# Tidal Peeling Events: low-eccentricity tidal disruption of a star by a
stellar-mass black hole

Chengcheng Xin Columbia University, Department of Astronomy, 550 West 120th
Street, New York, NY, 10027, USA  Zoltán Haiman  Columbia University,
Department of Astronomy, 550 West 120th Street, New York, NY, 10027, USA
Columbia University, Department of Physics, 550 West 120th Street, New York,
NY, 10027, USA  Rosalba Perna Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794-3800, USA  Center for Computational
Astrophysics, Flatiron Institute, New York, NY 10010, USA  Yihan Wang  Nevada
Center for Astrophysics, University of Nevada, 4505 S. Maryland Pkwy., Las
Vegas, NV 89154-4002, USA  Department of Physics and Astronomy, University of
Nevada, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA  Taeho Ryu  Max
Planck Institute for Astrophysics, Karl-Schwarzschild-Strasse 1, 85748
Garching, Germany  Physics and Astronomy Department, Johns Hopkins University,
Baltimore, MD 21218, USA

###### Abstract

Close encounters between stellar-mass black holes (BHs) and stars occur
frequently in dense star clusters and in the disks of active galactic nuclei
(AGNs). Recent studies have shown that in highly eccentric close encounters,
the star can be tidally disrupted by the BH (micro-tidal disruption event, or
micro-TDE), resulting in rapid mass accretion and possibly bright
electromagnetic signatures. Here we consider a scenario in which the star
might approach the stellar-mass BH in a gradual, nearly circular inspiral,
under the influence of dynamical friction on a circum-binary gas disk or
three-body interactions in a star cluster. We perform hydro-dynamical
simulations of this scenario using the smoothed particle hydrodynamics code
PHANTOM. We find that the mass of the star is slowly stripped away by the BH.
We call this gradual tidal disruption a “tidal-peeling event”, or a TPE.
Depending on the initial distance and eccentricity of the encounter, TPEs
might exhibit significant accretion rates and orbital evolution distinct from
those of a typical (eccentric) micro-TDE.

##  1 Introduction

Stars and their compact remnants, which include stellar-mass black holes
(BHs), are expected to be abundant in dense stellar clusters of all kinds
(Mackey et al., 2007; Strader et al., 2012), and they can also be found in the
disks of Active Galactic Nuclei (AGNs). Dynamical interactions between compact
objects and stars in clusters are frequently expected (Rodriguez et al., 2016;
Kremer et al., 2018). As a result, stars in a cluster will inevitably undergo
close encounters with stellar-mass BHs. These close encounters between stars
and BHs, which are of particular interest here, can lead to binary formation
or to tidal disruption of the star by the BH (the so-called micro-TDEs, Perets
et al. 2016).

Stars and stellar-mass BHs found in an AGN disk are likely the result of two
mechanisms:  (i) Capture from the nuclear star cluster (Artymowicz et al.,
1993), which consists mostly of massive stars (e.g. O- and B-type stars with
masses $\gtrsim$ 2-15$M_{\odot}$). These stars’ orbits will eventually align
with the AGN disk after a number of crossings of the disk (Yang et al., 2020).
(ii) In-situ formation: Gravitational instabilities in the outer parts of the
disk trigger star formation (Goodman, 2003; Dittmann & Miller, 2020), and
those stars, as well as their remnant compact objects, remain embedded in the
disk. The unusual disk environment causes stars to accrete and grow in mass
(Cantiello et al., 2021; Jermyn et al., 2021), which makes BH remnants a
common outcome upon their death. Once trapped in the AGN disk, BHs can go
through radial migration and undergo close encounters with stars or compact
objects (e.g., Tagawa et al., 2020). Therefore, micro-TDEs can also occur in
AGN disks, in addition to the stellar cluster environment.

Micro-TDEs are expected to be ultra-luminous events, and their expected
accretion rates and electromagnetic (EM) features have recently begun to be
investigated in more detail via smooth particle hydrodynamical (SPH)
simulations (Lopez et al. 2019; Kremer et al. 2021; Wang et al. 2021; Kremer
et al. 2022; Ryu et al. 2022) and moving-mesh (Ryu et al., 2023). Existing
studies have performed numerical experiments to investigate nearly parabolic
encounters with eccentricity $e\sim 1$. Kremer et al. (2022) recently
presented a variety of hydrodynamical simulations of the typical micro-TDE
with parabolic orbits to show that stars in vacuum can experience different
degrees of tidal disruption depending on pericenter distance and stellar mass,
while the peak luminosity of the EM emission might be super-Eddington when
pericenter distance is within $\sim 2R_{t}$, where $R_{t}=(M_{\rm BH}/M_{\rm
s})^{1/3}R_{\rm s}$ is the order-of-magnitude estimate of the tidal radius for
a star with mass $M_{\rm s}$ and radius $R_{\rm s}$ disrupted by a BH with
mass $M_{\rm BH}$.

On the other hand, low-eccentricity micro-TDEs in compact orbits are of
particular interest in this paper for the following reasons. First,
observational work has suggested that binaries in clusters have lower
eccentricity as they become more compact (Meibom & Mathieu, 2005; Hwang et
al., 2022). 3D hydro-simulations by Ryu et al. (2023) further suggest that
three-body interactions in clusters such as encounters between binary stars
and stellar-mass BHs can also lead to eventual close interactions between one
star in the original binary and the BH, where, in some cases, a low-
eccentricity micro-TDE in a close orbit can form if the star becomes bound to
the BH. Additionally, star-BH binaries in an AGN disk can become tightly bound
due to external torques exerted by the dynamical friction of the AGN disk gas.
Hydrodynamical simulations have shown that a circumbinary disk tends to shrink
the orbit of the binary within an AGN disk (Li et al., 2021; Kaaz et al.,
2021; Li & Lai, 2022) and drive it to low eccentricity, either $e\rightarrow
0$ or $e\rightarrow 0.45$, depending on the initial value (Muñoz et al., 2019;
D’Orazio & Duffell, 2021; Zrake et al., 2021).

Unlike the abrupt disruption that the star experiences in a parabolic TDE or
micro-TDE, lower-eccentricity micro-TDEs gradually strip mass from the star,
typically over many orbital times, analogous to the extreme-mass-ratio
inspiral of a white dwarf (WD) and an intermediate mass BH, in which the WD
loses mass periodically during the inspiral (Zalamea et al., 2010; Chen et
al., 2022). We call this “tidal-peeling event” (TPE) in this paper.

In this paper, we numerically model the general case of TPEs with SPH
simulations using PHANTOM, without including the low-density background gas
such as the AGN disk. We focus on exploring the BH mass accretion rate and
orbital evolution in TPEs under different assumptions for the initial mass of
the star, eccentricity and pericenter distance of the encounter.

We organize this paper as follows. We describe our simulation models, analysis
method and a resolution study in § 2, 3 and 4, respectively. In § 5, we show
the morphological evolution of the TPEs. Section § 6 illustrates our
prediction for the EM signatures of TPEs, based on the computation of the BH
mass accretion rates, stellar mass loss via tidal interactions and the orbit
evolution of the remnant. In § 7, we explore the effect of having more massive
stars undergoing TPEs. Finally, we discuss some implications of our results in
§ 8, and we summarize our conclusions in § 9.

