# Conditions for Clump Survival in High-z Disc Galaxies

Avishai Dekel1,2, Offek Tziperman1, Kartick Sarkar1, Omri Ginzburg1, Nir
Mandelker1, Daniel Ceverino3,4, Joel Primack2,5

1Racah Institute of Physics, The Hebrew University, Jerusalem 91904 Israel
2SCIPP, University of California, Santa Cruz, CA 95064, USA
3Departamento de Fisica Teorica, Facultad de Ciencias, Universidad Autonoma de
Madrid, Cantoblanco, 28049 Madrid, Spain
4CIAFF, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049 Madrid,
Spain
5Physics Department, University of California, Santa Cruz, Santa Cruz, CA
95064, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

We study the survival versus disruption of the giant clumps in high-redshift
disc galaxies, short-lived (S) versus long-lived (L) clumps and two L sub-
types, via analytic modeling tested against simulations. We develop a
criterion for clump survival, with or without their gas, based on a predictive
survivability parameter $S$. It compares the energy sources by supernova
feedback and gravitational contraction to the clump binding energy and losses
by outflows and turbulence dissipation. The clump properties are derived from
Toomre instability, approaching virial and Jeans equilibrium, and the
supernova energy deposit is based on an up-to-date bubble analysis. For
moderate feedback levels, we find that L clumps exist with circular velocities
$\sim\\!50\,{\rm km}\,{\rm s}^{-1}$ and masses $\\!\geq\\!\\!10^{8}M_{\odot}$.
They are likely in galaxies with circular velocities $\\!\geq\\!\\!200\,{\rm
km}\,{\rm s}^{-1}$, consistent at $z\\!\sim\\!2$ with the favored stellar mass
for discs, $\\!\geq\\!\\!10^{9.3}M_{\odot}$. L clumps favor disc gas fractions
$\\!\geq\\!\\!0.3$, low-mass bulges and redshifts $z\\!\sim\\!2$. The
likelihood of L clumps is reduced if the feedback is more ejective, e.g., if
the supernovae are optimally clustered, if radiative feedback is very strong,
if the stellar initial mass function is top-heavy, or if the star-formation-
rate efficiency is particularly high. A sub-type of L clumps (LS), which lose
their gas in several free-fall times but retain bound stellar components, may
be explained by a smaller contraction factor and stronger external
gravitational effects, where clump mergers increase the SFR efficiency. The
more massive L clumps (LL) retain most of their baryons for tens of free-fall
times with a roughly constant star-formation rate.

###### keywords:

dark matter — galaxies: discs — galaxies: evolution — galaxies: formation —
galaxies: haloes — galaxies: mergers

††pagerange: Conditions for Clump Survival in High-z Disc Galaxies–Conditions
for Clump Survival in High-z Disc Galaxies††pubyear: 2002

##  1 Introduction

The typical massive star-forming galaxies at the peak epoch of galaxy
formation, in the redshift range $z\\!\sim\\!1\\!-\\!3$, are dominated by
extended rotating and turbulent gas-rich discs (Genzel et al., 2006; Förster
Schreiber et al., 2006; Genzel et al., 2008; Tacconi et al., 2010; Genzel et
al., 2011; Tacconi et al., 2013; Genzel et al., 2014; Förster Schreiber et
al., 2018), each hosting several giant star-forming clumps (Elmegreen &
Elmegreen, 2005; Genzel et al., 2006, 2008; Guo et al., 2012, 2015; Fisher et
al., 2017; Guo et al., 2018; Huertas-Company et al., 2020; Ginzburg et al.,
2021). Most of these clumps are commonly assumed to form by gravitational disc
instability (Toomre, 1964)111Possibly triggered by compressive modes of
turbulence (Inoue et al., 2016, and work in progress).. This process has been
simulated both in isolated discs (Noguchi, 1999; Immeli et al., 2004b, a;
Bournaud, Elmegreen & Elmegreen, 2007; Genzel et al., 2008) and in a
cosmological setting (Dekel, Sari & Ceverino, 2009; Agertz, Teyssier & Moore,
2009; Ceverino, Dekel & Bournaud, 2010; Ceverino et al., 2012), particularly
in Mandelker et al. (2014), Mandelker et al. (2017), Ginzburg et al. (2021)
and Dekel et al. (2021). Given the high gas fraction in $z\\!\sim\\!2$
galaxies (e.g., Daddi et al., 2010; Tacconi et al., 2018), the typical clump
mass predicted by Toomre instability is on the order of a few percents of the
disc mass, which makes the clumps play an important dynamical role in the
evolution of these discs, a process dubbed “violent disc instability” (VDI)
(e.g., Dekel, Sari & Ceverino, 2009; Dekel & Burkert, 2014). Gas-rich clumps
are predicted to form with a mass near and below the characteristic Toomre
mass and preferably at relatively large disc radii (see Dekel et al., 2021,
for evidence in simulations and observations). Those clumps that survive
disruption by stellar and supernova feedback are predicted to migrate toward
the disc centre due to VDI-driven torques in a few disc orbital times,
corresponding to several hundred Megayears at $z\\!\sim\\!2$ (e.g., Dekel,
Sari & Ceverino, 2009; Ceverino, Dekel & Bournaud, 2010; Krumholz & Burkert,
2010; Cacciato, Dekel & Genel, 2012; Forbes, Krumholz & Burkert, 2012; Forbes
et al., 2014; Krumholz et al., 2018; Dekel et al., 2020b).

Simulations reveal that the clumps can be divided into two major types. One
consists of long-lived clumps (hereafter L clumps) that remain bound and live
for tens of clump free-fall times or more, allowing them enough time to
complete their inward migration. The other involves short-lived clumps
(hereafter S clumps), which are disrupted after a few free-fall times
(Mandelker et al., 2017). Using machine learning, the two types of clumps as
identified in the VELA-3 cosmological simulations (§LABEL:sec:sim_type) were
also identified in the large catalog of observed clumps from the CANDELS-HST
survey Guo et al. (2018), with consistent relative distributions of clump
masses, radial positions in the discs and host-galaxy masses (Ginzburg et al.,
2021).222A word of caution is that the observational estimates of clump
properties suffer from large uncertainties, which require careful
considerations of systematic errors, possibly using machine learning (Huertas-
Company et al., 2020).

According to a wide variety of existing simulations, the clump type depends on
certain clump properties at formation, on certain host-galaxy properties, and
on the assumed subgrid physical models for star formation and especially
feedback. Mandelker et al. (2017), based on the VELA-3 cosmological
simulations, found that L clumps tend to form above a threshold mass of
$\sim\\!10^{8}M_{\odot}$, while S clumps dominate at smaller clump masses. The
L clumps were found to be more compact, round and bound, while the S clumps
are more diffuse, elongated and they become unbound. In terms of galaxy
properties, the L clumps in VELA-3 tend to reside in more massive galaxies,
consistent with the predicted mass threshold for long-lived extended discs at
a halo mass of $M_{\rm v}\\!\sim\\!2\times 10^{11}M_{\odot}$ (Dekel et al.,
2020a). Based on isolated-galaxy simulations, the clump formation, mass and
longevity are correlated with a high gas fraction in the disc (Fensch &
Bournaud, 2021; Renaud, Romeo & Agertz, 2021).

Most importantly, the clump type is very sensitive to the subgrid feedback
model adopted in the simulation. The VELA-3 simulations, that assume a
moderate supernova feedback strength with a modest effective radiative
momentum driving of $2\\!-\\!3L/c$ (where $L$ is the clump luminosity), give
rise to clumps of the two types. On the other hand, the VELA-6 simulations
(Ceverino et al. in preparation), of the same suite of galaxies but with a
stronger feedback, turn out to produce mostly S clumps with a negligible
population of L clumps even at the massive end (see below). This recovers the
results from simulations that put in very strong winds (Genel et al., 2012),
or that include enhanced radiative feedback of $30\\!-\\!50L/c$ due to strong
trapping of infra-red photons and additional elements of feedback (Hopkins,
Quataert & Murray, 2012a; Oklopčić et al., 2017). Such simulations produced
only S clumps.

On the theory side, Dekel, Sari & Ceverino (2009), based on Dekel & Silk
(1986), crudely estimated that supernova feedback by itself may not have
enough power to disrupt the massive clumps. Murray, Quataert & Thompson (2010)
argued that momentum-driven radiative stellar feedback, enhanced by infra-red
photon trapping, could disrupt the clumps on a dynamical timescale, as it does
in the local molecular clouds. However, Krumholz & Dekel (2010) pointed out
that such an explosive disruption would be possible in the high-redshift giant
clumps only if the efficiency of star-formation rate (SFR) in a free-fall time
is as high as $\epsilon\\!\sim\\!0.1$, significantly larger than what is
implied by the observed Kennicutt-Schmidt relation in different types of
galaxies at different redshifts, namely $\epsilon$ of about one to a few
percent (Tacconi et al., 2010; Krumholz, Dekel & McKee, 2012; Freundlich et
al., 2013). Dekel & Krumholz (2013) proposed instead that outflows from high-
redshift clumps, with mass-loading factors $\eta$ of order unity to a few, are
driven by steady momentum-driven outflows from stars over many tens of free-
fall times. Their analysis is based on the finding from high-resolution 2D
simulations that radiation trapping is negligible because it destabilizes the
wind (Krumholz & Thompson, 2012, 2013). Each photon can therefore contribute
to the wind momentum only once, so the radiative force is limited to
$\sim\\!L/c$. Combining radiation, protostellar plus main-sequence winds, and
supernovae, Dekel & Krumholz (2013) estimated the total direct injection rate
of momentum into the outflow to be $\sim\\!2.5L/c$. The adiabatic phase of
clustered supernovae and main-sequence winds may double this rate (Gentry et
al., 2017). The predicted mass loading factors of order unity were argued in
Dekel & Krumholz (2013) to be consistent with the values deduced from the
pioneering observations of outflows from giant clumps (Genzel et al., 2011;
Newman et al., 2012). They concluded that most massive clumps are expected to
complete their migration prior to gas depletion. With the additional gas
accretion onto the clumps, they argued that the clumps are actually expected
to grow in mass as they migrate inward.

Dekel et al. (2021) studied the evolution of the properties of the massive
clumps during their migration through the disc. They developed an idealized
analytic “bathtub” model for a single clump, where clump gas turns into stars
and the clump exchanges gas and stars with the disc through accretion,
feedback-driven outflows and tidal stripping. The clump evolution is governed
by the balance between these processes under conservation of gas and stellar
mass, which lead to an analytic solution. The model reveals how the clump
evolution depends on the assumed efficiency of accretion, star formation and
outflows. The model predictions were confronted with simulation results, both
of isolated gas-rich galaxies at high resolution and of high-redshift galaxies
in their cosmological framework, exploring a variety of feedback strengths at
moderate levels. The theoretical predictions were confronted with the CANDELS
clump catalog of Guo et al. (2018) at $z\\!=\\!1.5\\!-\\!3$, indicating that
the observed gradients of clump properties as a function of distance from the
disc centre are consistent with the predicted migration of massive clumps and
the associated mild evolution of clump properties.

Our goal in this paper is to understand the conditions for clump survival
versus disruption by feedback, and thus seek the origin of the different clump
types in terms of clump and disc properties as well as feedback strength. We
do this first via analytic modeling. For clumps in their early stages of
evolution, we define a survivability parameter $S$, the ratio of (a fraction
of) the clump mass to the outflowing mass. The value of this quantity below or
above unity distinguishes between S and L clumps. This parameter can be
derived from basic clump properties based on the energy balance in the clump.
We consider the energy sources by supernova feedback and gravitational driving
of turbulence, and energy sinks by dissipation of turbulence and outflows, in
comparison with the clump gravitational binding energy. In this model, the
clump properties are deduced from Toomre marginal instability (as in Dekel,
Sari & Ceverino, 2009), with the kinematics of fully collapsed clumps obeying
Jeans equilibrium (Ceverino et al., 2012). The input by supernova feedback,
inspired by Dekel & Silk (1986), is based on up-to-date studies of bubble
evolution as in Dekel et al. (2019), fine-tuned in our ongoing work. The
gravitational driving of turbulence is estimated from the gravitational
collapse and clump encounters. The analytic model is then confronted with the
clumps in the VELA-3 and VELA-6 cosmological simulations.

Beyond the distinction between L and S clumps, we address a new sub-division
into two types of L clumps that will be identified in the simulations below.
One sub-type, to be termed LL, consists of clumps that keep a relatively high
gas fraction for many tens of clump free-fall times. The other sub-type, to be
termed LS, is made of clumps that lose most of their gas to outflows in
$\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!10$ free-fall times, but
keep long-lived bound stellar components that survive for many tens of free-
fall times. Our analytic model provides physical explanations for the basic
distinction between S and L clumps as well as for the sub-division into LS and
LL clumps.

The paper is organized as follows. In §2 to §7 we present the analytic model,
and in §8 and §9 we compare the model to simulations. In §2 we introduce the
survivability parameter $S$. In §3 we model the clump binding energy and the
energy drains by dissipation of turbulence and outflows. In §4 we model the
energy sources by supernova feedback and by gravity. In §5 we introduce a
correction for clumps that have not reached virial equilibrium. In §6 we use
the model to explain the division to S and L clumps and the sub-division to LS
and LL clumps. In §7 we identify the dependence of the clump type on the
different properties of the clump and the host disc as well as the feedback
strength. Then, In §8 we present the division to clump types in the
simulations. Finally, in §9, we confront the model predictions with the
simulation results. In §10 we summarize our conclusions and discuss them.

##  2 Condition for clump survival

###  2.1 A survivability parameter

We consider a clump of mass $M_{\rm c}(t)$, at time $t$ after its formation,
that has lost mass $M_{\rm out}(t)$ to outflows up until that time. For a
clump in virial equilibrium, we define a clump survivability parameter $S$
such that a criterion for significant mass loss through outflows by time $t$
is

$S(t)\equiv\frac{0.5\,M_{\rm c}(t)}{M_{\rm out}(t)}\leq 1\,.$ (1)

While $M_{\rm out}$ is expected to grow steadily in time, $M_{\rm c}$ may tend
to have a rather constant value after the early stages of clump buildup, or
decline for clumps that lose a significant fraction of their gas to outflows
(Dekel et al., 2021). When $S$ is above unity, the clump has retained most of
its mass, while when $S$ is below unity, the clump has lost most of its mass
to outflows. Eq. (1) is motivated by noticing that for a clump in virial
equilibrium, an instantaneous removal of half its mass (in a free-fall time)
makes it unbound with zero energy. We will soften the requirement for virial
equilibrium in §5 and generalize the definition of $S$ accordingly in eq.
(53).

However, a mass loss with $S\\!\leq\\!1$ does not necessarily imply a total
disruption. If the mass loss is adiabatic, over many free-fall times, part of
the system may remain bound. A criterion for destructive instantaneous mass
loss is thus $\dot{M}_{\rm out}(t)\,t_{\rm ff}\\!\geq\\!0.5M_{\rm c}(t)$,
where $t_{\rm ff}$ is the clump free-fall time. If we crudely approximate
$\dot{M}_{\rm out}(t)\\!\sim\\!M_{\rm out}(t)/t$, where $t$ is the time since
clump formation, the criterion for total disruption becomes

$\tau S(t)\leq 1\,,$ (2)

where $\tau\\!=\\!t/t_{\rm ff}$. Thus, at $\tau\\!\sim\\!1$, a value of
$S\\!\leq\\!1$ that implies removal of most of the gas also indicates a total
disruption, $\tau S\\!\leq\\!1$, namely an S clump. However, if half the mass
is lost only by a later time, $S(\tau)\\!\leq\\!1$ at $\tau$ well above unity,
the mass-loss would be totally destructive (S) only if $\tau S\\!\leq\\!1$ as
well, while a bound (stellar) remnant is expected (LS) if $\tau S\\!>\\!1$.

The quantities $M_{\rm c}$ and $M_{\rm out}$ that define $S$ in eq. (1) can be
measured from simulations, as we do below in one version of our analysis,
though the measured $M_{\rm out}$ may be rather uncertain, especially at early
times when the clump radius is likely to evolve rapidly.

###  2.2 The survivability by energy balance

As an alternative, we wish to predict $S$ from theoretical considerations,
where we express $M_{\rm c}$ and $M_{\rm out}$ by physical quantities that we
can estimate using physical arguments and based on an energy balance.

Assuming a spherical clump of mass $M_{\rm c}$ within radius $R_{\rm c}$, the
potential is characterized by a circular velocity $V_{\rm c}^{2}\\!=\\!GM_{\rm
c}/R_{\rm c}$. In virial equilibrium, the total energy of the gas per unit
mass is

$-E_{\rm vir}=-\frac{1}{2}V_{\rm c}^{2}=-\frac{1}{2}\frac{GM_{\rm c}}{R_{\rm
c}}\,.$ (3)

We tentatively assume that $V_{\rm c}$ and the gas mass $M_{\rm g}$ are
roughly constant during the early phase of clump evolution, before a
significant amount of the gas is ejected. This is motivated by analytic
modeling and simulations of clump evolution (Dekel et al., 2021), and it will
be tested further in simulations below.

