# Spin-induced offset stream self-crossing shocks in tidal disruption events

T. Jankovič1, C. Bonnerot2, 3, A. Gomboc1
1 Center for Astrophysics and Cosmology, School of Science, University of Nova
Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
2 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17,
DK-2100 Copenhagen Ø, Denmark
3 School of Physics and Astronomy & Institute for Gravitational Wave
Astronomy, University of Birmingham, Birmingham B15 2TT, UK Contact e-mail:
<EMAIL_ADDRESS>
(Last updated 2021 November 14; in original form 2021 November 14)

###### Abstract

Tidal disruption events occur when a star is disrupted by a supermassive black
hole, resulting in an elongated stream of gas that partly falls back to
pericenter. Due to apsidal precession, the returning stream may collide with
itself, leading to a self-crossing shock that launches an outflow. If the
black hole spins, this collision may additionally be affected by Lense-
Thirring precession that can cause an offset between the two stream
components. We study the impact of this effect on the outflow properties by
carrying out local simulations of collisions between offset streams. As the
offset increases, we find that the geometry of the outflow becomes less
spherical and more collimated along the directions of the incoming streams,
with less gas getting unbound by the interaction. However, even the most
grazing collisions we consider significantly affect the trajectories of the
colliding gas, likely promoting subsequent strong interactions near the black
hole and rapid disc formation. We analytically compute the dependence of the
offset to stream width ratio, finding that even slowly spinning black holes
can cause both strong and grazing collisions. We propose that the deviation
from outflow sphericity may enhance the self-crossing shock luminosity due to
a reduction of adiabatic losses, and cause significant variations of the
efficiency at which X-ray radiation from the disc is reprocessed to the
optical band depending on the viewing angle. These potentially observable
features hold the promise of constraining the black hole spin from tidal
disruption events.

###### keywords:

methods: numerical – black hole physics – hydrodynamics – relativistic
processes

††pubyear: 2023††pagerange: Spin-induced offset stream self-crossing shocks in
tidal disruption events–References

##  1 Introduction

A stellar tidal disruption event (TDE) occurs at the center of a galaxy when a
star is scattered in close proximity of a supermassive black hole. This highly
energetic event, which can outshine the host galaxy, is a consequence of the
star being disrupted due to the black hole’s tidal field. After the
disruption, approximately half of the debris escapes from the gravitational
potential of the black hole, while the other half returns to the black hole’s
vicinity. There it can form an accretion disc, which may emit radiation for
months to years (Rees, 1988).

By observing the light curve of a TDE it is possible to learn more about the
properties of the disrupted star and its initial orbit, and also about the
properties of the supermassive black hole and accretion physics (Komossa,
2015; van Velzen & et al., 2019). The information about the orbit of the
disrupted star offers a way to study the stellar dynamics in galactic centers.
The mass and size of the star would provide a way to determine which stars are
more prone to be disrupted to get insight into the initial mass function in
the proximity of black holes. The emitted signal also depends on the black
hole’s mass and spin, whose determination is crucial for the understanding of
the demographic of black holes as well as their growth and evolution. To date,
around $100$ events of this type have been observed, in different ranges of
the electromagnetic spectrum (e.g. Alexander et al. 2020; Saxton et al. 2021;
Petrushevska et al. 2023). This field will be revolutionized starting 2025
with the advent of the Vera Rubin Observatory, which will detect thousands of
new TDEs, thus increasing the sample size by one or two orders of magnitude
(Bricman & Gomboc, 2020).

The evolution of a TDE starts with the stellar disruption when the star is
deformed into an elongated stream of debris, due to the black hole’s tidal
field. Bound parts of the debris move on a range of elliptic orbits and return
to the proximity of the black hole with a fallback rate that depends on the
properties of the system (e.g. Lodato et al. 2009; Guillochon & Ramirez-Ruiz
2013; Bonnerot et al. 2022; Jankovič & Gomboc 2023). Debris with more negative
energy returns sooner and collides with the still in-falling stream due to the
relativistic apsidal precession. This collision leads to a self-crossing shock
that dissipates the kinetic energy of the debris, causing the formation of an
outflow, part of which can later circularize into an accretion disc (Sądowski
et al., 2016; Liptai et al., 2019; Bonnerot & Lu, 2020; Bonnerot et al.,
2021).

A global numerical simulation of the entire TDE process requires a resolution
higher than what is computationally feasible to date, mostly due to the
significantly larger longitudinal extent of the debris stream than its
transverse width (Bonnerot & Stone, 2021). Therefore, several numerical
studies focused on disruptions of stars on elliptic orbits or by less massive
black holes, in which the difference between transverse and longitudinal
extents is smaller (Hayasaki et al., 2013; Shiokawa et al., 2015; Bonnerot et
al., 2016; Clerici & Gomboc, 2020). This numerical limitation prompted local
studies of the self-crossing shock considering more physically realistic
parameters, such as those conducted by Jiang et al. (2016) and Lu & Bonnerot
(2019), which demonstrated that the shock launches a large-scale outflow.
These studies assumed that the colliding streams are similar, an assumption
that depends on the effect of the nozzle shock, caused by a strong vertical
compression of the in-falling stream component near the pericenter. The
validity of this assumption has been later confirmed by Bonnerot & Lu (2022),
who found that the nozzle shock leads to minimal net expansion of the receding
stream’s height and width. However, Jiang et al. (2016) and Lu & Bonnerot
(2019) mostly focused on stream collisions near non-rotating black holes and
did not study the impact of the black hole’s spin in a systematic way.

In general, the black hole’s spin and stellar angular momentum are not
aligned, and, when the tip of the stream returns to the black hole’s vicinity,
its orbit is inclined with respect to the black hole’s rotational plane. In
this configuration, a relativistic effect known as Lense-Thirring precession
makes the gas angular momentum vector precess around the black hole spin,
potentially resulting in a misalignment between the colliding streams in the
self-crossing region (Bonnerot & Stone, 2021). In most extreme cases the
streams can even miss each other entirely and several additional revolutions
may be necessary for them to successfully collide (Batra et al., 2021).
Bonnerot & Lu (2022) found that the width of the stream component receding
from the black hole evolves at a slower rate than it was previously thought
(Guillochon & Ramirez-Ruiz, 2015), which can make this effect significant even
for slowly spinning black holes.

In this paper, we carry out the first systematic study of the effect of the
black hole’s rotation on the self-crossing shock. We perform local smoothed-
particle hydrodynamics (SPH) simulations of collisions between two streams
offset in the direction perpendicular to the orbital plane of the debris. We
analyze the properties of the outflow in the local co-moving frame and also
study the subsequent evolution of the outflow in the black hole’s reference
frame. We find that the outflow becomes less spherical as the black hole’s
spin increases, which also reduces the gas density along certain directions,
leading to potentially observable features that could be used to constrain the
black hole’s spin.

This paper is organized as follows. In Section 2, we explain the numerical
setup, initial conditions and the numerical procedure. The results are
presented in Section 3 and discussed in Section 4. We summarize our main
conclusions in Section 5.

##  2 SPH simulations

We simulate the collision between the two stream components involved in the
self-crossing shock. The colliding streams are constructed by continuously
injecting particles close to the intersection point. We consider the effect of
the black hole’s rotation by off-setting the streams such that only a fraction
of the gas directly collides.

###  2.1 Initial conditions

Lense-Thirring precession due to the black hole’s spin induces a vertical
offset $\Delta z$ between the two incoming streams with cylindrical radii
$H_{1}$ and $H_{2}$. By comparing $\Delta z$ to the sum of cylindrical radii
of streams $H_{1}+H_{2}$, it is possible to characterize the different types
of interactions. $\Delta z>H_{1}+H_{2}$ corresponds to a situation where
streams do not interact with one another and self-crossing is avoided. A
complete collision would happen for $\Delta z=0$. For $0<\Delta z\leq
H_{1}+H_{2}$ only parts of the streams collide.

We study the effect of the black hole’s rotation by simulating collisions of
two streams offset by different values of $\Delta z$ with the smoothed-
particle hydrodynamics code phantom (Price & et. al., 2018). Following Lu &
Bonnerot (2019), this simulation is carried out in the frame co-moving with
the common tangential velocity of the two incoming streams. In this frame, the
streams collide head-on with a velocity $v$ equal to their original radial
velocity. This setup is valid both if the collision is prompt or if it happens
after many windings (Batra et al., 2021).

Our initial setup consisting of two offset incoming streams is illustrated in
Figure 1. The coordinate system is defined with unit vectors
$\textbf{{e}}_{\rm x}$, $\textbf{{e}}_{\rm y}$, $\textbf{{e}}_{\rm z}$ and the
origin is set to the intersection point. Therefore, gas particles of one
stream have velocities $\textbf{{v}}_{\rm 1}=(0,v,0)$, while the velocity of
particles in the other stream is $\textbf{{v}}_{\rm 2}=(0,-v,0)$. We use the
same value $\dot{M}$ for the mass inflow rate of the two streams,
$\dot{M}_{1}=\dot{M}_{2}=\dot{M}$. We consider a collision of two streams with
circular cross-sections and equal widths $H_{1}=H_{2}=H$, where we define
width as the cylindrical radius. This assumption is justified by the recent
work of Bonnerot & Lu (2022), who find that the nozzle shock leads to almost
no net expansion, such that the stream size remains the same to order unity,
i.e. $H_{1}\approx H_{2}$. Values of parameters $v$, $\dot{M}$, and $H$ are
set to one, which also defines our code units. Streams are also cold due to
adiabatic expansion after the disruption. Therefore, we choose a value of the
specific internal energy $u=10^{-5}$ (expressed in code units), such that the
collision is highly supersonic.