##  2 Simulation Methods

We perform SPH simulations of TPEs of stars by a $10M_{\odot}$ BH using
PHANTOM (Price et al., 2018). We run simulations for (4 stellar masses)
$\times$ (4 eccentricities) $\times$ (6 penetration factors) $=96$ models in
total, where the penetration factor $\beta$ is defined as the ratio between
the tidal radius and the pericenter distance, or $R_{t}/r_{p}$. We consider
main-sequence (MS) stars with four different masses, $M_{\rm s}=$ 1, 5, 10 or
15 $M_{\odot}$, and investigate the dependence of the initial eccentricities
of outcomes by considering $e_{0}=$ 0.0, 0.2, 0.4 and 0.6. We begin all
simulations by placing the star at the apocenter of the orbit. Finally, we
consider the following penetration factors $\beta=R_{t}/r_{p}$ = 1, 0.67, 0.5,
0.4, 0.33 and 0.25, which corresponds to the pericenter distances $r_{p}=$ 1,
1.5, 2, 2.5, 3 and 4 times the tidal radii. For simplicity, we introduce the
letter $\mathcal{M}(M_{\rm s},e_{0},\beta)$ to denote any specific model,
where $M_{\rm s}$ is given in units of $M_{\odot}$. We fix the BH mass in all
the simulation models at $M_{\rm BH}=10M_{\odot}$.

We first use the 1D stellar evolution code MESA (Paxton et al., 2019) to
generate the profile of each MS star with the core H fraction of 0.5, where we
assume solar abundances for composition, hydrogen and metal mass fractions
$X=0.74$ and $Z=0.02$ respectively (helium mass fraction $Y=1-X-Z$), and mean
molecular weight $\mu\sim 0.59$ (fully ionized gas). For the stellar masses
that we consider, MESA uses the OPAL and HELM table for the equation of state
(Paxton et al., 2019), which we adopt in the TPE simulations. We then take the
density and internal energy profile of MESA MS stars to start the simulations
in PHANTOM. We first map the 1D MESA model onto our 3D SPH grid and relax it
for a few stellar dynamical times ($t_{\rm dyn}=\sqrt{R_{s}^{3}/GM_{s}}$)
until it reaches hydrostatic equilibrium. $t_{\rm dyn}$ is typically 1 to a
few hours depending on the mass and radius of the star.

In the TPE simulations with PHANTOM, we use artificial viscosity varying
between ${a^{\rm AV}}_{\rm min}=0.1$ to ${a^{\rm AV}}_{\rm max}=1$. This is
the typical range for ${a^{\rm AV}}$ to evolve, which contributes to shock
capture (e.g. Coughlin et al., 2017). We adopt an equation of state that
includes radiation pressure assuming instantaneous local thermodynamic
equilibrium. This assumption is valid because the gas in our simulations is
expected to be optically thick. We employ $10^{5}$ SPH particles in each
simulation, which is justified in § 4, and each simulation uses up to 6,000
CPU hours on processor Intel Xeon Gold 6226 2.9 Ghz. For this resolution, the
smallest spatial scale within which accretion can be resolved is $r_{\rm
acc}=100r_{g}$, where $r_{g}=GM_{\rm BH}/c^{2}$. If a SPH particle falls
within the “accretion“ radius, it is accreted onto the BH. The particles are
removed from the simulation once accreted by the BH; the removed mass is added
to the mass of the sink particle.

##  3 Analysis

In this study, we focus on some key physical quantities, such as the amount of
mass lost in TPEs and the accretion rate, directly measured from our
simulation output. Also, we investigate their dependence on different initial
conditions – the mass of the star ($M_{s}$), the initial eccentricity
($e_{0}$), and the penetration parameter ($\beta$) that is inversely
proportional to the initial pericenter distance.

First, we measure the mass accretion onto the BH, $M_{\rm acc}$, by evaluating
the mass accreted onto the sink particle representing the BH. The BH accretion
rate $\dot{M}_{\rm BH}$ is computed as the finite difference of $M_{\rm acc}$
divided by the time difference ($\sim 0.4$ hours) between two adjacent outputs
of the simulation.

In a TPE, the star’s mass is slowly stripped by the BH, which leads to the
star being partially or totally disrupted. In past studies of TDEs or micro-
TDEs using numerical simulations (e.g., Mainetti et al., 2017; Kremer et al.,
2022), the mass bound to the star or BH is usually computed using an iterative
process described in Lombardi, Jr. et al. (2006). However, since the iteration
evaluates the specific binding energy of each particle, including a
gravitational potential term, it assumes spherical geometry for the remnant,
which is not always applicable in our TPE simulations, see Fig. 5 for example.
Additionally, in some TPEs, the remnant is not isolated as it is connected
with debris, for which the iterative process can lead to inaccurate
identification of the remnant. Alternatively, we define the mass of the
stellar remnant ($M_{\rm rem}$) as the total mass of particles within the
initial radius of the star (measured from the densest point in the star).

In addition to the stellar material lost to $M_{\rm acc}$, the star can also
lose mass to the surroundings when the stellar material is unbound during the
disruptions. We measure the fraction of total mass removed from the star,
$f_{\rm rm}$. The mass removed consists of mass accreted by the BH ($M_{\rm
acc}$) and mass ejected (total stellar mass minus remnant mass; $M_{\rm
s}$-$M_{\rm rem}$). Note that the mass removed from the star includes the mass
unbound to the remnant, but bound and not yet accreted by the BH. So $f_{\rm
rm}$=($M_{\rm s}$ \- $M_{\rm rem}$ \+ $M_{\rm acc}$)/$M_{s}$.

The orbital features of the stellar remnant can be described by the evolution
of the orbital separation ($r$), semi-major axis (SMA; $a$) and eccentricity
($e$) over time. We define $r$ to be the distance between the particle of the
highest density in the stellar remnant, typically at the core of the star
(small deviation can happen due to any oscillation in the star during the
disruption), and the position of the sink particle (BH). The SMA and the
eccentricity are calculated using the specific energy and specific angular
momentum of the binary, adapted from the calculation in Muñoz et al. (2019),
where the equation of motion of the binary are evaluated with the external
gravitational and accretion forces. In § 6, we evaluate the evolution of $a$
and $e$, as well as their change per each orbit around the BH.

##  4 Resolution tests for initial stellar profile

A typical choice for resolutions of hydro-simulations of TDEs or micro-TDEs is
$N\sim 10^{5}$ particles (e.g. Mainetti et al., 2017; Kremer et al., 2022). We
performed resolution tests to determine whether or not a higher resolution is
needed, by using PHANTOM to model the initial stellar profile using different
numbers of SPH particles $N=10^{5},2\times 10^{5},4\times 10^{5},8\times
10^{5},10^{6}$. In particular, we compare the radial density profiles of the
fully relaxed $1M_{\odot}$ star with the numbers of SPH particles given above
in Fig. 1. The gray region shows where the initial profile varies the most,
which occurs at the surface of the star. We find that different resolutions
only cause the density to fluctuate by $\sim$0.01%, which only takes place in
less than 1% of the SPH particles by mass and $\lesssim 0.2R_{\odot}$ by
radius. Overall, the density profiles for resolutions from $N=10^{5}$ to
$N=10^{6}$ particles show excellent agreement. Therefore, we run all TPE
simulations, starting from their stellar profiles, with particle number
$N=10^{5}$. As a comparison, we also depict the polytropic star with
$\gamma=4/3$ of the same mass using a purple dashed line.