The cumulative energy balance by time $t$, per unit gas mass, can be written
as

$E_{\rm sn}(t)+E_{\rm grav}(t)-E_{\rm dis}(t)=E_{\rm vir}+E_{\rm out}(t)\,,$
(4)

where all the quantities are positive. Except $E_{\rm vir}$, they are all
integrals of their rates over time. Each will be evaluated below as a function
of certain basic parameters in terms of $V_{\rm c}$ and $\tau$. The terms on
the left-hand side represent sources of energy gain and loss. The term $E_{\rm
sn}$ is the energy deposited in the gas by stars and supernovae. The term
$E_{\rm grav}$ represents the additional mechanical energy sources largely
associated with gravity, such as the gravity associated with the initial
collapse of the clump, continuous accretion, mergers, tidal effects and shear,
all capable of driving turbulence. The term $E_{\rm dis}$ represents the
dissipative losses of the turbulence, cascading down to small scales where it
thermalizes and can cool radiatively. The terms on the right-hand side
represent the energy required to bring the gas from virial equilibrium to
outflow. Here $E_{\rm vir}$ is the energy required for bringing the gas from
$-E_{\rm vir}$ to $E\\!=\\!0$, ready for escape, and $E_{\rm out}$ is the
extra kinetic energy carried by the outflow.

Using eq. (4), the survivability parameter $S$ as defined in eq. (1) can be
approximated by

$S\\!\simeq\\!\frac{f_{\rm g,c}^{-1}\,E_{\rm vir}}{E_{\rm out}}\,,$ (5)

where $f_{\rm g,c}$ is the gas fraction in the clump, starting somewhat below
unity and declining in time. This arises from eq. (3) and the relation of
$E_{\rm out}$ to $V_{\rm c}$,

$E_{\rm out}=\frac{1}{2}\frac{M_{\rm out}}{M_{\rm g}}V_{\rm
out}^{2}\simeq\frac{M_{\rm out}}{M_{\rm g}}V_{\rm c}^{2}\,,$ (6)

where the typical outflow velocity is crudely assumed to be near the escape
velocity, $V_{\rm out}^{2}\\!\simeq\\!2\,V_{\rm c}^{2}$. Thus, $S$ can be
expressed as

$S=\frac{f_{\rm g,c}^{-1}}{S^{\prime-1}-1}\,,\quad
S^{\prime}\equiv\frac{E_{\rm vir}}{E_{\rm sn}+E_{\rm grav}-E_{\rm dis}}\,.$
(7)

We note that the critical value for losing mass by outflows that corresponds
to $S\\!=\\!1$ is $S^{\prime}\\!=\\!1/(f_{\rm g,c}^{-1}+1)$. This is
$S^{\prime}\\!\sim\\!0.5$ for $f_{\rm g,c}\\!\sim\\!1$ and
$S^{\prime}\\!\sim\\!f_{\rm g,c}$ for $f_{\rm g,c}\\!\ll\\!1$, which are valid
respectively early and late in the clump evolution.

The cumulative specific energies that enter $S$ in eq. (7) will be evaluated
below based on theoretical arguments. They all turn out to be proportional to
$E_{\rm vir}\tau$, and can be written as

$E_{\rm sn}=f_{\rm sn}\,V_{\rm c,50}^{-1}\,E_{\rm vir}\,\tau\,,\quad E_{\rm
grav}=f_{\rm grav}\,E_{\rm vir}\,\tau\,,$ (8) $E_{\rm dis}=f_{\rm dis}\,E_{\rm
vir}\,\tau\,,\quad E_{\rm out}=f_{\rm out}\,E_{\rm vir}\,\tau\,,$ (9)

with the $f$ factors functions of basic clump parameters, and where $V_{\rm
c,50}\\!=\\!V_{\rm c}/50\,{\rm km}\,{\rm s}^{-1}$ is the main explicit clump
property that enters. The clump $V_{\rm c}$ (and $E_{\rm vir}$) will be
predicted assuming Toomre disk instability (Toomre, 1964). The feedback energy
$E_{\rm sn}$ will be estimated from the theory of supernova bubbles as a
function of the turbulence velocity dispersion $\sigma_{\rm c}$. The
dissipative energy loss $E_{\rm dis}$ can also be estimated as a function of
$\sigma_{\rm c}$. In turn, $\sigma_{\rm c}$ can be related to $V_{\rm c}$
assuming Jeans equilibrium and the degree of angular-momentum conservation
during clump formation. The value of $E_{\rm grav}$ can be crudely evaluated
during clump formation and at later times. In the following estimates we learn
that $f_{\rm sn}$ is of order unity or larger, $f_{\rm dis}\\!\sim\\!1$,
$f_{\rm grav}$ is of order unity or smaller, and $f_{\rm out}\\!\ll\\!1$.

Inserting eqs. (8) and (9) in $S^{\prime}$ from eq. (7) we obtain

$S^{\prime}=\frac{1}{(f_{\rm sn}\,V_{\rm c,50}^{-1}+f_{\rm grav}-f_{\rm
dis})\,\tau}\,,$ (10)

which serves as our operational expression for $S^{\prime}$ and $S$.
Interestingly, the key quantity $V_{\rm c}$ enters $S^{\prime}$ only through
the first term. The dominant factor in determining the value of $S^{\prime}$
is the supernova feedback represented by the first term in the denominator of
eq. (10), as $f_{\rm sn}$ varies with the feedback strength and $V_{\rm c}$
varies between massive and low mass clumps. The time dependence enters via
$\tau$ as a multiplicative factor for all the terms in the denominator, as
well as through possible time dependencies of each $f$ factor.

According to eq. (7), the time $\tau_{(S\\!=\\!1)}$ when $S$ equals unity is
when $S^{\prime-1}\\!=\\!1+f_{\rm g,c}^{-1}$, which from eq. (10) is at

$\tau_{(S\\!=\\!1)}=\frac{1+f_{\rm g,c}^{-1}}{f_{\rm sn}V_{\rm
c,50}^{-1}+f_{\rm grav}-f_{\rm dis}}\,.$ (11)

If the clump is to lose a significant fraction of its gas, a larger value of
$\tau_{(S\\!=\\!1)}$ indicates a later time for gas loss. Similarly, the time
that indicates full disruption, when $\tau S\\!=\\!1$, is given by

$\tau_{(\tau S\\!=\\!1)}=[f_{\rm sn}\,V_{\rm c,50}^{-1}+f_{\rm grav}-f_{\rm
dis}-f_{\rm g,c}^{-1}]^{-1}\,.$ (12)

##  3 Clump properties & energy drains

In the coming two sections we evaluate the relevant energies and the
associated $f$ factors that enter $S^{\prime}$ in eq. (10). In the current
section we address the relevant clump properties, using Toomre instability to
derive the binding energy and using Jeans equilibrium to relate rotation to
dispersion velocity, and evaluate the energy losses by outflows and
dissipation. In the following section we evaluate the energy gains by
supernovae and gravity.

###  3.1 Star formation rate and outflow energy

The outflow energy can be expressed as a function of the SFR, $\dot{M}_{\rm
sf}$, via the mass-loading factor

$\eta=\frac{\dot{M}_{\rm out}}{\dot{M}_{\rm sf}}\,.$ (13)

According to the simulations described below, $\eta$ is expected to be of
order unity. Following the standard convention, the SFR can be expressed via
the free-fall time in the clump, $t_{\rm ff}$, and the SFR efficiency
parameter, $\epsilon$, as

$\dot{M}_{\rm sf}=\epsilon\,\frac{M_{\rm g}}{t_{\rm ff}}\,.$ (14)

The clump free-fall time is

$t_{\rm ff}=\left(\frac{3\,\pi}{32\,G\,\rho}\right)^{1/2}\simeq 5.2\,{\rm
Myr}\,n_{2}^{-1/2}f_{\rm g,c}^{1/2}\,,$ (15)

where $\rho$ is the total density in the clump, $n\\!=\\!100\,{\rm
cm}^{-3}\,n_{2}$ is the corresponding gas number density and $f_{\rm
g,c}\\!\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!1$ is the gas
fraction in the clump in its early phase. We very crudely assumed here
$\rho\\!=\\!f_{\rm g,c}^{-1}\,m_{\rm p}\,n$. A value of $n\\!\sim\\!100\,{\rm
cm}^{-3}$ is expected if the clump has collapsed in 3D by a factor of a few
from a gas density of order $1\,{\rm cm}^{-3}$ in the disc.

The SFR efficiency per free-fall time, $\epsilon$, has to be of order a few
percent in order to match the Kennicutt-Schmidt law in different environments
and redshifts (e.g., Krumholz, Dekel & McKee, 2012). The effective value of
$\epsilon$ as defined for the whole clump mass and free-fall time may be
larger if the star formation actually occurs in dense sub-regions within the
clump, where the free-fall timescale is shorter. We denote
$\epsilon\\!=\\!0.03\,\epsilon_{.03}$.

Assuming a constant SFR and $M_{\rm g}$, using eqs. (13) and (14) one can
write

$M_{\rm out}=\eta\,\dot{M}_{\rm sf}\,t=0.03\,\epsilon_{.03}\,\eta\,M_{\rm
g}\,\tau\,,$ (16)

and express the outflow energy per unit gas mass from eq. (6) as

$E_{\rm out}=0.03\,\epsilon_{.03}\,\eta\,\tau\,V_{\rm c}^{2}\,.$ (17)

Comparing eq. (17) to eq. (3), assuming the fiducial values of $\eta$ and
$\epsilon$, we learn that $E_{\rm out}\\!\ll\\!E_{\rm vir}$ in the first few
free-fall times, and they become comparable at $\tau\\!\sim\\!16$.

###  3.2 Toomre clump binding energy

In order to evaluate the clump properties that enter $E_{\rm vir}$ in eq. (3),
we assume that the disc is in marginal Toomre instability (Toomre, 1964) with
$Q\\!=\\!1$. We assume that the clump mass is $\mu\,M_{\rm T}$, where $M_{\rm
T}$ is the Toomre mass corresponding to the fastest growing scale. Following
Dekel, Sari & Ceverino (2009), Ceverino, Dekel & Bournaud (2010) and Ceverino
et al. (2012), we define a key quantity, the cold mass fraction,

$\delta\equiv\frac{M_{\rm d}}{M_{\rm tot}}\,.$ (18)

Here $M_{\rm d}$ is the cold mass in the disk (gas and young stars), and
$M_{\rm tot}$ is the total mass encompassed by the sphere of the disc radius
$R_{\rm d}$ (including gas, stars and dark matter). Assuming a local power law
for the disc rotation curve, $V_{\rm d}(r)\\!\propto\\!r^{\alpha}$, with a
characteristic value $V_{\rm d}$, and a disc radial velocity dispersion
$\sigma_{r,{\rm d}}$, a Toomre parameter of $Q\\!=\\!1$ implies that

$\delta=\sqrt{2}(1+\alpha)^{1/2}\,\frac{\sigma_{r,{\rm d}}}{V_{\rm d}}\,.$
(19)

Then, the Toomre proto-clump radius relative to the disc radius is

$\frac{R_{\rm T}}{R_{\rm d}}=\frac{\pi}{4\,(1+\alpha)}\,\delta\,,$ (20)

and the Toomre clump mass relative to the disc mass is

$\frac{M_{\rm T}}{M_{\rm d}}=\left(\frac{R_{\rm T}}{R_{\rm
d}}\right)^{2}=0.025\,\delta_{.2}^{2}\,,$ (21)

where $\delta\\!=\\!0.2\,\delta_{.2}$, the density in the proto-clump is
assumed to be similar to the mean density in the disc, and $\alpha\\!=\\!0$ is
assumed. We thus have

$M_{\rm c}=\mu M_{\rm T}\quad{\rm and}\quad R_{\rm c,i}\\!=\\!\mu^{1/2}R_{\rm
T}\,,$ (22)

where $R_{\rm c,i}$ is the initial radius of the proto-clump patch in the
disc. The clump mass in terms of the cold-disc mass becomes

$M_{\rm c}=0.25\times 10^{8}M_{\odot}\,\mu_{.5}\,\delta_{.2}^{2}\,M_{\rm
d,9.3}\,,$ (23)

where $M_{\rm d}=10^{9.3}M_{\odot}\,M_{\rm d,9.3}$.

In order to obtain the clump circular velocity, we assume that the clump has
collapsed by a contraction factor $c$ from its initial radius to a final
radius $R_{\rm c}$,

$R_{\rm c,i}=cR_{\rm c}\,.$ (24)

Then, combining eq. (20) and eq. (24), the clump potential well is
characterized by

$V_{\rm c}^{2}=\frac{G\,M_{\rm c}}{R_{\rm
c}}=\frac{\pi}{4\,(1+\alpha)}\,c\,\mu^{1/2}\,\delta^{2}\,V_{\rm d}^{2}\,,$
(25)

recalling that $V_{\rm d}^{2}\\!=\\!G(\delta^{-1}M_{\rm d})/R_{\rm d}$.
Assuming hereafter a flat rotation curve, $\alpha=0$, we obtain for the
characteristic clump circular velocity

$V_{\rm c}\simeq 50\,{\rm km}\,{\rm
s}^{-1}\,c_{3}^{1/2}\,\mu_{.5}^{1/4}\,\delta_{.2}\,V_{{\rm d},200}\,,$ (26)

where $c\\!=\\!3\,c_{3}$, $\mu\\!=\\!0.5\mu_{.5}$,
$\delta\\!=\\!0.2\,\delta_{.2}$, and $V_{\rm d}\\!=\\!200\,{\rm km}\,{\rm
s}^{-1}V_{{\rm d},200}$. This could be inserted in eq. (3) for $E_{\rm vir}$.

The fiducial values of the parameters are crudely justified as follows. A
contraction factor of $c\\!\sim\\!3\\!-\\!5$ is expected for clump
virialization and rotation support (Ceverino et al., 2012). A galactic
circular velocity of $V_{\rm d}\\!\simeq\\!1.5V_{\rm v}\\!\simeq\\!200\,{\rm
km}\,{\rm s}^{-1}$ is expected for a galaxy in a halo of $M_{\rm
v}\\!\sim\\!3\times 10^{11}M_{\odot}$ at $z\\!\sim\\!2$, based on the virial
relation, eq. (62) below. This is above the threshold for long-lived disks at
all redshifts (Dekel et al., 2020a). This halo mass implies a stellar mass of
$M_{\rm s}\\!\sim\\!3\times 10^{9}M_{\odot}$ (Behroozi et al., 2019), and a
comparable gas mass for $f_{\rm g}\\!\simeq\\!0.5$ (see §7.1). A value of
$\delta\\!\sim\\!0.2$ can be expected for a gas fraction in the disc of
$f_{\rm g}\\!\sim\\!0.5$, given the contribution of the bulge and the dark
matter to $M_{\rm tot}$ in the denominator of $\delta$ (see §7.3). Eq. (26) is
in good agreement with the values obtained for clumps in the VELA-3
simulations (Mandelker et al., 2017, Fig. 8), where the typical galaxy masses
at $z\\!\sim\\!2$ are indeed comparable to the fiducial values mentioned
above, to be discussed further below.

###  3.3 Rotation and dispersion: Jeans equilibrium

Certain energy terms in eq. (4) depend on the clump velocity dispersion,
$\sigma_{\rm c}$, which characterizes the turbulence in the clumps, and is
significantly larger than the speed of sound of $\sim\\!10\,{\rm km}\,{\rm
s}^{-1}$ at a typical temperature of $\sim\\!10^{4}$K. In $\sigma_{\rm c}$ we
refer to the one-dimensional component of the velocity dispersion,
specifically the radial component. Following Ceverino et al. (2012), assuming
that the clump is in Jeans equilibrium, $\sigma_{\rm c}$ can be evaluated from
$V_{\rm c}$ and the clump contraction factor $c$, as follows.

Assuming cylindrical symmetry, the Jeans equation in the equatorial plane
reads

$V_{\rm c}^{2}=V_{\rm
rot,c}^{2}-\frac{r}{\rho}\frac{\partial(\rho\,\sigma_{\rm c}^{2})}{\partial
r}\simeq V_{\rm rot,c}^{2}+2\,\sigma_{\rm c}^{2}\,,$ (27)

where $V_{\rm rot,c}$ is the rotation velocity in the clump, and $\sigma_{\rm
c}$ is the radial velocity dispersion, which is assumed to be constant in the
second equality. The factor 2 is for an isothermal-sphere-like density
profile, $\rho\\!\propto\\!r^{-2}$. In practice, simulations show that for
entire discs this factor can vary between unity near the effective radius of
the gas and 4 at five effective radii, well outside the main body of the disc
(Kretschmer et al., 2021), indicating that variations may be expected within
the clumps as well. We note that with no rotation, $\sigma_{\rm
c}^{2}\\!\simeq\\!0.5V_{\rm c}^{2}$.

For a disc in marginal Toomre stability with $Q\\!=\\!1$, and a clump
contraction factor $c$, we obtain from eq. (19) to eq. (24) that the clump is
related to the disc via

$V_{\rm c}^{2}=\frac{\pi}{2}\,c\,\mu^{1/2}\,\sigma_{r,{\rm d}}^{2}\,,$ (28)

independent of $\alpha$. Assuming that angular momentum is conserved during
the collapse of the clump, one obtains (as in eq. 19 of Ceverino et al., 2012)
a second relation between clump and disc,

$V_{\rm rot,c}^{2}=\frac{\pi^{2}}{32}\,(1+\alpha)\,\mu\,c^{2}\sigma_{r,{\rm
d}}^{2}\,.$ (29)

Using eqs. (28) and (29), we obtain within the clump

$V_{\rm rot,c}^{2}=\frac{\pi}{16}\,(1+\alpha)\,c\,\mu^{1/2}\,V_{\rm
c}^{2}\simeq 0.2\,c\,\mu^{1/2}\,V_{\rm c}^{2}\,,$ (30)

where the second equality is for a flat rotation curve, $\alpha\\!=\\!0$. Note
that full rotation support, $V_{\rm rot,c}\\!=\\!V_{\rm c}$, is obtained for a
maximum contraction factor $c\\!=\\!5\,\mu^{-1/2}$. Inserting eq. (30) in eq.
(27) we obtain

$\frac{\sigma_{\rm c}^{2}}{V_{\rm c}^{2}}\simeq 0.5-0.1\,c\,\mu^{1/2}\equiv
0.5\,f_{\sigma}\,.$ (31)

The factor $f_{\sigma}$ defined here can range from $f_{\sigma}\\!=\\!1$ in
the case of no rotation to $f_{\sigma}\\!=\\!0.58$ when angular-momentum is
conserved during clump collapse, with the fiducial values $c\\!=\\!3$ and
$\mu\\!=\\!0.5$. This gives

$\sigma_{\rm c}\simeq 35\,{\rm km}\,{\rm s}^{-1}\,f_{\sigma}^{1/2}\,V_{\rm
c,50}\,.$ (32)

For $V_{\rm c}\\!=\\!50\,{\rm km}\,{\rm s}^{-1}$, a value of $\sigma_{\rm
c}\\!\simeq\\!35\,{\rm km}\,{\rm s}^{-1}$ is expected if there is no angular
momentum, and $\sigma_{\rm c}\\!\simeq\\!27\,{\rm km}\,{\rm s}^{-1}$ when
angular momentum is conserved, so we denote $\sigma_{\rm c}\\!=\\!30\,{\rm
km}\,{\rm s}^{-1}\,\sigma_{30}$.

