# Detecting Population III Stars through Tidal Disruption Events in the Era of
JWST and Roman

Rudrani Kar Chowdhury Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong  Janet N. Y. Chang Department of Physics, The
University of Hong Kong, Pokfulam Road, Hong Kong  Lixin Dai Department of
Physics, The University of Hong Kong, Pokfulam Road, Hong Kong  Priyamvada
Natarajan Department of Astronomy, Yale University, New Haven, CT 06511, USA
Department of Physics, Yale University, New Haven, CT 06520, USA  Black Hole
Initiative at Harvard University, 20 Garden St., Cambridge MA, 02138, USA

###### Abstract

The first generation metal-free stars, referred to as population III (Pop III)
stars, are believed to be the first objects to form out of the pristine gas in
the very early Universe. Pop III stars have different structures from the
current generation of stars and are important for generating heavy elements
and shaping subsequent star formation. However, it is very challenging to
directly detect Pop III stars given their high redshifts and short life-times.
In this paper, we propose a novel signature for detecting Pop III stars
through their tidal disruption events (TDEs) by massive black holes. We model
the emission properties and calculate the expected rates for these unique TDEs
in the early Universe at $z\sim 10$. We find that Pop III star TDEs have much
higher mass fallback rates compared to normal TDEs in the local universe and
are therefore rather luminous, rendering them feasible for detection. They
also have very long observed flare evolution timescale, making it more likely
to detect such TDEs during their rising phase. We further demonstrate that a
large fraction of the TDE emissions are redshifted to infrared wavelengths and
can be detected by the James Webb Space Telescope and the Nancy Grace Roman
Space Telescope. Lastly, the TDE rate sensitively depends on the black hole
mass function in the early Universe. We find a promising Pop III star TDE
detection rate of up to a few tens per year using the Nancy Grace Roman Space
Telescope.

##  1 Introduction

The first generation of stars are believed to form from the pristine gas a few
hundred million years after the Big Bang. These metal-free stars are commonly
referred to as population III (Pop III) stars. Pop III stars are believed to
form in collapsed dark matter mini-halos with masses of around
$10^{6}M_{\odot}$ at $z\sim 15-20$ (Abel et al., 2002; Bromm et al., 2002;
Yoshida et al., 2006). The initial mass function (IMF) of these short-lived
Pop III stars, is expected to be tilted to higher masses and hence they are
likely more massive than the current generation of forming stars (Population
I, or Pop I stars). While there is uncertainty at the present time regarding
the Pop III IMF, their masses are believed to lie within the range of
$30M_{\odot}-300M_{\odot}$. Pop III stars are expected to be key players in
the early Universe and they are believed to be responsible for re-ionizing the
inter galactic medium (IGM) (Sokasian et al., 2004; Johnson et al., 2007;
Kulkarni et al., 2014), and lifting the fog of the dark ages and jump-starting
structure formation. Moreover, Pop III stars produce metals in their core and
therefore play a crucial role in bridging the gap between the metal-free
primordial gas and higher-metallicity Population II (Pop II) and Pop I stars
by polluting the IGM through supernovae explosions (Chen et al., 2022). Hence,
as metal polluters that shape subsequent generations of star formation, Pop
III stars are an important population to detect directly as this will enable
constraining their properties and better understand the evolution of
subsequent generations of stellar populations.

A plethora of numerical studies have been carried out to track the formation
of Pop III stars in the first few billion years of the Universe (Abel et al.,
2002; Bromm et al., 2009; Park et al., 2021a; Saad et al., 2022; Prole et al.,
2023). In particular, recent studies show that Pop III stars do not form in
isolation, and that typically multiple Pop III stars can form inside a single
minihalo (Clark et al., 2011; Liu & Bromm, 2020; Park et al., 2021b). While it
is generally believed that the Pop III star formation rate peaks around $z\sim
10$ (Scannapieco et al., 2003; Jaacks et al., 2018), there has been
speculation that they could continue forming at lower redshifts if pristine
gas pockets are available in galaxies (Jimenez & Haiman, 2006; Liu & Bromm,
2020).

Direct observation of Pop III stars, however, turns out to be extremely
challenging. So far only a few Pop III like candidate objects have been
identified (Vanzella et al., 2020; Welch et al., 2022): detected through
serendipitous gravitational lensing (Rydberg et al., 2013; Vikaeus et al.,
2022b); in pair-instability supernovae (Hummel et al., 2012; Regős et al.,
2020); in gamma ray bursts (de Souza et al., 2011; Mesler et al., 2014; Lazar
& Bromm, 2022) and recent claimed detection with JWST NIRSpec observations
(Wang et al., 2022a).

In this paper, we propose and explore in detail a novel detection channel for
Pop III stars motivated by the unfolding JWST observations and evidence for
the possible existence of massive black holes (MBHs) formed via direct
collapse of gas in place at these early epochs (Bogdán et al., 2023; Natarajan
et al., 2024). Therefore, Pop III stars stand to be revealed via the
observation of their tidal disruption by MBHs in early galaxies. Recent JWST
observations in combination with Chandra X-ray observations, have revealed and
confirmed the presence of an accreting supermassive black hole (SMBH) in the
source UHZ1 at $z\approx 10.1$, a mere 470 Myrs after the Big Bang (Bogdán et
al., 2023) that is consistent with formation via direct collapse of gas.

Tidal Disruption Events (TDEs) are produced whenever a star approaches a
massive black hole (MBH) and comes within the distance at which the tidal
force of the black hole exceeds the self-gravity of the star. In such cases,
the star will be destroyed by the tidal force, which in turn will produce a
luminous tidal disruption event (TDE) (Rees, 1988; Evans & Kochanek, 1989). At
the present time about 100 TDEs of Pop I stars in the local Universe have been
observed (Gezari, 2021), and these are typically associated with powerful
emission observed in multiple wavebands including X-ray, ultraviolet (UV)
optical and radio (Auchettl et al., 2017; Alexander et al., 2020; Saxton et
al., 2020; van Velzen et al., 2021).

In this paper, we model the properties of Pop III star TDEs. As these first
stars are expected to exist primarily in high-redshift galaxies, their optical
and UV emission will be redshifted rendering them detectable in infrared (IR)
wavelengths. This study is extremely timely due to the recent launch and
successful data stream from the James Webb Space Telescope (JWST, Gardner et
al., 2006) and planning underway for the soon to be launched Nancy Grace Roman
Space Telescope (Roman, Spergel et al., 2015). JWST and Roman operate in IR
bands, covering the wavelength range of $600$ nm - $28000$ nm and $480$ nm
-$2300$ nm respectively. With the high resolution Near Infrared Camera
(NIRCam) and Near Infrared Spectrograph (NIRSpec) along with Mid-Infrared
Instrument (MIRI), JWST has already started to revolutionize our understanding
of the first galaxies and early MBHs (e.g., Castellano et al., 2022; Atek et
al., 2023; Yajima et al., 2023; Bogdán et al., 2023). On the other hand, the
planned extremely large field of view of Wide Field Instrument (WFI) on board
Roman is designed to observe a vast area of the sky, making it possible to
detect numerous faint objects simultaneously. One of the primary objectives of
Roman is to shed light on the matter distribution over a large range of
redshifts to understand the nature of dark matter and expansion history of the
universe. Both missions are hence expected to yield a large sample of
extremely faint high redshift sources.

Our paper is structured as follows: in Section 2, we introduce the basic
structure of Pop III stars, based on which we calculate the key TDE
parameters. We also introduce the fiducial model used to calculate the
emission properties of these Pop III TDEs. In Section 3, we present our main
results, predictions for the intrinsic and the observed emission spectra;
luminosities and the light curves of Pop III star TDEs. We also calculate
event rates and discuss the detection prospects for this new class of sources
with JWST and Roman. We conclude with a summary of our key findings and
implications of our results for unveiling the early Universe with these new
probes in Section 4.

##  2 Modelling Pop III TDEs

###  2.1 Mass-Radius Relation of Pop III Stars

We consider Pop III stars in the main sequence stage. Massive Pop III stars
are primarily radiation-pressure dominated, so their structures can be
approximated using a polytropic model with an index of $\gamma=4/3$ or
$n=\frac{1}{\gamma-1}=3$ (Bromm et al., 2001b). Throughout this work, we use
the mass-radius relationship of Pop III stars adopted from Bromm et al.
(2001b):

$R_{\star}\simeq
0.7R_{\odot}\left(\frac{M_{\star}}{M_{\odot}}\right)^{0.45}\left(\frac{Z}{10^{-9}}\right)^{0.09},$
(1)

where $M_{\star}$ is the stellar mass, $R_{\star}$ is the stellar radius, and
$Z$ is the metallicity of Pop III stars. When $Z$ increases, cooling through
metal lines is enhanced and the effective temperature of a Pop III star drops,
which means the average stellar density also decreases given
$\rho_{\star}\propto T^{3}$ for a polytropic star with $n=3$ (Fowler & Hoyle,
1964). Therefore, $R_{\star}$ increases with increasing $Z$. The value of the
metallicity $Z$ considered has varied from zero to a very small non-zero
number in previous treatments (Bromm et al., 2001b; Schaerer, 2003; Murphy et
al., 2021; Klessen & Glover, 2023). Many studies show that there likely exists
a critical metallicity at which the transition from metal-free Pop III stars
to metal-poor PopII stars occurs and that value is $\sim
10^{-3}-10^{-5}Z_{\odot}$ depending on the IMF of Pop III stars (Bromm et al.,
2001a; Schneider et al., 2002; Yoshida et al., 2004; Wise et al., 2012; Jaacks
et al., 2018). Furthermore, Jaacks et al. (2018) show that the mean gas
metallicity increases with decreasing redshift $z$ and reaches $\sim
10^{-5}Z_{\odot}$ at $z\sim 10$. Hereafter, in this work in order to probe the
metallicity dependence of our results, we calculate fiducial properties for
Pop III stars with two chosen values for the metallicity: $Z=10^{-5}$ and a
lower metallicity of $Z=10^{-9}$.