In our simulations, we use an adiabatic equation of state with $\gamma=4/3$.
This is motivated by a large trapping radius of the shocked gas and the fact,
that the gas is radiation pressure dominated. The trapping radius
$R_{\mathrm{tr}}$ defines the distance from the intersection point, beyond
which the photons generated by shocks diffuse away from the outflowing gas. Lu
& Bonnerot (2019) showed that for a non-spinning black hole $R_{\rm
tr}/H\approx 100\gg 1$, implying that the outflow evolves adiabatically until
large distances. However, the situation might be different for a rotating
black hole where the collision is offset. In this case, may be possible for
photons to rapidly diffuse away along specific directions where the density is
lower. This possibility is further discussed in Section 4.3. The importance of
the radiation pressure can be determined by comparing the radiation pressure
$P_{\gamma}$ to the gas pressure $P_{\mathrm{g}}$. For typical conditions in
the self-crossing region, the radiation pressure dominates with a high ratio
$P_{\gamma}/P_{\mathrm{g}}\approx 1000\gg 1$ (Jiang et al., 2016; Lu &
Bonnerot, 2019).

###  2.2 Numerical procedure

As illustrated in Figure 1, the streams are created by cyclically injecting
SPH particles from a cylinder, which is constructed in the following way. We
derive gas density in the stream from $\rho=\dot{M}/(\pi H^{2}v)$, which we
use to create a 3D cube with size $2H$ and total mass
$M_{\mathrm{cube}}=\rho(2H)^{3}$. The cube is populated by $N$ particles with
mass $m=10^{-3}$, where $N=M_{\mathrm{cube}}/m$, which also determines the
resolution of our simulations. For this choice of $m$, the smoothing length of
SPH particles is $h_{\mathrm{sl}}\approx(m/\rho)^{1/3}\approx 0.1H<H$, meaning
that the resolution is sufficiently high.111To assess the accuracy of our
study more thoroughly, we simulated stream collisions also for particle masses
$m=10^{-4}$ and $10^{-2}$. We found no significant differences when decreasing
the mass to less than $m=10^{-3}$, indicating that the resolution used in our
simulations is sufficiently high. We use a glass distribution, a distribution
where the distances between particles are equal, to avoid the noise resulting
from a random particle distribution. The glass distribution is reached by
relaxing the particle distribution inside the cube using periodic boundary
conditions, from which we then select a cylinder with radius $H$ and height
$2H$.

The particles inside this cylinder are not all evolved initially, but instead,
act as “ghost” particles (blue particles inside the grey regions) that are
progressively injected inside the computational domain from the surfaces at
$y_{\mathrm{inj}}=\pm 3H$ (black particles inside the orange regions in Figure
1). At each timestep $\Delta t$, this is done by shifting their position by
$\Delta y=\pm v\Delta t$, and injecting the ghost particles that have left the
cylinder from the top or bottom. The number of injected particles is $\Delta
N=\dot{M}\Delta t/m$. While particles initially possess mostly kinetic energy,
dissipation happens through a shock when the streams collide, leading to the
formation of an outflow, which we now investigate in detail.

Figure 1: Sketch showing our setup for the local simulations of collisions
between offset streams. The streams are created by injecting SPH particles
with opposite velocities $\textbf{{v}}_{1}$ and $\textbf{{v}}_{2}$ from two 3D
cylinders of radius $H$ and length $2H$ that are offset by $\Delta z$. This is
done by defining ghost particles inside these cylinders (blue points in the
grey regions) that are cyclically added to the computational domain, where
these particles (black points in the orange regions) are then hydrodynamically
evolved by the code.

##  3 Results

In this section we present the properties of the outflow from the self-
crossing shock produced by colliding streams. The collision is simulated in a
local co-moving frame according to the procedure described in Section 2,
considering different offsets between the streams.222Movies made from the
simulations are available online at https://github.com/tajjankovic/Spin-
induced-offset-stream-self-crossing-shocks-in-TDEs/tree/main/Movies. We
determine various properties of the outflow in the co-moving and in the black
hole’s reference frames.

###  3.1 Collision dynamics and shock heating rate

Figure 2 shows the 3D density of gas $\rho$ contained in slices in the $zy$
plane at $x=0$ for stream offsets $\Delta z=0$, $0.3H$ and $1.2H$. Gas
particles are injected at a time $t=0$ from the surfaces at $y_{\mathrm{inj}}$
(indicated with thick light blue segments), and the streams collide at $t=3$.
Trajectories of three different particles are shown as coloured lines, where
we use the same colour for particles injected at the same position and time,
but varying the vertical offset. In the $\Delta z=0$ case (left panel) all the
incoming gas is directly involved in the collision, which leads to the
formation of a shock that dissipates kinetic energy. Consequently, pressure
forces sharply increase and an outflow is launched from the collision region
that is symmetric both azimuthally and with respect to the equatorial plane at
$y=0$. After the gas has been forced to move from the collision region along
the equatorial plane at $\theta={\pi}/{2}$, its trajectories get further
deflected away from this plane by vertical pressure forces (see particles
trajectories in Figure 2), such that the outflowing gas moves along a wide
range of polar angles. The path of a particle becomes straight after it
reaches a radial distance $r\approx 3H$ from the intersection point, due to a
sharp reduction of pressure forces. Inside the outflow, the internal energy of
the gas gets converted back to kinetic energy until it reaches a terminal
velocity equal to the initial (pre-collision) velocity, as expected from
energy conservation. With time the outflow keeps expanding to reach a steady-
state, a state where the properties of the outflow do not change with time.

For $\Delta z>0$, only a fraction of the incoming streams directly collides.
However, the shock from this directly colliding gas affects also the non-
directly colliding parts of the stream. The reason is that the shock wave
launched by the direct collision propagates outward to affect even the gas
furthest away from the intersection point. This can be seen from the highest
density parts in Figure 2 (middle panel) for the $\Delta z=0.3H$ case, which
indicates the presence of shocked gas outside the directly colliding gas
region. The momentum of the non-directly colliding gas does not get entirely
cancelled in the $\textbf{{e}}_{\rm y}$ direction, causing the outflow to be
deflected away from the equatorial plane. This effect is the strongest for
$\Delta z=1.2H$, where we see that the trajectories of all three particles
remain almost aligned with the incoming stream direction (right panel of
Figure 2). Distinctions between the different panels of Figure 2 suggest that
stream collisions can be classified into two regimes, which we will refer to
as “strong collision” and “grazing collision”. In the strong collision regime,
the outflow is close to spherical, being only slightly deflected from the
equatorial plane. In the grazing collision regime, the outflow is primarily
collimated along the trajectories of the incoming streams, which only display
a low level of expansion.

$\Delta z/H=0.0$ $\Delta z/H=0.3$ $\Delta z/H=1.2$

Figure 2: Density slices in the $zy$ plane at $x=0$ and time $t=100$ for
values of the offset $\Delta z=0$ (left panel), $\Delta z=0.3H$ (middle panel)
and $\Delta z=1.2H$ (right panel). Coloured lines show the trajectories of
three particles, using the same colour for particles injected at the same
position and time, but varying the vertical offset.

The transition between the two regimes can be determined quantitatively and it
is a consequence of the interaction between the gas in the incoming streams
and the outgoing matter that has already passed through the self-crossing
shock. This interaction results in a deflection of the gas trajectories inside
the incoming streams, which reduces the amount of the incoming gas that
directly collides. This mechanism can be understood more precisely from Figure
3, where the index “1” refers to the gas from the incoming (upward-moving)
stream involved in the collision, and the index “2” refers to the shocked
outgoing (downward-moving) gas that causes the aforementioned deflection.
Indices “i” and “f” correspond to stream velocities before and after the
interaction between these two gas components, respectively. We estimate the
distance $d$ by which the incoming stream gets deflected (towards the left),
by modelling the interaction as an elastic collision confined to a single
deflection point (red point in Figure 3). This distance is obtained from
$d\approx h\,|v_{\mathrm{1z}}|/v_{\mathrm{1y}}$, where $h$ is the distance
between the deflection point and the equatorial plane, and the two speeds are
defined as
$v_{\mathrm{1z}}\equiv\textbf{{v}}_{\mathrm{1,f}}\cdot\textbf{{e}}_{\mathrm{z}}$
and
$v_{\mathrm{1y}}\equiv\textbf{{v}}_{\mathrm{1,f}}\cdot\textbf{{e}}_{\mathrm{y}}$.
Momentum conservation imposes that
$v_{\mathrm{1z}}=2\dot{M}_{\mathrm{2,d}}v_{\mathrm{2z}}/\dot{M}_{\mathrm{1,d}}$,
where $\dot{M}_{\mathrm{1,d}}$ and $\dot{M}_{\mathrm{2,d}}$ denote the mass
inflow rates of the two gas components involved in the deflection, and
$v_{\mathrm{2z}}\equiv\textbf{{v}}_{\mathrm{2,i}}\cdot\textbf{{e}}_{\mathrm{z}}$.
We note that not all the gas from the incoming stream is deflected, but only
that contained inside the circular segment at a distance $d$ from the edge of
the stream. Evaluating the area of the circular segment $A_{\mathrm{d}}\approx
2^{5/2}3^{-1}(d/H)^{3/2}H^{2}$ at the lowest order in $d/H\ll 1$,333We note
that this assumption is not entirely accurate, but we make it in order to find
a simple analytical equation for $d/H$. Furthermore, we have verified our
analytic approach by evaluating $A_{\mathrm{d}}$ without approximations and
calculating $d/H$ numerically. We found only minor differences in comparison
to the analytic estimate. and making use of flux conservation
$\dot{M}_{\mathrm{1,d}}=\dot{M}A_{\mathrm{d}}/(\pi H^{2})$, we find

$\displaystyle\frac{d}{H}$
$\displaystyle\approx\left(\frac{3\pi}{2^{3/2}}\frac{\dot{M}_{\mathrm{2,d}}}{\dot{M}}\frac{|v_{\mathrm{2z}}|}{v_{\mathrm{1y}}}\frac{h}{H}\right)^{2/5},$
$\displaystyle\approx
0.64\left(\frac{\dot{M}_{\mathrm{2,d}}}{0.5\dot{M}}\right)^{2/5}\left(\frac{|v_{\mathrm{2z}}|}{0.1v_{\mathrm{1y}}}\right)^{2/5}\left(\frac{h}{2H}\right)^{2/5}.$
(1)

The numerical values we used here are obtained from the simulations, which
lead to an evaluated distance $d\approx 0.64H$. This deflection causes an
effective decrease of the stream width, such that a direct collision is
eventually avoided if $\Delta z>2(H-d)\approx 0.7H$. This analytical condition
matches that found from the simulations to be in the grazing collision regime
defined above. In that case, even though some of the incoming gas directly
collides early on, at later times the streams end up passing above and below
each other without experiencing significant dissipation (see the rightmost
panel of Figure 2). After the collision, the streams therefore, display only a
low level of expansion while the trajectory of their center of mass is largely
unaffected.