###  3.4 Dissipation of Turbulence

Eq. (31) allows us to express $E_{\rm dis}$, the dissipative loss of the
turbulence per unit mass, as a function of $V_{\rm c}$. Assuming that the
timescale for turbulence decay is $\gamma\,t_{\rm ff}$, where $\gamma$ is a
factor of order unity, the dissipated energy by time $t$ is

$E_{\rm dis}=\frac{3}{2}\sigma_{\rm
c}^{2}\,\gamma^{-1}\,\tau=\frac{3}{4}f_{\sigma}\,\gamma^{-1}\,V_{\rm
c}^{2}\,\tau\,,$ (33)

where the turbulence is assumed to be isotropic with $\sigma_{\rm c}$ the one-
dimensional velocity dispersion. The corresponding dimensionless factor in eq.
(9) is

$f_{\rm dis}=1.5\,f_{\sigma}\,\gamma^{-1}\,.$ (34)

A comparison to eq. (3) indicates that $E_{\rm dis}$ becomes comparable to
$E_{\rm vir}$ at $\tau$ of order unity or a few, when $E_{\rm out}$ in eq.
(17) is still much smaller.

##  4 Sources: supernovae and gravity

Complementing the previous section, we evaluate in this section the energy
input to the clump gas by supernovae and by gravity.

###  4.1 Energy deposit by Supernova Feedback

Here, we evaluate $E_{\rm sn}(t)$, the energy per unit mass deposited in the
clump gas by supernovae bubbles till time $t$. While the feedback also
contains contributions from stellar winds and radiative pressure, we
tentatively limit the analysis here to the energy deposited in the ISM by
supernovae. The calculation is inspired by Dekel & Silk (1986), following the
revised treatment of supernova bubbles in Dekel et al. (2019) and ongoing work
(Tziperman, Sarkar, Dekel, in preparation). The energy is evaluated here as
the sum of the energies deposited by individual supernova bubbles. We assume a
constant SFR during the time relevant for clump evolution, motivated by Dekel
et al. (2021). The calculation is simplified by the realization that the
relevant time of one or more free-fall times is much larger than the fading
time of each bubble, $t_{\rm fade}$, and that overlap of active bubbles
younger than $t_{\rm fade}$ is negligible as long as they occur in random
positions within the clump. An alternative calculation could address super-
bubbles generated by supernovae that are clustered in subclumps, e.g., in the
spirit of Gentry et al. (2017) and Dekel et al. (2019).

####  4.1.1 An individual supernova bubble

According to the standard model for the evolution of a supernova bubble in a
uniform medium, the first important stage is the Sedov-Taylor adiabatic phase,
during which the shock radius grows as $R\\!\propto\\!t^{2/5}$ and its
velocity slows down as $V\\!\propto\\!t^{-3/5}$. The end of this stage is
commonly defined at the cooling time, $t_{\rm cool}$, when the bubble has lost
one third of its initial energy to radiation333This is when the radiated
energy is comparable to the kinetic energy and to the thermal energy, each
contributing one third of the supernova energy at $t_{\rm cool}$ (Kim &
Ostriker, 2015; Sarkar, Gnat & Sternberg, 2021, Figure A1).. For an assumed
cooling rate (Dekel et al., 2019, eq. 9), which is a fair approximation for a
solar-metallicity gas at temperatures in the range $10^{5-7.3}K$, the cooling
time is estimated to be (Dekel et al., 2019, eq. 12)

$t_{\rm cool}\simeq 3.95\,{\rm kyr}\,n_{2}^{-0.55}e_{51}^{0.22}\,,$ (35)

where $e_{51}$ is the initial supernova energy in units of $10^{51}\,{\rm
erg}$. The shell velocity and radius just prior to the cooling time are
$V_{\rm cool}\\!\simeq\\!340\,{\rm km}\,{\rm
s}^{-1}\,n_{2}^{0.13}\,e_{51}^{0.066}$ and $R_{\rm cool}\\!\simeq\\!3.53\,{\rm
pc}\,n_{2}^{-0.42}\,e_{51}^{0.29}$.

The second stage is the snow-plow phase, where a dense shell of cold gas is
mainly pushed by the pressure of the enclosed hot central volume. The shell
mass increases as it sweeps up the ambient gas, and it slows down as
$R\\!\propto\\!t^{2/7}$ and $V\\!\propto\\!t^{-5/7}$. The velocity of the
shock radius in the snow-plow phase is $V_{\rm s}\\!=\\!(2/7)\,R_{\rm cool}/t$
(note the discontinuity in the velocity at $t_{\rm cool}$, e.g., Sarkar, Gnat
& Sternberg, 2021). In the snow-plow phase, the total energy in the bubble is
commonly assumed to decline as $\propto t^{-4/7}$, but we adopt a slightly
steeper decline of $\propto t^{-0.7}$, based on more involved theoretical
arguments and spherical simulations (Dekel et al., 2019).

The bubble is regarded as faded away at time $t_{\rm fade}$, when its outward
velocity becomes comparable to the velocity dispersion outside it,
$\sigma_{\rm c}$. The energy is deposited until that time, and soon after the
bubble melts into the general clump medium. In analogy to eq. 18 of Dekel et
al. (2019), the fading time is evaluated to be

$t_{\rm fade}=73\,{\rm
kyr}\,n_{2}^{-0.37}\,e_{51}^{0.32}\,\sigma_{30}^{-7/5}\,,$ (36)

where $\sigma_{30}=30\,{\rm km}\,{\rm s}^{-1}\,\sigma_{\rm c}$. This fading
time is significantly smaller than the clump free-fall time of $\sim 5\,{\rm
Myr}$ (eq. 15), over which the clump evolves. The shell radius at that time is

$R_{\rm fade}=8.1\,{\rm
pc}\,n_{2}^{-0.37}\,e_{51}^{0.32}\,\sigma_{30}^{-2/5}\,,$ (37)

which is significantly smaller than the clump radius of $\lower
2.58333pt\hbox{$\;\buildrel>\over{\sim}\;$}\\!200\,{\rm pc}$. The energy
deposited in the medium by a single bubble until $t_{\rm fade}$ is

$e_{\rm fade}\simeq 8.7\times 10^{49}\,{\rm
erg}\,n_{2}^{-0.13}\,e_{51}^{0.93}\,\sigma_{30}^{0.98}\,.$ (38)

####  4.1.2 Multiple supernova bubbles

Following Dekel & Silk (1986), we write the total deposited energy by time $t$
as

$M_{\rm g}\,E_{\rm sn}(t)=\int_{0}^{N(t)}e(t-t_{*})\,dN(t_{*})\,,$ (39)

where $e(t^{\prime})$ is the energy deposited by a single supernova from its
birth to time $t^{\prime}$, and $N(t)$ is the cumulative number of supernovae
by time $t$. Assuming a constant SFR during the early phases of the clump
lifetime,

$N(t)=\nu\,\dot{M}_{\rm sf}\,t\,,$ (40)

with $\nu=0.01M_{\odot}^{-1}\nu_{.01}$ the number of supernovae per solar mass
of forming stars, determined by the stellar initial mass function. A change of
variables gives

$M_{\rm g}\,E_{\rm sn}(t)=\nu\,\dot{M}_{\rm
sf}\,\int_{0}^{t}e(t^{\prime})\,dt^{\prime}\,.$ (41)

For $t\\!\gg\\!t_{\rm fade}$, this simplifies to

$M_{\rm g}\,E_{\rm sn}(t)=e_{\rm fade}\,\nu\,\dot{M}_{\rm sf}\,t\,.$ (42)

Expressing the SFR via eq. (14) we obtain

$E_{\rm sn}(t)=\frac{1}{2}(51\,{\rm km}\,{\rm
s}^{-1})^{2}\,n_{2}^{-0.13}\,\nu_{.01}\,\epsilon_{.03}\,e_{51}^{0.93}\,\sigma_{30}^{0.98}\,\tau.$
(43)

Expressing $\sigma_{\rm c}$ by $V_{\rm c}$ using eq. (31) and eq. (32) we
obtain

$E_{\rm sn}(t)=\frac{1}{2}(55\,{\rm km}\,{\rm
s}^{-1})^{2}n_{2}^{-0.13}\nu_{.01}\,\epsilon_{.03}e_{51}^{0.93}f_{\sigma}^{0.49}V_{\rm
c,50}^{0.98}\tau.$ (44)

The corresponding dimensionless factor in eq. (8), ignoring the weak $n$
dependence and rounding the power $0.98$ to $1$, is

$f_{\rm
sn}=1.22\,\nu_{.01}\,\epsilon_{.03}\,e_{51}^{0.93}\,f_{\sigma}^{0.49}\,.$ (45)

The characteristic velocity for supernova feedback, $V_{\rm sn}$, is defined
by $E_{\rm sn}\\!=\\!0.5\,V_{\rm sn}^{2}$. For the fiducial values of the
parameters it is $V_{\rm sn}\\!\sim\\!50\,{\rm km}\,{\rm s}^{-1}\,\tau^{1/2}$.

Comparing to eq. (33), we find that $E_{\rm sn}(t)$ is comparable to $E_{\rm
dis}(t)$ (if $V_{\rm c,50}\\!\sim\\!1$), or larger (if $V_{\rm c,50}$ is
significantly smaller than unity). They are always larger than $E_{\rm
out}(t)$ of eq. (17). They both become comparable to $E_{\rm vir}$ of eq. (3)
at $t\\!\sim\\!t_{\rm ff}$. Note that $E_{\rm sn}$ is roughly proportional to
$V_{\rm c}$, while the other energies scale with $V_{\rm c}^{2}$.

In order to test the validity of neglecting bubble overlap, we estimate the
fraction of the clump volume that is occupied by active bubbles at a given
time. By inserting $t_{\rm fade}\\!\sim\\!70\,{\rm kyr}$ from eq. (36) in eq.
(40), with the fiducial values for $\nu$ and $\epsilon$, and using $R_{\rm
fade}\\!\sim\\!8\,{\rm pc}$ from eq. (37), the volume in active bubbles is
estimated to be $\sim\\!10^{6}\,{\rm pc}^{3}$. This is while the clump volume,
assuming a radius of $R_{\rm c}\\!\sim\\!300\,{\rm pc}$, is
$\sim\\!10^{8}\,{\rm pc}^{3}$. This implies that only a small fraction of the
clump is occupied by active bubbles, which justifies the neglect of bubble
overlap as long as the supernovae are spread over the bubble volume.

###  4.2 Gravitational Driving of Turbulence

During the formation of the clump, in the first one to a few free-fall times,
the contraction by a factor $c$ is associated with a deepening of the
potential from $-GM_{\rm c}/(cR_{\rm c})$ to $-GM_{\rm c}/R_{\rm c}$, namely a
gravitational energy gain of

$E_{\rm grav}=(1-c^{-1})\,V_{\rm c}^{2}\,.$ (46)

This is comparable to $E_{\rm vir}$, and to $E_{\rm dis}$ and $E_{\rm sn}$ at
$t\\!\sim\\!t_{\rm ff}$. Thus, at $\tau\\!\sim\\!1$, we expect in eq. (8)
$f_{\rm grav}\\!=\\!2\,(1-c^{-1})/\tau$, of order unity.

At later times, we expect the gravitational energy input rate to become
smaller, such that $E_{\rm grav}(t)$ becomes smaller than $E_{\rm dis}(t)$ and
$E_{\rm sn}(t)$. This implies that, with $E_{\rm grav}\\!=\\!f_{\rm
grav}\,E_{\rm vir}\,\tau$, we expect $f_{\rm grav}\\!\sim\\!1$ at
$\tau\\!\sim\\!1$, and $f_{\rm grav}\\!<\\!1$ later. It could in general be
much smaller than unity, except during episodes of mergers, intense accretion,
or excessive tidal effects and shear, when it could grow to $f_{\rm
grav}\\!\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!1$.

To get an estimate for $f_{\rm grav}$ long after $\tau\\!=\\!1$, we appeal to
gravitational clump encounters. For discs that are self-regulated at marginal
Toomre instability, Dekel, Sari & Ceverino (2009) estimated the timescale for
stirring turbulence by gravitational encounters between clumps to be on the
order of the disc dynamical time $t_{\rm d}$ or larger (their eq. 18 in §3).
This is the timescale for the encounters to generate a specific energy change
of order $\sigma_{r,{\rm d}}^{2}$, and it is estimated to be

$t_{\rm enc}\simeq 2\,\beta_{.2}^{-1}\,Q_{.67}^{4}\,t_{\rm d}\sim
10\,c_{3}^{3/2}\,t_{\rm ff}\,.$ (47)

Here $\beta\\!\sim\\!0.2$ is the instantaneous disc mass fraction in clumps,
and $Q\\!\simeq\\!0.67$ is the value of the Toomre parameter for marginal
instability in a thick disc. The second equality stems from the clump free-
fall time being $t_{\rm ff}\\!\sim\\!c^{-3/2}t_{\rm d}$. This indicates that
at $\tau$ significantly larger than unity, typically,

$f_{\rm grav}\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1\,c_{3}^{-3/2}\,,$ (48)

namely significantly smaller than unity, on the order of $0.1$. The value of
$f_{\rm grav}$ could be larger during episodes of excessive gravitational
interactions. This is especially true during clump mergers, and possibly also
in episodes of intense accretion, strong tidal effects and shear. While only a
fraction of the clump encounters lead to actual mergers, $f_{\rm grav}$ inside
the merging clumps is expected to be larger than estimated in eq. (48).

Based on eq. (46) and eq. (48), we crudely adopt

$f_{\rm grav}\sim\begin{cases}2(1-c^{-1})\ \ {\rm if}\ \tau\sim 1\\\ 0.3\ \ \
\ \ {\rm if}\ \tau>1,\ {\rm for\ low\ }c\\\ 0.1\ \ \ \ \ {\rm if}\ \tau>1,\
{\rm for\ high\ }c\,.\\\ \end{cases}$ (49)

##  5 Non-virialized clumps

We present here a modification to $S$ of eq. (1), that is especially relevant
for the S clumps that disrupt on a free-fall timescale. Such clumps may not
have reached an equilibrium before being disrupted, such that the binding
energy per unit mass in them is smaller than $E_{\rm vir}$ at virial
equilibrium. This introduces corrections to the expressions in the previous
sections, in particular the survivability parameter $S$. The binding energy
per unit mass, the gravitational energy gained during the collapse of the
clump and the energy needed for disrupting the clump, can be written as

$E_{\rm b}=\frac{GM_{\rm c}}{R_{\rm c}}-\frac{GM_{\rm c}}{R_{\rm
c,i}}=(1-p)V_{\rm c}^{2}\,,$ (50)

where $V_{\rm c}^{2}=GM_{\rm c}/R_{\rm c}$, with $V_{\rm c}$ and $R_{\rm c}$
the clump circular velocity and radius but not necessarily their virial
values. The factor $p$ should be $p\\!=\\!c^{-1}$ where $c=R_{\rm c,i}/R_{\rm
c}$, but we note that for $p\\!=\\!0.5$ the binding energy equals the virial
energy, and we are back to the analysis of virialized clumps. We therefore
define

$p=\begin{cases}c^{-1}\ \ {\rm if}\ c\\!\leq\\!2\\\ 0.5\ \ \ {\rm if}\
c\\!>\\!2\,.\\\ \end{cases}$ (51)

We note that $R_{\rm c}$ and $V_{\rm c}$ relate to their virial analogs as
$R_{\rm c}\\!=\\!2pR_{\rm c,v}$ and $V_{\rm c}^{2}\\!=\\!(2p)^{-1}V_{\rm
c,v}^{2}$.

The factor $p$ could be interpreted as the fraction of $M_{\rm c}$ that is
left bound after the instantaneous mass removal. This is because, if we equate
the energy change by the removal,

$\Delta E=\frac{GM_{\rm c}}{R_{\rm c}}-\frac{pGM_{\rm c}}{R_{\rm
c}}=(1-p)V_{\rm c}^{2}\,,$ (52)

with $E_{\rm b}$ of eq. (50), we see that this $p$ is the same as the $p$ in
eq. (51). This implies that in order to unbind the clump, one needs to remove
a fraction $(1-p)$ of its mass. We therefore generalize eq. (1) to

$S=\frac{(1-p)\,M_{\rm c}}{M_{\rm out}}\,.$ (53)

Now eq. (5) remains the same but with $E_{\rm vir}$ replaced by $E_{\rm b}$,
namely $V_{\rm c,v}^{2}$ replaced by $V_{\rm c}^{2}\\!=\\!(2p)^{-1}V_{\rm
c,v}^{2}$. Accordingly, as in eq. (6), we assume escape with $V_{\rm
out}^{2}\\!=\\!2V_{\rm c}^{2}$. Then, in eq. (7), the expression for $S$
remains the same, and $E_{\rm vir}$ is replaced by $E_{\rm b}$ in the
expression for $S^{\prime}$. Defining the “$f$” factors as in eq. (8) and eq.
(9), the expression for $S$, replacing $S^{\prime}$ of eq. (10), becomes

$S=\frac{f_{\rm g,c}^{-1}}{S^{\prime-1}-1},\ \
S^{\prime}=\frac{2(1-p)}{(f_{\rm sn}V_{\rm c,50}^{-1}+f_{\rm grav}-f_{\rm
dis})\tau}\,.$ (54)

The only difference is the factor $2(1-p)$ in $S^{\prime}$. One should also
note that $V_{\rm c}$ here is the circular velocity of the clump as is, even
when it is not in virial equilibrium.