Given the stellar structure outlined above, we can calculate the the average
stellar density $\rho_{\star}$ of Pop III stars based on Eq. 1. In Fig. 1, we
show the variation of $\rho_{\star}$ for Pop III stars as a function of their
mass in the typical mass range of $30-300M_{\odot}$ and compare that with the
case of Pop I main-sequence stars. For the latter, we assume that their masses
can extend to few hundreds of $M_{\odot}$ (Crowther et al., 2010; Rickard &
Pauli, 2023), and we adopt the mass-radius relation from Kippenhahn & Weigert
(1990):

$R_{\star}\sim\begin{cases}R_{\odot}\left(\frac{M_{\star}}{M_{\odot}}\right)^{0.8};&M_{\star}\leq
1M_{\odot}\\\
R_{\odot}\left(\frac{M_{\star}}{M_{\odot}}\right)^{0.57};&M_{\star}>1M_{\odot}\end{cases}$
(2)

It can be seen that for both types of stars, $\rho_{\star}$ drops as the
$M_{\star}$ increases. Furthermore, Pop III stars have
$\rho_{\star}\sim\rho_{\odot}$ for the lower metallicity case $Z\sim 10^{-9}$
and $\rho_{\star}\sim 0.1\rho_{\odot}$ as $Z$ approaches the upper limit of
$10^{-5}$.

Figure 1: Average density ($\rho_{\star}$) of Pop III stars with $Z=10^{-5}$
(red solid line) and $Z=10^{-9}$ (red dashed line) in the mass range of
$30-300M_{\odot}$ compared to Pop I stars (black solid line) in the mass range
of $0.1-300M_{\odot}$. All stars are in their main sequence stages. Pop III
stars typically have $\rho_{\star}$ $\sim 0.01-1\ \rho_{\odot}$ and those with
lower metallicities or lower masses are denser. The stellar density of
$1\rho_{\odot}$ , $0.1\rho_{\odot}$ and $0.01\rho_{\odot}$ are marked with
green dotted lines.

###  2.2 Tidal Disruption of Pop III Stars

A star is tidally disrupted by a MBH when it approaches within its tidal
disruption radius ($R_{T}$). To first order, $R_{T}$ can be calculated using

$R_{T}=R_{\star}\left(\frac{M_{\rm BH}}{M_{\star}}\right)^{1/3},$ (3)

where $M_{\rm BH}$ is the black hole (BH) mass. Therefore, for a fixed $M_{\rm
BH}$, $R_{T}$ only depends on the average stellar density
($R_{T}\propto\rho_{\star}^{-1/3}$). Fig. 2 shows the comparison of $R_{T}$
for Pop III stars (of different masses and metallicities) and with that for
solar type stars disrupted by MBHs with mass ranging from $M_{\rm
BH}=10^{5}-10^{9}M_{\odot}$. It can be seen that $R_{T}$ for Pop III stars at
very low metallicity ($Z\sim 10^{-9}$) are similar to that of the Sun, while
$R_{T}$ for Pop III stars at relatively high metallicities ($Z\sim 10^{-5}$)
are a few times larger, which is consistent with the density comparison seen
in Fig. 1.

Moreover, since $R_{T}\propto M_{\rm BH}^{1/3}$ while the gravitational radius
of a BH $R_{g}\equiv GM_{\rm BH}/c^{2}\propto M_{\rm BH}$, there exists a
upper limit of $M_{\rm BH}$ (called the Hill mass) beyond which an approaching
star is swallowed by the MBH as a whole. One can see in Fig. 2 that the high-
metallicity ($Z\sim 10^{-5}$) Pop III stars can be disrupted by MBHs with
masses up to $10^{9}M_{\odot}$. This can be used as an important indicator for
Pop III TDE detection, since Pop I main-sequence stars can only be disrupted
by MBHs with $M_{\rm BH}\lesssim 10^{8}M_{\odot}$ except in the extreme case
when the MBH has a close to maximal spin (Kesden, 2012). Although the
estimated mass of the BH in sources like UHZ1 (at $z\approx 10.1$) is
estimated to be $\sim 4\times 10^{7}M_{\odot}$, it is expected that even more
massive SMBHs could exist at these epochs although they are expected to be
extremely rare. Meanwhile, MBHs with masses less than that of UHZ1 could be
significantly more numerous at these early epochs.

Figure 2: Tidal disruption radius $R_{T}$ (scaled by the BH gravitational
radius $R_{g}$) of Pop III stars and a solar-type star as a function of BH
mass $M_{\rm BH}$. The red and blue solid or dashed lines represent Pop III
stars of mass $300M_{\odot}$ ($M_{300}$) and $30M_{\odot}$ ($M_{30}$)
respectively with metallicity $Z=10^{-5}$ ($Z_{-5}$) or $10^{-9}$ ($Z_{-9}$).
The black solid line shows a solar-type star for comparison. It can be seen
that a Pop III star, depending on its metallicity, has $R_{T}$ similar to or a
few times larger than that of the Sun. The green dotted line indicates
$R_{T}=R_{g}$, which gives the maximum mass of $M_{\rm BH}$ for a star to be
disrupted inside the MBH event horizon. Figure 3: Left panel: Debris mass
fallback time $t_{\rm fb}$ for tidal disruption of $300M_{\odot}$ ($M_{300}$,
red) and $30M_{\odot}$ ($M_{30}$, blue) Pop III stars with $Z=10^{-5}$
($Z_{-5}$, solid lines) and $Z=10^{-9}$ ($Z_{-9}$, dashed lines) compared to a
regular $1M_{\odot}$ star (black solid line). Right panel: Eddington ratio of
the peak debris mass fallback rate $\dot{M}_{\rm fb,peak}$ for the same stars
as in the left panel. Pop III stars have shorter fallback times and much
higher peak mass fallback rates compared to Pop I star TDEs.

After the star is disrupted, about half of the stellar debris is ejected from
the system and the rest remains bound to the MBH with a spread in its specific
binding energy. The rate that the bound stellar debris orbits back to the
pericenter is called the debris mass fallback rate ($\dot{M}_{\rm fb}$), which
can be calculated as (Evans & Kochanek, 1989; Phinney, 1989):

$\dot{M}_{\rm fb}(t)\approx\frac{1}{3}\frac{M_{\star}}{t_{\rm
fb}}\left(\frac{t}{t_{\rm fb}}\right)^{-5/3},$ (4)

where the fallback time $t_{\rm fb}$ is the orbital time of the most tightly
bound debris.

In this work we adopt the results from Guillochon & Ramirez-Ruiz (2013)
(hereafter GR13) who performed high-resolution hydrodynamical simulations to
study this disruption process. In particular, adopting polytropic stellar
models, GR13 provide the following fitting formulae for the debris fallback
time and the peak mass fallback rate:

$t_{\rm fb}=B_{\gamma}M_{6}^{1/2}m_{\star}^{-1}r_{\star}^{3/2}~{}\rm yr$ (5)

$\dot{M}_{\rm
fb,peak}=A_{\gamma}M_{6}^{-1/2}m_{\star}^{2}r_{\star}^{-3/2}~{}M_{\odot}\ \rm
yr^{-1}$ (6)

where $M_{6}=M_{\rm BH}/10^{6}M_{\odot}$ , $m_{\star}=M_{\star}/M_{\odot}$ and
$r_{\star}=R_{\star}/R_{\odot}$. Furthermore, for a poly-tropic star with
$\gamma=4/3$, the coefficients are:

$B_{4/3}=\frac{-0.38670+0.57291\sqrt{\beta}-0.31231\beta}{1-1.2744\sqrt{\beta}-0.90053\beta},$
(7)
$A_{4/3}=\rm{exp}\left[\frac{27.261-27.516\beta+3.8716\beta^{2}}{1-3.2605\beta-1.3865\beta^{2}}\right].$
(8)

Here $\beta\equiv R_{T}/R_{p}$ is called the penetration factor, where $R_{p}$
is the pericenter distance of the stellar orbit. A deep plunging orbit is
commonly associated with $\beta\gg 1$, whereas mild or partial disruption is
denoted by $\beta\sim 1$ & $\beta\lesssim 1$ respectively. GR13 showed that
stars with polytropic index $\gamma=4/3$ are fully disrupted when
$\beta\gtrsim 1.85$, and under this condition $\dot{M}_{\rm fb,peak}$ remains
similar. Hence we mark this as the critical penetration factor $\beta_{c}$ and
use $\beta=\beta_{c}=1.85$ for the calculations throughout this paper unless
otherwise specified. For $\beta=1.85$, $B_{4/3}\sim 0.08$, and $A_{4/3}\sim
3$.

