# Star-forming environments in smoothed particle magnetohydrodynamics
simulations II: Re-simulating isolated clumps to determine equivalence of
extracted clumps and parent simulations

James Wurster1 and Connar Rowan2,1
1Scottish Universities Physics Alliance (SUPA), School of Physics and
Astronomy, University of St. Andrews, North Haugh, St Andrews, Fife KY16 9SS,
UK
2Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks
Road, Oxford, OX1 3PU, UK
<EMAIL_ADDRESS>
(Submitted: Revised: Accepted: )

###### Abstract

What is the numerical reproducibility of a stellar system (including its
discs) when evolving only a sub-set of (partially-evolved) smoothed particle
hydrodynamics (SPH) particles? To investigate this, we modelled the evolution
of 29 star forming clumps that were extracted from our previous simulations
that investigated the formation and early evolution of low-mass star clusters.
These clumps were evolved using a three-dimensional smoothed particle
radiation magnetohydrodynamics code, where we included or excluded non-ideal
magnetohydrodynamics to match the cluster simulation. While star formation
proceeded as expected, we were unable to identically reproduce any of the
systems present at the end of the cluster simulations. However, the final
distributions of stellar mass, stellar system mass, disc mass, and disc radii
were reproduced statistically; unfortunately, the distribution of average
magnetic field strengths in the discs was not reproduced statistically, but
this may be a result of our updated algorithms governing the evolution of the
magnetic field. Therefore, given that our clumps yield stellar masses that are
statistically similar to those in the original low-mass star clusters, we have
demonstrated that we can statistically reproduce systems (aside from their
magnetic field strength) by evolving a subset of SPH particles. Therefore,
clumps such as these can be used as initial conditions to investigate the
formation of isolated stars from less-contrived initial environments.

###### keywords:

protoplanetary discs – stars: formation – turbulence – magnetic fields – MHD –
methods: numerical

††pagerange: Star-forming environments in smoothed particle
magnetohydrodynamics simulations II: Re-simulating isolated clumps to
determine equivalence of extracted clumps and parent simulations–A††pubyear:
2023

##  1 Introduction

High-resolution numerical simulations of isolated star formation are very
useful to investigate the star formation process and the associated physical
processes. These simulations are often initialised using idealised initial
conditions to minimise degeneracies when adding new physics. Although
modelling star formation in these pristine initial environments permits a
detailed investigation into the new physical process, parameters, or initial
conditions, it carries the risk of incorrect conclusions resulting from the
pristine environment itself. For example, the protostellar disc formation
community grappled with the magnetic braking catastrophe (e.g., Allen et al.,
2003; Mellon & Li, 2008) for over a decade until Tsukamoto et al. (2015b) and
Wurster et al. (2016); Wurster et al. (2022) robustly showed that the Hall
effect was required to prevent the catastrophe. However, when modelling low-
mass star clusters, where the star-forming clumps were influenced by their
larger-scale environment, it was discovered that the magnetic braking
catastrophe never even existed (e.g., Seifried et al., 2013; Wurster et al.,
2019)! This demonstrated the importance of using a realistic initial
environment.

Given the high resolution permitted when modelling isolated star formation,
these studies must continue. However, great care must be taken when setting
the initial environment. There have been many studies that investigated
various initial (free) parameters, including the amount and type of kinetic
energy, namely turbulence (e.g., Seifried et al., 2012, 2013; Joos et al.,
2013; Tsukamoto & Machida, 2013; Matsumoto et al., 2017; Wurster & Lewis,
2020a, b) and rotation (e.g., Wurster & Lewis, 2020a, b), the internal energy
(e.g., Tsukamoto et al., 2018), the magnetic field strength (e.g., Bate et
al., 2014; Wurster et al., 2016; Machida et al., 2018), the magnetic field
geometry, including fields misaligned with the rotation axis (e.g., Machida et
al., 2006; Joos et al., 2012; Lewis et al., 2015; Tsukamoto et al., 2017,
2018; Hirano & Machida, 2019; Hirano et al., 2020) and turbulent magnetic
fields (e.g., Gerrard et al., 2019), the comic ray ionisation rate (e.g.,
Wurster et al., 2018a, b; Kobayashi et al., 2023), and the inclusion/exclusion
of the various non-ideal magnetohydrodynamics processes (e.g., Tomida et al.,
2015; Tsukamoto et al., 2015a, b; Wurster et al., 2016, 2021). By varying the
parameter under investigation, these studies produced a variety of systems
with a variety of environmental characteristics. While these studies provided
great insights into the effect of these parameters, for each process, what is
the optimal and most realistic parameter? How should these parameters be
selected to produce a realistic star forming core?

To mitigate these questions, higher-mass simulations can be performed (e.g.,
Bate, 2012; Seifried et al., 2013; Wurster et al., 2019), where stars and
discs form and evolve in a clustered environment. In addition to providing
more realistic details about star and disc formation and evolution given their
less-contrived environment111i.e., while the large-scale simulation itself is
initialised by prescribed (and possibly idealised) initial conditions, star
formation itself starts from already evolved gas, hence our label of ‘less-
contrived.’, this may provide details on the star formation mechanism itself,
namely whether it occurs via core collapse (McKee & Tan, 2002, 2003),
competitive accretion (Zinnecker, 1982; Bonnell et al., 2001a, b), or
inertial-inflow (Padoan et al., 2020). The latter two models suggest that gas
from well-beyond the initial star-forming core ultimately joins and influences
the system; this indicates that modelling star formation in isolated cores
represents an incomplete environment, even if selecting the most reasonable
parameters. Although this provides evidence for the requirement of modelling
higher-mass systems, modelling higher-mass systems as in the above references
necessarily means doing so at lower numerical resolution. While the evolution
of stars and their discs will be more realistic and gas will be permitted to
flow in from great distances, many features will be under-resolved or not
resolved at all (Wurster et al., 2021). Nonetheless, this is a necessary
compromise.

As an alternative compromise to increasing mass and lowering resolution, it
should be possible to extract regions from higher-mass simulations to use as
initial conditions. Although this requires the existence of these higher-mass
simulations, a plethora have been run by various groups for various studies;
therefore, is it a ‘simple’ matter to mine these simulations for additional
data. As existing proofs-of-concept, several studies of various scales have
already investigated where star-forming gas originated (e.g., Lewis et al.,
2015; Lewis & Bate, 2017; Pelkonen et al., 2021; Arroyo-Chávez & Vázquez-
Semadeni, 2022; Collins et al., 2023). Wurster & Rowan (2023) likewise
determined where star-forming gas originated within the star cluster
simulations of Wurster et al. (2019) and then extracted that gas to construct
star-forming clumps. From an analysis of those initial star-forming clumps, we
concluded that stars were born from a wide variety of environments (clumps),
where there was a distribution of parameters (e.g., density, Mach number,
magnetic field strength and geometry, etc., …) both within and amongst the
clumps. We concluded that there was not a single universal star forming clump.
Furthermore, we developed those initial clumps such that they could be evolved
in future simulations.

In this paper, we evolve 29 star-forming clumps that we extracted from the
cluster simulations of Wurster et al. (2019, herein WBP2019) and analysed in
Wurster & Rowan (2023, herein Paper I). The goal of this study is to determine
how closely the evolution of the extracted clumps matches the evolution of the
same gas within the cluster simulations themselves. If the original systems
are reproduced, either exactly or statistically, then we submit that the
initial clumps as extracted in Paper I provide excellent initial conditions
for studies of isolated star formation, since the initial environment is more
realistic and we have greatly reduced the number of initial (and free)
parameters. A natural follow-up to this current study will be to re-simulate
these clumps at the high-resolution of typical isolated star formation
simulations. This will yield the best of both worlds by permitting star
formation to be numerically investigated in high-resolution simulations where
the initial conditions are realistic rather than pristine, idealised, and
contrived.