We next evaluate the potential modifications to the different energies in
terms of the clump parameters. The expression for $f_{\rm grav}$ remains the
same as in eq. (49). The expressions for $E_{\rm dis}$ and $E_{\rm sn}$, in
eq. (33) and eq. (43), depend on $\sigma_{\rm c}$. In order to relate
$\sigma_{\rm c}$ to $V_{\rm c}$, we used eq. (31), derived assuming Jeans
equilibrium (eq. 27), clump properties in Toomre instability with $Q\\!=\\!1$
(eq. 28), and conservation of angular momentum (eq. 29). If the clump is not
in equilibrium, the relation between $\sigma_{\rm c}$ and $V_{\rm c}$ becomes
uncertain. We found in Ceverino et al. (2012) that the final value of the
clump velocity dispersion, $\sigma_{\rm c,v}$, is comparable to
$\sigma_{r,{\rm d}}$ of the disc. We thus make the assumption that
$\sigma_{\rm c}$ is constant during the clump collapse. As in eq. (31), we
parameterize the relation between $\sigma_{\rm c}$ and $V_{\rm c}$ in terms of
the parameter $f_{\sigma}$, of order unity, $\sigma_{\rm c}^{2}/V_{\rm
c}^{2}\\!\equiv\\!0.5f_{\sigma}$. Then, the expressions for $E_{\rm dis}$ and
$E_{\rm sn}$ remain as in eq. (33) and eq. (44), and so are $f_{\rm dis}$ and
$f_{\rm sn}$ in eq. (34) and eq. (45). The value of $f_{\sigma}$ can be
determined either from $\sigma_{\rm c}\\!=\\!\sigma_{\rm c,v}$ (version 1) or
from $\sigma_{\rm c}\\!=\\!\sigma_{r,{\rm d}}$ (version 2), with comparable
results for typical clumps. Based on eq. (31) and eq. (28), we can have,
respectively,

$f_{\sigma}=\begin{cases}1-0.2\,c\,\mu^{1/2}\,,\ \ \sigma_{\rm
c}\\!=\\!\sigma_{\rm c,v}\\\ (4/\pi)\,c^{-1}\mu^{-1/2}\,,\ \ \sigma_{\rm
c}\\!=\\!\sigma_{r,{\rm d}}\,.\end{cases}$ (55)

We adopt hereafter version 2, $\sigma_{\rm c}\\!=\\!\sigma_{r,{\rm d}}$, with
version 1 yielding similar results.

##  6 Survivability of three clump types

Based on the previous sections, the “$f$” factors that enter $S^{\prime}$ in
eq. (10) or eq. (54) can be summarized as

$f_{\rm sn}\simeq
1.22\,\nu_{.01}\,\epsilon_{.03}\,e_{51}^{.93}\,f_{\sigma}^{.49}\ ,$ (56)
$f_{\rm dis}\simeq 1.5\,f_{\sigma}\,\gamma^{-1}\sim 1\,,$ (57) $f_{\rm
grav}\sim\begin{cases}2(1-c^{-1})\ \ {\rm if}\ \tau\sim 1\\\ 0.3\ \ \ \ \ {\rm
if}\ \tau>1,\ {\rm for\ low\ }c\\\ 0.1\ \ \ \ \ {\rm if}\ \tau>1,\ {\rm for\
high\ }c\,.\\\ \end{cases}$ (58)

The $V_{\rm c}$ that enters the first term in eq. (10) is given by eq. (26),
and $f_{\sigma}$ is given by eq. (55). For completeness,

$f_{\rm out}\simeq 0.06\,\epsilon_{.03}\,\eta\,.$ (59)

We note that for $V_{\rm c}\\!\simeq\\!50\,{\rm km}\,{\rm s}^{-1}$ and $f_{\rm
sn}\\!\simeq\\!1$, we have $E_{\rm sn}(t)\\!\simeq\\!E_{\rm dis}(t)$, and they
are both comparable to $E_{\rm vir}$ at $t\\!\simeq\\!t_{\rm ff}$, while
$E_{\rm out}(t)\\!\simeq\\!E_{\rm dis}(t)/16$ at any time.

###  6.1 Short-lived versus long-lived clumps

At the end of the formation of the clump, near $\tau\\!\sim\\!1$, we expect
$f_{\rm grav}\\!\sim\\!1$ based on eq. (46). Then, for the fiducial $f_{\rm
dis}\\!\sim\\!1$, eq. (10) simplifies to

$S^{\prime}\\!\simeq\\!\frac{V_{\rm c,50}}{f_{\rm sn}\,\tau}\,.$ (60)

This is to be multiplied by $2(1-p)$ if correcting for non-virialized clumps
as in eq. (54). This implies that the clump survival is mainly determined by
the competition between the energy deposited by feedback and the binding
energy of the clump. For moderate feedback strength, $f_{\rm sn}\\!\sim\\!1$,
and for massive clumps of fairly deep potential wells, $V_{\rm
c,50}\\!\sim\\!1$, we obtain $S^{\prime}\\!\simeq\\!1$. For $f_{\rm
g,c}\\!\sim\\!1$ this yields $S\\!\gg\\!1$. These clumps are thus expected to
keep most of their gas during the early phase of clump evolution. We term
these long-lived clumps (L). If, on the other hand, $f_{\rm sn}V_{\rm
c,50}^{-1}\\!>\\!2$, namely the feedback is stronger and/or the clump $V_{\rm
c}$ is lower, we get $S^{\prime}\\!<\\!0.5$, which implies $S\\!<\\!1$. In
this case $\tau S$ is also smaller than unity at the early times, so these
clumps are expected to disrupt on a free-fall timescale. We term these short-
lived clumps (S)

The conditions for L clumps can arise from a large $V_{\rm c}$ with the
fiducial values for supernova feedback, or from a weaker feedback. Based on
eq. (26), a large $V_{\rm c}$ can be obtained for either a massive disc with a
large $V_{\rm d}$ or a large-$\delta$ disc due to a high gas fraction or a
low-mass bulge and low central dark-matter mass. A high clump contraction
factor and a clump mass that is comparable to the Toomre mass can also
contribute to a large $V_{\rm c}$, but in a weaker way. Based on eq. (5), a
low $f_{\rm g,c}$ will increase $S$ and help the survivability. On the other
hand, the conditions for S clumps can be due to a smaller $V_{\rm c}$, namely
lower-mass clumps, or to stronger feedback that yields a larger $f_{\rm sn}$.

###  6.2 Two types of long-lived clumps

Among the L clumps that keep most of their gas until after the first few free-
fall times, we envision two sub-types. Certain L clumps, to be termed LL
clumps, may lose only a small fraction of their gas mass to outflows for tens
of free-fall times, keeping a non-negligible gas mass and SFR until late in
their inward migration at $\tau\sim 50\\!-\\!100$. Other L clumps, to be
termed LS clumps, may lose most of their gas mass within the first few or
$\sim$ten free-fall times, but unlike the S clumps that lose their gas more
rapidly, the LS clumps manage to form stellar components and keep them bound
for long lifetimes, showing only little gas and low SFR. These are the clumps
that, despite having $S$ drop to below unity by $\tau\\!\sim\\!10$, have $\tau
S\\!>\\!1$ in eq. (2), indicating that they do not fully disrupt due to the
rather slow outflow rate. The properties of the LS clumps are expected to be
in certain ways between those of the S clumps and the LL clumps. The existence
of LS clumps was hinted in simulations already in Mandelker et al. (2017),
where they were not analyzed in as much detail as the S and LL clumps. In the
simulation results described below we analyze this population of clumps as
well. Next, we try to consider a possible way to understand the origin of this
intermediate population of LS clumps.

A while after $\tau\\!\sim\\!1$, for L clumps that have survived the early
phase of clump formation, one may inquire what values of the parameters in eq.
(10) may distinguish between LS and LL clumps. It seems that the value of
$f_{\rm sn}V_{\rm c,50}^{-1}$ has a crucial role. Lets assume $f_{\rm
dis}\\!\sim\\!1$. Since at $\tau$ of a few or greater we expect $f_{\rm grav}$
to become smaller than unity, $S^{\prime}$ in eq. (10) is larger than in eq.
(60), which was roughly valid at $\tau\\!\sim\\!1$. If $f_{\rm
grav}\\!\ll\\!1$, and $f_{\rm sn}V_{\rm c,50}^{-1}\\!\sim\\!f_{\rm
dis}\\!\sim\\!1$, then in eq. (10) $S^{\prime}$ is close to unity and in eq.
(11) $\tau_{(S\\!=\\!1)}\\!\gg\\!1$. At $\tau$ of a few this corresponds to
$S\\!\gg\\!1$, namely an LL clump. If, on the other hand, $f_{\rm sn}V_{\rm
c,50}^{-1}$ is larger than unity, and $f_{\rm grav}$ is not too small, we
expect $S^{\prime}$ to decline to $\sim 1/\tau$, so as long as $f_{\rm g,c}$
has not dropped yet by a large factor (e.g. if it is declining slower than
$1/\tau$), we obtain $S\\!<\\!1$, namely significant mass loss. This would be
a non-disrupted LS clump with $\tau S\\!>\\!1$ because roughly $\tau S\propto
f_{\rm g,c}^{-1}$, a growing function of time.

For a numerical example, if the LS clumps have $f_{\rm sn}V_{\rm
c,50}^{-1}\\!\simeq\\!2$, $f_{\rm grav}\\!\simeq\\!0.3$ and $f_{\rm
dis}\\!\simeq\\!1.3$ (because of the higher $f_{\sigma}$), with $f_{\rm
g,c}\sim 0.67$, we get $S\\!\simeq\\!1.5/(\tau-1)$, which would cross the
critical value at $\tau_{(S\\!=\\!1)}\\!\simeq\\!2.5$. This is compared to
much larger values of $S$ and $\tau_{(S\\!=\\!1)}$ for LL clumps, where
typical values may be $f_{\rm sn}V_{\rm c,50}^{-1}\\!\simeq\\!1$, $f_{\rm
dis}\\!\simeq\\!1$ and $f_{\rm grav}\\!\ll\\!1$. The bimodality is emphasized
by the vanishing values obtained in the denominator of eq. (7) for LL clumps,
which could yield large values of $S^{\prime}$ near unity and therefore very
large values of $S$. For $\tau$ significantly larger than unity, these clumps
have $\tau S\\!>\\!1$, namely no total disruption, as expected for LS clumps.

Motivated by the simulation results described below, it is possible that the
distinction between LS and LL clumps may be associated with a difference in
collapse factor, with the LS clumps contracting less than the LL clumps. For a
given mass, a smaller $c$ would correspond to a larger $R_{\rm c}$ and
therefore a smaller $V_{\rm c}$ for LS clumps. Based on eq. (30), this will be
associated with a weaker rotation support. The larger $R_{\rm c}$ in the LS
clumps may be associated with a larger $f_{\rm grav}$ due to more intense
clump mergers and stronger tidal effects. These may increase the SFR
efficiency $\epsilon$ (e.g., via shocks), and thus increase $f_{\rm sn}$. The
combination of a larger $f_{\rm sn}$, a smaller $V_{\rm c}$ and a larger
$f_{\rm grav}$ can significantly lower $S^{\prime}$ in eq. (10) for LS
compared to LL clumps, thus generating a bimodal distribution of L clumps.
This possible explanation will be tested in the simulations.

The distinction between the LS clumps and S clumps could be addressed in terms
of $\tau_{(\tau S\\!=\\!1)}$, given in eq. (12). Assuming for simplicity
$f_{\rm grav}\\!=\\!f_{\rm dis}\\!=\\!1$, namely eq. (60), and $f_{\rm
g,c}\\!=\\!1$, we obtain

$\tau_{(\tau S\\!=\\!1)}\\!\sim\\![f_{\rm sn}\,V_{\rm c,50}^{-1}-1]^{-1}\,.$
(61)

For $f_{\rm sn}\,V_{\rm c,50}^{-1}$ significantly larger than unity, we get a
small $\tau_{(\tau S\\!=\\!1)}$, namely an S clump. For $f_{\rm sn}\,V_{\rm
c,50}^{-1}$ only slightly above unity, we get $\tau_{(\tau
S\\!=\\!1)}\\!\gg\\!1$, namely an LS clump.

##  7 Thresholds for clump survival

We now analyze the distinction between S and L clumps in terms of the implied
thresholds for the relevant physical quantities. Appealing to the critical
value $S\\!\sim\\!1$ at $\tau\\!\sim\\!1$, we investigate the critical
condition $S^{\prime}\\!\sim\\!V_{\rm c,50}/(f_{\rm sn}\,\tau)\\!\sim\\!0.5$,
based on eq. (60), in terms of the relevant disc and clump properties. These
predictions will be compared to simulations in the following sections.

###  7.1 Galaxy velocity and mass

Assuming tentatively a moderate feedback, $f_{\rm sn}\\!\simeq\\!1$, one of
the strong dependences of $V_{\rm c}$ in eq. (26) is on $V_{\rm d}$, meaning
that L clumps below and near the Toomre mass are expected in galaxies above a
threshold disc rotation velocity. With the fiducial values of
$c_{3}\\!\sim\\!\delta_{.2}\\!\sim\\!\mu_{.5}\\!\sim\\!1$ in eq. (26) and eq.
(60), we typically expect L clumps of $V_{\rm c}\\!>\\!25\,{\rm km}\,{\rm
s}^{-1}$ in discs of $V_{\rm d}\\!>\\!100\,{\rm km}\,{\rm s}^{-1}$. Note that
these velocities could double if $f_{\rm sn}\\!\sim\\!2$ or if $S$ is
considered at $\tau\\!\sim\\!2$, implying that our model predictions should be
considered as semi-qualitative estimates only.

We recall that discs tend to be long lived, not disrupted by mergers in an
orbital time, when they reside in halos more massive than a threshold mass of
$M_{\rm v}\\!\sim\\!2\\!\times\\!10^{11}M_{\odot}$ (Dekel et al., 2020a). This
is also the mass range where extended rings survive instability-induced inward
mass transport due to a massive central bulge (or a central cusp of dark
matter) (Dekel et al., 2020b). Such a bulge forms by a wet compaction event
(Zolotov et al., 2015), which typically occurs near a similar critical mass
(Tomassetti et al., 2016). This mass is slightly smaller but in the ball park
of the “golden mass” of most effective galaxy formation (Dekel, Lapiner &
Dubois, 2019).

Above this critical mass, the disc is supported by rotation, so $V_{\rm d}$
can be assumed to be comparable to and slightly larger than the halo virial
velocity $V_{\rm v}$, e.g., $V_{\rm d}\\!\simeq\\!1.5\,V_{\rm v}$. Then
$V_{\rm d}$ that enters eq. (26) can be related to the halo mass via the halo
virial relation (e.g., Dekel & Birnboim, 2006, appendix),

$M_{{\rm v},11.3}\simeq V_{{\rm v},120}^{3}(1+z)_{3}^{-3/2}\simeq
1.37\,V_{{\rm d},200}^{3}(1+z)_{3}^{-3/2}\,.$ (62)

Here, the virial quantities $M_{\rm v}$ and $V_{\rm v}$ are measured in units
of $10^{11.3}M_{\odot}$ and $120\,{\rm km}\,{\rm s}^{-1}$ respectively, and
$(1+z)$ is normalized to $3$ ($z\\!=\\!2$). At $z\\!\sim\\!2$, the typical
stellar mass is related to the halo mass as

$M_{\rm s}\simeq 2\times 10^{9}M_{\odot}\,M_{{\rm v},11.3}\,f_{\rm sv}\,,$
(63)

with $f_{\rm sv}\\!\simeq\\!1$ assumed hereafter (Moster, Naab & White, 2018;
Behroozi et al., 2019, from galaxy-halo abundance matching). The corresponding
disc gas mass is

$M_{\rm d}\\!\sim\\!2\times 10^{9}M_{\odot}\,M_{{\rm v},11.3}\,f_{\rm gs}\,,$
(64)

where $f_{\rm gs}\\!\sim\\!1$ is the gas-to-stellar mass ratio, such that the
gas-to-baryonic fraction is $f_{\rm g}\\!=\\!(f_{\rm
gs}^{-1}+1)^{-1}\\!\sim\\!0.5$. Then, from eq. (62),

$V_{{\rm d},200}\simeq 0.9\,M_{{\rm v},11.3}^{1/3}\,(1+z)_{3}^{1/2}\simeq
0.9\,M_{{\rm d},9.3}^{1/3}\,f_{\rm gs}^{-1/3}\,(1+z)_{3}^{1/2}.$ (65)

Thus, for masses above the threshold mass for discs, we expect $V_{\rm
d}\\!>\\!100\,{\rm km}\,{\rm s}^{-1}$ at all redshifts, so
$S^{\prime}\\!>\\!0.5$ in eq. (10), namely L clumps.