For Pop III stars, one can then use the mass-radius relationship (Eq. 1) and
rewrite Eq. 5 and 6 as:

$t_{\rm fb}\simeq
9\left(\frac{B_{4/3}}{0.08}\right)\left(\frac{m_{\star}}{300}\right)^{-0.3}\left(\frac{Z}{10^{-5}}\right)^{0.1}M_{6}^{0.5}~{}\rm
day$ (9) $\begin{split}\dot{M}_{\rm fb,peak}\simeq 2.7\times
10^{3}\left(\frac{A_{4/3}}{3}\right)\left(\frac{m_{\star}}{300}\right)^{1.3}\left(\frac{Z}{10^{-5}}\right)^{-0.1}\\\
M_{6}^{-0.5}~{}M_{\odot}\ \rm yr^{-1}\end{split}$ (10)

We plot $t_{\rm fb}$ and $\dot{M}_{\rm fb,peak}$ as functions of $M_{\rm BH}$
for various Pop III stars compared to the Sun in Fig. 3. It can be seen that
$t_{\rm fb}$ for Pop III stars are usually around a few days, which is shorter
compared to that of a solar-type star wherein $t_{\rm fb}$ $\sim$ a few tens
of days for the case of $M_{\rm BH}=10^{6}M_{\odot}$. Furthermore, Pop III
stars have extremely high $\dot{M}_{\rm fb,peak}$ which exceeds that of Pop I
star TDEs by several orders of magnitude. For example, when $M_{\rm
BH}=10^{6}M_{\odot}$, for Pop III star TDEs $\dot{M}_{\rm fb,peak}$ can reach
$10^{4-6}\dot{M}_{\rm Edd}$, where $\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/\eta
c^{2}$ is the Eddington accretion rate of the BH with $L_{\rm Edd}$ being the
Eddington luminosity, $c$ the speed of light and $\eta$ is the radiative
efficiency with a nominal value of 0.1. This hyper-Eddington debris mass
fallback rate is a consequence of the very large masses of Pop III stars
coupled with their short debris mass fallback timescales.

After the peak of the flare, the debris mass fallback rate drops with time
following $t^{-5/3}$. At late times, the fallback rate should transit from
super-Eddington to sub-Eddington. We can calculate the timescale over which
the fallback rate stays super-Eddington using $\dot{M}_{\rm fb}=\dot{M}_{\rm
Edd}$:

$t_{\rm Edd}\approx
38.5\left(\frac{m_{\star}}{300}\right)^{3/5}\left(\frac{t_{\rm fb}}{10~{}\rm
day}\right)^{2/5}M_{6}^{-3/5}\ \rm yr$ (11)

This timescale is shown in Fig. 4. It is clearly seen that Pop III TDEs have
super-Eddington fallback rates for a much longer duration (up to hundreds of
years) compared to a solar type star TDE (a few years) considering $M_{\rm
BH}=10^{6}M_{\odot}$.

Figure 4: The Eddington timescale defined as the timescale that the debris
mass fallback rate stays above the Eddington accretion rate. It is seen that
Pop III star TDEs can stay super-Eddington for very long timescales (up to
hundreds of years).

###  2.3 TDE Optical/UV Emission Model

The extremely high debris mass fallback rate seen in Fig. 3 means that even if
just a small fraction of the debris reaches the vicinity of the MBH, a super-
Eddington accretion flow could result, which will launch powerful winds due to
the large radiation pressure (Dai et al., 2018; Bu et al., 2023). Furthermore,
outflows are also expected to be powered by debris stream collisions (Shiokawa
et al., 2015; Bonnerot & Rossi, 2019; Lu & Bonnerot, 2020).

It has been proposed that these outflows are responsible for producing the
optical/UV emission observed from TDEs in the local universe (Loeb & Ulmer,
1997; Strubbe & Quataert, 2009; Lodato & Rossi, 2011; Roth et al., 2016;
Metzger & Stone, 2016; Roth et al., 2020; Dai et al., 2021). There have also
been sophisticated numerical simulations studying super-Eddington BH
accretion, outflow and emission (Sadowski & Narayan, 2016; Dai et al., 2018;
Jiang et al., 2019; Thomsen et al., 2022). However, no simulation has been
done yet to study super-Eddington accretion flows with Eddington ratios as
high as $10^{5-7}$. Given lack of guidance from simulations, we adopt an
analytical model proposed by Strubbe & Quataert (2009, hereafter SQ09) to
calculate the emission properties of Pop III star TDEs, while making a few
changes to adapt model parameters consistent with more recent studies.

Figure 5: The time that the photo-sphere transits from expansion to recession
($t_{\rm edge}$) as a function of $M_{\rm BH}$ for various types of stars.
Colour scheme is the same as Fig. 3. $t_{\rm edge}$ is longer for the Pop III
stars compared to solar-type stars.

We first provide a brief summary of the SQ09 model below. SQ09 assume that
during the early, super-Eddington phase a sizable fraction of the fallback
material constitutes an outflow. Furthermore, they also assume that a fraction
of the wind kinetic energy is converted to thermal energy, which in turn
produces black-body radiation. This assumption appears to hold even when the
outflows are powered by debris stream collisions, and it is supported by
recent simulations (Dai et al., 2018; Zanazzi & Ogilvie, 2020; Andalman et
al., 2022; Thomsen et al., 2022). SQ09 assume a spherical wind geometry and a
constant wind velocity, under these circumstances, the wind density profile
can be approximated as:

$\rho(r)=\begin{cases}\dot{M}_{\rm wind}/(4\pi r^{2}v_{\rm wind}),&r\lesssim
R_{\rm edge}\\\ 0,&\text{outside}\end{cases}$ (12)

Here the wind mass rate $\dot{M}_{\rm wind}$ is assumed to be a constant
fraction of the debris mass fallback rate so that $\dot{M}_{\rm out}\equiv
f_{\rm out}\dot{M}_{\rm fb}$. We use a fiducial value $f_{\rm out}=0.5$
(instead of $f_{\rm out}=0.1$ used by SQ09), inspired by recent numerical
simulations of super-Eddington accretion flows (e.g. Dai et al., 2018; Jiang
et al., 2019; Thomsen et al., 2022). We recognize that the exact value of
$f_{\rm out}$ has a dependence on the Eddington ratio, but our results do not
significantly vary when $f_{\rm out}>0.5$. $R_{\rm edge}\equiv v_{\rm wind}t$
denotes the edge of the wind. The wind velocity $v_{\rm wind}$ is assumed to
be the escape velocity $v_{\rm esc,L}\equiv\sqrt{2GM_{\rm BH}/R_{L}}$ at the
wind-launching radius $R_{L}$, which is set to be $2R_{p}$ (the
circularization radius). Therefore, we have $v_{\rm esc,L}=\sqrt{GM_{\rm
BH}/R_{p}}$, which can be further simplified as:

$v_{\rm esc,L}=0.1\beta^{1/2}m_{\star}^{1/6}r_{\star}^{-1/2}M_{6}^{1/3}c$ (13)

Furthermore, SQ09 calculate the temperature of the wind by assuming that the
gas thermal energy density and kinetic energy density are similar at the wind-
launching site:

$aT_{L}^{4}=\frac{1}{2}\rho_{\rm fb,L}v_{\rm esc,L}^{2}.$ (14)

Here $a=\sigma/4$ with $\sigma$ being the Stefan-Boltzmann constant, $T_{L}$
is the gas temperature at $R_{L}$, and $\rho_{\rm fb,L}$ is the gas density at
$R_{L}$ which can be calculated as $\rho_{\rm fb,L}\equiv\dot{M}_{\rm
fb}/(4\pi R_{L}^{2}v_{\rm esc,L})$. It is further assumed that the wind
expands adiabatically so that its temperature scales with gas density:

$T\propto\rho^{1/3}.$ (15)

The photosphere radius of the wind, $R_{\rm ph}$, is located where

$R_{\rm ph}\kappa_{s}\rho(R_{\rm ph})\sim 1,$ (16)

where $\kappa_{s}$ is the electron scattering opacity which is taken to be
$0.35\ \rm cm^{2}g^{-1}$ assuming the Hydrogen abundance for Pop III stars is
$\sim 75\%$, while rest of the mass is mostly dominated by Helium (Bromm et
al., 2009).

Initially, the wind gas density is very high due to the large fallback rate,
so $R_{\rm edge}\kappa_{s}\rho(R_{\rm edge})\gg 1$ and the photo-sphere almost
coincides with the edge of the wind, which gives :

$R_{\rm ph}\approx R_{\rm edge}=1.3\times
10^{3}f_{v}\beta^{1/2}m_{\star}^{1/6}r_{\star}^{-1/2}M_{6}^{-2/3}\left(\frac{t}{\rm{day}}\right)~{}R_{s}$
(17)

where $R_{s}$ is the Schwarzschild radius of a non-spinning black hole. The
effective temperature of the photo-sphere at this stage is given by:

$\begin{split}T_{\rm ph}\approx 1.5\times
10^{4}f_{v}^{-1/3}m_{\star}^{-7/72}r_{\star}^{1/24}M_{6}^{5/36}\\\
\left(\frac{t}{\rm day}\right)^{-7/36}\left(\frac{t_{\rm fb}}{\rm
yr}\right)^{-1/18}\rm K\end{split}$ (18)

As $R_{\rm edge}\kappa_{s}\rho(R_{\rm edge})\propto\dot{M}_{\rm wind}/R_{\rm
edge}$ decreases with $t$, after a certain time, $R_{\rm
edge}\kappa_{s}\rho(R_{\rm edge})$ drops to 1 and the photo-sphere starts to
recede afterwards. This transition time is calculated to be:

$t_{\rm edge}\approx 9.6f_{\rm
out,0.5}^{3/8}f_{v}^{-3/4}B_{\gamma}^{1/4}M_{6}^{1/8}R_{p,3R_{s}}^{3/8}m_{\star}^{1/8}r_{\star}^{3/8}~{}\rm
day$ (19)

where $f_{\rm out,0.5}\equiv f_{\rm out}/0.5$, and $f_{v}=v_{\rm wind}/v_{\rm
esc,L}$ (this fraction is set as $1$ throughout this paper unless otherwise
specified). As we will show later, $t_{\rm edge}$ is also the time when the
flare bolometric luminosity reaches the peak. We plot $t_{\rm edge}$ for Pop
III stars and the Sun in Fig. 5. It is clearly seen that the transition occurs
on timescales of a few tens of days for Pop III stars as compared to a few
days for the Sun in case of $M_{\rm BH}=10^{6}M_{\odot}$. This leads to larger
photo-sphere radii and higher peak luminosities for Pop III TDEs.

