# Simulating the collapse of rotating primordial gas clouds to study the
survival possibility of Pop III protostars

Shubham P. Raghuvanshi  Harish-Chandra Research Institute (HRI), Chhatnag
Road, Jhusi, Prayagraj, 211019, Uttar Pradesh, India  Jayanta Dutta Harish-
Chandra Research Institute (HRI), Chhatnag Road, Jhusi, Prayagraj, 211019,
Uttar Pradesh, India

(Accepted in Astrophysical Journal)

###### Abstract

It has been argued that the low-mass primordial stars ($m_{\rm PopIII}\,\leq
0.8\,M_{\odot}$) are likely to enter the main sequence and hence possibly be
found in the present-day Galaxy. However, due to limitations in existing
numerical capabilities, current three-dimensional (3D) simulations of disk
fragmentation are capable of following only a few thousands of years of
evolution after the formation of the first protostar. In this work we use a
modified version of Gadget-2 smoothed particle hydrodynamics(SPH) code to
present the results of non-linear collapse of the gas clouds associated with
various degrees of initial solid body rotation (parameterized by $\beta$)
using a piecewise polytropic equation of state. The 3D simulations are
followed till the epoch when 50$M_{\odot}$ of mass has been accreted in
protostellar objects, which is adequate enough to investigate the dynamics of
the protostars with the surrounding gaseous medium and to determine the mass
function, accretion rate and survival possibility of these protostellar
objects till present epoch. We found that evolving protostars that stay within
slow-rotating parent clouds can become massive enough due to accretion in the
absence of radiative feedback, whereas $10-12\%$ of those formed within a
fast-rotating clouds ($\beta\geq 0.1$) have the possibility to get ejected
from the gravitational bound cluster as low mass stars.

–Pop III stars – Smoothed particle hydrodynamics – Gadget-2 – sink particles

##  1 Introduction

The age of the Universe and the expected time at which the very first stars
formed makes direct observations a difficult prospect (see recent survey,
e.g., Frebel et al., 2019; Schauer et al., 2020; Suda et al., 2021; Hartwig et
al., 2022; Finkelstein et al., 2022). Theoretical prediction of $\Lambda$CDM
model show that the entire process is led by gravitational collapse of dark
matter halos as a consequence of hierarchical structure formation (see the
latest results, e.g., Springel et al., 2020; Wang et al., 2020; Bohr et al.,
2020; May & Springel, 2021; Latif et al., 2022). At the time of collapse the
primordial gas in halos is very hot and remains spread out due to its high
pressure (Barrow et al., 2017; Chon et al., 2018; Barkana, 2018). Gas cools by
radiating away energy and collapse to form a thin rotating circumstellar disk
that grows over time and fragments due to gravitational and spiral-arm
instability (Inoue & Yoshida, 2020; Wollenberg et al., 2020; Chiaki & Yoshida,
2022). Some of the fragments that go on to become stars are not isolated and
continue to interact with the surrounding gas. This interaction leads to an
increase in mass of fragments as well as changes in their orbits. This leads
to a very basic question of what is the final fate of these evolving fragments
in the cluster. Do they merge with the central star (Kulkarni et al., 2019;
Klessen, 2019), or do they move away from the cluster after their dynamical
interaction with each other and with the surrounding gas (Sharda et al., 2019;
Sugimura et al., 2020)?

It may happen that a fraction of them can either become massive due to rapid
accretion (Umeda et al., 2016; Woods et al., 2017; Fukushima et al., 2020)
such that the resulting stars explode as (pair-instability) supernovae (Whalen
et al., 2014; Welsh et al., 2019; Jeon et al., 2021) or collapse to blackholes
(Madau & Rees, 2001; Matsumoto et al., 2015) or may lead to form supermassive
blackhole (SMBH: Alister Seguel et al., 2020; Herrington et al., 2022). There
might also exist a fraction that remain as low mass protostars and hence can
survive to the present day provided their mass remain as low as 0.8
$M_{\odot}$ (Marigo et al., 2001; Ishiyama et al., 2016; Susa, 2019; Dutta et
al., 2020a). Thus, the mass function of these fragments remains unclear and
needs more investigation (see review by Haemmerlé et al., 2020). While it is
possible to run detailed simulations of a few systems, it is difficult to
explore a wide range of parameter space with this approach. When the numerical
integration over density regime is computed beyond the formation of the first
protostellar core, the collapse tends to be chaotic and highly nonlinear and
becomes difficult to follow the dynamical system for long a time. As a
consequence, current simulations lack the ability of following the evolution
of fragments over a sufficient number of orbital revolutions within the disk.

In this paper, we aim to develop a model, building upon our earlier work
(Dutta, 2016a) on the fragmentation of the unstable disk centred within the
rotating collapsing gas clouds, using modified version of the Gadget-2 SPH
simulations and piecewise polytropic equation of state, in order to put some
upper bound on the final mass of the protostars after long time evolution of
the gas. In the next section §2 we describe in detail the initial conditions,
implementation of polytropic index profile in the mathematical model and the
modified numerical scheme. The details of dynamics are outlined in §3 with an
emphasis on fragments that stay below the critical mass for surviving until
present-day. We summarise work in §4.

##  2 Numerical Methodology

We start our discussion by considering uniform density spheres of gas with
number density $n=10^{4}\text{cm}^{-3}$ and temperature $T=250$ K, with
initial solid body rotation. The gas density is represented by SPH particles.
The clouds are numerically designed to model the local thermodynamic
equilibrium (LTE) conditions for primordial gas and to study the effects of
rotation on the time scales associated with the collapse and subsequent
fragmentation.

###  2.1 Initial condition

The gas clouds are modelled with approximately 2 million SPH particles, each
with mass $m_{\rm gas}=4.639\times 10^{-4}$ $M_{\odot}$, uniformly distributed
inside a sphere of radius equal to the Jeans radius at LTE i.e. $R$ = $R_{\rm
J}$ $\approx$ 0.857 pc with total mass $M$ = $M_{\rm J}$ $\approx$
940$M_{\odot}$. This numerical set-up allows us to follow the collapse
accurately for about 10 orders of magnitude in density up to the number
density $5\times 10^{14}\text{cm}^{-3}$ till the formation of first central
sink (i.e., central hydrostatic core), and about 4 orders of magnitude in size
up to about 10 AU. The mass resolution for $N_{\rm ngb}=100$ SPH neighbours is
about 0.04639$M_{\odot}$. This implies that the rotating gas clouds are
numerically well-resolved up to a critical number density $n_{\rm
crit}=2.02\times 10^{15}\text{cm}^{-3}$, given by

$n_{crit}=\left(\frac{3}{4\pi}\right)\left(\frac{5K_{B}T}{G}\right)^{3}\left(\frac{1}{\mu
m_{p}}\right)^{4}\left(\frac{1}{m_{\rm gas}N_{\rm ngb}}\right)^{2}\,,$ (1)

for temperature $T=1300$ K. Here $\mu$ is the hydrogen mass fraction of the
gas, $m_{\rm p}$ is the mass of the proton and all other symbols have usual
meaning. The free fall time over sound crossing time $t_{\rm ff}/t_{\rm
sc}\sim 0.1$ for the clouds confirms the validity of the initial conditions
for some degree of gravitational collapse.

Initial velocities are assigned to the SPH particles depending upon the
angular velocity $(\Omega)$ of the clouds in addition to the thermal
distribution of velocities and no internal turbulent motion. In absence of
internal turbulent motions, the degree of rotation (i.e., strength of
centrifugal support) of clouds is modelled by estimating the rotational energy
over the total gravitational potential energy (quantified by the parameter
$\beta=R^{3}\Omega^{2}/3GM$, as depicted in Sterzik et al., 2003). We also
model the distribution of angular momentum that originates either from
distortion of the clouds or from their nonaxisymmetric nature due to
differential rotation between the high and low- density regimes (Larson, 1984;
Meynet & Maeder, 2002). The gravitational forces from the dark matter are
negligible compared to the self gravity of the gas on the length scales of our
simulation, therefore for the sake of simplicity, we do not consider dark
matter or the expansion of space itself in our simulations.

Although our calculation is based on the initial condition that assumes a
spherical cloud with uniform density distribution, it is to be noted that
molecular clouds are in general very irregular in shape. This is because in
reality the formation of molecular clouds due to continual accumulation of
baryonic matter within a dark matter halo encounters different types of forces
occurring simultaneously (Larson, 1972; Shu, 1977; Bodenheimer & Boss, 1981).
For example, the interplay between the self-gravity of the cloud and the
internal pressure of the infalling gas can introduce asymmetries or
inhomogeneities in the clouds, which will be amplified during the collapse at
a later time. That is why we see in the cosmological simulation that the
initial density distribution of molecular clouds before it reaches LTE are
non-homogeneities in nature and rather closer to the non-singular isothermal
sphere (Suto & Silk, 1988; Omukai & Nishi, 1998).

