# Collisions of red giants in galactic nuclei

Taeho Ryu1,2, Pau Amaro Seoane3, 4, 5, 6, Andrew M. Taylor7, Sebastian T.
Ohlmann8,
1 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748
Garching, Germany
2 Physics and Astronomy Department, Johns Hopkins University, Baltimore, MD
21218, USA
3Universitat Politècnica de València, Spain
4Max Planck Institute for Extraterrestrial Physics, Garching, Germany
5Higgs Centre for Theoretical Physics, Edinburgh, UK
6Kavli Institute for Astronomy and Astrophysics, Beijing 100871, China
7DESY, Zeuthen, Germany
8Max Planck Computing and Data Facility, Garching, Germany E-mail: tryu@mpa-
garching.mpg.de

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In stellar-dense environments, stars can collide with each other. For
collisions close to a supermassive black hole (SMBH), the collisional kinetic
energy can be so large that the colliding stars can be completely destroyed,
potentially releasing an amount of energy comparable to that of a supernova.
Such violent collisions, which we call BH-driven disruptive collisions (BDCs),
have been examined mostly analytically, with the non-linear hydrodynamical
effects being left largely unstudied. Using the moving-mesh hydrodynamics code
AREPO, we investigate high-velocity ($>10^{3}$ km/s) collisions between 1M⊙
giants with varying radii, impact parameters, and initial approaching
velocities, and estimate their observables. Very strong shocks across the
collision surface efficiently convert $\gtrsim 10\%$ of the initial kinetic
energy into radiation energy. The outcome is a gas cloud expanding
supersonically, homologously, and quasi-spherically, generating a flare with a
peak luminosity $\simeq 10^{41}-10^{44}$ erg/s in the extreme UV band ($\simeq
10$ eV). The luminosity decreases approximately following a power-law
$t^{-0.7}$ initially, then $t^{-0.4}$ after $t\simeq$10 days at which point it
would be bright in the optical band ($\lesssim 1$eV). Subsequent, and possibly
even brighter, emission would be generated due to the accretion of the gas
cloud onto the nearby SMBH, possibly lasting up to multi-year timescales. This
inevitable BH-collision product interaction can contribute to the growth of
BHs at all mass scales, in particular, seed BHs at high redshifts.
Furthermore, the proximity of the events to the central BH makes them a
potential tool for probing the existence of dormant BHs, even very massive
ones which cannot be probed by tidal disruption events.

###### keywords:

††pubyear: 2022††pagerange: Collisions of red giants in galactic
nuclei–LABEL:lastpage

##  1 Introduction

Dynamical interactions between stars in stellar-dense environments, e.g.,
globular clusters and galactic centers, play a crucial role in driving the
evolution of the host and determining its thermodynamic state (Hut et al.,
1992). If the stellar density is sufficiently high, stars can collide with
relative velocities comparable to the dispersion velocity of the host. In
globular clusters, up to 40% of main-sequence stars in the core would undergo
a collision during the lifetime of the cluster (Hills & Day, 1976). For
clusters with very high number densities ($\gtrsim 10^{7}\;\mathrm{pc}^{-3}$),
a star may suffer multiple such collisions (Dale & Davies, 2006).

Galactic centers are extreme environments where stars are densely packed
(e.g., $10^{6}-10^{7}\;\mathrm{pc}^{-3}$ for nuclear clusters, Neumayer et al.
2020 and references therein) around a supermassive black hole (SMBH). Because
the relative velocity between stars near the SMBH is roughly the Keplerian
speed $\propto r^{-0.5}$, stars near the BH would collide at very high speeds
(e.g., $v_{\rm rel}\gtrsim 2000$km/s within $\simeq 0.1\;\mathrm{pc}$ around a
$10^{7}\;{\rm M}_{\odot}$ BH). If the kinetic energy of the collision
($\gtrsim 10^{50}$ erg for a collision between two stars with mass
$\;M_{\star}=1\;{\rm M}_{\odot}$ and $v_{\rm rel}\gtrsim 2000$ km/s) is
greater than the binding energy of the stars ($10^{48}-10^{49}$ erg for
$\;M_{\star}=1\;{\rm M}_{\odot}$), the stars would be completely destroyed,
leaving behind an expanding gas cloud. If even a small fraction of the
collisional kinetic energy is converted into radiation, the high-velocity
collision can generate a bright electromagnetic transient from the Galactic
nucleus region.

The total rates of such events between main-sequence stars have been estimated
to be $10^{-4}-10^{-5}$ yr-1 galaxy-1 (Rose et al., 2020; Amaro Seoane, 2023b;
Rose et al., 2023) if the core is fully relaxed to the Bahcall-Wolf power-law
$\propto r^{-7/4}$ (Bahcall & Wolf, 1976)111While the Bahcall-Wolf solution is
a mathematically correct solution when all stars have the same mass, in the
realitistic situation where the stellar mass distribution is inhomogeneous,
the slope can be steeper (Alexander & Hopman, 2009; Preto & Amaro-Seoane,
2010).. The rate for collisions between giants could be higher due to larger
cross-sections (Amaro Seoane, 2023b). However, if collisions continuously
deplete the inner part of the stellar-density cusp, the rate would become
smaller, e.g., $\simeq 10^{-5}-10^{-7}$ yr-1 galaxy-1 for main-sequence stars,
depending on the assumption of the stellar influx into the center (Balberg &
Yassur, 2023). Since these powerful collisions essentially destroy stars in
galactic center environments, these events can affect the frequency of other
types of nuclear transients. For example, Balberg & Yassur (2023) suggests
that high-velocity collisions can almost completely suppress extreme mass-
ratio inspirals.

High-velocity collisions between main-sequence stars (e.g., Benz & Hills,
1987, 1992; Lai et al., 1993; Rauch, 1999; Freitag & Benz, 2005) have been
studied using numerical simulations, focusing on the mass ejection and the
impact of such collisions on the thermodynamic state of the host, rather than
their observation signatures. Recently, Amaro Seoane (2023b) analytically
investigated the observables of high-velocity collisions between stars of
various types in galactic nuclei. They found that the peak luminosity of high-
velocity collisions can be as high as $10^{44}\eta_{\rm rad}$ erg/s. Here,
$\eta_{\rm rad}$ is one of the determining factors which measures how
efficiently the initial kinetic energy is converted into radiation energy. If
$\eta_{\rm rad}$ is of order unity, the peak luminosity can be comparable to
different types of nuclear transients, such as tidal disruption events.
However, $\eta_{\rm rad}$ in their work was left as a free parameter because
evaluating $\eta_{\rm rad}$ involves non-linear hydrodynamics effects such as
shocks, which cannot be done analytically.

In this paper, we investigate the hydrodynamics of high-velocity collisions,
or black hole-drive disruptive collisions (BDCs), between $1\;{\rm M}_{\odot}$
giants and numerically estimate the radiation conversion efficiency and their
observables, using the moving-mesh hydrodynamics code AREPO (Springel, 2010;
Weinberger et al., 2020; Pakmor et al., 2016). In the simulations, we consider
collisions with $v_{\rm rel}=10^{4}$ km/s between two identical $1\;{\rm
M}_{\odot}$ giants with four different radii ($R_{\star}=10\;{\rm R}_{\odot}$,
$20\;{\rm R}_{\odot}$, $50\;{\rm R}_{\odot}$, and $100\;{\rm R}_{\odot}$),
four impact parameters ($b=0.04\;R_{\star}$, $0.2\;R_{\star}$,
$0.4\;R_{\star}$, and $0.8\;R_{\star}$), and three initial approaching
velocity ($v_{\rm rel}=10^{4}$ km/s, $5\times 10^{3}$ km/s, and $2.5\times
10^{3}$ km/s). The largest approaching speed corresponds to roughly the
largest relative velocity for stellar collisions near the BH, i.e., the
Keplerian velocity at the smallest possible distance from the BH where at
least two stars exist for a typical stellar density around a massive BH
assuming the Bahcall-Wolf power law: $r\simeq 10^{-5}\;\mathrm{pc}$ for
$10^{5}\;{\rm M}_{\odot}$ BH, $\simeq 10^{-4}\;\mathrm{pc}$ for $10^{6}\;{\rm
M}_{\odot}$ BH, and $\simeq 10^{-3}\;\mathrm{pc}$ for $10^{7}\;{\rm
M}_{\odot}$ BH. Because collisions with lower relative velocities are expected
to create fainter transients, our simulations with the largest $v_{\rm rel}$
would provide an upper limit for the luminosity and total radiated energy of
these events.

This paper is organized as follows. We describe our methods in § 2, including
the code description (§ 2.1), stellar models (§ 2.2), and initial conditions
(§ 2.3). Then we present our results in § 3 and discuss astrophysical
implications for the collisions in § 4. Finally, we summarize and conclude in
§ 5.

##  2 Methods

###  2.1 Code

We perform a suite of 3D hydrodynamic simulations of BDCs between red giants
using the massively parallel gravity and magnetohydrodynamics moving-mesh code
AREPO (Springel, 2010; Pakmor et al., 2016; Weinberger et al., 2020). The code
inherits advantages of the two widely used hydrodynamical schemes, the
Lagrangian smoothed particle method and the Eulerian finite-volume method,
allowing for an accurate treatment of supersonic flows and shock capturing
without introducing an artificial viscosity, and low advection errors. We use
the ideal equation of states that takes into account radiation pressure
assuming local thermodynamic equilibrium,

$\displaystyle P=\frac{\rho k_{\rm B}T}{\mu m_{\rm
p}}+\frac{4\sigma}{3c}T^{4},$ (1)

where $P$ is the total pressure, $\rho$ the density, $k_{\rm B}$ the Boltzmann
constant, $T$ the temperature, $\mu=0.62$ the mean molecular weight, $m_{\rm
p}$ the proton mass, and $\sigma$ the Stefan-Boltzmann constant.

Figure 1: Evolution of a $1\;{\rm M}_{\odot}$ star in a Hertzsprung–Russell
diagram. The color bar shows the age of the star. The four star symbols mark
the four giant models adopted for collision experiments: (from smallest to
largest symbols) $\;R_{\star}\simeq 9$, 20, 50, and 100$\;{\rm R}_{\odot}$.

