# Close encounters of black hole - star binaries with stellar-mass black holes

Taeho Ryu1,2, Ruggero Valli1, Rüdiger Pakmor1, Rosalba Perna3,4, Selma E. de
Mink1,5, Volker Springel1
1 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748
Garching, Germany
2 Physics and Astronomy Department, Johns Hopkins University, Baltimore, MD
21218, USA
3 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794-3800, USA
4 Center for Computational Astrophysics, Flatiron Institute, New York, NY
10010, USA
5 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science
Park 904, 1098XH Amsterdam, The Netherlands E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Dynamical interactions involving binaries play a crucial role in the evolution
of star clusters and galaxies. We continue our investigation of the
hydrodynamics of three-body encounters, focusing on binary black hole (BBH)
formation, stellar disruption, and electromagnetic (EM) emission in dynamical
interactions between a BH-star binary and a stellar-mass BH, using the moving-
mesh hydrodynamics code AREPO. This type of encounters can be divided into two
classes depending on whether the final outcome includes BBHs. This outcome is
primarily determined by which two objects meet at the first closest approach.
BBHs are more likely to form when the star and the incoming BH encounter first
with an impact parameter smaller than the binary’s semimajor axis. In this
case, the star is frequently disrupted. On the other hand, when the two BHs
encounter first, frequent consequences are an orbit perturbation of the
original binary or a binary member exchange. For the parameters chosen in this
study, BBH formation, accompanied by stellar disruption, happens in roughly 1
out of 4 encounters. The close correlation between BBH formation and stellar
disruption has possible implications for EM counterparts at the binary’s
merger. The BH that disrupts the star is promptly surrounded by an optically
and geometrically thick disk with accretion rates exceeding the Eddington
limit. If the debris disk cools fast enough to become long-lived, EM
counterparts can be produced at the time of the BBH merger.

###### keywords:

black hole physics – gravitation – stellar dynamics

††pubyear: 2022††pagerange: Close encounters of black hole - star binaries
with stellar-mass black holes–LABEL:lastpage

##  1 Introduction

Dynamical interactions between stars and the compact objects they leave behind
play an important role in dense environments, such as globular and nuclear
star clusters and disks of Active Galactic Nuclei (AGNs). On global, large
scales they can influence cluster thermodynamics (Hut et al., 1992), while on
local, small-scales close interactions can alter the original birth
composition of isolated stars and binaries.

Dynamical formation of binary black holes (BBHs) is one of the leading
pathways (e.g., Downing et al., 2010; Portegies Zwart & McMillan, 2000;
Samsing et al., 2014; Rodriguez et al., 2015; Antonini et al., 2016; Askar et
al., 2017; Banerjee, 2018; Perna et al., 2019; Fragione et al., 2019; Di Carlo
et al., 2019; Rodriguez et al., 2019; Arca Sedda et al., 2020; Mapelli et al.,
2021) to forming the binaries which have been observed via gravitational wave
(GW) emission at their merger by the LIGO and Virgo observatories (The LIGO
Scientific Collaboration et al., 2021). Subsequent dynamical encounters
between BBHs and tertiary BHs can further influence the orbital parameters of
the binaries, and hence their timescale to merger by GW radiation (e.g., Trani
et al., 2019; Samsing et al., 2020; Wang et al., 2021; Arca Sedda et al.,
2021).

Despite the relatively high fractions of stars and compact objects in
binaries, hydrodynamic simulations of close encounters involving binaries have
begun only recently. Lopez et al. (2019) and Ryu et al. (2022, Paper 1 in the
following) studied close encounters between BBHs and single stars. They found
that, in addition to altering the spin of the accreting BHs, tidal disruption
events (TDEs) can have a significant impact on the binary BH’s orbit, in ways
which can be quantitatively different than the case of pure scattering. The EM
signatures produced by these close encounters can also differ significantly
from those of TDEs by isolated BHs: depending on the geometry of the
encounter, the accretion rate can display periodic modulations with the
orbital period. Detections of such events can provide constraints on the
formation of BBH mergers (Samsing et al., 2019).

More recently, Ryu et al. (2023, Paper 2 in the following) performed the first
investigation of close encounters between binary stars and single BHs. Their
hydrodynamic simulations showed a variety of possible outcomes, from full
disruptions of both stars, to a full disruption of one star and a partial
disruption of the other, to dissociation into bound and unbound single stars.
Among these cases of dissociation, interesting outcomes include the formation
of a runaway star, and of a fast-moving BH that accretes the tidally disrupted
debris of the other star. In other outcomes, the binary stars are dissociated,
and one of the stars is exchanged with the intruding BH, resulting in the
formation of an X-ray binary.

Here we extend the line of investigation begun in Paper 1 and Paper 2 by
performing a suite of hydrodynamic simulations of nearly parabolic close
encounters between BH-star binaries and single BHs. Similarly to Paper 2, we
use the moving-mesh code AREPO (Springel, 2010; Pakmor et al., 2016;
Weinberger et al., 2020), whose quasi-Lagrangian approach to hydrodynamics is
well adapted to the problem. Our study aims to elucidate how such encounters
can lead to a variety of outcomes, including EM transients due to the
disruption of the star and the formation (via member exchange) of tight BBHs
surrounded by debris material, potentially leading to a situation in which the
BBH merger could be accompanied by an EM counterpart.

Our paper is organized as follows: § 2 presents the estimate for the rate of
this type of encounters in globular clusters. § 3 describes the details of the
numerical simulations and the initial conditions. We present our simulation
results in § 4, with particular emphasis on a classification of the outcomes
and its dependence on encounter parameters. We discuss these results in the
context of their possible EM counterparts in § 5, and we finally summarize our
work and conclude in § 6.

##  2 Encounter rate in globular clusters

We begin by estimating the encounter rate of three-body interactions between
BH-star binaries and single stars in globular clusters. Following Paper 2, we
first calculate the differential rate of a single BH encountering a BH-star
binary as ${\rm d}\mathcal{R}/{\rm d}N_{\bullet}\simeq n\Sigma v_{\rm rel}$.
Here, $n$ is the binary number density in the vicinity of the BH, $n\simeq
f_{\rm b}n_{\rm s}$, where $f_{\rm b}$ is the non-interacting star - BH binary
fraction, $f_{\rm b}\simeq 10^{-4}-10^{-5}$ (Morscher et al., 2015; Kremer et
al., 2018), and $n_{\rm s}$ is the number density of stellar-mass objects near
the center of the clusters. The variable $v_{\rm rel}$ represents the relative
velocity between the binary and the BH while $\Sigma$ is the encounter cross-
section. For $v_{\rm p}=\sqrt{{2G(M_{\bullet}+\;M_{\star})}/{r_{\rm
p}}}\gg\sigma$, we can write $\Sigma\simeq\uppi
G(M_{\bullet}+\;M_{\star})r_{\rm p}/\sigma^{2}$ where $\sigma$ is the velocity
dispersion. Adopting our results that strong encounters occur when $r_{\rm
p}<a$, and assuming that this relation applies to binaries of any size and
mass ratio, we can approximate $\Sigma\simeq\uppi
G(M_{\bullet}+\;M_{\star})a\sigma^{-2}$. Then, we find that ${\rm
d}\mathcal{R}/{\rm d}N_{\bullet}$ can be expressed as

$\displaystyle\frac{{\rm d}\mathcal{R}}{{\rm d}N_{\bullet}}$
$\displaystyle\simeq\frac{n\uppi G(M_{\bullet}+\;M_{\star})a}{\sigma},$
$\displaystyle\simeq 4\times 10^{-13}\;\mathrm{yr}^{-1}\left(\frac{f_{\rm
b}}{10^{-4}}\right)\left(\frac{n_{\rm s}}{10^{5}{\rm
pc}^{-3}}\right)\left(\frac{M_{\bullet}+\;M_{\star}}{20\,{\rm
M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{a}{100\,{\rm
R}_{\odot}}\right)\left(\frac{\sigma}{15\;\mathrm{km}\sec}\right)^{-1}.$ (1)