In addition, as the gas cloud possesses small angular momentum, it soon
experiences a differential rotation between the layers of the gas due to which
the cloud becomes slightly centrally condensed. As a consequence, non-
axisymmetric features are likely to appear at a very initial phase of
formation of molecular clouds and even tend to wind up to form trailing spiral
patterns at later stages of collapse. Larson 1984 Larson (1984) has elaborated
the fact that these tiny irregularities during the formation of the molecular
cloud can certainly develop non-radial gravitational forces in a non-
axisymmetric mass distribution. The associated gravitational/tidal torques
that transfer angular momentum outward on an orbital time-scale is another
reason for the non-homogeneities, which we see in cosmological simulations.
Besides turbulent motion viscosity, and even magnetic fields can also play a
role in shaping the clouds during its initial dynamical phase (Truelove et
al., 1998; McKee & Tan, 2002; Krumholz & McKee, 2005).

###  2.2 Modeling Polytropic equation of state

In order to account for various heating and cooling processes happening
simultaneously, which become important at certain number densities during the
collapse, we model the thermal behaviour for the gas clouds (following the
discussion in Jappsen et al., 2005) with a piecewise polytropic equation of
state

$T_{i}(n)=a_{i}n^{\gamma_{i}-1}\text{,}i=1,2,3\,.$ (2)

Here the polytropic index $\gamma_{i}$ changes values in a piecewise constant
manner as a function of the number density $n$. The constant of
proportionality $a_{i}$, which is initially determined from the thermal
conditions of the gas is also rescaled in order to maintain the continuity of
temperature across the certain intermediate densities $n_{\rm int}$

$T_{i}(n_{\rm int})=T_{i+1}(n_{\rm int})$ (3)

according to the following equation

$a_{i+i}=a_{i}n_{\rm int}^{\gamma_{i}-\gamma_{i+1}}$ (4)

The intermediate densities and the values for the polytropic index in
different density intervals are chosen carefully in order to reproduce the
temperature-density profile resulting from the primordial chemistry (Dutta et
al., 2015; Pallottini et al., 2017; Bovino et al., 2019). Furthermore as the
fast rotating clouds have larger time scales associated with the collapse and
tend to have lower rates of compressional heating, they are significantly
colder than their slow rotating counterparts (Dutta, 2016a). Therefore all the
values of the polytropic index also depend on the degree of rotation of the
clouds. Table 1 summarizes the chosen values for the polytropic indices for
all the clouds, where the piecewise polytropic index profile is divided in to
three regions separated by intermediate densities $n_{\rm
int}=10^{9},10^{12}\text{cm}^{-3}$.

In order to implement general polytropic process in the publicly available
Gadget-2, we have added a polytropic index variable that controls the rate of
change of entropic function same as the adiabatic index in original code. In
addition to this, we have identified the original adiabatic index variable in
the code with the quantity $1+1/C_{V}$, where $C_{V}$ is specific heat at
constant volume for the gas. The implementation is explained in detail in the
appendix section. This is necessary in order to accurately model the thermal
and chemical evolution of gas. This also reduces the computational cost and
allows the simulations to be followed for a long period of time.

| $\gamma_{1}$ | $\gamma_{2}$ | $\gamma_{3}$
---|---|---|---
$\beta$ |  $10^{4}$-$10^{9}\textrm{cm}^{-3}$ |  $10^{9}$-$10^{12}\textrm{cm}^{-3}$ |  $10^{12}$-$10^{14}\textrm{cm}^{-3}$
(0.0, 0.04) | 1.1362 | 0.9874 | 1.0363
(0.05, 0.09) | 1.0684 | 1.0865 | 1.0422
(0.10, 0.12) | 1.0408 | 1.1174 | 1.0591
(0.13, 0.15) | 1.0214 | 1.1326 | 1.0781
Table 1: The values of the polytropic index $\gamma$ used at three density
intervals (represented by the number density in cm-3) of the collapsed gas for
different degrees of rotational support parametrized by $\beta$

###  2.3 Simulation details

Once the central density reaches the critical value given by equation 1, the
total mass enclosed in single kernel volume $\left(m_{\rm gas}N_{\rm
ngb}\right)$ becomes greater than the local Jeans mass which limits the
density resolution for SPH simulations. Furthermore, the adaptive time steps
for the integration near the critical value become of the order of 0.01 year,
which is too small to be able to follow the simulations for any reasonable
amount of time after the formation of the central core and hence no
fragmentation can be seen.

To circumvent this problem we search among all the processors for the highest
density particle every ten time steps after the number density($n$) reaches
$5\times 10^{13}\text{cm}^{-3}$ for the first time. Since the particles are
distributed over a number of processors according to the domain decomposition,
we broadcast the information of this highest density particle to all the
processors to check for neighbours in their own domain. We then dynamically
replace the entire region centralised at the highest density particle with
$n\geq 5\times 10^{14}\text{cm}^{-3}$ and $T\geq 1300$ K by non gaseous sink
particles upon satisfaction of sink formation criteria as given in Bate &
Bonnell (1997), i.e., the particle is on current time step and the divergence
of both the velocity and acceleration are negative in the vicinity of this
particle. The total potential energy within two smoothing lengths is greater
than the sum of thermal and rotational kinetic energies and that this region
is also virialy unstable.

The sink particles are formed from about 50 neighbouring gas particles within
one smoothing length and thereafter interact with the rest of the gas only
gravitationally. A sink can accrete the gas particles falling into an
accretion radius $r_{\rm acc}$ that we fix to be about 8 AU. This is done
provided the particle is on the current time step, and the total energy of a
gas particle relative to the sink is negative i.e. the particle is
gravitationally bound to the candidate sink. In addition, the specific angular
momentum of the gas particle around the sink is less than what is required to
form a circular orbit.

When a gas particle is accreted, its mass and linear momentum are added to the
sink particle and the location of the sink particle is shifted to occupy the
centre of mass of the two. The accreted gas particles are removed from the
simulation and their effect is taken into account using appropriate boundary
conditions near the accretion region. In addition to the accretion radius, we
also define an outer accretion radius $r_{\rm outeracc}=1.25r_{\rm acc}$ such
that the gas particles falling into this outer accretion radius are evolved
only gravitationally till they reach the accretion radius and are possibly
accreted by the candidate sink. The gas particles may also leave the outer
accretion region in the course of their motion. We prevent sink particles to
be formed within $2r_{\rm outeracc}$ of each other in order to restrain the
counterfeit formation of sink particles from the gas, which eventually would
have been accreted by the candidate sink.

As a check, we also keep track of the global quantities of the gaseous system
such as total energy, angular momentum and entropy throughout the simulation.
The sink particles are usually created at protostellar density and
temperature, and subsequently identified with growing protostars. The
gravitational softening for the sink particles is set to be equal to $r_{\rm
acc}$ while for the gas particles we use variable gravitational softening
length, which is proportional to their SPH smoothing length. This greatly
improves the time taken for the simulations to run. Besides, following the
discussion in Clark et al. (2011), we also implement a constant external
pressure boundary in addition to vacuum and periodic boundary conditions in
Gadget-2.. To this end, we modify the original SPH momentum equation,

$\frac{dv_{i}}{dt}=-\sum_{j}m_{j}\left[f_{i}\frac{P_{i}}{\rho_{i}^{2}}\nabla_{i}W_{ij}(h_{i})+f_{j}\frac{P_{j}}{\rho_{j}^{2}}\nabla_{i}W_{ij}(h_{j})\right]$
(5)

by subtracting the contribution of the external pressure, $P_{\rm
ext}=2.5\times 10^{6}K_{B}\text{K}\text{cm}^{-3}$ from both $P_{\rm i}$ and
$P_{\rm j}$. All the other symbols have usual meaning.

Figure 1:  Radially binned mass-averaged profile of the number density (top
left), free-fall time over sound-crossing time (top right), temperature
(bottom left) and the radial velocity (bottom right) in the spherical collapse
at various epoch when the central number density reaches a peak $\sim$$10^{7}$
cm-3 (red), $10^{10}$ cm-3 (green) and $10^{13}$ cm-3 (blue) for the first
time. The solid line represents the case with no rotation
$\left(\beta_{0}=0\right)$, dashed for $\beta_{0}=0.05$ and dot-dashed for
$\beta_{0}=0.1$. The black dashed line in the density plot shows the power-law
($n\propto r^{-2.2}$) dependence. The gas distribution continues to remain
self-similar at different epochs during the initial phase of collapse.