###  2.2 Stellar model

We adopt the internal structure of giants evolved using the 1D stellar
evolution code MESA (version r22.05.1 Paxton et al., 2011; Paxton et al.,
2013) to model giants in 3D. The star has an initial mass $\;M_{\star}=1\;{\rm
M}_{\odot}$ and a metallicity of $Z=0.02$. We treat the mixing processes and
winds following Choi et al. (2016). More specifically, we model convection
using the mixing length theory with a mixing length parameter of $1.81$. We
adopt the Ledoux (1947) criterion to determine the boundary of the convective
regions an exponential overshoot prescription (Herwig, 2000) with parameters
$f=0.016$ and $f_{0}=0.008$ at the top of the core and $f=0.0174$,
$f_{0}=0.0087$ at the bottom of the hydrogen-burning shell. Semiconvection is
treated following Langer et al. (1983) with an efficiency factor of 0.1. We
allow the star on the red giant branch to lose mass via wind following the
prescription from Reimers (1975) with scaling factor of $0.1$.

Figure 1 shows the evolution of the $1\;{\rm M}_{\odot}$ star in a
Hertzsprung–Russell diagram until it reaches the tip of the red-giant branch.
We take the giants at four different evolutionary stages where their radii are
$\;R_{\star}\simeq 10$, $20$, $50$, and $100\;{\rm R}_{\odot}$ (indicated by
the star symbols in the figure).

We construct 3D giants from the 1D giant models using the method developed in
Ohlmann et al. (2017) with $10^{6}$ cells. Modeling the entire giant with gas
cells is computationally expensive given very steep density gradients. So
instead, we model the inner part of the star with a point particle,
representing effectively the core. Furthermore, we place gas cells on top of
it such that the internal structure above the core matches with the MESA model
while the entire star stays in hydrostatic equilibrium. The point particle
interacts only gravitationally with gas: it only gravitationally pulls the
envelope which is cancelled by the pressure gradient of the gas when the star
is in isolation. We choose that the size of the region modelled using a point
particle is 5% of the stellar radius (“point particle radius”). The point
particle radius is in fact greater than the size of the core ($R\simeq
0.02\;{\rm R}_{\odot}$). This choice is justified by the fact that the mass of
the core is effectively the same as the enclosed mass within $\simeq
0.05\;R_{\star}$ (vertical dotted lines), as illustrated in Figure 2. This
means the total binding energy inside our 3D giants is essentially the same as
what we would have had when the point particle radius were exactly the core
radius. With this choice of the point radius, while we reduce computational
costs significantly, we lose only a small fraction of the total energy budget
inside the star.

We then relax the 3D stars fully in isolation, which usually takes 5 - 10
stellar dynamical times ($\sqrt{\;R_{\star}^{3}/G\;M_{\star}}$). Figure 3
shows the radial density of the fully relaxed stars above the point particle
(top panel) and their errors (bottom panel) relative to the MESA models. The
relative errors of the density of the inner part of the stars, where most of
the mass is concentrated, are less than a few %. Although the errors at the
surface are relatively large, the deviation of such small masses at the
surface, corresponding to the plateau at the end of each line in Figure 2,
should not affect our results.

We performed resolution tests for nearly head-on collisions between giants
with $\;R_{\star}=100\;{\rm R}_{\odot}$ with different resolutions. The choice
of the collision parameters are motivated by the fact that the impact of the
shock in such a collision is the strongest (see Figure 8), which requires the
highest resolution. We first constructed giants with $N=2.5\times 10^{5}$,
$5\times 10^{5}$, $10^{6}$, $2\times 10^{6}$ and $4\times 10^{6}$ cells and
performed the collision experiments. We find that the results have already
converged very well when $N\geq 10^{6}$: the conversion factor $\eta_{\rm
rad}$, defined in Equation 8, differs by less than 1%. In fact, the difference
in $\eta_{\rm rad}$ between $N\leq 5\times 10^{5}$ and $N=10^{6}$ is already
reasonably small, $\lesssim 20\%$ for $N=2.5\times 10^{5}$ and $\lesssim 10\%$
for $N=5\times 10^{5}$ relative to the case with $N\geq 10^{6}$. Furthermore,
we confirmed that the total energy is conserved within $\lesssim 1\%$ until
the end of the simulations.

Figure 2: Enclosed mass as a function of radius for the four giants with $\;R_{\star}=10\;{\rm R}_{\odot}$ (black), $20\;{\rm R}_{\odot}$ (red), $50\;{\rm R}_{\odot}$ (blue), $100\;{\rm R}_{\odot}$ (green). The vertical dotted lines, sharing the same color, indicate the size of the region modelled using a point particle. Although the point particle size is greater than the size of the core ($R\simeq 0.02\;{\rm R}_{\odot}$), given the flat mass-radius relation between the core radius and the point particle radius, we essentially retain the total energy budget inside the star above the core with significantly low computational costs. Figure 3: The radial density profile (top) of the giants with four different radii relaxed for five - ten stellar dynamical times and the relative error with respect to the MESA models (bottom), as a function of radius from the core. The dashed grey lines in the top panel show the density profiles of the MESA models. The density profiles of the 3D stars match well with the MESA models within a few $\%$ except for those at the stellar surface.  Model number | Mass | Radius | $v_{\rm rel}$ | Impact parameter $b$
---|---|---|---|---
- | $\;{\rm M}_{\odot}$ | $\;{\rm R}_{\odot}$ | 1000 km/s | $\;R_{\star}$
1 | 1 | 10 | 10 | 0.04
2 | 1 | 10 | 10 | 0.2
3 | 1 | 10 | 10 | 0.4
4 | 1 | 10 | 10 | 0.8
5 | 1 | 10 | 5 | 0.04
6 | 1 | 10 | 2.5 | 0.04
7 | 1 | 20 | 10 | 0.04
8 | 1 | 50 | 10 | 0.04
9 | 1 | 100 | 10 | 0.04
Table 1: Initial parameters: (from left to right) model number, stellar mass,
stellar radius, relative velocity $v_{\rm rel}$ at infinity, and impact
parameter $b$.

###  2.3 Initial conditions

We place two identical stars, initially separated by $10\;R_{\star}$, on a
hyperbolic orbit with some relative velocity at infinity $v_{\rm rel}$. So it
takes $10\;R_{\star}/v_{\rm rel}\simeq(0.1-1)$ days, depending on
$\;R_{\star}$ and $v_{\rm rel}$, until the two stars collide. We note that the
time is measured since collision in this paper: accordingly, the initial time
of the simulations is $t\simeq-(0.1-1)$ days. Those stars are embedded in a
low-density background medium with density of
$10^{-18}\;\mathrm{g}/\;\mathrm{cm}^{3}$ and temperature of $10^{4}$ K. The
background density is comparable to the density of the interstellar medium at
the Galactic center ranging between $10^{5}$ to $10^{6}$ particles
per$\;\mathrm{cm}^{3}$ (Gillessen et al., 2019) at Galactic center distances
that dominate the collision rate (see Amaro Seoane, 2023b). We discuss the
impact of the background density and temperature on the properties of
collision products in §4.1. Our fiducial model is the near-head-on collision
between the two $10\;{\rm R}_{\odot}$ giants initially approaching towards
each other at $v_{\rm rel}=10^{4}$km/s with an impact parameter
$b=0.04\;R_{\star}$. Here, $b=0.04\;R_{\star}$ is the smallest possible impact
parameter given the softening length of the point particle: in other words,
the gravity of the point particles becomes inaccurate at the closest approach
distance with $b<0.04\;R_{\star}$. For this giant, we additionally consider
off-axis collisions with larger impact parameters, $b=$0.2, 0.4, and
0.8$\;R_{\star}$, and two additional $v_{\rm rel}=2500$ and $5000$ km/s, to
study the dependence of the impact parameter and the collision velocity,
respectively. For larger giants, we only consider the near head-on collisions
with $v_{\rm rel}=10^{4}$ km/s. The initial parameters of the models are
summarized in Table 1.

Figure 4: Density $\rho$ (top), temperature $T$ (top-middle), mach number
$\mathcal{M}$ (bottom-middle), and speed $v$ (bottom) of gas in a nearly head-
on ($b=0.04R_{\star}$) collision between two giants with $\;R_{\star}=10\;{\rm
R}_{\odot}$ at four different times, $t=-0.07$ days (before collision), 0 days
(at collision), 1 days and 30 days (after collision). The red dots in each
panel indicate the location of the cores. The white contour line in the top
panel for $\rho$ show the location of the photosphere at which the radially
integrated optical depth $\simeq 1$ and those in the bottom-middle panel for
$\mathcal{M}$ the boundaries at $\mathcal{M}\simeq 1$. The arrows in the
bottom panels indicate the direction of gas motion. Initially the two stars
start to move towards each other with $v_{\rm rel}=10^{4}$ km/s (left). At
collision, very steep pressure gradients are built up at the collision surface
and strong shocks are created when the incoming gas collides with the pressure
barrier (left-middle). The gas bounces off and expands quasi-spherically and
homologously at supersonic speeds (right-middle  and right).

Figure 5: Spherically averaged density (top-left), temperature (top-right),
radial velocity (bottom-left) and optical depth (bottom-right), of the
expanding cloud produced in a collision between two $10\;{\rm R}_{\odot}$
giants, as a function of radius from the collision point at five different
times. The averages of the temperature and radial velocity are mass-weighted.
The optical depth is estimated by radially integrating $\kappa\rho{\rm d}r$
inwards from $20\times$ the distance of the core from the collision point. The
three vertical lines show the distance from the BH with mass $10^{6}$,
$10^{7}$, and $10^{8}\;{\rm M}_{\odot}$ at which the Keplerian velocity is the
same as the initial relative velocity. In other words, by the time the outer
edge reaches the distances, the gas would meet the central BH. The dots
indicate the locations of the cores at the five times, sharing the same colors
with the lines. The vertical bars indicate the location of the outer edge
where the expansion velocity is the maximum.

Figure 6: Average density $\overline{\rho}$ (top-left), mass-weighted average
of temperature $\overline{T}$ (top-right), the peak expansion speed $v^{\rm
r}_{\rm peak}$ (bottom-left) and the surface average of the optical depth
$\tau$ to the center(bottom-right) of the cloud in all models, as a function
of time since collision. The average density and temperature are estimated
using the cells within a radius containing $75\%$ of the cloud mass.  Figure
7: Same as Figure 4, but for off-axis collisions ($b=0.2$, $0.4$, and
$0.8\;R_{\star}$) between two giants with $\;R_{\star}=10\;{\rm R}_{\odot}$ at
$t\simeq 5$ days since collision.