Figure 1: The radial density profile of a fully relaxed $1M_{\odot}$ star in
PHANTOM, using $N=10^{5},2\times 10^{5},4\times 10^{5},8\times 10^{5},10^{6}$
SPH particles. The density is normalized to the core density $\rho_{\rm c}$.
Different resolutions yield converging initial density profiles for the star,
despite a small surface layer (R$>0.8R_{\odot}$; gray region), containing
$\lesssim$0.9% of stellar mass. This justifies our choice to use $N=10^{5}$
particles throughout the simulations. As a sanity check, we overlay the
analytical solution of $4/3$-polytrope (purple dahsed line).

##  5 Morphology of TPE

Figure 2: $\mathcal{M}(1,0.4,1)$ – Tidal peeling morphology of a $1M_{\odot}$
star and a $10M_{\odot}$ BH, where the orbit is initially a low-eccentricity
inspiral ($e_{0}=0.4$), and the pericenter distance between star and BH is 1
tidal radius ($r_{p}=2.2R_{\odot}$; $\beta=1$). The color bar shows the
projection of log-scale column density in the x-y plane. We overlay the
initial equipotential surface of the binary to show that the stellar material
fills up the Roche Lobe around the BH, and the star loses mass through the
Lagrangian points. The initial orbital period is quoted in parathesis,
specifically, $P\approx 5.7$ hours in this model. We show six time frames of
the event that demonstrate the tidal “peeling” process, until the star is
completely disrupted by the BH. The star orbits around the BH and passes
through the pericenter four times until it is torn apart by the BH.

Figure 3: $\mathcal{M}(1,0.4,1)$ – Same snapshots of the simulation as in Fig.
2 on the x-z plane, or edge-on view of the orbit and the accretion disk.

The stars in our TPE simulations encounter the BH in low-eccentricity
($e=0-0.6$) and ultra-compact ($\beta=0.25-1$) orbits. Depending on the
initial conditions, the mass of the star can be slowly peeled by the BH, and
stellar material is lost on the timescale of many orbital periods. In general,
TPEs will have novel morphological evolution, e.g. distinct morphology from
that seen in TDEs or micro-TDEs, and in particular, 1) gradual tidal stripping
and formation of spirals, 2) possible debris-star interactions, and 3)
efficient circularization of debris into an accretion disk. Each of these is
demonstrated in the following examples.

Fig. 2 shows a typical morphology of a TPE, where the column density of the
gas particles is shown in the color bar and the BH is represented by the green
dot. In this example (Model $\mathcal{M}(1,0.4,1)$; recall the definition in §
3), the $1M_{\odot}$ star on an eccentric orbit with $\beta=1$ is “peeled” due
to the tidal influence of the BH, which continues for four orbits before the
star is totally disrupted (at the $\sim 4^{\rm th}$ orbits). The snapshots are
taken at $t=0,4.9,12.0,18.2,23.5$ and $36.3$ hours since the onset of the
simulation, where the orbital period is $P\approx 5.7$ hrs. Some stellar
debris circularizes and forms an accretion disk around the BH, while some
becomes unbound and are ejected into infinity, including mass lost through the
“L3” point; we show the initial equipotential surface of the binary in each
panel. This can be more clearly seen in Fig. 3 that shows the edge-on view of
Fig. 2. The disk is initially smaller than the pericenter distance of the
orbit for a short period of time, before it inflates and puffs up later on due
to radiation pressure and shock heating, similar to the findings of Wang et
al. (2021).

Generally, tidal peeling is more violent for smaller orbital separations. All
of our TPE simulations result in super-Eddington BH accretion rates. However,
a significant fraction of the star being tidally disrupted, leaving most of
the dense stellar material around the BH, results in large optical depth that
likely will delay and dim the EM emission from a TPE. In reality, the
luminosity could be modulated by several mechanisms such as jet emission or
wind outflow from the accretion disk, which are not included in this study.
Additionally, in some configurations, such as $\mathcal{M}(1,0.6,0.67)$ in
Fig. 4, the star intersects with its own tidal streams periodically, which
will form a shock front that further modifies the luminosity from the TPE. In
this model, the remnant remains intact for many orbits. In the second panel,
the star encounters the tail of its own stream formed in the last orbit,
leaving behind a hot ploom near the star as seen in the last panel. Although
these phenomena cannot be resolved in our simulations, in the following
sections, we will qualitatively discuss their implications for the overall EM
signature of the TPEs in addition to the accretion rates that we measure
directly from the simulations.

Figure 4: $\mathcal{M}(1,0.6,0.67)$ – Tidal peeling of the same BH-star binary
described in Fig. 2, but with initial eccentricity $e_{0}=0.6$ and
$\beta=0.67$. The initial orbital period (in parenthesis) is 19.2 hours.

Finally, TPEs from the interaction of BHs with more massive stars are
considered since stars near the galactic center (Genzel et al., 2003; Levin,
2003; Paumard et al., 2006) and those formed in an AGN disk (Levin, 2003;
Goodman & Tan, 2004) are also thought to be preferentially massive, and they
offer morphology different from TPEs with a solar-like star. In Fig. 5, we
demonstrate the TPE between a $5M_{\odot}$ star and the BH in circular orbit
with the initial separation of one tidal radius. The surface of this star is
almost in contact with the BH, $a=r_{p}\approx 1.3R_{s}$. Compared to a solar-
like star in the same initial orbit, a more massive star experiences more
rapid tidal peeling. As a result, the spirals formed from the disrupted
material are more closely packed, compared to those in Fig. 2. The snapshots
of the TPE are taken at $t=0,0.88,1.77,2.66,3.54$ and $4.43$ hours, and this
TPE model has orbital time $P\approx 1$ hr. The massive star is totally
disrupted within the first orbit, and the stellar material eventually
circularizes into a smooth disk.

Figure 5: $\mathcal{M}(5,0.0,1)$ – Tidal peeling of BH-star with a higher
stellar mass, $M_{\rm s}=5M_{\odot}$, initially circular orbit ($e_{0}$=0) and
pericenter distance equal to the tidal radius $\beta=1$. The initial orbital
period of the binary is $\sim 1$ hour. The star is completely disrupted soon
after the beginning of the simulation.

##  6 Accretion rate and orbital evolution of TPEs

Figure 6: Time evolution of key physical quantities characterizing TPEs, for
the models $\mathcal{M}(1,0.4,1)$ (left) and $\mathcal{M}(1,0.6,0.67)$
(right). The six panels, from top to bottom, show the (i) mass accreted by the
BH, (ii) accretion rate in Eddington units, (iii) the fraction of mass removed
from the star, (iv) the separation between the remnant and the BH, (v) the
evolution of the semi-major axis and (vi) the eccentricity. The pericenter and
apocenter passages are labeled with red-dashed and blue-solid lines,
respectively. The hatched regions represent total disruption of the star.