The rest of the paper is organised as follows. In Section 2, we describe our
methods and how they contrast to those in WBP2019. In Section 3 we summarise
our initial conditions as generated in Paper I. We present our results in
Section 4 and provide further discussion in Section 5. We conclude in Section
6.

##  2 Methods

We solve the equations of self-gravitating, radiation non-ideal
magnetohydrodynamics (MHD) using the three-dimensional smoothed particle
hydrodynamics (SPH) code sphNG. This code originated from Benz (1990), but is
under continual development; it has since been substantially modified to
include a consistent treatment of variable smoothing lengths (Price &
Monaghan, 2007), individual time-stepping and sink particles (Bate et al.,
1995), magnetic fields (for a review, see Price, 2012), and non-ideal MHD
(Wurster et al., 2014, 2016; Wurster, 2016, 2021) using the single-fluid
approximation. For stability of the magnetic field, we use the Børve et al.
(2001) source-term approach, and maintain a divergence-free magnetic field
using constrained hyperbolic/parabolic divergence cleaning (Tricco & Price,
2012; Tricco et al., 2016).

sphNG has continued to be updated since 2017, when we began the simulations
published in WBP2019. We intentionally chose to use the current222Current as
of the beginning of this project in 2022. version of sphNG for this study
since it includes the most up-to-date algorithms. We acknowledge that this
means that any differences when comparing the current results to those in
WBP2019 will be a combined result of the extracted cores being isolated from
the remainder of the cloud and the different algorithms. We considered using
the same version of sphNG, however, science should always move forwards, and
we want our results to be as current as possible, even with the reduced
ability to compare to the source data. The three main differences are as
follows.

In this study, we model radiation as flux-limited diffusion (Whitehouse et
al., 2005; Whitehouse & Bate, 2006), whereas WBP2019 used the combined
radiative transfer and diffuse interstellar medium model of Bate & Keto
(2015). The latter method uses flux-limited diffusion for the optically thick
dense regions and a thermodynamics model for the optically thin, low density
material. While this worked well in WBP2019 which includes a component of the
interstellar medium (ISM), when we tested it on small clumps in the current
study, the thermodynamics model substantially and artificially heated the
boundaries of the clump, causing it to dissipate. Since the clumps (and cores)
in the original model would be shielded from the ISM, we are justified in
using only the flux-limited diffusion model. We note that the Bate & Keto
(2015) model may work for our larger clumps, but we intentionally chose to use
the same radiation algorithm for the entire suite.

In this study, we use the artificial resistivity algorithm outlined in Price
et al. (2018) rather than the algorithm presented in Tricco & Price (2013).
The current algorithm is less resistive (Wurster et al., 2017), thus the
resulting discs are less influenced by numerical processes. In ideal MHD
simulations, we thus expect to form smaller discs in the current study due to
less artificial resistivity.

In this study, we use v2.1 of Nicil (Wurster, 2021) rather than v1.2.1
(Wurster, 2016) to calculate the non-ideal coefficients. The new version uses
a reduced chemical network, whereas the previous version used two proxy
elements. In general, resistivity is slightly higher in v2.1 than v1.2.1,
suggesting slightly larger discs should form in the non-ideal clumps in the
current study.

##  3 Initial conditions

The method to generate our initial conditions is described in detail in Paper
I. Briefly, at the end each simulation in WBP2019 (i.e., at
$t_{\text{final}}=276$ kyr), we identify the existing sink particles. We then
identify all the particles associated with each sink, where we define an
associated particle to be either bound to the sink or within a critical radius
of $r_{\text{min}}=50$ au. Moving backwards in time, we repeat this process
and continually associate new particles with the sink as necessary. Once a
dump file is reached where the star does not exist (i.e., the dump file before
it formed), then we create a ‘pseudo-sink’; we add all the particles in the
pseudo-sink to the list, and add close and bound particles one final time. We
then track these particles back in time until the maximum density decreases to
$\rho_{\text{max}}=10^{-16}$ g cm-3; at this time, we augment these particles
with all the particles within 4$h$ of any associated particle, where $h$ is
the smoothing length of a particle. This is defined as our clump, and the time
this occurs is defined as the extraction time, $t_{\text{extract}}$. The
centre of mass is shifted to the origin and the velocity is shifted to remove
any bulk velocity.

We then embed the clump into a warm background medium whose density is 30
times lower than the average density at the boundary of the clump. The thermal
energy of the background is set so that, on average, the background and clump
are in pressure equilibrium. Finally, the magnetic field of the background is
uniform and is equal to the average magnetic field of the boundary; this
action is performed component-wise. The box itself is periodic, and is
approximately twice the size of the clump in each direction.

As in WBP2019, we include sink particles with accretion radius
$r_{\text{acc}}=0.5$ au. When a gas particle reaches $\rho=10^{-5}$ g cm-3
near the end of the second collapse phase, it is replaced with a sink
particle. All particles within 0.25 au of the centre of the sink are
automatically accreted, and all particles within 0.5 au are tested to
determine if they meet the criteria to be accreted. Since we resolve the
opacity limit, each sink particle represents one star or brown dwarf and the
properties of the sink reflect the properties of the star.

For this study, we maintain the numerical resolution of WBP2019, which is
$10^{-5}$ M⊙ for each equal-mass SPH particle.

Each clump is evolved for a length of time equal to
$t_{\text{final}}-t_{\text{extract}}$, therefore the total runtime of each
simulation is different. Due to limited computational resources, for this
study, we chose to investigate the clumps extracted from models N05 and I05 in
WBP2019, which were initialised with a normalised mass-to-flux ratio of
$\mu_{0}=5$; this value matches the mass-to-flux ratio used in many of our
previous studies of isolated star formation (e.g., Wurster et al., 2016,
2018a, 2018c; Wurster et al., 2022). Therefore, we evolve the 17 clumps
extracted from N05 and the 12 clumps extracted from I05, which, as with the
parent simulation, are evolved employing non-ideal MHD and ideal MHD,
respectively. We explicitly note that there was a stellar merger in I05,
therefore, the clump construction process of 11 clumps began at
$t_{\text{final}}$, while the construction process of the final star began at
$t<$ $t_{\text{final}}$.

For reference, Fig. 1 shows gas cloud in model N05 from WBP2019 at two
different scales at $t_{\text{final}}$.

Figure 1: Average gas density of cloud N05 from WBP2019 at two different
scales at the final time of $t_{\text{final}}=276$ kyr. The black dots
represent sink particles, and have a radius of 350 and 100 times the accretion
radius of the sink for the top and bottom panel, respectively. The cloud
extends to $r\sim 0.2$ pc (top), however, the dense structures and star
formation are confined to a region of $r\sim 0.07$ pc (bottom). No stellar
region exists in isolation, thus any region that has contracted has had its
gas replenished.

###  3.1 Definitions

For simplicity and clarity, we re-state the important definitions here that we
will use throughout the paper; these are the same as in Paper I.

  * •

Parent simulation or parent cloud: The simulations presented in WBP2019. Each
cloud initially contained 50 M⊙ of gas.

  * •

Progenitor sink particle or progenitor star: The sink particle in the parent
simulation that was used to construct the clump.

  * •

Progenitor system: The hierarchical system to which the progenitor star
belongs at $t_{\text{final}}$. This is to help guide the analysis and is not
used in the construction of the clumps.