Figure 3: Density slice in the $zy$ plane at $x=0$ and time $t=7$, which
illustrates the interaction between the gas in the incoming (upward-moving)
streams (indices “1”) and the outgoing (downward-moving) matter (indices “2”)
that has already passed through the self-crossing shock, for a value of the
offset $\Delta z=1H$. The interaction (denoted by a red point) occurs at a
distance $h$ from the equatorial plane and causes a deflection of the incoming
stream by a distance $d$. Vector arrows represent gas velocities, where we use
index “i” and “f” for components before and after the deflection,
respectively.

When the two streams collide a shock is formed and the kinetic energy of the
gas is dissipated due to shock heating. We obtain an analytic estimate of the
shock heating rate $\dot{E}$ from

$\dot{E}\approx\xi(\Delta z)\dot{M}\Delta u,$ (2)

where we estimate the jump in specific internal energy $\Delta u$ from the
Rankine-Hugoniot conditions as $\Delta u={2}v^{2}(\gamma+1)^{-2}=0.37v^{2}$,
using $\gamma=4/3$. In Equation (2) we introduce a factor $\xi(\Delta
z)={A_{\mathrm{coll}}(\Delta z)}/{A_{\mathrm{0}}}\in[0,1]$, which represents
the fraction of the stream surfaces that directly collides initially.
$A_{\mathrm{coll}}$ is calculated as the intersection between the surfaces of
two circular streams with a radius equal to $H$ that are offset by $\Delta z$,
whereas $A_{\mathrm{0}}=\pi H^{2}$ is the cross-section of a stream. When
$\Delta z=0$, the streams overlap and $A_{\mathrm{coll}}=A_{\mathrm{0}}$.
However, as $\Delta z$ increases, the intersection between the streams
surfaces decreases, and is reduced to zero when $\Delta z=2H$.

We have simulated 19 stream collisions corresponding to values of the offset
$\Delta z\in[0,1.8H]$,444For larger values of $\Delta z$, stream interactions
are no longer physical due to the smoothing length of SPH particles, which
extends outside of the stream boundaries. This has the effect of artificially
enhancing the level of interaction between the streams, while for such large
offsets we would physically expect the streams to be largely unaffected. with
an increment of $0.1H$. The comparison between values of $\dot{E}$ from
simulations (red points) and values calculated from Equation (2) (black dashed
line) is shown in Figure 4. For $\Delta z\approx 0$ the values of $\dot{E}$
from simulations are lower than the analytic estimate. We attribute this to
boundary effects — values of the energy generated in shocks from simulations
are lower because there is no external pressure outside of streams, leading to
an expansion that adiabatically reduces the internal energy of the outflow. As
the offset increases the shock heating rate becomes larger than the analytic
estimate. This is because the gas that does not directly collide is still
affected by the self-crossing shock due to shockwave propagation (e.g. see
middle panel of Figure 2). At $\Delta z\approx 0.7H$ we notice a sharp decline
in $\dot{E}$, which marks the transition from the strong collision to the
grazing collision regime. For $\Delta z\gtrsim 0.7H$, energy dissipation is
lower than analytically predicted due to the effect described above (see
Equation (3.1)) that causes an effective decrease of the streams widths.

Figure 4: Shock heating rate $\dot{E}$ as a function of the offset $\Delta z$
expressed in units of the stream width $H$. $\dot{E}$ is normalized by
$\dot{M}_{\mathrm{tot}}v^{2}$, where $\dot{M}_{\mathrm{tot}}=2\dot{M}$ is the
total inflow rate in the collision region and $v$ is the incoming stream
velocity. The dashed line is calculated from Equation (2), while the red dots
represent values obtained directly from simulations.

###  3.2 Angular dependence of the outflow rate

We now calculate the dependence of the outflow rate $\dot{M}_{\rm out}$ on the
direction, specified by the polar and azimuthal angles $\theta$ and $\phi$. To
this aim, we first construct a spherical map, where each pixel covers the same
surface area $A$, using the pixelation scheme of the HEALPix555Current link to
the HEALPix website: https://healpix.sourceforge.io/. software (Górski et al.,
2005; Zonca et al., 2019). We calculate the outflow rate $\dot{M}_{\rm
out}=\mathrm{d}m/\mathrm{d}t$, where $\mathrm{d}m$ is the mass of gas that
crossed a given pixel during a time interval $\mathrm{d}t$. We determine
$\mathrm{d}m$ and $\mathrm{d}t$ by comparing two simulation snapshots at
different times, $t_{1}=100$ and $t_{2}=150$, at which the steady-state is
reached. With this procedure, we are able to construct HEALPix maps of the
outflow rate. We use these maps to calculate the normalized mass flux

$F=(\dot{M}_{\rm out}/\dot{M}_{\mathrm{tot}})/A_{\mathrm{}},$ (3)

through individual pixels, where the total outflow rate through the whole
sphere is $\dot{M}_{\mathrm{tot}}=2\dot{M}=2$ due to mass conservation.

In Figure 5 we show spherical projections of the normalized mass flux
distributions for $\Delta z/H=0,0.3,0.6,0.8,1.1,1.4$. For $\Delta z=0$ we can
see that the outflow rate is higher near the equatorial plane, consistently
with the findings by Lu & Bonnerot (2019).666We also notice a slight increase
of $F$ at polar angles within $\sim 10^{\circ}$ degrees of the poles compared
to the work by Lu & Bonnerot (2019), which is likely a a consequence of the
colliding with itself near this location. However, this difference is
restricted to a small fraction of the outflowing gas and therefore does not
affect our results. As the offset between the streams increases, the outflow
is more collimated and aligned with the direction of the incoming streams, as
explained in Section 3.1. In Figure 5, we also see a clear distinction between
the two regimes we defined above. In the strong collision regime ($\Delta
z\lesssim 0.7H$, first three panels), the outflow is partially deflected from
the equatorial plane and spans a wide range of polar directions. In the
grazing collision regime ($\Delta z\gtrsim 0.7H$, last three panels), the
outflow spans only a narrow range of $\theta$ and $\phi$ with substantially
higher values of the mass flux.

One purpose of calculating the normalized mass flux $F$ is related to the
method of simulating the subsequent phase of accretion disc formation in TDEs,
as described in Bonnerot & Lu (2020) and Bonnerot et al. (2021) who model the
outflow from the self-crossing shock through an injection of SPH particles
from the intersection point. $F$ is a quantity that satisfies the condition
$\int_{0}^{\pi}\int_{0}^{2\pi}F(\theta,\phi)\sin{\theta}\mathrm{d}\phi\mathrm{d}\theta=1$
when integrated over a unit sphere, which makes it the appropriate probability
distribution to model the outflow using particle injection. For a given
direction determined by $\theta$ and $\phi$ we obtain $F$ from a HEALPix map
of $\dot{M}_{\rm out}$ (see Equation 3). It is obtained directly for the
values of $\Delta z$ we have simulated, while for the others we rely on linear
interpolation. For example, a HEALPix map for $\Delta z=0.22\,H$ is obtained
by linearly interpolating between $\Delta z=0.2\,H$ and $\Delta
z=0.3\,H$.777We tried to obtain an analytic expression for the angular
distribution of the outflow rate by fitting spherical harmonics functions to
HEALPix maps of $\dot{M}_{\rm out}$. However, we found this procedure not
suitable due to a high number of independent spherical harmonics functions
needed to produce a good fit. We have made the HEALPix maps obtained from our
simulations and the code to extract $F$ from values of $\Delta z$, $\theta$
and $\phi$ publicly available to the community at
https://github.com/tajjankovic/Spin-induced-offset-stream-self-crossing-
shocks-in-TDEs.git for use in future research.

Figure 5: Filled contours of spherical (Mollweide) projections of the
normalized mass flux $F$ for the gas outflowing from the self-crossing region
in terms of $\theta$ (values along the edges of projections) and $\phi$
(values along the horizontal directions) for different $\Delta z$. We use
different colorbar limits for the strong (top left, top right and middle left
panels) and the grazing (middle right, bottom left and bottom right panels)
collision regimes, due to the large difference in the gas density. Projections
are smoothed with a Gaussian symmetric beam implemented in HEALPix.