##  3 Result

###  3.1 Overview

We provide an overview of the evolution of the collision product using our
fiducial model, e.g., head-on collision between the two $10\;{\rm R}_{\odot}$
giants. We present in Figure 4 (from top to bottom) the density $\rho$, the
temperature $T$, the Mach number $\mathcal{M}$, and the speed in the mid-plane
at four different times in our fiducial model.

Initially, the two stars approach at $v_{\rm rel}\simeq 10^{4}$ km/s ($1^{\rm
st}$ column). Since their first contact, the envelopes are continuously
compressed due to the converging motion. Along the contact surface (the
pronounced narrow feature across the center in the $2^{\rm nd}$ column, dubbed
“shock surface”), pressure gradients are built up and the temperature is
raised above $10^{7}$ K due to adiabatic compression. As the later incoming
gas collides supersonically with the pressure wall, shocks are created. Some
of the very hot gas in the shock surface escapes radially perpendicular to the
collision axis (or along the shock surface) with an opening angle of $\simeq
30^{\circ}$ and speeds of a few thousands km/s, which is not particularly high
compared to the rest. At the strongest compression, a significant fraction of
the kinetic energy is converted into heat energy ($\gtrsim 30\%$), which is
already a few orders of magnitude greater than the total binding energy of the
stars. When the pressure gradient exceeds the ram pressure, the compressed gas
bounces off and expands quasi-spherically and homologously at supersonic
speeds (see $3^{\rm rd}$ and $4^{\rm th}$ column panels in Figure 4). On top
of the expanding motion, the converted heat energy continuously drives the
outer part of the gas cloud to expand by the PdV work, meaning that some of
the heat energy is converted back into kinetic energy. At the same time, the
outer edge of the cloud supersonically collides with the background medium.
This has two effects. First, mass piles up at the boundary between the gas
cloud and the background medium, reducing the kinetic energy of the expansion
front. Second, shocks are created, which dissipates the kinetic energy of the
expansion front to heat energy. As a result of both effects, the expansion
front slows down.

###  3.2 Evolution of expanding cloud - parameter dependence

####  3.2.1 Fiducial case

To describe the evolution of the expanding gas more quantitatively we show in
Figure 5 the spherically-averaged density $\rho$ and (mass-weighted)
temperature $T$, the expansion speed $v^{\rm r}$, and the area-weighted
average of the optical depth $\tau$ over the solid angle for our fiducial
model as a function of distance from the collision point at five
logarithmically sampled times between $1$ and 30 days after collision. The
density $\rho$ (top-left) and the temperature $T$ (top-right) of the inner
regions of the expanding gas cloud are nearly constant. As the cloud expands
adiabatically, the overall level of $\rho$ and $T$ drops while maintaining its
slope: $\rho\simeq 10^{-8}\;\mathrm{g}/\;\mathrm{cm}^{3}$ at $t\simeq 1$ day
to $10^{-12}\;\mathrm{g}/\;\mathrm{cm}^{3}$ at $t\simeq 30$ days, and $T\simeq
2\times 10^{5}$ K at $t=1$ day to $5\times 10^{3}$ K at $t\simeq 30$ days, at
which point the cloud is cooler than the background medium. $\rho$ and $T$
outside the flat region decay towards the outer edge with a different
steepness: the density drops following a power law of $\propto r^{-\lambda}$
with $\lambda\simeq 12-13$ upon collision, gradually decreasing to
$\lambda\simeq 8$ at $t\simeq 30$ days. But the temperature decays rather like
$\propto r^{-1}$ at $1\lesssim t\lesssim 30$ days. The decaying slopes of
$\rho$ and $T$ depend on $\;R_{\star}$, $b$, and $v_{\rm rel}$, but the
dependence of the slope of $T$ is generally stronger. d$\ln\rho$/d$\ln r$ is
almost the same, independent of $\;R_{\star}$ whereas -d$\ln T$/d$\ln r$ tends
to be larger for larger $\;R_{\star}$ (e.g., $\lambda\simeq 2-3$ for
$\;R_{\star}=100\;{\rm R}_{\odot}$). d$\ln T$/d$\ln r$ is steeper for larger
$b$ (e.g., $\lambda\simeq 2-3$ for $b=0.8\;R_{\star}$), while d$\ln\rho$/d$\ln
r$ is only slightly less steeper for larger $b$ (e.g., $\lambda\simeq 12$ for
$b=0.8\;R_{\star}$). The dependence of the slopes on $v_{\rm rel}$ is
relatively weak: $\lambda$ for $\rho$ is almost same for $2500{\rm km/s}\leq
v_{\rm rel}\leq 10^{4}{\rm km/s}$ and $\lambda$ for $T$ is slightly larger for
smaller $v_{\rm rel}$ (e.g., $\lambda\simeq 1-1.5$ for $v_{\rm rel}=2500$
km/s).

As shown in the bottom-left panel of Figure 5, the cloud expands homologously,
i.e., $v^{\rm r}\propto r$ or constant $v^{\rm r}$ at the same mass
coordinate, which is also found in all other models. Right after the
collision, the maximum expansion velocity at the outer edge is greater than
the initial relative velocity by a factor of $\simeq 5$ and stays constant.
The period of time with a constant peak $v^{\rm r}$ is very brief for this
particular model ($\lesssim 0.1$ days). However the constant maximum $v^{\rm
r}$ phase is longer for collisions with larger $\;R_{\star}$, which is
illustrated in the bottom-right panel of Figure 6. After the constant maximum
$v^{\rm r}$ phase, the peak expansion velocity continuously decreases due to
the interactions with the background medium.

The gas cloud is initially optically thick. The optical depth to the center is
$\tau\gtrsim 10^{5}$ at $t\simeq 1$ day, as demonstrated in the bottom-right
panel of Figure 5. As it expands and cools, $\tau$ decreases following a
power-law of $t^{-7/3}$ (see the bottom-right panel of Figure 6), indicating
that the entire cloud will become optically thin within 7 - 8 months,
consistent with the analytic estimate by Amaro Seoane (2023b). The nearly flat
$\tau$ inside the cloud indicates that the transition from optically thick to
completely optically thin may be prompt.

####  3.2.2 Comparison between models

To further demonstrate the dependence of the stellar radius $R_{\star}$, the
impact parameter $b$ and the initial relative velocity $v_{\rm rel}$, we
compare in Figure 6 the evolution of the same four quantities, shown in Figure
5 between different models. For a proper comparison, we estimate
$\overline{\rho}$ as the average volume within a distance enclosing 75% of the
gas mass222Note that the radius enclosing 75% of the cloud mass corresponds to
the radius inside which $\rho$ and $T$ are constant, coinciding with the
distance of the cores from the collision point. and $\overline{T}$ as the
mass-weighted average of $T$ within the same volume. As shown in the top
panels, $\overline{\rho}$ and $\overline{T}$ decrease over time, following a
power-law of $t^{-3}$, and $t^{-1}$, respectively, almost independently of
$R_{\star}$ and $b$ except for $\overline{T}$ with $v_{\rm rel}=2.5\times
10^{3}$ km/s. The $t^{-3}$ power-law for $\overline{\rho}$ is expected from an
homologous expansion: $\rho\propto(v^{\rm r}t)^{-3}\propto t^{-3}$. As the
$t^{-1}$-scaling relation for $\overline{T}$ suggests, the total (radiation +
gas) internal energy at a given mass coordinate decreases like
$t^{-1}$333Total specific energy $=4\sigma T^{4}/[3c\rho]+k_{\rm B}T/[\mu
m_{\rm p}]\propto t^{-1}$ because $T\propto t^{-1}$ and $\rho\propto t^{-3}$..
The significant deviation from the $t^{-1}$ power-law for $v_{\rm rel}=2500$
km/s indicates that there is continuous energy exchange between gas at
different mass shells. Unlike other cases where the radiation energy is
dominant, in this case, the gas internal energy is comparable to the radiation
energy and the total internal energy drops like $\propto t^{-4/3}$, resulting
in a non-power law decay curve for $\overline{T}$. Although each of the two
quantities, $\overline{\rho}$ and $\overline{T}$, tends to follow a single
power-law, the degree to which their magnitudes depend on $R_{\star}$, $b$,
and $v_{\rm rel}$ is different. $\overline{\rho}$ has a very weak dependence
on $b$ and $\;R_{\star}$. $\overline{T}$ is insensitive to $b$ and weakly
depends on $\;R_{\star}$: only a factor of 1.5 greater for
$\;R_{\star}=100\;{\rm R}_{\odot}$ than that $\;R_{\star}=10\;{\rm
R}_{\odot}$.

$v^{\rm r}_{\rm peak}$ stays constant upon collision at $(3-6)\times v_{\rm
rel}$. The constant-$v^{\rm r}_{\rm peak}$ phase lasts longer for the case
involved with stronger shocks (e.g., larger $\;R_{\star}$ for given $b$ and
$v_{\rm rel}$). Eventually, $v^{\rm r}_{\rm peak}$ decreases over time because
of the interactions with the background medium, following a power-law of
$t^{-1/3}$ for all models. In particular, the peak expansion speed with
varying $\;R_{\star}$ tends to asymptote to a single value at later times. As
$b$ and $v_{\rm rel}$ decrease, $v^{\rm r}_{\rm peak}$ is smaller at a given
time. But the difference is at most by a factor of 3 for the collision
parameters considered.

As explained for our fiducial model above, the optical depth is initially high
at collision, $\tau>O(10^{6})$. The optical depth for most cases gradually
decreases as the gas cloud expands, following a power-law $t^{-7/3}$, which is
expected from the scaling relations of $\rho$ and $v^{\rm r}_{\rm peak}$:
$\tau\propto\rho R_{\rm peak}\propto t^{-3}t^{2/3}\propto t^{-7/3}$ where
$R_{\rm peak}$ is the location of the peak expansion speed $\simeq v^{\rm
r}_{\rm peak}t\propto t^{2/3}$. The deviation from the $t^{-7/3}$ power-law
relation becomes more significant as the collisions happen at lower $v_{\rm
rel}$ and higher $b$.