In galaxies below the critical mass for discs, the velocity dispersion is
significant, with $V_{\rm d}/\sigma_{r,{\rm d}}\\!\sim\\!1$, such that in
Jeans equilibrium the rotation velocity $V_{\rm d}$ is smaller than $V_{\rm
v}$ by a factor of a few (Dekel et al., 2020a, b; Kretschmer et al., 2021).
This reduces $V_{\rm d}$ further, well beyond the $M_{\rm d}^{1/3}$ dependence
of eq. (65), such that $V_{\rm d}$ drops below $100\,{\rm km}\,{\rm s}^{-1}$
immediately below the threshold mass for discs even at high redshifts. This
brings $S^{\prime}$ to below 0.5, thus not allowing L clumps in galaxies of
$M_{\rm v}\\!<\\!10^{11}M_{\odot}$, namely $M_{\rm d}\\!<\\!10^{9}M_{\odot}$.
Indeed, in the VELA-3 simulations there are hardly any long-lived clumps in
disks of $M_{\rm d}\\!<\\!2\times 10^{9}M_{\odot}$ (Mandelker et al., 2017,
section 6.2.1), associated with $M_{\rm v}\\!<\\!2\times 10^{11}M_{\odot}$
haloes at $z\\!\sim\\!2$.

###  7.2 Clump mass

The clump circular velocity $V_{\rm c}$ that enters $S^{\prime}$ can be
expressed in terms of the clump mass $M_{\rm c}$. Using eq. (65) and eq. (23)
in eq. (26), we obtain

$V_{\rm c,50}\simeq c_{3}^{1/2}\,\mu_{.5}^{-1/12}\,\delta_{.2}^{1/3}\,f_{\rm
gs}^{-1/3}\,(1+z)_{3}^{1/2}\,M_{\rm c,7.5}^{1/3}\,,$ (66)

where $M_{\rm c}=10^{7.5}M_{\odot}\,M_{\rm c,7.5}$. The value of
$\delta/f_{\rm gs}$, both referring to the galaxy gas disc, is expected to be
similar for different galaxies (§7.3), so we are left with no explicit
dependence on the galaxy mass. The dependence on $\mu$ became negligible, so
for a fixed $c$, $V_{\rm c}$, and therefore $S^{\prime}$, is determined by the
absolute value of $M_{\rm c}$. It implies that when sampling clumps in
galaxies of different masses, one expects a threshold for survival at a
critical clump mass, with $V_{\rm c}\\!\sim\\!50\,{\rm km}\,{\rm s}^{-1}$
corresponding to $M_{\rm c}\\!\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!10^{8}M_{\odot}$. On the other
hand, we expect no clear threshold as a function of the clump mass relative to
the disc mass, via $\mu$ or $M_{\rm c}/M_{\rm d}$.

This is indeed comparable to the transition clump mass from short-lived to
long-lived clumps in the cosmological VELA-3 simulations at $M_{\rm
c}\\!\sim\\!10^{8}M_{\odot}$ (Mandelker et al., 2017, Fig. 8), where the
massive galaxies with clumps have $M_{\rm v}\\!\sim\\!(2\\!-\\!9)\times
10^{11}M_{\odot}$ at $z\\!=\\!2$, when most clumps are detected. Indeed, in
these simulations there is only a minor distinction between long-lived and
short-lived clumps as a function of $M_{\rm c}/M_{\rm d}$.

###  7.3 Cold fraction and gas fraction

The clump circular velocity $V_{\rm c}$ of eq. (26) that enters $S^{\prime}$
also depends strongly on the cold fraction $\delta\\!=\\!M_{\rm d}/M_{\rm
tot}$, with a preference for L clumps at higher values of $\delta$.

The threshold in $\delta$ can be translated to a threshold in gas fraction
with respect to the baryons, for given stellar (disc plus bulge) and dark
matter components, and for the typical clumps of a given fraction $\mu$ of the
Toomre mass. We assume four components within the disc radius: a gas disc, a
stellar disc, a bulge and dark matter, and describe the relevant relations
between them by three structure parameters as follows. Let $f_{\rm db}$ be the
ratio of dark-matter to baryonic mass (and adopt a fiducial value of $f_{\rm
db}\\!=\\!1$, but it could vanish in dark-matter-deficient cores). Let $f_{\rm
bd}$ be the stellar bulge to disc ratio (fiducial value $f_{\rm bd}\\!=\\!1$,
but it could vanish in a bulge-less disc). Let $f_{\rm cs}$ (between zero and
unity) be the fraction of the stellar disc that adds to the gas in
constituting the cold disc mass $M_{\rm d}$, which participates in the
instability and therefore enters $\delta$ (fiducial $f_{\rm cs}\\!=\\!0$,
ranging from zero to unity). The gas fraction is then

$f_{\rm g}\frac{M_{\rm g}}{M_{\rm g}+M_{\rm s}}=\frac{(1+f_{\rm db})(1+f_{\rm
bd})\,\delta-f_{\rm cs}}{1+f_{\rm bd}-f_{\rm cs}}\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}0.4\,\delta_{.2}\,,$ (67)

in which the last equality is for the fiducial values $f_{\rm db}\\!=\\!f_{\rm
bd}\\!=\\!1$ and $f_{\rm cs}=0$. We then learn from the condition
$S^{\prime}\\!>\\!0.5$ that, with the other parameters at their fiducial
values, the threshold for clump survival is $f_{\rm g}\\!\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!0.4$, which is in the ball-park
of the observed values at $z\\!\sim\\!2$ (Tacconi et al., 2010; Daddi et al.,
2010; Tacconi et al., 2018).

Relevant possible deviations from the fiducial values of the structure
parameters are likely to decrease the threshold value of $f_{\rm g}$ and thus
lead to increased survival. As a first example, if stars contribute to the
cold disc, say $f_{\rm cs}\\!=\\!0.5$, for a survival threshold at
$\delta\\!=\\!0.2$ the corresponding gas-fraction threshold is at $f_{\rm
g}\\!\sim\\!0.2$. Secondly, a smaller bulge-to-disc ratio, $f_{\rm
bd}\\!<\\!1$, with the other parameters fixed, leaves the threshold at $f_{\rm
g}\\!\simeq\\!0.4\,\delta_{.2}$. Thirdly, for a smaller dark-matter fraction,
$f_{\rm db}\\!<\\!1$, with the other parameters fixed, the threshold gas
fraction becomes smaller. For instance, with no dark matter, the threshold is
at $f_{\rm g}\\!\simeq\\!0.2\,\delta_{.2}$. In these three cases, for a given
$\delta\\!\sim\\!0.2$ threshold for survival, the gas fraction threshold would
be equal or lower than for the fiducial structural cases $f_{\rm cs}\\!=\\!0$
and $f_{\rm dn}\\!=\\!f_{\rm bd}\\!=\\!1$, with a value

$f_{\rm g}\\!\sim\\!(0.2\\!-\\!0.4)\,\delta_{.2}\,.$ (68)

We note that the clump circular velocity $V_{\rm c}$ has an additional
dependence on gas fraction through $V_{\rm d}\\!\propto\\!f_{\rm gs}^{-1/3}$,
which somewhat weakens the overall $f_{\rm g}$ dependence of $V_{\rm c}$ to
roughly $V_{\rm c}\\!\propto\\!f_{\rm g}^{2/3}$.

It is possible that the clump contraction factor $c$ is somewhat larger for a
higher gas fraction in the disc, because more dissipation is permitted. If so,
then the $\delta$ threshold for survival, based on eq. (26), would become
lower by a factor $c_{3}^{-1/2}$, and the $f_{\rm g}$ threshold would decrease
accordingly, thus increasing the clump survivability.

Indeed, a comparable gas-fraction threshold for the survival of massive clumps
at $f_{\rm g}\\!\sim\\!0.3$ has been found in simulations of isolated galaxies
of a given mass and a small bulge (Fensch & Bournaud, 2021). We note that the
$f_{\rm g}$ dependence in eq. (26), stemming from the $\delta$ dependence, is
stronger than the mass dependencies and slightly stronger than the dependence
on the energy per supernovae in $S^{\prime}\\!\propto\\!f_{\rm
sn}^{-1}\\!\propto\\!e^{-0.93}$. This is consistent with the finding of Fensch
& Bournaud (2021).

One can note in eq. (66) that when averaged over galaxies of different masses,
and allowing $M_{\rm c}$ to be at it’s critical value for survival, the
dependence on the gas fraction that stems from the product
$\delta^{1/3}\,f_{{\rm gs}}^{-1/3}$, becomes significantly weaker than in the
discussion above for a fixed $V_{\rm d}$ and a typical clump of a given $\mu$.

###  7.4 Feedback strength

The supernova energy in $S^{\prime}$, via $f_{\rm sn}$, is almost linear with
$e_{51}$. This is about unity for a typical supernova, but the energies from
supernovae that cluster strongly in a star-forming cloud add up in a super-
linear way (Gentry et al., 2017), which may lead to a larger effective value
of $e_{51}$, representing both thermal and kinetic feedback. This would lead
to clump disruption when all other parameters are at their above fiducial
values, including massive clumps of about a Toomre mass.

On top of the supernova feedback, there is feedback from stellar winds and
radiative feedback from massive stars, which may or may not be boosted by
infrared photon trapping at high gas densities. These additional feedback
mechanisms should be modeled in detail. Here, for very qualitative results, we
tentatively model them very crudely by an increase in the energy per supernova
$e_{51}$. This would translate to a value of $f_{\rm sn}$ somewhat above
unity.

It turns out that in a cosmological simulation suite, VELA-6, similar to
VELA-3 but with a stronger feedback, even the clumps more massive than
$\sim\\!10^{8}M_{\odot}$ tend to disrupt after one to several disc dynamical
times (see below, and Ceverino et al. in preparation), while clumps of similar
masses tend to survive in VELA-3 for many free-fall times (Mandelker et al.,
2017). The feedback in VELA-3 included thermal feedback from supernovae,
stellar winds, and radiative feedback from massive stars with no photon
trapping, with a total injected momentum of $\sim\\!3\,L/c$. In comparison,
the feedback in VELA-6 includes additional kinetic feedback from supernovae to
represent strongly clustered supernovae, and stronger radiative feedback form
massive stars due to infrared photon trapping at high gas densities. In our
current simplistic model based on energetics, the associated enhanced feedback
strength may very crudely be referred to as an increase in the effective
$e_{51}$, bringing it to a value of a few, thus increasing the disruptive term
$f_{\rm sn}$ that enters $S^{\prime}$ in eq. (7). The same may be
qualitatively true for other analyses and simulations that incorporated very
strong winds or enhanced radiative feedback due to strong IR trapping, at a
momentum level of tens of $L/c$ (Murray, Quataert & Thompson, 2010; Genel et
al., 2012; Hopkins, Quataert & Murray, 2012b; Oklopčić et al., 2017), in which
all clumps were indeed disrupted on a few free-all timescales. Some of these
simulations also had low gas fractions, which contributed to the poor
survivability as discussed in §7.3. We note that several arguments have been
put forward against the plausibility of such extremely strong radiative
feedback (Krumholz & Dekel, 2010; Krumholz & Thompson, 2012, 2013; Dekel &
Krumholz, 2013).

###  7.5 Star-formation efficiency

The disruptive factor $f_{\rm sn}$ is linear with $\epsilon$, the SFR
efficiency per free-fall time in the star-forming region. This efficiency is
assumed to have a given fixed value under normal conditions, at the level of a
few percent, based on observations and theory (e.g., Krumholz, Dekel & McKee,
2012). The actual value for our giant clumps would depend on whether the star-
forming region is regarded as the whole clump or smaller and denser sub-clumps
that may cluster together to form the giant clump or be fragments of the giant
clump. A value of $\epsilon\\!\sim\\!0.1$, which may be valid during mergers
and strong tidal effects and therefore be associated with a high $f_{\rm
grav}$, would strengthen the disruption power, while a value of
$\epsilon\\!\sim\\!0.01$ would increase the survivability.

The parameter $\nu$, the number of supernovae per one solar mass of forming
stars, depends on the stellar initial mass function (IMF). For a Chabrier IMF
the common value is $\nu\\!\simeq\\!0.01M_{\odot}^{-1}$ or lower (Botticella
et al., 2017). A top-heavy IMF will make $\nu$ larger, which may tip the
balance in favor of disruption.

###  7.6 Redshift dependence

A redshift dependence enters $V_{\rm c}$ in eq. (26) through $\delta$ and
$V_{\rm d}$. If $\delta$ is proportional to $f_{\rm g}$ (see §7.3), then the
dependence of $V_{\rm c}$ on gas fraction at a given disc mass is roughly
$\propto\\!f_{\rm g}^{2/3}$ ($\propto\\!f_{\rm g}$ at a given halo or baryonic
mass). The average gas fraction is rising from $f_{\rm g}\\!\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!0.1$ at $z\\!=\\!0$ to $f_{\rm
g}\\!\sim\\!0.5$ at $z\\!\sim\\!2$ (Tacconi et al., 2010; Daddi et al., 2010;
Tacconi et al., 2018). This can crudely be described as a power low steeper
than $f_{\rm g}\\!\propto\\!(1+z)^{2}$ in this range. For given disc mass and
gas fraction (or a given halo mass), $V_{\rm d}\\!\propto\\!(1+z)^{1/2}$.
Together, the redshift dependence is crudely $V_{\rm
c}\\!\propto\\!(1+z)^{2}$. This strong redshift dependence implies that long-
lived clumps are expected at $z\\!\sim\\!2$ and above and not at much lower
redshifts. The growth of $f_{\rm g}$ with redshift becomes much milder above
$z\sim 2$, so the preference for surviving clumps increases with redshift at a
slower pace at higher redshifts. Naturally, a necessary condition for L clumps
is being hosted in a disc within a halo above the threshold mass for long-
lived discs (Dekel et al., 2020a), which become rare at very high redshifts,
when the Press-Schechter mass of typical haloes is much smaller than the
threshold mass for discs. Thus, while the clump frequency per disc should keep
increasing with redshift, the total number of clumps should peak near
$z\\!\sim\\!2$.

In the case of a clump mass at it’s critical value $M_{\rm
c}\\!\simeq\\!10^{8}M_{\odot}$ in a mixture of galaxy masses, based on eq.
(66), the dependence of $V_{\rm c}$ on gas fraction becomes weaker, so crudely
$V_{\rm c}\\!\propto\\!(1+z)^{1/2}$. This yields a slightly increased
survivability near the critical clump mass at higher redshift, as long as the
galaxy is sufficiently massive to allow a long-lived disc.

##  8 Clump types in simulations

###  8.1 The VELA simulations

####  8.1.1 The simulations

Figure 1:  Face-on views of the surface density of cold mass in VELA-3
simulated discs showing giant clumps. Shown are V07 at $z\\!=\\!2.33$ (left)
and V19 at $z\\!=\\!4.56$ (right). The cold mass is comprised of cold gas
($T\\!<\\!1.5\\!\times\\!10^{4}~{}{\rm K}$) and young stars ($<\\!100~{}\,{\rm
Myr}$). The disc radii are $R_{\rm d}\\!=\\!11.8$ and $3.2\,{\rm kpc}$
respectively. In both cases the disc half-thickness is $H_{\rm
d}\\!\sim\\!R_{\rm d}/6$. The baryonic masses of the discs are $M_{\rm
d}/(10^{10}M_{\odot})\\!\sim\\!3.50$ and $1.35$, and the fractions of cold-
mass are $28\%$ and $33\%$ respectively. Both discs show giant clumps with
masses $M_{\rm c}\\!>\\!10^{8}M_{\odot}$.

We use here the VELA suite of zoom-in cosmological simulations, 34 galaxies
with halo masses $10^{11}-10^{12}M_{\odot}$ at $z\\!\sim\\!2$, which have been
used to explore many aspects of galaxy formation at high redshift (e.g.,
Ceverino et al., 2014; Moody et al., 2014; Zolotov et al., 2015; Ceverino,
Primack & Dekel, 2015; Inoue et al., 2016; Tacchella et al., 2016a, b;
Tomassetti et al., 2016; Ceverino et al., 2016a, b; Mandelker et al., 2017;
Dekel et al., 2020a, b; Ginzburg et al., 2021; Dekel et al., 2021). The
simulations utilize the ART code (Kravtsov, Klypin & Khokhlov, 1997; Kravtsov,
2003; Ceverino & Klypin, 2009), which follows the evolution of a gravitating
$N$-body system and the Eulerian gas dynamics with an AMR maximum resolution
of $17.5-35\,{\rm pc}$ in physical units at all times. The dark-matter
particle mass is $8.3\times 10^{4}M_{\odot}$ and the minimum mass of stellar
particles is $10^{3}M_{\odot}$. The code incorporates gas and metal cooling,
UV-background photoionization and self-shielding in dense gas, stochastic star
formation, stellar winds and metal enrichment, and feedback from supernovae.

The thermal feedback model assumes that each stellar particle represents a
single stellar population and injects the luminosity from supernovae and
stellar winds as thermal heating, at a constant heating rate over $40\,{\rm
Myr}$, the lifetime of the lightest star that explodes as a core-collapse
supernova. (Ceverino & Klypin, 2009). The model adds non-thermal radiative
pressure in regions where ionizing photons from massive stars are produced and
trapped. In the VELA-3 version of the feedback model (RadPre in Ceverino et
al., 2014), the radiative pressure is acting in cells that contain stellar
particles younger than $5\,{\rm Myr}$ and whose gas column density exceeds
$10^{21}\,{\rm cm}^{-2}$, as well as in these cell’s closest neighbors. In
VELA-3 , the momentum driving efficiency is $\sim 3L/c$, which is comparable
to the supernova contribution, but lower than in certain other simulations
where the feedback is very strong (e.g., Genel et al., 2012; Oklopčić et al.,
2017). Further details regarding the feedback and other sub-grid models in
VELA-3 can be found in Ceverino et al. (2014) and Mandelker et al. (2017,
M17).