###  2.4 Check for self-similarity solution

In this section we quickly check the gas distributions at different epochs of
time associated with the runway phase of collapse of the clouds for various
degrees of initial solid body rotation. For non rotating
$\left(\beta=0\right)$ clouds, the density remains spherically symmetric
throughout and follows a well-known power law profile $r^{-2.2}$ while the
central density increases monotonically with time as $\propto 1/(t-t_{\rm
ff})^{2}$. The initial phase of collapse is likely to remain self-similar at
different epochs of free-fall time. This means that the collapsing gas
distribution is invariant, i.e., looks similar in every scale of density
regime. This can be seen from Figure 1, consistent with the conventional
studies (Shu, 1977; Suto & Silk, 1988; Omukai & Nishi, 1998) in which the gas
distribution is a self-similar corresponding to $\gamma_{\rm eff}\sim 1.09$.
Clouds with various rotational support also follow roughly the same power-law
density profile (Matsumoto & Hanawa, 1999; Machida et al., 2008b). However as
collapsed gas gets redistributed and accumulated near the centre of mass of
the cloud, the degree of rotational support also increases. This causes the
density and its gradient to be slightly lower and gas temperatures to be
little lower near the centre for clouds with higher degree of rotation (Saigo
et al., 2008; Bromm & Yoshida, 2011; Meynet et al., 2013; Dutta, 2015).

##  3 Result

Figure 2:  Logarithmically scaled distributions of the number density of the
gas clouds in the equatorial plane (plane of rotation of the circumstellar
disk) for sixteen realizations $\beta\equiv$ 0.0 - 0.15 at an epoch of time
when 50 $M_{\odot}$ has been accreted into the sinks in total. The white dots
are the evolving sinks particles in the simulation and the circle shows the
size of the central region in AU. As shown in the images, the fast-rotating
clouds are likely to develop a small $N$-body system that has a noticeable
dense spiral-arms like structure in which protostellar mass evolves
substantially on larger length scales as compared to their lower rotating
counterparts.

Figure 3:  Temperature distributions of the gas clouds in the equatorial plane
(plane of rotation of the circumstellar disk) for sixteen realizations
$\beta\equiv$ 0.0 - 0.15 at an epoch of time when 50 $M_{\odot}$ gas has been
accreted into the sinks in total. The evolving sinks are shown by the white
dots whereas the circle represents the size of the central region in AU. Since
the gas is centrally condensed for slow-rotating clouds,- the temperature near
the central region becomes high enough. There is also temperature gradient
throughout the layers of the spiral-arms for fast-rotating clouds, which makes
them prone to fragmentation due to interplay between gravitational torque and
pressure gradient.

The transport of angular momentum to smaller scales results in the formation
of rotationally supported spiral-arm, the so-called circumstellar disk or
disk-like structures, around the central hydrostatic core. Till this point the
collapse has been studied rigorously (Greif et al., 2012; Hirano et al., 2014;
Dutta, 2016b; Riaz et al., 2018). Please see Appendix for runaway collapse
phase. Here we follow the simulation further till the epoch of time when
$50M_{\odot}$ of mass are accreted in total onto the dynamically created sink
particles as a consequence of instabilities within the spiral-arm and its
fragmentation. We study the dynamics associated with multiple sinks and their
interaction with ambient gas.

###  3.1 Gas distribution during disk fragmentation

Figure 2 & 3 shows the snapshots of logarithmically scaled densities and the
temperature distribution of the gas in the equatorial plane of the
circumstellar disk for the sixteen values of rotation parameter $\beta$ = 0.0
- 0.15. All the images understandably reflect the fact that the collapsed gas
in the spiral-arm becomes unstable by accreting mass from the surrounding and
hence prone to fragmentation. This may also be the consequences of interplay
between the gravitational torque and pressure gradient throughout the layers
of the spiral-arms. In all these snapshots, the white dots represent the sink
particles in the simulation and the circle shows the scale in AU at which the
fragmentation takes place. The size of the circle generally keeps on extending
with increasing $\beta$-parameter. For example, the central region for
$\beta=0.02$ is approximately 600 AU, whereas for $\beta=0.1$ it is $\sim$
3000 AU. The snapshots also reflect the fact that all the clouds with non zero
rotation form a small $N$-body protostellar system immersed in the ambient
medium near the centre of the clouds. As expected, the fast-rotating clouds
are likely to develop noticeable dense spiral-arms in which protostellar mass
evolves substantially, and in all probability to fragment reasonably more,
say, $N\sim 8-11$ for $\beta>0.05$, as seen in the Fig. 2 & 3. This satisfies
the justification for conservation of angular momentum during the gas
evolution (Hirano & Bromm, 2018a). In contrast, the slow-rotating clouds
contain a relatively small number of protostars, $N\sim 4-7$, indicating the
possibility of having a high accretion rate, as predicted in theoretical
calculations (Machida & Doi, 2013; Liu et al., 2021). We have limited our
calculations up to this epoch because it is extremely difficult to follow
simulations for the fast-rotating clouds above $\beta>0.15$ beyond this stage
of evolution.

Another feature to be noted in Figure 2 is that the gas distribution is highly
complicated and non-linear in nature that can be thought of a supersonic and
compressible flow coupled to the accreting sinks in the gravitationally bound
protostellar system. Due to interaction with the ambient medium, the evolving
sinks experiences a strong friction, also known as drag forces (Dutta et al.,
2020b), which can change the movement and orbit of the sinks (similarly what
is seen in the X-ray binaries system Bobrick et al., 2017; Park et al., 2021).

###  3.2 Evolution of the sinks

The fragmentation of the rotating unstable spiral-arms within the
circumstellar disk has significant implications on the final mass of the
evolving sinks. Figure 4 & 5 shows the mass of the sinks in our simulation as
a function of time. The first feature to be noted is that all clouds are
inclined to fragment on an estimated mass scale that evolve up to $\sim$0.001
- 20 $M_{\odot}$ depending on the strength of rotational support of the parent
clouds. Because the fragmentation takes place as a consequence of the
gravitational instability, the characteristic mass scale may be substantially
smaller. Second, as the gas continues to collapse to higher densities, the
spiral-arms keep on developing instabilities that heralds in the successive
formation of secondary sinks within the circumstellar disk. We see that most
of the fragmentation takes place within $\sim$100 - 200 years from the
formation of the central core for clouds with $\beta\leq 0.05$. Higher is the
degree of rotation, longer is the time $t_{\rm frag}$ taken by the gas to
become gravitationally unstable to fragmentation. This is expected as the
sinks within slow rotating clouds begin to be Jeans unstable much earlier due
to very strong accretion rate, $\sim 1M_{\odot}/yr$ (as seen in Figure 5). In
addition, we see that sinks moving with lower radial velocity within dense
ambience are likely to have high accretion rate. Thus, even if the sinks had
low mass at the time of formation, their mass can be increased approximately
by an order of magnitude relative to their initial mass. As a consequence,
sinks now face more gravitational drag due to increase of mass, and therefore
they are likely to change their orbits. This also triggers the sinks to be
more centrally condensed and continue to accrete gas to end up as massive
protostars within a few thousands of years of evolution. We also see that
sinks for low $\beta$-values are quite strongly bound gravitationally.
However, clouds with higher degree of rotation tend to fragment more
vigorously due to spiral-arm instabilities on larger scales and contain both
low and high-mass sinks (some of which even have quite a high radial velocity
as compared to the escape velocity of the cloud). For example, a number of
protostars with $m_{*}<1\,M_{\odot}$ are formed for the clouds with $\beta\geq
0.1$. This is consistent with the probability of existence of the smallest
fragmentation scale $\sim$$0.03$ AU with $\sim$$0.01\,M_{\odot}$ (Becerra et
al., 2015; Hirano et al., 2017). We conclude that the formation of the sinks
and their dynamical interaction with the ambience depend on the history of
collapse (i.e., evolution history of chemical/thermal changes, turbulence and
angular momentum conservation).

Figure 4:  Mass evolution of the sinks in the simulations are shown as a
function of time since the formation of first central hydrostatic core all the
way upto the epoch when 50 $M_{\odot}$ has been accreted by the sinks in
total, for fifteen instances of the clouds parametrized by $\beta\equiv$ 0.008
- 0.15. Depending on the rotation of the clouds, most of the sinks are formed
within a few hundreds to thousands years of evolution from the formation of
the central hydrostatic core. Note that fast rotating clouds tend to fragment
more to form sinks over a span of mass 0.001 - 20 $M_{\odot}$ depending on the
strength of rotational support.

Figure 5:  Different panels show the time evolution of the mass accretion rate
by the sinks within the rotating gas clouds quantified by the parameter
$\beta$, similar to Figure 4, i.e., once roughly $\sum{m_{\rm
sink}}\,\sim\,50\,M_{\odot}$ has been accreted by the sinks in total. Sinks
formed out of the slow-rotating clouds are expected to accrete the ambient gas
at comparatively higher rate (say, $dm/dt\sim 10^{-1}-10^{0}$ yr-1), whereas
accretion for those formed in fast-rotating clouds span over roughly four
order of magnitude, and could be as low as $dm/dt\sim 10^{-4}$ yr-1.

