# The correlations of stellar tidal disruption rates with properties of
massive black holes and their host galaxies

Yunfeng Chen School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100101, China;
<EMAIL_ADDRESS>Kavli Institute for Astronomy and Astrophysics, and School of
Physics, Peking University, Beijing, 100871, China<EMAIL_ADDRESS>Qingjuan
Yu Kavli Institute for Astronomy and Astrophysics, and School of Physics,
Peking University, Beijing, 100871, China<EMAIL_ADDRESS>Youjun Lu National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101,
China<EMAIL_ADDRESS>School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, China Qingjuan Yu

###### Abstract

Stars can be either disrupted as tidal disruption events (TDEs) or swallowed
as a whole by massive black holes (MBHs) at galactic centers when they
approach sufficiently close to these MBHs. In this work, we investigate the
correlations of such stellar consumption rates with both the MBH mass $M_{\rm
BH}$ and the inner slope of the host galaxy mass density distribution
$\alpha$. We introduce a simplified analytical power-law model with a power-
law stellar mass density distribution surrounding MBHs and separate the
contributions of two-body relaxation and stellar orbital precession for the
stellar orbital angular momentum evolution in nonspherical galaxy potentials.
The stellar consumption rates derived from this simplified model can be well
consistent with the numerical results obtained with a more realistic treatment
of stellar distributions and dynamics around MBHs, providing an efficient way
to estimate TDE rates. The origin of the correlations of stellar consumption
rates with $M_{\rm BH}$ and $\alpha$ are explained by the dependence of this
analytical model on those MBH/host galaxy properties and by the separation of
the stellar angular momentum evolution mechanisms. We propose that the strong
positive correlation between the rates of stellar consumption due to two-body
relaxation and $\alpha$ provides one interpretation for the overrepresentation
of TDEs found in some rare E+A/poststarburst galaxies. We find high TDE rates
for giant stars, up to those for solar-type stars. The understanding of the
origin of the correlations of the stellar consumption rates will be necessary
for obtaining the demographics of MBHs and their host galaxies via TDEs.

Galaxy dynamics (591); Gravitational wave astronomy (675); High energy
astrophysics (739); Supermassive black holes (1663); Tidal disruption (1696);
Transient sources (1851); Time domain astronomy (2109).

##  1 Introduction

Tidal disruption event (TDE) happens when a star travels too close to a
massive black hole (MBH) so that the tidal force exerted on the star by the
MBH surpasses the star’s self-gravity, causing the star to be ripped apart and
disrupted, accompanied by luminous flares due to subsequent accretion of the
stripped stellar material (e.g., Hills, 1975; Rees, 1988). TDEs may frequently
occur at the centers of galaxies, as observations have shown the ubiquitous
existence of MBHs (with mass ${\sim}10^{6}$–$10^{10}{M_{\odot}}$) at galactic
nuclei (e.g., Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al.
2002; Kormendy & Ho 2013; Graham 2016). These TDEs provide rich
electromagnetic signals, which help to study the relativistic effects,
accretion physics, formation of radio jets and interior structure of torn
stars, etc. (Komossa, 2015; Alexander, 2017; Gezari, 2021).

TDEs illuminate those dormant MBHs, and the event rates in individual galaxies
depend on the MBH properties (e.g., Magorrian & Tremaine 1999, hereafter MT99;
Wang & Merritt 2004; Ivanov et al. 2005; Kesden 2012; Fialkov & Loeb 2017).
Therefore, they can serve as powerful probes of the MBH demographics, such as
the MBH mass, spin, binarity, and the occupation fraction of MBHs in galaxies
(Gezari, 2021). Depending on the event rate, the consumption of intruding
stars may provide an important channel for the growth of MBHs in the centers
of galaxies (Freitag & Benz, 2002; Yu, 2003; Brockamp et al., 2011; Alexander,
2017) and even has significant effects on the MBH spin evolution (Zhang et
al., 2019), especially for those relatively less massive ones.

TDEs are first discovered as energetic transients in the soft X-ray band by
archival searches of the ROSAT All-Sky Survey data (Bade et al., 1996; Grupe
et al., 1999; Komossa & Greiner, 1999; Greiner et al., 2000) and later found
via searches by multi-wavelength (e.g., UV, optical, X-ray, and gamma-ray)
observations (Alexander, 2017; Gezari, 2021). Over the past decade,
statistically significant samples of TDEs have begun to accumulate, largely
owing to the rapid advancement of the optical time domain surveys, such as PTF
(Law et al., 2009; Rau et al., 2009), ASAS-SN (Shappee et al., 2014), Pan-
STARRS (Chambers et al., 2016), and ZTF (Bellm et al., 2019). For example, a
recent census by Gezari (2021) listed $56$ TDE candidates, among which around
two thirds are discovered by the wide-field optical time domain surveys. Based
on different samples of TDE candidates, the TDE event rate is estimated to be
${\sim}10^{-5}$–$10^{-4}{\rm\,yr^{-1}}{\rm\,gal^{-1}}$ (e.g., Donley et al.
2002; Esquej et al. 2008; Maksym et al. 2010; Wang et al. 2012; van Velzen &
Farrar 2014; Khabibullin & Sazonov 2014; Auchettl et al. 2018; van Velzen
2018).

Some recent studies on the TDE host galaxies reveal intriguingly that TDEs
prefer those rare E+A/poststarburst galaxies (Arcavi et al., 2014; French et
al., 2016, 2017; Law-Smith et al., 2017; Graur et al., 2018; Hammerstein et
al., 2021). After removing the accountable selection effects (e.g., MBH mass,
redshift completeness, strong active galactic nucleus presence, bulge colors
and surface brightness), the remaining factor of the overrepresentation of
TDEs in those rare subclass of galaxies is ${\sim}$25–48 (Law-Smith et al.,
2017). The underlying mechanism(s) responsible for this preference is still
unclear. Some proposed explanations include binary MBH formation due to a
recent galaxy merger, stellar orbital perturbations induced by what has also
caused the starburst in the past, a unique population of stars such as an
evolved population of A giants which are susceptible for disruptions, and high
central stellar density distributions or concentrations (e.g., see Alexander
2017; Law-Smith et al. 2017; French et al. 2020; Gezari 2021). To properly
interpret these observations and thereby distinguish different dynamical
mechanisms occurring in the galactic nuclei to produce TDEs, it is necessary
to have a thorough understanding of these mechanisms, especially on the
dependence of those theoretically predicted rates on the properties of the
central MBHs and their host galaxies.

In a spherical stellar system composed of a central MBH and surrounding stars,
the tidal radius for a given type of stars and the MBH event horizon determine
the size of the loss cone in the phase space of the stellar energy and angular
momentum, in the sense that a star at a given energy can be consumed (either
tidally disrupted or swallowed as a whole) by the MBH when its angular
momentum falls below a critical value. The rate of stellar consumptions is
determined by the refilling rate of low-angular-momentum stars into the loss
cone, since stars initially in the loss cone will be quickly consumed (e.g.,
within an orbital period). The two-body relaxation process of stars can set a
lower limit for the refilling rate of low-angular-momentum stars into the loss
cone. In realistic stellar systems, some other mechanisms may also work or
even play a dominant role in refilling the loss cone, and therefore enhance
the stellar consumption rates. Possible mechanisms include the resonant
relaxation (Rauch & Tremaine, 1996; Hopman & Alexander, 2006), massive
perturbers which may accelerate the two-body relaxation process (Perets et
al., 2007), binary MBHs or recoiled MBHs (Ivanov et al., 2005; Chen et al.,
2011; Stone & Loeb, 2011), as well as the stellar orbital precession within
nonspherical galaxy gravitational potentials (MT99; Yu 2002; Vasiliev 2014).

In an earlier work (Chen et al. 2020b, hereafter CYL20), a statistical study
is conducted on the cosmic distributions of stellar tidal disruptions by MBHs
at galactic centers, due to the combined effects of two-body relaxation and
orbital precession of stars in triaxial galaxy potentials; and the statistical
results reveal the correlations of the stellar consumption rate with both the
MBH mass $M_{\rm BH}$ and the inner slope of the galaxy surface brightness
profile $\gamma$. A negative correlation between the stellar consumption rate
(per galaxy) with the MBH mass (e.g., Wang & Merritt 2004; Stone & Metzger
2016) is found to exist only for $M_{\rm BH}\lesssim 10^{7}{M_{\odot}}$, and
the correlation becomes positive for $M_{\rm BH}\gtrsim 10^{7}{M_{\odot}}$. At
a given MBH mass $M_{\rm BH}$, the stellar consumption rate is higher in
galaxies with larger $\gamma$ (steeper inner surface brightness profile) if
$M_{\rm BH}\lesssim 10^{7}{M_{\odot}}$, but insensitive to $\gamma$ for MBHs
with larger masses. In triaxial galaxy potentials, the phase space of the loss
cone described for spherical galaxies can be replaced by the phase space of a
general loss region to incorporate the stars that can precess onto low angular
momentum orbits. The dichotomic trends of the stellar consumption rate at
different MBH mass ranges can be explained by that the dominant fluxes of
stellar consumption have different origins of the stellar low-angular-momentum
orbits.