Assuming more than a few tens of single stellar-mass black holes exist in
dense clusters at the present day (Morscher et al., 2015; Askar et al., 2018;
Kremer et al., 2018), and $\simeq$150 globular clusters in the Milky Way
(Harris, 2010), the rate of strong three-body encounters per Milky Way-like
galaxy is,

$\displaystyle\mathcal{R}$ $\displaystyle\simeq 6\times
10^{-9}\;\mathrm{yr}^{-1}\left(\frac{N_{\bullet}}{15000}\right)\left(\frac{f_{\rm
b}}{10^{-4}}\right)\left(\frac{n_{\rm s}}{10^{5}{\rm
pc}^{-3}}\right)\left(\frac{M_{\bullet}+\;M_{\star}}{20\,{\rm
M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{a}{100\,{\rm
R}_{\odot}}\right)\left(\frac{\sigma}{15\;\mathrm{km}\sec}\right)^{-1}.$ (2)

Two outcomes that can be produced in this type of encounters are EM transients
due to disruption of the star and formation of BBHs. In particular, because
BBHs would likely form in $\simeq 25\%$ of all encounters (see § 4.4), the
rate $\mathcal{R}$ of BBH-forming events would be
$O(10^{-9})\;\mathrm{yr}^{-1}$. The rate for encounters involving massive
stars would be relatively high, compared to that for low-mass stars before all
the massive stars turn into compact objects. However, as discussed in §5.3,
the total number of this type of encounters over the full cluster lifetime
would be higher for less massive stars because of their longer lifetime and
higher abundance. Note that $f_{\rm b}$ depends on cluster parameters such as
the initial binary fraction and the cluster age (Morscher et al., 2015), and
calculating $\mathcal{R}$ requires a detailed modeling of cluster evolution,
as well as the star formation history. Thus, for a more precise estimate of
$\mathcal{R}$ a more careful consideration of cluster evolution history is
required.

##  3 Simulation details

###  3.1 Numerical methods

Our numerical methods and setup are essentially the same as in Paper 2. We
perform a suite of 3D hydrodynamic simulations of close encounters using the
massively parallel gravity and magnetohydrodynamics moving-mesh code AREPO
(Springel, 2010; Pakmor et al., 2016; Weinberger et al., 2020), which combines
advantages of the two conventional hydrodynamical schemes, the Eulerian
finite-volume method and the Lagrangian smoothed particle method, such as
shock capturing without introducing an artificial viscosity, low advection
errors, an efficient tracking of supersonic flow, and an automatically
adaptive adjustment of spatial resolution. We use the HELMHOLTZ equation of
state (Timmes & Swesty, 2000) which accounts for radiation pressure, assuming
local thermodynamic equilibrium. We include $8$ isotopes ($\mathrm{n}$,
$\mathrm{p}$, ${}^{4}\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{14}\mathrm{N}$,
${}^{16}\mathrm{O}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, Pakmor et al.
2012). We follow the advection of the elements which are then used for the
update of the thermodynamics quantities (e.g., pressure). We do not follow the
nuclear reactions, which should be fine given the short duration of the
simulations and the reaction rates expected for the temperatures and densities
that occur in our simulations.

Figure 1: Top panel: The radial density profile of the main-sequence stars
with $M_{\star}=10\,{\rm M}_{\odot}$ (red) relaxed for five stellar dynamical
times. Bottom panel: The relative error with respect to the MESA model, as a
function of mass. The dashed grey line in the top panel indicates the profiles
for the MESA model, which are just sitting below the solid line.

###  3.2 Stellar model

The initial state of the star was taken as an evolved main-sequence (MS) star
computed using the stellar evolution code MESA (version r22.05.1) (Paxton et
al., 2011; Paxton et al., 2013, 2015, 2018, 2019; Jermyn et al., 2023). The
star has an initial mass of $10\,{\rm M}_{\odot}$ and a metallicity $Z=0.006$,
which is lower than solar and consistent with what is found for globular
clusters (e.g., VandenBerg et al., 2013), whose high stellar density
facilitates dynamical interactions.111Note that the exact value of the
metallicity does not affect our main results because the two most often cases
in our simulations are fly-bys of stars around the BHs, or near-collisional
disruptions of stars, and the outcomes of these two cases are not
significantly affected by a slight change in the stellar internal structure
due to a different metallicity.. Convection is modeled according to the mixing
length theory with a mixing length parameter of $1.5$. We use the Ledoux
(1947) criterion to determine the boundary of the convective regions and
include an exponential overshoot prescription (Herwig, 2000) with parameters
$f=0.014$ and $f_{0}=0.004$. We treat semiconvection as in Langer et al.
(1983, 1985) assuming it is fully efficient. Wind mass loss is modeled with
the prescription from Vink et al. (2001). We evolve the star until about
halfway through the main sequence, which we choose as the time when the
central hydrogen mass fraction drops to 0.3, at which point the star has
developed a $2.6\,{\rm M}_{\odot}$ convective helium core with a radius of
$0.8\,{\rm R}_{\odot}$. The stellar radius is $R_{\star}=5.4\,{\rm
R}_{\odot}$, and the central density is $\simeq 11\,{\rm g\,cm^{-3}}$. Since
we evolve the stellar model as a single star, we neglect possible past
interactions that could have affected the structure. For example, if the black
hole binary system formed through binary evolution, the star may have accreted
mass (e.g., Renzo & Götberg, 2021). If the system formed through dynamical
capture, the star may have lost mass. We expect such effects to be small and
not affect our results significantly.

We use the MESA stellar model as initial state for the AREPO simulation. After
mapping the 1D MESA model into a 3D AREPO grid with $N\simeq 5\times 10^{5}$
cells, the 3D single star is first relaxed. It usually takes up to five
stellar dynamical time until it is fully relaxed. The stellar dynamical time
is defined as $\sqrt{R_{\star}^{3}/GM_{\star}}$ where $R_{\star}$ and
$M_{\star}$ are the radius and mass of the star, respectively. Note that we
increase the resolution for each single star by almost a factor of 2 compared
to that in Paper 2, showing the results were converged with $N\gtrsim
2.5\times 10^{5}$. This is to more conservatively guarantee the convergence of
our results, and to better resolve stars that may be partially disrupted
during encounters. In addition to that, the resolution becomes finer over time
by adopting refinement near the BHs (§ 3.4): some simulations with violent
interactions have $10^{7}$ cells at the end of the simulations. The density
profiles of the relaxed stars considered in our simulations are depicted in
Figure 1. As shown in the figure, the relative difference of our 3D star with
respect to the MESA model is $1\%$ for the inner region up to $2\,{\rm
M}_{\odot}$. The match is better than $10\%$ throughout most of the star,
except for the surface. We expect that this is sufficient for the aims of this
study.

Figure 2: Schematic diagrams for the initial configuration of the BH-star
binary (blue solid circle and red solid star) and single BH (black circle) for
a prograde case with an inclination angle $i<90^{\circ}$ and phase angle
$\phi=0^{\circ}$, projected onto the $x-y$ plane (left) and the $x-z$ plane
(right). The arrows indicate the instantaneous direction of motion. The open
symbols, on the same circle with the solid symbols, indicate the case with
$\phi>0^{\circ}$.

###  3.3 Black holes

As in Paper 2, we model the BH using a sink particle assuming it is not
rotating initially. It only interacts gravitationally with gas and grows in
mass via accretion of gas. We set the gravitational softening length of the BH
($\simeq 0.01\,{\rm R}_{\odot}$) to be ten times the minimum softening length
of the cells of the stars.