###  6.1 Overview using two examples

Fig. 6 demonstrates six key features of mildly eccentric TPEs for the case of
the 10$M_{\odot}$ BH and the $1M_{\odot}$ star. This figure presents two
models – $\mathcal{M}(1,0.4,1)$ (left): initial eccentricity ($e_{0}$) is 0.4
and initial pericenter distance $r_{p}/R_{t}$=1 ($\beta=1$), and
$\mathcal{M}(1,0.6,0.67)$ (right): a more eccentric and less compact model
with $e_{0}$=0.6 and $r_{p}/R_{t}$=1.5 ($\beta=0.67$). We show the time
evolution of (i) the mass accreted onto the BH ($M_{\rm acc}$), (ii) the mass
accretion rate ($\dot{M}_{\rm BH}$) in Eddington luminosity $\dot{M}_{\rm
Edd}=L_{\rm Edd}/0.1c^{2}$, (iii) the fraction of mass removed from the star
($f_{\rm rm}$), (iv) the orbital separation ($r$), (v) the evolution of the
SMA normalized to its initial value ($a/a_{0}$) and (vi) the evolution of
eccentricity ($e$). The bottom four panels of Fig. 6 reflect the properties of
the stellar remnant and are therefore only computed before total disruption;
the time after total disruption of the star is labeled with hatched lines.
Finally, we show the times of pericenter and apocenter passages with red-
dashed lines and blue-solid lines, respectively.

In the first model, $\mathcal{M}(1,0.4,1)$, the mass of the BH grows
monotonically with time, while the accretion rate increases until a plateau
around 5 hours ($\sim P$), exceeding the Eddington limit by more than seven
orders of magnitude. In fact, the values of $\dot{M}_{\rm BH}$ that we find
are typically super-Eddington within the first few orbits of disruption, if
$r_{p}$ within $\sim 3R_{t}$. In this model, the stellar remnant orbits around
the BH on a $\sim 5$hr orbital timescale, during which the binary separation
shrinks and the fraction of stellar mass removed becomes larger until the star
gets totally disrupted after approximately 4 orbital times. The large
fluctuations in $a$ and $e$ indicate that the star-BH orbit is not Keplerian
due to tidal effects and shocks, resulting in the dissipation of orbital
energy and asymmetric mass loss.

For an initially less compact binary, e.g. $\mathcal{M}(1,0.6,0.67)$ (right-
hand side of Fig. 6), the stellar remnant does not undergo total disruption in
the first few orbits. In fact, the mass accretion rate spikes after each
pericenter passage (minima in $r$) with a small time delay, while the peak
level decreases over time. Similar observations have been reported in
simulations of binary stars, where the peak of mass transfer rate is found
shortly after each binary orbit’s pericenter (Lajoie & Sills, 2011). $a$ and
$e$ show fluctuations unique to TPEs, discussed further in § 8, indicating
non-Keplerian orbital evolution, even for a slightly tidally disrupted star’s
orbit.

###  6.2 Dependence on the initial conditions

In this section, we investigate the dependence of the six key quantities above
on different initial conditions, namely $M_{s}$, $e_{0}$, and $\beta$,
providing characteristics of the EM emission of TPEs. We measure these
quantities during the first three orbits of the remnant around the BH, from
one apocenter to the next (between blue solid lines in Fig. 6). In particular,
we compute the change per-orbit of mass accreted onto the BH, the BH accretion
rate, and the fractional stellar mass removed, which are denoted by $M_{\rm
acc,a}$, $\dot{M}_{\rm BH,a}$ and $f_{\rm rm,a}$, respectively. This allows us
to take into account any enhancements in $\dot{M}_{\rm BH}$ during each orbit,
including the peaks near the pericenters as seen in the right-hand side of
Fig. 6. We also evaluate the total change in SMA ($\Delta a/a_{0}$) and
eccentricity ($\Delta e$) each orbit.

Figure 7: The change of mass accretion onto the BH per orbit, $M_{\rm acc,a}$,
for TPEs with a $1M_{\odot}$ star, as a function of initial $e_{0}$ and
$\beta$, evaluated for the first three orbits of the stellar remnant around
the BH. We show the pericenter distances corresponding to each $\beta$ in the
parentheses. The darker end of the color bar represents larger $M_{\rm acc,a}$
values, which decrease as the initial orbit becomes wider and more eccentric.
In the most compact configurations, the star is totally disrupted (crosses),
while in the least compact orbits, zero mass is accreted by the BH (open
circles). The onset of mass transfer is analytically expected to occur when
$r_{p}\approx R_{t}$ (red dotted line).

In comparison with a typical TDE or micro-TDE, where the star is on a
parabolic orbit and more than half of its mass can be lost at the first
pericenter passage (e.g. Mainetti et al., 2017; Bartos et al., 2017; Yang et
al., 2020; Kremer et al., 2022), in a TPE, the star typically loses mass to
the BH more gradually over many orbits around the BH. The degree of mass loss
from the star and the mass accretion onto the BH can be different, depending
on the choices of $M_{s}$, $e_{0}$, and $\beta$.

Fig. 7 shows the orbital change of mass accretion, $M_{\rm acc,a}$, of TPEs
with the $1M_{\odot}$ star and the $10M_{\odot}$ BH, under different
assumptions for $\beta$ (x-axis) and $e_{0}$ (y-axis). In the most compact
models ($\beta=1$), the star gets totally disrupted within the first three
orbits, which are denoted with crosses. This is roughly consistent with the
analytical expectation that the star undergoes tidal disruption when the
pericenter distance of the orbit is comparable to the tidal radius, i.e.
$r_{p}/R_{t}\sim 1$ (red dot-dashed line). More generally, $M_{\rm acc,a}$ is
larger for initially more compact orbits, meaning smaller $r_{p}$ (larger
$\beta$) and smaller $e_{0}$. The latter is equivalent to having smaller
initial orbital separation, since we initially place the star at the apocenter
distance $r_{\rm apo}=a_{0}(1+e_{0})$. However, we see a smaller dependence of
$M_{\rm acc,a}$ on the initial eccentricity than on initial pericenter
distance. The amount of mass accreted by the BH inevitably increases over time
once mass transfer begins, resulting in the highest values of $M_{\rm acc,a}$
in the third orbit. In the models with the largest pericenter distances,
$r_{p}/R_{t}\gtrsim 3$, there is no mass accretion onto the BH in the first
three orbits, denoted by the open circles.

Figure 8: Similar to Fig. 7, but we show the orbit-averaged fraction of mass
removed from the star and the BH, $f_{\rm rm,a}$ = $M_{\rm rm}/M_{s}$, where
$M_{\rm rm}$ = $M_{s}-M_{\rm rem,s}+M_{\rm acc,BH}$, where $M_{s}$ is the
total mass of the star, $M_{\rm rem,s}$ the remnant mass and $M_{\rm acc,BH}$
the mass accreted onto the BH. The crosses represent total disruption. The red
dot-dashed lines again represent the onset of mass transfer limit at
$r_{p}\approx R_{t}$.