  * •

Associated clump: The complete list of SPH particles that will ultimately
accrete onto the sink, come within the critical distance of the sink, or
become bound to the sink; associated clumps were created in Paper I, but are
only used in this section of this paper.

  * •

Extracted clump or clump: The final list of SPH particles that includes the
associated particles plus all their neighbours. This definition does not
include the background particles. This is the clump to which we are referring
throughout the rest of this paper. This is referred to as an augmented clump
in Paper I.

  * •

Extraction time: The time at which the maximum density in the associated clump
decreases to $\rho_{\text{max}}\leq 10^{-16}$ g cm-3, defining the initial
location of the associated clump.

  * •

N05: The name of the parent simulation in WBP2019 that was initialised with
$\mu_{0}=5$ and evolved using non-ideal MHD.

  * •

I05: The name of the parent simulation in WBP2019 that was initialised with
$\mu_{0}=5$ and evolved using ideal MHD.

###  3.2 Simulations names

The clumps are named as XcY, where X is the name of the parent simulation and
Y is a number based upon the order in when the clump was extracted, from
earliest to latest.

The progenitor systems at $t_{\text{final}}$ are identified as X [Z]. At
$t_{\text{final}}$, N05 has four isolated stars, one binary system, one triple
system, and two systems of four stars which are identified as $Z=$ (s1, s2,
s3, s4, b1, t1, q1, q2), respectively. At the same time, I05 has five isolated
stars and three binary systems which are identified as $Z=$ (s1, s2, s3, s4,
s5, b1, b2, and b3+), respectively; the latter includes a plus sign since a
star previously merged with a star in that system. Note that all these systems
are dynamically evolving; they form before $t_{\text{final}}$ but are not
guaranteed to survive beyond $t_{\text{final}}$.

##  4 Results

###  4.1 General evolution and large-scale structure

Fig. 2 shows the evolution of six clumps extracted from N05.

Figure 2: The evolution of six clumps extracted from N05, with the time of
extraction increasing from top to bottom. The times in each panel are zeroed
to the beginning of the evolution of the respective clump. The black dots
represent the sink particles and have radii 350 and 100 times that of the
accretion radius for the top two and bottom four rows, respectively. Note that
the rows use different spatial and density scales to better highlight the
features. Star formation proceeds rapidly (within $\sim$10-20 Myr) as the
entire clump gravitationally contracts.

Since most stars in WBP2019 are in hierarchal systems at $t_{\text{final}}$,
it is unsurprising that some initial clumps appear similar since gas can be
associated with several progenitor stars (first column; see also Fig. 14 and
fig. A1 in Paper I). As a demonstration, we have intentionally selected N05c2
and N05c3 where the progenitors are in a tight binary system (and part of the
widely separated quadruple system N05 [q1]). The top two rows show the global
evolution of N05c2 and N05c3, where the two evolutions appear qualitatively
similar, particularly the structures near the dense core. The most notable
difference is that N05c3 is smaller at all times, which can be accounted for
by its lower initial mass (1.61 compared to 1.81 M⊙). Moreover, 1.60 M⊙ of gas
is common to both clumps, therefore, N05c3 is effectively a subset of N05c2.

When comparing N05c10 and N05c11 where the progenitors were also in a binary
system (N05 [b1], which is less tight than above), the two clumps demonstrate
nearly identical large-scale evolution (the latter is not shown). In this
case, the clumps were extracted at the same time with a very large overlap in
SPH particles (0.75 M⊙ of gas is common, where the clump masses are 0.83 and
0.78 M⊙ for N05c10 and N05c11, respectively).

Therefore, since nearly all the hierarchal systems in WBP2019 were formed by
capture, the difference in evolution of clumps with progenitors from the same
system depends on the initial separation of the progenitor stars, and how much
different gas they interacted with. The tables in Appendix A show the
percentage of common gas amongst clumps whose progenitors are in a bound
system at $t_{\text{final}}$. It is also more likely that younger clumps
spawned from the same progenitor system will have similar evolutions compared
to older clumps since there is less time for their histories to diverge.

Despite the initial turbulence, the evolution here proceeds similarly to that
of simulations of isolated star formation. Here, the maximum gas density
continually increases throughout the first hydrostatic core phase until the
first star is formed within $\sim$10-20 kyr. During this time, the clump is
globally gravitationally collapsing and its volume is decreasing. Although
this is in agreement with simulations of isolated star formation, it disagrees
with the results of WBP2019 where star forming clumps or cores never detach
from their surroundings (c.f. Fig. 1). This disagreement suggests that our
results will be minimally impacted by the new numerical algorithms discussed
in Section 2.

For a direct comparison between our extracted clumps and the region around the
progenitor star for the six clumps, see Fig. 3.

Figure 3: The average gas density in the $xz$-plane of six clumps at the final
time (top; centred on the first star to form) compared to the region from the
parent simulation centred on the progenitor star (bottom). The black dots
represent the sink particles, and have a radius 100 times that of the
accretion radius. The large-scale filamentary structure present in the parent
simulations is not reproduced by the evolving the extracted clumps.

At this final time, the parent clouds (bottom) are filamentary, with most of
the stars located within the filaments or at intersections of filaments; the
progenitor to N05c15 is being fed by at least two different filaments.
However, the surrounding gas in the extracted clumps (top) does not contain
the same well-defined filamentary structure. There are a few potential
striations feeding N05c2 and N05c3, and slightly better defined striations
feeding N05c15; the latter are over-densities that are rotating about the star
and only slowly feeding the star rather than well-defined filaments. In all
cases, the striations in the extracted clumps extend from the centre to the
edge of the clump, suggesting that they are shaped by larger-scale processes
than contained within our clumps. Therefore, our extraction method as
developed in Paper I does not reproduce the large-scale filamentary
structures, or even the localised filamentary structures around the final
star(s).

####  4.1.1 Stellar evolution as a proxy for clump evolution

To quantitatively evaluate the evolution of the clumps compared to their
progenitors, we will use stellar (sink) evolution as a useful proxy. Figs. 4
and 5 show the mass evolution of the individual stars in the clumps and the
progenitor systems for N05 and I05, respectively, where the horizontal axis
spans the evolved lifetime of the clump.

Figure 4: Mass evolution of the stars in all 17 clumps extracted from N05 and
from N05 itself. In each panel, we compare the evolution of the progenitor
star (red) and the remaining stars in the progenitor system (green) to the
first star to form in the clump (blue) along with the remainder of its stars
(grey). The time axis is zeroed to the beginning of the evolution of the
parent cluster N05, thus the lines representing the stars in the extracted
clumps are shifted by the extraction time of the clump. The time axes spans
from the extraction time of the clump to $t_{\text{final}}$, thus is different
in each panel; the vertical axis varies to best highlight the stellar mass
evolution. The primary star in the clumps typically follows the mass evolution
of one star in the progenitor system. The masses and mass evolution of
individual stars in the clumps appears to better agree with the progenitor
systems than when comparing the discs and disc systems. Figure 5: Mass
evolution of the stars in all 12 clumps extracted from I05 and from I05
itself, as in Fig. 4. The legend is the same as in Fig. 4 and given only in
that figure for clarity.

If the stellar evolution is reproduced exactly, then in each panel, the red
and blue lines will match; if the stellar system evolution is reproduced
exactly, then additionally the green and grey lines will also match. We
clearly see that we do not identically reproduce any system.