###  3.3 Subsequent evolution of the outflowing gas

The outflow from the self-crossing shock can be either unbound and escape or
bound and return to the black hole’s vicinity, where it may form an accretion
disc. The fate of the outflow can be determined by transforming the gas
properties from the local co-moving frame to the black hole’s reference frame,
and calculating its specific energy $\epsilon$ and specific angular momentum
$\ell$ as well as determining its trajectories. We use the description of the
self-crossing shock presented in Lu & Bonnerot (2019) and Bonnerot & Stone
(2021), adopting a Newtonian description of the black hole’s gravity for
simplicity.

We consider disruptions of a star with mass $M_{\star}$ and radius $R_{\star}$
by a black hole with mass $M_{\mathrm{bh}}$. The star is initially on a
parabolic orbit, with an angular momentum vector inclined with respect to the
black hole spin vector, and characterized by the impact parameter
$\beta=R_{\mathrm{t}}/R_{\mathrm{p}}$, where $R_{\mathrm{p}}$ is the
pericenter distance and
$R_{\mathrm{t}}=R_{\star}(M_{\mathrm{bh}}/M_{\star})^{1/3}$ is the tidal
radius. The star has an initial specific angular momentum
$\ell_{\star}=\sqrt{2GM_{\mathrm{bh}}R_{\mathrm{p}}}$. Disruption by the black
hole’s tidal forces induces an energy spread of the debris
$\Delta\epsilon=GM_{\mathrm{bh}}R_{\star}/R_{\mathrm{t}}^{2}$ (Stone et al.,
2013). After the disruption, bound debris moves on a range of elliptic orbits,
the most bound one having a specific energy $-\Delta\epsilon$ and eccentricity
$e=1-\left(2/\beta\right)\left(M_{\mathrm{bh}}/M_{\star}\right)^{-1/3}$. We
make a choice that the streams have the same energy $-\Delta\epsilon$, and we
discuss the consequences of the streams having different energies in Section
4.1. Following its pericenter passage, the most bound debris now moving away
from the black hole collides (for a low enough vertical offset) with the
still-infalling gas at a distance given by the intersection radius

$R_{\mathrm{int}}=R_{\rm p}\frac{1+e}{1-e\cos(\Delta\phi/2)},$ (4)

where $\Delta\phi\approx 3\pi R_{\mathrm{g}}/R_{\mathrm{p}}$ is the apsidal
precession. $R_{\mathrm{int}}$ determines the position in the black hole’s
reference frame of the local co-moving frame where the above simulations were
carried out. We assume that this position is point-like, due to the condition
$H\ll R_{\mathrm{int}}$ (Bonnerot & Lu, 2022), and located along the $y$ axis
at $z=0$.

Defining $\textbf{{e}}_{\mathrm{z}}$ along the angular momentum of the in-
falling stream component, Lense-Thirring precession can deflect the receding
stream either in the $+\textbf{{e}}_{\mathrm{z}}$ or
$-\textbf{{e}}_{\mathrm{z}}$ direction from the orbital plane. We focus here
on the situation where the deflection is along the
$-\textbf{{e}}_{\mathrm{z}}$ direction, since the other case leads to similar
dynamics as long as the value of $\Delta z/H$ is the same. In the black hole’s
reference frame, the stream receding from the black hole has velocity
$\textbf{{v}}_{1}^{\mathrm{int}}$, while $\textbf{{v}}_{2}^{\mathrm{int}}$ is
the velocity of the in-falling gas. The two stream components have the same
magnitude
$v_{\mathrm{int}}=|\textbf{{v}}_{1}^{\mathrm{int}}|=|\textbf{{v}}_{2}^{\mathrm{int}}|$.
We determine the speed of the colliding flows at the intersection point as
$v_{\mathrm{int}}=\left(2GM_{\mathrm{bh}}/R_{\mathrm{int}}-2\Delta\epsilon\right)^{1/2}$
and calculate the intersection angle measured between the two velocity vectors
$\cos\psi=\textbf{{v}}_{1}^{\mathrm{int}}\cdot\textbf{{v}}_{2}^{\mathrm{int}}/v_{\mathrm{int}}^{2}$,
as in Dai et al. (2015)

$\cos{\psi}=\frac{1-2e\cos(\Delta\phi/2)+e^{2}\cos\Delta\phi}{1-2e\cos(\Delta\phi/2)+e^{2}}.$
(5)

We use $v_{\mathrm{int}}$ and $\psi$ to determine the radial
$v_{\mathrm{r}}=\textbf{{v}}_{1}^{\mathrm{int}}\cdot\textbf{{e}}_{\mathrm{y}}=-\textbf{{v}}_{2}^{\mathrm{int}}\cdot\textbf{{e}}_{\mathrm{y}}=v_{\mathrm{int}}\sin(\psi/2)$
and tangential
$v_{\mathrm{t}}=\textbf{{v}}_{1}^{\mathrm{int}}\cdot\textbf{{e}}_{\mathrm{x}}=\textbf{{v}}_{2}^{\mathrm{int}}\cdot\textbf{{e}}_{\mathrm{x}}=v_{\mathrm{int}}\cos(\psi/2)$
velocity components. In the local co-moving frame, moving with a velocity
equal to $v_{\mathrm{t}}$, streams collide with velocities $\pm
v_{\mathrm{r}}$ and a shock is formed that launches an outflow (see Section
3.2). We fix the velocity of the outflow to $|v_{\mathrm{r}}|$ (same as the
collision speed) as found in the simulations. Therefore, the velocity along an
arbitrary direction given by the polar and azimuthal angles $\theta$ and
$\phi$ in the co-moving frame (corresponding to an arbitrary pixel in the flux
map of Figure 5) can be determined from
$\textbf{{v}}_{\mathrm{out}}(\theta,\phi)=v_{\mathrm{r}}(\sin\theta\cos\phi\textbf{{e}}_{\mathrm{x}}+\cos\theta\textbf{{e}}_{\mathrm{y}}+\sin\theta\sin\phi\textbf{{e}}_{\mathrm{z}})$.
The velocity components of the outflow in the black hole’s frame are then
obtained via the Galilean transformation
$\textbf{{v}}_{\mathrm{out}}^{\prime}(\theta,\phi)=\textbf{{v}}_{\mathrm{out}}(\theta,\phi)+v_{\mathrm{t}}\textbf{{e}}_{\mathrm{x}}$.

We calculate the specific energy
$\epsilon=-GM_{\mathrm{bh}}/R_{\mathrm{int}}+v_{\mathrm{out}}^{\prime 2}/2$
and the specific angular momentum
$\bm{\ell}=\textbf{{R}}_{\mathrm{int}}\times\textbf{{v}}_{\mathrm{out}}^{\prime}$
of the outflow in the black hole’s reference frame, where
$\textbf{{R}}_{\mathrm{int}}=R_{\mathrm{int}}\textbf{{e}}_{\mathrm{y}}$ is the
radial vector to the intersection point. We define a 2D grid of $\theta$ and
$\phi$, and determine $\epsilon$ and $\bm{\ell}$ for every $(\theta,\phi)$
pair. We obtain the normalized flux $F$ (see Equation (3)) for these angles by
interpolating from HEALPix maps, which allows us to calculate the flux-
weighted distributions of $\epsilon$ and
$\ell_{\mathrm{z}}=\bm{\ell}\cdot\textbf{{e}}_{\mathrm{z}}$ shown in Figure 6
for different $\Delta z/H$. We note that negative values of
$\ell_{\mathrm{z}}/\ell_{\star}$ correspond to a part of the outflow that is
counter-rotating (moving on retrograde orbits) with respect to the initial
stellar trajectory. The stream collision induces a spread in the distributions
of $\epsilon$ and $\ell_{\mathrm{z}}$ around the values before the collision,
$-\Delta\epsilon$ and $\ell_{\star}$, respectively. In the strong collision
regime, where $\Delta z\lesssim 0.7H$, the offset distributions are very
similar to the $\Delta z/H=0$ case, displaying a sharp cut-off, where the
outflow is maximally accelerated or decelerated during the collision. In the
grazing regime, where $\Delta z\gtrsim 0.7H$, the outflow is more collimated,
and the distributions are narrower with a pronounced peak around pre-collision
values. The widths of the distributions are no longer determined by a sharp
cut-off, but instead gradually decrease due to the decrease in the mass flux.

Figure 6: Distributions of orbital energy $\epsilon$ (upper panel) and angular
momentum $\ell_{\mathrm{z}}$ projected along the initial stellar angular
momentum (lower panel) for different values of the offset $\Delta z/H$,
assuming $M_{\star}=0.5\,\mathrm{M_{\odot}}$,
$R_{\star}=0.46\,\mathrm{R_{\odot}}$, $M_{\mathrm{bh}}=2.5\times
10^{6}\,\mathrm{M_{\odot}}$ and $\beta=1$. Each bin of the histograms is
weighted by the total flux (mass fraction) within its indicated specific
energy or angular momentum range. $\epsilon$ and $\ell_{\mathrm{z}}$ are
normalized by the values before the collision $\Delta\epsilon$ and
$\ell_{\star}$, respectively. The integral of distributions is normalized to
1.