###  3.3 Histogram of mass function and radial velocity

Here we try to understand the basic properties of the sinks, such as mass,
radial velocity and rotational velocity, as reflected in Figure 6. On the
other hand, Figure 7 depicts the histogram of the mass function (Top) and the
ratio of the radial velocity of the sink particles to the local escape
velocity of the cloud (Bottom) at the end of the simulations, i.e., when
$50M_{\odot}$ has been accreted in total. The newly formed sinks in their
parent clouds tend to have a wide range of velocities, the typical value of
both radial ($v_{\rm rad}$) and rotational ($v_{\rm rot}$) component lies
within a span of roughly $\sim$$0.01-25$ km s-1. This is consistent with the
previous studies (e.g., Greif et al., 2011; Dutta, 2016a). In addition, we see
that in the absence of radiative feedback, relatively high-mass sinks are
likely to be formed irrespective of the rotation of the clouds. Another
interesting aspect from the simulations is that a tiny fraction of low-mass
sinks for $\beta$ $\equiv$ 0.1 - 0.15 move with relatively high radial
velocity as compared to others. They can therefore directly travel at the
periphery at later stage of evolution, and can even go away from potential
well of the gaseous system with their radial velocity exceeding the escape
velocity ($v_{\rm escape}\equiv 10-12$ km s-1, as seen in Dutta et al.,
2020a). Due to high velocity, the ejected sinks can accrete negligible mass
from the surrounding. There are subtle deviations between the rotationally
supported clouds. From the analysis, it is clear that the mass function for
fast rotating clouds peaks at lower mass value, and also the central sink can
get quite massive for fast rotating clouds due to larger time scale for
fragmentation. The radial velocities for these sinks can also get 2-3 times
their escape velocity depending on the strength of the cloud’s rotation.

Figure 6:  Properties of the individual evolving sinks formed as a result of
disk fragmentation within their parent clouds, parameterised by a wide range
of rotational support, are shown in different panel: radial velocity (Top
Left), rotational velocity (Top Right), ratio of radial velocity to the escape
velocity (Bottom Left) and their mass (Bottom Right) as a function of radial
distance from the center of the cloud. This is done at an epoch of time when
the multiple system contains a mass roughly $\sum{m_{\rm
sink}}\,\sim\,50\,M_{\odot}$. This epoch corresponds to a few thousands of
years after the formation of the central hydrostatic core. As discussed in our
previous study Dutta (2016a), it is evident that clouds with higher rotational
support tend to fragment more with a range of mass span over low to high,
whereas slowly rotating clouds produce comparatively high-mass sinks.

Figure 7:  Various panels show the histograms for the mass of the sink
particles (Top) and the ratio of magnitude of velocities of the sink particle
to the escape velocity of the cloud (Bottom) at end of the simulations, i.e.,
at the epoch when the multi-scaled $N$-body protostellar system attains a
total mass of $\sum{m_{\rm sink}}\,\sim\,50\,M_{\odot}$. It is clear that the
slow-rotating clouds (e.g., $\beta$ $\equiv$ 0.008, 0.01) are inclined to
fragment on a mass scale $\sim$0.001 - 20 $M_{\odot}$ within the central
regime $\sim$20 - 100 AU. On the contrary, fast rotating clouds are likely to
contain a few low-mass sinks that move faster towards the outer end and
accrete a negligible mass from surrounding gaseous medium.

###  3.4 Survival possibility

Here comes a very important issue to address: the survival possibility of such
evolving sinks that may be identified as primordial protostars. Note that the
lifetime of a star is inversely proportional to the mass that it contains, say
a combination of hydrogen, helium and other higher metallicity gas (see e.g.,
Binney & Tremaine, 2008, for a detailed calculation of stellar dynamics). In
this scenario, a star can survive for billions of years provided its accretion
rate remains very minimal so that estimated final mass would be as low as
$\sim 0.8$ M⊙ (Komiya et al., 2009; Kirihara et al., 2019). In Figure 8, we
plot the mass of the sinks that get ejected from the multi-scaled N -body
cluster for rotating clouds parametrized by $\beta$. As expected, the
ejections are likely to happen for the higher rotating gas clouds (e.g.
$\beta>0.1$, as shown in Figure 6, the overall mass distribution of the
protostars). These realisations show that due to conservation of angular
momentum, a fraction of protostars are spread out to the outer periphery of
the clouds out of which only a tiny fraction (10 - 12%) are being ejected as
low mass stars. As they continue to have negligible accretion rate for a long
period of time, their final mass remains as low as 0.8$M_{\odot}$, and hence
likely to survive till today. The dotted horizontal line in Figure 8 reflects
the threshold mass($0.8M_{\odot}$) for the ejected protostars. The others can
be the massive ones. There is hence a high possibility that a fraction of the
sinks within the fast rotating clouds can escape the cluster as low-mass
sinks.

There are considerable chances that they can be evolved as main-sequence stars
before entering ZAMS to survive for billions of years till the present epoch.
This also confirms the theoretical prediction of the existence probability of
first generation stars (it may be either Pop III/Pop II stars or extremely
metal poor stars) if they would have contained very low mass ($\leq 1$ M⊙)
before entering as a ZAMS (Andersen et al., 2009). Following the recent study
by Dutta et al. (2020a) of the Bondi-Hoyle accretion flow, one can estimate
the mass-velocity relations for the protostars in which an initial high speed
ensures that the mass accretion is relatively smaller. There is hence a good
possibility of their existence even in our Galaxy (either in bulge or in
halo). Recent state-of-the art observational tools and wide-range surveys have
suggested the existence of extremely metal-poor stars that could be the
possible candidates of low-mass Pop III/Pop II stars (e.g. Komiya et al.,
2010; de Bennassuti et al., 2017; Husain et al., 2021). Therefore searching
for more such low-mass stars and where they are located in our Galaxy have
become one of the primary interests in present-day observations (Johnson,
2015; El-Badry et al., 2018; Griffen et al., 2018; Susa, 2019; Liu & Bromm,
2020).

Figure 8:  Mass of the sinks that get ejected from the multi-scaled $N$-body
protostellar system is plotted as a function of the rotation parameter $\beta$
of the clouds. The dotted horizontal line indicates the threshold
mass($0.8M_{\odot}$) for the ejected protostars. Only a few sinks are capable
of overcoming the strong gravitational drag of the dense gaseous medium and
can go away from the potential well of the cluster with radial velocities
greater than the escape speed. However, it is noted that only a tiny fraction
of them ($\sim$ 10 - 12%) continue to have a negligible accretion rate for a
long period of time to end up as low-mass sinks ($\leq 0.8\,M_{\odot}$). One
aspect is that they are likely to remain in the main sequence for billions of
years to be survived till present epoch.

##  4 Summary and Discussion

In this work, we performed a suite of 3D simulations using our modified
version of Gadget-2 SPH code to follow the gravothermal evolution in a number
of primordial gas clouds associated with a degree of rotation that spans over
roughly two order of magnitude. The heating and cooling phenomenon that arises
during chemical and thermal evolution of the collapsing gas is approximated
with a piecewise polytropic equation of state appropriate for the primordial
chemistry. Below we outline the main points related to the modification in the
SPH simulations and summary of results along with open issues.

###  4.1 Code modification

  * •

In addition to the “adiabatic” and “isothermal” modes available in the
publicly available version of Gadget-2, we added a “polyttropic” mode in the
code, enabling of which in the makefile evolves the gas system with a general
polytropic equation of state(EOS). This is done by introducing a polytropic
exponent, and appropriately modifying the formulas for internal energy,
entropy and rate of change of entropy.

  * •

We have carried out sink particle technique in the original Gadget-2,
following the discussion in Bate & Bonnell (1997) along with the sink boundary
conditions near the accretion surface. We have also defined an outer accretion
radius $r_{\rm outeracc}\approx 1.25r_{\rm acc}$, for the sinks, such that the
gas particles falling into this outer accretion radius are evolved only
gravitationally. Sink particles can be activated in code by enabling the sink
mode in the makefile. In this mode, the simulations run till 50 unit of mass
(for example, here we use 50 $M_{\odot}$) has been accreted into the sinks in
total, and writes the data related to all the sinks (e.g. mass, position,
velocity, internal spin) in their individual text files.

###  4.2 Summary of results

  * •

Irrespective of the cloud’s rotation, most of the sinks start to accrete from
the ambient gaseous medium while orbiting the central region. In the absence
of radiation mechanism, the continued accretion results in the increase of
mass of the evolving sinks, which are likely to turn out to be the high mass
protostars, or even massive one, depending on the dynamical evolution. More
the mass they accrete, the more the friction they experience due to
gravitational drag, which in turn slows their velocity and constrain them to
either change their orbital movement or stop. This is more noticeable in slow-
rotating clouds where protostars are formed within a few hundreds of AU from
the central region and keep on evolving in the absence of radiative feedback
from them.