We see similar trends in the fraction of stellar material removed from the
star (or $f_{\rm rm,a}$; Fig. 8). Tidal peeling can remove stellar mass slowly
over a few orbital times, which can be seen from the persistent increase of
$f_{\rm rm,a}$ over the first three orbits. Generally, a larger fraction of
the star is removed when the initial orbit has smaller $r_{p}$ and $e_{0}$,
and as time goes on. Note that even in the widest binaries
($r_{p}/R_{t}\gtrsim 3$), a small amount of stellar mass is removed under
tidal effects, which is beyond the analytical prediction (e.g. Zalamea et al.,
2010) for the onset of mass loss (red dot-dashed line), although the mass
accretion onto the BH can be zero (as seen in Fig. 7). Finally, we again
observe larger variations in $f_{\rm rm,a}$ due to $r_{p}$ than due to
$e_{0}$.

Figure 9: Mass accretion rate onto the BH, $\dot{M}_{\rm BH}$ as a function of
initial eccentricity and penetration factor (the corresponding pericenter
distances are quoted in parentheses), taken at the 1st to 3rd pericenter
passages. $\dot{M}_{\rm BH}$ are given in units of $\dot{M}_{\rm Edd}\sim
2.2\times 10^{-7}M_{\odot}$/yr for a $10M_{\odot}$ BH. The open circles
represent zero accretion rate. The open circles represent mass accretion rate
of zero. The crosses represent total disruption.

Fig. 9 shows that typically $\dot{M}_{\rm BH,a}$ range from $\sim 10^{4}$ to
$10^{8}$ times the Eddington accretion rate of the BH. The values of
$\dot{M}_{\rm BH,a}$ are overall higher when the initial binary orbit is more
compact and less eccentric, although, like in Fig. 7 and 8, the impact of the
initial value of $r_{p}$ is larger than the impact of $e_{0}$. Like the trend
in both BH mass accretion and fraction of stellar mass loss, the values of
$\dot{M}_{\rm BH,a}$ tend to increase over time, except in some models with
$e_{0}$=0.6, e.g. $\mathcal{M}(1,0.6,0.67)$ in Fig. 6, where the tidal
influence is the weakest due to the large initial separation between the star
and the BH.

Most TPE models in our simulations indicate partial disruption of the star,
which suggests EM emission from TPEs persisting over many orbital times.
Although we only simulate the first few orbital times of TPEs in this work, we
investigate the orbital evolution of the stellar remnant during this time, and
we attempt to find patterns in the evolutions of the SMA and the eccentricity
that could predict whether the binary separation widens or becomes more
compact. Future work should investigate the long-term behavior of star-BH
TPEs, in order to determine (1) the full duration of their EM emission, and
(2) whether or not the star will be eventually totally disrupted by the BH.

Figure 10: The change of semi-major axis ($\Delta a$) normalized by its
initial value ($a_{0}$) during the first, second and third orbit around the
BH, given in percentages. The yellow colors represent (near) zero changes in
SMA during the orbit, while the redder (bluer) points represent orbit
expanding (shrinking).

In Fig. 10, we demonstrate the variations in SMA ($\Delta a$) per orbit
evaluated during the first three orbits of TPEs with the $1M_{\odot}$ star
around the BH. We investigate the change in $\Delta a$, normalized by the
initial SMA $a_{0}$ of each model, due to different initial conditions $\beta$
and $e_{0}$. The color bars show percentage values of $\Delta a/a_{0}$, which
typically fluctuate within $\sim$4%. We observe that in most models, $\Delta
a$ remains roughly zero (yellow points), corresponding to very small variation
in $a$ during one orbit, meaning that the orbital separation at one apocenter
is not too different from the next one. The redder points in Fig. 10
correspond to the models where the orbits are widening ($\Delta a>0$); the
bluer points corresponds to shrinking orbits ($\Delta a<0$). There is a lack
of overall trends that dictates whether $\Delta a$ increases or decreases with
the two initial conditions, except that the most compact orbits tend to decay.

Figure 11: The change of eccentricity ($\Delta e$) during the first, second
and third orbit around the BH. The yellow colors represent (near) zero changes
in eccentricity during the orbit, while the redder (bluer) points represent
orbit becoming more (less) eccentric.

Fig. 11 shows the change of eccentricity $\Delta e$ in the first three orbits
for the same models in Fig. 10. Most models show small variations in $\Delta
e$ (yellow points), except for the initially circular models (bottom points)
and the most compact models with different $e_{0}$ (points in the first
column), which is consistent with the behaviors in $\Delta a$. The stars in
these models are the most tidally influenced by the BH, where $\Delta e$ shows
significant fluctuations in all three orbits – some orbits become more
eccentric then later circularize, and vice versa.

###  6.3 Sources of luminosity

In a TPE, the super-Eddington accretion onto the BH powers outflow from the
accretion disk. The EM emission from the TPE is delayed by the photon
diffusion time ($\tau_{\rm diff}$), which dilutes the emission from the
accretion disk. From our simulation, $\tau_{\rm diff}=\tau H/c\sim 10^{5}$
years, similar to the photon diffusion time in the sun. In this relation,
$H\sim 1.5R_{\odot}$ is the thickness of the accretion disk formed from the
TPE. $\tau$ is the optical depth to electron scattering, computed assuming
fully ionized gas as,

$\tau=\int_{r}^{\infty}\rho(r^{\prime})\frac{\sigma_{T}}{m_{p}}dr^{\prime}\approx
10^{11},$ (1)

where $\sigma_{T}$ is the electron scattering cross-section. $\rho$ is the
3-dimensional density of the accretion disk taken directly from our
simulations, which is typically very high since a large fraction of the star
is stripped to form the disk in a TPE. Overall, the photon diffusion time
$\tau_{\rm diff}$ is much longer compared to the viscous timescale of the
accretion disk (eq. 4 in D’Orazio et al., 2013),

$\displaystyle\tau_{\rm visc}$ $\displaystyle\simeq
1060\left(\frac{\mathcal{M}}{10}\right)^{2}\left(\frac{0.01}{\alpha}\right)t_{\rm
orb}\approx 20{\rm\ days},$ (2)

where $\mathcal{M}$ is the Mach number, $\alpha$ is the Shakura-Sunyaev
viscosity parameter, and $t_{\rm orb}$ is the orbital time that is typically a
few hours. However, given the super-Eddington accretion rate of a TPE, a
relativistic jet may be launched and break out from the disk, possibly
allowing the TPE to shine through. Since $\dot{M}_{\rm BH}\gg\dot{M}_{\rm
Edd}$, there could be strong accretion disk outflow that might also modify the
EM emission of a TPE. If the TPE is embedded in an AGN disk, the star and the
BH will accrete mass from the disk. We use the calculations in Tagawa et al.
(2020) to estimate that the mass accretion rates onto the star and the BH are
both approximately $10^{3}\dot{M}_{\rm Edd}$, with the BH’s accretion rate
$\sim 5$% of the star’s. We assume that the TPE is located at $r\sim 10^{-2}$
pc to the central massive BH of mass $10^{6}M_{\odot}$, where the disk density
is $\rho_{\rm AGN}\sim 10^{12}M_{\odot}/{\rm pc}^{3}$ and the aspect ratio is
of $\simeq 10^{-3}$. The accretion rates from the AGN disk are also super-
Eddington, although they are still few orders of magnitude lower than
$\dot{M}_{\rm BH}$ in the TPE. Modeling these aspects of TPEs would require
higher resolution, radiative transfer, and/or perhaps a different numerical
code that can include the low-density background AGN disk, which could be
addressed in future work.