A further quantitative understanding of the origin of these correlations is of
great importance as it provides the key to distinguish the dominant
mechanism(s) of TDEs under different circumstances. Apart from that, the
current and future TDE observations shed new light on the demographic studies
of the MBH population, especially the mass function and the occupation
fraction of MBHs at the low-mass end (e.g., Stone & Metzger 2016; Fialkov &
Loeb 2017); the correlations of the stellar consumption with the MBH/galaxy
properties cannot be ignored in a proper interpretation of the observational
results, and should be accompanied with a quantitative understanding of the
origin. To understand the origin of these correlations, we employ an
analytical model considering power-law stellar distribution under the
Keplerian potential of the central MBH. We first verify that this simplified
model provides a fairly good approximation when evaluating the event rate of
stellar consumption due to the two mechanisms, i.e., two-body relaxation and
stellar orbital precession in nonspherical potentials. Then we identify the
dominant factor(s) responsible for the slope and the scatter of each
correlation.

The paper is organized as follows. In Section 2, we first briefly describe the
galaxy sample used in this study. Then with the sample galaxies, we show the
correlations of the stellar consumption rates with the MBH mass and the galaxy
inner stellar distribution, as obtained in CYL20. In Section 3, we construct
the analytical model and obtain the approximated expression for the stellar
consumption rate due to either mechanism and inspect the contribution from
different terms in the approximated expressions to these correlations. In
Section 4, we explore the possibility of using such correlations to explain
the observed overrepresentation of TDEs in those rare E+A/poststarburst
galaxies. Based on the quantitative correlations constructed from the
analytical model, we also generalize the discussion from the consumption rates
of solar-type stars to those of other different types of stars.

##  2 Galaxy sample and rate correlations

###  2.1 Galaxy sample

In this study, we adopt the two observational samples of early-type galaxies
given by Lauer et al. (2007) and Krajnović et al. (2013) to investigate the
correlation between the stellar consumption rate by the central MBH and either
the MBH mass or the host galaxy inner stellar distribution. For both samples,
high-spatial-resolution observations of the galaxy surface brightness by $HST$
are available, which are crucial for TDE studies since stars from the inner
region of the host galaxy (e.g., inside the influential radius of the central
MBH) contribute significantly to the stellar consumption (MT99). Note that
these two galaxy samples have also been adopted by CYL20 to study the cosmic
distributions of the stellar tidal disruptions and by Chen et al. (2020a) to
study the properties of the cosmic population of binary MBHs as well as their
gravitational wave emissions.

For those early-type galaxies, their surface brightness profiles $I(R)$ can be
well described by the Nuker law (Lauer et al., 1995). The best-fitting Nuker-
law parameters for the sample galaxies can be found in each source paper. When
calculating the stellar disruption/consumption rate, the mass-to-light ratio
$M/L$ of the galaxy is also required, since $M/L$ and the surface brightness
profile together determine the mass density distribution of the galaxy if it
is spherically distributed. For galaxies in Lauer et al. (2007), we obtain
their $M/L$ following Section 2.1 of Stone & Metzger (2016), i.e., their
Equations (4)-(5). For galaxies in Krajnović et al. (2013), we note that their
$r$-band $M/L$ can be found in Cappellari et al. (2013), but their Nuker-law
parameters were fitted using the $V$-band observations. In this study, we
assume that the $V$-band $M/L$ for these galaxies are the same as their
$r$-band values. We estimate the MBH masses $M_{\rm BH}$ for the sample
galaxies based on empirical scaling relations. For galaxies in Lauer et al.
(2007), we follow Section 2.1 of Stone & Metzger (2016), i.e., adopting the
$M_{\rm BH}$–$\sigma_{\rm e}$ relation from McConnell & Ma (2013) when
$\sigma_{\rm e}$ is available in Lauer et al. (2007), while adopting the
$M_{\rm BH}$–$L_{V}$ relation from McConnell & Ma (2013) otherwise. For
galaxies in Krajnović et al. (2013), we adopt the $M_{\rm BH}$–$\sigma_{\rm
e}$ relation from McConnell & Ma (2013), with $\sigma_{\rm e}$ taken from
Cappellari et al. (2013). When studying the stellar consumption due to loss-
region draining, we assume that each sample galaxy has a triaxial shape
characterized by $p_{\rho}=0.9$ and $q_{\rho}=0.8$, where $p_{\rho}$ and
$q_{\rho}$ represent the medium-to-major and minor-to-major axis ratios of the
galaxy mass density distribution.111We adopt this shape
($p_{\rho}=0.9,q_{\rho}=0.8$) as a representative of generic triaxial stellar
systems. The corresponding triaxiality parameter is
$T_{\rho}=(1-p_{\rho}^{2})/(1-q_{\rho}^{2})\simeq 0.53$. As seen from Figure 7
of Chen et al. (2020a), the loss region (i.e., the stellar reservoir for the
refilling of the loss cone in triaxial systems) changes abruptly when
$T_{\rho}$ is very close to $0/1$ or $q_{\rho}$ is very close to $1$, while it
changes little with other $T_{\rho}$ and $q_{\rho}$ values. Therefore, the
results obtained in this study will hold even when the host galaxy has a
different shape, as long as the shape is not very close to perfectly spherical
or axisymmetric ones.

###  2.2 Stellar consumption rate correlations

In a spherical stellar system, the rate of stellar consumption by the central
MBH due to two-body relaxation can be evaluated by solving the Fokker-Planck
equation and the solution can be found in MT99 (see also Lightman & Shapiro
1977; Cohn & Kulsrud 1978), i.e.,

${F^{\rm
lc}}(\mathcal{E})d\mathcal{E}=\frac{4\pi^{2}\bar{f}(\mathcal{E})P(\mathcal{E})\bar{\mu}(\mathcal{E})J_{\rm
c}^{2}(\mathcal{E})d\mathcal{E}}{\ln R_{0}^{-1}(\mathcal{E})}.$ (1)

Here $\bar{f}(\mathcal{E})$ is the “isotropized” distribution function (see
Eq. 21 of MT99), $P(\mathcal{E})\equiv P(\mathcal{E},J^{2}=0)$ is the orbital
period of a test particle with “specific binding energy” $\mathcal{E}$
(hereafter abbreviated as energy) and zero “specific angular momentum” $J$
(hereafter abbreviated as angular momentum), $\bar{\mu}(\mathcal{E})$ is the
orbital-averaged diffusion coefficient of the dimensionless angular momentum
$R(\mathcal{E})\equiv J^{2}(\mathcal{E})/J_{\rm c}^{2}(\mathcal{E})$ in the
limit $R\rightarrow 0$, with $J_{\rm c}(\mathcal{E})$ denoting the angular
momentum of a star on a circular orbit with energy $\mathcal{E}$,
$R_{0}(\mathcal{E})$ marks the value of $R(\mathcal{E})$ at which the
distribution function falls to zero. $R_{0}(\mathcal{E})$ is given by

$R_{0}(\mathcal{E})=R_{\rm
lc}(\mathcal{E})\times\left\\{\begin{split}&\exp(-q)&q\geq 1\\\
&\exp(-0.186q-0.824\sqrt{q})&q<1\end{split},\right.$ (2)

where $R_{\rm lc}(\mathcal{E})\equiv J^{2}_{\rm lc}(\mathcal{E})/J^{2}_{\rm
c}(\mathcal{E})$ represents the relative size of the loss cone at energy
$\mathcal{E}$ in the phase space, with $J_{\rm lc}$ denoting the loss-cone
angular momentum and $q(\mathcal{E})\equiv
P(\mathcal{E})\bar{\mu}(\mathcal{E})/R_{\rm lc}(\mathcal{E})$ representing the
ratio of the change of $R(\mathcal{E})$ for radial orbits during an orbital
period to $R_{\rm lc}(\mathcal{E})$ (MT99).

The stellar consumption rate due to loss-region draining in a nonspherical
potential at a sufficiently long consumption time $T$ can be approximated by

${F^{\rm drain}}(\mathcal{E})d\mathcal{E}=4\pi^{2}f(\mathcal{E})J_{\rm
lc}^{2}(\mathcal{E})\exp\left[-\frac{T}{P(\mathcal{E})}\frac{J_{\rm
lc}^{2}(\mathcal{E})}{J_{\rm lr}^{2}(\mathcal{E})}\right]d\mathcal{E},$ (3)

where in triaxial galaxies $J_{\rm lr}(\mathcal{E})$ corresponds to the
characteristic angular momentum at energy $\mathcal{E}$ below which stars can
precess into the loss cone, and it is determined by the nonspherical shape of
the host galaxy (see Eq. 9 in CYL20 and Eq. 52 in MT99). Due to the draining,
stars initially inside the loss region are gradually depleted, and stars
initially outside the loss region can be diffused into it due to two-body
relaxation. As shown in CYL20, in triaxial galaxies, the loss-region refilling
rate can be obtained by generalizing the analysis done for spherical systems,
i.e., by replacing $J_{\rm lc}$ in Equations (1)–(2) with the angular momentum
$J_{\rm lr}$ characterizing the size of the loss region. When the stars
initially in the loss region are depleted significantly at $T\gg
P(\mathcal{E}){J_{\rm lr}^{2}(\mathcal{E})}/{J_{\rm lc}^{2}(\mathcal{E})}$,
the loss-region refilling rate due to two-body relaxation can be larger than
the loss-region draining rate given by Equation (3) and thus dominate the
stellar consumption rate.