The VELA-6 version of the radiative feedback (based on RadPre_IR in Ceverino
et al., 2014) includes moderate trapping of infrared photons once that gas
density in the cell exceeds $300\,{\rm cm}^{-3}$. In addition, the VELA-6
feedback model includes momentum injection from the expanding supernova shells
and stellar winds (Ostriker & Shetty, 2011). A momentum of
$10^{6}M_{\odot}\,{\rm km}\,{\rm s}^{-1}$ per star more massive than
$8M_{\odot}$ is injected at a constant rate over $40\,{\rm Myr}$. The VELA-6
model also includes a factor of three boost in the injected momentum due to
clustering of supernovae (Gentry et al., 2017), which is implemented in the
form of non-thermal pressure as described in Ceverino et al. (2017). Further
details regarding the feedback and other sub-grid models in VELA-6 can be
found in Ceverino et al. (2014), Ceverino et al. (2017) and Ceverino et al.
(in preparation).

M17 have analyzed the clumps in the VELA-3 galaxies as well as in counterparts
that lacked the radiation pressure feedback (VELA-3 and VELA-2 respectively).
They found that with radiation pressure a significant fraction of the clumps
of $M_{\rm c}\\!\leq\\!10^{8}M_{\odot}$ are disrupted on timescales of a few
free-fall times, while most of the more massive clumps survive. The migration
inward of these long-lived clumps produced galacto-centric radial gradients in
their properties, such as mass, stellar age, gas fraction and sSFR, which were
found in Dekel et al. (2021) to be consistent with an analytic model and with
observed clumps, distinguishing them from the short-lived clumps. However, in
M17 the analyzed output times where separated coarsely by $\sim\\!100\,{\rm
Myr}$, which prevented a detailed study of the evolution of individual clumps.
In the current study, we have re-simulated several of the VELA-3 galaxies,
saving about ten snapshots per disc crossing time, with fine timesteps that
can be as short as $\sim\\!5\,{\rm Myr}$. Among these galaxies, five have L
clumps (V07, V08, V19, V26, V27), and four others have only S clumps (V11,
V12, V14, V25).

As in M17, the disc is defined as a cylinder with radius $R_{\rm d}$ and half
height $H_{\rm d}$, containing 85% of the cold gas
($T\\!<\\!1.5\\!\times\\!10^{4}~{}{\rm K}$) and the young stars (ages
$<\\!100\,{\rm Myr}$) within $0.15R_{\rm v}$. The disc stars are also required
to obey a kinematic criterion, that their specific angular momentum parallel
to that of the disc, $j_{\rm z}$, is at least $70\%$ of the maximal possible
value it could have given its galacto-centric distance, $r$, and orbital
velocity, $v$, namely $j_{\rm max}\\!=\\!vr$.

The VDI phase typically follows a dramatic event of wet compaction into a
star-forming “blue nugget”, that tends to occur when the halo mass is above
$M_{\rm v}\\!\sim\\!10^{11.3}M_{\odot}$, and could happen at different
redshifts, typically above $z\\!=\\!1$ (Zolotov et al., 2015; Tomassetti et
al., 2016; Tacchella et al., 2016a, b, figure 2). An extended, long-lived
clumpy disc, evolving to a clumpy ring, typically develops after the major
compaction event, above the threshold mass where the disc is not disrupted by
frequent mergers (Dekel et al., 2020a, b).

Figure 1 shows face on views of two simulated disks, V07 at $z\\!=\\!2.33$ and
V19 at $z\\!=\\!4.56$, shortly after the start of their VDI phase. The figure
shows the surface density of the cold mass, integrated over $\pm\\!R_{\rm d}$
perpendicular to the disc, where $R_{\rm d}\\!\simeq\\!12$ and $3.2\,{\rm
kpc}$ for V07 and V19 respectively. While these two discs are at very
different redshifts, with different masses and sizes, both show giant star-
forming clumps with masses $M_{\rm c}\\!>\\!10^{8}M_{\odot}$.

The gas fractions in the VELA-3 discs are somewhat lower than estimated in
typical observed galaxies at similar redshifts. This has been discussed in
detail in several papers which used these simulations (e.g., Zolotov et al.,
2015; Tacchella et al., 2016b; Mandelker et al., 2017). While the VELA
simulations are state-of-the-art in terms of high-resolution AMR hydrodynamics
and the treatment of key physical processes at the subgrid level, they are not
perfect in terms of their treatment of star-formation and feedback, much like
other simulations. Star-formation tends to occur too early, leading to lower
gas fractions later on. The stellar masses at $z\\!\sim\\!2$ are a factor of
$\sim\\!1.5\\!-\\!2$ higher than inferred for haloes of similar masses from
abundance matching (e.g., Rodríguez-Puebla et al., 2017; Moster, Naab & White,
2018; Behroozi et al., 2019). However, for the purposes of the present study,
the relatively low gas fractions during the peak VDI phase would only
underestimate the actual accretion of fresh gas onto clumps during their
migration, providing a lower limit on clump survival. The effect of gas
fraction on clump properties and survival in simulated isolated discs is
further discussed in Fensch & Bournaud (2021). In VELA-6 with the stronger
feedback, the agreement with the stellar-to-halo mass ratio deduced from
observations becomes better, but this comes at the expense of more dynamical
destruction on the clump scales, which may or may not be more realistic. We do
not attempt in this paper to decide between the different feedback models, but
rather to study the effect of each on the survival and disruption of the giant
clumps.

####  8.1.2 Clump analysis

Clumps are identified in 3D and followed through time following the method
detailed in M17. Here we briefly summarize the main features. Clumps are
searched for within a box of sides $4R_{\rm d}$ in the disc plane and height
$4H_{\rm d}$ centered on the galaxy centre. Via a cloud-in-cell interpolation,
the mass is deposited in a uniform grid with a cell size of
$\Delta\\!=\\!70\,{\rm pc}$, two-to-three times the maximum AMR resolution. We
then smooth the cell’s density, $\rho$, into a smoothed density, $\rho_{\rm
w}$, using a spherical Gaussian filter of FWHM $\\!=\\!{\rm min}(2.5\,{\rm
kpc},0.5R_{\rm d})$, defining a density residual
${\delta}_{\rm{\rho}}\\!=\\!(\rho\\!-\\!\rho_{\rm w})/\rho_{\rm w}$. Performed
separately for the cold mass and the stellar mass, we adopt at each point the
maximum of the two residual values. Clumps are defined as connected regions
containing at least 8 grid cells with a density residual above
${\delta}_{\rm{\rho}}^{\rm min}\\!=\\!10$, making no attempt to remove unbound
mass from the clump. We define the clump centre as the baryonic density peak,
and the clump radius, $R_{\rm c}$, as the radius of a sphere with the same
volume as the clump. The clump mass, however, is the mass contained in the
cells within the connected region. Ex-situ clumps, which joined the disc as
minor mergers, are identified by their dark matter content and the birth place
of their stellar particles. They are not considered further here, where we
focus on the in-situ clumps.

The SFR in the clumps is derived from the mass in stars younger than $30\,{\rm
Myr}$, which is sufficiently long for fair statistics and sufficiently short
for ignoring the stellar mass loss. Outflow rates from the clumps are measured
through shells of radii $(R_{\rm c},R_{\rm c}+100\,{\rm pc})$. The gas outflow
rate is computed by $\dot{M}_{\rm out}\\!=\\!\Delta^{-1}\sum_{i}V_{r}\,m_{i}$,
with $\Delta\\!=\\!100\,{\rm pc}$, where the sum is over cells within the
shell with $V_{r}\\!>\\!0$ and a 3D velocity larger than the escape velocity
from the clump, $V^{2}\\!>\\!2GM_{\rm c}/R_{\rm c}$, and where $m_{i}$ is the
gas mass in the cell. The gas accretion rate is computed in analogy to the
outflow rates but with $V_{r}\\!<\\!0$ and no constraint on $V^{2}$. We note
that this calculation involves large uncertainties.

Individual clumps, that contain at least 10 stellar particles, are traced
through time based on their stellar particles. For each such clump at a given
snapshot, we search for all “progenitor clumps” in the preceding snapshot,
defined as clumps that contributed at least $25\%$ of their stellar particles
to the current clump. If a given clump has more than one progenitor, we
consider the most massive one as the main progenitor and the others as having
merged, thus creating a clump merger tree. If a clump in snapshot $i$ has no
progenitors in snapshot $i\\!-\\!1$, we search the previous snapshots back to
two disc crossing times before snapshot $i$. If no progenitor is found in this
period, snapshot $i$ is declared the initial, formation time of the clump, and
for that clump $t$ is set to zero at that time.444If the mass weighted mean
stellar age of the clump at its initial snapshot is less than the timestep
since the previous snapshot, we set the initial clump time to this age rather
than to zero. This introduces an uncertainty of a few Megayears in the clump
age. When tracing the evolution of a clump we refer to the main progenitor and
consider the mergers to be part of the accretion onto the clump. The clump
lifetime $\tau_{\rm c}$ is the age of the clump at the last snapshot when the
clump is still identified.

Figure 2:  S vs L clumps. The distribution of lifetimes in units of free-fall
times, $\tau_{\rm c}\\!=\\!t_{\rm c}/t_{\rm ff}$, for clumps in VELA-3 (blue
solid line, normalized to unity). The minimum clump mass is set in this figure
to $2\times 10^{7}M_{\odot}$. Left: Number of clumps. Middle: Weighted by
clump mass. Right: Number at each snapshot, to be compared to observations. We
see in the right panel a natural bimodality into S and L clumps, which we
separate at $\tau_{\rm c}\\!\sim\\!7$. Also shown is the distribution of
lifetimes in the VELA-6 simulations with the stronger feedback (dashed red),
which is derived from the available coarser output timesteps of
$\sim\\!100\,{\rm Myr}$. The comparison with VELA-3 is performed using the
same coarse timesteps in VELA-3 (dashed blue). The VELA-3 coarse-timestep
histogram is independently normalized to unity, and the corresponding VELA-6
histogram is based on the same normalization as VELA-3 (the number of clumps
from the coarse timesteps is lower than it is for the fine timesteps). We see
that the coarse-timestep distributions are offset toward larger $\tau_{\rm c}$
(for reasons that are briefly discussed in the text), but the comparison of
VELA-3 and VELA-6 using the coarse timesteps is fair. The VELA-6 simulations
show less clumps than VELA-3, which are predominantly S clumps.

Figure 3:  LS versus LL long-lived clumps in VELA-3. The distribution of
$\tau_{\rm gas}$, the time when the clump have lost 90% of the gas it had at
$\tau\\!=\\!4$ (if $\tau_{\rm c}\\!<\\!4$ then $\tau_{\rm gas}$ is set to
equal $\tau_{\rm c}$). The minimum clump mass is set in this figure to
$2\times 10^{7}M_{\odot}$. Left: Number of clumps. Middle: Weighted by clump
mass. Right: Number at each snapshot, as observed. For the S clumps
($\tau_{\rm c}\\!<\\!7$) we have $\tau_{\rm gas}\\!\simeq\\!\tau_{\rm c}$.
There is a natural division of the L clumps ($\tau_{\rm c}\\!>\\!7$) at
$\tau_{\rm gas}\\!\simeq\\!25$ to LS clumps that lose most of their gas
moderately early and LL clumps that keep most of their gas for tens of free-
fall times.

Figure 4:  Images of gas (left) and stars (right) in examples of the 3 clump
types at four times. Top: An LL clump at $\tau\\!=\\!2,4,9,65$, surviving both
in stars and gas for tens of free-fall times. Middle: An LS clump at
$\tau\\!=\\!4,8,10,25$, surviving in stars but losing all gas by
$\tau\\!\sim\\!10$. Bottom: An S clump at $\tau\\!=\\!1,3,4,5$, losing all its
gas and dispersing its stars by $\tau\\!\sim\\!5$.

###  8.2 Three clump types

The clumps for analysis were selected as follows. From the nine galaxies that
were re-simulated with fine output timesteps, output snapshots were selected
where the galaxies are discs with an axial ratio $R_{\rm d}/H_{\rm
d}\\!>\\!3$. Clumps were selected for analysis only in snapshots when their
baryonic mass is above a threshold mass $M_{\rm c}\\!>\\!10^{7}M_{\odot}$,
with the additional requirement that their initial gas mass was $M_{\rm
g}\\!>\\!10^{6}M_{\odot}$ at the snapshot when they were first identified.

As found in M17, the clumps in the VELA-3 simulations can be divided into two
major classes based on their lifetimes. This is demonstrated in Fig. 2, which
shows (in solid blue) the probability distribution of clump lifetimes in units
of clump free-fall times, the latter being the mass-weighted average of
$t_{\rm ff}$ over the clump lifetime. The distributions are shown in three
different ways, corresponding to (a) clump number where each clump is counted
once (left), (b) clump mass where each clump is counted once and the mass is
the average over the clump lifetime (middle), and (c) clump number where each
clump is counted at every snapshot in which it appears (right). The latter
allows direct comparison to the distribution of clumps in observations. Here,
the lifetimes were determined using the fine timesteps as opposed to the
coarse timesteps used in M17. We see a bimodality into short-lived and long-
lived clumps, which is emphasized in the right panel. We term them S and L
clumps respectively, and separate here near a clump lifetime of $\tau_{\rm
c}\\!=\\!t_{\rm c}/t_{\rm ff}\\!\sim\\!7$.555Any choice on the order
$\tau_{\rm c}\\!\sim\\!10$ would make sense. For example, M17 used $\tau_{\rm
c}\\!=\\!20$. The difference between the medians of the normalized $\tau_{\rm
c}$ for S and L clumps, $\sim\\!1.9$dex, reflects a large difference of
$\sim\\!1.4$dex in $t_{\rm c}$, aided by a smaller difference of
$\sim\\!0.5$dex in $t_{\rm ff}$ (see Fig. 7).

For a quick comparison of the clumps in the VELA-3 and VELA-6 simulations,
with different feedback strengths, we appeal to a clump analysis using the
coarser output timesteps of $\sim\\!100\,{\rm Myr}$, which are the data
currently available for VELA-6. Figure 2 shows in dashed lines the
distributions of clump lifetimes using the coarse timesteps. The dashed VELA-3
histogram (blue) is normalized to unity, and the dashed VELA-6 histogram (red)
is normalized based on VELA-3. We note in passing that for VELA-3 the
distribution using the coarse timesteps is different from the fine-timestep
distribution, for several reasons. First, the overall number of clumps is
lower in the coarse-timestep analysis, despite having more galaxies in this
analysis. This is because the fine timesteps allow more clumps to be
identified, and because more zero-lifetime clumps are excluded in the analysis
using coarse timesteps (termed ZLC in M17). Second, around $\tau_{\rm
c}\\!\sim\\!10$ there are more clumps at a given $\tau_{\rm c}$ in the coarse-
timestep analysis. This indicates that the corresponding histogram is shifted
towards larger $\tau_{\rm c}$ values, despite the natural tenancy to measure
larger lifetimes with finer timesteps. This shift is partly due to the way
ages are assigned to S clumps at the first snapshot, especially if they are
identified in one snapshot only, or because $t_{\rm ff}$ is somehow
underestimated when averaged over less snapshots. The above comparison between
the the fine and coarse timesteps is not relevant for our current aim to
qualitatively compare between the clumps in VELA-3 and VELA-6 using the same
coarse timesteps. We find that in VELA-6, with the stronger feedback, there
are significantly fewer clumps, and a significantly smaller fraction of L
clumps. This indicates that clump survival is a strong function of the
feedback strength, as predicted by our model. A more detailed comparison
between the clumps in the VELA-3 versus the VELA-6 simulations is deferred to
Ceverino et al. (in preparation).

We next realize that the L clumps in VELA-3 can be divided into two sub-
classes based on the way they lose their gas to outflows. This is demonstrated
in Fig. 3, which shows for the S and L clumps in VELA-3 the distribution of
$\tau_{\rm gas}$, the time when the clump has lost 90% of the gas mass it had
at $\tau\\!=\\!4$. The latter is typically the time when the gas mass reaches
a maximum. If $\tau_{\rm c}\\!<\\!4$ we set $\tau_{\rm gas}\\!=\\!\tau_{\rm
c}$. Among the L clumps, one can identify a bi-modality into two types,
separated near $\tau_{\rm gas}\\!\sim\\!25$, which we term LS and LL clumps
respectively (the first L for the stars, the second S or L for the gas). The
LL clumps are the long-lived clumps analyzed in M17 (sometimes together with
the LS clumps) and in Dekel et al. (2021), which keep a significant fraction
of their gas and a non-negligible SFR for many tens of free-fall times, till
they complete their inward migration. The LS clumps, which we analyze here in
detail, are in certain ways between the S clumps and the LL clumps. They lose
most of their gas on a timescale of $\sim\\!10$ free-fall times, while keeping
a long-lived bound stellar component with only little SFR.

Figure 5:  The probability distribution of the clump properties $M_{\rm c}$,
$R_{\rm c}$ and $V_{\rm c}$, averaged over the early phase $\tau\\!<\\!4$, for
the three clump types. Each clump is counted once. The medians are marked.
There is a trend with mass from S through LS to LL, with the LL clumps
typically above $10^{8}M_{\odot}$. The LL clumps tend to have smaller radii.
As a result there is a clear trend with binding energy, via $V_{\rm c}$, from
the loosely bound S clumps of $V_{\rm c}\\!\sim\\!20\,{\rm km}\,{\rm s}^{-1}$,
through LS, to the strongly bound LL clumps of $V_{\rm c}\\!\sim\\!50\,{\rm
km}\,{\rm s}^{-1}$.