##  7 Massive Stars

Figure 12: Mass accreted by the BH ($M_{\rm acc,a}$) normalized to stellar
mass, the fraction of mass removed ($f_{\rm rm,a}$), and accretion rates onto
the BH ($\dot{M}_{\rm BH,a}$) as function of (1) stellar mass and penetration
factor at fixed initial $e_{0}$ = 0.4 (left column), and (2) stellar mass and
eccentricity at fixed pericenter $\beta=0.67$. These are evaluated in the
first orbit of the simulation. As in previous figures, cross indicates full
disruption. In general, $M_{\rm acc,a}$, $f_{\rm rm,a}$ and $\dot{M}_{\rm
BH,a}$ decreases for larger initial separation and eccentricity. The more
massive stars are, the more likely to be completely disrupted due to larger
stellar radius compared to tidal radius. There is a lack of trend in $M_{\rm
acc,a}$, $f_{\rm rm,a}$ and $\dot{M}_{\rm BH,a}$ depending on $M_{s}$ on for
$M_{s}>1M_{\odot}$, indicating that the stellar structures are not
significantly different for those stars. Figure 13: Initial density profiles
of stars used in TPEs – $M_{s}=1,5,10$ and $15M_{\odot}$ as labeled in the
legend. The log-scale density (y-axis) is normalized by the core density of
the star, which is a function of radius (bottom x-axis; normalized by stellar
radius). The dashed line (top x-axis) of corresponding colors indicate the
ratio of stellar radius to the tidal radius of each star.

Due to different stellar physics in massive stars, we investigate the behavior
of TPEs where stars more massive than solar mass are involved, $M_{s}=5,10$,
and $15M_{\odot}$. Fig. 12 shows (i) the properties of TPEs depending on the
initial stellar mass and initial pericenter distance, at fixed $e_{0}=0.4$
(left panels), and (ii) the same properties depending on initial $M_{s}$ and
$e_{0}$, at fixed $r_{p}$ (right panels). From top to bottom, we show the
change in mass accretion onto the BH, the fraction of mass removed from the
star, and mass accretion rate per orbit. The crosses indicate that more
massive stars are more likely to undergo total disruption given the same
initial orbital configurations. In Fig. 13, we see that this is because a more
massive star’s radius is closer to the pericenter, even though its density
profile is steeper. Here we show the density profiles of the initial stars
$M_{s}=1,5,10$ and $15M_{\odot}$, as labeled. The dashed lines (top x-axis)
represent the ratio between stellar radius and tidal radius, which is larger
for a more massive star.

In Fig. 12, we normalize the BH mass accretion by the initial mass of the
star, $M_{\rm acc,a}/M_{\rm s}$ in the top two panels. Therefore, any change
in $M_{\rm acc,a}/M_{\rm s}$, as well as in $f_{\rm rm,a}$ and $\dot{M}_{\rm
BH}$, with the initial $M_{s}$ (along the y-axes) reflects different interior
structures of the stars due to different masses. There are minimal changes in
$M_{\rm acc,a}/M_{\rm s}$, $f_{\rm rm,a}$ and $\dot{M}_{\rm BH}$ along the
$M_{s}$ axis, at any fixed $r_{p}$ or $e_{0}$, especially for $M_{s}\geq
5M_{\odot}$. This indicates that the stellar interiors, mainly the envelopes
that are responding to the tidal stripping of the BH, are not significantly
different for different stellar masses, unless the core of the star is also
disrupted, i.e. the cases of total disruptions.

Overall, these three quantities show more variation due to different initial
$r_{p}$ and $e_{0}$, compared to the effect of stellar mass. At fixed $e_{0}$,
$M_{\rm acc,a}/M_{\rm s}$, $f_{\rm rm,a}$ and $\dot{M}_{\rm BH}$ decrease as
the initial pericenter distance becomes wider, where $M_{\rm acc,a}/M_{\rm s}$
and $\dot{M}_{\rm BH}$ reduce to zero (open circles) even for more massive
stars. Similarly, at fixed $r_{p}$, these quantities decrease as $e_{0}$ gets
larger, due to the fact that elliptical orbits with larger eccentricities
(given the same pericenter distances) are longer orbits. Consistent with the
$M_{s}=1M_{\odot}$ cases, the impact of $r_{p}$ is overall more significant
than the impact of $e_{0}$. Generally, having a more massive star in the TPE
results in more mass accretion onto the BH and higher accretion rates. Our
figures show the fractions of star lost or accreted by the BH, which indicates
the importance of different stars’ interior structures.

Figure 14: Comparing the rate of stellar mass loss from our simulation,
$\mathcal{M}(1,0.2,0.67)$ (black), to that predicted by the analytical
solution (red) in Zalamea et al. (2010).

Finally, as a sanity check, we evaluate the mass loss rate of a $1M_{\odot}$
star from the analytical solution described in Zalamea et al. (2010), and
compare this solution to our simulation results. This analytical solution
predicts the rate of mass loss of a white dwarf (WD) when it is tidally
disrupted by a SMBH, which can be directly applied to our TPE scenario.
Zalamea et al. (2010) predicts that an outer shell of the star with thickness
$\Delta R$ is removed at each tidal stripping, as long as $R_{s}<2R_{t}$,
where $\Delta R=R_{s}-R_{t}\ll R_{s}$. The only differences being (i) our
stellar density profile describes a solar-like MS star that is governed by
gas+radiation pressure, instead of a WD governed by electron degeneracy
pressure, and (ii) the pericenter is much closer to the tidal radius, since we
have a stellar-mass BH rather than a SMBH. Adopting these changes, the
analytical calculation of the stellar mass loss rate ($\dot{M}_{\rm loss}$;
red) from our simulation is shown Fig. 14, along with the mass loss rate we
evaluate from the simulation output (black). This figure shows reasonable
consistency between the two, where the analytical solution is roughly half the
simulation results at first. However, the analytical solution shows a slower
drop in amplitude.

##  8 Discussions

###  8.1 Comparing TPEs to micro-TDEs, TDEs by intermediate-mass and
supermassive BHs

Figure 15: Accretion rates of the BH in TPE model $\mathcal{M}(1,0.4,1)$
evaluated using (i) mass accreted onto the sink particle (top panel) and (ii)
using mass accretion calculation in eq. 3, following Kremer et al. (2022)
(bottom panel). The bottom panel adopts three choices of power-law index:
$s=0,0.5$ and 1.  Figure 16: Same comparison as Fig. 15, but for parabolic
micro-TDE model $\mathcal{M}(1,1,1)$ from Kremer et al. (2022). Similar to
Fig. 15, the accretion rates from the sink and disk methods show roughly
consistent results.