When the loss-region refilling rate due to two-body relaxation dominates the
stellar consumption rate, compared with the rate obtained through two-body
relaxation in spherical systems, the correction of the stellar consumption
rates for triaxial galaxies can be mainly due to the change of the logarithm
term of $\ln R_{0}^{-1}(\mathcal{E})$ in Equations (1)–(2); and compared to
the estimates for spherical systems, the average stellar consumption rates in
triaxial galaxies are increased roughly by a similar factor of $f^{\rm
tri}\sim$3–5 for low-mass MBHs (with masses ranging from $M_{\rm BH}\sim
10^{5}$–$10^{7}M_{\odot}$; see Figure 3 or 4 in CYL20). This rate increasing
factor is almost independent of the MBH mass for low-mass MBHs (when $T\gtrsim
1$ Gyr; see Fig. 5 of CYL20). That is, the consumption rate–MBH mass
correlation obtained by only considering two-body relaxation in spherical
systems is similar to that by considering loss-region draining and refilling,
except for a normalization difference by a factor of $f^{\rm tri}\sim 3-5$
(see Fig. 3 of CYL20). Thus, below for the purpose of this work on discussing
the changing tendencies of the stellar consumption rates with properties of
MBHs and host galaxies and for simplicity, we use Equations (1)–(2) obtained
due to the two-body relaxation for spherical systems to analyze parts of the
origin of the correlations.

The origin of the correlations of the stellar consumption rates with
properties of MBHs and host galaxies can be analyzed through the functions of
${\mathscr{F}^{\rm lc}}=\int{F^{\rm lc}}(\mathcal{E})d\mathcal{E}$ or
${\mathscr{F}^{\rm drain}}=\int{F^{\rm drain}}(\mathcal{E})d\mathcal{E}$,
where ${\mathscr{F}^{\rm lc}}$ is the total rate of stellar consumption due to
two-body relaxation obtained by assuming that galaxies are spherical, and
${\mathscr{F}^{\rm drain}}$ is the total rate of stellar consumption due to
the draining of the loss region obtained by assuming that galaxies are non-
spherical. By taking into account that at some $\mathcal{E}$, ${F^{\rm
drain}}(\mathcal{E})$ can become subdominant compared to the corresponding
loss region refilling rate due to two-body relaxation, the stellar consumption
rate (obtained by the integration over $\mathcal{E}$) in a triaxial galaxy,
$\sim\int\max[f^{\rm tri}{F^{\rm lc}}(\mathcal{E}),{F^{\rm
drain}}(\mathcal{E})]d\mathcal{E}$, is expected to be larger than $\max(f^{\rm
tri}{\mathscr{F}^{\rm lc}},{\mathscr{F}^{\rm drain}})$, as well as being lower
than the sum of $f^{\rm tri}{\mathscr{F}^{\rm lc}}+{\mathscr{F}^{\rm drain}}$.
Thus the stellar consumption rate is expected to be $\sim{\mathscr{F}^{\rm
drain}}$ if ${\mathscr{F}^{\rm drain}}\gg f^{\rm tri}{\mathscr{F}^{\rm lc}}$,
and $\sim f^{\rm tri}{\mathscr{F}^{\rm lc}}$ if ${\mathscr{F}^{\rm drain}}\ll
f^{\rm tri}{\mathscr{F}^{\rm lc}}$; As to be seen below (e.g., Figs. 1-2),
both of those two cases cover a significantly large parameter space of host
galaxy properties, so that the functions of ${\mathscr{F}^{\rm lc}}$ and
${\mathscr{F}^{\rm drain}}$ can be used to analyze the origin of the
correlations of the stellar consumption rates. We estimate those stellar
consumption rates of ${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm drain}}$
for each sample galaxy according to Equations (1)–(3). We refer to CYL20 for
some detailed procedures to calculate the relevant quantities in these
equations.

Figure 1 shows the dependence of the stellar consumption rate due to two-body
relaxation ${\mathscr{F}^{\rm lc}}$ on the MBH mass $M_{\rm BH}$ (left panel)
and on the inner slope of the galaxy stellar number/mass density distribution
$\alpha$ (right panel), respectively. The stellar consumption rate of each
galaxy is evaluated based on Equation (1). As seen from this figure,
${\mathscr{F}^{\rm lc}}$ has a negative correlation with $M_{\rm BH}$, and a
positive one with $\alpha$. We conduct linear fittings to the
$\log{\mathscr{F}^{\rm lc}}-\log M_{\rm BH}$ relation (left panel) and the
$\log{\mathscr{F}^{\rm lc}}-\alpha$ relation (right panel), respectively. The
best-fitting relations are shown by the red dashed line in each panel, with
the corresponding slope and intercept labeled in the panel. Similarly, in
Figure 2, we show the dependence of the stellar consumption rate due to loss-
region draining in nonspherical potentials ${\mathscr{F}^{\rm drain}}$ on the
mass of the central MBH $M_{\rm BH}$ and on the inner slope of the galaxy
stellar number/mass density distribution $\alpha$, where we set
$T=10{\rm\,Gyr}$. In contrast to the case of two-body relaxation,
${\mathscr{F}^{\rm drain}}$ has a positive correlation with $M_{\rm BH}$ and a
mildly negative correlation with $\alpha$. The best-fitting relations are also
shown by the red dashed lines in both panels.

Figure 1: Dependence of the stellar consumption rate by the central MBH due to
two-body relaxation ${\mathscr{F}^{\rm lc}}$ on the mass of the central MBH
$M_{\rm BH}$ (left panel) and on the inner slope of the galaxy stellar
number/mass density distribution $\alpha$ (right panel). Each open circle
represents a galaxy in the sample. The red dashed lines in the left and the
right panels represent the best fits to the correlation between
${\mathscr{F}^{\rm lc}}$ and $M_{\rm BH}$ and the correlation between
${\mathscr{F}^{\rm lc}}$ and $\alpha$, respectively. The best fit form is
obtained by using the linear least square regression algorithm and is also
labeled in each panel. This figure shows the slopes and scatters of both the
correlation between ${\mathscr{F}^{\rm lc}}$ and $M_{\rm BH}$ and that between
${\mathscr{F}^{\rm lc}}$ and $\alpha$. Figure 2: Legends are the same as those
for Fig. 1, except that the stellar consumption is due to the draining of the
loss region in nonspherical potentials. This figure shows the positive
correlation between ${\mathscr{F}^{\rm drain}}$ and $M_{\rm BH}$, and the
mildly negative correlation between ${\mathscr{F}^{\rm drain}}$ and $\alpha$.

As seen from Figures 1 and 2, we have ${\mathscr{F}^{\rm
lc}}\gtrsim{\mathscr{F}^{\rm drain}}$ at $M_{\rm BH}\lesssim
10^{7}{M_{\odot}}$ and ${\mathscr{F}^{\rm lc}}\lesssim{\mathscr{F}^{\rm
drain}}$ at $M_{\rm BH}\gtrsim 10^{7}{M_{\odot}}$. Correspondingly, we apply
the correlations of ${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm drain}}$
to interpret the correlation tendencies of the stellar consumption rates in
those low-mass and high-mass ranges of MBHs, respectively.

Note that the fits shown in Figures 1 and 2 cover the whole MBH mass range, as
it is a clear way to show the effects of the same mechanism through a large
mass range. If the fit is limited to only a small mass range of $M_{\rm
BH}\lesssim 10^{7}{M_{\odot}}$, specifically the correlation between
${\mathscr{F}^{\rm lc}}$ and $M_{\rm BH}$ could appear quite mild or may not
be negative due to the small mass range and large rate scatters. In general,
the other correlation tendencies are found not to be affected significantly
even if limiting the fit mass range to either $M_{\rm BH}\lesssim
10^{7}{M_{\odot}}$ or $M_{\rm BH}\gtrsim 10^{7}{M_{\odot}}$. In this work, for
connecting the stellar consumption rates with the underlying mechanisms and
for simplicity, we use the fit results of the whole mass range below.

##  3 The power-law stellar distribution model

Table 1: Best-fit parameters and their $1\sigma$ uncertainties for the different correlations shown in Figures 1–2, 4, and 6.  |  $\log_{10}Y=b_{M}+k_{M}\log_{10}M_{\rm BH,8}$ |  |  $\log_{10}Y=b_{\alpha}+k_{\alpha}\alpha$
---|---|---|---
$Y$ | $b_{M}$ | $k_{M}$ |  | $b_{\alpha}$ | $k_{\alpha}$
F^lc | -4.64(0.04) | -0.28(0.04) |  | -6.61(0.18) | 1.37(0.12)
σ_h,200^3 M_BH,8^-1 | -0.40(0.04) | -0.47(0.04) |  | -2.35(0.20) | 1.37(0.13)
ζ(α) | -0.11(0.01) | -0.06(0.01) |  | -1.02(0.01) | 0.63(0.00)
ϕ(ω) | -0.72(0.02) | 0.26(0.02) |  | 0.57(0.11) | -0.90(0.07)
F^drain | -3.54(0.03) | 0.90(0.03) |  | -2.67(0.31) | -0.64(0.21)
χ(α) | 0.26(0.01) | 0.16(0.01) |  | 1.18(0.04) | -0.64(0.03)
κ(M_BH,σ_h) | -0.37(0.04) | 0.69(0.04) |  | -0.71(0.29) | 0.20(0.19)
(R_lr,T/0.1)^25α-15 | -0.18(0.01) | 0.01(0.01) |  | 0.12(0.02) | -0.21(0.02)
Figure 3: Comparisons of the calculated stellar consumption rates due to the
two-body relaxation and those obtained approximately by adopting the simple
power-law model described in Section 3, where the galaxies are assumed to be
spherical. This figure verifies that the simple power-law model is effective
in estimating the stellar consumption rates by the central MBHs due to the
two-body relaxation, and therefore can be used to inspect the correlations of
the stellar consumption rate with both the MBH mass and the inner stellar
distribution of the galaxy.