  * •

Clouds with higher degree of rotational support tend to fragment more
vigorously at far distance from the central region and on larger time scales.
After a few thousand years of evolution, a small fraction of these sinks still
remain as low mass protostellar objects. In this case, they remain loosely
bound to each other gravitationally and may possess radial velocities larger
than the escape velocity of the clouds. Thus there is considerable
plausibility of these protostars to be ejected from their parent clouds and
remain on the main-sequence for a long time.

  * •

The detailed calculation arguably confirms the long-time conjectures of two
completely opposite views regarding the mass function of the first generation
of stars – whether they are high-mass or low-mass. Based on our simulations,
we have corroborated the fact that the primordial stars might have been formed
in a broad range of masses depending on dynamical history of the collapse and
initial configuration of the accreting protostars. By looking at the results
one can anticipate the fact that the ejected protostars are likely to be
survived till present epoch provided their mass remains as low as $m_{\rm
PopIII}\leq 0.8M_{\odot}$. The lowest probable mass range of the protostars
from our model is comparable to the recent observations (Schlaufman et al.,
2018).

We conclude that the survival rate of a primordial star depends on the initial
degree of rotation of the parent gas clouds. Hence simulating long term
evolution of the primordial protostellar system is indispensable in order to
have an estimation of the initial mass function(IMF) of the Pop III
protostars. However, one needs to perform a more rigorous investigation along
with inclusion of more sophisticated simulations such as magnetic field,
radiative feedback and primordial chemistry.

###  4.3 Effect of magnetic field

A number of studies have confirmed the impact of primordial magnetic fields on
the thermodynamics of Pop III star formation (Sur et al., 2010) and on the 21
cm emission line of atomic hydrogen (Schleicher et al., 2009). Other magneto-
hydrodynamics (MHD) simulations (for example, Price & Bate, 2007; Machida et
al., 2008a; Schleicher et al., 2010) show the contribution from so-called
magnetic flux (Maki & Susa, 2007) and magnetic braking (Meynet et al., 2011).
Besides, magnetic fields can also be amplified by orders of magnitudes over
their initial cosmological strengths by a combination of small-scale dynamo
action and field compression (Doi & Susa, 2011; Sur et al., 2012; Turk et al.,
2012; Machida & Doi, 2013). Magnetic fields may also provide support against
fragmentation (Peters et al., 2014) and outflow (Hirano & Machida, 2019).
Hence, it is important to note that the inclusion of magnetic fields may
substantially influence the gas dynamics, especially the disk evolution and
fragmentation phenomenon (see e.g., McKee et al., 2020; Stacy et al., 2022).

###  4.4 Effect of radiative feedback

The radiation-hydrodynamics (RHD) simulations on the other hand show that the
radiation from protostars can significantly change the accretion phenomenon
(Whalen et al., 2004, 2008; Wise & Abel, 2008; Wise et al., 2012; Susa et al.,
2009) and can even evaporate the disk (Hosokawa et al., 2011; Johnson et al.,
2013; Hirano & Yoshida, 2013). In general we expect feedback to lower the
accretion of mass and metals onto protostars (Suzuki, 2018) and hence our
estimates can be thought of as being upper bounds on these masses (Barkana,
2016; Barrow et al., 2017; Chon et al., 2018). However, it is noted that the
radiative feedback becomes important approximately after $\sim$$10^{4}$ year
has elapsed from the time of formation of the first protostar. Here, the
results from our calculations clearly demonstrate that protostars with $v_{\rm
r}\geq v_{\rm esc}$ are able to escape clusters within a few times $10^{4}$
year. Hence they can accrete negligible bulk of mass, which implies that
radiation emitted from the surface of these stars are unlikely to have an
impact on their mass accretion process. On the other hand, it is obvious that
feedback effects need to be included for those protostars roaming around with
$v_{\rm r}\ll v_{\rm esc}$. However, analyzing the fate of these protostars is
beyond the scope of the present study.

###  4.5 Effect of primordial chemical network

In a realistic collapse scenario, various chemical species go through the
numerous reactions that happen concurrently. This results in the formation of
a number of other molecules depending on the local thermodynamic state of the
gas (see wonderful reviews by Loeb & Barkana, 2001; Ciardi & Ferrara, 2005).
The thermodynamic balance among these primordial chemical species then
determines the net rate of compressional heating and radiative cooling in the
gas. Therefore a detailed knowledge of several possible chemical reactions and
mass fractions of all the chemical species is required in order to specify the
overall chemo-thermal state of the gas (Omukai & Yoshii, 2003; Ripamonti,
2007; Dutta, 2015). Hence one has to follow the entire network in detail in
order to determine the thermodynamic evolution of the gas, which seems to be
fairly complicated to follow in the simulations even for primordial gas with a
relatively simpler chemical network of Hydrogen and Helium.

###  4.6 Combined 3D large-scale simulation with Bondi-Hoyle semi-numerical
approach

As discussed above, the protostellar system can numerically be considered as a
classical $N$-body problem in which the evolving protostars accretes the
ambient gas and coupled with gravity while orbiting around the central gas
cloud. However, in reality, this complex phenomenon is indeed difficult to
simulate in full 3D simulations (Bagla & Khandai, 2009; Stacy et al., 2010) as
there is a huge difference of density gradient between the layers of spiral
arms. This results in a substantial computational cost. However, with some
approximation, the protostellar system can be modeled using semi-analytical
calculations with the help of Bondi-Hoyle accretion (Bondi & Hoyle, 1944;
Bondi, 1952) that may provide an satisfactory outcome for the study of stellar
dynamics (see e.g., Edgar, 2004; Lee et al., 2014; Bobrick et al., 2017; Xu &
Stone, 2019). It is therefore important to focus on combining the 3D
simulation along with Bondi-Hoyle semi-numerical approach (Dutta et al.,
2020a; Park et al., 2021; Keto & Kuiper, 2022) to study the long-term
evolution of gas, especially the instabilities that grow during the build-up
of the circumstellar disk and complex fragmentation process of its spiral-arms
(in preparation).

###  4.7 Evidences for metal-poor stars

In recent past, a number of studies have predicted that there is a substantial
feasibility of finding the first generation of stars, that could be the
possible candidate for Pop III stars or extremely metal poor(EMP) stars, in
the present-day local universe (White & Springel 2000; Tumlinson 2010;
Schneider et al. 2012; Gibson et al. 2013; Bland-Hawthorn et al. 2015; Komiya
et al. 2016; Ishiyama et al. 2016). The search for EMP stars or very metal
poor stars have been a cutting-edge research now-a-days, and with the advent
of new ultra-modern telescope and futuristic surveys there have been a number
of pioneering observational studies that provide a crucial information about
the early Universe (Dawson et al., 2004; Eisenstein et al., 2005; Tegmark et
al., 2006; Furlanetto et al., 2006; Cooke et al., 2009; Bagla et al., 2009;
Caffau et al., 2011, 2012; Ahn et al., 2012; Dawson et al., 2013; Rydberg et
al., 2013; MacDonald et al., 2013; Frebel et al., 2014; Mirocha et al., 2017).
There have been numerous observational analysis on the evolution of baryonic
matter both from cosmological viewpoint (Wells & Norman, 2021) and statistical
interpretation on how the Universe has evolved into the present state (please
see recent studies by Planck Collaboration et al., 2011, 2014, 2016, 2020, for
a preliminary understanding of the evolution). Besides, Pop III stars can also
be major origins of both merging binary blackholes and EMP stars, as shown in
Tanikawa et al. (2022). Recent studies have also shown the possibility of
detecting primeval galaxies at higher redshift $z\geq 10$ through JWST
observation (Jeon & Bromm, 2019; Riaz et al., 2022). This may constrain the
mass function of Pop III stars. Interestingly, Hubble has just detected a very
old magnified star of mass $\sim 50-100M_{\odot}$ around redshift 6.2 (Welch
et al., 2022). With the help of state-of-the-art observations from Atacama
Large Millimeter/submillimeter Array, Tokuoka et al. (2022) has also confirmed
the possible systematic rotation in the mature stellar population of a $z\sim
9.1$ Galaxy.

## Acknowledgements

All the simulation are run on the HPSC cluster at Harish-Chandra Research
Institute (HRI) at Prayagraj. We thank Jasjeet Singh Bagla, Athena Stacy,
Sharanya Sur and Sukalpa Kundu for their constructive comments on the
manuscript. This research has made use of NASA’s Astrophysics Data System
Bibliographic Services.