####  3.2.3 Fitting formulae

Combining all the scaling relations, we find that the average density
$\overline{\rho}(t)$, mass-weighted average of temperature $\overline{T}$ peak
expansion velocity $v^{\rm r}_{\rm peak}(t)$, size of the outer edge $R_{\rm
peak}(t)$ and radial expansion speed $v^{\rm r}_{\rm peak}(r,t)$ after $t>5$
days can be well-described by the following analytic expressions,

$\displaystyle\overline{\rho}(t)$ $\displaystyle=6\times 10^{-10}{\rm
g/cm}^{3}\left(\frac{t}{1{\rm day}}\right)^{-3}\left(\frac{v_{\rm
rel}}{10^{4}{\rm km/s}}\right)^{-3},$ (2) $\displaystyle\overline{T}(t)$
$\displaystyle=1.5\times 10^{5}K\left(\frac{t}{1{\rm
day}}\right)^{-1}\tan^{-1}(\sqrt{\;R_{\star}/10\;{\rm R}_{\odot}})$
$\displaystyle\hskip 14.45377pt\mathrm{for}~{}v_{\rm rel}\geq
5000~{}\mathrm{km/s}~{}\mathrm{and}~{}b\lesssim 0.4\;R_{\star},$ (3)
$\displaystyle\tau(t)$ $\displaystyle=2.5\times 10^{5}\left(\frac{t}{1{\rm
day}}\right)^{-7/3},$ $\displaystyle\hskip 14.45377pt\mathrm{for}~{}v_{\rm
rel}>5000~{}\mathrm{km/s}~{}\mathrm{and}~{}b\lesssim 0.4\;R_{\star},$ (4)
$\displaystyle v^{\rm r}_{\rm peak}(t)$ $\displaystyle=50000{\rm
km/s}\left(\frac{t}{1{\rm day}}\right)^{-1/3}\left(\frac{v_{\rm
rel}}{10^{4}{\rm
km/s}}\right)^{0.7}\left(\frac{b/R_{\star}+5}{5}\right)^{-4},$ (5)
$\displaystyle R_{\rm peak}(t)$ $\displaystyle=6.5\times
10^{14}\left(\frac{t}{1{\rm day}}\right)^{2/3}\left(\frac{v_{\rm
rel}}{10^{4}{\rm
km/s}}\right)^{0.7}\left(\frac{b/R_{\star}+5}{5}\right)^{-4},$ (6)
$\displaystyle v^{\rm r}(r,t)$ $\displaystyle=\begin{cases}v^{\rm r}_{\rm
peak}\left(\frac{r}{R_{\rm peak}}\right)\hskip 7.22743pt{\rm for}~{}r\leq
R_{\rm peak},\\\ 0\hskip 49.14307pt{\rm for}~{}r>R_{\rm peak},\end{cases}$ (7)

where the expression for $R_{\rm peak}$ is found by analytically integrating
$v^{\rm r}(r,t)$ over time. Note that $\overline{\rho}$ decays faster than
that expected from the expression $3M_{\rm gas}/(4\pi R_{\rm peak}^{3})\simeq
t^{-2}$ because $\overline{\rho}$ follows the homologous relation whereas the
peak expansion speed slows down so the outer edge expands slower than that
expected for homologous expansion.

Note that we do not include the term describing the dependence on
$\;R_{\star}$ in most of the expressions above because of their very weak
$\;R_{\star}$-dependence. On the other hand, the omission of the $v_{\rm
rel}$-dependence in Equation 3.2.3 for $\overline{T}$ is because of too small
number of models with varying $v_{\rm rel}$ for reliable fitting. Instead, we
have specified the range of $v_{\rm rel}$ where the equation is valid.

###  3.3 Stellar core

The cores move almost synchronously with the bulk of the gas. The orbit of the
cores are barely affected by the collision: they remain unbound after
collision and move away from each other at a speed almost same as the incoming
speed. The distances from the collision point in our fiducial model at five
different times are marked with circles in Figure 5.

The mass bound to the cores is larger for larger $v_{\rm rel}$ and smaller
$b$. But it is overall insignificant. For $b\leq 0.2\;{\rm R}_{\odot}$ and
$v_{\rm rel}\geq 5000$ km/s, the bound mass is less than $6\times
10^{-6}\;{\rm M}_{\odot}$. It is $\simeq 2\times 10^{-3}\;{\rm M}_{\odot}$ for
the model with $v_{\rm rel}=2500$ km/s and that with $b=0.4\;{\rm R}_{\odot}$
and $\simeq 3\times 10^{-2}\;{\rm M}_{\odot}$ for the model with $b=0.8\;{\rm
R}_{\odot}$.

###  3.4 Conversion factor

In this section, we investigate how much heat energy is created in collisions,
which is closely related to the amount of energy that can be radiated away and
potentially observed. We first define the conversion factor $\eta_{\rm rad}$
as the ratio of the total radiation energy to the initial kinetic energy,

$\displaystyle\eta_{\rm rad}(t)=\frac{\int
aT(t)^{4}dV}{\int\rho(t=0)v(t=0)^{2}dV},$ (8)

where $a$ is the radiation constant and $dV$ is the volume element of each
cell. Using $\eta_{\rm rad}$ one can estimate the total radiation energy as
$\simeq 0.25\eta_{\rm rad}\;M_{\star}v_{\rm rel}^{2}$ for equal-mass
collisions. To distinguish gas that initially belonged to the stars from the
background gas, we employ a selection condition using a passive scalar. The
passive scalar is an artificial scalar quantity initially assigned to each
cell which then evolves via advection without affecting the evolution of
hydrodynamics quantities. The initial values of the passive scalar of the
cells in the stars are one and that of the background cells is zero. So
depending on the mass exchange (or mixing) between the cells, the passive
scalar varies between zero (vacuum cells) and one (cells originally in the
stars). We perform the integration over cells with the passive scalar $\gtrsim
0.1$. The value of $\eta_{\rm rad}$ is largely unaffected by the choice of the
threshold of the passive scalar, provided that it is greater than 0.

Figure 8: The ratio of the radiation energy to the initial kinetic energy
$\eta_{\rm rad}$ as a function of time, measured since collision, for all our
models.

We show $\eta_{\rm rad}$ for all our models in Figure 8 before the radiation
energy in the optically thin gas becomes dominant. It is generally found that
$\eta_{\rm rad}$ dramatically increases at collision to $\eta_{\rm rad}\simeq
0.1-0.8$, meaning a significant fraction of the initial kinetic energy is
converted into heat energy. The maximum conversion factors are summarized in
Table 2. Then as the cloud expands and cools, $\eta_{\rm rad}$ decreases down
to $\lesssim 10^{-2}$. We see three clear post-peak trends of $\eta_{\rm
rad}$. First, $\eta_{\rm rad}$ is larger when larger stars collide.
Additionally, $\eta_{\rm rad}$ is approximately $\propto R_{\star}$ at any
given time: $(1-2)\simeq 10^{-3}$ for $\;R_{\star}=10\;{\rm R}_{\odot}$,
$\simeq(3-4)\times 10^{-3}$ for $\;R_{\star}=20\;{\rm R}_{\odot}$,
$\simeq\times 10^{-2}$ for $R_{\star}=50\;{\rm R}_{\odot}$, and $\simeq
2\times 10^{-2}$ for $\;R_{\star}=100\;{\rm R}_{\odot}$ at $t\simeq 3$ days.
We attribute this positive correlation between $\eta_{\rm rad}$ and
$R_{\star}$ to the fact that for the same relative velocity, larger (cooler)
stars collide at higher $\mathcal{M}$, resulting in stronger shocks over a
wider contact surface ($\propto R_{\star}$). Second, $\eta_{\rm rad}$ is
almost the same when $b\lesssim 0.2\;R_{\star}$, while $\eta_{\rm rad}$ begins
to decrease with $b$ when $b\gtrsim 0.2\;R_{\star}$. This trend is somewhat
expected given that as $b$ increases, the mass of gas that is shocked at
collision decreases. Lastly, $\eta_{\rm rad}$ decreases with $v_{\rm rel}$
because of lower $\mathcal{M}$ collisions for given sound speed (i.e., the
same star). $\eta_{\rm rad}$ at $v_{\rm rel}=5000$ km/s is almost the same as
that at $v_{\rm rel}=10000$ km/s, but $\eta_{\rm rad}$ at $v_{\rm rel}=2500$
km/s is lower by a factor of $\simeq 2$ than that for our fiducial case. The
overall levels of the conversion factors that we obtain are comparable to what
Amaro Seoane (2023b) imposed in order for their analytical model to match with
the observed object ZTF19acboexm (see their Figure 9).

We can also define the conversion factor for the ram pressure of gas moving at
supersonic speeds,

$\displaystyle\eta_{\rm
ram}(t)=\frac{\int\rho(t)v(t)^{2}dV}{\int\rho(t=0)v(t=0)^{2}dV},$ (9)

where the integration in the denominator is carried out over cells for which
the passive scalar $>0.1$, and that in the numerator the integration is
carried out only over cells with supersonic speeds, $\mathcal{M}\geq 1$. As
illustrated in the $3^{\rm rd}$ column panels of Figure 4 and 7, almost all
the gas is supersonically expanding. As a result, $1-\eta_{\rm
ram}\simeq\eta_{\rm rad}$.

Figure 9: Bolometric luminosity $L$ (top), blackbody temperature $T_{\rm BB}$
(middle), and blackbody radius $R_{\rm BB}$ (bottom), estimated for stellar
collisions using Equations 10 and 11. The dotted grey horizontal lines in the
bottom panel indicate the distances of the collision from the black hole with
$M_{\rm BH}=10^{6}\;{\rm M}_{\odot}$ ($\simeq 10^{14}\;\mathrm{cm}$) and
$10^{7}\;{\rm M}_{\odot}$ ($\simeq 10^{15}\;\mathrm{cm}$). The magenta guide
lines show the power-law that describes the quantity shown in the last two
panels.

###  3.5 Observables

We estimate the luminosity $L$, blackbody radius $R_{\rm BB}$, and temperature
$T_{\rm BB}$, using the radiation energy and the local cooling time $t_{\rm
cool}$. We first construct a spherical grid with an extremely small opening
polar angle ($\theta\simeq 10^{-10}$ radians) to avoid the singularity at the
poles, radially extending out to near the outer boundary of the domain. The
grid in the radial direction is logarithmically divided, i.e., constant
$\Delta r/r$ where $\Delta r$ is the cell size at $r$, while that in the
$\theta$ and $\phi$ directions are linearly divided, i.e., constant
$\Delta\theta$ and $\Delta\phi$. The number of grids in $r$, $\theta$, and
$\phi$ are (800, 600, 600), which we confirmed to give converging estimates
for the observables. We then identify the photosphere at which the optical
depth $\tau\simeq 1$. $\tau$ is integrated along each $r$-path with the
opacity found using an OPAL opacity table for Solar metallicity (Iglesias &
Rogers, 1996). The photospheric area is,

$\displaystyle A_{\rm BB}=\int_{0}^{2\pi}\int_{\simeq
0}^{\simeq\pi}r(\tau=1)^{2}\sin\theta drd\theta d\phi,$ (10)

which gives the effective size of the emitting region or blackbody radius
$R_{\rm BB}=(A_{\rm BB}/4\pi)^{1/2}$.

