (Section 4.1) and
truncated spiral shocks near the ISCO.

The radiation efficiency $L/\dot{M}c^{2}$ is time-dependent, with typical
values $>10\%$ during the strong outflow stage, and drops to $\sim 5\%$ in gas
circularization stage (Figure 12). The luminosity does not tightly follow the
accretion rate, especially when the outflow rate is high because the shocks,
not accretion, dominate emission. Despite the super-Eddington fallback rate,
the bolometric luminosity is sub-Eddington for the black hole mass.

The outflows produced by the stream-stream collisions are optically thick. As
material propagates away from the black hole, the average $R_{\rm th}$
increases to the order of $\sim 100r_{\rm s}$, and we estimate the photosphere
expands to the order of $\sim 10^{14}\rm cm$. The hot post-shock gas at
$<r_{\rm SI}$ has typical temperature $\sim 10^{5}-10^{6}$K, but the average
gas temperature near $R_{\rm th}$ drops to $\sim 10^{4}$K (Figure 16). The
photosphere can be highly aspherical, especially when circularization
concentrates more gas near the orbital plane. The photosphere size and
temperature may show orders of magnitude variation (Figure 17, Figure 18),
potentially leading to a strong viewing angle dependence for the SED even
before a super-Eddington disk forms.

When the fallback and returning stream are both thin and maintain sufficient
density contrast with surrounding gas, $\sim 1-2\dot{M}_{\rm Edd}$ of outflow
produced in stream-stream collision is unbound. The average radial velocity is
at the order of $0.1c$, slightly exceeds the local escape velocity (Figure 8).
The integrated kinetic energy through spherical shells $\gtrsim 150r_{\rm s}$
carried by the unbound gas reaches $\sim 10^{49}\rm erg$. For smaller
collision radius (larger black hole mass for the same $\beta$, or larger
$\beta$ for the same black hole mass) and higher fallback rate, the collisions
tend to drive higher unbound mass flux and kinetic luminosity. After the
strong outflow stage, the circularization stage rarely produces large amounts
of unbound gas in these simulation, but bound gas may still reach large radius
before falling back to the inner flow region.

When studying TDE systems, including bound-bound and bound-free opacity is
essential to capture the optical band emission. The classical Kramer’s law
type of free-free opacity can be a few orders of magnitude smaller than the
opacity functions including the atomic process (for example, OPAL and TOPs
opacity). During circularization stage, some gas in the post-shock flow can be
heated to $\sim 10^{6}$K close to the black hole. There might be directions
where gas density is low and therefore optically thin to these high energy
photons, for example, near the polar region and away from the stream-stream
collision site. These results suggest it is possible to produce soft X-ray
band emission prior to the optical peak.

We would like to thank Phil Chang and Maria Renee Meza for supportive
collaboration. We are grateful to stimulating discussions and helpful insights
from Kate Alexander, Ryan Chornock, Brenna Mockler, Ann Zalbludoff, Daichi
Tsuna, Wenbin Lu, Enrico Ramirez-Ruiz, Jane Lixin Dai, Clémont Bonnerot, Phil
Hopkins. XH is supported by the Sherman Fairchild Postdoctoral Fellowship at
the California Institute of Technology. This work used Stampede 2 at Texas
Advanced Computing Center through allocation AST150042 from the Advanced
Cyberinfrastructure Coordination Ecosystem: Services $\&$ Support (ACCESS)
program, which is supported by National Science Foundation grants 2138259,
2138286, 2138307, 2137603, and 2138296. Resources supporting this work were
also provided by the NASA High-End Computing (HEC) Program through the NASA
Advanced Supercomputing (NAS) Division at Ames Research Center. Support for
this work was provided by the National Aeronautics and Space Administration
under TCAN grant 80NSSC21K0496 and the National Science Foundation under grant
2307886. The Center for Computational Astrophysics at the Flatiron Institute
is supported by the Simons Foundation.

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##  Appendix A Stream Vertical Structure Near Pericenter

Figure 20 shows the vertical density profile of the streams for different
resolutions discussed in Section 3.3. Increasing resolution leads to more
compression and higher central density. At this time, the stream is expanding
vertically, with typical velocity $v_{\theta}\lesssim 0.01c$. We compare the
radiation pressure $P_{\rm rad,\theta\theta}$ and gas pressure $P_{\rm gas}$
and find the stream is radiation pressure dominated. The expansion velocity
from level five to seven does not follow a linear trend, suggesting there
might be unresolved physics. It is also likely that the stream evolves to
slightly different dynamical stages when changing resolution, resulting
different velocity profiles.