After the photo-sphere starts to recede, the photo-sphere radius and
temperature at this later stage can be estimated following Eq. 15 and Eq. 16
and given as:

$\displaystyle\begin{split}R_{\rm ph}&=\kappa_{s}\dot{M}_{\rm out}/(4\pi
v_{\rm wind})\\\ &\approx 5.4f_{\rm out,0.5}f_{v}^{-1}\left(\frac{\dot{M}_{\rm
fb}}{\dot{M}_{\rm Edd}}\right)R_{p,3R_{s}}^{1/2}R_{s}\end{split}$ (20)

$T_{\rm ph}\approx 2.5\times 10^{5}f_{\rm
out,0.5}^{-1/3}f_{v}^{1/3}\left(\frac{\dot{M}_{\rm fb}}{\dot{M}_{\rm
Edd}}\right)^{-5/12}M_{6}^{-1/4}R_{p,3R_{s}}^{-7/24}~{}\rm K$ (21)

where $R_{\rm p,3R_{s}}=R_{p}/3R_{s}$.

Specifically for Pop III stars, using their mass-radius relationship (Eq. 1),
we have:

$\begin{split}t_{\rm edge}\simeq 85f_{v}^{-3/4}\left(\frac{f_{\rm
out}}{0.5}\right)^{3/8}\left(\frac{B_{\gamma}}{0.08}\right)^{1/4}\left(\frac{\beta}{1.85}\right)^{-3/8}\\\
\left(\frac{m_{\star}}{300}\right)^{15/44}\left(\frac{Z}{10^{-5}}\right)^{3/44}M_{6}^{-1/8}\
\rm day\end{split}$ (22)

Before $t_{\rm edge}$:

$R_{\rm ph}\propto M_{6}^{1/3}m_{\star}^{-2/33}z_{\star}^{-1/22},$ (23)
$T_{\rm ph}\propto M_{6}^{1/9}m_{\star}^{-2/33}z_{\star}^{-1/264}.$ (24)

After $t_{\rm edge}$:

$\displaystyle R_{\rm ph}\propto m_{\star}^{28/33}z_{\star}^{3/22},$ (25)
$\displaystyle T_{\rm ph}\propto
M_{6}^{2/9}m_{\star}^{-4/11}z_{\star}^{-17/264}.$ (26)

Figure 6: SEDs of the Pop III star tidal flares in the rest frame of the host
galaxy. The default parameter set used for plotting this figure and the
subsequent ones is: $M_{\rm
BH}=10^{6}M_{\odot},~{}M_{\star}=300M_{\odot}~{}(M_{300}),~{}Z=10^{-5}~{}(Z_{-5}),~{}f_{\rm
out}=0.5,~{}f_{v}=1$, and $\beta=\beta_{c}=1.85$ (the red solid curve) unless
specified otherwise. (a) The evolution of the SED at different epochs: $t_{\rm
edge}/2$, $t_{\rm edge}$ (peak luminosity), $2\times t_{\rm edge}$ and
$4\times t_{\rm edge}$. (b) The SEDs at $t=t_{\rm edge}$ for various Pop III
stars of different masses or metallicities and a solar-type star. (c) The
dependence of the SED (at $t=t_{\rm edge}$) on $M_{\rm BH}$. (d) The
dependence of the SED (at $t=t_{\rm edge}$) on other parameters including the
mass outflow fraction ($f_{\rm out}$), the ratio between wind velocity and
escape velocity ($f_{v}$) and the stellar orbital penetration parameter
$\beta$ (with $\beta_{c}=1.85$ being the critical value for full tidal
disruption).

We note that this first-order calculation in SQ09 assumes that the outflows
start to be launched around the peak of the debris mass fallback rate and they
ignore the outflows produced prior. The luminosity increases in the initial
phase as the wind builds up and therefore the characteristic rising time for
the flare will be $t_{\rm edge}$ instead of $t_{\rm fb}$. While this
assumption can cause issues for a solar-type star TDE which has $t_{\rm
edge}\lesssim t_{\rm fb}$, it works well for Pop III star TDEs in which
$t_{\rm edge}\gg t_{\rm fb}$. Therefore, Pop III star TDE flares should rise
on timescales of $t_{\rm edge}$, i.e., a few weeks to months in the intrinsic
TDE frame.

Furthermore, SQ09 also consider disk emission after the fallback rate drops to
the sub-Eddington level. At this phase, the gas can radiate efficiently and a
thin accretion disk is expected to form, which produces emission mainly in the
X-ray and EUV bands. However, for Pop III star TDEs, as seen from Fig. 4, the
fallback rate stays super-Eddington for a very long time (tens to hundreds of
years). Therefore, in this work, we focus only on the emission produced from
the winds launched in the super-Eddington phase and ignore the disk emission
at very late times.

##  3 Results

###  3.1 Pop III Star TDE Intrinsic Emission Spectrum and Luminosity

We first calculate the TDE luminosity and spectral energy distribution (SED)
in the rest frame of the host galaxy based on the properties of the photo-
sphere described in Section 2.3. Assuming blackbody emission, the energy
spectrum and bolometric luminosity can be computed using:

$\nu L_{\nu}=4\pi^{2}R_{\rm ph}^{2}\nu B_{\nu}(T_{\rm ph}),$ (27)
$L=4\pi\sigma R_{\rm ph}^{2}T_{\rm ph}^{4}.$ (28)

We plot the emission SEDs in Fig. 6 and explore the dependence on various
model parameters. The default set of parameters for a fiducial Pop III star
TDE model is: $M_{\rm
BH}=10^{6}M_{\odot},~{}M_{\star}=300M_{\odot},~{}Z=10^{-5},~{}f_{\rm
out}=0.5,~{}f_{v}=1$ and $\beta=\beta_{c}=1.85$. We stick to these parameters
for calculations unless specified otherwise.

Figure 7: Optical (left) and UV (right) light curves of the Pop III star TDEs
around a MBH with $M_{\rm BH}=10^{6}M_{\odot}$ in comparison to that that of a
solar-type star. The color and line styles for different stars are the same as
in Fig. 3. For the fiducial Pop III star TDE model (the red solid curve), we
mark the four epochs used in Fig. 6 (a) (i.e., $t_{\rm edge}/2$, $t_{\rm
edge}$, $2\times t_{\rm edge}$ and $4\times t_{\rm edge}$) using light blue
dots. The late-time light curves are fitted with power-law functions that are
plotted using cyan, dot-dashed lines. We see that the Pop III star TDEs have
much longer rise and decay timescales compared to standard TDEs of normal
stars. Furthermore, for a Pop III star TDE, while its optical light curves
follow the debris fallback rate relatively closely and peaks around $t_{\rm
edge}$, its UV light curve peaks at a later time and decays much more slowly
due to the temperature evolution of the flare.

We illustrate the evolution of the emission from a single TDE in Fig. 6(a),
which shows the SEDs at different epochs for the fiducial case. It is seen
that initially the luminosity increases and the peak of the SED shifts
slightly towards lower frequency, as a result of increasing $R_{\rm ph}$ and
decreasing $T_{\rm ph}$ during this phase. After $t_{\rm edge}$, $R_{\rm ph}$
recedes while $T_{\rm ph}$ increases, so the SED evolves in reverse, i.e., the
luminosity decreases and the SED peak shifts towards higher frequency.

Fig. 6(b) shows the SEDs at $t=t_{\rm edge}$ (the peak of the flare) from the
TDEs of Pop III stars of different masses and metallicities in comparison to
that of a solar type star, all around a $10^{6}M_{\odot}$ MBH. It can be
immediately noticed that a significant fraction of the emission energy resides
in the UV/optical wavelength regimes for all cases. The SEDs of Pop III star
TDEs, compared to that of the solar-type TDE, have lower peak temperatures
(with the SED peaking in the optical band instead of UV) and slightly larger
luminosities. However, the mass or metallicity of a Pop III star does not make
a significant difference to its tidal flare emission. Increasing the
metallicity leads to a slightly smaller luminosity and increasing the stellar
mass shifts the SED towards slightly longer wavelengths.

Fig. 6(c) shows the impact of $M_{\rm BH}$ on the SED. It can be seen that
more massive BHs produce more luminous flares. Interestingly, the recent
detection of sources such as UHZ1 (Bogdán et al., 2023) and GNz-11 (Maiolino
et al., 2023) suggest that massive black holes of $M_{\rm BH}\gtrsim
10^{6-7}M_{\odot}$ at $z\geq 10$ could be much more abundant than previously
estimated. Therefore, it is likely that some Pop III TDEs can produce very
luminous flares with intrinsic bolometric luminosity larger than $10^{45}\
\rm{erg\ s^{-1}}$.