We attempt to bracket the range of realistic radiated luminosity from the
collision event by employing two different methods, each of which places
different weights on the contribution from the gas cloud layers (the inner
regions or outer regions near the photosphere) within the identified
photosphere. Our estimates should be accurate at an order-of-magnitude level.
However, for more accurate modeling of light curves, we will carry out
detailed non-equilibrium radiation transport calculations in future follow-up
work dedicated to estimating light curves and spectra.

In both methods, the total luminosity for each radial path is estimated by
summing the contributions from the cells with the local cooling time $t_{\rm
cool}$ shorter than the evolution time $t$ within the photosphere. Here,
$t_{\rm cool}$ is defined as $h_{\rho}\tau(1+u_{\rm gas}/u_{\rm rad})/c$ where
$h_{\rho}$ is the density moment scale height inside the photosphere and
$u_{\rm rad}$ ($u_{\rm gas}$) the radiation (gas thermal) energy. However, the
difference between the two methods is the assumption for how most of the
radiation energy is radiated away. In one method, we assume that the total
radiation energy within the photosphere is radiated away over a time
comparable to the cooling time at the base of the cloud. Under this
assumption, the inner regions tend to dominate the luminosity. We first
integrate the total radiation energy along the radial path and divide it by
the cooling time at the base of the cloud $t_{\rm cool,max}$, i.e., the
longest cooling time which is no longer than $t$, or

$\displaystyle L_{1}=\int_{0}^{2\pi}\int_{\simeq
0}^{\simeq\pi}\left[\int_{r(t_{\rm cool}=t)}^{r(\tau=1)}aT^{4}r^{2}\sin\theta
dr\right]t_{\rm cool,max}(\theta,\phi)^{-1}d\theta d\phi.$ (11)

In the second, we assume that the radiation energy of each cell is radiated
away over the local cooling time. So the total luminosity is estimated,

$\displaystyle L_{2}=\int_{0}^{2\pi}\int_{\simeq 0}^{\simeq\pi}\int_{r(t_{\rm
cool}=t)}^{r(\tau=1)}aT^{4}t_{\rm cool}(r,\theta,\phi)^{-1}r^{2}\sin\theta
drd\theta d\phi.$ (12)

In this method, the outer regions near the photosphere dominate the
luminosity. As stressed before, the evolution of the hydrodynamics quantities
for optically thin gas (i.e., outer region near the photosphere) in our
simulations is intrinsically less accurate than those for optically thick gas.
Hence, $L_{1}$ should be considered more consistent with our hydrodynamical
scheme. We find that the shapes of the $L_{1}$ and $L_{2}$ lightcurves are
very similar. However, $L_{1}$ is consistently smaller than $L_{2}$ by a
factor of $\simeq 10$. For this reason, we present $L_{1}$ and the resulting
blackbody temperature $T_{\rm BB,1}=(L_{1}/\sigma A_{\rm BB})^{1/4}$ in this
section and those from Equation 12 in Appendix A.

Figure 9 shows $L_{1}$ (top), $T_{\rm BB,1}$ (middle), and $R_{\rm BB}$
(middle) as a function of time measured since collision for all our models.
Note that the luminosity and the blackbody temperature differ depending on the
assumption of radiation (Equations 11 and 12), but $R_{\rm BB}$ is independent
of the assumption. The luminosity increases dramatically to its peak at
collision. The peak luminosity is $L_{1}\gtrsim 10^{41}-10^{43}$ erg/s
($L_{2}\gtrsim 10^{42}-10^{44}$), which is higher for larger $R_{\star}$ and
smaller $b$ and higher $v_{\rm rel}$, which has the same trend as $\eta_{\rm
rad}$. The temperature at peak is $T_{\rm BB,1}\simeq 10^{5}$ K. Because
$T_{\rm BB}\propto L^{1/4}$, $T_{\rm BB,2}$ is greater than $T_{\rm BB,1}$ by
less than a factor of 2. We summarize peak $L$ and $T_{\rm BB}$ at peak for
all our models in Table 2. Subsequently both $L$ and $T_{\rm BB}$, independent
of the assumption for the diffusion time (so both $L_{1}$ and $L_{2}$),
decrease following a power-law $\propto t^{-\xi}$ with $\xi$ slightly
differing at early and late times. $L$ at $t\lesssim 5$ days reveals a
decaying curve with $\xi\simeq 0.7-0.8$, followed by a slower decay with
$\xi\simeq 0.4$ at $t\gtrsim 5$ days. $L$ therefore decreases by a factor of
10 for the first 5 days. The decay in $L$ for the next 30 days is relatively
small, by only by a factor of a few. The change in $\xi$ for $T_{\rm BB}$ is
very mild: $\xi\simeq 0.6$ at $t\lesssim 5$ days and $\simeq 0.5$ at $t\gtrsim
5$ days. $T_{\rm BB}$ decreases from $\simeq(1-2)\times 10^{5}$ K at collision
to $10^{4}$ K at $5-15$ days, $(4-6)\times 10^{3}$ K at $30$ days. This means
the collision will be bright in extreme UV at collision which shifts to
optical on a time scale of a month. Lastly, $R_{\rm BB}$ increases to $\simeq
10^{15}$cm in 30 days, approximately following power-law growth of $\propto
t^{0.8}$.

The light curves from our simulations reveal some differences from that
analytically predicted by Amaro Seoane (2023b). Assuming a constant $\eta$
comparable to the minimum $\eta_{\rm rad}$ shown in Figure 8, their analytic
model predicts a peak luminosity consistent with the numerically integrated
peak luminosity shown in Figure 9. However, the luminosity from their analytic
model peaks at a few days after collision and subsequently decays faster. We
attribute these discrepancies to the difference in the way of calculating the
luminosity. In their analytic model, the luminosity was estimated under the
assumption that $\eta$ does not change over time and the total radiation
energy within the gas cloud is radiated away instantaneously on a time scale
comparable to the longest possible photon cooling time at any given time
(e.g., based on the optical depth to the center). On the other hand, in this
work, we take into account the time-dependent contributions (e.g., adiabatic
loss of energy due to expansion) of the cloud.

The observables estimated in this section are driven by stellar collisions.
But given the fact that these collisions occur near a SMBH, the expanding gas
cloud and the nearby BH would very likely interact, generating a possibly even
brighter flare, which we discuss in § 4.2.

Model number | $\;R_{\star}$ | $v_{\rm rel}$ | $b$ | $\eta_{\rm peak}$ | $L_{\rm peak,1}$ | $L_{\rm peak,2}$ | $T_{\rm BB,peak,1}$ | $T_{\rm BB,peak,2}$
---|---|---|---|---|---|---|---|---
- | $\;{\rm R}_{\odot}$ |  $10^{3}$ km/s | $\;R_{\star}$ | - |  $10^{43}$erg/s |  $10^{43}$erg/s |  $10^{5}$K |  $10^{5}$K
1 | 10 | 10 | 0.04 | 0.68 | 9.6 | 1.3 | 3.0 | 2.1
2 | 10 | 10 | 0.2 | 0.58 | 9.3 | 1.3 | 2.9 | 2.0
3 | 10 | 10 | 0.4 | 0.40 | 6.5 | 1.0 | 2.7 | 1.9
4 | 10 | 10 | 0.8 | 0.13 | 4.2 | 0.7 | 2.7 | 1.7
5 | 10 | 5 | 0.04 | 0.68 | 2.1 | 0.3 | 2.2 | 1.4
6 | 10 | 2.5 | 0.04 | 0.54 | 0.5 | 0.1 | 1.0 | 1.0
7 | 20 | 10 | 0.04 | 0.75 | 13 | 2.0 | 2.7 | 1.7
8 | 50 | 10 | 0.04 | 0.75 | 26 | 3.4 | 2.2 | 1.4
9 | 100 | 10 | 0.04 | 0.72 | 26 | 3.4 | 1.7 | 1.1
Table 2: Peak conversion factor $\eta$, luminosity at peak $L_{\rm peak}$ and
blackbody temperature at peak $T_{\rm BB,peak}$ for each model, using Equation
11 ($L_{\rm peak,1}$ and $T_{\rm BB,peak,1}$) and Equation 12 ($L_{\rm
peak,2}$ and $T_{\rm BB,peak,2}$)

##  4 Discussion

###  4.1 Interaction of gas cloud with interstellar medium

In this work, we simulated BDCs of giants surrounded by a medium with a
constant density of $10^{-18}$g/cm3 and temperature of $10^{4}$ K. As the
cloud expands, it collides inelastically with the background medium, which
results in the continuous decrease in the kinetic energy of the expansion
front. In addition, the collision between the outer edge of the cloud and the
background medium can create shocks, converting the kinetic energy into heat
energy. The net effect is the deceleration of the gas cloud, deviating from an
homologous behavior, which is also found from our simulations where the
velocity of the outer edge decreases following $t^{-1/3}$. This impact of the
surrounding medium would be faster if the colliding stars were initially
embedded in a denser medium. For example, the rising slope of $\eta_{\rm rad}$
would be less steep for the case with lower-density background gas. Given the
supersonic motion of the cloud, how the cloud expands would not be
significantly affected by the temperature of the background medium for a given
background density. However the evolution of $\eta_{\rm rad}$ would be changed
depending on the background temperature. In fact, we performed extra
simulations with different background temperatures ($100-5000$ K), showing
that while the expansion properties of the cloud (e.g., $\overline{\rho}$,
$\overline{T}$, and $v_{\rm peak}^{\rm r}$) are almost independent of the
background temperature, $\eta_{\rm rad}$ tends to be lower at the local
minimum and increases more slowly afterward for a lower background
temperature.