We follow the same procedure for accretion described in Paper 2. However, we
significantly improve the resolution near the BH using refinement (see § 3.4),
which leads to more accurate estimates of the accretion rate with stricter
conditions for accretion than in Paper 2. We search for cells bound to the BH
(i.e. negative orbital energy relative to the BH) within $10^{3\,}r_{\rm g}$
(c.f., $1.5\times 10^{4}\,r_{\rm g}$ in Paper 2) where $r_{\rm
g}=GM_{\bullet}/c^{2}$ is the gravitational radius of the BH and $M_{\bullet}$
denotes the mass of the BH. We still apply the same inverse-distance kernel
(Monaghan & Lattanzio, 1985) to put more weight onto closer cells. Although
the change in the momentum and the mass of the BH due to accretion is taken
into account, our simulations do not include potential radiative feedback
produced by accretion.

###  3.4 Mesh refinement

The simulation code can adjust the local mesh resolution by adaptively
splitting or merging cells if certain prescribed refinement criteria are
satisfied (for more details, see § 6 in Springel 2010). We apply the
refinement technique to cells in the vicinity of each BH to better resolve the
stream structure there. At every time step, the code refines cells near a BH
if all of the following conditions are met:

  1. 1.

the distance from the BH fulfils $r<5000~{}r_{\rm g}$,

  2. 2.

the cell density is $\rho>2\times 10^{-4}\;\mathrm{g}\;\mathrm{cm}^{-3}$,

  3. 3.

the cell mass exceeds $>6\times 10^{22}\;\mathrm{g}$,

  4. 4.

and $\Delta d/r>0.26$ for $500~{}r_{\rm g}<r<5000~{}r_{\rm g}$ and $\Delta
d>500\,r_{\rm g}$ for $r<500\,r_{\rm g}$, where $\Delta d$ is the cell size.

The refinement radius in condition (i) is chosen to be larger than the
accretion radius ($1000~{}r_{\rm g}$ in this work) to ensure that gas streams
inside the accretion radius are well resolved. Condition (ii) is designed to
apply the refinement only to the cells that represent “real” gas, not vacuum
regions. Criterion (iii) avoids a runaway creation of low-mass cells. Finally,
the resolution limit imposed through condition (iv) guarantees that there are
at least $O(10^{2})$ cells within the accretion radius. On the other hand, at
every time step, the code can also derefine cells within $r<5000\,r_{\rm g}$
around each BH if the cell mass is $<1.5\times 10^{22}\;\mathrm{g}$, meaning
the mass of cells within $r<5000\,r_{\rm g}$ never becomes smaller than this
mass resolution limit.

We ran a few simulations with five different resolution limits within the
range $0.05\leq\Delta d\leq 0.4$. This confirmed that the global evolution of
the systems, such as their final interaction outcomes, is not affected by the
refinement. The accretion rate has converged when the cell size fulfils
$\Delta d/r<0.3$. Note that the number of cells within a volume at distance
$r$ increases approximately by a factor of 8 when $\Delta d$ decreases by a
factor of 2.

###  3.5 Star – black hole binary

Before we carry out our encounter experiments, we relax binaries consisting of
a fully relaxed star and a BH for $10$ stellar dynamical times. We
parameterize the binary’s semimajor axis $a$ using an approximate analytic
estimate of the Roche lobe radius (Eggleton, 1983),

$\displaystyle\frac{r_{\rm
RL}}{a}=\frac{0.49\,q^{2/3}}{0.6\,q^{2/3}+\ln(1+q^{1/3})},$ (3)

where $r_{\rm RL}$ is the volume averaged Roche lobe radius of the star,
$q=M_{\star}/M_{\bullet}$ is the mass ratio, and $a$ is the orbital
separation. We define $a_{\rm RL}\equiv a(R_{\rm RL}=R_{\star})$ as the
separation at which the star fills its Roche lobe. For $q=0.5$ and $r_{\rm
RL}=R_{\star}$, $a_{\rm RL}\simeq 3.12\,R_{\star}\simeq 16.9\,{\rm
R}_{\odot}$.

We have performed this binary relaxation process for every binary with
different orbital parameters (3 different binaries in total). The semimajor
axis and the eccentricity of the relaxed binaries differ by less than 1% from
their initial values.

Model number | Model name | $a$ | $b$ | $\phi~{}[^{\circ}]$ | $i$ | $t_{\rm p}$ | $P$ | $v_{\rm orb}$
---|---|---|---|---|---|---|---|---
Unit | $a_{\rm RL}$ | $\,{\rm R}_{\odot}$ | - | $\,{\rm R}_{\odot}$ | ∘ | ∘ | hours | days | ${\rm km\,s^{-1}}$
1 | $a4b2\phi 0i30$ | 4 | 67.5 | 2 | 67.5 | 0 | 30 | 50 | 12 | 291
2 | $a4b1\phi 0i30$ | 4 | 67.5 | 1 | 33.8 | 0 | 30 | 18 | 12 | 291
3 | $a4b1/2\phi 0i30$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 30 | 6.3 | 12 | 291
4 | $a4b1/4\phi 0i30$ | 4 | 67.5 | 1/4 | 8.45 | 0 | 30 | 2.3 | 12 | 291
5 | $a4b2\phi 180i30$ | 4 | 67.5 | 2 | 67.5 | 180 | 30 | 50 | 12 | 291
6 | $a4b1\phi 180i30$ | 4 | 67.5 | 1 | 33.8 | 180 | 30 | 18 | 12 | 291
7 | $a4b1/2\phi 180i30$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 30 | 6.3 | 12 | 291
8 | $a4b1/4\phi 180i30$ | 4 | 67.5 | 1/4 | 8.45 | 180 | 30 | 2.3 | 12 | 291
9 | $a4b2\phi 0i150$ | 4 | 67.5 | 2 | 67.5 | 0 | 150 | 50 | 12 | 291
10 | $a4b1\phi 0i150$ | 4 | 67.5 | 1 | 33.8 | 0 | 150 | 18 | 12 | 291
11 | $a4b1/2\phi 0i150$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 150 | 6.3 | 12 | 291
12 | $a4b1/4\phi 0i150$ | 4 | 67.5 | 1/4 | 8.45 | 0 | 150 | 2.3 | 12 | 291
13 | $a4b2\phi 180i150$ | 4 | 67.5 | 2 | 67.5 | 180 | 150 | 50 | 12 | 291
14 | $a4b1\phi 180i150$ | 4 | 67.5 | 1 | 33.8 | 180 | 150 | 18 | 12 | 291
15 | $a4b1/2\phi 180i150$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 150 | 6.3 | 12 | 291
16 | $a4b1/4\phi 180i150$ | 4 | 67.5 | 1/4 | 8.45 | 180 | 150 | 2.3 | 12 | 291
17 | $a2b1/2\phi 0i30$ | 2 | 33.7 | 1/2 | 8.44 | 0 | 30 | 1.2 | 4.1 | 412
18 | $a2b1/2\phi 180i30$ | 2 | 33.7 | 1/2 | 8.44 | 180 | 30 | 1.2 | 4.1 | 412
19 | $a2b1/2\phi 0i150$ | 2 | 33.7 | 1/2 | 8.44 | 0 | 150 | 1.2 | 4.1 | 412
20 | $a2b1/2\phi 180i150$ | 2 | 33.7 | 1/2 | 8.44 | 180 | 150 | 1.2 | 4.1 | 412
21 | $a6b1/2\phi 0i30$ | 6 | 101 | 1/2 | 25.3 | 0 | 30 | 12 | 22 | 238
22 | $a6b1/2\phi 180i30$ | 6 | 101 | 1/2 | 25.3 | 180 | 30 | 12 | 22 | 238
23 | $a6b1/2\phi 0i150$ | 6 | 101 | 1/2 | 25.3 | 0 | 150 | 12 | 22 | 238
24 | $a6b1/2\phi 180i150$ | 6 | 101 | 1/2 | 25.3 | 180 | 150 | 12 | 22 | 238
25 | $a4b1/2\phi 0i0$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 0 | 6.3 | 12 | 291
26 | $a4b1/2\phi 0i60$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 60 | 6.3 | 12 | 291
27 | $a4b1/2\phi 0i120$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 120 | 6.3 | 12 | 291
28 | $a4b1/2\phi 180i0$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 0 | 6.3 | 12 | 291
29 | $a4b1/2\phi 180i60$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 60 | 6.3 | 12 | 291
30 | $a4b1/2\phi 180i120$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 120 | 6.3 | 12 | 291
31 | $a4b1/2\phi 45i30$ | 4 | 67.5 | 1/2 | 16.9 | 45 | 30 | 6.3 | 12 | 291
32 | $a4b1/2\phi 90i30$ | 4 | 67.5 | 1/2 | 16.9 | 90 | 30 | 6.3 | 12 | 291
33 | $a4b1/2\phi 135i30$ | 4 | 67.5 | 1/2 | 16.9 | 135 | 30 | 6.3 | 12 | 291
34 | $a4b1/2\phi 225i30$ | 4 | 67.5 | 1/2 | 16.9 | 225 | 30 | 6.3 | 12 | 291
35 | $a4b1/2\phi 270i30$ | 4 | 67.5 | 1/2 | 16.9 | 270 | 30 | 6.3 | 12 | 291
36 | $a4b1/2\phi 315i30$ | 4 | 67.5 | 1/2 | 16.9 | 315 | 30 | 6.3 | 12 | 291
Table 1: The initial model parameters for encounters between a circular binary
($q=0.5$) with total mass of $30\,{\rm M}_{\odot}$ and a $10\,{\rm M}_{\odot}$
BH. The model name (second column) contains the information of key initial
parameters: for the model names with the format $a(1)b(2)\phi(3)i(4)$, the
numerical values encode (1) the initial semimajor axis of the binary $a/a_{\rm
RL}$, (2) the impact parameter $b$, (3) the initial phase angle $\phi$ in
degrees, and (4) the initial inclination angle $i$ in degrees. Here, $a_{\rm
RL}\simeq 3.12\,R_{\star}\simeq 16.9\,{\rm R}_{\odot}$ is the separation when
the star in the binary fills its Roche lobe. The last three columns show the
dynamical time $t_{\rm p}$ at pericenter (see its definition in the text), in
units of hours, the orbital period $P$ of the binary, and the relative
velocity $v_{\rm orb}$ of the binary members.