At a fixed $\Delta z/H$ and $M_{\mathrm{bh}}$ the flux-weighted distributions
of $\epsilon$ and $\ell_{\mathrm{z}}$ distributions have the same shape. This
can be shown by calculating analytically the maximum spread in energy
$\Delta\epsilon_{\mathrm{max}}$ and angular momentum
$\Delta\ell_{\mathrm{max}}$ after the collision. By introducing a unit vector
$\textbf{{e}}_{\mathrm{r,max}}$, which points radially from the collision
point in the direction where the gas has the largest energy and angular
momentum, it is possible to derive
$\Delta\epsilon_{\mathrm{max}}=v_{\mathrm{r}}v_{\mathrm{t}}\textbf{{e}}_{\mathrm{r,max}}\cdot\textbf{{e}}_{\mathrm{\phi}}$
and
$\Delta\ell_{\mathrm{max}}=R_{\mathrm{int}}v_{\mathrm{r}}\textbf{{e}}_{\mathrm{r,max}}\cdot\textbf{{e}}_{\mathrm{\phi}}$.
Therefore, in a disruption of a star with $M_{\star}=0.5\,\mathrm{M_{\odot}}$,
$R_{\star}=0.46\,\mathrm{R_{\odot}}$888Stellar radius is obtained from fitting
formulae for the zero-age main-sequence radii as functions of their masses
presented in Tout et al. (1996) assuming solar metallicity. on a $\beta=1$
orbit by a $M_{\mathrm{bh}}=2.5\times 10^{6}\,\mathrm{M_{\odot}}$ black hole,
$\Delta\ell_{\mathrm{max}}/\Delta\epsilon_{\mathrm{max}}=R_{\mathrm{int}}/v_{\mathrm{t}}\approx
3.3\ell_{\star}/\Delta\epsilon$, which is in agreement with the ratio obtained
by comparing the two panels of Figure 6. The fate of the outflow can be
understood by considering the spread of flux-weighted distributions of
$\epsilon$ and $\ell_{\mathrm{z}}$, around the values before the collision,
which depend on the black hole mass. In disruptions by black holes with
$M_{\mathrm{bh}}\in[10^{6},10^{7}]\,\mathrm{M_{\odot}}$, approximations
$e\approx 1$, and $0<\Delta\phi\lesssim\pi/2$ are valid. By expanding
$\cos(\Delta\phi/2)$ to the second order in Taylor series in Equation
$\eqref{eq:Rint}$, we find $R_{\mathrm{int}}\propto M_{\mathrm{bh}}^{-1}$.
Additionally, we use approximations $v_{\mathrm{int}}\approx v_{\mathrm{r}}$
and $v_{\mathrm{t}}\approx\ell_{\star}/R_{\mathrm{int}}$ to derive
$v_{\mathrm{r}}\propto M_{\mathrm{bh}}$ and $v_{\mathrm{t}}\propto
M_{\mathrm{bh}}^{5/3}$, respectively. By assuming a collision with no offset,
where $\textbf{{e}}_{\mathrm{r,max}}\cdot\textbf{{e}}_{\mathrm{\phi}}=1$, we
obtain $\Delta\epsilon_{\mathrm{max}}/\Delta\epsilon\propto
M_{\mathrm{bh}}^{7/3}$ and $\Delta\ell_{\mathrm{max}}/\ell_{\star}\propto
M_{\mathrm{bh}}^{-2/3}$, meaning that there is more unbound gas and less gas
on retrograde orbits as $M_{\mathrm{bh}}$ increases.

From the flux-weighted distributions of $\epsilon$ and $\ell_{\mathrm{z}}$ we
calculate the distribution of the pericenter distances of the outflow. As
$\Delta z/H$ increases, the distributions become narrower around the stellar
pericenter $\approx R_{\mathrm{p}}$ due to the more stream-like outflow.
However, even for $\Delta z=1.8H$ that is the most grazing collision we
considered, the spread remains of order unity with $\Delta
R_{\mathrm{p}}\approx R_{\mathrm{p}}\gg R_{\star}$, implying that the shocked
stream has expanded significantly compared to its pre-collision state. This
expansion along with the resulting differences in apsidal precession angles
will likely affect the subsequent gas evolution, as we discuss in Section 4.2.

The unbound outflow has a positive energy, while gas with a negative energy is
bound. The mass fractions $f_{\mathrm{unb}}$ of unbound gas for disruptions of
a $M_{\star}=0.5\,\mathrm{M_{\odot}}$ star as a function of $M_{\mathrm{bh}}$
for $\Delta z/H=0$, 0.6, 0.8, 1.2 are shown in Figure 7.999For
$M_{\star}=0.5\,\mathrm{M_{\odot}}$ and $M_{\mathrm{bh}}\gtrsim 6.2\times
10^{6}\,\mathrm{M_{\odot}}$ the initial angular momentum $\ell_{\star}$ is
lower than the angular momentum of the marginally bound parabolic orbit in the
Schwarzschild spacetime $\ell_{\mathrm{mb}}=4R_{\mathrm{g}}c$. For clarity, we
calculate $f_{\mathrm{unb}}$ also for higher $M_{\mathrm{bh}}$ even if all of
the gas is swallowed by the black hole. We see, that for
$M_{\mathrm{bh}}\lesssim 2\times 10^{6}\,\mathrm{M_{\odot}}$ there is only
bound gas, due to a smaller spread in energy. As the black hole mass
increases, the fraction of unbound gas $f_{\mathrm{unb}}$ increases up to 50%,
which is reached for $M_{\mathrm{bh}}\gtrsim 3\times
10^{7}\,\mathrm{M_{\odot}}$. The dependence on $\Delta z/H$ in the strong
collision regime (blue solid and red dashed lines) is very weak due to very
similar flux-weighted distributions $\epsilon$ and $\ell_{\mathrm{z}}$ (see
Figure 6). The change in $f_{\mathrm{unb}}$ is most apparent in the transition
from the strong to the grazing collision regime (green dotted and orange dash-
dotted lines).

Figure 7: The mass fraction $f_{\mathrm{unb}}$ of the unbound gas for
different stream offsets $\Delta z/H$ and black hole masses $M_{\mathrm{bh}}$,
for $M_{\star}=0.5\,\mathrm{M_{\odot}}$, $R_{\star}=0.46\,\mathrm{R_{\odot}}$
and $\beta=1$.

At a fixed $M_{\mathrm{bh}}$ there is less unbound outflow for higher $\Delta
z/H$. This can be better understood from Figure 8 where we show spherical
projections of the normalized mass flux $F$ from the self-crossing
region101010Figure 8 is similar to Figure 5, except that it is rotated by an
angle $\phi=-90^{\circ}$, meaning that the center and the boundary of Figure 8
are in the $-\textbf{{e}}_{\mathrm{z}}$ and $+\textbf{{e}}_{\mathrm{z}}$
direction, respectively. From Figure 5 one might expect that the stream
components for $\Delta z/H=1.2$ would appear symmetric with respect to the
horizontal axis. However, the stream moving toward
$-\textbf{{e}}_{\mathrm{z}}$ direction is located on the boundary of the
Figure 8 and therefore appears to be more extended due to projection effects.
and highlight the zero-energy contours — the unbound gas is located inside the
grey borders for $M_{\mathrm{bh}}=2\times 10^{6}\,\mathrm{M_{\odot}}$ (solid
line), $3\times 10^{6}\,\mathrm{M_{\odot}}$ (dashed line) and $5\times
10^{6}\,\mathrm{M_{\odot}}$ (dotted line). In the strong collision regime (two
uppermost panels), the contour for $M_{\mathrm{bh}}=3\times
10^{6}\,\mathrm{M_{\odot}}$ already encloses a large fraction of the unbound
outflow. However, in the grazing collision regime (lowermost panel), the
unbound region extends over the outflow only for $M_{\mathrm{bh}}=5\times
10^{6}\,\mathrm{M_{\odot}}$, which is consistent with the black hole mass
above which the unbound fraction becomes significant (see Figure 7).

Figure 8: Filled contours of spherical (Mollweide) projections of the
normalized mass flux $F$ for the gas outflowing from the self-crossing region
for $\Delta z/H=0$ (top row), 0.6 (middle row) and 1.2 (bottom row). We use
different colourbar limits for the strong and the grazing collision regime,
due to the large difference in the gas density. Projections are smoothed with
a Gaussian symmetric beam implemented in HEALPix. Grey lines represent the
zero-energy contours for $M_{\mathrm{bh}}\,=2$, 3 and $5\times
10^{6}\,\mathrm{M_{\odot}}$, which are denoted by solid, dashed and dotted
line styles, respectively. We use values $M_{\star}=0.5\,\mathrm{M_{\odot}}$,
$R_{\star}=0.46\,\mathrm{R_{\odot}}$ and $\beta=1$. We note that flux
projections are rotated by an angle $\phi=-90^{\circ}$ in comparison to Figure
5. Purple, red, and blue lines denote contours corresponding to a value of $F$
equal to 70% of its maximal value. We use these contours to select the outflow
segments for which we calculate the trajectories shown in Figure 9.

In Figure 9 we show an isometric view of 3D trajectories for 100 outflow
segments, selected from contours where $F$ is equal to 70% of its maximal
value, in a way that the distances between neighboring segments are
approximately equal. There are two such contours, located at the northern and
southern hemispheres, denoted by the red and blue lines, respectively, in
Figure 8. For $\Delta z/H>0$ these contours delimit two distinct regions
(trajectories of segments from the northern and southern contours denoted by
red and blue lines, respectively, in Figure 9), where the outflow is more
collimated. For $\Delta z/H=0$ these contours are denoted by purple lines to
emphasize, that the bulk of the outflow is contained in a single region
between the two contours and launched towards the equatorial plane
(trajectories of segments denoted by purple lines in Figure 9). In Figure 9
the green “$\star$” and the black “$\bullet$” symbols represent the collision
point and the black hole, respectively, while arrows indicate the direction of
the outflow from the self-crossing region. We study the deflection along the
$-\textbf{{e}}_{\mathrm{z}}$ direction and note that the results are symmetric
with respect to the orbital plane of the in-falling gas if the receding gas is
deflected along the $+\textbf{{e}}_{\mathrm{z}}$ direction. We calculate
trajectories for values of the offset $\Delta z/H=0$ (top row), 0.6 (middle
row), 1.2 (bottom row) and black hole masses
$M_{\mathrm{bh}}=10^{6}\,\mathrm{M_{\odot}}$ (left column), $2\times
10^{6}\,\mathrm{M_{\odot}}$ (right column) in a disruption of a
$M_{\star}=0.5\,\mathrm{M_{\odot}}$ star. Most elliptic trajectories are
integrated until the outflow segments reach the pericenter. For hyperbolic
trajectories and highly energetic elliptic trajectories (with
$|\epsilon|<0.001\Delta\epsilon$), we integrate orbits only for radial
distances $r<3\times 10^{4}\,\mathrm{R_{\odot}}$.