Although the deviation from an homologous expansion was only found near the
outer edge for the duration of our simulations, as an order-of-magnitude
estimate, the motion of the entire gas cloud would become completely deviated
from an homologous expansion when the swept-up mass is comparable to the mass
of the cloud,

$\displaystyle t_{\rm non-homologous}$ $\displaystyle\simeq\frac{3M_{\rm
gas}}{4\pi\rho_{\rm ISM}(v^{\rm r})^{3}},$ $\displaystyle\simeq 1100~{}{\rm
days}\left(\frac{M_{\rm gas}}{2\;{\rm M}_{\odot}}\right)\left(\frac{\rho_{\rm
ISM}}{10^{-18}\;\mathrm{g}/\;\mathrm{cm}^{3}}\right)^{-1}\left(\frac{v^{\rm
r}}{10^{4}{\rm km/s}}\right),$ (13)

where $v^{\rm r}$ is the expansion speed and $\rho_{\rm ISM}$ the density of
the background medium. At the same time, this also means that the scaling
relations for the homologous expansion found from our simulations would be
applied to the evolution of the homologously expanding part of the collision
product, independent of the existence of the background medium.

###  4.2 Interaction of gas cloud with supermassive black hole

In addition to the burst caused by the stellar collision (see § 3.5), there
would be a subsequent burst due to accretion onto the nearby SMBH. As a
result, the overall shape of the luminosity would be that the stellar
collision creates the first peak with $L\gtrsim 10^{42}$ erg/s which decays,
followed by a sharp rise to Eddington due to accretion onto the BH, possibly
remaining at that level for up to years until the captured gas is accreted
onto the BH. We will examine the observables from the BH-cloud interaction by
considering two cases, 1) Case 1. no-decelerating expansion (§4.2.1) and 2)
Case 2. decelerating expansion (§4.2.2). And we will discuss the astrophysical
implications for BHs in §4.2.3.

####  4.2.1 Case 1. no-decelerating expansion

We first assume that the entire gas cloud expands homologously and the
expansion speed of the outer edge is $v^{\rm r}\simeq\psi v_{\rm rel}$ with
$\psi\simeq 3-6$ (see Figure 6). The gas cloud starts to interact with the BH
when the size of the expanding gas cloud becomes comparable to the distance to
the BH $R_{\rm BB}$ for given $v_{\rm rel}$,

$\displaystyle R_{\rm BH}\simeq\frac{GM_{\bullet}}{v_{\rm
rel}^{2}}=10^{15}~{}{\rm cm}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{-2}.$
(14)

The time difference between the first collision-driven burst and the
subsequent accretion-driven burst would be set by the time $\tau_{\rm BH}$ at
which the outer edge of the cloud reaches the BH, $R_{\rm BH}-R_{\rm
Sch}\simeq R_{\rm BH}\simeq R_{\rm peak}$, where $R_{\rm Sch}$ is the
Schwarzschild radius,

$\displaystyle\tau_{\rm BH}\simeq\frac{R_{\rm BH}}{v^{\rm r}}\simeq 3~{}{\rm
days}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{-3}.$
(15)

To zeroth order, the part of the cloud that is within the Bondi radius $R_{\rm
Bondi}\simeq 2GM_{\bullet}/(v^{\rm r})^{2}$ from the BH would be
gravitationally captured by the BH and subsequently accreted onto the BH.
Assuming a Bondi–Hoyle accretion (Bondi & Hoyle, 1944; Bondi, 1952), the
luminosity $L_{\rm Bondi}$ with radiative efficiency $\epsilon$ can be
estimated,

$\displaystyle L_{\rm Bondi}\simeq\frac{4\pi\epsilon G^{2}M_{\bullet}^{2}\rho
c^{2}}{(v^{\rm r})^{3}},$ $\displaystyle\simeq 3\times 10^{47}~{}{\rm
erg/s}\left(\frac{\epsilon}{0.1}\right)\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-1}\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{3},$
(16)

which is super-Eddington for $M_{\bullet}<3\times 10^{8}\;{\rm
M}_{\odot}(v_{\rm rel}/10^{4}{\rm km~{}s}^{-1})^{1.5}$. Note that $L_{\rm
Bondi}$ has no dependence on $t$ given the scaling relations for $\rho$
($\propto t^{-3}$, Equation 2) and $v^{\rm r}(r=R_{\rm BH})$ ($\propto
t^{-1}$, Equation 7): $L_{\rm Bondi}\propto\rho(v^{\rm r})^{-3}\propto t^{0}$.
Super-Eddington accretion may be possible if the gas is optically thick and
the trapping radius $R_{\rm tr}=(L_{\rm Bondi}/L_{\rm
Edd})(GM_{\bullet}/\epsilon c^{2})$ is smaller than the Bondi radius
(Begelman, 1979). The ratio of the two radii is,

$\displaystyle\frac{R_{\rm tr}}{R_{\rm Bondi}}\simeq 300\left(\frac{t}{1{\rm
day}}\right)^{-2}\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{-1},$ (17)

suggesting super-Eddington accretion would be possible at $t\gtrsim 20~{}{\rm
days}(v_{\rm rel}/10^{4}{\rm kms}^{-1})^{-0.5}$. Here, we caution that the
time ratio in Equation 17 is estimated under the assumption that the global
accretion flow is not affected by any accretion feedback, which is highly
uncertain. Assuming a black body, the temperature at the Bondi radius if
$L\simeq L_{\rm Edd}$ is,

$\displaystyle T_{\rm Bondi}(t)\simeq 10^{5}K\left(\frac{t}{3{\rm
day}}\right)^{-1}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{3/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{-1/2},$ (18)

and at the onset of the BH-gas interaction (or $t\simeq\tau_{\rm BH}$),

$\displaystyle T_{\rm Bondi}(t=\tau_{\rm BH})\simeq
10^{5}K\left(\frac{\psi}{5}\right)\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-1/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right).$
(19)

If $L\simeq L_{\rm Bondi}$,

$\displaystyle T_{\rm Bondi}$ $\displaystyle\simeq 2.4\times
10^{4}K\left(\frac{t}{150{\rm
day}}\right)^{-1}\left(\frac{\epsilon}{0.1}\right)^{1/4}$
$\displaystyle\times\left(\frac{M_{\bullet}}{5\times 10^{8}\;{\rm
M}_{\odot}}\right)^{1/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{-5/4},$ (20)

and at $t=\tau_{\rm BH}$,

$\displaystyle T_{\rm Bondi}(t=\tau_{\rm BH})$ $\displaystyle\simeq 2.4\times
10^{4}K\left(\frac{\psi}{5}\right)\left(\frac{\epsilon}{0.1}\right)^{1/4}$
$\displaystyle\times\left(\frac{M_{\bullet}}{5\times 10^{8}\;{\rm
M}_{\odot}}\right)^{-3/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{-7/4}.$ (21)

Because $R_{\rm Bondi}$ increases faster than $R_{\rm peak}$,

$\displaystyle\frac{R_{\rm Bondi}}{R_{\rm peak}}\propto t,$ (22)

as the most optimistic case, the entire gas cloud could be ultimately captured
by the BH in a time $\tau_{\rm capture}$ at which $R_{\rm Bondi}\simeq R_{\rm
peak}$,

$\displaystyle\tau_{\rm capture}\simeq 40~{}{\rm
days}\left(\frac{\psi}{5}\right)\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{-3}.$
(23)

Then the maximum duration of the Eddington luminosity may be set by,

$\displaystyle\tau_{\rm acc}\lesssim\frac{M_{\rm gas}\epsilon c^{2}}{L_{\rm
Edd}}\simeq 9~{}{\rm years}\left(\frac{\epsilon}{0.1}\right)\left(\frac{M_{\rm
gas}}{2\;{\rm M}_{\odot}}\right)\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-1},$ (24)

where $M_{\rm gas}$ is the mass of the gas cloud, i.e., total mass of the two
collided stars. Here, we assumed that the entire gas would be accreted onto
the BH. However, radiation pressure from super-Eddington accretion would be
strong enough to generate outflow. For such a case, only a fraction of the gas
cloud would end up accreting and $\tau_{\rm acc}$ would be shorter than
estimated above.

Figure 10: Time $\tau_{\rm BH}$ to the accretion-driven burst since the peak
collision-driven luminosity for different collision velocities $v_{\rm rel}$,
as a function of black hole mass. The lines illustrate Case 2. decelerating
expansion (§ 4.2.1) where the entire cloud expands homologously up to 0.5 days
since collision (dashed), then the outer edge starts decays like $t^{-1/3}$
(solid) due to interactions with a background medium, using Equation 7. The
less steep diagonal bars demarcate the range of $\tau_{\rm BH}$ for the case
where the gas cloud continuously expands homologously with the outer edge
moving at $(3-6)\times v_{\rm rel}$ (Case 1. no-decelerating expansion (§
4.2.2), corresponding to the peak expansion speed upon collision in our
simulations (see the bottom-right panel of Figure 6).  Figure 11: Bondi
luminosity due to free-fall accretion of the collision product onto a black
hole for different collision velocities $v_{\rm rel}$, as a function of black
hole mass. The solid line is for the decelerating peak expansion speed (Case
2. decelerating expansion, § 4.2.1) and dashed lines for the non-decelerating
expansion speed (Case 1. no-decelerating expansion (§ 4.2.2). The grey dashed
diagonal line indicates the Eddington luminosity.

####  4.2.2 Case 2. decelerating expansion

Now we examine the observables from interactions between decelerating
expanding cloud with $v^{\rm r}_{\rm peak}\propto t^{-1/3}$ and the SMBH,
using Equations 2–7. For this case, $\tau_{\rm BH}$ has a different dependence
on $M_{\bullet}$ and $v_{\rm rel}$,

$\displaystyle\tau_{\rm BH}\simeq 3~{}{\rm
days}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{3/2}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{-3}\left(\frac{b/R_{\star}+5}{5}\right)^{6},$ (25)

We show in Figure 10 the range of $\tau_{\rm BH}$ for three different
collision velocities $v_{\rm rel}$ as a function of $M_{\bullet}$ assuming a
non-decelerating expansion speed (thick diagonal bars, $\psi=3-6$) and a
decelerating expansion speed (solid lines). The interaction onset time would
be longer generally if the expansion of the cloud slows down. Depending on
$M_{\rm BH}$ and $v_{\rm rel}$, the second burst could happen over a wide
range of time. For example, if a collision with $v_{\rm rel}\gtrsim 2500$ km/s
occurs in the Galactic center (with $M_{\bullet}\simeq 4\times 10^{6}\;{\rm
M}_{\odot}$ GRAVITY Collaboration et al., 2019), the second accretion-driven
burst would occur after the collision in less than a day to $6-7$ months
depending on the location of the collision from the BH. For very massive black
holes ($M_{\rm BH}>10^{8}\;{\rm M}_{\odot}$), $\tau_{\rm BH}$ can be more than
tens of years.