###  3.6 Initial conditions

Following the same terminology as in Paper 2, we refer to quantities with a
subscript containing $b-\bullet$ as those relating to the orbit between a
binary and a single BH. We assume a parabolic encounter with eccentricity
$1-e_{b-\bullet}=10^{-5}$ between a single $10\,{\rm M}_{\odot}$ BH with a
binary consisting of a $20\,{\rm M}_{\odot}$ BH and a $10\,{\rm M}_{\odot}$
star. The exact choices of the system parameters are somewhat arbitrary, but
BHs with such masses have been found in X-ray binaries (Binder et al., 2021,
e.g.). Encounters between objects of comparable masses are expected in the
dense centers of young mass-segregated star clusters. We later discuss
potential effects of different masses and orbits of encountering objects in §
5.3, based on our simulation results. We consider three semi-major axes for
the initial binary systems: $a/a_{\rm RL}=2$, $4$ and $6$, corresponding to an
orbital period of 4, 12, and 22 days, respectively. We assume the binaries are
circular at the start of our simulations. This is primarily to simplify the
initial conditions, but this may not be unreasonable given that close binaries
are often found to be circular (Almeida et al., 2017).

The distance between the binary’s center of mass and the BH at the first
closest approach $r_{\rm p,b-\bullet}$ is parameterized using the impact
parameter $b$, i.e., $r_{\rm p,b-\bullet}=0.5\,ba$ where $a$ is the binary
semimajor axis. We consider $b=1/4$, $1/2$, $1$, and $2$ for $a/a_{\rm RL}=4$,
and $1/2$ for $a/a_{\rm RL}=2$ and $6$. The binary’s angular momentum
direction is always along the $z$-axis in our simulations. We illustrate the
initial configuration of the stellar binary and the BH in Figure 2.

We investigate the dependence of encounter outcomes on key encounter
parameters, that is inclination angle $i=0$, $30^{\circ}$, $60^{\circ}$,
$120^{\circ}$ and $180^{\circ}$, $b=1/4$, 1/2, 1 and 2, and the phase angle
$\phi=0^{\circ}-180^{\circ}$ with $\Delta\phi=45^{\circ}$. We define $\phi$ as
the initial angle between the line connecting the two members in the binary
and the coordinate $x-$axis (see Figure 2). We start by studying the
dependence on the two phase angles of the binary ($\phi=0^{\circ}$ and
$180^{\circ}$) by fixing all the other parameters. To achieve this, we
initially rotate the binary while the initial separation between the center of
mass of the binary and the BH is fixed at $r=5\,a$. This allows us to examine
the outcomes from the first contact of the single BH with a different member
of the binary. However, given the relatively high computational costs, instead
of simulating encounters with every combination of $i$ and $b$, we perform
simulations for the encounters of the intermediate-size binaries ($a/a_{\rm
RL}=4$) with different combinations of $b=1/4$, 1/2, 1 and 2, and
$i=30^{\circ}$, $150^{\circ}$, and $\phi=0^{\circ}$ and $180^{\circ}$. For the
smallest and largest binaries ($a/a_{\rm RL}=2$ and 6), we only consider
$i=30^{\circ}$ and $150^{\circ}$ while $b=1/2$. In addition, we further
examine the dependence of $i$ on the outcome properties by considering $i=0$,
$60^{\circ}$, $120^{\circ}$ and $180^{\circ}$ (for $b=1/2$). Last, we also
study the impact of the phase angle $\phi$ on the encounter outcomes by
simulating encounters with six additional phase angles ($\phi=45^{\circ}$,
$90^{\circ}$, $135^{\circ}$, $225^{\circ}$, $270^{\circ}$, and $315^{\circ}$).

In Table 1, we summarize the initial parameters considered in our simulations.
Each of the models is integrated in time up to a few $100\,t_{\mathrm{p}}$ as
needed to identify the final outcomes. Here, $t_{\rm p}=\sqrt{r_{\rm
p}^{3}/GM}$ is the dynamical time at $r=r_{\rm p}$, where $M$ is the total
mass of the three objects ($40\,{\rm M}_{\odot}$). The value of $t_{\rm p}$
for each model is given in Table 1.

The total computational cost for each run varies, mainly depending on how long
the interactions last until the final outcomes are produced. Using 200 - 300
CPU-cores of the Intel Xeon CascadeLake-AP processor (Xeon Platinum 9242), the
total compute time per run has been around 70000 - 100000 core hours.

Figure 3: An example of a BBH-forming encounter, Model 20.$a2b1/2\phi
180i150$, showing the density distribution in the binary orbital plane at a
few different times in units of $t_{\rm p}$. The color bar gives the
logarithmic density in ${\rm g}\,{\rm cm}^{-3}$. The time is measured since
the expected pericenter passage between the binary’s center of mass and the
single BH. At $t/t_{\rm p}\simeq-16$ (top-$1^{\rm st}$), the binary (star -
green dot) and the single BH (yellow dot) approach each other. At $t/t_{\rm
p}=-3.21$ (top-$3^{\rm rd}$), the incoming BH strongly encounters with the
star in the binary, followed by the collision of the star (top-$4^{\rm th}$).
The BH that disrupted the star is gravitationally captured by the other BH,
forming a merging binary with $a\simeq 6.70\,{\rm R}_{\odot}$ and $e\simeq
0.943$, corresponding to $t_{\rm GW}\simeq 10^{4}\;\mathrm{yr}$ (bottom
panels).