The explanation to the correlations shown in Figures 1 and 2 obtained by using
Equations (1)-(3) is not explicit, since many of the terms in these equations
are calculated numerically for individual galaxies. In the following
subsections, we employ a power-law model to obtain approximate expressions for
the stellar consumption rates due to both the mechanisms. By saying “power-law
model” we mean that the stellar number/mass density distribution of the galaxy
can be described by a single power law and only the Keplerian potential of the
central MBH is considered. We investigate the relative contributions of
different terms in this simplified model to the stellar consumption rate and
thus figure out the dominant factors that lead to these two correlations. As a
counterpart, we define the “full model” as the one evaluating the stellar
consumption rates through Equations (1)–(3). Below, we describe the details of
the power-law model.

In the simplified model, we assume that the distribution of stars around the
central MBH follows a single power law, i.e., $n_{\ast}(r)\propto r^{-\alpha}$
where $n_{\ast}(r)$ is the spatial number density of stars at radius $r$ and
$\alpha$ is the power-law index. We define the influential radius of the MBH
$r_{\rm h}$ to be the radius within which the enclosed stellar mass equals the
MBH mass, i.e., $M_{\rm h}\equiv M_{\ast}(r\leq r_{\rm h})=M_{\rm BH}$. We
make the power-law assumption based on the finding that stellar diffusion in
the energy space gradually drives the distribution of stars in a system
containing a central MBH towards a power-law cusp (Bahcall & Wolf, 1976). In
practice, we adopt the Nuker law (Lauer et al., 1995) as a generic description
of the surface brightness profiles of the galaxies. The break radii of
realistic galaxies $R_{\rm b}$ are typically much larger than the influential
radii of the MBHs $r_{\rm h}$, ensuring that the power-law density
distribution a fairly good approximation for stars inside $r_{\rm h}$.
Therefore, the spatial number density of stars surrounding the MBH could be
formulated as

$n_{\ast}(r)=\frac{(3-\alpha)N_{\rm h}}{4\pi r_{\rm
h}^{3}}\left({\frac{r}{r_{\rm h}}}\right)^{-\alpha},$ (4)

where $N_{\rm h}$ is the total number of stars enclosed in $r_{\rm h}$. The
total number of stars enclosed in any given radius $r$ is $N_{\ast}(\leq
r)=N_{\rm h}(r/r_{\rm h})^{3-\alpha}$. If we assume identical mass of stars,
we also have $M_{\ast}(\leq r)=M_{\rm h}(r/r_{\rm h})^{3-\alpha}$, with
$M_{\ast}(\leq r)$ being the total stellar mass enclosed in radius $r$.
Throughout this paper, we make the approximation that $\alpha=\gamma+1$, where
$\gamma$ is the inner slope of the Nuker-law surface brightness profile. It
turns out to be a good approximation for sample galaxies with $\gamma\gtrsim
0.2$. For galaxies with $\gamma\lesssim 0.2$, the approximation tends to
overestimate the inner slope of the host galaxy stellar number/mass density
profile. However, the main conclusions obtained in this paper do not change by
a small number of such galaxies in the sample.

In the power-law model, we ignore the self-gravity of the stellar system and
assume that the potential is mainly contributed by the central MBH. We make
this assumption based on the fact that the loss-cone consumption rate due to
either mechanism peaks around the influential radius of the central MBH if we
convert energy to radius based on the radial energy profile for stars in
circular orbits, i.e., $\mathcal{E}(r)$. For example, for most of the sample
galaxies, $\mathcal{E}{F^{\rm lc}}(\mathcal{E})$ peaks around $r_{\rm h}$
within a factor of $2$, while $\mathcal{E}{F^{\rm drain}}(\mathcal{E})$ peaks
around $r_{\rm h}$ within a factor of $3$ when the consumption time is fixed
to $T=10{\rm\,Gyr}$. Adopting a different $T$ does not significantly affect
our results. Under the Keplerian potential of the central MBH, we evaluate the
ergodic distribution function based on the Eddington’s formula. The resulting
differential energy distribution of stars $N(\mathcal{E})d\mathcal{E}$ (i.e.,
the number of stars in the stellar system with energy in the range of
$\mathcal{E}\rightarrow\mathcal{E}+d\mathcal{E}$) is expressed by

$N(\mathcal{E})d\mathcal{E}=\xi(\alpha)N_{\rm
h}\left(\frac{\mathcal{E}}{\mathcal{E}_{\rm
h}}\right)^{\alpha-3}d\ln\mathcal{E},$ (5)

where

$\xi(\alpha)=\frac{\sqrt{\pi}}{8}(3-\alpha)2^{3-\alpha}\frac{\Gamma(\alpha+1)}{\Gamma(\alpha-\frac{1}{2})},$
(6)

and $\mathcal{E}_{\rm h}\equiv\sigma_{\rm h}^{2}/2$ where $\sigma_{\rm
h}^{2}\equiv GM_{\rm BH}/r_{\rm h}$.

###  3.1 Two-body relaxation mechanism

Figure 4: Dependence of the terms $\sigma^{3}M_{\rm BH}^{-1}$,
$\zeta(\alpha)$, and $\phi(\omega)$ in the approximated stellar consumption
rate due to the two-body relaxation in the power-law model ${\mathscr{F}^{\rm
lc,PL}}$ (Eq. 10) on the MBH mass $M_{\rm BH}$ (left panels) and the inner
slope of the host galaxy mass density distribution $\alpha$ (right panels).
From top to bottom, the three inspected terms are ${\sigma_{\rm h}^{3}M_{\rm
BH}^{-1}}$, $\zeta(\alpha)$ (Eq. 11), and $\phi(\omega)$ (Eq. 12),
respectively. In the top two panels, $\sigma_{\rm h,200}$ and $M_{\rm BH,8}$
are $\sigma_{\rm h}$ and $M_{\rm BH}$ in unit of $200{\rm\,km\,s^{-1}}$ and
$10^{8}{M_{\odot}}$, respectively. The best-fit is shown by the red dashed
line in each panel, and the best fit form is also marked there with $Y$
representing the quantity of $\sigma_{\rm h,200}^{3}M_{\rm BH}^{-1}$,
$\zeta(\alpha)$, and $\phi(\omega)$ in the top, middle, and bottom panels,
respectively. Note that for the convenience of direct comparisons of the
scatters with Fig. 1, all the panels here share the same decades along the
$y$-axis with the panels in Fig. 1. This figure aims to reveal the dominant
contributors to the slopes and scatters of both the correlation between
${\mathscr{F}^{\rm lc}}$ and $M_{\rm BH}$ and that between ${\mathscr{F}^{\rm
lc}}$ and $\alpha$. See Section 3.1 for more details.

We first consider the stellar consumption due to the two-body relaxation by
assuming the galaxy gravitational potential are spherical. When the Keplerian
potential of the central MBH dominates, we have $P(\mathcal{E},J^{2})\simeq
P(\mathcal{E})\simeq 2\pi GM_{\rm BH}/(2\mathcal{E})^{3/2}$; and the
differential energy distribution can be expressed as $N(\mathcal{E})\simeq
4\pi^{2}f(\mathcal{E})P(\mathcal{E})J_{\rm c}^{2}(\mathcal{E})$ (i.e., see Eq.
5 of MT99). In this case, Equation (1) can be approximated as

${F^{\rm
lc}}(\mathcal{E})d\mathcal{E}\simeq\frac{N(\mathcal{E})\bar{\mu}(\mathcal{E})d\mathcal{E}}{\ln
R_{0}^{-1}(\mathcal{E})}\simeq\frac{N(\mathcal{E})}{t_{\rm
r}(\mathcal{E})}\frac{d\mathcal{E}}{\ln R_{0}^{-1}(\mathcal{E})},$ (7)

where in the latter expression we have used the inverse of the relaxation
timescale $1/t_{\rm r}(\mathcal{E})$ to approximate the orbital-averaged
diffusion coefficient of the angular momentum, $\bar{\mu}(\mathcal{E})$. The
relaxation timescale $t_{\rm r}(\mathcal{E})$ can be expressed as (see Bar-Or
et al. 2013 and Eq. 1 of Alexander 2017)