The Bonding luminosity is still independent of $t$ and has the same
$M_{\bullet}-$ and $v_{\rm rel}-$dependence as the case with the no-
decelerating expansion, but it is roughly a factor of 3 greater at given
$M_{\rm BH}$ and $v_{\rm rel}$,

$\displaystyle L_{\rm Bondi}\simeq 10^{48}~{}{\rm
erg/s}\left(\frac{\epsilon}{0.1}\right)\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-1}\left(\frac{v_{\rm rel}}{10^{4}{\rm km/s}}\right)^{3},$
(26)

which is further illustrated in Figure 11. While the expression for the
blackbody temperature at the Bondi radius has the same dependence on
$M_{\bullet}$ and $v_{\rm vel}$ as Equations 18 and 4.2.1, because of the
different expression for $\tau_{\rm BH}$, $T_{\rm Bondi}(t=\tau_{\rm BH})$ is
written differently,

$\displaystyle T_{\rm Bondi}(t=\tau_{\rm BH})\simeq$
$\displaystyle\begin{cases}8\times
10^{4}K\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-3/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{2.1}\left(\frac{b/R_{\star}+5}{5}\right)^{-6},\\\ \hskip
180.67499pt\textrm{for $L=L_{\rm Edd}$},\\\ 6\times
10^{3}K\left(\frac{\epsilon}{0.1}\right)^{1/4}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{-5/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{2.8}\left(\frac{b/R_{\star}+5}{5}\right)^{-6},\\\ \hskip
180.67499pt\textrm{for $L=L_{\rm Bondi}$}.\\\ \end{cases}$ (27)

We compare $T_{\rm Bondi}$ at the onset of the accretion-drive burst (so
$T_{\rm Bondi}$ at $t=\tau_{\rm BH}$) in Figure 12 between the non-
decelerating expansion case (thick bars) and the decelerating expansion case
(lines). For low-mass black holes, $T_{\rm Bondi}$ is quite similar, e.g.,
$10^{5}$ K for $M_{\bullet}=10^{5}-10^{6}\;{\rm M}_{\odot}$. However, because
of a steeper decline for the decelerating expansion case ($T_{\rm
Bondi}\propto M_{\bullet}^{-3/4}-M_{\bullet}^{-5/4}$, Equation 4.2.2) than for
the no-decelerating expansion case ($T_{\rm Bondi}\propto
M_{\bullet}^{-1/4}-M_{\bullet}^{-3/4}$, Equations 19 and 4.2.1), $T_{\rm
Bondi}$ for the decelerating expansion case is generally lower for high-mass
BHs: for $M_{\bullet}=10^{9}\;{\rm M}_{\odot}$, $T_{\rm Bondi}\simeq
10-10^{3}$ K for the decelerating expansion case whereas $T_{\rm Bondi}\simeq
10^{3}-10^{4}$ K for the no-decelerating expansion case.

For the decelerating expansion case, the Bondi radius increases faster,

$\displaystyle\frac{R_{\rm Bondi}}{R_{\rm peak}}\propto t^{4/3},$ (28)

which leads to a smaller $\tau_{\rm capture}$,

$\displaystyle\tau_{\rm capture}\simeq 11~{}{\rm
days}\left(\frac{M_{\bullet}}{10^{7}\;{\rm
M}_{\odot}}\right)^{3/4}\left(\frac{v_{\rm rel}}{10^{4}{\rm
km/s}}\right)^{-2.5}\left(\frac{b/R_{\star}+5}{5}\right)^{-3}.$ (29)

The duration of the accretion process would be the same as Equation 24.

Figure 12: Temperature from the Eddington-limited Bondi-luminosity at the
onset of the accretion of the collision product onto a black hole for
different collision velocities $v_{\rm rel}$, as a function of black hole
mass. As before in Figure 10, the lines show the case where the cloud
undergoes a non-decelering expansion up to 0.5 days, followed by an
deceleration of the outer edge like $t^{-1/3}$ at 0.5 days since collision due
to interactions with a background medium, using Equations 2-7. The diagonal
bars indicate the range of $\tau_{\rm BH}$ when the entire gas cloud expands
without being decelerated with the peak expansion speed of $(3-6)\times v_{\rm
rel}$. The power-laws are analytically derived in Equations 18 ($\propto
M_{\bullet}^{-1/4}$) and 4.2.2 ($\propto M_{\bullet}^{-3/4}$ and $\propto
M_{\bullet}^{-5/4}$).  Figure 13: Parameter space for disruptive events of a
giant with $\;M_{\star}=1\;{\rm M}_{\odot}$ and $\;R_{\star}=10\;{\rm
R}_{\odot}$ in terms of the distance from the BH for varying BH masses. The
region dubbed “Black hole“ is defined by the Schwarzschild radius, $r_{\rm
Sch}=2GM_{\bullet}/c^{2}$. If the pericenter distance of a star is smaller
than a few times outside $r_{\rm Sch}$, the star would be directly captured by
the black hole. If the separation is smaller than the stellar radius, they
would collide (“black hole-star collision”). When the pericenter distance is
smaller than the tidal radius $r\leq r_{\rm
t}=(M_{\bullet}/M_{\star})^{1/3}R_{\star}$, stars are tidally destroyed by the
BH (“tidal disruption”). Finally, disruptive collisions happen between the
distance at which the Keplerian velocity is greater than the stellar escape
velocity, $r\leq r_{\rm collision}=(M_{\bullet}/M_{\star})R_{\star}$ and the
tidal radius $r\leq r_{\rm t}$. The white diagonal lines in the region for
disruptive collisions corresponds to the collision velocity for given BH mass
and radius.

####  4.2.3 Astrophysical implication for black holes

The possibility of the accretion of at least some fraction of the expanding
cloud onto the SMBH in proximity can have significant implications for the
growth of BHs in the cosmic landscape. While several mechanisms for massive BH
formation have been proposed, the precise mechanism for growing BH seeds at
extremely high redshifts remains uncertain (see for reviews Colpi & Dotti,
2011; Inayoshi et al., 2020). The proposed mechanisms include 1) rapid growth
of the remnants of the Population III stars via super-Eddington accretion
(e.g., Volonteri & Rees, 2005; Haiman & Loeb, 2001; Ryu et al., 2016; Lupi et
al., 2016; Sassano et al., 2023), the direct collapse of supermassive self-
gravitating objects (e.g., Omukai & Nishi, 1998; Yoshida et al., 2008; Zwick
et al., 2023), growth of BHs in a runaway process (e.g., Devecchi et al.,
2012; Stone et al., 2017; Tagawa et al., 2020; Rizzuto et al., 2023). In
principle, as long as a BH is more massive than colliding stars, the velocity
of stars around the BH can be large enough that stellar collisions can be
completely disruptive. Hence, the accretion of gas produced in stellar
collisions onto a nearby BH can provide another venue for the growth of
stellar-mass BHs to massive BHs, in particular see BHs at high redshift.

However, disruptive collisions are not the only growth mechanism for BHs in
stellar-dense environments. We show in Figure 13 the regions around BHs in
which several events possibly contributing to their growth, i.e., disruptive
collisions, tidal disruption events, BH-star collisions, and direct captures
by BHs, can occur. When the distance from the BH is less than a few times
greater than the Schwarzschild radius $r_{\rm Sch}=2GM_{\bullet}/c^{2}$
(dubbed “direct capture” radius), the star would directly fall into the BH
(e.g., $r<2r_{\rm Sch}$ for parabolic orbits). If the closest approach
distance between the BH and a star is smaller than the stellar radius,
$r\lesssim\;R_{\star}$, they collide, during which the BH would
gravitationally capture a fraction of the star and accrete. When a star orbits
at a distance greater than both the stellar radius and the direct capture
radius, and smaller than the so-called tidal radius, $r_{\rm
t}=(M_{\bullet}/M_{\star})^{1/3}R_{\star}$, very strong BH’s tidal forces
disrupt the star, creating debris, some of which would end up accreting onto
the BH. This event is called tidal disruption event (Hills, 1988; Rees, 1988).
Finally, the region for disruptive collisions between giants may be
characterized by the two distances, the distance within which the Keplerian
velocity around the BH exceeds the stellar escape speed, $r_{\rm
collision}=(M_{\bullet}/M_{\star})R_{\star}$ and the tidal radius.

As shown in Figure 13, all four star-destroying events can contribute to the
growth of stellar-mass and intermediate-mass BHs. However, for SMBHs with
$r_{\rm Sch}>\;R_{\star}$, only three events, namely, disruptive collisions,
tidal disruptions, and direct captures, can feed the BHs. For very massive BHs
(e.g., $M_{\bullet}>10^{9}\;{\rm M}_{\odot}$), disruptive collisions would be
the dominant and likely only observable transient among those considered here
that lead to the mass growth of the BHs (Amaro Seoane, 2023a).

This has an interesting implication for the detection of dormant BHs. TDEs
have been considered a unique signpost for the existence of dormant SMBHs.
However, because there is a maximum BH mass capable of disrupting stars,
$M_{\bullet}$ at which $r_{\rm t}$ equals to the direct capture radius, TDEs
can not be used to detect very massive quiescent black holes. However,
disruptive stellar collisions can occur near BHs at all mass scales, which
would make these events a promising tool to probe the existence of very
massive dormant BHs which cannot be probed by other transients. In particular,
if the luminosity due to the interaction of the collision product with the BH
is Eddington-limited, an inference of the BH mass would potentially be
possible from the observed radiated light curve.

Which type of the events is dominant at different mass ranges would depend on
the stellar density, the accretion efficiency and occurrence rates, which is
beyond the scope of our paper. We will examine this aspect in more detail in
our future work.

###  4.3 Particle acceleration

In this work we have conducted numerical hydrodynamical simulations that
confirm, following stellar collision event, the formation of strong shocks.
These shocks arise due to the high velocity of the outflow, and its impact
with the surrounding ISM in the galactic nucleus environment. These shock
waves subsequently compress and heat the surrounding ISM gas.