Finally, Fig. 6(d) shows how various other parameters, namely, $f_{\rm out}$
(outflow fraction), $f_{v}$ (ratio between wind velocity and escape velocity)
and $\beta$ (stellar orbital penetration parameter), affect the SED. One can
see that overall the choice of these parameters only mildly impacts the SED.
We note that increasing $f_{\rm out}$ or decreasing $f_{v}$ generally both
lead to a larger peak luminosity. A lower $\beta$ means the star is only
partially disrupted, which will reduce the outflow mass, so the effect induced
is similar to having a lower $f_{\rm out}$. Increasing $\beta$ beyond
$\beta_{c}$ only slightly increases the peak fallback rate so it barely also
affects the peak luminosity of the flare.

Next we consider the evolution of the flare flux in specific wavelength bands.
Fig. 7 shows the optical ($430-750$ THz) and UV ($750-3\times 10^{4}$ THz)
light curves of TDEs of Pop III and solar-type stars by a MBH of $M_{\rm
BH}=10^{6}M_{\odot}$. For the fiducial Pop III star model, we also mark the
epochs ($t=t_{\rm edge}/2,t_{\rm edge},2t_{\rm edge},4t_{\rm edge}$) on the
light curves. One can see that both the optical and UV luminosities increase
in the initial phase when $R_{\rm ph}$ traces the edge of the wind. However,
the behavior of the UV and optical light curves after $t_{\rm edge}$ are
different. For a solar-type star, since the peak of the flare SED stays in the
UV regime, both the UV and the optical light-curve reach the peak around
$t_{\rm edge}$ and decay afterwards. However, for a Pop III star, the TDE
flare SED shifts from UV to optical bands around $t_{\rm edge}$ and shifts
back to UV band afterwards. Therefore, while the tidal flare optical light
curve still peaks at $t_{\rm edge}$, the UV light curve continues to rise for
tens to few hundreds of days after $t_{\rm edge}$. Furthermore, at late times,
while the optical luminosity decays rather closely following the debris mass
fallback rate ($\propto t^{-5/3}$), the UV light curve has a much shallower
slope ($\propto t^{-0.57}$) due to the temperature evolution in this phase.
This also means that the UV light curve has an evolution timescale longer than
that of the optical light curve.

In summary, the bolometric luminosity of a Pop III star TDE in the rest frame
of the host galaxy increases with a larger $M_{\rm BH}$; a smaller
metallicity; a higher outflow fraction, or a slower wind velocity, which all
likely promote the potential observability of these TDEs. Furthermore, a
higher Pop III mass will additionally serve to lengthen the evolution
timescale of these light curves. We calculate the properties of the observed,
and redshifted TDE emission in the next section.

###  3.2 Pop III Star TDEs: Observed Fluxes and Light Curves

Figure 8: The observed fluxes of TDEs of Pop III stars of mass $300M_{\odot}$
($M_{300}$, left panel) and $30M_{\odot}$ ($M_{30}$, middle panel) with
different metallicities of $10^{-5}$ ($Z_{-5}$, top panel) and $10^{-9}$
($Z_{-9}$, bottom panel) compared to a solar type star (right panel) at
different epochs ($t_{\rm edge}/2,t_{\rm edge},2t_{\rm edge},4t_{\rm edge}$).
Green dotted line shows the sensitivity limit of F150W filter of NIRCam.
Magenta dotted line represents the sensitivity limit of F106 filter of WFI,
which has the best sensitivity limit of all the four filters (F106, F129,
F158, F184) used in the High Latitude Wide Area Survey. Red shaded regions
indicate JWST NIRcam bands. It can be seen that a large part of the fluxes are
in the NIR wavelengths and lie above the detection limit of JWST’s NIRCam and
Roman’s WFI.

So far we have established that the emission from Pop III TDEs mostly resides
in UV/optical wavebands in the rest frame of the TDE host galaxy. In the
subsequent calculations, we adopt a canonical value of redshift $z=10$ for Pop
III star TDEs and calculate the red-shifted TDE emissions observed at $z=0$.

With $\nu_{\rm e}$ and $\nu_{\rm o}$ denoting the rest-frame frequency and the
observed frequency, respectively, the two are related as:

$\nu_{\rm e}=(1+z)\nu_{\rm o}.$ (29)

Therefore, the observed Pop III star TDE SED should be redshifted following
$\nu_{o}=\nu_{e}/(1+z)$ and peaks around or slightly below $10^{14}$ Hz, which
corresponds to $\lambda\sim 10^{3}$ nm. This means that a large fraction of
the Pop III star TDE emission should be redshifted to the near-infrared (NIR)
regime, and such events can be potentially detected by the Near Infrared
Camera (NIRCam) on JWST covering the wavelength range of $600-5000$ nm as well
as the Wide Field Instrument (WFI) on Roman with a waveband of $480-2300$ nm.

The observed specific flux, $F_{\nu}(\nu_{\rm o}$), is related to the rest-
frame specific luminosity, $L_{\nu}(\nu_{\rm e})$, following (Hogg et al.,
2002)

$F_{\nu}(\nu_{\rm o})=\frac{1+z}{4\pi D_{L}^{2}}L_{\nu}(\nu_{\rm e}).$ (30)

Here $D_{L}$ denotes the luminosity distance which can be expressed as

$D_{L}=(1+z)D_{c},$ (31)

where

$D_{c}=\frac{c}{H_{0}}\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}}}$

is the comoving distance, $\Omega_{M},\Omega_{\Lambda},H_{0}$ is the matter,
energy density parameter and Hubble constant at the current time.

Figure 9: The dependence of the observed flux at $t=t_{\rm edge}$ on other
parameters, such as the MBH mass ($M_{\rm BH}$), penetration factor ($\beta$),
mass outflow fraction ($f_{\rm out}$) and wind velocity ($f_{v}$). Here we
stick to the fiducial model of a Pop III star with
$M_{\star}=300M_{\odot},Z=10^{-5}$. The shaded regions, magenta and green
dotted lines are the same as in Fig. 8.

We calculate the observed fluxes of Pop III TDEs at different epochs
throughout the event for different stellar models using Eq. 30. The results
are plotted in Fig. 8. It can be seen that for all models the fluxes in the
NIR band stay above the detection limit of the JWST NIRCam ($\sim 11$ nJy
corresponding to the F150W filter) and Roman WFI ($\sim 20.9$ nJy
corresponding to F106 filter) throughout these epochs. Also no significant
differences in the flux level are observed due to different masses or
metallicities of Pop III stars. Moreover, we note that the observed flux from
the TDE of a disrupted solar-type star is also detectable by NIRcam and WFI.

Additionally, we have examined the dependence of observed flux on other model
parameters and the results are shown in Fig. 9. Consistent with the results
illustrated in Fig. 6 (c) and (d), one can see that the NIR flux increases
significantly with increasing $M_{\rm BH}$ and moderately with decreasing
$f_{v}$, but is barely affected by $f_{\rm out}$ and $\beta$. In particular,
the NIR flux becomes undetectable even around peak luminosity when $M_{\rm
BH}<10^{5}M_{\odot}$, unless the wind is exceptionally slow ($f_{v}\lesssim
0.1$). In summary, in all models, the observed Pop III star TDE NIR flux
around flare peak stays significantly above the detection limit of NIRCam and
WFI, as long as $M_{\rm BH}\gtrsim 10^{6}M_{\odot}$. Therefore, our results
demonstrate that the detection of Pop III star TDEs at $z\sim 10$ with both
JWST or Roman is feasible.

Figure 10: We show the NIR light curves of Pop III star TDEs in comparison to
that of a solar-type star. The NIR flux integrated are in the JWST NIRcam
band. All TDEs happen around a MBH with $M_{\rm BH}=10^{6}M_{\odot}$ at
$z=10$. Color schemes and line styles are the same as Fig. 7. Furthermore, for
the default model of Pop III star
($M_{\star}=300M_{\odot}=M_{300},~{}Z=10^{-5}=Z_{-5}$), we also show the
observed NIR light curve when the TDE happens at $z=15$ (red thin line) or
when the MBH is a UHZ1-like source with $M_{\rm BH}=4\times 10^{7}M_{\odot}$
at $z=10.1$ (orange line). The late-time light curves are compared to a power-
law decay pattern of $t^{-5/3}$ (cyan dot-dashed line). Green and magenta
dotted solid lines denote the flux limit of NIRCam (F150W) and WFI (F106)
respectively. It is noticed that observed NIR light curves evolve very slowly
due to cosmological time dilation.

Next we calculate the NIR light curves of Pop III star TDEs as would be
observed in the JWST NIRcam band. The NIR light curves are shown in Fig. 10,
where one can see that the NIR flux can exceed both the JWST NIRCam and Roman
WFI detection limits even when the TDE originates at $z=15$. Moreover, at such
high redshifts, the time dilation effect due to cosmological redshift is very
strong, since $t_{\rm o}=(1+z)t_{\rm e}$. Taking a Pop III star TDE at $z=10$
as an example, the observed NIR light curve (above the detection limit) rises
on timescales $\sim 100-1000$ days and decays over $10^{3}-10^{4}$ days. At
late time, the NIR flux still decays following a pattern relatively close to
the classical mass fallback rate ($\propto t^{-5/3}$).