Figure 9: Isometric views of 3D trajectories in the black hole’s reference
frame of 100 outflow segments selected from contours where $F$ is equal to 70%
of its maximal value for $\Delta z/H=0$, 0.6, 1.2 and
$M_{\mathrm{bh}}=10^{6}\,\mathrm{M_{\odot}}$, $2\times
10^{6}\,\mathrm{M_{\odot}}$ for $M_{\star}=0.5\,\mathrm{M_{\odot}}$,
$R_{\star}=0.46\,\mathrm{R_{\odot}}$ and $\beta=1$. We use red and blue
colours for segments located on the northern and southern contours,
respectively (see Figure 8). For $\Delta z/H=0$ we use only purple lines to
illustrate that the bulk of the outflow is contained inside a single region,
while for offset collisions the outflow is contained inside two regions. Most
elliptic trajectories are integrated for radial distances smaller than the
pericenter distance, while hyperbolic trajectories are integrated for radial
distances $r<3\times 10^{4}\,\mathrm{R_{\odot}}$ (we use the same condition
for highly energetic elliptic trajectories with
$|\epsilon|<0.001\Delta\epsilon$). The green “$\star$” symbol and the black
“$\bullet$” symbol represent the collision point and the black hole,
respectively. Arrows indicate the direction of the outflow movement from the
self-crossing region.

In a disruption by a $M_{\mathrm{bh}}=10^{6}\,\mathrm{M_{\odot}}$ black hole
(left panels of Figure 9) all trajectories are elliptical, since there is no
unbound gas (see Figure 7). For $\Delta z/H=0$ the outflow from the self-
crossing region is quasi-spherical. Only a minor degree of spherical asymmetry
is present due to a higher amount of gas moving on prograde, than on
retrograde orbits. As the offset between streams increases, the geometry of
the outflow becomes less spherically symmetric and more stream-like, which is
especially apparent for $\Delta z/H=1.2$ (see also Figure 8). Increasing the
black hole mass to $M_{\mathrm{bh}}=2\times 10^{6}\,\mathrm{M_{\odot}}$ (right
panels of Figure 9) the outflow is less spherically symmetric and trajectories
extend to larger distances due to the larger energy spread induced by the
stream collision.

###  3.4 Dependence of collision properties on black hole’s spin

We now relate the parameters used to simulate offset collisions to physical
parameters, particularly the black hole spin. Due to Lense-Thirring
precession, several revolutions, or windings of the stream around the black
hole may be necessary for a collision to be successful. We denote the number
of such windings by $N_{\mathrm{w}}$, where $N_{\mathrm{w}}=0$ corresponds to
a prompt collision, i.e. after the first pericenter passage of the stream
following a stellar disruption. In the following, we use an analytical
treatment similar to that of Batra et al. (2021), to evaluate the properties
of offset collisions.

During the pericenter passage, the angular momentum vector of the debris L,
inclined by an angle $i\in[0,\pi]$ with respect to the black hole’s spin
vector S, changes by $\Delta\textbf{{L}}=\Delta\bm{\Phi}\times\textbf{{L}}$,
where

$\Delta\bm{\Phi}=\bm{\Omega}_{\mathrm{p}}\Delta t\approx\sqrt{2}\pi
a\left(\frac{R_{\mathrm{g}}}{R_{\mathrm{p}}}\right)^{3/2}\bm{\hat{s}}$ (6)

is the precession angle, $\bm{\hat{s}}=\textbf{{S}}/|\textbf{{S}}|$ is the
unit vector aligned with the direction of S and $a$ is the dimensionless spin
parameter. In Equation (6),
$\bm{\Omega}_{\mathrm{p}}=2aR_{\mathrm{g}}^{2}c/(R_{\mathrm{p}}^{3})\bm{\hat{s}}$
is the precession frequency (Nixon & King, 2012) and $\Delta
t\approx\pi\sqrt{R_{\mathrm{p}}^{3}/(2GM_{\mathrm{bh}})}$ is the time the
debris spends at the pericenter.

The in-falling and receding streams, therefore, follow different orbital
planes, which intersect along a line $L_{\mathrm{LT}}$ at an angle $\gamma$
from the major axis of the initial orbit. We define this angle as increasing
in the direction opposite from the stream’s orbital motion and restrict its
range to $\gamma+\Delta\phi/2\in[0,2\pi]$ for clarity. Note that
$L_{\mathrm{LT}}$ coincides with the projection $\textbf{{S}}_{\mathrm{p}}$ of
the black hole spin vector along the orbital plane of the in-falling stream
since $\Delta\textbf{{L}}\propto\textbf{{S}}\times\textbf{{L}}$. Due to Lense-
Thirring precession, L gets shifted by an angle (Stone et al., 2019)

$\Delta\Omega=|\Delta\bm{\Phi}|\sin i.$ (7)

We calculate $\Delta z$ from the projected distance between the intersection
point and $L_{\mathrm{LT}}$

$\Delta
z=R_{\mathrm{int}}\sin\left(\gamma+(N_{\mathrm{w}}+1/2)\Delta\phi\right)\Delta\Omega.$
(8)

By convention, $\Delta z>0$ corresponds to deflections along the direction of
L and the opposite direction for $\Delta z<0$. During each pericenter passage
the orbital plane of the stream changes, which shifts
$\textbf{{S}}_{\mathrm{p}}$, leading to a rotation of $L_{\mathrm{LT}}$ by an
angle $\Delta\gamma=-N_{\mathrm{w}}|\Delta\bm{\Phi}|\cot i$. This formula can
be understood from the fact that if $i=\pi/2$, S coincides with
$\textbf{{S}}_{\mathrm{p}}$ and $L_{\mathrm{LT}}$ does not change. If
$i\approx 0$, $L_{\mathrm{LT}}$ changes the most due to orbital plane
precession because S is the furthest from being initially contained in the
orbital plane. Taking into account this effect leads to a more accurate
expression for the vertical offset

$\Delta
z^{\prime}=R_{\mathrm{int}}\sin\left(\gamma+\Delta\gamma+(N_{\mathrm{w}}+1/2)\Delta\phi\right)\Delta\Omega.$
(9)

As long as its evolution is dominated by the black hole’s tidal force, the
stream is confined between two orbital planes, inclined by an angle $\alpha$,
that intersect along a line, which we refer to as $L_{\mathrm{H}}$. During its
first approach, the in-falling stream is in hydrostatic equilibrium until the
tidal force takes over, which we assume happens at the semi-minor axis of the
most bound debris
$b_{\mathrm{orb}}=\ell_{\star}\left(2\Delta\epsilon\right)^{-1/2}$, where the
stream width is $H_{0}$. This implies (see Bonnerot et al. 2022) that
$L_{\mathrm{H}}$ is along the major axis of the initial orbit, and that
$\alpha=H_{0}/b_{\mathrm{orb}}$. We calculate the width of colliding streams
$H_{\mathrm{int}}$ from the projected distance between the intersection point
and $L_{\mathrm{H}}$, which yields

$H_{\mathrm{int}}=\left|R_{\mathrm{int}}\sin\left((N_{\mathrm{w}}+1/2)\Delta\phi\right)\right|\alpha.$
(10)

The width of the in-falling stream decreases until it reaches a minimal value
near pericenter, where a nozzle shock occurs that makes the stream bounce
back. By imposing an absolute value in Equation (10) we assume that this
bounce is perfect, i.e. that it instantaneously reverses the sign of the
velocity component orthogonal to the orbital plane. This assumption is likely
oversimplified (see Bonnerot & Lu 2022) but we nevertheless adopt it here for
simplicity, keeping in mind that it may affect our results, especially for a
large number of windings.

The streams collide after $N_{\mathrm{w}}$ orbital windings if the condition
$|\Delta z|<2H_{\mathrm{int}}$ is met111111We take the absolute value of
$\Delta z$ because we compare the width of streams to the distance between
their centers of mass. (Batra et al., 2021). In Figure 10, we show $\Delta
z/H_{\mathrm{int}}$ as a function of $a$ and $\gamma$ for $N_{\mathrm{w}}=0$
(top left panel) and $N_{\mathrm{w}}\leq 1$ (top middle panel), 4 (top right
panel) and 50 (bottom panels). We calculate $\Delta z$ from Equation (8)
(first four panels) and Equation (9) (the last panel) for
$M_{\star}=0.5\,\mathrm{M_{\odot}}$, $R_{\star}=0.46\,\mathrm{R_{\odot}}$,
$M_{\mathrm{bh}}=2.5\times 10^{6}\,\mathrm{M_{\odot}}$, $\beta=1$,
$i=30^{\circ}$ and $H_{\mathrm{0}}=5\,\mathrm{R_{\odot}}$ (for these
parameters $\Delta\phi=36^{\circ}$). In the case of prompt collisions
($N_{\mathrm{w}}=0$), the vertical offset is antisymmetric along the
horizontal line at $(\gamma+\Delta\phi/2)/\pi=1$, with $\Delta
z/H_{\mathrm{int}}<0$ and $\Delta z/H_{\mathrm{int}}>0$ for
$(\gamma+\Delta\phi/2)/\pi>1$ and $(\gamma+\Delta\phi/2)/\pi<1$, respectively.
For values of $(\gamma+\Delta\phi/2)/\pi=0$, 1 and 2, the projected distance
between the intersection point and $L_{\mathrm{LT}}$ equals zero, which
results in a self-crossing collision with $\Delta z=0$. At a fixed $\gamma$ in
offset collisions, $\Delta z$ increases with $a$ until a prompt collision is
avoided, meaning that it will occur for a larger number of windings
$N_{\mathrm{w}}>0$. The lowest value of $a\approx 0.1$, for which the
collision can be avoided, is reached for $(\gamma+\Delta\phi/2)/\pi=1/2$ and
$3/2$, that is when the projected distance from the intersection point to
$L_{\mathrm{LT}}$ (see Equation (8)) is the largest.