The shocks formed in these stellar collisions provide an environment highly
conducive to efficient particle acceleration. As particles interact with the
turbulent magnetic fields expected close to the shock front, they can gain a
significant fraction of the free energy available from the differential flow
speeds (in the shock’s rest frame, the upstream towards the shock with
velocity $V$ and the downstream moves away from the shock at velocity $V/4$).
This process of diffusive particle acceleration at shocks, an example of first
order Fermi acceleration, is expected to result in the generation of a power-
law spectrum of non-thermal particles up to very high energies (Blandford &
Ostriker, 1978; Bell, 1978).

A fraction of the energy in the accelerated particle population produced by
stellar collisions will subsequently be radiated via non-thermal emission
through various energy loss processes (see Matthews et al., 2020; Orlando et
al., 2021, for revieww in the context of active galactic nuclei jets and
supernovae, respectively). For instance, the accelerated electrons will
produce synchrotron radiation as they spiral around the magnetic fields also
generated during the collision. This emission is expected to be detectable in
the radio, and potentially the X-ray, bands. In addition, the interaction
between accelerated protons and the surrounding gas can generate gamma-ray
emission through processes like inelastic proton-proton collisions.

The non-thermal radiation emitted by the accelerated particles produced in
stellar collisions offers valuable diagnostics into the physical processes at
play during violent stellar collision events. By analyzing the observed non-
thermal radiation, we can gain a clearer understanding of shock front
environment. Ultimately, these insights will elucidate on the dynamics of the
collision itself. Our numerical hydrodynamics simulations, coupled with
theoretical estimates for the production of non-thermal particles not included
in our numerical description, provide can provide insights into particle
acceleration in stellar collisions. This will be addressed elsewhere in a
separate work.

##  5 Conclusion and Summary

In this work, we investigate the hydrodynamics of black hole-drive disruptive
collisions (BDCs) between giants in galactic nuclei and their observational
signatures using two state-of-the-art codes, the 3D moving-mesh hydrodynamics
code AREPO and the 1D stellar evolution code MESA. The initial conditions of
our simulations involved two identical $1\;{\rm M}_{\odot}$ giants with
different radii, initial relative speeds, and impact parameters. This work
complements to the analytical calculations presented by Amaro Seoane (2023b),
which is generally consistent with each other. We improve the estimates of the
events’ observables by accurately taking into account the realistic stellar
internal structure and non-linear hydrodynamics effects.

When two stars collide with exceedingly large kinetic energy, very strong
shocks are created along the contact surface. The two stars are fully
destroyed and merged into an homologously, quasi-spherically, and
supersonically expanding gas cloud. The maximum expansion speed of the cloud
is larger than the initial relative velocity of the stars by a factor of
$3-6$. The expansion speed at a given mass coordinate stays the same, but the
outer edge of the cloud slows down because of the interaction with the
background medium. As it expands, the overall level of its density and
temperature drops following a power-law $\propto t^{-3}$ and $\propto t^{-1}$,
respectively, becoming optically thin within a few hundred days. At any given
time of evolution up to 30 days, the density and temperature of the inner
regions of the cloud remain relatively constant, rapidly decaying towards the
outer edge, following a power-law: $\rho(r)\propto r^{-8}-r^{-12}$ and
$T(r)\propto r^{-1}-r^{-2}$. These quantities exhibit weak dependencies on the
stellar radius within $10-100\;{\rm R}_{\odot}$ and the impact parameter
within $b\simeq 0.4\;{\rm R}_{\odot}$. But the dependence on the collision
velocity is relatively strong. We provide fitting formulae for the average
cloud density, temperature, maximum expansion speed, and optical depth
(Equations 2\- 7), which would be useful for analytic estimates for these
BDCs.

One of the key findings of our study is to numerically estimate the amount of
radiation energy converted from the initial kinetic energy, which plays a
crucial role in determining the observable properties of the collisions. The
overall trend of the conversion efficiency, defined as the ratio of the
converted radiation energy to the initial kinetic energy, is such that it
peaks at $\gtrsim 0.1$ at collision, decays to $10^{-4}-10^{-2}$ within 10
days, and then gradually increases. The efficiency reaches $10^{-2}-10^{-1}$
in one month since the collision. But its magnitude depends on various
factors, including the stellar radius, impact parameter, and collision
velocity. More specifically, a collision between larger stars colliding at a
higher speed with a smaller impact parameter tends to result in greater
conversion efficiency.

We estimate the luminosity, the blackbody radius, and the blackbody
temperature, using the converted radiation energy and local cooling time
within the gas cloud. The peak luminosity can reach values exceeding $10^{42}$
erg/s and exhibits the similar dependence wtih the conversion efficiency. Over
time, the luminosity decays following a power-law of $t^{-0.8}$ at early times
and $t^{-0.4}$ after 10 days since collision. The blackbody radius increases
almost linearly with time ($\propto t^{0.8}$), while the temperature
decreases, following a power-law of $t^{-0.5}-t^{-0.6}$. The collision events
would initially produce bursts of extreme ultraviolet ($\simeq 10$eV)
gradually shifting to optical ($\simeq 0.1$eV), with temporal evolution
spanning from days to weeks. These events can be observed by ongoing (e.g.,
ZTF Bellm et al. 2019444https://www.ztf.caltech.edu, ASSA-SN Kochanek et al.
2017555https://www.astronomy.ohio-state.edu/asassn) and future (e.g., LSST
Ivezić et al. 2019666https://www.lsst.org and ULTRASAT Shvartzvald et al.
2023777https://www.weizmann.ac.il/ultrasat) surveys. More detailed radiation
transport calculations will be carried out in our follow-up project, with
which the detection rate for each survey will be estimated.

In addition to the burst resulting from the stellar collision itself, a
subsequent burst occurs due to the accretion of the gas cloud onto the
supermassive black hole in the galactic center in $5(M_{\bullet}/10^{7}\;{\rm
M}_{\odot})$ days for $v_{\rm rel}=10^{4}$ km/s since collision. Assuming
Bondi accretion, the accretion luminosity can easily exceed the Eddington
limit as well as the luminosity from the stellar collision. Because the Bondi
radius expands faster than the gas cloud, the entire cloud would be
gravitationally captured in the black holes’s potential in
$11(M_{\bullet}/10^{7}\;{\rm M}_{\odot})^{3/4}$ days and subsequently accrete
onto the black hole. It would take $\lesssim 9(M_{\bullet}/10^{7}\;{\rm
M}_{\odot})^{-1}$ years if the entire cloud was accreted. Therefore, the
overall luminosity curve would include a peak from the collision event,
followed by a rise to the Eddington luminosity. This heightened luminosity can
be sustained for up to 10 years.

Although the estimate of the time scales and luminosity due to gas-black hole
interactions are still on the order-of-magnitude level, this aspect indicates
very important implications. The possibility of the gas accretion onto the
black hole at all mass scales in proximity subsequently after the collision
suggests that the collision can provide another mechanism for black hole
growth. Tidal disruption events have been proposed as a tool to detect dormant
black holes, mostly up to $10^{8}\;{\rm M}_{\odot}$. However, because
disruptive stellar collisions can occur near very massive dormant ones
($>10^{9}\;{\rm M}_{\odot}$), such collisions can be a potentially promising
tool to probe the existence of very massive dormant black holes.

Finally we demonstrate the conversion of kinetic energy into radiation energy,
providing insights into the efficiency of particle acceleration in these
collisions. The resulting bursts of ultraviolet and optical emission indicate
the generation of high-energy particles, highlighting the importance of
particle acceleration processes in understanding the observational signatures
of such events.

While this study, to our knowledge, is the first detailed hydrodynamics
calculations of BDCs between giants, there are a few caveats in our modelling
that will be improved in our future work. First, the assumption for local
thermodynamic equilibrium is only valid for optically thick gas. This means
the evolution of the collision product at early times is accurate, but as the
gas cloud becomes optically thin, our treatment of radiation pressure becomes
inaccurate. As remarked in § 3.5, This would affect the shape of the
lightcurves at late times. We will perform detailed non-equilibrium radiation
transport calculations for the late time evolution in our follow-up project
using our hydrodynamics calculations at early times when our assumption for
local thermodynamic equilibrium is valid. This will significantly improve the
light curve modelling. Second, there are several physical impacts that we have
not considered yet, such as, magnetic fields, recombination, and the existence
of non-thermal particles. Using the machinery that we built for this work, We
will explore their impacts in a series of studies dedicated to investigating
the impact of each physics.

BDCs will offer insights into many astrophysical aspects that can not be
provided by other transients, such as the stellar dynamics and potential
particle acceleration in galactic nuclei and globular clusters, black hole
growth, and detection of dormant black holes.

## Acknowledgements

TR is grateful to Luc Dessart and Re’em Sari for constructive comments on the
manuscript, Hans-Thomas Janka for fruitful discussions for similarities and
dissimilarities of these events with core-collapse supernovae, and Ruggero
Valli for providing an MESA input file for creating the giants used for the
simulations. This research project was conducted using computational resources
(and/or scientific computing services) at the Max-Planck Computing & Data
Facility. The authors gratefully acknowledge the scientific support and HPC
resources provided by the Erlangen National High Performance Computing Center
(NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) under
the NHR project b166ea10. NHR funding is provided by federal and Bavarian
state authorities. NHR@FAU hardware is partially funded by the German Research
Foundation (DFG) – 440719683. In addition, some of the simulations were
performed on the national supercomputer Hawk at the High Performance Computing
Center Stuttgart (HLRS) under the grant number 44232. PAS acknowledges the
funds from the “European Union NextGenerationEU/PRTR”, Programa de Planes
Complementarios I+D+I (ref. ASFAE/2022/014).

## Data Availability

Any data used in this analysis are available on reasonable request from the
first author.

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##  Appendix A Luminosity estimate

Figure 14 show the luminosity $L_{2}$ (top) estimated using Equation 12 and
the resulting blackbody temperature $T_{\rm BB}$ (bottom), as a function of
time measured since collision for all our models.

Figure 14: Bolometric luminosity $L_{2}$ (top) and blackbody temperature
$T_{\rm BB}$ (bottom) using Equations 10 and 12. As in Figure 9, the magenta
guide lines in the bottom panel show the power-law that describes the
quantity.


We are indebted to Prof. M. M.S. Nasser and Dr. Harri Hakula for kindly
providing several numerical results for this paper. We also thank K. Zarvalis
for his help with the figures.

We are grateful to the referee for many valuable suggestions.

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