Furthermore, inspired by the recent discovery of UHZ1, we include the case of
a Pop III star disrupted by a MBH with $M_{\rm BH}=4\times 10^{7}M_{\odot}$
hosted by galaxy at $z=10.1$ (orange line). As expected, the observed NIR flux
is significantly higher in this case, which increases the chance of observing
such events in current and upcoming NIR surveys.

Interestingly, given the observed Pop III star TDE flares rise over a few
years, there is still a prospect of identifying such TDEs detected during the
rising phase as transients, if multiple detections are made within the typical
operation time of surveys. However, during the flare decay phase, as the
evolution timescale is very long, these events will have almost near constant
brightness during the limited operation time and therefore they are more
likely to be mis-categorized as AGNs based on their photometry. One promising
method to distinguishing Pop III star TDEs from TDEs of Pop I stars or AGNs is
through spectroscopic follow-ups. Metal lines should be present in the spectra
of either Pop I star TDEs or AGNs even at $z\sim 10$ given that gas is likely
no longer metal free (Yang et al., 2023). However, if a Pop III star TDE
occurs in a previously quiescent galaxy, a metal-free spectrum should be
produced, since all gas supplied to the MBH likely originates from the tidally
disrupted Pop III star.

###  3.3 Rates of Pop III Star TDEs

Figure 11: Different forms of IMF for the Pop III stars as listed in Table 1.
It should be clarified that Jaacks et al. (2018) calculated Pop III IMF only
within the mass range of $1M_{\odot}-150M_{\odot}$, and Greif et al. (2011)
calculated the IMF only within the mass range of $0.1M_{\odot}-10M_{\odot}$.
For this study, we extrapolated all three IMFs to the range of
$30M_{\odot}-300M_{\odot}$.

We now estimate the intrinsic rates of Pop III TDEs in the early Universe. We
adopt the approach in Pfister et al. (2020, 2022) to calculate the TDE rate
through two-body scattering. The differential TDE rate is given by:

$\begin{split}\frac{d^{2}\Gamma}{d\log m_{\star}d\log\beta}=8\pi^{2}GM_{\rm
BH}\frac{R_{T}}{\beta}\Phi(m_{\star})m_{\star}\\\
\times\int_{0}^{E_{m}}\zeta(q,f,\beta,m_{\star})\,dE\ \end{split}$ (32)

Here $\Phi(m_{\star})$ is the stellar initial mass function (IMF), $E$ is the
specific orbital energy of the star with the maximum value at $E_{m}=GM_{\rm
BH}/R_{T}$ corresponding to the orbit with $R_{T}$ as the closest approach,
and $\zeta(q,f,\beta,m_{\star})$ is a function of the loss-cone filling factor
$q$, the stellar distribution function $f$, the stellar mass $m_{\star}$ and
the penetration parameter $\beta$. The TDE of a Pop III star can be induced by
the scattering between a Pop III star and a normal star or that between two
Pop III stars. However, at the redshift range that we are considering, Pop III
stars contribute to only a few percent of total stellar mass (Magg et al.,
2022). Therefore, we only include the scattering of Pop III stars by normal
stars into our calculation. Further details of the terms in Eq. 32 and the
rate calculation are given in Appendix A.

IMF | $\Phi(m_{\star})$ | $\alpha$ | $m_{c}$ | $\Gamma[\rm gal^{-1}yr^{-1}]$ | $\dot{N}_{\rm max}[\rm Mpc^{-3}yr^{-1}]$
---|---|---|---|---|---
Greif et al. (2011) | $\propto m_{\star}^{-\alpha}$ | $-0.17$ | $-$ | $1.8\times 10^{-8}$ | $1.1\times 10^{-9}$
de Bennassuti et al. (2014) | $\propto m_{\star}^{\alpha-1}\exp{(-m_{c}/m_{\star})}$ | $-1.35$ | $20M_{\odot}$ | $4.1\times 10^{-8}$ | $2.5\times 10^{-9}$
Jaacks et al. (2018) | $\propto m_{\star}^{-\alpha}\exp{(-m_{c}^{2}/m_{\star}^{2})}$ | $0.17$ | $4.47M_{\odot}$ | $2.0\times 10^{-8}$ | $1.2\times 10^{-9}$
Table 1: Different IMFs of Pop III stars suggested by different works in the
literature. $\Phi(m_{\star})$ describes the IMF equation, with $\alpha$ and
$m_{c}$ denoting the slope and cut-off mass, respectively. $\Gamma$ represents
the rate of Pop III star TDE from a single galaxy with $M_{\rm
BH}=10^{6}M_{\odot}$. $\dot{N}_{\rm max}$ represents the Pop III star TDE
volumetric rate computed using the BHMF from the the merger-driven model.
Figure 12: The black hole mass function at $z=10$ for different growth models
(left: Bondi-Hoyle-Lyttleton; middle: super-Eddington; right: merger driven).
The BHMFs are adopted from Trinca et al. (2022) with permission. The red
points in all the three panels denote the BHMF inferred from the detection of
UHZ1.

We summarize the assumptions used for the TDE rate calculation as below:

  * •

Ranges of parameters: We adopt the MBH mass range of $M_{\rm
BH}=[10^{5}M_{\odot},10^{8}M_{\odot}]$, Pop III stellar mass range of
$m_{\star}=[30M_{\odot},300M_{\odot}$], normal star in the range of
$m_{\star}=[0.1M_{\odot},10M_{\odot}$], and $\beta$ in the range
$[\beta_{c}=1.85,R_{T}/R_{s}]$ to calculate the total TDE rate by integrating
Eq. 32.

  * •

IMF of the stellar populations: We adopt the general Kroupa IMF (Kroupa, 2001)
for the normal main-sequence stars. However, the IMF for Pop III stars is
model-dependent and not constrained very well yet. Different IMFs have been
proposed in the literature, and a few examples are shown in Table 1 and Fig.
11. However, using different Pop III star IMFs bring negligible effects to TDE
rates, as explained in Appendix A.

  * •

Mass fraction of Pop III stars: Magg et al. (2022) showed that at $z\sim 10$
Pop III stars has a mass fraction between $(1-15)\%$ among all stellar
population depending on the timescale at which Pop III stars transit to Pop II
stars. We adopt a Pop III star mass fraction of $7\%$ throughout the galaxy
for our calculation.

  * •

Stellar density distribution: As there lacks studies of the structures of
galaxies for different stellar populations at these extremely high redshifts,
we follow the classical papers such as Wang & Merritt (2004); Stone & Metzger
(2016); Pfister et al. (2020) and assume isothermal stellar density
distributions in the Keplerian potential for both Pop III and normal stars,
which gives

$\rho(r)=\rho_{0}(r/r_{\rm inf})^{-\alpha}$ (33)

where $\rho(r)$ is the stellar density, $\alpha=2$, and $r_{\rm inf}=GM_{\rm
BH}/2\sigma^{2}$, with $\sigma$ being the velocity dispersion of the host
galaxy.

Under these assumptions, the TDE rate of Pop III stars from a single galaxy
hosting $10^{6}M_{\odot}$ MBH is calculated and listed in Table 1. This
intrinsic TDE rate is $\Gamma\sim 10^{-8}~{}{\rm gal^{-1}~{}yr^{-1}}$, which
is barely affected by the choice of Pop III star IMF.

One can further compute the volumetric rate of Pop III star TDEs using:

$\begin{split}\dot{N}=\iiint\frac{dn}{d\log M_{\rm
BH}}\times\frac{d^{2}\Gamma}{d\log m_{\star}d\log\beta}\\\ \times d\log
m_{\star}~{}d\log\beta~{}\,d\log M_{\rm BH}\end{split}$ (34)

here $\Phi({M_{\rm BH}})=dn/d\log M_{\rm BH}$ denotes the black hole mass
function (BHMF), which is defined as the number of MBHs with masses between
$\log M_{\rm BH}$ and $\log M_{\rm BH}+d\log M_{\rm BH}$ in unit co-moving
volume. We note that the BHMF at high redshifts sensitively depends on the
seeding scenarios and growth channels of MBHs, much of which is currently
uncertain. In this work, we adopt three different BHMFs from Trinca et al.
(2022), that consider not only light, intermediate and heavy MBH seeds but
also different processes responsible for their growth. Fig. 12 shows the BHMF
at $z=10$ for these three growth models, namely, Eddington-limited Bondi-
Hoyle-Lyttleton accretion, super-Eddington accretion and merger-driven growth
(Hoyle & Lyttleton, 1941; Bondi, 1952). We focus on the MBHs with $M_{\rm
BH}\geq 10^{5}M_{\odot}$ for this work, since the observed Pop III star TDE
flux goes below the threshold of NIRCam and WFI when $M_{\rm
BH}<10^{5}M_{\odot}$ (Fig. 9). It is seen that the BHMF from the merger-driven
growth channel generally has higher values in this $M_{\rm BH}$ range than the
other two BHMF choices. The volumetric Pop III star TDE rates using these
three different BHMFs are calculated and shown in Fig. 13, which are around
$10^{-10}-10^{-9}\rm Mpc^{-3}yr^{-1}$. Out of the three, not surprisingly, the
BHMF based on the merger-driven growth model produces the highest integrated
volumetric TDE rate $\dot{N}_{\rm max}\approx 10^{-9}\rm Mpc^{-3}yr^{-1}$.

We also include the BHMF inferred from calibrating models with the detection
of UHZ1 in Fig. 12, and note that the BHMF taken from Trinca et al. (2022) (as
well as other current BHMF models) underestimate the number density of massive
black holes in all the growth models. Hence it is possible that the actual Pop
III TDE rate is higher than our estimate. An updated BHMF at high redshift
from additional observations of a population of $z\sim 10$ BHs will be crucial
for more accurately constraining the Pop III star TDE rate.