In most cases, all the stars in a progenitor system form after the extraction
time of a given clump. However, there are exceptions, such as the progenitors
to N05c9, N05c1, and N05c2. The first case represents a star that rapidly
proceeds from nearby diffuse gas to birth and only interacts with the low-
density gas of the circumtriple disc, while the latter two cases represent
stars that for much of their life are unbound thus do not interact with the
same gas.

In most cases in the clumps extracted from N05, the star in the extracted
clump forms before its progenitor and has more mass than its progenitor star;
there is no clear trend for the stars that form in the clumps extracted from
I05. In most cases in all models, the first star to form in the extracted
clump reasonably well follows the mass evolution of one star in the progenitor
system. In some cases (e.g., N05c1,2,8,10), this is the progenitor star
itself, suggesting that our extraction algorithm can reproduce that star.
However, in many cases (e.g. N05c3,6,15), the first star follows the evolution
of a different star in the progenitor system. In many of these latter cases,
the oldest star in the clump follows the evolution of an older star in the
progenitor system, suggesting that its core is at least partially included in
the clump and is already somewhat evolved at extraction time of the clump.

In most cases, the evolutionary tracks of the clump and cloud stars are
similar, suggesting a similar accretion mechanism in all simulations, at least
on the scales near the star itself. One notable star, however, is the
progenitor to N05c9. This star is ejected from its system in the parent
simulation, and its accretion is halted. In the clump simulation, however,
only a single star forms and it does not need to compete with additional stars
for material. These two evolutions demonstrate the effect of dynamical
interaction and the competition for material. It is likely that if the
progenitor never interacted with the three other stars in N05, it would also
have grown to a large mass.

In summary, the first star to form in the clump has a similar evolution to one
of the stars in the progenitor system. Therefore, our extraction algorithm
permits us to reasonably reproduce the accretion history of one star in the
progenitor system, even if it is not the progenitor star. However, we are less
able to reproduce the accretion history of an entire progenitor system.

###  4.2 Final states

In Figs. 2 and 3, we have shown how the large-scale gas evolution differs
between the clumps and the parent clouds, in part due to the non-existence and
existence of filaments, respectively. As previously stated, this is a result
of the extraction method in Paper I extracting clumps based upon the gas that
was associated with a progenitor star and disc and not the surrounding
filaments. Despite the lack of filaments, our algorithm design meant we were
able to reasonably reproduce the mass evolution of some stars (Figs. 4 and 5).
However, how well is the progenitor system (including discs) reproduced? Figs.
6 and 7 compare the stars and discs of all the clumps extracted from N05 and
I05, respectively, to their progenitors at the end of the simulation.

Figure 6: The average gas density in the $xz$-plane of all 17 clumps extracted
from N05 at the final time (top row of each panel; centred on the first star
to form) compared to the region from the parent simulation centred on the
progenitor star (bottom row of each panel). The black dots represent the sink
particles and have a radius 10 times that of the accretion radius. The systems
that form from the clumps reproduce their progenitor systems with varying
degrees of accuracy, with better accuracy for the progenitor low-order
systems.

Figure 7: The average gas density in the $xz$-plane of all 12 clumps extracted
from I05 at the final time (top row of each panel; centred on the first star
to form) compared to the region from the parent simulation centred on the
progenitor star (bottom row of each panel). The black dots represent the sink
particles and have a radius 10 times that of the accretion radius. The
progenitor to I05c9 merged with the progenitor to I05c8 prior to the final
time; we have evolve this clump to $t_{\text{final}}$ but have not included a
comparison image. The progenitors of I05c8 and I05c10 are present in the
middle of their respective panels, but cannot be seen since the gas density is
too low. The systems that form from the clumps reproduce their progenitor
systems with varying degrees of accuracy, with better accuracy for the
progenitors with well-defined discs.

Although some systems are in good qualitative agreement with their progenitor
systems, our clumps clearly do not form a one-to-one reproduction of their
progenitor.

For a quantitative comparison between the stars and discs from the extracted
clumps to the progenitor stars and discs, Fig. 8 compares the stellar system
mass, disc mass, disc-to-star mass ratio, the disc radius, and the magnetic
field strength of the discs.

Figure 8: From top to bottom: mass of the stellar system, mass of the disc,
ratio of the disc mass to the stellar mass, disc radius enclosing 63.2 per
cent of the mass, and magnetic field strength of the disc where the bars span
95 per cent of the field strengths in the disc. The circles represent
circumstellar discs, the x’s represent circumbinary discs, the triangles
represent circumsystem discs about three stars, and squares represent
circumsystem discs about four stars. Although multiple discs exist per system,
often hierarchically, we have selected the discs that best represent the
system (e.g., see Figs. 3 – 7). The symbols to the left of the vertical lines
represent stars and discs in the parent clouds while the symbols to the right
represent stars and discs in the extracted clumps. I05c6 and I05c11 have
progenitors that are only 426 au apart but not bound, however, we have
included both progenitors for the comparisons of I05c6 and I05c11. The
progenitor to I05c9 merged prior to the end of the simulation, hence the lack
of points on the left-hand side of the line. There is a varying degree of
agreement amongst the models, but this suggests that discs properties may be
reasonably reproduced even if their morphological features are not (e.g., see
Figs. 6 and 7).

In this figure, we plot the properties of the highest-order disc(s) rather
than the highest order system. For example, N05 [q1] is a quadruple system,
but clearly there is no circumsystem disc about all four stars; thus, we
instead plot the properties of the two circumstellar discs and one
circumbinary disc. On the left-hand side of each thin grey line, we plot the
properties of the stars and discs associated with the progenitor star (system)
from the parent simulation, and on the right-hand side, we plot the properties
of the stars and discs from the extracted clumps.

From Figs. 3 – 7, some clumps reasonably reproduce their progenitor systems,
although never exactly. There is typically better agreement when the
progenitor is in a lower-order system (i.e., isolated or a binary) extracted
at a later time (i.e., is a younger clump). Thus for young low-order systems,
it appears that the clump extraction method of Paper I is reasonable to
reproduce the system. However, given the preceding results and figures, in
general, it is clear that we cannot extract gas and have it robustly and
identically evolve as when it is embedded in the parent cloud. Given the chaos
and non-linearity of star formation, and the presence of large-scale
processes, this is not surprising.

In the remainder of this section, we discuss the stellar and disc properties
and distributions at the final time. We must be cautious about small-number
statistics since we are only evolving 17 (12) non-ideal (ideal) MHD clumps,
not all of which form discs. We also note that these are dynamically evolving
systems, so the time of analysis may also play a role in our conclusions. In
the following histograms, we do not normalised the distributions, and all
histograms have bin widths of 0.5 dex. We plot the parent distribution twice,
once where each object in the parent simulation is counted only once
(parent-U) and once where we repeatedly count stars and discs each time they
appear in a progenitor clump (parent-RC); i.e., every point in Fig. 8 is
included once in the parent-RC histogram. The mean and standard deviation of
the histograms for the non-ideal and ideal simulations are given in Tables 1
and 2, respectively. We also compare each pair of histograms within each
magnetic prescription using the two-sample Kolmogorov-Smirnov test (K-S test)
and present the $p$-value in these tables; values larger than 0.05 suggest
that the two distributions are compatible with being drawn from the same (but
unknown) distribution.