The orbit of the star in a micro-TDE is typically expected to be parabolic
when tidally disrupted by the BH. From a recent study of hydro-simulations of
micro-TDEs (e.g. Kremer et al., 2022), they are likely ultra-luminous
transients, similar to our finding for TPEs. We find that similar to micro-
TDEs, TPEs have super-Eddington accretion rates, up to $\sim
10^{8}\dot{M}_{\rm Edd}$, which is in order of magnitude comparable to that of
“normal” micro-TDEs, see Figure 11 in Kremer et al. (2022). However, the
method that Kremer et al. (2022) use to measure the accretion rate by assuming
that some disk mass is accreted by the BH within the viscous time, or eq. 3,
is different from our method of using a sink particle to measure BH accretion
rate. They assume:

$\dot{M}_{\rm BH}\propto\Bigg{(}\frac{M_{\rm disk}}{t_{\rm
visc}}\Bigg{)}\Bigg{(}\frac{R_{\rm in}}{R_{\rm disk}}\Bigg{)}^{s}.$ (3)

In this relation, we choose an accretion disk with radius $R_{\rm
disk}=R_{t}=2.2R_{\odot}$ that includes particles within the initial Roche
Lobe radius around the BH, see the last panel of Fig. 2. $M_{\rm disk}$ is the
disk mass, which eventually reaches $0.8M_{s}$. $t_{\rm visc}$ is the viscous
timescale that we adopt from eq. 2, but using Mach number $\mathcal{M}=3$ and
$\alpha=0.1$. $R_{\rm in}$ is the inner edge of the disk – we choose $R_{\rm
in}=10r_{\rm sch}$, where $r_{\rm sch}=2GM_{\rm BH}/c^{2}$. Finally, the
choice of power-law index $s$ account for different levels of mass loss due to
outflows. In Fig. 15, we compare the accretion rates computed with eq. 3 to
that found with the sink particle, on a TPE model $\mathcal{M}(1,0.4,1)$. The
top panel shows the mass accretion rate onto the sink particle ($\dot{M}_{\rm
BH,sink}$), and the bottom panel shows the accretion rate from the disk
calculation ($\dot{M}_{\rm BH,disk}$) assuming three choices for the power-law
index $s=0,0.5,1$. $\dot{M}_{\rm BH,disk}$ is overall comparable to
$\dot{M}_{\rm sink}$, while it rises earlier – some mass falls within $R_{\rm
disk}$ instantaneously after the simulation begins. We perform the same
comparison for a parabolic micro-TDE model that was reported in Kremer et al.
(2022), with $M_{\rm BH}=10M_{\odot}$, $M_{\rm s}=1M_{\odot}$, $e_{0}=1$ and
$\beta=1$, see Fig. 16. We adopt a disk with radius $R_{\rm disk}\sim
3.7R_{\odot}$, value used by Kremer et al. (2022), and $t_{\rm visc}$
evaluated with Mach number $\mathcal{M}=1$. The two methods again yield
similar accretion rates.

Despite having similar accretion rates, the orbital periods are generally
shorter for a TPE, which are between a few to few tens of hours, compared to
periods of days to weeks for a micro-TDE. Some micro-TDE models in Kremer et
al. (2022), such as the model with a more massive $M_{s}=5M_{\odot}$ star and
a $10M_{\odot}$ BH, show multiple passages and therefore periodic accretion
onto the BH just like in TPE. However, the orbital period in this model is
$\sim$4 days, significantly longer than TPE periods, so we will be able to
distinguish that from a TPE. But generally, stars in most micro-TDEs undergo
tidal stripping only once, leaving very different morphological evolution,
accretion and orbital signatures compared to a TPE.

Another important comparison should be made between TPEs and tidal disruptions
of a solar-like star by an intermediate-mass BH (IMBH). Recent work by Kıroğlu
et al. (2022) find, using hydro-simulations, that in all cases where a
1$M_{\odot}$ star is disrupted by a IMBH, the stellar remnant is eventually
ejected to be unbound, either after the first pericenter or after many
pericenter passages. In our TPE simulations, all stars remain in a binary with
the BH, or are eventually completely disrupted by the BH. If the star survives
for many pericenter passages with IMBH, then the star is only partially
disrupted and the accretion rate increases with the number of orbits. This is
also not the case in TPEs, see the RHS of Fig. 6, where $\dot{M}_{\rm BH}$
decreases with the number of orbits. Finally, the orbital periods of tidal
disruptions by IMBH typically span a wide range, from 10s of hours to 10
thousand years. The shortest-period events with comparable periods to TPEs
correspond to the lowest BH mass ($M_{\rm BH}\lesssim$10$M_{\odot}$) and
smallest pericenter distance ($r_{p}/R_{t}\ll 1$). Therefore, these events are
basically the micro-TPEs in Kremer et al. (2022) and their similarities and
differences to TPEs are already mentioned above.

The best indicator that a micro-TDE is present in an AGN, rather than a TDE of
a solar-like star by a SMBH, is if the mass of the SMBH is above the Hill’s
limit $\gtrsim 10^{8}M_{\odot}$, beyond which the Schwarzschild radius of the
BH is greater than the tidal radius. However, micro-TDEs or TPEs have
distinguishable signatures even if they exist near a smaller SMBH. First, the
spectra of micro-TDEs and TDEs are expected to be very different, because the
remnant produced in micro-TDEs tend to be optically thick – this is even more
so the case in TPEs, which lead to a hotter accretion disk that cools less
efficiently (Wang et al., 2021) and result in emission in the higher-end of
X-rays. Additionally, like in most micro-TDEs, SMBH in a TDE typically will
disrupt the star once and strip $\sim$half to all of its mass, while partial
disruptions are more common in TPEs. Partial disruptions in TDEs, however,
will have periodic flares on a yearly scale, such as recently observation of
repeated bursts in AT2018fyk (Wevers et al., 2022), much longer than the
expected periods of micro-TDEs and TPEs.

Overall, our simulations show that TPEs are novel transient phenomena that can
be distinguished from other ultra-luminous transients such as micro-TDEs,
tidal disruptions by IMBHs and SMBHs, and partial disruptions in sTDEs.

###  8.2 Simulation caveats

Theoretical investigations of TPEs have many important implications, such as
understanding interactions in compact star-BH binaries in star clusters or AGN
disks and observations of ultra-luminous transient events especially those
near the galactic center. Our results offer first-hand understanding of TPEs
with simulations, while they should be treated as numerical experiments rather
than accurate physical descriptions of TPEs in a cluster or embedded in an AGN
disk. We list the following caveats of our simulations that should be improved
in the future. First, we start the simulations with already very compact
orbits, while in reality, they should be expected at the end of some dynamical
process such as a long AGN-disk mediated inspiral or interactions between
multiple stars or compact objects in a star cluster. Since the BH and star
should approach each other from a much larger distance, we might expect the
star to have already been partially disrupted by the BH, although no mass will
be accreted by the BH beyond the separation of $r_{p}\sim 3R_{t}$, as shown in
our results. The binary could, however, accrete from the external AGN gas, if
embedded in an AGN disk. In future work, we will investigate the effect of
torques from the circumbinary gas on binary, which can shrink the orbital
separation. Additionally, one could also include the low-density AGN disk gas
as a background of the TPE simulations, instead of using vacuum. This is
challenging with SPH simulations, but could instead be feasible with grid-
based codes. Finally, it is also important to add radiation outflows from the
optically-thick accretion disk and shock properties due to the relative motion
of the star/BH and the debris, in order to more accurately describe TPEs.
Future work should perform simulations or make analytical predictions for TPEs
considering all of these additional factors above.