$t_{\rm r}(\mathcal{E})\simeq\eta\frac{Q^{2}P_{\rm
c}(\mathcal{E})}{N_{\ast}(\leq r)\ln Q}\simeq\frac{2\pi\eta GM_{\rm
BH}Q^{2}}{N_{\rm h}\sigma_{\rm h}^{3}\ln
Q}\left(\frac{\mathcal{E}}{\mathcal{E}_{\rm h}}\right)^{3/2-\alpha},$ (8)

where $\eta\simeq 1/8$, $Q\equiv M_{\rm BH}/\langle m_{\star}\rangle\simeq
N_{\rm h}$ with $\langle m_{\star}\rangle$ representing the mean mass of
stars, and $P_{\rm c}(\mathcal{E})$ is the orbital period of a circular orbit
at energy $\mathcal{E}$. To get the rightmost expression in Equation (8), we
assume the dominance of the Keplerian potential of the central MBH and use
$r/r_{\rm h}\simeq(\mathcal{E}/\mathcal{E}_{\rm h})^{-1}$. Substituting
Equations (5) and (8) into Equation (7), we have

$\mathcal{E}{F^{\rm lc}}(\mathcal{E})\simeq\frac{\xi(\alpha)\sigma_{\rm
h}^{3}\ln Q}{2\pi\eta GM_{\rm BH}\ln
R_{0}^{-1}(\mathcal{E})}\left(\frac{\mathcal{E}}{\mathcal{E}_{\rm
h}}\right)^{2\alpha-9/2}.$ (9)

If we omit the variance of $\ln R_{0}^{-1}(\mathcal{E})$, which is a mildly
decreasing function of $\mathcal{E}$ in the range
$\mathcal{E}\gtrsim\mathcal{E}_{\rm h}$, then $\mathcal{E}{F^{\rm
lc}}(\mathcal{E})$ is a decreasing function of $\mathcal{E}$ as long as
$\alpha<9/4$, as seen from Equation (9). If the inner stellar distribution is
core-like, i.e., $\alpha$ is small, $\mathcal{E}{F^{\rm lc}}(\mathcal{E})$
decreases relatively sharply with increasing $\mathcal{E}$, whereas if the
inner stellar distribution is cusp-like with large $\alpha$,
$\mathcal{E}{F^{\rm lc}}(\mathcal{E})$ decreases relatively mildly with
increasing $\mathcal{E}$.

In real galaxies, $\mathcal{E}{F^{\rm lc}}(\mathcal{E})$ peaks at
$\mathcal{E}_{\rm peak}\equiv\omega\mathcal{E}_{\rm h}$ where $\omega\simeq
1$. For $\mathcal{E}<\mathcal{E}_{\rm peak}$, the self-gravity from the
stellar system cannot be neglected and Equation (9) does not work any more. As
a matter of fact, $\mathcal{E}{F^{\rm lc}}(\mathcal{E})$ is an increasing
function of $\mathcal{E}$ at $\mathcal{E}\leq\mathcal{E}_{\rm peak}$, instead
of a decreasing function as described by Equation (9) for
$\mathcal{E}>\mathcal{E}_{\rm peak}$. Therefore, the stellar consumption rate
can be divided into two parts separated by $\mathcal{E}_{\rm peak}$. For the
inner part with $\mathcal{E}\geq\mathcal{E}_{\rm peak}$, the rate can be
approximated by integrating ${F^{\rm lc}}(\mathcal{E})$ over the range
$\mathcal{E}\geq\mathcal{E}_{\rm peak}$ using Equation (9). For those real
galaxies in our sample, we find that the fraction of the stellar consumption
rate contributed by the inner part (i.e., with
$\mathcal{E}\geq\mathcal{E}_{\rm peak}$) spans $0.1\sim 0.8$, and this
fraction tends to be larger for a galaxy with a larger $\alpha$. Therefore, to
compensate for the fraction of the rate contributed by those stars with
$\mathcal{E}\leq\mathcal{E}_{\rm peak}$ which are omitted in the integration,
we introduce a fudge factor ${f_{\omega}}$ by integrating ${F^{\rm
lc}}(\mathcal{E})$ over the range $\mathcal{E}\geq{f_{\omega}}\mathcal{E}_{\rm
peak}={f_{\omega}}\omega\mathcal{E}_{\rm h}$ using Equation (9). The resulting
stellar consumption rate in the power-law model is given by

$\displaystyle{\mathscr{F}^{\rm lc,PL}}$ $\displaystyle\simeq$
$\displaystyle\int_{{f_{\omega}}\omega\mathcal{E}_{\rm
h}}^{\infty}\mathcal{E}{F^{\rm
lc}}(\mathcal{E})d\ln\mathcal{E}\simeq\frac{\zeta(\alpha)\sigma_{\rm
h}^{3}({f_{\omega}}\omega)^{2\alpha-\frac{9}{2}}\ln Q}{2\pi\eta GM_{\rm BH}\ln
R_{0}^{-1}(\omega\mathcal{E}_{\rm h})}$ (10) $\displaystyle=$
$\displaystyle\zeta(\alpha)\phi(\omega)\frac{\sigma_{\rm h}^{3}}{M_{\rm
BH}}\frac{\ln Q}{2\pi G\eta},$

where $\zeta(\alpha)$ and $\phi(\omega)$ are defined as

$\zeta(\alpha)=\frac{2}{9-4\alpha}\cdot\xi(\alpha),$ (11)

and

$\phi(\omega)=\frac{({f_{\omega}}\omega)^{2\alpha-\frac{9}{2}}}{\ln
R_{0}^{-1}(\omega\mathcal{E}_{\rm h})}.$ (12)

We find that ${f_{\omega}}=1/2$ works well empirically. Therefore, we set
${f_{\omega}}=1/2$ in the following analysis if not stated specifically.

We compare the stellar consumption rates calculated from Equations (1)–(2) and
those obtained approximately from the power-law model, as shown in Figure 3.
Apparently, the rates estimated by using the simple power-law model are well
consistent with those calculated from the full model (see Eqs. 1–3 and Chen et
al. 2020b), which verifies the effectiveness of the simple power-law model in
obtaining the correct rate and suggests the validity of using this simple
model to inspect the correlations between ${\mathscr{F}^{\rm lc}}$ with both
$M_{\rm BH}$ and $\alpha$. According to Equation (10), both the correlation
between ${\mathscr{F}^{\rm lc}}$ and $M_{\rm BH}$ and that between
${\mathscr{F}^{\rm lc}}$ and $\alpha$ are controlled by three terms, i.e.,
${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$, $\zeta(\alpha)$, and $\phi(\omega)$.
Figure 4 shows the dependence of the above three terms on the MBH mass $M_{\rm
BH}$ (left panels) and the inner slope of the host galaxy mass density
distribution $\alpha$ (right panels), separately. As seen from this figure,
$\sigma_{\rm h}^{3}M_{\rm BH}^{-1}$ negatively correlates with $M_{\rm BH}$,
$\phi(\omega)$ positively correlates with $M_{\rm BH}$, while the term
$\zeta(\alpha)$ only weakly correlate with $M_{\rm BH}$; both $\sigma_{\rm
h}^{3}M_{\rm BH}^{-1}$ and $\zeta(\alpha)$ positively correlate with $\alpha$,
while $\phi(\omega)$ negatively correlates with $\alpha$, To describe the
relative contribution of each term to the ${\mathscr{F}^{\rm lc}}-M_{\rm BH}$
relation and the ${\mathscr{F}^{\rm lc}}-\alpha$ relation quantitatively, we
conduct a linear least square fitting to the data points shown in each panel
of Figure 4. The best fit is shown by the red dashed line in each panel and
its form is also labeled there. The best-fit parameters with their $1\sigma$
uncertainties are also listed in Table 1.

We identify the main factors that lead to the ${\mathscr{F}^{\rm lc}}-M_{\rm
BH}$ relation and the ${\mathscr{F}^{\rm lc}}-\alpha$ relation quantitatively
based on the fitting results (see Tab. 1 and legends in Figs. 1 and 4).

  * •

For the ${\mathscr{F}^{\rm lc}}-M_{\rm BH}$ relation, the best fit gives a
slope of $-0.28$ (left panel of Fig. 1). The best fits shown in the left
panels of Figure 4 (from top to bottom) have slopes of $-0.47$, $-0.06$, and
$0.26$, respectively. The sum of these three slopes is $-0.27$, which can
approximately account for that found in the left panel of Figure 1. Among the
three terms, ${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$ dominates the total slope.
The contribution from $\phi(\omega)$ can only cancel about half the amount of
${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$, and $\zeta(\alpha)$ contributes the
least to the total slope. Apart from the slope, the data points in the top
left panel of Figure 4 have comparable scatters as those in the left panel of
Figure 1, and are substantially larger than those in the middle left or the
bottom left panel of Figure 4. Therefore, the intrinsic scatter of the
${\mathscr{F}^{\rm lc}}-M_{\rm BH}$ relation is also dominated by term
${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$, i.e., by the intrinsic scatter of the
correlation between ${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$ and $M_{\rm BH}$,
which is in turn determined by the scatter of the correlation between
$\sigma_{\rm h}$ and $M_{\rm BH}$.