| N: pU | N: pRC | N: clump |  $\mu\pm\sigma$: pU |  $\mu\pm\sigma$: pRC |  $\mu\pm\sigma$: clump | K-S: pU & pRC | K-S: pU & clump | K-S: pRC & clump
---|---|---|---|---|---|---|---|---|---
Stellar mass | 17 | 49 | 31 |  0.16M⊙$\pm$0.37dex |  0.20M⊙$\pm$0.37dex |  0.19M⊙$\pm$0.30dex | 0.95 | 0.47 | 0.67
Stellar system mass | 11 | 27 | 21 |  0.18M⊙$\pm$0.50dex |  0.29M⊙$\pm$0.51dex |  0.23M⊙$\pm$0.43dex | 0.90 | 0.67 | 0.025
Disc mass | 10 | 23 | 20 |  0.018M⊙$\pm$0.51dex |  0.023M⊙$\pm$0.61dex |  0.025M⊙$\pm$1.0dex | 0.96 | 0.31 | 0.33
Disc radius | 10 | 23 | 20 |  37.9R⊙$\pm$0.28dex |  49.4R⊙$\pm$0.27dex |  44.8R⊙$\pm$0.30dex | 0.70 | 0.92 | 0.44
Disc magnetic field | 10 | 23 | 20 |  0.025G$\pm$0.47dex |  0.037G$\pm$0.47dex |  0.010G$\pm$0.32dex | 0.76 | 0.003 |  2.3$\times 10^{-6}$
Table 1: The properties of the distributions for the non-ideal models. In the header, pU (pRC) represents parent-U (parent-RC) for brevity. The average and standard deviation are calculated using the logarithm of the values for consistency with the histograms. The final three columns are the $p$-values from the two-sample K-S test, where values above 0.05 suggest that the two distributions are compatible with being drawn from the same distribution.  | N: pU | N: pRC | N: clump |  $\mu\pm\sigma$: pU |  $\mu\pm\sigma$: pRC |  $\mu\pm\sigma$: clump | K-S: pU & pRC | K-S: pU & clump | K-S: pRC & clump
---|---|---|---|---|---|---|---|---|---
Stellar mass | 11 | 22 | 23 |  0.20M⊙$\pm$0.43dex |  0.22M⊙$\pm$0.39dex |  0.15M⊙$\pm$0.35dex | 1 | 0.16 | 0.046
Stellar system mass | 10 | 15 | 15 |  0.19M⊙$\pm$0.47dex |  0.20M⊙$\pm$0.46dex |  0.22M⊙$\pm$0.42dex | 1 | 0.94 | 0.89
Disc mass | 8 | 12 | 14 |  0.0065M⊙$\pm$0.81dex |  0.0067M⊙$\pm$0.73dex |  0.0027M⊙$\pm$1.1dex | 1 | 0.44 | 0.31
Disc radius | 8 | 12 | 14 |  19.3R⊙$\pm$0.22dex |  18.9R⊙$\pm$0.19dex |  20.3R⊙$\pm$0.29dex | 1 | 0.91 | 0.53
Disc magnetic field | 8 | 12 | 14 |  0.099G$\pm$0.39dex |  0.092G$\pm$0.41dex |  0.022G$\pm$0.25dex | 1 | 0.001 | 0.0004
Table 2: The properties of the distributions for the ideal models.

Fig. 9 shows the distribution of stellar masses for both non-ideal and ideal
MHD simulations.

Figure 9: Histogram showing the distribution of stellar masses in the parent
(red and green) and clump (blue) distributions. For the red parent model
(parent-RC), we repeatedly count stars such that they are included once for
every progenitor system in which they appear, while the green distribution
(parent-U) includes each star only once. All distributions span a similar
range and are qualitatively similar, despite each distribution having a
difference number of members.

The parent-RC and clump distributions for the non-ideal MHD models each have a
different number of members, with 49 and 31 stars, respectively. Given how we
have double counted the parent population, this shows that we have
unintentionally suppressed star formation in our extracted clumps, therefore,
we have removed some of the material or large-scale processes that promote
star formation in the parent cloud (e.g., inertial inflows). However, this
conclusion is unique to the non-ideal MHD models since both the parent-RC and
clump distributions of the ideal MHD models have nearly the same number of
stars, with 22 and 23 stars, respectively. Given these two contradictory
results, we cannot offer a conclusion on whether or not star formation is
suppressed in the clumps compared to the parent clouds.

Independent of the number of members in the non-ideal MHD distributions, they
all span a similar range and are qualitatively similar, with the $p$-value
from the K-S test suggesting that they are compatible with being drawn from
the same distribution. The ideal MHD clump distribution is slightly narrower
than the parent distributions, and the K-S test suggests that the clump and
parent-RC distributions are not compatible with being drawn from the same
distribution, although the clump and parent-U distributions are. This weaker
and non-agreement for parent-U and parent-RC, respectively, is likely due to
the small number statistics and the clumps distribution containing one lower
mass star while parent-U (parent-RC) contains one (two) higher mass star(s);
this reinforces the caution of our quantitive results due to our small
numbers. This shows that our extraction process reasonably and statistically
reproduces the stellar distribution for non-ideal MHD, and, in the clumps, we
do not form stars that are either much more or much less massive than in the
parent cloud; the average stellar mass is lower in the ideal MHD clumps than
their parents, however, it is statistically reproduced when comparing with
parent-U. Therefore, our extraction process yields clumps that can be evolved
to statistically reproduce the parent-U distribution of stellar masses,
despite stars from each clump not exactly forming stars with the same masses
as in the progenitor system, as discussed with the evolutionary tracks shown
in Figs. 4 and 5.

When we instead look at the total stellar mass in a system (see Fig. 10), we
see that the non-ideal MHD distributions are qualitatively similar, except
that the parent-RC distribution also has several more massive systems.
However, this most massive bin contains the system N05 [q2] in quadruplicate,
thus skewing the conclusion, which is readily seen when comparing to the
parent-U distribution. The results of our K-S test show that the clump and
parent-U distributions are compatible with being drawn from the same
distribution whereas, due to the inclusion of the most massive system four
times in parent-RC, the clump and parent-RC distributions are not compatible
with being drawn from the same distribution. The top row of Fig. 8 reinforces
that the difference between the clump and parent distributions is from the
older, higher order systems (i.e., N05 [q1,q2]); however, the slightly younger
triple system, N05 [t1], has reasonable agreement with the stellar masses,
even if the clump and parent systems do not contain the same number of stars.
Thus, age of the system is likely the primary contribution to any disagreement
between the distributions, followed by multiplicity.

For the ideal MHD models, we again see reasonable agreement between the
distributions and the K-S test again concludes that the clump distribution is
compatible with being drawn from the same distributions as the parent-U and
parent-RC distributions. Additionally, Fig. 8 shows reasonable agreement for
most of the systems. Unlike the non-ideal MHD models, there is greater
disagreement from the ‘middle-aged’ systems I05 [7-11] rather than the older
systems; multiplicity is not high enough for a comment.

Therefore, from the distributions and a direct qualitative comparison, it is
reasonable to conclude that we can reproduce the stellar system mass from our
extraction algorithm in a statistical sense; deviations in direct comparisons
between the clumps and parent typically increases with age and multiplicity of
the system. Again, we offer caution due to small numbers and greater impact of
outliers.

Figure 10: Histogram showing the distribution of stellar system masses for the
parent (with repeated-counting in red; with unique counting in green) and
clump (blue) distributions. The parent clouds can form slightly more massive
systems, however, this is not a robust conclusion given the small number of
systems in each distribution and how system masses were double-counted in the
parent distribution.

The disc mass distributions are shown in Fig. 11. There is more deviation
between the parent and clump distributions, both in shape and spread, than
compared to the stellar and stellar system masses; for these distributions,
the standard deviation can be $\lesssim$ dex. Despite this, all four
$p$-values are greater than 0.05. There is better agreement when comparing the
distributions of their radii (Fig. 12); all four $p$-values for the disc radii
are greater than those for the disc masses.