##  Appendix A Implementation of polytropic equation of state in Gadget-2

The standard model of thermodynamic behaviour of the primordial gas clouds has
been well-understood (see for e.g., Abel et al., 2002; Yoshida et al., 2006;
Glover & Savin, 2009; Turk et al., 2009). The primordial chemical network
primarily contains numerous concurrent reactions between hydrogen and helium.
For example, at low densities $(n_{H}\approx 1-10^{4}\text{cm}^{-3})$, the
hydrogen atoms combine with the free electrons to produce hydrogen molecule
ion $H^{-}$ which in turn combines with the hydrogen atoms to form a small
abundance of $H_{2}$. The gas is then cooled through $H_{2}$ rotational and
vibrational line emission up to temperature of 200K. However, the small
abundances of $H_{2}$ is not sufficient to cool the gas further and the gas
begins to heat up with increase in density up to a number density of about
$10^{8}\text{cm}^{-3}$. At this stage of collapse, the hydrogen atoms are
converted to molecules via three body reactions, which again cool the gas
through line emissions. The cloud becomes optically thick to the strongest of
$H_{2}$ emission lines beyond number density $10^{11}\text{cm}^{-3}$
(Ripamonti & Abel, 2004; Clark et al., 2011; Dutta et al., 2015). However, if
we are only interested in the thermodynamic evolution of the gas and not the
detailed balance between abundant chemical species, then we can make
substantial simplifications and use a general polytropic equation of state for
all the chemical processes that involve transfer of heat, (in the line of
discussion Jappsen et al., 2005):

$T=a\rho^{\eta-1}\,,$ (A1)

where $\eta$ is the polytrpoic index. Therefore following the above
discussion, we use a polytropic equation of state with a piecewise constant
profile for a polytropic index appropriate for thermodynamic evolution of
primordial gas clouds. Where the polytropic index $\eta$ changes its values at
certain critical number densities, according to the following relations. The
publicly available version of Gadget-2 can be used to examine the numerically
evolved ideal gas system with adiabatic equation of state

$P=A\rho^{\gamma},\,$ (A2)

where $A$ is constant and $\gamma=C_{P}/C_{V}$ is the adiabatic index. The
internal energy per unit mass, u, of the gas as calculated in the code is

$u=\left(\frac{A}{\gamma-1}\right)\rho^{\gamma-1}$ (A3)

In order to model the general polytropic process that can involve the heat
transfer as well, we write the following identity from the the first law of
thermodynamics

$NK_{B}C\Delta T=P\Delta V+NK_{B}C_{V}\Delta T,\,$ (A4)

where C is a rate of heat added to the system (for adiabatic process C=0), N
is the total number of gas particles and the other symbols have their usual
meaning. From the above equation and ideal gas law $PV=NK_{B}T$ we can derive
the polytropic equation of state as

$P=B\rho^{\eta},\,$ (A5)

where $B$ is constant and $\eta=\left(C-C_{P}\right)/(C-C_{V})$ is the
polytropic index. Unlike the adiabatic index $\left(\gamma\right)$ the
polytropic index $\left(\eta\right)$ can be greater, smaller or equal to 1,
and the two are related by.

$\gamma=\eta+\frac{C}{C_{V}}\left(1-\eta\right)$ (A6)

Therefore in the code we choose $\gamma=1+1/C_{V}=7/5$ for diatomic gas and
replace $\gamma$ with $\eta$ at appropriate places, for example the internal
energy formula in A3 is modified to be

$u=\left(\frac{B}{\gamma-1}\right)\rho^{\eta-1}$ (A7)

With these modification the code can handle general polytropic processes where
the temperature can increase $\left(\eta>1\right)$ or decrease
$\left(\eta<1\right)$ with density and does not require special treatment for
the isothermal $\left(\eta=1\right)$ case.

##  Appendix B Runaway collapse phase

Figure 9:  Initial (runway) collapse phase has been shown for the gas clouds
associated with various degrees of initial solid body rotation just before the
formation of the central core. The physical properties (estimated as
logarithmically scaled radially averaged) of gravitational collapse, such as
gas distribution (Top Panel), velocity structure (Middle Panel) and accretion
phenomenon (Bottom Panel) are plotted as function of the radius. The collapse
remains self similar at different scales and satisfies the power-law density
profile $n\sim r^{-2.2}$ irrespective of the rotation of the clouds, (in
agreement of previous studies Bromm & Larson, 2004; Glover, 2005; Yoshida et
al., 2006). It also shows that the slow rotating clouds have higher degree of
compressional heating and are therefore hotter compared to fast rotating
counterparts. Once the primordial gas gets redistributed near the centre of
mass of the cloud, the ratio $v_{\rm rot}(r)/v_{\rm kep}(r)$, where $v_{\rm
kep}=\sqrt{GM_{\rm enc}(r)/r}$ and $M_{\rm enc}(r)$ is the mass enclosed
inside the sphere of region of radius $r$, indicates that the clouds with
higher degree of rotational support go through more efficient phase of angular
momentum transport (as shown in Greif et al., 2012; Stacy & Bromm, 2014;
Dutta, 2016b; Hirano & Bromm, 2018b).

In this section we describe the gas distributions, velocity profile and time
scales associated with the initial phase of collapse for the clouds for
various degrees of initial solid body rotation till the formation of the
central core in our simulations, as illustrated in Figure 9. All the physical
quantities are radially averaged within logarithmic binned as calculated from
our simulations.

The estimated free-fall time over sound crossing time justifies the
gravitational collapse of primordial gas, for which the density distribution
obey the power-law profile with $n\sim r^{-2.2}$ irrespective of the
rotational strength of the clouds. This also confirms that collapse is a self-
similar (Susa et al., 1998) process. Therefore the density profile of the
collapsed clouds at the outer part is nearly the same as that of the inner
regime till the core is formed at the center of clouds (see, e.g., Meynet et
al., 2013; Stacy et al., 2013; Latif & Schleicher, 2015; Dutta, 2016b, for
detailed discussion). Due to high strength of rotation, the radial velocities
are considerably lower, as expected for clouds with $\beta\geq 0.05$. So the
radial component of velocity is less dominating and gradually becomes
comparable to the rotational component near the centre of mass till about 100
AU. This is due to the fact that the infalling gas loses angular momentum near
the centre as it gets accreted by the central core. The loss of angular
momentum near the centre is compensated by transport of angular momentum
farther from the centre. Hence, the angular momentum transport is more
noticeable for the clouds with higher degree of rotational support. This is
evident from the distribution of the rotational component of the velocities.
This also vindicates the formation of larger rotationally supported spiral-
arms in the clouds with fast rotating clouds. The flow outside the spiral-arms
remains sub keplerian i.e. $v_{\rm rot}(r)\leq v_{\rm kep}(r)$, as seen in the
middle panel of Figure 9.

In order to quantify the effects of rotation on the accretion phenomenon and
associated time scales, we estimate the mass accretion rate, $\dot{M}(r)=4\pi
r^{2}\rho(r)v_{\rm rad}(r)$ and the accretion time $t_{\rm acc}=M_{\rm
enc}(r)/4\pi\rho v_{\rm rad}(r)r^{2}$, and plot them as a function of radial
distance for different value of the $\beta$-parameter. As can be seen from
figure 9, the mass accretion rate reaches a maximum value of about 0.1
$M_{\odot}$ per year at about 20 AU from the centre of mass. For distances
smaller than this the sound crossing time tends to become comparable to the
free fall time which decreases the mass accretion rate near the centre.
Because the accretion phenomenon is directly related to the instability within
the gas, we also check the degree of Jeans instability in our clouds by
measuring the two quantities, $t_{\rm acc}(r)/t_{\rm ff}(r)$ and $t_{\rm
frag}(r)/t_{\rm ff}(r)$ as a function of the radial distance. Here, the
fragmentation time, $t_{\rm frag}(r)$, has been estimated by the ratio of
Jeans mass and the mass accretion rate of the core (Dutta et al., 2015). This
can provide an approximation of the fragmentation of the gas for different
strength of rotation of the clouds. As the infalling gas gets redistributed
near the centre of mass of the cloud, the rotational support near the centre
also increases, heralding the formation of spiral-arm or disk-like structure.
These spiral-arms are likely to become unstable and prone to fragmentation by
accreting mass near its boundaries. The fragmentation time scales and sizes of
the disks are proportional to their degree of rotation and are considerably
larger for fast rotating clouds $\beta\geq 0.05$ than their slowly rotating
counterparts for the fixed polytropic index profile.