Figure 13: The differential Pop III star TDE volumetric rates as a function of
$M_{\rm BH}$ for the three different BHMFs in Fig. 12. One can see that the
BHMF through the merger-driven growth channel overall produces the highest TDE
volumetric rate.

###  3.4 Pop III Star TDE Predictions for JWST and Nancy Grace Roman Space
Telescope

In this section we estimate the total number of Pop III star TDEs that stand
to be detected by telescope JWST and Roman. For a first-order estimate of the
upper limit of the detection numbers, we ignore factors such as survey
strategies and limitations. The total number can then be approximated using:

$N=\dot{N}\times V\times T.$ (35)

Here $\dot{N}$ is the volumetric TDE rate obtained in the previous section,
$T$ is duration of the observational survey, and $V$ is the co-moving volume
from which the event can be detected. $V$ can be computed using the following
equation:

$V=\frac{4\pi[d(z_{\rm max})^{3}-d(z_{\rm min})^{3}]}{3}\times f_{\rm sky},$
(36)

where $d(z)$ is the co-moving distance dependent on redshift $z$, and $f_{\rm
sky}$ is the sky coverage fraction of the telescope of survey.

For the case of Pop III star TDEs, we adopt $z_{\rm min}=10$, when the Pop III
star population reaches peak, and $z_{\rm max}=15$, when both the Pop III
stars and MBHs should have started to form. More importantly, we have
predicted in Fig. 10 that Pop III star TDEs at $z=15$ are still bright enough
to be detected by JWST and Roman. Moreover, for simplicity, we stick to a
constant $\dot{N}=1.2\times 10^{-9}\rm\ Mpc^{-3}\ yr^{-1}$ (based on the Pop
III star IMF by Jaacks et al. (2018) and the BHMF from the merger-driven
growth model), and note that this likely gives an upper limit to the detected
number.

JWST Cycle 1 has various deep surveys, such as COSMOS-Web with a total area of
0.54 $\rm deg^{2}$ and duration of 255 hours (Casey et al., 2023), and the
JWST Advance Deep Extragalactic Survey (JADES) with its sky coverage of
$236~{}\rm arcmin^{2}$ and a total duration of $426$ hours considering both
the deep and medium modes (Eisenstein et al., 2023). Using both the COSMOS-Web
survey and the deep mode of the JADES survey, which can both probe galaxies at
$z>10$, the total detected number of Pop III TDEs in one year is about
$5\times 10^{-4}$. Throughout the 10 year expected lifespan of JWST, even if
we assume COSMOS-Web like surveys are continuously conducted, the total
expected detection number is still only $(10\ {\rm yr}/255\ {\rm hr})\times
5\times 10^{-4}\sim 0.2$ event in total. Therefore, the chance of detecting
Pop III star TDEs using JWST is slim. If such TDEs were detected, it would
indicate the BHMF at $z>10$ is much larger than currently estimated, which
would pose an interesting challenge in our understanding of the formation
efficiency and evolution of MBHs in the early Universe. With the recent
discovery of UHZ1 and its interpretation as an OBG arising from a heavy
initial seed by Natarajan et al. (2024), it appears that BH seed formation in
the early Universe can occur via multiple seeding pathways hence rendering the
process significantly more efficient than previously believed.

A much more promising telescope for detecting Pop III star TDEs is the
upcoming Nancy Grace Roman Space Telescope, which is designed to have a
sensitivity similar to JWST (Fig. 10) and conduct wide-field surveys over a
very large FOV of $0.281~{}\rm deg^{2}$ (Mosby et al., 2020). Hence, from the
High Latitude Wide Area Roman survey that aims to cover a total sky area of
$\sim 2000$ deg2 (Wang et al., 2022b), the number of detected Pop III TDEs can
reach $\lesssim 60$ in one year. Furthermore, there is another proposed
survey, the Next-generation All-sky Near-infrared Community surveY (NANCY)
(Han et al., 2023), which plans to perform an all-sky scan. This survery
strategy will lead to the detection of $\sim 1000$ Pop III TDEs in a single
year! Once a large sample of Pop III star TDEs are observed, the number can be
used to put a strong constraints on the properties and mass fraction of Pop
III stars as well as the BHMF at $z\gtrsim 10$ which will provide the much
needed insights into the efficiency of early BH formation.

##  4 Summary and Discussion

Pop III stars are the first-generation stars which are expected to form from
metal-free primordial gas. Their formation and properties have been under
study with multiple theoretical investigations, but their direct detection
remains a challenging open question. The primary focus of this work is to
investigate the prospect of detecting Pop III stars through the flares
produced when these stars are tidally disrupted by MBHs, utilizing telescopes
such as JWST and Roman that can probe the high redshift Universe. We summarize
the key features of our assumed model and main findings therefrom below and
show a schematic illustration in Fig. 14.

Figure 14: Schematic diagram of Pop III TDEs. In a galaxy at $z\gtrsim 10$, a
Pop III star is scattered by other stars and gets tidally disrupted by the
MBH. A large fraction of the fallback debris material is converted to a
powerful wind with outflowing mass rate $\dot{M}_{\rm out}=f_{\rm
out}\dot{M}_{\rm fb}$, which produces a luminous optical/UV flare. At $z=0$,
the flare emissions are redshifted to the NIR wavelength and can be detected
by JWST and the Nancy Roman Space Telescope.

  * •

Pop III stars have tidal disruption radii around or a few times larger than
that of a solar-type star. Very massive Pop III stars with
$M_{\star}>100M_{\odot}$ can possibly be disrupted by MBHs with $M_{\rm BH}$
up to $\sim 10^{9}M_{\odot}$.

  * •

The debris mass fallback rates for Pop III star TDEs can stay hyper-Eddington
for long periods. For example, for a typical MBH with $M_{\rm
BH}=10^{6}M_{\odot}$, $\dot{M}_{\rm fb}$ can reach a peak value of
$10^{4}-10^{6}\dot{M}_{\rm Edd}$ and stays super-Eddington for $10-100$ years,
with its peak value . In the more extreme case of a UHZ1-like source with
$M_{\rm BH}\gtrsim 10^{7}M_{\odot}$, $\dot{M}_{\rm fb,peak}$ drops to around
$100-10^{5}\dot{M}_{\rm Edd}$ and $\dot{M}_{\rm fb}$ can still stay super-
Eddington for more than a few years.

  * •

We adopt the super-Eddington outflow model proposed by Strubbe & Quataert
(2009) and only consider the emission produced by the outflow in this phase
while predicting potential observational signatures.

  * •

In the rest frame of the host galaxy, Pop III star TDEs mainly produce
UV/optical emission (Fig. 6), with the bolometric luminosity increasing almost
linearly with $M_{\rm BH}$. These tidal flares have much longer evolution
timescales compared to typical Pop I star TDEs detected currently (Fig. 7).

  * •

As Pop III star TDEs are expected to mostly occur at high redshifts ($z\gtrsim
10$), a large fraction of the emission is redshifted to the NIR band and the
observed flux exceeds the detection limit of JWST NIRCam and Roman WFI (Fig. 8
and Fig. 9). Moreover, due to the time dilation effect, the observed NIR
flares evolve even more slowly (rising in 100-1000 days and decaying over
10-100 years) (Fig. 10).

  * •

The volumetric rate of Pop III star TDEs is insensitive to Pop III star IMF,
but sensitively depends on the BHMF at high redshifts which in turn depends on
the seeding and growth models of MBHs.

  * •

We obtain an upper limit on Pop III star TDE volumetric rate of $\sim
10^{-9}\rm Mpc^{-3}~{}yr^{-1}$ (Fig. 13), based on the current BHMF model. We
note that the recent detection of overmassive black holes at high redshifts
indicate that the density of MBHs at $z\sim 10$ and hence our estimated TDE
rates could be lower than the actual numbers.

  * •

The high NIR luminosity and long duration of the Pop III star tidal flares
make these events detectable by the Roman Space telescope and we predict up to
a few tens of events per year with its wide field instrument. However, the
probability of detecting a Pop III TDE using JWST is low due to its narrower
FOV.

Our results under the SQ09 model are rather robust against the choice of Pop
III star mass, metallicity, and other model parameters such as the penetration
parameter of the stellar orbit, outflow fraction and speed, etc. However, it
should be acknowledged that this 1D analytical model cannot explain the
observed X-ray emissions produced in Pop I star TDEs in the local universe.
For the latter, simulations of super-Eddington disks around SMBHs with
$\dot{M}_{\rm acc}\sim(1-10)\dot{M}_{\rm Edd}$ reveal that an optically thin
funnel can form in the polar region through which X-rays can leak out (Dai et
al., 2018; Jiang et al., 2019; Thomsen et al., 2022). However, it is
previously shown that the funnel can diminish when $\dot{M}_{\rm acc}$ reaches
$>100\dot{M}_{\rm Edd}$ (Sadowski & Narayan, 2016). Therefore, Pop III star
TDEs with extremely high fallback rates likely cannot produce strong X-ray
emissions, unless when relativistic jets are produced and pointing towards the
observer. Moreover, we note that the realistic structures of the massive stars
can affect the TDE debris mass fallback rate calculation (Law-Smith et al.,
2020). Therefore, future detailed modeling of Pop III star structures will be
useful to improve our calculation.