Figure 11: Histogram showing the distribution of disc masses for the parent
(with repeated-counting in red; with unique counting in green) and clump
(blue) distributions. The clumps yield a wider distribution of disc masses,
and contain larger discs since clumps have fewer large-scale processes that
can truncate a disc. Figure 12: Histogram showing the distribution of disc
radii for the parent (with repeated-counting in red; with unique counting in
green) and clump (blue) distributions. Both parent and clump distributions are
similar, although the clumps contain larger discs since clumps have fewer
large-scale processes that can truncate a disc.

This is reinforced in Figs. 6 and 7 which renders the final discs in the
evolved clumps to their progenitors. These K-S values are reflected in the
comparison of the disc masses and radii shown in the second and fourth rows of
Fig. 8, respectively. When comparing the resulting disc mass and radius to
their progenitors, there is generally good agreement between the radii for
both the ideal and non-ideal MHD models, whereas there can be a factor of
$\sim$10 between the disc masses (although this can be a few dex in the
extreme cases).

Although our extraction process includes all the gas that becomes bound to the
parent star (which also includes all the gas in the progenitor disc), it does
not include the gas that may interact but never becomes bound. Therefore, both
the clump and the parent cloud contain different SPH particles that will
interact differently and hence have different effects on the disc morphology.
In addition to different external interactions, our distributions are
dependent on the particular snapshot in time since dynamical interactions are
continually stripping and feeding the discs. Therefore, the distribution of
discs properties is a less reliable comparison than stellar properties.

Despite the qualitative differences, for both the disc mass and radius
distributions, the K-S test suggests that the clump distribution is compatible
with being drawn from the same distributions as the parent-RC and parent-U
distributions.

Finally, Fig. 13 shows the distributions of average magnetic field strengths
in the discs at the final time. For both ideal and non-ideal MHD, there is a
systematic shift that the discs in the clumps have a lower magnetic field
strength; the average field strength is $\sim$3 times lower for the non-ideal
MHD models and $\sim$4 times lower for ideal MHD. All four $p$-values are less
than 0.05, indicating that the clump distributions are not compatible with
being drawn from the same distributions as either parent distributions. This
shift is reasonable for the non-ideal MHD models since the clumps were evolved
with a newer, slightly more resistive version of Nicil. However, all clumps
were evolved with the new artificial resistivity algorithm that is less
resistive suggesting that the discs in the clumps (at least for ideal MHD)
should be slightly stronger. Therefore, it is possible that the clump or cloud
size has a direct effect on the evolution of the large-scale magnetic field
and thus ultimately how that field affects the field in the discs; exploring
this further is out of the scope of the current work. Given our previous
comments about chaotic evolution of the discs, and the different numerical
algorithms governing artificial and physical resistivity, we cannot reach any
robust conclusions regarding the magnetic field in extracted versus progenitor
discs other than to say that the clump distribution is not representative of
the parent distributions.

Figure 13: Histogram showing the distribution of disc magnetic field strengths
for the parent (with repeated-counting in red; with unique counting in green)
and clump (blue) distributions. The distributions are systematically lower in
the disc clumps than the parent clumps and are not drawn from the same
distributions as per the results of our K-S test. We cannot determine the
cause given that disc evolution is chaotic and that the clumps and parent
clouds were evolved with different physical and artificial resistivity
algorithms.

In most non-ideal MHD cases, the average field strength is only slightly lower
in the clump than its progenitor, with the exception of N05 [q2]. However, in
all non-ideal MHD cases, the ranges of magnetic field strengths in the discs
have a large overlap between the clump and progenitor discs; see the bottom
row of Fig. 8, where the vertical bars span 95 per cent of the field strengths
in the discs. The magnetic field strength in most discs spans $\sim$dex,
although there is an even larger spread in N05 [q1], which promotes the
overlap. Despite the overlap, our previous comments about the effect of the
improved physical and artificial resistivity comments remain true. There is
more variation amongst the average and range of magnetic field strengths in
the ideal MHD discs, both within the progenitor and clump disc distributions
and between the two distributions. This reinforces that non-ideal MHD
processes help to regulate the evolution of the magnetic field.

Given the influence that the magnetic field strength has on the disc
properties in simulations of isolated star formation (e.g., Wurster et al.,
2016; Lewis & Bate, 2018; Wurster & Bate, 2019), it is surprising that there
is statistical agreement amongst disc radii and masses while the magnetic
fields are much weaker in the discs; however, the importance of the magnetic
field on disc properties is reduced in cluster simulations (e.g., Seifried et
al., 2013; WBP2019), suggesting that the evolution of the clumps is more
analogous to that of star formation in clusters rather than in isolation. For
a better understanding of the effect of the magnetic field, future studies
should test the evolution of the clumps using the older versions of the
physical and artificial resistivity algorithms; for further discussion on the
effect of physical and artificial resistivity, see Wurster (2016, 2021) and
Wurster et al. (2017); Wurster et al. (2018d), respectively. Therefore, this
suggests that our decision to use the most up-to-date algorithms as discussed
in Section 2 may have been incorrect.

##  5 Discussion

As discussed in Section 4 and shown in Fig. 2, well-defined filaments do not
form in our clumps. The gas is gravitationally collapsing on to the dense
core, although in some cases, this flow is redirected by a striation rather
than retaining a somewhat radial collapse. In other cases, the gas retains or
gains angular momentum and spirals around the core before accreting onto it.
These gas flows are different than in the parent clouds.

Several studies of isolated low-mass star formation have included decaying
turbulence (e.g., Seifried et al., 2012, 2013; Joos et al., 2013; Lewis &
Bate, 2018; Wurster & Lewis, 2020a, b; Hennebelle, 2021). All of these results
include structured remnants, where the filamentary flows are better defined
for studies with higher rms Mach numbers and higher initial masses. All of our
initial cores are supersonic with $1.5\lesssim\mathcal{M}\lesssim 3$, however,
they are also low-mass with $M_{0}\lesssim 2$ M⊙. Despite the supersonic Mach
number and that the parent cloud formed well-defined filaments, our extracted
clumps behave like isolated star forming clumps. Therefore, whether modelling
isolated star formation in idealised initial cores or in clumps extracted from
larger-scale simulations, processes on even larger scales are required to
generate and sustain filaments. Future studies are required to investigate the
gas flow in both the parent simulation and the extracted clumps to determine
the impact and importance of the filaments, and to determine why stars with
similar mass distributions can be formed with and without the presence of
filaments.

From our analysis of the extracted clumps, it is clear that, even when
accounting for all the gas that is ever associated with a progenitor star, the
original system cannot be identically reproduced. This includes the
multiplicity, the stellar masses, and the discs. Therefore, large-scale
processes are irrevocably lost during the extraction procedure. Additionally,
even when evolving two similar clumps (e.g., N05c2 and N05c3, where the former
includes nearly all the gas in the latter plus an additional $\sim$12 per
cent), we see different evolutions and final states. It has been previously
and repeatedly shown that different initial realisations of turbulence produce
different results (e.g., Liptai et al., 2017; Geen et al., 2018), and we can
extend this conclusion by stating the even slightly different subsets of gas
from a parent simulations will produce different results; however, in all
cases, the slightly different realisations or subsets yield results that are
statistically similar. This reinforces that star formation is a non-linear,
chaotic process and any sort of change (including removing so-called
‘unimportant’ gas from a simulation or slightly changing an initial turbulent
field) will prevent exact reproducibility. However, star formation may be
statistically robust to these small changes.