Figure 6:  Time evolution of clump mass, $M_{\rm g}$, $M_{\rm s}$ and $M_{\rm
c}\\!=\\!M_{\rm g}+M_{\rm s}$, for gas, stars and total, as a function of
$\tau\\!=\\!t/t_{\rm ff}$. Shown are the medians and $1\sigma$ scatter for
each of the three types. For the S clumps (red), the solid curve and shaded
area refer to the small fraction of clumps that survive until the given time,
while in the dashed curve the disrupted clumps are included with zero mass.
The gas mass is also shown relative to its value at $\tau\\!=\\!4$, near its
peak, and via the gas fraction $f_{\rm g}\\!=\\!M_{\rm g}/M_{\rm c}$. Also
shown is $\mu$, the clump mass relative to the Toomre mass. The S clumps lose
their gas in the first few free-fall times. The LS clumps lose most of their
gas to outflows by $\tau\\!\sim\\!10$, but keep a bound stellar component. The
LL clumps keep their gas and stars for tens of free-fall times.

Figure 7:  Evolution of clump size, velocity, and free-fall time at $\tau$.
The density contrast $c$ is measured in two alternative ways, first by
estimating the initial clump radius from the Toomre analysis (left), and
second from the 3D density contrast between clump and disk (middle). The LL
clumps tend to be smaller ($R_{\rm c}\\!\sim\\!300\,{\rm pc}$), more tightly
bound $V_{\rm c}\\!\sim\\!50\,{\rm km}\,{\rm s}^{-1}$, and with short free-
fall times ($t_{\rm ff}\\!\sim\\!5\,{\rm Myr}$). The S clumps are more diffuse
and loosely bound ($R_{\rm c}\\!\sim\\!400\,{\rm pc}$, $V_{\rm
c}\\!\sim\\!20\,{\rm km}\,{\rm s}^{-1}$, $t_{\rm ff}\\!\sim\\!23\,{\rm Myr}$).
The contraction factor is somewhat higher for the LL clumps. The difference in
clump shape between the clump types is marginal, with the S clumps slightly
more elongated.

Figure 4 presents images of gas and stellar surface density in typical clumps
of the three types in VELA-3. The LL clump shown at the top at times
$\tau\\!=\\!2,4,9,65$ reveals a bound clump both in stars and gas all the way
to $\tau\\!=\\!65$. On the other hand, in the LS clump shown in the middle at
$\tau\\!=\\!4,8,10,25$, the gas has disappeared by $\tau\\!=\\!25$, while the
significant stellar component remains largely unchanged till later times. The
S clump shown at $\tau\\!=\\!1,3,4,5$ have lost all its gas already by
$\tau\\!=\\!5$, and the stellar component becomes dilute at that time, to
disappear soon thereafter.

The three clump types differ by many of their properties beyond the difference
in life-time and gas depletion time that have been used for their
classification. A few of these properties are shown in Fig. 5, through their
distributions in VELA-3 during the first 4 free-fall times of each clump. In
terms of clump mass, we see a weak systematic decrease of mass from LL to LS
clumps, and a larger decrease into the S clumps, with medians $M_{\rm
c}\\!\sim\\!10^{8.13},10^{8.0},10^{7.7}M_{\odot}$, the latter biased up by the
selection threshold of $M_{\rm c}\\!>\\!10^{7}M_{\odot}$. In terms of clump
radius, the LL clumps tend to be compact, with a median radius of $R_{\rm
c}\\!\sim\\!320\,{\rm pc}$ (possibly affected by an effective threshold due to
the simulation resolution), with the LS and S clumps both tending to be more
diffuse, with medians at $R_{\rm c}\\!\sim\\!400\,{\rm pc}$. As a result, the
clump virial velocity $V_{\rm c}$ tends to be the largest for LL clumps,
intermediate for LS clumps, and smallest for S clumps, with medians of $V_{\rm
c}\\!\sim\\!50,33,22\,{\rm km}\,{\rm s}^{-1}$ for LL , LS and S respectively.

The differences and similarities between the three clump types are explored in
more detail by the time evolution of the relevant clump properties, for which
the median and $\pm 34$ percentiles are shown as a function of time $\tau$ in
Figs. 6 to 11. These detailed properties will serve us in the comparison with
the analytic model of §2 to §7.

####  8.2.1 mass

Figure 6 presents the evolution of clump gas and stellar mass. Most of the S
clumps disrupt within the first one or two free-fall times, so the S clumps
referred to by the solid curve and shaded area represent the minority that
survive for a little longer. The evolution of gas mass $M_{\rm g}$ is used for
the basic classification of the L clumps into LL and LS clumps via $\tau_{\rm
gas}$, which is to be used in the following figures. Referring to the medians
of $M_{\rm g}(t)$, even the surviving S clumps have a median gas mass lower by
a factor of $2\\!-\\!3$ than the L clumps. The LS clumps, after reaching a
peak near $\tau$ of a few, show a steep decline where half the gas mass is
lost by $\tau\\!\sim\\!8$ and practically no gas is left after
$\tau\\!\sim\\!15$, while the LL clumps tend to keep most of their gas till
$\tau\\!\sim\\!20$, with some of the LL clumps gaining gas mass during their
evolution. This relative constancy of gas mass is associated with a roughly
constant SFR. The stellar mass $M_{\rm s}$ for the two L types is similar till
$\tau\\!\sim\\!10$, growing till $\tau\\!\sim\\!5$ and then gradually
flattening off. At later times, the $M_{\rm s}$ of the LL clumps keeps growing
roughly linearly with time, reflecting a constant SFR, while the $M_{\rm s}$
of the LS clumps is more constant due to the vaishing SFR (though its median
is slowly rising, perhaps due to the larger masses of the clumps that survive
longer, or due to accretion). The gas fraction provides an alternative
quantity for the classification, with similar results to $M_{\rm g}$. All the
clumps start with $f_{\rm g}\\!\sim\\!0.6\\!-\\!0.8$ and the LS and LL clumps
differ after $\tau\\!\sim\\!4$, reaching $f_{\rm g}\\!=\\!0.5$ near
$\tau\\!\sim\\!8$ and $\sim\\!15$ respectively, with the LS clumps dropping to
negligible gas fractions after $\tau\\!\sim\\!15$.

The median initial total mass of the L clumps at $\tau$ of a few is $M_{\rm
c}\\!\sim\\!10^{8}M_{\odot}$, while for the S clumps it is smaller by a factor
of $\sim\\!3$. For the two types of L clumps, $M_{\rm c}$ is similar till
$\tau\\!\sim\\!5$. After this time, while the median mass of the LL clumps is
rather constant and even slowly rising, the mass of the LS clumps is gradually
declining by a factor of $\sim\\!3$ due to gas loss by outflows to a minimum
mass near $\tau\\!\sim\\!20$, before it is slowly rising at later times. This
could be due to stellar accretion or contamination by background disc stars.
From $\mu(t)$, we learn that the median LL and LS clumps are of $\sim\\!20\%$
and $\sim\\!10\%$ of the Toomre mass, respectively.

Figure 8:  Evolution of SFR, sSFR and the SFR efficiency per free-fall time
$\epsilon$ as measured over the whole clump. The SFR naturally follows the gas
mass, so it is low for the S clumps, high for the LL clumps and roughly
constant at $\\!\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!\\!1M_{\odot}\,{\rm yr}^{-1}$,
and dropping for the LS clumps after a peak at $\tau\\!\sim\\!5$. The SFR
efficiency is high, especially for the S and LS clumps, because it refers to
the whole clump volume, while the star-forming region is actually more
compact.

####  8.2.2 size and binding

Figure 7 highlights a basic continuous trend of $V_{\rm c}$ as a function of
clump type, with peak median values in the initial phase
$\tau\\!=\\!0\\!-\\!10$ of $V_{\rm c}\\!\sim\\!50,\,33$ and $20\,{\rm
km}\,{\rm s}^{-1}$ for LL , LS and S clumps, respectively. This will directly
affect the survivability parameter $S^{\prime}$, and will turn out to be the
main distinguishing feature between the S and L clumps, as predicted in §6.1.
The difference in $V_{\rm c}^{2}\\!=\\!GM_{\rm c}/R_{\rm c}$ may be due to a
difference in $M_{\rm c}$ or $R_{\rm c}$ or both. Figure 7 shows that during
the early phase the median radius is $R_{\rm c}\\!\sim\\!400\,{\rm pc}$ for
the S and LS clumps, while it is $R_{\rm c}\\!\sim\\!300\,{\rm pc}$ for the LL
clumps, with some clumps smaller than $200\,{\rm pc}$ (where the radius is
probably overestimated due to resolution). This implies that the lower $V_{\rm
c}$ for S clumps is predominantly due to the lower $M_{\rm c}$. As for the LS
versus LL clumps, since the peak of $M_{\rm c}$ in the early phase differs by
less than $0.2$ dex, the difference by a factor of almost two in the median
$V_{\rm c}$ stems mostly from the similar difference in $R_{\rm c}$. This may
be the basic difference between the LS and LL clumps, as discussed in §6.2.
This difference in $R_{\rm c}$ is seen in the concentration $c$ as derived
from the Toomre mass at $\tau\\!\lower
2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!10$ (but less so in $c$ as
estimated from the density contrast).

Figure 7 also shows a continuous trend in clump free-fall time, with a value
at $\tau\\!=\\!5$ of $t_{\rm ff}\\!\sim\\!21,14$ and $6\,{\rm Myr}$ for S , LS
and LL clumps, respectively. For the S clumps, the large $t_{\rm ff}$,
reflecting a low density, is both due to the low $M_{\rm c}$ and the high
$R_{\rm c}$. On the other hand, the difference between the two types of L
clumps is mostly due to the difference in $R_{\rm c}$ and $c$.

As in M17, the clump shape elongation is measured via the axis ratio
$\ell_{1}/\ell_{3}$, the distribution of which for the three clump types at
$\tau<10$ is shown in Fig. 7. The median S clumps are slightly more elongated
than the L clumps, $\ell_{1}/\ell_{3}\\!\sim\\!0.3$ versus $\sim\\!0.4$, with
some of the L clumps as round as $\ell_{1}/\ell_{3}\\!\sim\\!0.8$. The LS and
LL clumps are not significantly different in shape.

####  8.2.3 SFR

Figure 8 presents the evolution of SFR and sSFR and the associated SFR
efficiency per free-fall time $\epsilon$. For all types, the early-phase sSFR
is high, at the level of $\sim\\!20\,{\rm Gyr}^{-1}$, characteristic of a
starburst in an initial stellar-poor clump. The median SFR in the S clumps is
lower by a factor of three compared to the L clumps. Naturally the following
decline in sSFR is associated with the decline in $M_{\rm g}$, which occurs in
the S clumps first, then in the LS clumps, and later in the LL clumps. Figure
8 then shows an apparently surprising result concerning the SFR efficiency
$\epsilon\\!=\\!\dot{M}_{\rm sf}\,t_{\rm ff}\,/M_{\rm g}$, where in the early
phase it is $\epsilon\\!\sim\\!0.08$ and $0.055$ for the LS and LL clumps,
respectively. This is mostly due to the larger $t_{\rm ff}$ for LS clumps,
while $\dot{M}_{\rm sf}$ and $M_{\rm g}$ are similar. This is surprising
because the SFR efficiency is argued based on observations to be in the same
ballpark in all environments and redshifts (e.g. Krumholz, Dekel & McKee,
2012). One way to interpret this is that the SFR in the LS clumps actually
occurs in regions denser than the average, near the clump centre or in sub-
clumps, but this is not reproduced in the simulated clumps in which sub-clumps
are not properly resolved.

Another possibility is that the higher $\epsilon$ is due to a more efficient
mode of bursty star formation in the LS clumps, e.g., due to shocks induced by
clump mergers, intense accretion, or strong tidal effects. An inspection of
Fig. 3 of Krumholz, Dekel & McKee (2012), which is an accumulation of star-
forming regions in galaxies in different environments and redshifts, and
focusing on high-redshift galaxies (blue symbols), one can see that the
starbursts (open symbols), which are likely to represent mergers, have
$\epsilon$ values that are systematically higher than those of the normal
discs (filled symbols), by a factor of a few. Referring to the discussion in
§6.2, these inferred mergers can be associated with an increase in $f_{\rm
grav}$ and $f_{\rm sn}$ compared to the LL clumps, which makes $S^{\prime}$
smaller and thus can explain the early disruption of the LS clumps. Such an
excess of mergers, accretion and tidal effects can be associated with the
larger radii of the LS clumps, introducing a larger cross-section for mergers
and capture of gas within the clump Hill sphere, as well as stronger tidal
effects.

Figure 9:  Evolution of outflow and inflow. Shown are the cumulative masses in
outflow and inflow, as well as the corresponding parameters $\eta$ and $\psi$
and the outflow velocity $V_{\rm out}$. The outflows are most efficient for
the S clumps and least efficient for the LL clumps, with medians in the early
phase of $\eta\\!\sim\\!1.2,0.3,0.2$ respectively. The inflows are more
similar between the clump types.

Figure 10:  Evolution of clump internal kinematics, the rotation velocity
$V_{\rm rot}$, the radial velocity dispersion $\sigma_{r}$, and the rotation
support parameter $R\\!=\\!V_{\rm rot}^{2}/V_{\rm c}^{2}$. The LL clumps have
a larger rotation component, consistent with the higher contraction factor.

Figure 11:  Evolution of disc mass $M_{\rm d}$ and mass fraction $\delta$, and
clump position in the disc in the radial and vertical directions. The disc
fraction for galaxies hosting LL clumps is somewhat higher, indicating a
stronger disc instability with more massive clumps. The radial migration rate
for LS clumps seems higher in terms of $\tau$, but this is an artifact of the
longer free-fall times for these clumps. There is no significant difference in
the effective $z$ positions of the different clump types.

####  8.2.4 outflow and inflow

Figure 9 shows the evolution of gas outflows from and inflows into the clumps.
During the early phase, the outflowing mass is significantly smaller than the
clump mass, as predicted in §3.1. As expected, the outflows in the S clumps
are the strongest, with a median larger by a factor of $\sim\\!3$ compared to
the L clumps. The outflow is larger by a factor of $\sim\\!2$ in the LS clumps
compared to the LL clumps, leading to a smaller $S$ in eq. (1). The outflow
mass loading factor varies systematically in a corresponding way, with a
median of $\eta\\!\sim\\!1.2,0.3,0.2$ for S , LS and LL clumps respectively at
$\tau\\!<\\!10$. The inflow parameter $\psi$ is defined following eq. 9 of
Dekel et al. (2021) (where it is termed $\alpha$),

$\psi=2\,\dot{M}_{\rm in},t_{\rm d}/M_{\rm c}\,.$ (69)

Here $\dot{M}_{\rm in}$ is measured in analogy to $\dot{M}_{\rm out}$, in a
shell of radii $(R_{\rm c},R_{\rm c}+100\,{\rm pc})$, requiring
$V_{r}\\!<\\!0$ but not constraining the absolute value of $V$. The inflowing
mass during the early phase is also larger in the LS clumps than in the LL
clumps by a factor $\sim\\!2$, with the S clumps similar to and slightly
smaller than the LS clumps. The associated value of $\psi$ is larger for the
LS clumps than the LL clumps until $\tau\\!\sim\\!0.7$, with the S clumps
similar to the LL clumps. The inflowing mass is consistent with more mergers
and intense accretion involved in the early phases of the LS clumps, and an
associated higher $f_{\rm grav}$, leading to a lower $S^{\prime}$.

####  8.2.5 kinematics

Figure 10 shows that at the early times the clumps are not rotation supported,
with the LL clumps somewhat more supported by rotation than the LS clumps.
With $V_{\rm rot}$ the mass-weighted rotation velocity, the rotation-support
parameter is $R\\!=\\!V_{\rm rot}^{2}/V_{\rm c}^{2}\\!\sim\\!0.15$ and $0.1$
for the LL and LS clumps respectively. This is consistent with the larger
contraction factor $c$ for LL clumps, as predicted in eq. (31) in §3.3, if
angular momentum is conserved during clump collapse. Both $V_{\rm rot}$ and
$\sigma_{r}$ are larger for the LL clumps, though the difference in $V_{\rm
rot}$ is slightly larger. We note in passing that the LL clumps gradually
become more rotation supported at later times.

####  8.2.6 disc

Figure 11 refers to the clump positions in the host galactic discs. The
initial values of $r/R_{\rm d}$ and $|z|/H_{\rm d}$ (at $\tau\\!\sim\\!2$) are
similar for the different clump types, indicating that the differences among
the types are not largely due to the position of formation within the disc.
The two types of L clumps migrate radially inwards at a rather slow pace, in
the ball park of the various theoretical predictions (Dekel, Sari & Ceverino,
2009; Krumholz & Burkert, 2010; Krumholz et al., 2018; Dekel et al., 2020b).
The non-zero values of $|z|/H_{\rm d}$ and their large scatter reflect the
fact that the clumps oscillate about the disc plane. While the median position
of the LS clumps is at $|z|\\!\sim\\!\\!0.3H_{\rm d}$, the LL clumps get
somewhat closer to the disc mid-plane by $\tau\\!\sim\\!5$, and keep a median
distance of $|z|\\!\sim\\!\\!0.2H_{\rm d}$ or less till $\tau\\!\sim\\!30$.

The median LS discs tend to be more massive than the LL disks, but the disc
mass fraction compared to the total mass including stars and dark matter,
$\delta$, is somewhat larger for the LL clumps. This is consistent with a
shorter inward migration time in units of disc dynamical time for the LL
clumps (Dekel, Sari & Ceverino, 2009). The weak opposite trend in $r(t)/R_{\rm
d}$ in terms of $\tau$ is probably because $t_{\rm ff}$ is larger for the LS
clumps.