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11institutetext:  Software Engineering Group, University of Würzburg, Germany
✉ Corresponding author: 11email<EMAIL_ADDRESS>
# Early Explorations of Lightweight Models for Wound Segmentation on Mobile
Devices

Vanessa Borst(✉)  Timo Dittus  Konstantin Müller  Samuel Kounev

###### Abstract

The aging population poses numerous challenges to healthcare, including the
increase in chronic wounds in the elderly. The current approach to wound
assessment by therapists based on photographic documentation is subjective,
highlighting the need for computer-aided wound recognition from smartphone
photos. This offers objective and convenient therapy monitoring, while being
accessible to patients from their home at any time. However, despite research
in mobile image segmentation, there is a lack of focus on mobile wound
segmentation. To address this gap, we conduct initial research on three
lightweight architectures to investigate their suitability for smartphone-
based wound segmentation. Using public datasets and UNet as a baseline, our
results are promising, with both ENet and TopFormer, as well as the larger
UNeXt variant, showing comparable performance to UNet. Furthermore, we deploy
the models into a smartphone app for visual assessment of live segmentation,
where results demonstrate the effectiveness of TopFormer in distinguishing
wounds from wound-coloured objects. While our study highlights the potential
of transformer models for mobile wound segmentation, future work should aim to
further improve the mask contours111This is an extended version of our KI 2024
paper, available at the Springer website..

###### Keywords:

Medical Image Segmentation Mobile AI Deep Learning.

##  1 Introduction

Chronic wounds, such as diabetic foot ulcers, affect a significant proportion
of the world’s population and present a major challenge to healthcare systems
worldwide. In the United States alone, it is estimated that more than 6
million people suffer from chronic wounds each year [23]. In addition to
diabetes, age is a significant risk factor, with a disproportionate impact on
the elderly. This is evidenced by a prevalence of 7.8% in German nursing home
residents [24] compared to 1% in the German population [9]. As the world’s
population continues to age and the incidence of conditions such as diabetes
increases, the prevalence of chronic wounds is expected to soar in the coming
years. This increase poses a significant challenge to healthcare providers in
providing appropriate care to those affected. From the patient’s perspective,
the need for frequent visits to specialised wound care facilities to manage
chronic wounds can be a significant burden. Access to such clinics may be
limited, particularly in remote areas, and mobility issues exacerbate the
challenge. There is an urgent need for innovative solutions to alleviate these
burdens and improve access to wound care services.

Telemedicine and automation offer promising ways to address these challenges
by improving efficiency and accessibility. Automated wound size monitoring has
the potential to eliminate the need for patients to visit clinics solely for
monitoring purposes, thereby reducing the associated logistical challenges and
relieving the burden on healthcare professionals. Furthermore, automated
approaches provide timely and accurate information on wound status,
facilitating proactive intervention. Mobile semantic segmentation emerges as a
key technology in realizing these advancements, providing real-time feedback
and segmentation masks directly to users, thereby circumventing latency issues
associated with server-based processing. In addition, deploying models
directly on smartphones ensures enhanced privacy and security by keeping
sensitive medical data local. In addition, it enables offline functionality,
which is crucial for applications in remote areas with limited network
coverage. Despite the advantages of mobile segmentation, there are also
significant challenges associated with this endeavour. To be effective, such
solutions must include small, lightweight models that can be seamlessly
deployed on smartphones. At the same time, such models must be robust in the
face of challenging conditions such as different backgrounds, varying lighting
conditions, and different camera settings, as mobile wound monitoring is
intended to take place in the patient’s home environment.

However, despite the necessity for compact yet resilient models, the
investigation into mobile wound segmentation remains limited to date. While
several methods have been proposed for wound segmentation utilizing feature-
engineering-based machine learning [27, 35, 14], deep learning [20, 16, 4, 8],
or even both [31, 15], they often lack emphasis on lightweight architectures
and mobile suitability. Some publications suggest that the analyzed
architecture(s) could be used on mobile devices due to its efficiency, but
such indications often rely solely on metrics such as parameter count as
mobile performance indicator, lacking real-world deployment [33, 20]. Of the
few approaches deployed to a mobile device in practice, many apply traditional
image processing rather than deep learning [25, 30, 7], while others evaluate
only single architectures [19], or require manual annotation outlines from
users [1, 2]. To the best of our knowledge, a systematic comparison of
different neural network-based wound segmentation techniques on mobile devices
is still pending. In contrast, the number of proposed architectures for mobile
segmentation beyond the wound context has increased significantly. To address
this discrepancy, this work conducts initial investigations into the
applicability of different architectures from non-wound domains for mobile
wound segmentation, including a visual comparison in real-world usage.

##  2 Approach

###  2.1 Task Definition and Requirements

This study aims to assess the efficacy of lightweight models for mobile wound
segmentation by systematically comparing approaches from beyond the wound
domain. In particular, we focus on their ability to perform live segmentation
from smartphone camera streams, for which we consider the following
requirements paramount:  (i) accuracy: High segmentation performance for
accurate analysis of wound area  (ii) efficiency: Small number of parameters
to enable mobile processing  (iii) robustness: Effective performance under
varying conditions for use in non-standardized home environments  (iv)
practicality: Effective segmentation in real-world deployment to enhance
medical practice  .

###  2.2 Model Selection

The number of proposed models for segmentation on mobile devices has increased
significantly beyond wound segmentation. Notable advances include CNN-based
methods, such as encoder-decoder architectures with lightweight backbones such
as MobileNets [11, 10] combined with different decoders, ENet [21], UNeXt
[29], or Fast-SCNN [22], which have proven successful in mobile environments.
In addition, vision transformer-based models, such as MobileFormer [3],
MobileViT [17], Seaformer [32], or TopFormer [38], have been explored
increasingly. From this spectrum, our model selection aims to cover different
network types (CNN, vision transformer) and application domains (general-
purpose, (bio)medicine), while setting reasonable limits on training effort.
Therefore, we limit the inclusion of architectures to four, prioritizing the
assessment of different-sized variants of selected methods where feasible.
Specifically, we selected TopFormer [38], a representative (hybrid) vision
transformer among recent general-purpose state-of-the-art (SOTA) methods, and
UNeXt [29], a compact CNN-based SOTA technique for medical segmentation. The
selection of these two SOTA models was based on the following criteria:  (i)
competitive performance on respective benchmark datasets  (ii) publication in
prestigious journals or conferences  (iii) efficient resource utilization, as
indicated by reported parameter counts and GFLOPs  (iv) code availability on
GitHub  . Additionally, ENet [21], a widely referenced general-purpose CNN for
real-time semantic segmentation from 2016, was included due to its remarkable
efficiency in terms of trainable parameters and promising outcomes in medical
segmentation studies [37, 28], given its size. Lastly, to establish a baseline
for wound segmentation, we integrated UNet [26], a prominent model in
biomedical segmentation. Table 1 outlines the selection, with parameters
reported in million (M) for binary segmentation (one out-channel).

Table 1: Overview of the included segmentation techniques.

UNet [26] ENet [21] UNeXt [29] TopFormer [38] Base Small Base Small Tiny
Params 31.03M 0.35M 1.47M 0.25M 5.03M 3.01M 1.39M Type CNN CNN CNN Hybrid CNN-
ViT Domain Biomedicine General purpose Medicine General purpose

##  3 Experimental Setup

###  3.1 Dataset

We perform experiments on a combination of two prominent datasets: the Foot
Ulcer Segmentation Challenge 2021 (FuSeg) [34] and the Diabetic Foot Ulcer
Challenge 2022 (DFUC 2022) [13, 36] datasets. FuSeg encompasses 1,210 foot
ulcer images, each with a resolution of 512 × 512 pixels, sourced from 889
patients’ medical visits spanning from October 2019 to April 2021. Meanwhile,
the DFUC 2022 dataset comprises 4,000 foot ulcer images with a resolution of
640 × 480 pixels, captured approximately 30 to 40 cm away from the ulcer. Our
custom dataset, Combined Foot Ulcers (CFU), integrates FuSeg and DFUC 2022,
comprising images with mixed resolutions of either 512 × 512 or 640 × 480
pixels. Due to the privacy of test set annotations (200 out of 1,210 for FuSeg
and 2,000 out of 4,000 for DFUC 2022), only publicly accessible training and
validation images are utilized for CFU. To prevent data leakage, duplicate
images within FuSeg and DFUC 2022 were identified and removed. A similarity
analysis, employing the perceptual hashing algorithm [18], was conducted.
Image sets with identical raw bytes or perceptual hash values exhibiting a
Hamming distance of 11 or less were deemed duplicates. For illustrative
purposes, a number of examples are provided in Figure 2. Eliminating duplicate
image-annotation pairs, we obtained a final dataset with 2,887 samples after
discarding 123 pairs. Next, these samples were randomly partitioned into
training (60%), validation (20%), and test (20%) set.

###  3.2 Evaluation Protocol

To assess model performance, we employed established segmentation metrics,
such as Dice score (DSC), Intersection over Union (IoU), precision (Prc), and
recall (Rec), utilizing the micro-averaging method while excluding the
background class. Our analysis covered two training modalities: training
models from scratch and utilizing pre-training on either the ImageNet [6] or
Cityscapes [5] dataset. To facilitate this, we downloaded pre-trained weights
where available: ENet (Cityscapes), UNet (ImageNet), and TopFormer (ImageNet).
Since pre-trained weights were unavailable for UNeXt, we trained both variants
ourselves for 100 epochs on Cityscapes, using the best checkpoints as initial
weights.