Model number | Model name | BBH? | STAR? | Binary type | $a$ | $e$ | $\log_{10}t_{\rm GW}$ | $v$ | Single type | $v$
---|---|---|---|---|---|---|---|---|---|---
- | - | - | - | - | $\,{\rm R}_{\odot}$ | - | yr | km/s | - | km/s
1 | $a4b2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 76.2 | 0.796 | - | - |  | -
2 | $a4b1\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 55.3 | 0.347 | - | - |  |
3 | $a4b1/2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 110 | 0.352 | - | - |  | -
4 | $a4b1/4\phi 0i30^{\star\star}$ | - | - | - | - | - | - | - | - | -
5 | $a4b2\phi 180i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 63.3 | 0.1 | 12.2 | 56.0 | $\star$ | $167$
6 | $a4b1\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 257 | 0.685 | 13.3 | 61.0 | - | -
7 | $a4b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 114 | 0.518 | 12.5 | 31.3 | - | -
8 | $a4b1/4\phi 180i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 41.9 | 0.727 | 9.91 | 60.5 | $\star$ | $183$
9 | $a4b2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 62.0 | 0.156 | - | 57 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 172
10 | $a4b1\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 42.2 | 0.724 | - | 67.8 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 203
11 | $a4b1/2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 50.3 | 0.722 | - | 57.1 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 170
12 | $a4b1/4\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 50 | 0.731 | - | 57.8 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 172
13 | $a4b2\phi 180i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 63.3 | 0.161 | - | 52.4 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 157
14 | $a4b1\phi 180i150^{\star}$ | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 66.1 | 0.857 | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | -
15 | $a4b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 12 | 0.661 | 8.031 | 38.6 | - | -
16 | $a4b1/4\phi 180i150^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 109 | 0.310 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
17 | $a2b1/2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 18.0 | 0.396 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
18 | $a2b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 61.0 | 0.406 | 11.7 | 64.0 | - | -
19 | $a2b1/2\phi 0i150$ | No | Destroyed | - | - | - | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20),\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 60.8, 234
20 | $a2b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 6.70 | 0.943 | 4.36 | 47.6 | - | -
21 | $a6b1/2\phi 0i30^{\star\star}$ | - | - | - | - | - | - | - | - |
22 | $a6b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 354 | 0.787 | 13.2 | 69.1 | - | -
23 | $a6b1/2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 76.0 | 0.728 | - | 65.7 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 199
24 | $a6b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 149 | 0.972 | 8.64 | 79.2 | - | -
25 | $a4b1/2\phi 0i0^{\star\star}$ | - | - | - | - | - | - | - |  |
26 | $a4b1/2\phi 0i60^{\star}$ | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 101 | 0.725 | 11.5 | - | $\star$ | -
27 | $a4b1/2\phi 0i120$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 68 | 0.488 | - | 42.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 126
28 | $a4b1/2\phi 180i0$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 370 | 0.831 | 13 | 38.6 | - | -
29 | $a4b1/2\phi 180i60$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 155 | 0.713 | 12.3 | 38.0 | - | -
30 | $a4b1/2\phi 180i120$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 94.6 | 0.774 | 11.0 | 94.5 | - | -
31 | $a4b1/2\phi 45i30^{\star\star}$ | - | - | - | - | - | - | - |  |
32 | $a4b1/2\phi 90i30$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 37 | 0.301 | - | 75.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 227
33 | $a4b1/2\phi 135i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 48.0 | 0.816 | 9.556 | 60.6 | $\star$ | 181
34 | $a4b1/2\phi 225i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 58.4 | 0.555 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
35 | $a4b1/2\phi 270i30$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 64.5 | 0.460 | - | 69.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 232
36 | $a4b1/2\phi 315i30$ | No | Destroyed | - | - | - | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20),\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 93.1, 209

Table 2: The outcomes of each model: the model number (first column), the
model name (second column), whether a BBH forms (third column), and whether
the star survives or is destroyed (fourth column). The next five columns show
the type of final binary product and its properties (semimajor axis,
eccentricity, GW-driven merger time scale only for BBHs, and ejection
velocity). The last two columns indicate the type of singles as a final
product and their ejection velocity. The star symbol (⋆) at the end of some
model names indicates the case where the three objects form a quasi-stable
triple (see its definition in the main text) and the final outcomes are not
determined until the end of simulations. For this case, the properties of the
inner binary are provided. The double star symbols (⋆⋆) indicate cases where
the three objects do not form any quasi-stable object and the final outcomes
are not determined. The symbol $\leavevmode\hbox to8.61pt{\vbox
to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ indicates the
$20\,{\rm M}_{\odot}$ BH initially in the binary, $\leavevmode\hbox
to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip
3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ marks the
$10\,{\rm M}_{\odot}$ single incoming BH, and $\star(10)$ stands for the star
initially in the binary. Arrows indicate the active BH in the BBHs; double-
headed arrows signify that both BHs are active at the end of the corresponding
simulation.

Figure 4: The orbital properties of the final binaries, for binary black holes
(top panels) and black hole – star binaries (bottom panels). The left panels
compare the initial semimajor axis with the final one, and the right panels
show the semimajor axis and the eccentricity of the final binaries. The black
diagonal lines in the left panels depict the cases where the size of the
initial binaries and final binaries are identical. The grey curves in the top-
right panels indicate the gravitational wave-driven merger time scales of 14
Gyr, 1 Gyr and 1 Myr, respectively, for a binary consisting of $10\,{\rm
M}_{\odot}$ and $20\,{\rm M}_{\odot}$ black holes. In the top panels, the
solid (hollow) markers indicate BBHs that would (not) merge in a Hubble time.
On the other hand, the hollow markers in the bottom panels indicate the models
where the final outcome is an unstable triple and, for this case, the orbital
properties of the inner binary are presented.

Figure 5: An example of a non-BBH-forming encounter, Model 17.$a2b1/2\phi
0i30$. We depict the density distribution in the binary orbital plane at a few
different times in units of $t_{\rm p}$. At $t/t_{\rm p}\simeq-16$
(top-$1^{\rm st}$), the binary (star - green dot) and the single BH (yellow
dot) approach each other. At $t/t_{\rm p}=-3.21$ (top-$3^{\rm rd}$), the two
BHs encounter, resulting in the ejection of the initially single BH, while the
initial binary orbit is significantly perturbed to become an interacting
binary (bottom panels). Because of periodic interactions at pericenter, the
binary orbit continues to evolve until the end of the simulation. The
semimajor axis and eccentricity measured at the end of the simulation are
$\simeq 18\,{\rm R}_{\odot}$ and $\simeq 0.4$, respectively.

##  4 Results

###  4.1 Classification of outcomes

The outcomes of three-body encounters between BH-star binaries and single BHs
can be divided into three classes, depending on the final products.

  1. 1.