###  8.3 Detectability of TPEs as transients in AGNs

The AGNs are extremely dynamical locations to host luminous transients.
Identifying TPEs among different transient events in AGNs will require careful
examination of their EM signatures. AGNs around heavy SMBHs ($M_{\rm
SMBH}\gtrsim 10^{8}M_{\odot}$) are shown to be the ideal place for identifying
micro-TDEs (Yang et al., 2022) and other transients alike. In order to observe
TPEs, they need to outshine the AGN disk. Since our results show that TPEs
result in super-Eddington accretion onto the BH, there could be super-luminous
jet launching from the BH. Therefore, the EM emissions from TPEs can be
subject to jet modulation, among many other mechanism such as accretion disk
outflows and shocks, as mentioned in § 6.3. Even though the accretion disk
formed from the stellar remnant is optically thick, and the AGN can also trap
the radiation, the emissions from TPEs can be more visible if (i) the jet can
eject gas from the circumbinary disk (Tagawa et al., 2022), and (ii) stellar-
mass BHs can open cavities in the AGN disk (Kimura et al., 2021) – both of
these will reduce the opacity of the surrounding gas. Finally, if the AGN does
not launch any jets, then TPEs can outshine the AGN more easily in the radio
or in the gamma rays.

Here, we focus on the existing observational signatures of two micro-TDE
candidates observed in AGNs that might also indicate TPE origins. Micro-TDE
candidates in AGNs with a SMBH too massive for tidal disruption of a solar-
type star (ASASSN-15lh and ZTF19aailpwl; Yang et al., 2022), have peak
luminosity $L_{\rm peak}\approx 5\times 10^{45}$ erg s-1 and $L_{\rm
peak}\approx 10^{45}$ erg s-1. Yang et al. (2022) hypothesize that the higher
peak luminosity of ASASSN-15lh indicates a micro-TDE, unless it is a result of
tidal disruption of a star more massive than solar. From our simulations, we
see that TPEs with a more massive star also produce higher accretion rates.
The observations of ZTF19aailpwl show a longer rise time than a typical TDE,
indicating a more gradual tidal disruption than a TDE with SMBH, e.g. produced
by micro-TDEs with low eccentricity such as a tidal peeling event. Finally,
the rate of micro-TDEs are expected to be low in AGNs, at roughly $2$ Gpc-3
yr-1 (Yang et al., 2022), and even lower in star clusters or stellar triple
systems with BHs, while these predictions have large uncertainties. Only the
brightest events are expected to be eventually observed, since the emission of
most weaker micro-TDEs and TPEs will be dimmed significantly by the
surrounding AGN gas. The mechanism that the emission from an event like the
TPE propagates through an AGN disk is analogous to the propagation of GRB
afterglow in a dense medium (Perna et al., 2021; Wang et al., 2022).
Therefore, bright TPEs might have observational signatures similar to that of
ultra-long GRBs.

##  9 Summary

In this paper, we perform the first hydro-simulations of TPEs with the SPH
simulation code PHANTOM to investigate their morphology, accretion signature
and orbital evolution. We explore a range of initial conditions, including
stellar mass, initial eccentricity and penetration factor, which make up 96
simulation models in total. We examine the impacts of these initial parameters
on the behaviors of TPEs.

First, we observe the “tidal peeling” feature from our simulations where a
solar-like or massive star is slowly and periodically tidally disrupted by a
stellar-mass BH and its mass is slowly removed over many orbits. Due to low
eccentricity, the orbital periods of TPEs are generally shorter ($P\sim{\rm
few}$-few 10s of hours) compared to the micro-TDEs and TDEs. In the most
compact orbits, $r_{p}\approx R_{t}$, the star gets completely disrupted very
quickly, after $\sim$1-4 orbits; otherwise, the star ends up being partially
disrupted. Out of the three initial conditions, the penetration factor has the
largest effect on the accretion and orbital signatures of interest, namely
mass accreted onto the BH, accretion rate, the fraction of mass removed from
the star, the orbital separation, semi-major axis and eccentricity. As the
orbit becomes more compact, there is more mass accreted by the BH, higher
accretion rate and higher fraction of mass removed from the star. Lower
eccentricity has a similar effect, since lower $e_{0}$ means that the orbit is
shorter (recall that the star is placed at the apocenter at the start of the
simulations). A few models with higher eccentricities show a periodic
fluctuation in $\dot{M}_{\rm BH}$ that peaks after each pericenter passage.

The orbital separation, semi-major axis and eccentricity demonstrate less
obvious trends, especially when $\beta<1$ (less compact systems). It is clear
from the fluctuations in $a$ and $e$ that the orbit of a star in a TPE
deviates from Keplerian due to the tidal influence and possibly also shocks
from the stellar remnant encountering the tidal streams. In the most compact
configurations, $\beta=1$, the orbital separation always shrinks regardless of
the choice of $e_{0}$ and $M_{s}$, so both the semi-major and eccentricity
decrease with the number of orbits. In these cases, the star is always
completely disrupted at the end, consistent with the analytical limit of onset
mass loss of tidal stripping at $\beta=1$ (e.g. Zalamea et al., 2010).
Finally, if there is a more massive star in the TPE, the stellar radius is
larger and, at fixed $\beta$, it is closer to the tidal radius. Therefore, the
disruption is more rapid and total disruption of the star is more common.
There is higher mass loss from the star as well as more accretion by the BH.
However, for stars more massive than $1M_{\odot}$, the fraction of the initial
stellar mass lost or accreted by the BH does not vary significantly due to
different stellar masses. This indicates the similarity in the stellar
structures of the more massive stars.

The resulting accretion rates of TPEs are typically highly super-Eddington,
$\dot{M}_{\rm BH}\sim 10^{4-8}\dot{M}_{\rm Edd}$. However, since the accretion
disk formed from the dense stellar material around of the BH is extremely
opaque, the emission from TPEs will be affected by photon diffusion. Other
mechanisms might exist to modulate the luminosity of the TPE, other than the
BH accretion rate, such as relativistic jet launching from the BH and shocks
due to relative motion of the star remnant and the tidal streams. A jet might
empty a cocoon of low-density region around the TPE, possibly allowing the
emission to be less affected by the thick accretion disk or AGN disk. Our
results are also subject to a few caveats due to the limitations of our
simulations. Future work should address more realistic aspects of TPEs, such
as the radiation for the hot accretion disk, shocks, binary inspiral from a
farther separation, and/or AGN background gas.

Finally, better theoretical understanding of TPEs is highly motivated by the
existing observations of abnormal flaring events from AGNs, such as SASSN-15lh
and ZTF19aailpwl, that can not be well explained by AGN variability, or other
luminous transients such as TDEs by SMBHs. AGNs are extremely dynamical
playgrounds for interacting stars and compact objects. Our results suggest
that identifying TPEs among many different ultra-luminous transients can be
feasible due to its unique accretion signatures and orbital evolution that we
find in this work.

## Acknowledgements

ZH acknowledges support from NASA grant 80NSSC22K082. RP acknowledges support
by NSF award AST-2006839. YW acknowledges support from Nevada Center for
Astrophysics. CX acknowledges the support from the Department of Astronomy at
Columbia University for providing computational resources for this research.

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