  * •

For the ${\mathscr{F}^{\rm lc}}-\alpha$ relation, the best fit gives a slope
of $1.37$ (right panel of Fig. 1). The best fits shown in the right panels of
Figure 4 (from top to bottom) give slopes of $1.37$, $0.63$, and $-0.90$,
respectively. The sum of these three slopes is $1.10$, which is close to that
found for the ${\mathscr{F}^{\rm lc}}-\alpha$ relation (right panel of Fig.
1), again suggests that the correlation between ${\mathscr{F}^{\rm lc}}$ and
$\alpha$ can be explained approximately by the combined effect of these three
terms. Among the three terms, ${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$ dominates
the contribution to the total slope. While $\zeta(\alpha)$ and $\phi(\omega)$
contribute to the total slope in close amounts but different signs so as to be
cancelled much. As seen also from the right panels of Figure 4, the scatters
among the data points in the top panel dominate over those in the middle or
bottom panels. Therefore, we can conclude that both the slope and the scatter
of the correlation between ${\mathscr{F}^{\rm lc}}$ and $\alpha$ are dominated
by the correlation between ${\sigma_{\rm h}^{3}M_{\rm BH}^{-1}}$ and $\alpha$.

###  3.2 Loss-region draining in nonspherical potentials

Figure 5: Legends are similar to that for Fig. 3, but the stellar consumption
rates are due to the draining of loss region in nonspherical potentials.
Figure 6: Dependence of the terms in the approximated stellar consumption rate
due to draining of the loss region in nonspherical potentials in the power-law
model ${\mathscr{F}^{\rm drain,PL}}$ (Eqs. 16 and 23) on the MBH mass $M_{\rm
BH}$ (left panels) and the inner slope of the host galaxy mass density
distribution $\alpha$ (right panels). From top to bottom, the three inspected
terms are $\chi(\alpha)$ (Eq. 21), ${\kappa(M_{\rm BH},\sigma_{\rm h})}$ (Eq.
22), and ${(R_{{\rm lr},T}/0.1)^{\frac{2}{5}\alpha-\frac{1}{5}}}$ (Eq. 15),
respectively. The best fit to the data is shown by the red dashed line in each
panel, and the best fit form is also marked there with $Y$ representing
$\chi(\alpha)$, ${\kappa(M_{\rm BH},\sigma_{\rm h})}$, and ${(R_{{\rm
lr},T}/0.1)^{\frac{2}{5}\alpha-\frac{1}{5}}}$, respectively. For the
convenience of direct comparisons of the scatters with Figure 2, all the
panels here share the same decades along the $y$-axis with the panels in
Figure 2. This figure aims to reveal the dominant contributors to the slopes
and scatters of the correlations either between ${\mathscr{F}^{\rm drain}}$
and $M_{\rm BH}$ or between ${\mathscr{F}^{\rm drain}}$ and $\alpha$. See
Section 3.2 for more details.

We now consider the correlation between ${\mathscr{F}^{\rm drain}}$ and
$M_{\rm BH}$ or $\alpha$ in galaxies with nonspherical mass distributions. We
define the loss-region draining timescale $\tau(\mathcal{E})\equiv
P(\mathcal{E})J_{\rm lr}^{2}(\mathcal{E})/J_{\rm lc}^{2}(\mathcal{E})$, which
characterizes how long the reservoir of loss-region stars at energy
$\mathcal{E}$ can sustain the consumption by the central MBH. Similar as in
the loss-cone case, we define the dimensionless loss-region angular momentum
$R_{\rm lr}(\mathcal{E})\equiv J_{\rm lr}^{2}(\mathcal{E})/J_{\rm
c}^{2}(\mathcal{E})$, which characterizes the relative size of the loss region
at energy $\mathcal{E}$ in the phase space. In generic triaxial galaxies,
$R_{\rm lr}(\mathcal{E})$ can reach the order of $\sim$0.1 at energies
satisfying $\mathcal{E}\lesssim\mathcal{E}_{\rm h}$ and decreases with
increasing $\mathcal{E}$ for $\mathcal{E}>\mathcal{E}_{\rm h}$. The draining
timescale can be then expressed as $\tau(\mathcal{E})=P(\mathcal{E})R_{\rm
lr}(\mathcal{E})/R_{\rm lc}(\mathcal{E})$. Therefore, the flux of stars being
drained from the loss region into the loss cone to be consumed can be
expressed as

${F^{\rm drain}}(\mathcal{E})d\mathcal{E}\simeq\frac{N(\mathcal{E})R_{\rm
lr}(\mathcal{E})d\mathcal{E}}{\tau(\mathcal{E})}\exp\left[-\frac{T}{\tau(\mathcal{E})}\right],$
(13)

where we have again made the approximation $P(\mathcal{E},J^{2})\simeq
P(\mathcal{E})\equiv P(\mathcal{E},J^{2}=0)$. Assuming the dominance of the
Keplerian potential of the central MBH, we have $P(\mathcal{E})\simeq 2\pi
GM_{\rm BH}/(2\mathcal{E})^{3/2}$ and $J_{\rm c}(\mathcal{E})\simeq GM_{\rm
BH}/(2\mathcal{E})^{1/2}$. Define the consumption radius $r_{\rm consp}\equiv
J^{2}_{\rm lc}/2GM_{\rm BH}$, then the loss-region draining timescale can be
approximated as

$\tau(\mathcal{E})\simeq\frac{\pi(GM_{\rm BH})^{2}R_{\rm
lr}(\mathcal{E})}{4\sqrt{2}\mathcal{E}^{5/2}r_{\rm consp}}.$ (14)

Since typically $\mathcal{E}{F^{\rm drain}}(\mathcal{E})$ is an increasing
function of $\mathcal{E}$, the total stellar consumption rate at the draining
time $T$ due to loss-region draining is dominated by the flux of stars at
$\mathcal{E}\simeq\mathcal{E}_{T}$, with $\mathcal{E}_{T}$ satisfies
$\tau(\mathcal{E}_{T})=T$, which gives

$\mathcal{E}_{T}=\left[\frac{\pi(GM_{\rm BH})^{2}R_{\rm
lr}{{}_{,T}}}{4\sqrt{2}Tr_{\rm consp}}\right]^{2/5},$ (15)

where $R_{{\rm lr},T}=R_{\rm lr}(\mathcal{E}_{T})$. At energy
$\mathcal{E}<\mathcal{E}_{T}$ where the loss region has not been exhausted
yet, $\mathcal{E}{F^{\rm drain}}$ increases with $\mathcal{E}$ in a manner
that $\mathcal{E}{F^{\rm drain}}\sim\mathcal{E}^{\alpha-1/2}$ in the power-law
model. Therefore, the total stellar consumption rate due to the loss-region
draining can be evaluated as

$\displaystyle{\mathscr{F}^{\rm drain,PL}}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{2\alpha-1}f_{T}\mathcal{E}_{T}{F^{\rm
drain}}(\mathcal{E}_{T})$ (16) $\displaystyle=$
$\displaystyle\frac{2\xi(\alpha)}{2\alpha-1}\frac{f_{T}N_{\rm h}R_{{\rm
lr},T}}{T}\left(\frac{\mathcal{E}_{T}}{\mathcal{E}_{\rm
h}}\right)^{\alpha-3},$

where $\xi(\alpha)$ is given by Equation (6), $f_{T}$ is a fudge factor
introduced to make the rates estimated by the power-law model consistent with
those by the full model, and $f_{T}$ changes from $0.78$ at $T=0.1{\rm\,Gyr}$
to $0.56$ at $T=10{\rm\,Gyr}$. Since we analyze the rate correlations at
$T=10{\rm\,Gyr}$, we adopt $f_{T}=0.56$ below.

The definition of the consumption radius $r_{\rm consp}$ naturally leads to a
separatrix in the MBH mass $M_{\rm BH}$, below which $r_{\rm consp}=r_{\rm
t}=(f_{\rm t}M_{\rm BH}/m_{\star})^{1/3}R_{\star}$, with $m_{\star}$ and
$R_{\star}$ being the mass and radius of the star and $f_{\rm t}$ being a
dimensionless factor, while above which $r_{\rm consp}=2r_{\rm swl}=8GM_{\rm
BH}/c^{2}$ ($r_{\rm swl}\equiv 4GM_{\rm BH}/c^{2}$; see Eq. 1 in CYL20). For
MBHs with masses being below the separatrix, setting $m_{\star}$ and
$R_{\star}$ to be the solar mass and solar radius and $f_{\rm t}=1$, we have

$\displaystyle\frac{\mathcal{E}_{T}}{\mathcal{E}_{\rm
h}}=\left[\frac{\pi(GM_{\rm BH})^{2}R_{{\rm lr},T}}{Tr_{\rm consp}\sigma_{\rm
h}^{5}}\right]^{\frac{2}{5}}$ $\displaystyle\simeq 1.24\left[\frac{M_{\rm
BH}}{10^{8}{M_{\odot}}}\right]^{\frac{2}{3}}\left[\frac{\sigma_{\rm
h}}{200{\rm\,km\,s^{-1}}}\right]^{-2}\left[\frac{R_{{\rm
lr},T}}{0.1}\right]^{\frac{2}{5}}\left[\frac{T}{10{\rm\,Gyr}}\right]^{-\frac{2}{5}}.$
(17)