BBH-forming encounters: This class refers to encounters in which a BBH
emerges. In this case, the impact parameter is mostly $\lesssim 1/2-1$. The
incoming single BH frequently interacts first with the star by the time the
binary’s center of mass and the single BH arrive at pericenter (models with
“Yes” in the fifth column in Table 2). In this situation, the star in the
binary nearly collides with the incoming single BH. We show one example for
this type of encounter in Figure 3. The incoming single BH loses a significant
amount of its kinetic energy and is gravitationally captured by the other BH
initially in the binary. Because of the member exchange due to a violent star-
removing encounter, the size of the final binary is not necessarily correlated
with the size of the initial binary. To illustrate this, we compare in the
top-left panel of Figure 4 the final $a$ of the BBHs with the semimajor axis
of the initial BH-star binaries. The final $a$ covers over a wide range of
values and is not necessarily comparable to $a$ of the initial binary. These
violent interactions can lead to the formation of merging BBHs, as illustrated
in the top-right panels: the GW-driven merger time scale of 5 out of 14 final
BBHs is less than a Hubble time. Note that the absolute magnitude of the
binding energy of the merging BBHs is much larger than the typical kinetic
energy of stars in both globular and nuclear clusters ($\simeq\sigma^{2}$
where $\sigma$ is the velocity dispersion). This suggests that subsequent
interactions with other stars would not dissociate these “hard” binaries, but
rather make them more compact (Heggie, 1975) and more eccentric (Valtonen &
Karttunen, 2006), which would facilitate their mergers. In addition, the
disruption of the star prior to the BBH formation means that at least one
member of the BBH is frequently surrounded by gas upon binary formation. When
the BBH is compact, both BHs accrete gas.

  2. 2.

Non-BBH-forming encounters: In this class, the outcomes are member exchanges
between the two BHs or perturbations of the initial binary’s orbit (models
with “No” in the fifth column of Table 2). This mostly occurs when the two BHs
interact at the first contact between the binary and the single BH. We show
one example for this type of encounter in Figure 5, resulting in an orbit
perturbation. The impact of the encounters is relatively weak compared to the
BBH-forming encounters. As shown in the bottom-left panel of Figure 4, the
final value of $a$ scatters within less than a factor of 2 around the initial
$a$. The eccentricity of the final BH-star binary is widely distributed
between 0.1 and 0.9 (bottom-right panel), similarly to those of final BBHs
(top-right panel). In this type of encounters, EM transient phenomena, such as
tidal disruption events, collisions, or interacting binaries, can be created
(e.g., Model 17. $a2b1/2\phi 0i30$). In addition, the single BHs are ejected
at $\gtrsim 60\,{\rm km\,s^{-1}}$, comparable to or higher than the escape
velocity of globular clusters (i.e., tens of ${\rm km\,s^{-1}}$; Gnedin et al.
2002; Antonini & Rasio 2016).

  3. 3.

Undetermined: This class refers to cases where final outcomes are not
determined (12 models in total, Models with “-” in the fifth column in Table 2
and with superscript $\star$ or $\star\star$). Among these 12 models, there
are eight encounters (models designated with superscript $\star$) in which the
three objects form an unstable hierarchical triple, which we define as a
triple where the outer binary is on a very large eccentric orbit so that the
pericenter distance of the outer binary is smaller than the semimajor axis of
the inner binary. In the table, we provide the orbital parameters of the inner
binary. In the rest (models with superscript $\star\star$), interactions
become extremely prolonged so that a final outcome has not (yet) emerged.

From now on, we will focus on the first two classes, i.e., BBH-forming and
Non-BBH-forming encounters. These types of final outcomes and their properties
are summarized in Table 2.

Figure 6: Schematic diagram showing three dominant configurations resulting in
the perturbation of the original binary’s orbit in our simulations. The black
arrows indicate the direction of motion of objects, and long grey arrows the
trajectory of the incoming BH (black circle). In the first configuration (top
panels), for the prograde encounter with $\phi<90^{\circ}$ and $b\lesssim 1$,
the incoming BH undergoes a strong encounter with the BH (blue circle)
initially in the binary (small closest approach distance compared to the
binary semimajor axis), quickly turns around and advances to the left (top-
right panel). This quick turn-around motion gives a momentum kick (green
arrow) to the blue circle to the right with respect to the star (red circle).
The orbit of the initial binary is perturbed. The second configuration
(bottom-left panel) is a distant fly-by where the incoming BH does not
significantly interact with any of the binary members, and this happens when
$b\gtrsim 1$. The last configuration (bottom-right panel) shows the case where
the incoming BH passes through the binary without strong interactions with any
of the binary members (e.g., $b\simeq 1/4$ and $a/a_{\rm RL}=4$).

Figure 7: Schematic diagram showing the dominant configuration resulting in an
exchange of binary members. The black arrows indicate the direction of motion
of objects, and long grey arrows (right panel) the trajectory of the incoming
BH (black circle). In retrograde encounters with $\phi<90^{\circ}$, like
configuration 1 for the orbit perturbation (Figure 6), the incoming BH
strongly interacts with the BH (blue circle) initially in the binary, quickly
turns around and advances to the right. This motion results in a momentum kick
(green arrow) to the blue circle to the left with respect to the star (red
circle). The initially single BH gravitationally captures the star and forms a
binary.

###  4.2 Dynamical processes

The two most crucial factors to determine the outcomes for the parameter space
considered in our simulations are 1) the types of objects that meet at the
closest encounter (BH - BH or BH - star), and 2) the net direction of the
momentum kick relative to the bystander object (i.e., the object in the binary
that does not interact with the incoming BH at the first closest approach)
imparted by the interaction between the two meeting objects. As explained in
the previous section, the first aspect substantially affects the chances of
the survival of the star. The second aspect determines which objects end up in
the final binary (i.e., member exchange, binary perturbation) and how the
final binary’s orbit looks like.

For the non-BBH-forming encounters with $b\simeq 1/2-1$, the most frequent
outcomes are either a member exchange or a perturbation of the original
binary. The latter happens in retrograde encounters. This case can be
categorized into three configurations, which are illustrated in Figure 6. In
the first configuration (top panels), the incoming BH strongly interacts with
the other BH and turns around at a small pericenter distance compared to the
binary semimajor axis. The initially single BH is rapidly ejected from the
system in the direction roughly opposite to the incoming direction. This
interaction imparts a momentum kick to the BH that perturbs the binary orbit.
In the other two configurations, the incoming BH either moves around or passes
through the binary, without significant interactions with any of the binary
members (bottom panels).

The dominant channel for member exchange is depicted in Figure 7. For the non-
BBH-forming encounters in a prograde orbit, the two BHs meet first and pass
through their points of closest approach. Like the first configuration for
orbit perturbation, their relative motion gives a momentum kick to the motion
of the BH originally in the binary, relative to the star. The momentum kick
gives an additional acceleration in the BH’s receding motion from the star.
The initially single BH, after turning around the other BH, moves in a similar
direction with the star and gravitationally captures it.

For the BBH-forming encounters, the star and the initially single BH undergo
close encounters, naturally resulting in a tidal disruption event or stellar
collision. Both events can also impart a momentum kick to the disrupting BH.
In our simulations, the momentum kick is not large enough to prevent the two
BHs from forming a bound pair. For example, if the star and the incoming BH
undergo a head-on collision, the incoming BH dramatically slows down and forms
a merging BBH with the other BH (e.g., Model 20. $a2b1/2\phi 180i150$). We
caution that the head-on collision between the two equal-mass objects in our
simulations is an extreme case yielding a dramatic drop in the BH’s kinetic
energy. The net effect of such star-removing events on the motion of the
disrupting BH and the subsequent formation of a BBH depends on the mass ratio,
relative velocity, and the direction of the momentum kick.

###  4.3 Binary black hole formation

Typical semimajor axes of BBHs formed in the BBH-forming encounters range
within $10-400\,{\rm R}_{\odot}$ while eccentricities vary within $0.1-0.97$.
Correspondingly, the GW-driven merger time scales of those merging binaries
are in the range $10^{4}-10^{13}\;\mathrm{yr}$. Five of our models among these
encounters are merging BBHs with $a\simeq 7-150\,{\rm R}_{\odot}$ and $e\simeq
0.6-0.97$. As explained in § 4.2, the dominant formation channel is the
gravitational capture of the incoming BH by the BH originally in the binary
after strong interactions between the incoming BH and the star. Naturally, a
disruption event or a collision precedes the BBH formation. As a result,
either the disrupting BH or both BHs accrete matter by the time they form a
stable binary.