Figure 10: $\Delta z/H_{\mathrm{int}}$ as a function of the black hole spin’s
magnitude and orientation specified by $a$ and $\gamma$ for
$M_{\star}=0.5\,\mathrm{M_{\odot}}$, $R_{\star}=0.46\,\mathrm{R_{\odot}}$,
$M_{\mathrm{bh}}=2.5\times 10^{6}\,\mathrm{M_{\odot}}$, $\beta=1$,
$i=30^{\circ}$, and $H_{\mathrm{0}}=5\mathrm{R_{\odot}}$. We calculate $\Delta
z$ from Equation (8) (first four panels) and Equation (9) (the last panel) for
a number of windings $N_{\mathrm{w}}=0$ (top left panel) and
$N_{\mathrm{w}}\leq 1$, 4, 50.

When $N_{\mathrm{w}}$ increases, successful collisions become possible in new
regions of the parameter space where they were avoided for lower
$N_{\mathrm{w}}$. If $N_{\mathrm{w}}\Delta\phi<\pi$, these regions exhibit a
similar trend as in the case of prompt collisions (see top middle and top
right panels), but are progressively shifted towards lower values of $\gamma$
with increasing $N_{\mathrm{w}}$. This can be understood by considering the
change in $\gamma$ of regions where $\Delta z=0$. For these parts of the
parameter space, the projected distance from the intersection point to
$L_{\mathrm{LT}}$ does not change with $N_{\mathrm{w}}$ (the argument of the
sine function in Equation (8) is constant), meaning that the regions shift
towards lower values of $\gamma$ by a factor of $-N_{\mathrm{w}}\Delta\phi$
when $N_{\mathrm{w}}$ increases. These new regions are also shifted towards
higher values of $a$. This can be inferred by considering the shift of the
left-most boundary of a new region, where $\Delta z/H_{\mathrm{int}}=2$ for a
value of $\Delta z$ that is maximized, and therefore independent of
$N_{\mathrm{w}}$. As long as $H_{\mathrm{int}}$ increases with
$N_{\mathrm{w}}$, which is the case for
$(N_{\mathrm{w}}+1/2)\Delta\phi/\pi<1/2$ (implying $N_{\mathrm{w}}<3$ from
Equation (10)), a successful collision can happen for a larger $a$, thus
shifting the boundary to higher spin values when $N_{\mathrm{w}}$ is
increased. We also notice that the boundaries between regions with a different
$N_{\mathrm{w}}$ all intersect at two locations:
$(\gamma+\Delta\phi/2)/\pi=\Delta\phi/(2\pi)\approx\\{0.1,1.1\\}$ and
$a\approx 0.25$. This is because for $\gamma=0$, $\Delta z/H_{\mathrm{int}}$
does not depend on $N_{\mathrm{w}}$, and $\Delta z/H_{\mathrm{int}}=2$ is
therefore reached for the same value of $a$, independent of $N_{\mathrm{w}}$.
We find that collisions are possible for the majority of the parameter space
for $N_{\mathrm{w}}\leq 4$. The rest of the parameter space corresponds to
$N_{\mathrm{w}}\Delta\phi>\pi$ and is covered cyclically from highest to
lowest $\gamma$, creating thin horizontal bands of alternate colours with
$|\Delta z|/H_{\mathrm{int}}\approx 2$ (bottom left panel of Figure 10).

When taking into account the rotation of $L_{\mathrm{LT}}$ (see Equation (9)),
regions of constant $\Delta z/H_{\mathrm{int}}$ are shifted towards higher
values of $\gamma$ by $|\Delta\gamma|$ and the shift is greater as $a$
increases (see the right bottom panel in Figure 10). In addition, for
$N_{\mathrm{w}}>4$ and $a\gtrsim 0.5$, the projected distance from
$L_{\mathrm{LT}}$ to the intersection point is significantly affected,
resulting in a wider range of $\Delta z/H_{\mathrm{int}}$ in these regions
that replace the thin horizontal bands described above. However, the rest of
the parameter space is only marginally affected, implying that the streams
mostly collide with a similar offset as in the case when the rotation of
$L_{\mathrm{LT}}$ is not taken into account (bottom left panel of Figure 10).

The dependence on $i$, $H_{0}$ and $M_{\mathrm{bh}}$ can be understood by
taking the ratio of Equations (8) and (10), which gives

$\displaystyle\frac{|\Delta z|}{H_{\mathrm{int}}}$ $\displaystyle=\xi\pi
a|\sin i|\left(\frac{R_{\mathrm{g}}}{R_{\mathrm{p}}}\right)^{3/2}\frac{\
\ell_{\star}}{\sqrt{\Delta\epsilon}H_{0}}$ $\displaystyle\approx\xi a|\sin
i|\left(\frac{M_{\mathrm{bh}}}{10^{6}\,\mathrm{M_{\odot}}}\right)^{3/2}\left(\frac{H_{\mathrm{0}}}{5\,\mathrm{R_{\odot}}}\right)^{-1}$
(11)

where we assume $R_{\star}=0.46\,\mathrm{R_{\odot}}$, $\beta=1$, and introduce
$\xi$ as the ratio of the two sine functions. By assuming $\xi=1$, which is
motivated by the fact that $\xi\approx 0$ only for specific values of $\gamma$
and $N_{\mathrm{w}}$, we determine that for most values of $\gamma$,
collisions become more likely with decreasing $M_{\mathrm{bh}}$, increasing
$H_{0}$, and values of $i$ closer to 0 or $\pi$. In Figure 10, these changes
would be reflected by an overall shift towards larger values of $a$ such that
more of the parameter space corresponds to $|\Delta z|<2H_{\mathrm{int}}$.
Additionally, the vertical space between regions with $\Delta z=0$ would be
reduced when decreasing $M_{\mathrm{bh}}$ due to lower apsidal precession.

We conclude that the relationship between $\Delta z/H_{\mathrm{int}}$,
$\gamma$ and $a$ is highly complicated. When $a\approx 0$, we find that stream
collisions typically have low offsets, meaning that most collisions are
strong. However, when $a\gtrsim 0.1$, we discover that there can be both
strong and grazing collisions for all values of $a$ depending on the exact
value of $\gamma$, as given by the projection of the black hole spin vector on
the in-falling stream’s orbital plane. The trends with other parameters $i$,
$H_{0}$, and $M_{\mathrm{bh}}$ are however more tractable as shown with
Equation (3.4).

##  4 Discussion

###  4.1 Effects of the different properties of the colliding streams

Our choice of the stream collision configuration assumes that the streams have
a uniform density profile and that they are identical, meaning that they have
the same $H$, $\dot{M}$, and velocities. We now explore possible deviations
from this assumption and the consequences on the outflow properties found in
Section 3. Despite not leading to significant net expansion, the study by
Bonnerot & Lu (2022) suggests that the nozzle shock can modify the width of
the colliding stream components by a factor of a few. As a consequence, a part
of the receding gas could avoid a direct collision and the denser in-falling
stream would be less affected, resulting in a decrease in energy dissipation.
Additionally, the streams may reach the intersection point with different
$\dot{M}$ (or velocities) due to the slight difference in orbital energy
between them (e.g. Jankovič & Gomboc 2023). In this case, the kinetic energy
associated with the radial motion would be less efficiently dissipated
resulting in a more asymmetric outflow, where the outflow component from the
stream with higher $\dot{M}$ is more collimated and expands at a faster rate.
In addition, Rossi et al. (2021) find that due to different $\dot{M}$ the
intersection point does not remain fixed but instead evolves with time.

It is expected that the density distribution inside the colliding streams is
not uniform. Instead, the density is the highest at the stream’s center of
mass and decreases toward the stream’s edge (Jiang et al., 2016; Bonnerot &
Lu, 2022). As a consequence, the mass of the colliding stream parts would be
different in the collisions of streams with a uniform and a non-uniform
density distribution for the same $\Delta z/H$ (even if streams have the same
$\dot{M}$, $H$ and velocity), which would affect energy dissipation. This
effect is likely most prominent for $\Delta z\lesssim 2H$, where the mass of
directly colliding gas would be significantly lower for a non-uniform density
profile, implying that a lower vertical offset may be required to reach the
same collision strength as in the uniform case. For $\Delta z/H\ll 1$ we
expect that this effect does not significantly change the outcome of
collisions because mass of colliding gas is less affected by the density
distribution.

In Section 3.3 we evaluate the outflow properties only in the case of the most
bound debris. In typical situations ($M_{\mathrm{bh}}\approx
10^{6}\,\mathrm{M_{\odot}}$, $M_{\star}\approx 1\,\mathrm{M_{\odot}}$,
$\beta\approx 1$), apsidal precession is high enough for the intersection
radius to be significantly lower than the stream apocenter distance with
$R_{\mathrm{int}}\lesssim a_{\mathrm{min}}$, where
$a_{\mathrm{min}}=R_{\star}(M_{\mathrm{bh}}/M_{\star})^{2/3}/2$ is the semi-
major axis of the most bound debris. Because the trajectories at this location
are not strongly affected by changing the orbital energy, we do not expect the
collisions to be qualitatively modified.