How gas flows differ between the clump and the parent cloud suggests that
different star formation mechanisms may be prevalent on different scales. On
large scales, gas is funnelled towards the stars through the filaments, thus
suggesting the inertial inflow model (Padoan et al., 2020); in agreement, in
Paper I, we argued that star formation was likely proceeding via the internal
inflow model in the parent cloud itself, and that a clump was never a well-
defined, isolated region. On the system scale, the multiple stars and
dynamical interactions suggest that the stars are competing for the local gas,
which is evidence for the competitive accretion model (Zinnecker, 1982;
Bonnell et al., 2001a, b); this is notable in N05c9 which does not need to
compete for gas and grows to 0.3 M⊙ compared to its progenitor which does need
to compete for gas and is ejected after failing to do so. Therefore, we
postulate that star formation (or rather the formation of clumps where stars
will ultimately form) proceeds via the inertial inflow model on the large-
scale, while stars grow via competitive accretion within a clump.

###  5.1 Hydrodynamics

The pure hydrodynamic cloud (named Hyd) in WBP2019 formed 19 stars; these
progenitor stars yielded several massive clumps (Paper I), up to $\gtrsim 20$
per cent of the parent cloud. Therefore, evolving these hydrodynamic clumps is
much more computationally expensive than evolving the clumps extracted from
N05 and I05, despite the absence of magnetic fields. Since the goal of Paper I
is to produce low(ish)-mass clumps that will form lower-order systems that can
ultimately be run at higher resolution, our goal was not met with most of the
clumps in Hyd.

Nonetheless, we evolved a few of the hydrodynamic clumps. We found that star
formation proceeded very efficiently, and too many stars were formed to
perform a useful comparison as above. This is a result of the reasonably high
stellar density in Hyd, which meant that most progenitor stars interacted with
several other stars and large quantities of gas, yielding massive clumps.
Given the mass of these clumps, it is unsurprising that they behave more like
clusters and produce many stars than clumps like and produce few stars.

Therefore, a limitation of our extraction algorithm is that it cannot well
reproduce stellar systems in clumps that have been extracted from a dense
stellar cluster. Given the weak magnetic fields in models N20 and I20 (non-
ideal and ideal MHD models, respectively, initialised with normalised mass-to-
flux ratios of 20) from WBP2019, it would be interesting to determine if our
method can reproduce those systems given the weak magnetic regulation on star
formation in those models. Answering that question is beyond the scope and
resources of this project.

###  5.2 Future work

The work presented in this paper and in Paper I has many potential spin-off
studies, including the following.

####  5.2.1 Clump extraction parameters

As shown and discussed above, our extracted clumps do not form filaments given
our extraction algorithm developed in Paper I. Thus, a next step would be to
test the various parameters of that algorithm to determine if we could extract
clumps that would from filaments without the influence of the remaining cloud.
Although we expect that this would be possible (e.g., by adjusting our
boundness criteria, the radius to which we search for particles, and/or the
radius where we unconditionally add particles our clumps), we are not
confident that it could be done while continuing to extract reasonably sized
($\lesssim 5$ M⊙) clumps. However, this should be explored in future work.

Additionally, it would be beneficial to explore the effect of extracting the
clumps at lower density values (e.g.) $\rho_{\text{max}}=10^{-17}$ g cm-3 or
at the beginning of the parent simulation. We can also investigate merging the
extracted clumps of stars in a bound system. In addition to exploring the
effect that these changes have on the initial clumps themselves (as in Paper
I), it would be important to determine how these clumps evolve and if a set of
parameters allows for near identical reproduction of a system, or if all
parameters simply yield statistical agreement.

####  5.2.2 Resolution

We have demonstrated that our extracted clumps statistically reproduce the
stellar and discs systems from the parent clusters. Thus, we have demonstrated
that these clumps are excellent initial conditions for simulations of isolated
star formation. Therefore, these clumps can be re-resolved at the higher
resolutions typically used in isolated star formation simulations. Using these
clumps as initial conditions will reduce the number of free-parameters
required of typically isolated star formation simulations, yielding less
contrived results. Moreover, given the higher resolution, we will be able to
investigate features (e.g., outflows) that are not resolved at the resolutions
typically used in cluster simulations (Wurster et al., 2022). Therefore,
running several of these clumps at higher resolution will greatly contribute
to our understanding of the star formation process within its chaotic
environment. Although these clumps cannot reproduce filaments using the
current extraction algorithm with its default parameters, these clumps are
still a preferred initial condition since filaments generally do not form in
isolated star formation simulations (e.g., see the papers listed in the
Introduction).

An alternative method to generating high-resolution data of stellar systems
from less-contrived initial conditions would be to employ live particle
splitting (Kitsionas & Whitworth, 2002, 2007) in the parent simulations
themselves. This would permit the stellar systems to be resolved at high
resolution while still being influenced by the (lower-resolution) larger-scale
processes. However, this could still lead to long run-times. For example, if
we re-resolved 6 M⊙ of the 50 M⊙ (e.g., the sum of the masses of the largest
clumps extracted from each system in N05) by splitting each SPH particle into
13 low-mass child particles, then the total number of particles in the parent
simulation would increase from $\sim$$5\times 10^{6}$ to $\sim$$12\times
10^{6}$, and the lower-mass particles would have smoothing lengths $\sim$2.4x
smaller than their lower-resolution counterparts. Between the increased number
of particles and smaller smoothing lengths, we could predict an increase in
runtime by a factor of $\sim$5\. Although this is much cheaper than running
the entire simulation at high resolution, this would still be prohibitively
expensive given the code and the computer architecture the parent simulations
were ran on. An additional consideration to the long run times is the
interaction of particles of the same type (e.g., gas) but of different mass
(i.e., resolution). Given the dynamics of the stellar systems, SPH particles
will be ejected from the systems, thus an algorithm must be developed to de-
resolved these ejected particles to their original resolution. If not, then
the interpenetration of SPH particles of the same type but of different masses
will lead to numerical instabilities.

####  5.2.3 Use as initial conditions

One of the aims of this project is to develop less-contrived initial
conditions for simulations of isolated star formation. Given the qualitative
and quantitative results presented here, we argue that our initial conditions
are less-contrived than those typically used in studies of isolated star
formation and will hence be useful in investigating – at high resolution as
per Section 5.2.2 – the star formation process and the early evolution of its
environment. We admit that the statistical agreement is not as strong as we
would like between the clumps and the parent distributions, however, as
discussed in Section 1, most studies in the literature produce a range of
results by varying even a single parameter. Therefore, the results from our
clumps are reasonable given the agreement with the parent distributions and
the improved algorithm, and is consistent with the literature by producing a
variety of results.

We admit that it is not practical to use lists of particles as initial
conditions, particularly for meshed algorithms. Thus, as a future project, we
propose parametrising the initial clumps presented in Paper I. This future
project will aim to produce a suite of star forming cores where each core will
have self-consistent parameters (e.g., mass, radius, initial turbulent Mach
number, etc…,) and (radial) profiles for (e.g.) density, temperature,
velocity, and the magnetic field. Therefore, it will become a simple selection
of parameterised core which can then be readily initialised in any
astrophysical code. This will remove the need to select individual parameters
and idealised profiles in order to produce more realistic star formation
simulations.