###  4.4 Dependence of outcomes on parameters

We examine the dependence of outcomes on a few key encounter parameters, phase
angle $\phi$, impact parameter $b$, inclination angle $i$, and semimajor axis
$a$, by varying one parameter at a time, keeping the rest of them fixed. Our
simulations suggest that the two most important parameters that affect the
formation of BBHs in this scenario of three-body encounters are the impact
parameter and the phase angle.

  1. 1.

Phase angle $\phi$: this is found to be one of the key parameters that
separates BBH-forming encounters from non-BBH-forming encounters. For the
former, very likely outcomes are BBHs, frequently accompanied by a disruption
of the star. On the other hand, for the latter, frequent outcomes are
eccentric BH-star binaries produced via member exchange or weak tidal
perturbations of the initial stellar orbit. In addition, even for the BBH-
forming encounters, the direction of the encounter between the initially
isolated BH and the star at the first closest approach relative to the other
BH determines the size of the semimajor axis of the BBH: if the momentum kick
imparted on the encountering BH is given such that it adds to the encountering
BH’s momentum, a large binary forms (e.g., Model 6. $a4b1\phi 180i30$ and
Model 22. $a6b1/2\phi 180i30$). Although our study is not appropriate for rate
estimates, the dependence of outcomes on the phase angle may indicate that
roughly $\simeq 25\%$ of these three-body encounters between objects of
similar mass with $b\lesssim 1$ may possibly lead to BBH formation with a high
chance of creating EM transients.

  2. 2.

Impact parameter $b$: in general, the initial binary and the single BH can
interact significantly (member exchange or stellar collisions) at the first
closest approach when $r_{\rm p}\lesssim a$, which is also found in Ryu et al.
(2022); Ryu et al. (2023). Fly-by only occurs at $r_{\rm p}>a$ (Models 1.
$a4b2\phi 0i30$, and Model 9. $a4b2\phi 0i150$). For this case, the initial
binary orbit is weakly perturbed, resulting in a 10 - 20% change in the
semimajor axis. Relatively weak interactions also take place when the impact
parameter is too small compared to the size of the binary, i.e., $r_{\rm
p}<a/8$ (e.g., Model 4. $a4b1/4\phi 0i30$, and Model 16. $a4b1/4\phi
180i150$), as the single BH penetrates through the binary without interacting
strongly with any of the binary members (see the bottom-right panel of Figure
6).

  3. 3.

Inclination angle $i$: prograde encounters tend to result in strong
interactions between the first two encounter objects, frequently leading to
outcomes that involve member exchange (e.g., models with $\phi=0^{\circ}$ and
$i=30^{\circ}$) or stellar collisions (e.g., models with $\phi=180^{\circ}$
and $i=150^{\circ}$). This is because the relative velocity between the two
encountering objects is smaller, implying a larger gravitational focusing
cross section ($\propto v^{2}/\sigma^{2}$ where $\sigma$ is a typical relative
velocity at infinity). A typical configuration for member exchange in prograde
encounters is drawn in Figure 7. On the other hand, the first interactions in
retrograde orbits are relatively weak due to the large relative speed between
the two encountering objects. As a result, frequent outcomes are perturbations
of the initial binary orbit, as depicted in Figure 6.

  4. 4.

Semimajor axis $a$: given the same pericenter distance relative to $a$
($r_{\rm p}\simeq 0.25a$) for the simulations with varying $a$, the type of
the final outcomes does not show a strong dependence on $a$. However, the size
of the final binary is closely correlated with that of the initial binary,
e.g., $a\gtrsim 76\,{\rm R}_{\odot}$ of final binaries in models with
$a/a_{\rm RL}=6$ (or $a=101\,{\rm R}_{\odot}$) and $a\lesssim 61\,{\rm
R}_{\odot}$ in models with $a/a_{\rm RL}=2$ (or $a=32\,{\rm R}_{\odot}$).

Figure 8: Face-on (left) and edge-on (right) density distribution of disks
around the BH that disrupts the star at the first closest approach in four
selected models with $i=30^{\circ}$ or $150^{\circ}$, for Model 6. $a4b1\phi
180i30$ ($1^{\rm st}$ row), Model 15. $a4b1/2\phi 180i150$ ($2^{\rm nd}$ row),
Model 18. $a2b1/2\phi 180i30$ ($3^{\rm rd}$ row), and Model 22. $a6b1/2\phi
180i30$ ($4^{\rm th}$ row), at the end of the simulations. The white
horizontal bar at the bottom-left corner of each panel shows the spatial
scale, $4\,{\rm R}_{\odot}$, except for the second row of panels where it is
$2\,{\rm R}_{\odot}$.

Figure 9: Profiles of the structure of the disks in simulations with
$i=30^{\circ}$ or $150^{\circ}$ where a BBH forms, including the four models
shown in Figure 8: The aspect ratio, defined as the ratio of the density scale
height to the cylindrical radius $r$ (top-left), the ratio of the mass-
weighted average of the azimuthal velocity along the midplane within the scale
height to the Keplerian velocity $v_{\rm kep}$ (top-right), the average
density along the midplane within the scale height (bottom-left), and the
mass-weighted average of the temperature along the midplane within the scale
height (bottom-right). All the reported quantities are measured at the end of
the simulations. Figure 10: The accretion rates of the initially single BHs
that fully destroy the star in BBH-forming simulations with
$\phi=180^{\circ}$, and $i=30^{\circ}$ or $150^{\circ}$.

###  4.5 Accretion

Our simulations show that stars can be disrupted in three-body interactions
between BH-star binaries and single BHs via strong interactions with very
small impact parameters, i.e., collisions. In such events, a merging BBH can
subsequently form, and at least one of the BHs is surrounded by an accretion
disk which can create EM transient phenomena. To zero-th order, the disk
structure and features of the accretion rate can be imprinted onto light
curves of such events.

The refinement scheme adopted for the simulations allows us to resolve the gas
structure down to $0.01\,{\rm R}_{\odot}\simeq 10^{3}\,r_{\rm g}$ from the BH.
Although the regions that we can resolve are still too far from the BH to be
directly related to the accretion process, we can provide an accurately
resolved large scale structure of the disks formed in star-destroying events,
which can be used as initial conditions for detailed disk simulations. Here,
we define a disk as a group of gas cells tightly bound to the BH and
coherently orbiting in the azimuthal direction. The outer edge of the disk is
defined as the radius containing 99% of the total bound mass orbiting at a
velocity exceeding 1% of the local Keplerian speed $v_{\rm
kep}(r)=\sqrt{G[M(<r)+M_{\bullet}]/r}$, where $M(<r)$ is the mass enclosed
within $r$.

We show in Figure 8 both the face-on (left panels) and edge-on (right panels)
density distributions of the disks around the BH that destroys the star at the
first encounter in four example models, and in Figure 9 the radial profiles of
the aspect ratio, the density, the temperature, and the rotational velocity
for all models where an accretion disk forms. The aspect ratio $h/r$ is
defined as the ratio of the first-moment density scale height, averaged over a
given cylindrical radius, to the cylindrical radius. Here, we excluded Model
20. $a2b1/2\phi 180i150$ in this analysis because the BH in that model is
surrounded by a nearly spherical gas cloud, not by a disk. But we provide the
accretion rate for that model also, shown in Figure 10.