It is also worth mentioning that at the high redshifts we are considering, Pop
III stars only occupy a few percent of the total stellar population, while Pop
I and Pop II stars dominate the stellar mass and therefore should produce much
higher TDE rates. Although the focus of this work is on Pop III star TDEs, we
have shown that the NIR flux of Pop I star TDEs at $z\sim 10$ also exceeds the
detection limit of JWST and Roman. Pop III star and Pop I star TDEs, however,
can be distinguished by their different evolution timescales. Furthermore, the
Pop I star TDEs observed so far typically produce metal lines including C, N,
O, Mg, and Fe lines in their optical or UV spectra (Leloudas et al., 2019;
Blagorodnova et al., 2019; Gezari, 2021; Charalampopoulos et al., 2022), and
such signatures cannot be produced by Pop III star TDEs. Pop I and Pop II star
TDEs at high redshifts are worth further investigation, which is beyond the
scope of this paper.

Another interesting aspect worth considering is the detection of TDEs at high
redshifts magnified through gravitational lensing. Recent works show promising
results of finding extremely faint high-$z$ objects, including a number of Pop
III star candidates, faint stars, distant galaxies and black holes with their
observed fluxes largely boosted by nearby foreground lensing clusters (Kelly
et al., 2018; Kaurov et al., 2019; Schauer et al., 2022; Vikaeus et al.,
2022a; Bogdán et al., 2023; Meena et al., 2023; Diego et al., 2023). Hence,
the effect of lensing should also enhance the observed TDE fluxes, which will
bring more high-$z$ (Pop III) TDEs into view. While it has been shown that the
chance of having a high-$z$ galaxy lensed is roughly $\sim 10^{-3}$ (Saha et
al., 2024), the probability of having lensed transients yet to be studied in
detail.

With the advent of JWST, the number of detected quasars with $M_{\rm
BH}\gtrsim 10^{9}M_{\odot}$ at $z>6$ has increased by a large number (Wang et
al., 2023; Yang et al., 2023; Natarajan et al., 2024), which offers
interesting insights into the models of BH seed formation. TDEs including Pop
III TDEs could also be one of the channels responsible for the rapid growth of
BH seeds at very high redshifts. This was previously proposed by Pfister et
al. (2021) who showed that IMBHs at high redshifts can accumulate their masses
through TDEs at a similar rate as through gas accretion.

In summary, in this work we compute the properties of a brand new class of
high-redshift sources, tidally disrupted Pop III stars, and demonstrate that
they could be viably detected in upcoming wide field IR surveys. The prospect
of observing such TDEs has only become promising due to the recent launch of
JWST and the expected launch of the Nancy Grace Roman Space Telescope.

We thank A. Amruth, A. Bera, C. Bottrell, T. Kwan, S. Kimura, T. Matsumoto, L.
Thomsen, N. Yoshida, Z. Zhang for helpful discussions and A. Trinca for
providing the data on BHMF. RKC, NYC and LD acknolwedge the support from the
National Natural Science Foundation of China (HKU12122309) and the Hong Kong
Research Grants Council (HKU17304821, HKU17314822, HKU27305119). PN
acknowledges support from the Gordon and Betty Moore Foundation and the John
Templeton Foundation that fund the Black Hole Initiative (BHI) at Harvard
University where she serves as one of the PIs.

##  Appendix A Calculation of TDE Rates

For a distribution of Pop III stars with density profile following
$\rho\propto r^{-\alpha}$ and mass distribution expressed by $\Phi_{\rm
PopIII}(m_{\star})$, the TDE rate can be calculated using the loss cone
dynamics. We give a summary of the methodology as below and refer the readers
to Pfister et al. (2020, 2022) for more details. The differential TDE rate can
be expressed in terms of stellar mass ($m_{\star}$), black hole mass ($M_{\rm
BH}$) and the penetration parameter ($\beta$) such that:

$\frac{d^{2}\Gamma}{d\log m_{\star}d\log\beta}=8\pi^{2}GM_{\rm
BH}\frac{R_{T}}{\beta^{2}}\Phi_{\rm
PopIII}(m_{\star})m_{\star}\int_{0}^{E_{m}}\zeta(q(E,m_{\star}),f(E),\beta,m_{\star})\,dE\
$ (A1)

The specific binding energy $E$ is integrated from 0 to the maximum energy
$E_{m}=GM_{\rm BH}/R_{T}$. The function $\zeta(q,f,\beta,m_{\star})$ is
described as follows:

$\displaystyle\begin{split}\zeta(q,f,\beta,m_{\star})={}&\frac{f\log^{2}e}{1+q^{-1}\xi(q)\ln(1/R_{\rm{lc}}(m_{\star}))}\bigg{[}1-2\sum^{\infty}_{m=1}\frac{e^{-\alpha^{2}_{m}q/4}}{\alpha_{m}}\frac{J_{0}(\alpha_{m}\beta^{-1/2})}{J_{1}(\alpha_{m})}\bigg{]}\end{split}$
(A2)

where $R_{lc}$ describes the fraction of stars in the loss cone:

$R_{lc}(m_{\star})=\frac{L_{\rm lc}^{2}}{L_{\rm
circ}^{2}}\sim\frac{4ER_{T}(m_{\star})}{GM_{\rm BH}}$ (A3)

and

$\xi(q)=1-4\sum^{\infty}_{m=1}\frac{e^{-\alpha^{2}_{m}q/4}}{\alpha_{m}^{2}}$
(A4)

$J_{0}$ and $J_{1}$ are the Bessel functions of the first kind, with general
notation as:

$J_{\alpha}(x)=\sum^{\infty}_{m=0}\frac{(-1)^{m}}{m!\,\hat{\Gamma}(m+\alpha+1)}\bigg{(}\frac{x}{2}\bigg{)}^{2m+a}$
(A5)

where $\hat{\Gamma}(x)$ is the gamma function, and $\alpha_{m}$ is the $m$th
zero of the Bessel function $J_{0}$.

If we assume the density follows an isothermal profile with $\alpha=2$ (Eq
33), the distribution function $f(E)$ takes an analytical form:

$f(E)=(2\pi\sigma_{\rm inf}^{2})^{-3/2}\frac{\rho_{0}}{\langle
m_{\star}\rangle}\frac{\gamma(\alpha+1)}{\gamma(\alpha-1/2)}\bigg{(}\frac{E}{\sigma_{\rm
inf}^{2}}\bigg{)}^{\alpha-3/2}$ (A6)

where $\sigma_{\rm inf}=(GM_{\rm BH}/r_{\rm inf})^{1/2}$ is the velocity
distribution at the BH influence radius $r_{\inf}$, $\langle m_{\star}\rangle$
is the average stellar mass of the Pop III stellar mass function $\Phi_{\rm
PopIII}(m_{\star})$, $\gamma$ is the Euler Gamma function, and $\rho_{0}$ is
the central density:

$\rho_{0}=r_{\rm Pop{III}}\times M_{\rm
BH}\frac{3-\alpha}{4\pi}\left(\frac{\sigma_{\rm inf}^{2}}{GM_{\rm
BH}}\right)^{3}$ (A7)

Here $r_{\rm Pop{III}}=0.07$ denotes the mass fraction of Pop III stars among
all stellar populations in a galaxy at $z=10$ (Magg et al., 2022). The loss
cone filling factor $q$ can also be described as follow:

$q(E,m_{\star})=\nu(m_{\star})\bigg{(}\frac{E}{\sigma_{\rm
inf}^{2}}\bigg{)}^{\alpha-4},$ (A8)

with

$\begin{split}\nu(m_{\star})&=\frac{8\sqrt{\pi}}{3}(3-\alpha)\frac{\gamma(\alpha+1)}{\gamma(\alpha-1/2)}\\\
&\bigg{[}\frac{5}{32(\alpha-1/2)}+\frac{3I_{B}(1/2,\alpha)-I_{B}(3/2,\alpha)}{4\pi}\bigg{]}\bigg{(}\frac{G\langle
m_{\rm scat}^{2}\rangle}{\sigma_{\rm inf}^{2}\langle m_{\star}\rangle
R_{T}(m_{\star})}\bigg{)}\ln\Lambda.\end{split}$ (A9)

Here $\ln\Lambda=\ln(0.4M_{BH}/m_{\star})$ is the coulomb logarithm, $\langle
m_{\star}\rangle=\int{m_{\star}\Phi_{\rm PopIII}(m_{\star})dm_{\star}}$, and
$m_{\rm scat}$ represents the total population of stars inside a galaxy that
are available for scattering. To simplify the calculation, we assume all Pop
III star TDEs are produced by the scattering between a Pop III star and a Pop
I star star, so that

$\langle m_{\rm scat}^{2}\rangle=\int m_{\star}^{2}\Phi_{\rm
norm}(m_{\star})dm_{\star}.\\\ $ (A10)

Furthermore, $I_{B}$ is defined as

$I_{B}(\frac{n}{2},\alpha)=\int_{0}^{1}{t^{-\frac{n+1}{2}}(1-t)^{3-\alpha}B(t,\frac{n}{2},\alpha-\frac{1}{2})dt}$
(A11)

with $B$ as the incomplete Euler Beta function.

As discussed in Sec 3.3, the volumetric rate of TDE is very mildly affected by
the IMF of Pop III stars. This is mainly because that total TDE rate scales
with $\langle m_{\rm scat}^{2}\rangle/\langle m_{\star}\rangle^{2}$, which
does not differ much when the average is taken over the same mass range
($M_{\star}=30M_{\odot}-300M_{\odot}$) for different IMFs.

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