Substituting Equation (17) into Equation (16), the stellar consumption rate
due to the loss-region draining when the MBH mass is below the separatrix can
be approximated in the power-law model as

$\displaystyle{\mathscr{F}^{\rm drain,PL}}\simeq
1.24^{\alpha-3}\frac{2\xi(\alpha)}{2\alpha-1}\times
10^{-3}f_{T}{\rm\,yr^{-1}}\left[\frac{M_{\rm
BH}}{10^{8}{M_{\odot}}}\right]^{\frac{2\alpha-3}{3}}$
$\displaystyle\left[\frac{\sigma_{\rm
h}}{200{\rm\,km\,s^{-1}}}\right]^{6-2\alpha}\left[\frac{R_{{\rm
lr},T}}{0.1}\right]^{\frac{2\alpha-1}{5}}\left[\frac{10{\rm\,Gyr}}{T}\right]^{\frac{2\alpha-1}{5}}.$
(18)

On the other hand, when the MBHs have masses being above the separatrix, we
have

$\displaystyle\frac{\mathcal{E}_{T}}{\mathcal{E}_{\rm
h}}=\left[\frac{\pi(GM_{\rm BH})^{2}R_{{\rm lr},T}}{Tr_{\rm consp}\sigma_{\rm
h}^{5}}\right]^{\frac{2}{5}}\simeq$ $\displaystyle 0.735\left[\frac{M_{\rm
BH}}{10^{8}{M_{\odot}}}\right]^{\frac{2}{5}}\left[\frac{\sigma_{\rm
h}}{200{\rm\,km\,s^{-1}}}\right]^{-2}\left[\frac{R_{{\rm
lr},T}}{0.1}\right]^{\frac{2}{5}}\left[\frac{T}{10{\rm\,Gyr}}\right]^{-\frac{2}{5}}.$
(19)

Substituting Equation (19) into Equation (16), the corresponding stellar
consumption rate due to the loss-region draining when the MBH mass is above
the separatrix can be approximated in the power-law model as

$\displaystyle{\mathscr{F}^{\rm drain,PL}}\simeq
0.735^{\alpha-3}\frac{2\xi(\alpha)}{2\alpha-1}\times
10^{-3}f_{T}{\rm\,yr^{-1}}\left[\frac{M_{\rm
BH}}{10^{8}{M_{\odot}}}\right]^{\frac{2\alpha-1}{5}}$
$\displaystyle\left[\frac{\sigma_{\rm
h}}{200{\rm\,km\,s^{-1}}}\right]^{6-2\alpha}\left[\frac{R_{{\rm
lr},T}}{0.1}\right]^{\frac{2\alpha-1}{5}}\left[\frac{10{\rm\,Gyr}}{T}\right]^{\frac{2\alpha-1}{5}}.$
(20)

For the convenience of the following analysis, we define

$\chi(\alpha)=\left\\{\begin{split}1.24^{\alpha-3}\cdot\frac{2\xi(\alpha)}{2\alpha-1},\quad
r_{\rm t}\geq 2r_{\rm swl},\\\
0.735^{\alpha-3}\cdot\frac{2\xi(\alpha)}{2\alpha-1},\quad r_{\rm t}<2r_{\rm
swl}.\end{split}\right.$ (21)

and

$\displaystyle\kappa(M_{\rm BH},\sigma_{\rm h})=\left\\{\begin{split}M_{\rm
BH,8}^{\frac{2}{3}\alpha-1}\sigma_{\rm h,200}^{6-2\alpha},\quad r_{\rm t}\geq
2r_{\rm swl},\\\ M_{\rm BH,8}^{\frac{2}{5}\alpha-\frac{1}{5}}\sigma_{\rm
h,200}^{6-2\alpha},\quad r_{\rm t}<2r_{\rm swl}.\end{split}\right.$ (22)

With these definitions, the stellar consumption rate due to the loss-region
draining in the power-law model can be expressed as

$\displaystyle{\mathscr{F}^{\rm drain,PL}}$ $\displaystyle\simeq$
$\displaystyle 10^{-3}f_{T}{\rm\,yr^{-1}}\chi(\alpha)\kappa(M_{\rm
BH},\sigma_{\rm h})\times$ (23) $\displaystyle\left[\frac{R_{{\rm
lr},T}}{0.1}\right]^{\frac{2\alpha-1}{5}}\left[\frac{10{\rm\,Gyr}}{T}\right]^{\frac{2\alpha-1}{5}}.$

In the following, we fix the consumption time to $T=10{\rm\,Gyr}$.

Similarly as done for the case of the two-body relaxation, we compare the
stellar consumption rates calculated from Equation (3) and those obtained
approximately from the power-law model (Eq. 23) as shown in Figure 5. As seen
from the figure, the rates evaluated based on these two approaches are
generally consistent with each other which again verifies the effectiveness of
the power-law model in estimating the stellar consumption rate due to the
loss-region draining in nonspherical potentials.

According to Equation (23), both the correlation between ${\mathscr{F}^{\rm
drain}}$ and $M_{\rm BH}$ and that between ${\mathscr{F}^{\rm drain}}$ and
$\alpha$ are controlled by the terms of $\chi(\alpha)$, $\kappa(M_{\rm
BH},\sigma_{\rm h})$, and $\left(\frac{R_{{\rm
lr},T}}{0.1}\right)^{\frac{2\alpha-1}{5}}$, and Figure 6 shows the dependence
of these three terms on $M_{\rm BH}$ (left panels) and $\alpha$ (right
panels), separately. As done for those correlations shown in Figure 4, we also
conduct linear least square fittings to the data points shown in each panel of
Figure 6. The best fit is indicated by the red dashed line in each panel and
the best fit form is also labeled there. The best-fit parameters with their
$1\sigma$ uncertainties are also listed in Table 1.

We use the fitting results to identify the main factors that contribute to the
correlation between ${\mathscr{F}^{\rm drain}}$ and $M_{\rm BH}$ and that
between ${\mathscr{F}^{\rm drain}}$ and $\alpha$ (see Tab. 1 and legends in
Figs. 2 and 6).

  * •

As seen from the left panel of Figure 2, the ${\mathscr{F}^{\rm
drain}}$–$M_{\rm BH}$ correlation has a slope of $0.90$. While according to
the left panels of Figure 6, the correlations between those three inspected
terms (in logarithm) and $\log M_{\rm BH}$ have slopes of $0.16$, $0.69$, and
$0.01$, respectively. Overall, these three relations combined can well account
for the correlation between ${\mathscr{F}^{\rm drain}}$ and $M_{\rm BH}$.
Among the three terms, ${\kappa(M_{\rm BH},\sigma_{\rm h})}$ dominates the
correlation, which scales as $\sim\sigma_{\rm h}^{3}$ and $\sim\sigma_{\rm
h}^{3}M_{\rm BH}^{2/5}$ when $M_{\rm BH}$ is below and above the separatrix,
respectively, if setting $\alpha=3/2$. In addition, the term $\chi(\alpha)$
contributes minorly to the correlation. By comparing the data points in the
left panels of Figure 6 and the left panel of Figure 2, it is clearly revealed
that the scatter of the correlation between ${\mathscr{F}^{\rm drain}}$ and
$M_{\rm BH}$ is also dominated by ${\kappa(M_{\rm BH},\sigma_{\rm h})}$.

  * •

As seen from the right panel of Figure 2, there exist considerable scatters
among the data points. Given the scatters, the correlation between
${\mathscr{F}^{\rm drain}}$ and $\alpha$ appears to be mild. A negative slope
($-0.64$) is returned when the linear square regression is applied to the data
points, suggesting a decrease of about a factor of $4$ from $\alpha=1$ to
$\alpha=2$. From the right panels of Figure 6, the best-fit slopes of the
three inspected terms (in logarithm) against $\alpha$ are $-0.64$, $0.20$, and
$-0.21$, respectively. Again, these three relations combined together can well
account for that correlation between ${\mathscr{F}^{\rm drain}}$ and $\alpha$.
Among the three terms, $\chi(\alpha)$ dominates the correlation. The scatter
of the correlation is largely contributed by the term of ${\kappa(M_{\rm
BH},\sigma_{\rm h})}$.

##  4 Discussions

###  4.1 Overrepresentation of TDEs in E+A/poststarburst galaxies

Some recent studies towards TDE host galaxies suggest that TDEs prefer to
happen in those rare E+A/poststarburst galaxies (Arcavi et al., 2014; French
et al., 2016, 2017; Law-Smith et al., 2017; Graur et al., 2018; Hammerstein et
al., 2021). The underlying (physical) explanations for the preference are
still unclear, and there may be alternative characteristics to distinguish TDE
host galaxies from normal ones (Law-Smith et al., 2017). Here we discuss
whether the preference can be explained by the correlations between the
stellar consumption rate and the inner slope of the galaxy stellar number/mass
density distribution. Since TDE flares can only be observed when the MBH mass
is below the Hills’ mass (Hills, 1975), in which regime the two-body
relaxation mechanism dominates over the stellar orbital precession mechanism
(CYL20), we focus on the correlation between ${\mathscr{F}^{\rm lc}}$ and
$\alpha$.