We find that the disks are thick and pressure-supported, and mostly confined
within $r\simeq 30\,{\rm R}_{\odot}$. In general, the aspect ratio $h/r$ (top-
left panel of Figure 9) is comparable to or greater than order unity up to the
outer edge of the disks. $h/r$ declines from $h/r\simeq 3-5$ to $h/r\simeq 1$
outwards. The rotational velocity $v^{\phi}$ near the mid-plane is sub-
Keplerian ($v^{\phi}/v_{\rm kep}\simeq 0.1-0.6$), indicating the disk is not
rotationally supported. The velocity ratio remains the same out to the outer
disk edge. The density of the inner region stays flat at
$\rho\simeq(0.1-5)\;\mathrm{g}\;\mathrm{cm}^{-3}$ up to 0.1 - 0.2 of the disk
size, then declines steeply following a $r^{-4}$ power-law. On the other hand,
the temperature does not show such flatness at $r\lesssim\,{\rm R}_{\odot}$,
but continuously decreases following a $r^{-1}$ power-law.

Finally, we present in Figure 10 the accretion rate of the initially single
BHs that fully destroy the star at the first closest encounter. The general
trend is that, upon disruption or collision, the accretion rate dramatically
increases up to $\dot{M}\simeq(10^{-6}-10^{-5})\,{\rm M}_{\odot}\,{\rm
s}^{-1}$ and it takes around 80-100 hours until $\dot{M}$ declines by a factor
of 100 from its peak. When the binary is eccentric and the pericenter distance
is sufficiently close, a periodic perturbation from the other BH at periastron
results in periodic bursts on a time scale $\simeq$ the orbital period (e.g.,
Model 15. $a4b1/2\phi 180i150$, and Model 20. $a2b1/2\phi 180i150$). Although
the accretion rate is super-Eddington, the total accreted mass is at most
$0.1\,{\rm M}_{\odot}$ ($\lesssim 0.4\%$) until the end of the simulation, and
the magnitude of the BH spin driven by accretion can be as large as $0.01$.

We have to caution that such extremely high accretion rates for stellar-mass
BHs would result in strong outflows (e.g., Sądowski et al., 2014), which would
regulate the accretion rate. Although we have realized a significant
improvement in resolving gas motions near the BHs compared to Paper 2 thanks
to using refinement, since feedback from the BHs is not included in our
simulations it is likely that our accretion rates are overestimated.
Nonetheless, if the luminosity is mostly driven by accretion, the features
revealed in the accretion rate (e.g., periodic bursts) could possibly be
imprinted in the light curves.

Figure 11: The evolution of $a$ and $e$ of the five merging BBHs formed in
three-body interactions due to GW emission. The markers depict $a$ and $e$ of
the final merging binary black holes. The four grey horizontal lines indicate
the semimajor axes at which the rest-frame GW frequency (twice the orbital
frequency) is $f_{\rm GW}=10^{-4}$, $10^{-2}$, $1$, and $10^{2}$ Hz,
respectively.

##  5 Discussion

###  5.1 Formation of merging binary black holes

Our simulations show that close three-body encounters between a BH-star binary
and a single BH can create a merging BBH (see the top-left panel of Figure 4).
One possibly dominant formation process we identified is the close interaction
between the star and the incoming BH at the first closest approach, resulting
in a stellar disruption, followed by the formation of a BBH. 5 out of 11 BBHs
formed in our simulations would merge in a Hubble time via GW emission. The
semimajor axes of the merging BBHs are $\lesssim 114\,{\rm R}_{\odot}$ and
their eccentricities are quite high, $0.66\lesssim e\lesssim 0.97$. If the
required conditions are met ($r_{\rm p}\simeq 0.5\,a$, encounters between the
star and the incoming BH at the first closest approach), this type of
encounters can form, albeit likely rarely, a very compact eccentric BBH:
$t_{\rm GW}\simeq 10^{4}$ yr in Model 20. $a2b1/2\phi 180i150$.

To see whether the merging BBHs can have residual eccentricities when they
enter the frequency band of LIGO (10 Hz to 10kHz), we evolve the five binaries
assuming their orbits evolve purely via GW emission until $t_{\rm GW}=P$,
where $P$ is the binary orbital period. We solve Equations 5.6 and 5.7 in
Peters (1964) simultaneously using a 4th-order Runge-Kutta method with an
adoptive step size of $10^{-3}t_{\rm GW}$. As a sanity check, we confirmed
that our numerical solutions are consistent with the analytic solution
(Equation 5.11 in Peters 1964) within fractional errors of $\lesssim 10^{-8}$.
Figure 11 shows the evolution of $a$ and $e$ of the five merging BBHs,
starting from those found in our simulations (marked as scatters near the top-
left corner). As shown in the figure, by the time the BBHs enter the LIGO
frequency band, their residual eccentricities would be very small
($e<10^{-5}$).

Nonetheless, the circumbinary gas produced by the disruption of the star may
affect the (at least early) orbital evolution, which may hence deviate from
the purely GW-driven evolution considered above. The gas-binary interaction
and resulting binary evolution remains an active topic of study. A growing
number of numerical works have suggested that a binary surrounded by a
circumbinary disk can expand (e.g., Miranda et al., 2017; Muñoz et al., 2019;
Duffell et al., 2020) and can be driven into an eccentric orbit (e.g., Zrake
et al., 2021; D’Orazio & Duffell, 2021), depending on the disk and binary
parameters, as opposed to the predictions from the commonly held picture of
surrounding gas driving binaries into shrinking circular orbits (e.g.,
Armitage & Natarajan, 2002). However, given the limited parameter space
explored in previous work, it is not straightforward to predict the evolution
of our unequal-mass, very eccentric BBHs surrounded by a possibly misaligned
disk, based on the results from the previous work.

The remaining 6 BBHs with GW-driven merger timescales longer than a Hubble
time are hard binaries in typical stellar cluster environments. This means
that those binaries could become potential GW event candidates via weak
interactions with other objects and a few strong interactions like the ones
considered in this study.

###  5.2 Electromagnetic counterparts of binary black hole merger

The close association of BBHs and stellar disruptions can have important
implications for EM counterparts of BBH mergers. At the time the BBH forms,
there would be a prompt EM transient phenomenon due to the stellar disruption.
The very high accretion rate (Figure 10), along with the accretion-driven BH
spin and magnetic field of debris inherited from the star, suggests that a jet
can be launched. For such a case, the luminosity powered by the jet would
track the accretion rate as $\propto\dot{M}c^{2}$ with an uncertain efficiency
factor. We also found that both BHs can be surrounded by the stellar debris
and accrete, possibly suggesting that both BHs may be able to launch jets
simultaneously, potentially leading to a unique observational signature.

In addition to the prompt EM emission, the existence of the surrounding gas
when the BBH forms may result in a possible EM counterpart at the time of
merger. This is a quite similar situation as found in Ryu et al. (2022) where
an initially hard BBH encounters with a single star and becomes surrounded by
gas debris after disrupting the star. Perna et al. (2016) studied the
evolution of an initially hyper-Eddington accretion disk which cools and shuts
down the magnetorotational instability before the disk material is fully
accreted. Under these conditions, the “dead disk” is expected to survive until
the BBHs merge, and to heat up and re-ignite during the merger process, hence
yielding a possible EM counterpart to the GW event.

###  5.3 Varieties of encounters

Although we consider three-body encounters between a circular binary and a
single object with similar masses (the largest mass ratio is 0.5), there could
be a variety of these types of events involving, e.g., initially eccentric
binaries and a wide range of masses of encountering objects.

Encounters involving massive stars (i.e., $10\,{\rm M}_{\odot}$) are likely to
occur during the early evolutionary stages of star clusters unless there is
another episode of star formation, since stars with mass $>10\,{\rm
M}_{\odot}$ would collapse to compact objects in tens of Myrs. Therefore, over
the full cluster lifetime, the overall rate would indeed be higher for
encounters involving less massive MS stars because such binaries would survive
longer. Using Monte Carlo simulations of globular clusters, Kremer et al.
(2018) showed that up to 10 detached BH-MS binaries can exist in clusters at