###  3.3 Implementation Details

We implemented this approach using Python 3.10.9 and PyTorch 2.0.1 on a server
with two partitioned Nvidia A100 GPUs, reserving a large-sized instance with
40 GB VRAM for our experiments. All models were trained end-to-end using the
AdamW optimizer with an initial learning rate of 0.0001, a batch size of four,
and the binary cross entropy (BCE) loss. We used a ReduceLROnPlateau scheduler
conditioned on the Dice score, with a minimum learning rate of 0.000001. All
models were trained for 200 epochs from scratch and for at least 130 epochs in
the pre-training setting. For both, the best checkpoints, according to the
mean IoU on the validation set, were used for the ensuing test set evaluation.
All images were resized to 512 × 512 pixels and normalized during training. We
applied thresholding to annotated masks to eliminate out-of-class pixels and
used the following data augmentations for each training set sample: Gaussian
blur with a kernel size of 25 and a randomly sampled standard deviation from a
uniform distribution of [0.001, 2.0], random affine transforms (incl.
translations by up to 12.5%, rotations by up to 180 degrees, scaling between
50% and 150% of the original size, and shear of up to 22.5 degrees), color
jitter, as well as random horizontal and vertical flips, each with a 50%
probability. Details about the augmentations can be found in the example
configuration file of Listing 0.A.

##  4 Evaluation

###  4.1 Results

We compare the selected models in Table 2, where we report the micro-averaged
metric scores of each variant for both training modalities. Our findings
reveal that, without pretraining, UNeXt-B demonstrates the most favorable
segmentation performance in terms of DSC and IoU scores, effectively balancing
parameter count and segmentation quality. Notably, all lightweight models,
except UNeXt-S, achieve results within a similar size range as the larger-
sized UNet, with the base variants of UNeXt and TopFormer, and ENet even
slightly surpassing UNet. Pretraining considerably enhances performance across
all models regarding IoU and DSC metrics. Specifically, all lightweight
networks show comparable performance, with IoU scores ranging from 69% to 72%
and DSC values from 82% to 84%, except for UNeXt-S, which performs worse.

Table 2: Comparison of all models on the CFU dataset. The micro-averaged
metrics (in %) were calculated on the test set. Parameters are reported in
millions.

Without pretraining With pretraining Model IoU DSC Prc Rec IoU DSC Prc Rec
Params$\downarrow$ U-Net 65.69 79.30 81.57 77.14 74.69 85.56 87.74 83.39
31.03M TopFormer-B 66.10 79.59 86.63 73.61 72.41 84.00 84.56 83.44 5.03M
TopFormer-S 63.36 77.57 83.31 72.57 71.56 83.42 85.95 81.04 3.01M UNeXt-B
68.96 81.63 88.49 75.76 70.12 82.43 87.40 78.01 1.47M TopFormer-T 62.51 76.93
84.27 70.77 69.62 82.09 83.90 80.36 1.39M ENet 67.27 80.43 89.21 72.23 71.71
83.52 89.98 77.93 0.35M UNeXt-S 59.16 74.34 77.52 71.41 64.13 78.14 87.52
70.58 0.25M

###  4.2 Real-World Deployment

To assess the models’ real-world capabilities, we developed a Flutter
prototype. For seamless integration within the app, all PyTorch models except
UNet were converted to TorchScript by tracing. The deployed networks performed
live segmentation of camera streams, with center-cropped images to 224 × 224
pixels for efficient processing. Following preliminary trials, a prediction
threshold of 0.75 was established for the derivation of binary masks, with the
objective of achieving more precise segmentations and reduced jitter during
inference. We rigorously tested the architectures on an OnePlus 7 Pro Android
phone from 2019. All models ran smoothly without stalling, demonstrating the
feasibility of mobile segmentation using Flutter. To enable visual
assessments, we devised six distinct scenes and captured live segmentations
via screenshots, utilizing the variants with pretraining. The scenes
encompass:  (i) a neutral environment devoid of objects  (ii) two diverse
wound images not from the CFU dataset  (iii) two scenes featuring everyday
objects with colors resembling wounds  (iv) a scene with two everyday objects,
one of which has a wound-like color  . The resulting segmentation masks
generated by the models are depicted in Figure 1. Visually, ENet exhibits the
worst results among the tested models, with pixelated and holey masks as well
as inaccuracies such as falsely identifying everyday objects, including the
blue sponge. In contrast, the TopFormer and UNeXt variants show more promising
results, identifying wounds with fewer imperfections, albeit with somewhat
indistinct edges and contours. Notwithstanding the recognition difficulties
encountered by the UNeXt models when presented with objects of wound-like
colours, there is an overall improvement for both architectures compared to
ENet. Overall, the TopFormer-based networks appear to be superior, as they
proved quite effective at distinguishing between wounds and wound-coloured
objects. Notably, the smallest variant (TF-T) performs exceptionally well in
our example scenes, while the CNN-based UNeXt-B, despite exhibiting promising
performance in Table 2, displays significantly poorer performance.

Original

TopFormer-B

TopFormer-S

UNeXt-B

TopFormer-T

ENet

UNeXt-S

Figure 1: Live segmentation of different model variants in our mobile app,
arranged in descending order of parameter count.

##  5 Conclusion and Future Work

In conclusion, our preliminary investigations into lightweight neural networks
demonstrate their potential for mobile wound segmentation. The analyzed
architectures, especially TopFormer, yield promising results in real-world
application scenarios, although some refinements are still necessary, such as
clearer edge detection. Future work will include extending the comparison to
further promising architectures such as MBSNet [12], MobileFormer [3], or
Seaformer [32]. Furthermore, retraining the networks on an expanded dataset
with more images, including those featuring objects with wound-similar colors,
is expected to enhance the robustness and generalization ability of the
models, thereby improving the accessibility and effectiveness of future wound
care.

####  5.0.1 Acknowledgements

We thank Adrian Strutt for conducting preliminary experiments.

####  5.0.2

The authors have no competing interests to declare.

##  Appendix 0.A Appendix

(a) Examples from FuSeg

(b) 0111.png

(c) 0386.png

(d) Examples from DFUC

(e) 100475.jpg

(f) 100511.jpg

(g) 0161.png

(h) 0163.png

(i) 101502.jpg

(j) 101941.jpg

(k) 0318.png

(l) 0415.png

(m) 100480.jpg

(n) 100516.jpg

Figure 2: Examples of identical or highly similar image pairs from both
datasets.

⬇

architecture:

arch_type: "ENetV2"

model_name: "ENetV2-CFU-512-pretrained"

in_size: 512

use_pretrained: True

settings:

max_epochs: 200

batch_size: 4

random_seed: 42

full_determinism: True

mixed_precision: True

torch_compile: False

num_workers: 4

pin_memory: True

checkpoints:

save_mode: "best"

metric: "iou"

min_epoch: 30

save_sample_segmentations: False

transforms_backend: "torchvision"

optimizer:

type: "AdamW"

lr: 0.0001

loss:

type: "default"

scheduler:

type: "reduce_lr_on_plateau"

metric: "dice"

mode: "max"

min_lr: 0.000001

factor: 0.5

verbose: True

early_stop: False

metrics:

tracked:

- "dice"

- "iou"

- "precision"

- "recall"

averaging_method: "micro"

ignore_background_class: True

dataset:

type: "CFU"

transforms:

train:

- to_tensor: True

- pad_and_resize:

size: 512

- gaussian_blur:

kernel_size: 25

sigma: (0.001, 2.0)

- threshold:

threshold: 127

pixel_min: 0

pixel_max: 255

apply_to_mask: True

apply_to_image: False

- color_jitter:

brightness: 0.4

contrast: 0.5

saturation: 0.25

hue: 0.01

- normalize:

mean: [ 0.5, 0.5, 0.5 ]

std: [ 0.5, 0.5, 0.5 ]

- random_horizontal_flip:

p: 0.5

- random_vertical_flip:

p: 0.5

- random_affine:

degrees: (-180, 180)

translate: (0.125, 0.125)

scale: (0.5, 1.5)

shear: 22.5

fill: -1.0

mask_fill: 0

val:

- to_tensor: True

- pad_and_resize:

size: 512

- threshold:

threshold: 127

pixel_min: 0

pixel_max: 255

apply_to_mask: True

apply_to_image: False

- normalize:

mean: [ 0.5, 0.5, 0.5 ]

std: [ 0.5, 0.5, 0.5 ]

test:

- to_tensor: True

- pad_and_resize:

size: 512

- threshold:

threshold: 127

pixel_min: 0

pixel_max: 255

apply_to_mask: True

apply_to_image: False

- normalize:

mean: [ 0.5, 0.5, 0.5 ]

std: [ 0.5, 0.5, 0.5 ]

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