Figure 12:  Time evolution of $S$ (solid) and $\tau S$ (dashed), the
quantities that evaluate clump survivability against gas loss and total
disruption, respectively, for the three clump types. Left: $S$ as measured
directly from the simulations from $M_{\rm out}$, eq. (53). Right: $S$ as
measured from the physical parameters assuming energy balance, eq. (54)
(version 2 with $\gamma\\!=\\!1$). Values larger than 100 are set to 100.
During the first few dynamical times, the median $S$ values are significantly
smaller for the S clumps compared to the L clumps, consistent with the
expected disruption of the S clumps compared to the survival of the L clumps.
The median $S$ for the LS clumps declines to below unity after
$\tau\\!\sim\\!5\\!-\\!10$, while for the LL clumps it remains above unity
till $\tau\\!\sim\\!20$ and beyond. The median values of $\tau S$ distinguish
between the S and LS clumps, consistent with the destiny of complete
disruption for the S clumps and only gas removal for the LS clumps, leaving
behind bound stellar LS clumps.

Figure 13:  The probability distributions of the survivability parameter $S$
for the three clump types, as in Fig. 12 but during $\tau\\!=\\!3\\!-\\!5$.
Left: $S$ measured directly from the simulations, eq. (53). Most of the L
clumps have $S\\!>\\!1$ and most of the S clumps have $S\\!<\\!1$, with the
medians near $3$ and $0.5$ respectively. Right: $S$ measured from the physical
parameters, eq. (54) (version 2 with $\gamma\\!=\\!1$). Here all the LL clumps
have $S\\!\gg\\!1$, the LS clumps have $S$ distributed about unity, and the
vast majority of the S clumps have $S\\!\ll\\!1$, with the medians $\gg\\!1$,
$\sim\\!1$ and $\sim\\!0.1$ respectively.

##  9 Theory versus simulations

Using the simulations, we compute the survivability parameter $S$ as a
function of time since clump formation for the simulated clumps of the three
types. We compute $S$ in two alternative ways, which we term $S_{\rm dir}$ and
$S_{\rm mod}$. First, directly from the simulations, we determine $S$ from the
measured $M_{\rm out}(t)$, $M_{\rm c}$ and the correction factor $p$, using
the definition of $S$ in eq. (53). Alternatively, we compute $S$ from eq.
(54), based on our model assuming energy conservation and the different
sources of energy gain and loss, using the measured $V_{\rm c}$ and the
factors $f_{\rm sn}$ (eq. 56), $f_{\rm grav}$ (eq. 58) and $f_{\rm dis}$ (eq.
57).

Figure 12 shows the evolution of $S$ and of $\tau S$ for the three clump types
as computed by the two methods. Recall that $S$ is supposed to measure the
clump survivability against losing its gas mass by supernova feedback, thus
distinguishing between S and L clumps by $S$ values below and above unity,
respectively. The product $\tau S$ is expected to measure the clump
survivability against total disruption versus partial mass loss followed by a
bound stellar system, thus distinguishing between S and LS clumps, by $\tau S$
values below and above unity, respectively. Inspecting the medians, we see
that during the first few dynamical times $S$ is significantly smaller in the
S clumps compared to the L clumps, as expected. The median of $S$ for the LS
clumps, which starts well above unity at early times, declines to below unity
after $\tau\\!\sim\\!5\\!-\\!10$, while for the LL clumps it remains above
unity till $\tau\\!\sim\\!20$ or further, as anticipated. The median values of
$\tau S$ indeed distinguish between the S clumps that will disrupt completely
and the LS clumps that will keep their stars bound, as expected.

Viewing the survivability parameter from a different angle, Fig. 13 shows the
probability distributions of $S$ for the three clump types at the early phase
$\tau\\!=\\!3\\!-\\!5$. The values of $S$ as measured directly from the
simulations yield that most of the L clumps have $S\\!>\\!1$ and most of the S
clumps have $S\\!<\\!1$, with the medians near $3$ and $0.5$ respectively. The
calculation of $S$ based on the model assuming conservation of energy yields
that all the LL clumps have $S\\!\gg\\!1$, the LS clumps have $S$ distributed
about unity, and the vast majority of the S clumps have $S\\!\ll\\!1$, with
the medians near $\gg\\!1$, $1$ and $0.1$ respectively. We conclude that, in
general, $S$ serves its purpose as a predictor of clump types.

Comparing the measured $S$ and $\tau S$ by the two methods, we note that the
medians of these quantities indeed distinguish between the clump types in both
cases. We note however that in the direct calculation using eq. (53) the
median values of $\tau S$ for the S clumps are slightly above unity. This may
reflect the uncertainty in measuring $M_{\rm out}$. In particular, our
assumption that $V_{\rm out}^{2}\\!\simeq\\!2V_{\rm c}^{2}$ that enters eq.
(6) may be inaccurate. If we parametrize $V_{\rm out}^{2}=f_{\rm esc}\,2V_{\rm
c}^{2}$, we obtain $S_{\rm mod}\\!\propto\\!f_{\rm esc}$. A value of $f_{\rm
esc}\\!\sim\\!2$, or a range of values for the different clump types, as in
Fig. 9, may reduce the apparent differences between $S_{\rm dir}$ and $S_{\rm
mod}$. Such refinements of the model are deferred to future work.

##  10 Conclusion

Energetics considerations lead to a physical criterion for the survival versus
disruption of the giant clumps that dominate high-redshift galactic discs.
Depending on the disc and clump properties, the model predicts populations of
short-lived clumps (S) that lose their gas in a few free-fall times during
formation and long-lived clumps (L) that keep their gas for longer periods.
The latter are of two types, those that lose most of their gas to outflows in
$\\!\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!\\!10$ free-fall
times but keep gas-deficient, long-lived, bound stellar clumps (LS), and those
that keep most of their baryons for tens of free-fall times and remain star
forming roughly at a constant rate (LL).

Our model introduces a survivability parameter $S$, which can predict the
level of survivability against gas loss at a given time based on the clump and
disc properties and physical parameters that characterize the SFR and
feedback, distinguishing between S and L clumps. The quantity $\tau S$, where
$\tau\\!=\\!t/t_{\rm ff}$, then predicts the survivability against total
disruption by rapid gas loss, distinguishing S from LS clumps. These
quantities are defined as

$S=\frac{0.5\,M_{\rm c}}{M_{\rm out}(t)}\,,\quad\tau S\simeq\frac{0.5\,M_{\rm
c}}{\dot{M}_{\rm out}t_{\rm ff}}\,.$ (70)

Thus, the distinction between the three clump types based on the survivability
parameter $S$ during the first few clump free-fall times is as summarized in
Table 1, namely S clumps have $S\\!<\\!1$ and $\tau S\\!<\\!1$, LS clumps have
$S\\!\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!1$ but $\tau
S\\!\lower 2.58333pt\hbox{$\;\buildrel>\over{\sim}\;$}\\!1$, and LL clumps
have $S\\!>\\!1$ and $\tau S\\!>\\!1$. The model successfully reproduces the
distribution of clump properties in cosmological simulations, and provides a
physical basis for the existence of the three clump types in these
simulations.

The model considers the balance between the energies supplied by supernova
feedback and by gravitational interactions against the energy required for
overcoming the clump binding energy, the energy carried away by outflows, and
the dissipative losses of turbulence. The supernova input is based on a
generalization of the analysis by Dekel & Silk (1986). The clumps are assumed
to form by Toomre instability within VDI discs (Dekel, Sari & Ceverino, 2009).
If they manage to contract sufficiently, the clumps are assumed to reach
virial and Jeans equilibrium (Ceverino et al., 2012).

Three clump types
---
Type | Gas loss | Stars | $S$ | $\tau S$
Short-lived S | $\sim\\!t_{\rm ff}$ | none | $S\\!<\\!1$ | $\tau S\\!<\\!1$
Long-lived stars LS | $\sim\\!10t_{\rm ff}$ | bound | $S\\!\lower 2.58333pt\hbox{$\;\buildrel<\over{\sim}\;$}\\!1$ | $\tau S\\!\lower 2.58333pt\hbox{$\;\buildrel>\over{\sim}\;$}\\!1$
Long-lived LL | $\gg\\!t_{\rm ff}$ | bound | $S\\!>\\!1$ | $\tau S\\!>\\!1$
Table 1:  The three clump types are distinguished by the survivability
parameter during the first few clump free-fall times. $S\\!<\\!1$ predicts
significant gas loss, while $\tau S\\!<\\!1$ predicts rapid gas loss that
leads to total disruption.

We obtain $S\\!\simeq\\!(S^{\prime-1}\\!-\\!1)^{-1}$, namely the critical
value of $S^{\prime}$ corresponding to $S\\!=\\!1$ is $S^{\prime}=0.5$. For a
simple interpretation of the main results, considering only the balance
between the deposited supernova energy and the clump binding energy, we
approximate near the first free-fall time

$S^{\prime}\simeq\frac{E_{\rm vir}}{E_{\rm sn}}\simeq\frac{V_{\rm
c,50}}{e_{51}}\,.$ (71)

The linear dependence on the clump circular velocity $V_{\rm c}$ results from
the fact that $E_{\rm vir}\propto V_{\rm c}^{2}$ while $E_{\rm sn}\propto
V_{\rm c}$ (via its dependence on the velocity dispersion in the clump ISM).
The survivability is thus a strong function of the clump circular velocity
(which is correlated with the clump mass) and the feedback strength as
formulated via $e_{51}$. In turn, a Toomre analysis connects the clump
circular velocity to the disc properties via

$V_{\rm c}\propto f_{\rm g}\,V_{\rm d}\,.$ (72)

This introduces strong gas-fraction dependence and disc mass dependence for
the clump survival via $S$.

If the strength of ejective feedback is at a moderate level, we find that L
clumps are likely to exist, with a binding energy that is characterized by a
circular velocity $V_{\rm c}\\!\sim\\!50\,{\rm km}\,{\rm s}^{-1}$, at or above
a clump threshold mass $\sim\\!10^{8}M_{\odot}$. The L clumps tend to form in
discs of $V_{\rm d}\\!\geq\\!200\,{\rm km}\,{\rm s}^{-1}$, corresponding at
$z\\!\sim\\!2$ to stellar masses above $\sim\\!10^{9.3}M_{\odot}$ and galactic
halo masses above $\sim\\!10^{11.3}M_{\odot}$, which are typically required
for long-lived discs against disruption by mergers in an orbital time (as
predicted by Dekel et al., 2020a). L clumps are predicted to favor gas
fractions $\\!\geq\\!\\!0.3$ and more so when the central mass of bulge and
dark matter is small. The L clumps prefer to form at high redshifts, peaking
at $z\\!\sim\\!2$.

On the other hand, the likelihood of L clumps is severely reduced if the
ejective feedback is stronger, e.g., if the effective energy per supernova is
well above the standard value due to clustering of supernovae, if very strong
radiative feedback is included, if the stellar initial mass function is top-
heavy, or if the star-formation-rate efficiency is higher than commonly
assumed on average. Simulations with varying levels of ejective feedback
indeed permit L clumps with different efficiencies.

The division of the L clumps into two sub-populations is demonstrated in the
simulations. The model may explain the existence of LS clumps by a smaller
contraction factor during formation and stronger external gravitational
effects, where clump mergers may increase the SFR efficiency. This is compared
to the more compact LL clumps that retain most of their baryons for tens of
free-fall times.

A word of caution concerns the fact that we have modeled the overall stellar
feedback based on the supernova feedback component. This is a first crude
treatment that should be improved by detailed considerations of the other
feedback mechanisms, such as radiative feedback and stellar winds.
Furthermore, our current treatment ignored the clustering of supernovae, which
is expected to increase the deposited feedback energy in low-mass star
clusters and to decrease it in massive clusters (Gentry et al., 2017). These
should be improved in future studies.

Our findings emphasize a challenging general feedback puzzle. While the model
successfully predicts that for a moderate feedback strength the massive clumps
are long-lived, when implemented in simulations (such as VELA-3) this moderate
feedback leads to an overestimate of the stellar-to-halo mass ratio as
estimated from observations via abundance matching (Rodríguez-Puebla et al.,
2017; Moster, Naab & White, 2018; Behroozi et al., 2019). On the other hand,
simulations with stronger feedback (such as VELA-6), which better match the
stellar-to-halo mass ratio, practically fail to reproduce long-lived clumps.
If the observed massive clumps are long-lived, as indicated, for example, by
their not-so-young stellar ages and the gradients in clump properties that are
consistent with VDI-driven clump radial migration (Dekel et al., 2021), then
one is challenged to come up with more sophisticated feedback mechanisms. The
feedback should be more preventive and less ejective, such that it efficiently
suppresses star formation while it is less destructive in terms of ejecting
gas from the clumps.

## Acknowledgments

This work was supported by the Israel Science Foundation Grants ISF 861/20
(AD) and 3061/21 (NM), and by Germany-Israel DIP grant STE1869/2-1 GE625/17-1
(AD). OG is supported by a Milner Fellowship. DC is a Ramon-Cajal Researcher
and is supported by the Ministerio de Ciencia, Innovación y Universidades
(MICIU/FEDER) under research grant PID2021-122603NB-C21. The cosmological
simulations were performed at the National Energy Research Scientific
Computing centre (NERSC), Lawrence Berkeley National Laboratory, and at NASA
Advanced Supercomputing (NAS) at NASA Ames Research Centre. Development and
analysis have been performed in the computing cluster at the Hebrtew
University.

## DATA AVAILABILITY

Data and results underlying this article will be shared on reasonable request
to the corresponding author.

## References

  * Agertz, Teyssier & Moore (2009) Agertz O., Teyssier R., Moore B., 2009, MNRAS, 397, L64
  * Behroozi et al. (2019) Behroozi P., Wechsler R. H., Hearin A. P., Conroy C., 2019, MNRAS, 488, 3143
  * Botticella et al. (2017) Botticella M. T. et al., 2017, A&A, 598, A50
  * Bournaud, Elmegreen & Elmegreen (2007) Bournaud F., Elmegreen B. G., Elmegreen D. M., 2007, ApJ, 670, 237
  * Cacciato, Dekel & Genel (2012) Cacciato M., Dekel A., Genel S., 2012, MNRAS, 421, 818
  * Ceverino et al. (2016a) Ceverino D., Arribas S., Colina L., Rodríguez Del Pino B., Dekel A., Primack J., 2016a, MNRAS, 460, 2731
  * Ceverino, Dekel & Bournaud (2010) Ceverino D., Dekel A., Bournaud F., 2010, MNRAS, 404, 2151
  * Ceverino et al. (2012) Ceverino D., Dekel A., Mandelker N., Bournaud F., Burkert A., Genzel R., Primack J., 2012, MNRAS,
  * Ceverino & Klypin (2009) Ceverino D., Klypin A., 2009, ApJ, 695, 292
  * Ceverino et al. (2014) Ceverino D., Klypin A., Klimek E. S., Trujillo-Gomez S., Churchill C. W., Primack J., Dekel A., 2014, MNRAS, 442, 1545
  * Ceverino, Primack & Dekel (2015) Ceverino D., Primack J., Dekel A., 2015, MNRAS, 453, 408
  * Ceverino et al. (2017) Ceverino D., Primack J., Dekel A., Ka ssin S. A., 2017, MNRAS, 467, 2664
  * Ceverino et al. (2016b) Ceverino D., Sánchez Almeida J., Muñoz Tuñón C., Dekel A., Elmegreen B. G., Elmegreen D. M., Primack J., 2016b, MNRAS, 457, 2605
  * Daddi et al. (2010) Daddi E. et al., 2010, ApJ, 713, 686
  * Dekel & Birnboim (2006) Dekel A., Birnboim Y., 2006, MNRAS, 368, 2
  * Dekel & Burkert (2014) Dekel A., Burkert A., 2014, MNRAS, 438, 1870
  * Dekel et al. (2020a) Dekel A., Ginzburg O., Jiang F., Freundlich J., Lapiner S., Ceverino D., Primack J., 2020a, MNRAS, 493, 4126
  * Dekel & Krumholz (2013) Dekel A., Krumholz M. R., 2013, MNRAS, 432, 455
  * Dekel, Lapiner & Dubois (2019) Dekel A., Lapiner S., Dubois Y., 2019, arXiv e-prints
  * Dekel et al. (2020b) Dekel A. et al., 2020b, MNRAS, 496, 5372
  * Dekel et al. (2021) Dekel A., Mandelker N., Bournaud F., Ceverino D., Guo Y., primack J., 2021, arXiv e-prints, arXiv:2107.13561
  * Dekel, Sari & Ceverino (2009) Dekel A., Sari R., Ceverino D., 2009, ApJ, 703, 785
  * Dekel et al. (2019) Dekel A., Sarkar K. C., Jiang F., Bournaud F., Krumholz M. R., Ceverino D., Primack J. R., 2019, arXiv e-prints
  * Dekel & Silk (1986) Dekel A., Silk J., 1986, ApJ, 303, 39
  * Elmegreen & Elmegreen (2005) Elmegreen B. G., Elmegreen D. M., 2005, ApJ, 627, 632
  * Fensch & Bournaud (2021) Fensch J., Bournaud F., 2021, MNRAS, 505, 3579
  * Fisher et al. (2017) Fisher D. B. et al., 2017, ApJ, 839, L5
  * Forbes, Krumholz & Burkert (2012) Forbes J., Krumholz M., Burkert A., 2012, ApJ, 754, 48
  * Forbes et al. (2014) Forbes J. C., Krumholz M. R., Burkert A., Dekel A., 2014, MNRAS, 438, 1552
  * Förster Schreiber et al. (2006) Förster Schreiber N. M. et al., 2006, ApJ, 645, 1062