###  4.2 Circularization of the outflow

The debris from TDEs can form an accretion disc if its energy is efficiently
dissipated. While the self-crossing shock provides an initial source of
dissipation that can affect the gas trajectories, it is usually not enough on
its own to cause significant circularization. As demonstrated by Bonnerot & Lu
(2020) and Bonnerot et al. (2021), even when the self-crossing shock is strong
and results in a large-scale outflow, additional secondary shocks closer to
the black hole are necessary to promptly complete disc formation. The
efficiency of this process appears to be enhanced by the presence of
outflowing gas on retrograde orbits, as it leads to strong head-on collisions
near the pericenter.

For the offset self-crossing shock we focus on here, we expect a qualitatively
similar evolution towards disc formation only in the strong collision regime,
where the fraction of retrograde gas is significant. The evolution towards
disc formation may be slower in the grazing collision regime because there is
less gas on retrograde orbits. However, as explained in Section 3.3, even the
most grazing collision we considered with $\Delta z=1.8H$ leads to significant
stream expansion associated with an order unity spread in pericenter $\Delta
R_{\mathrm{p}}\approx 10R_{\mathrm{g}}\approx R_{\mathrm{p}}$. Consequently,
some of the gas in the expanded stream has very short pericenter distances,
leading to strong apsidal precession and collisions close to the black hole,
where the energy dissipation is enhanced. In addition, the collision at the
next winding will be strong despite the vertical offset induced by spin
because of the increased stream width. This indicates that a significant delay
in the accretion disc formation will occur only if the streams completely miss
each other. In this case, several revolutions may take place before an offset
collision occurs (see Section 3.4 and Batra et al. (2021)), after which we
expect an accretion disc to promptly assemble.

###  4.3 Observational features

For $\Delta z/H\approx 0$ the outflow from the self-crossing region is close
to spherical as seen in Figure 5. In this case, the photons produced by the
self-crossing shock suffer from large adiabatic losses, making the associated
radiation likely undetectable (Bonnerot & Lu, 2020). As $\Delta z/H$
increases, the outflow becomes more collimated and aligned with the direction
of the incoming streams. Along other directions where the gas density is
lower, radiation may be able to more efficiently emerge from the system
without suffering as many adiabatic losses.

In order to quantify this effect we evaluate the trapping radius
$R_{\mathrm{tr}}$, defined as the distance at which photons decouple from the
gas to start diffusing away. $R_{\mathrm{tr}}$ is calculated by equating the
dynamical timescale of the gas $t_{\mathrm{dyn}}\approx
R_{\mathrm{}}/v(R_{\mathrm{}})$ to the diffusion timescale of photons
$t_{\mathrm{diff}}\approx\tau R_{\mathrm{}}/c$, where $R$ is the radial
distance from the point of self-crossing, $v(R_{\mathrm{}})$ is the velocity
of the gas and $\tau$ is the optical depth, integrated from $R$ to infinity.
Therefore, $R_{\mathrm{tr}}$ can be expressed from

$c/v(R_{\mathrm{tr}})=\int_{R_{\mathrm{tr}}}^{\infty}\rho\kappa_{\mathrm{s}}\mathrm{d}R,$
(12)

where $\kappa_{\mathrm{s}}=0.34\,\mathrm{cm^{2}g^{-1}}$ is the electron
scattering opacity and $\rho$ is the gas density. The density is taken from
the simulation with $\Delta z/H=1.2$, and its value in physical units is
obtained by assuming $H=10\,\mathrm{R_{\odot}}$, $v_{\mathrm{r}}=0.01c$ and
$\dot{M}=0.7\,\mathrm{M_{\odot}\,yr^{-1}}$. $R_{\mathrm{tr}}$ is evaluated for
different values of $\theta$ and $\phi$, by computing the integral of Equation
(12) from outward in until its value reaches the local ratio of the speed of
light to the gas velocity. In Figure 11 we show the density of gas contained
in a slice in the $zy$ plane at $x=0$ for $\Delta z/H=1.2$, where we overplot
the trapping surface with a green line. Dashed blue lines indicate values of
$\tau$ obtained by integration from $R=15H$ (dashed grey circle) to infinity
along several directions. We see that the trapping surface follows the gas
density, resulting in lower trapping radii along the equatorial plane of the
colliding streams. This indicates that the radiation produced at the self-
crossing shock may emerge closer to the intersection point for an offset
collision than an aligned one, potentially leading to a more luminous signal
as the radiation suffers fewer adiabatic losses. A more careful determination
of the emerging radiation requires radiation-hydrodynamics simulations, which
we defer to future research.

Figure 11: A density slice of the outflow (in the local reference frame) in
the $zy$ plane at $x=0$ at time $t=150$ for $\Delta z=1.2H$. Length units are
expressed in units of the stream widths $H=10\,R_{\odot}$. Green solid line
represents the location of the trapping surface defined in Equation (12). We
show values of the optical depth $\tau$, integrated from $R=15H$ (grey dashed
circle) to infinity, along several directions marked by blue dashed lines.

In the later stages of a TDE, when the accretion disc has formed, X-ray
radiation from the disc can be absorbed and then reprocessed to the optical
band by the outflow (Roth et al., 2016; Lu & Bonnerot, 2019). For more offset
collisions, we expect the solid angle covered by the outflow to decrease due
to a deviation from sphericity (see Figure 9), such that X-ray photons may
only promptly escape along specific viewing angles while being efficiently
reprocessed along others. This effect could explain the variety of optical to
X-ray luminosity ratios found observationally (e.g. van Velzen et al. 2021).

##  5 Summary

After a star is disrupted, the part of the elongated stream of debris that
passed pericenter collides with the still-infalling gas due to the
relativistic apsidal precession, leading to a self-crossing shock. For
rotating black holes, Lense-Thirring precession causes a misalignment between
the colliding streams. We study the impact of this effect on the outflow from
the self-crossing shock by locally simulating the collision between two
streams with widths $H$ offset by a distance $\Delta z$. We analyze the gas
evolution during the collision, the geometry of the outflow, and its
subsequent dynamics, as well as consequences on accretion disc formation and
observational signatures. Our main conclusions are as follows.

  1. (i)

Based on the amount of dissipated kinetic energy of streams during the
collision and the outflow geometry, we identify two regimes of collisions. In
the strong collision regime ($\Delta z\lesssim 0.7H$), the collision results
in higher dissipated energy, and the outflow remains close to spherical. In
the grazing collision regime ($\Delta z\gtrsim 0.7H$), less energy gets
dissipated and the outflow is collimated along the directions of the incoming
streams.

  2. (ii)

We have made the outflow properties found from simulations available online at
https://github.com/tajjankovic/Spin-induced-offset-stream-self-crossing-
shocks-in-TDEs.git. The provided code returns the mass flux normalized by the
total outflow rate (see Equation (3)) given a direction of motion and a
vertical offset $\Delta z$ specified by the user.

  3. (iii)

The fractions of unbound gas and gas on retrograde orbits decrease with
$\Delta z/H$ due to a lower dissipated energy. Increasing the black hole mass
also causes more of the outflow to be unbound, but less of it to move on
retrograde orbits.

  4. (iv)

When $a\gtrsim 0.1$, we analytically predict that there can be both strong and
grazing collisions for all values of $a$ depending on the projection of the
black hole’s spin vector on the in-falling stream’s orbital plane.

  5. (v)

Even in the most extreme grazing encounter, the outflow components are wider
than streams before the collision, which likely causes strong collisions later
on, with some of the gas colliding closer to the black hole, increasing the
energy dissipation. As a consequence, a significant delay in accretion disc
formation would likely occur only if the collision is entirely prevented.

  6. (vi)

The radiation produced at the self-crossing shock may emerge closer to the
intersection point for an offset collision than an aligned one, potentially
leading to a more luminous signal as the radiation suffers fewer adiabatic
losses. Similarly, deviation from outflow sphericity may allow X-ray radiation
generated by the accretion disc to emerge along specific directions without
being absorbed.

We provided the first systematic study of the outflow properties from the
self-crossing shock produced by colliding streams that are offset due to the
black hole’s rotation. In the future, we aim to quantify the radiative output
of these interactions and explore the impact they may have on the ensuing
accretion disc formation, as these processes hold promise to provide
constraints on the black hole spin from observed TDEs.

## Acknowledgements

This research was supported by the Slovenian Research Agency grants P1- 0031,
I0-0033, J1-8136, J1-2460 and the Young Researchers program. This project has
also received funding from the European Union’s Horizon 2020 Framework
Programme under the Marie Sklodowska-Curie grant agreement no. 836751. We
acknowledge the use of phantom for detailed hydrodynamical simulations and
SPLASH for the visualization of the output (Price & et. al., 2018). We
gratefully acknowledge the HPC RIVR consortium (www.hpc-rivr.si) and EuroHPC
JU (eurohpc-ju.europa.eu) for funding this research by providing computing
resources of the HPC system Vega at the Institute of Information Science
(www.izum.si). Some of the results in this paper have been derived using the
healpy and HEALPix package (Górski et al., 2005; Zonca et al., 2019). We thank
GWverse (gwverse.tecnico.ulisboa.pt/), which funded T.J.’s visit to the Niels
Bohr Academy within the eCOST Action CA16104 as the Short Term Scientific
Mission, and made possible the start of this study.

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