Figure 20: The vertical stream density (the first row), $\theta$ direction
velocity (the second row) and pressure (the third row) at $r=4.75r_{\rm s}$
(the gray circle in Figure 11), $t=26.2$h. The blue, orange and green lines
corresponds to level five, six and seven. In the third row, the solids lines
are the $\theta\theta$ component of radiation pressure tensor, the dashed
lines are gas pressure.

Compared to recent studies focusing on the pericenter compression (Bonnerot &
Lu, 2022; Bonnerot et al., 2022; Coughlin, 2023), the central density in
Figure 20 is few orders of magnitude lower. Bonnerot & Lu (2022) proposes that
the compression is near adiabatic until a shock forms near the orbital plane.
As the shock propagates outward, it quickly re-expands the stream to roughly
the width before pericenter passage. The stream optical depth is sufficently
large so that losses due to radiative diffusion might be negligible.

Figure 21: The vertical stream pressure (the upper panel) and estimated
timescales (the lower panel) at $r=4.75r_{\rm s}$ (the gray circle in Figure
11), $t=13.1$h for level five (blue lines) and level six (orange lines). In
the upper panel, the solid line are the $\theta\theta$ component of radiation
pressure tensor. The dotted lines are fitted curve that scales as
$\propto\rho^{4/3}$. In the lower panel, the solid lines are estimated
vertical diffusion timescale, and the dashed lines are estimated vertical
dynamical timescale.

Despite the different approaches, we find roughly consistent large optical
depth across the stream as in Bonnerot & Lu (2022). We estimate the vertical
dynamical time $t_{\rm diff}=H_{\rm z}/v_{\rm z}\approx
r\Delta\theta/v_{\theta}$, where $r\Delta\theta$ and $v_{\theta}$ are distance
to the orbital plane and expansion velocity in the $\theta$ direction
(positive corresponds to expansion). The vertical diffusion timescale can be
estimated as $t_{\rm diff}=\tau_{\rm z}H_{\rm
z}/c\approx\kappa\rho(r\Delta\theta)^{2}/c$, where $\kappa=(\kappa_{\rm
s}+\kappa_{\rm a})$ is the total opacity from scattering and absorption.
Comparing these two timescales, the radiation diffusion is relatively slow in
the most of region within the stream, except for central region with highest
density. The radiation pressure roughly follows the adiabatic prediction
$\propto\rho^{4/3}$ in the regions where the diffusion timescale is longer
than the dynamical timescale, except for the central region. The radiation
pressure is also enhanced compared to adiabatic assumption in the central
region, where the most of dissipation happens.

One of the differences with Bonnerot & Lu (2022) is that we do not capture the
shock launched near the orbital plane that rebounds the stream vertically.
Hence, the velocity near the mid-plane is potentially lower than the velocity
that would be driven by the rebounding shock, resulting in a slower dynamical
time. The difference could related to the lower resolution in our global
simulations. To understand the details of the role of advective and diffusive
radiation flux, a localized high resolution study is required in the future
work.

Steinberg & Stone (2024) present a series localized pericenter studies in
addition to their global simulation, finding that the typical global
resolution cannot resolve the pericenter shock, yielding an artificially large
stream scale height, but having limited effect on longer term global
evolution. However, including the heating injected by recombination can
significantly inflate the stream after pericenter passage, leading to a more
expanded stream. Coughlin (2023) also propose that the recombination may
increase stream entropy near the pericenter passage. Both works suggest that
the dissipation associated with recombination is unlikely to drive a
significant transient in early time. An earlier study Kasen & Ramirez-Ruiz
(2010) suggested that it may play an important role to generate extra
radiation during the nozzle shock. Understanding this effect and its interplay
with radiation is an interesting topic for future work, but would require more
localized simulations focused in the pericenter dynamics.

In this work, we fix the $\dot{M}=10\dot{M}_{\rm Edd}$ when studying the
resolution effect. For future work with a time-dependent $\dot{M}$, resolving
the pericenter optical depth might be more challenging. Since the optical
depth $\tau\propto\rho\propto\dot{M}$, more resolution is likely required to
avoid over-cooling when the $\dot{M}$ is sub Eddington at the earliest time
when fallback commences. More aggressive use of AMR may be needed to resolve
this in the future.