##  6 Summary and conclusion

In Paper I, we extracted a star forming clump for every star that formed in
the low-mass star cluster formation simulation of WBP2019. In this paper, we
evolved and analysed the 17 (12) clumps that were extracted from the star
cluster seeded with a normalised mass-to-flux ratio of 5 that employed non-
ideal (ideal) MHD. We maintained the numerical resolution of WBP2019, but we
used a newer version of our SPH code to exploit the changes in the six years
since that study began. We evolved the extracted clumps from their extraction
time to the final time modelled in WBP2019, thus each clump was evolved for a
different length of time.

Our main conclusions are as follows:

  1. 1.

The first star to form in a clump typically follows the mass evolution of a
star in the progenitor system, but not necessarily its own progenitor. The
mass accretion rates onto individual stars is similar in both clumps and the
parent cloud, suggesting a similar (competitive) accretion mechanism on small
scales.

  2. 2.

Stellar systems including their discs are not identically reproduced. Although
the clump contains all the gas that will become bound to the progenitor star,
it does not contain all the gas that will interact with the star, thus leading
to different evolutions.

  3. 3.

The distributions of stellar mass and stellar system mass are similar between
the clumps and the parent clouds, and we have shown that the clump and parent
distributions are drawn from the same underlying distribution. Therefore, we
have statistically reproduced these distributions.

  4. 4.

The distributions of disc mass and radii are less similar between the clumps
and the parent clouds in a qualitative sense, but agree in a statistical
sense. However, discs are dynamically evolving through external interactions,
thus this weak (qualitative) agreement is likely a result of a combination of
the time we selected to analyse the discs and small number statistics.

  5. 5.

The large-scale filamentary structure observed in the parent clouds is not
reproduced in the extracted clumps. Therefore, the production and maintenance
of filaments occurs through processes on larger scales than those present in
star-forming clumps, even if they are seeded with decaying turbulence. Despite
this, the distributions of the stellar and stellar systems properties are
statistically reproduced, suggesting that well-defined filaments may not be
responsible for the final properties of a stellar system. Future investigation
is required.

From our analysis, we reiterate the common consensus that star formation in a
chaotic process. Important physical processes occur on many scales, so if
large-scale processes are removed when extracting a clump of gas, then the
progenitor system cannot be identically reproduced. Therefore, we demonstrated
that the gas in the initial clump is continually modified by external gas in
the parent system, and that its profile at extraction time is not enough to
predict its future evolution.

We have also shown that systems can be statistically reproduced. Therefore, we
conclude that our goal of generating low-mass clumps that can be used as
initial conditions for high-resolution simulations of star formation has been
achieved. Although messier than the traditional, idealised initial conditions,
modelling star formation in extracted clumps will provide greater insight into
the star formation process and the associated physical processes given the
more realistic initial environment.

## Acknowledgements

We would like to thank the referee for comments that somewhat improved the
quality of this manuscript. We would like to thank Ian A Bonnell for useful
discussions. JW would also like to thank all his collaborators for many years
of helpful and inspiring discussions that have greatly contributed to the
quality of his scientific and numerical works; he would like to specifically
thank Daniel J Price, Matthew R Bate, Ian A Bonnell, Terrence Tricco, and
Rebecca Nealon. JW acknowledges support from the University of St Andrews. CR
acknowledges support from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program under grant
agreement No 638435 (GalNUC). This work was performed using the DiRAC Data
Intensive service at Leicester, operated by the University of Leicester IT
Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk).
The equipment was funded by BEIS capital funding via STFC capital grants
ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1.
DiRAC is part of the National e-Infrastructure. Several figures were made
using splash (Price, 2007).

## Data availability

The data underlying this article and Wurster et al. (2019) will be available
upon reasonable request.

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##  Appendix A Initial clumps

The top and bottom panels of Fig. 14 respectively show the initial clumps for
the 17 and 12 non-ideal and ideal MHD clumps that we evolved. The clumps are
extracted over $\sim$60 kyr, extracted from a range of regions, and have
masses of $0.29<M_{\text{clump}}$/M⊙$<2.11$. As discussed in Section 4.1,
there are similarities between clumps extracted from the same progenitor
system, however, these clumps are not identical and can be extracted over an
extended period of time (e.g., 28 kyr for N05 [q1]).

Figure 14: Average gas density of clumps extracted from N05 (top) and I05
(bottom). Each panel includes the clump name and name of the progenitor
system, the mass of the clump, and the extraction time (as per WBP2019). The
clumps have not been re-centred, thus are a true indication of their
extraction location in the parent cloud. Clump mass typically decreases with
increasing extraction time, even for stars that originated in the same
progenitor system at $t_{\text{final}}$. Clouds from the same progenitor
system are qualitatively similar, but also generally vary in mass and
extraction time.

Tables 3 and 4 give mass of the gas common to the clumps extracted from the
same progenitor systems in N05 and I05, respectively. The table with I05 [b3+]
includes the star that merged with I05c8 and the table with I05 [s2,s4]
compares the two spatially close but unbound stars in I05. In most systems,
the smaller clumps are subsets of the most massive clump. Exceptions are N05
[q1] where N05c1 is somewhat isolated, and I05 [b1] where the two stars are
independent despite being bound.

N05 [q1]
N05c1 N05c2 N05c3 N05c4 N05c1 2.11 0.55 0.49 0.42 N05c2 1.81 1.60 1.32 N05c3
1.61 1.31 N05c4 1.32

N05 [q2]
N05c5 N05c6 N05c7 N05c9 N05c5 2.04 1.78 1.80 1.10 N05c6 1.79 1.78 1.09 N05c7
1.81 1.09 N05c9 1.10

N05 [t1]
N05c12 N05c14 N05c15 N05c12 1.35 1.14 1.05 N05c14 1.14 1.04 N05c15 1.05

N05 [b1]
N05c10 N05c11 N05c10 0.83 0.75 N05c11 0.78

Table 3: Each table presents the mass of the gas (in units of M⊙) common to
the clumps extracted from the progenitor systems in N05. The clump mass is
given on the diagonal; the matrix is symmetric, so only the upper diagonal is
completed for clarity. Although N05 [q1] is a quadruple system, N05c1 has a
small fraction of gas in common with N05c2-4 thus is relatively isolated;
N05c3-4 are sub-sets of N05c2-3. Each clump in N05 [q2] is a subset of each
more-massive, older clump, and all clumps are a subset of N05c5. Each clump in
N05 [t1] is a subset of each more-massive, older clump, and all clumps are a
subset of N05c12.

I05 [b1]

|  I05c1 |  I05c2
---|---|---
I05c1 | 1.85 | 0.29
I05c2 |  | 1.73

I05 [b2]

|  I05c3 |  I05c4
---|---|---
I05c3 | 1.97 | 1.86
I05c4 |  | 1.89

I05 [b3+]

|  I05c7 |  I05c8 |  I05c9
---|---|---|---
I05c7 | 1.22 | 1.10 | 0.68
I05c8 |  | 1.18 | 0.67
I05c9 |  |  | 0.68

I05 [s2,s4]

|  I05c6 |  I05c11
---|---|---
I05c6 | 0.82 | 0.35
I05c11 |  | 0.37

Table 4: Mass of the gas (in units of M⊙) common to the clumps extracted from
the stars in I05. The clump mass is given on the diagonal; the matrix is
symmetric, so only the upper diagonal is completed for clarity. Despite the
stars in I05 [b1] being bound, only $\sim$16 per cent of the gas is common
between the clumps, therefore they are mostly independent. I05c9 merged with
I05c8; nearly all of the gas in I05c9 is in the other two clumps, therefore,
I05c7,8 also represent the merged star. Although I05c6 and I05c11 are not
bound, nearly all the gas in I05c11 is also in I05c6, therefore, the latter
should well represent both progenitors.