To explore whether the overrepresentation can be explained by the high central
stellar density of the galaxies, Law-Smith et al. (2017) identified a region
in the Sersic index and MBH mass parameter space which contained ${\sim}2\%$
of their reference catalog galaxies but $\geq 60\%$ of those TDE host
galaxies. This means that the averaged TDE rate in those high-Sersic-index
galaxies should be higher than that in their low-Sersic-index counterparts by
a factor of ${\sim}25-48$. From the right panel of Figure 1, the mean value of
$\log{\mathscr{F}^{\rm lc}}$ increases by $1.37$ when $\alpha$ is increased
from $1$ to $2$, or equivalently when $\gamma$ (the inner slope of the galaxy
surface brightness profile) is increased from $0$ to $1$. As also seen from
the panel, the intrinsic scatter of the correlation has no significant
dependence on $\alpha$. Therefore, the averaged consumption rate
$\langle{\mathscr{F}^{\rm lc}}\rangle$ is increased by a factor of ${\sim}24$
from $\alpha=1$ to $\alpha=2$, which is within a factor of $2$ of the analysis
by Law-Smith et al. (2017). Given the limited number of TDEs in the
observational analysis (i.e., 5–10 events), it suggests that the correlation
between ${\mathscr{F}^{\rm lc}}$ and $\alpha$ may be responsible for the
overrepresentation reported by recent studies towards TDE host galaxies.

The overrepresentation may also be contributed by draining of the loss-region
stars in nonspherical potentials, as those poststarburst systems may also have
a higher degree of asymmetry in their shapes as compared with generic
galaxies. The presence of A type stars in the nuclear regions of those
E+A/poststarburst galaxies indicates their relatively young ages and dynamical
states (e.g., French et al. 2017; Law-Smith et al. 2017), and therefore a
relatively full loss region and a larger draining rate. As the loss region
draining rate ${\mathscr{F}^{\rm drain}}$ decreases with increasing time $T$,
we can define the time $T_{\rm eq}$ at which the loss region draining rate
${\mathscr{F}^{\rm drain}}(T=T_{\rm eq})$ equals to the loss cone refilling
rate $f^{\rm tri}{\mathscr{F}^{\rm lc}}$. We have ${\mathscr{F}^{\rm
drain}}(T)\gtrsim f^{\rm tri}{\mathscr{F}^{\rm lc}}$ at $T\lesssim T_{\rm eq}$
and ${\mathscr{F}^{\rm drain}}(T)\lesssim f^{\rm tri}{\mathscr{F}^{\rm lc}}$
at $T\gtrsim T_{\rm eq}$. By using the scaling dependence of
${\mathscr{F}^{\rm drain,PL}}$ on time $T$ shown in Equation (23) (i.e.,
$\propto T^{(1-2\alpha)/5}$) and the scaling dependence of ${\mathscr{F}^{\rm
drain}}$ and ${\mathscr{F}^{\rm lc}}$ on $M_{\rm BH}$ shown at
$T=10{\rm\,Gyr}$ in Figures 1–2, we can obtain the time $T_{\rm eq}$ by

$\left(\frac{2\alpha-1}{5}\right)\log\left(\frac{T_{\rm
eq}}{10{\rm\,Gyr}}\right)\simeq 1.10-\log f^{\rm tri}+1.18\log M_{\rm bh,8}.$
(24)

For example, we have $T_{\rm eq}\simeq 10{\rm\,Gyr}({f^{\rm tri}\over
3})^{-0.4}({M_{\rm BH}\over 3\times 10^{7}M_{\odot}})^{2.95}$ if $\alpha=3/2$
in Equation (24). Compared with the stellar consumption rate in a system with
age $\sim 10{\rm\,Gyr}$, the enhancement of stellar consumption rate due to
the draining of the loss-region stars in a young system with dynamical age $T$
can be estimated by a factor of ${{\mathscr{F}^{\rm
drain}}(T)\over{\mathscr{F}^{\rm drain}}[\min(10{\rm\,Gyr},T_{\rm
eq})]}\simeq[{\min(10{\rm\,Gyr},T_{\rm eq})\over T}]^{\frac{2\alpha-1}{5}}$.
If assuming the age $T=0.1{\rm\,Gyr}$, and $f^{\rm tri}\simeq 3$ and
$\alpha=3/2$ in Equation (24), the factor is $\sim 6$ for $M_{\rm BH}\gtrsim
3\times 10^{7}M_{\odot}$ (for which $T_{\rm eq}\gtrsim 10{\rm\,Gyr}$) and
smaller for lower MBH masses (e.g. $\sim 2$ for $M_{\rm BH}=10^{7}M_{\odot}$).
Those estimation obtained from the above analytical scaling fittings are
consistent with those shown in Figure 5 of CYL20.

Overall, the TDE event rate in the E+A galaxies may be enhanced by a factor of
several tens as compared with generic galaxies. Among them, the effect due to
larger $\alpha$ or higher central densities of E+A galaxies dominates the rate
enhancement, while draining of the loss-region stars in these dynamically
young systems may also contribute a factor of a few, depending on the
dynamical age of the system and the MBH mass. We expect that the above
explaination to the overrepresentation of the TDEs due to the different
contributions will be testable through a significant accumulation of TDE
observations along with observations of their host galaxy properties.

###  4.2 Generalization for different types of stars

In the above analysis, we assume a single stellar population with solar mass
and solar radius. In reality, however, the stellar system is composed of a
spectrum of stars with different masses $m_{\ast}$ and radii $R_{\ast}$. For
each species in the stellar system, the mass separatrix of MBHs that can
tidally disrupt or directly swallow low-angular-momentum stars is determined
by a comparison of $r_{\rm t}$ and $2r_{\rm swl}$, as mentioned in Section
3.2. Different types of stars have different $M_{\rm BH}$ separatrixes between
TDEs and direct capture events, which depend on stellar mass and radius by a
scaling factor of
$(m_{\ast}/M_{\odot})^{-\frac{1}{2}}(R_{\ast}/R_{\odot})^{\frac{3}{2}}$. In
this subsection, we discuss how the mass spectrum may affect the stellar
consumption rate estimations and their correlation tendencies and how our
above scaling analysis can imply for the rates of different types of stars.

We discuss the effects of the mass spectrum on the stellar consumption rate
estimations and their correlation tendencies through their effects on
${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm drain}}$ as follows.

  * •

${\mathscr{F}^{\rm lc}}$: The overall stellar consumption rate due to two-body
relaxation, ${\mathscr{F}^{\rm lc}}$, is not affected significantly by the
generalization of the single-mass stellar population assumption (MT99). As
shown in Appendix A in MT99, if replacing the single-mass stellar population
with an old stellar population (e.g., the Kroupa initial mass function between
$0.08M_{\odot}$ and $1M_{\odot}$), ${\mathscr{F}^{\rm lc}}$ will increase by a
factor of $\sim 1.66$, due to the combined effects of the increased stellar
number density and the decreased stellar diffusion rate. The consumption rate
of a given type of stars is then roughly proportional to its number fraction
among all the stars, if the mass segregation effect is not significant.
However, if the mass segregation effect is important, then the consumption
rate of high-mass stars may be further enhanced and that of low-mass stars may
be weakened.

For each species in the stellar system, the consumption rate due to two-body
relaxation varies with the changes of the number fraction of the species and
the logarithmic terms $\ln Q$ and $\ln R_{0}^{-1}(\mathcal{E})$ in Equations
(9)–(10). If we ignore the generally weak variations of the two logarithmic
terms, the ${\mathscr{F}^{\rm lc}}$ rate correlation tendencies due to two-
body relaxation apply effectively to disrupted or swallowed stars of different
types, including giant stars (with $m_{\ast}{\sim}M_{\odot}$,
$R_{\ast}{\sim}10$–$1000R_{\odot}$) and white dwarfs (with
$m_{\ast}{\sim}M_{\odot}$, $R_{\ast}{\sim}0.01R_{\odot}$).

  * •

${\mathscr{F}^{\rm drain}}$: The stellar consumption rate due to loss-region
draining, ${\mathscr{F}^{\rm drain}}$, is affected by relaxing the single-type
stellar population assumption in the following three aspects. (a)
${\mathscr{F}^{\rm drain}}$ is proportional to the number density of stars
(corresponding to the $N_{\rm h}$ term in Eq. 16). If the single stellar
population with solar mass and radius considered above in Section 3.2 is
replaced by an old population of stars (e.g., the Kroupa initial mass function
between $0.08M_{\odot}$ and $1M_{\odot}$), the change of the total stellar
number density will lead to an increase of ${\mathscr{F}^{\rm drain}}$ by a
factor of $5.33$. The rate of each given species is proportional to the number
fraction of the given species. (b) ${\mathscr{F}^{\rm drain}}$ can also depend
on mass and radius of each type of stars via their different loss-cone size or
the consumption radius $r_{\rm consp}$. According to Equations (15) and (16),
we have ${\mathscr{F}^{\rm drain}}\propto{r_{\rm consp}}^{{-2\alpha+6}\over
5}$, which is $\propto(m_{\ast}/M_{\odot})^{\frac{2\alpha-6}{15}}$
$(R_{\ast}/R_{\odot})^{\frac{-2\alpha+6}{5}}$ for MBHs with mass below the
mass separatrix between TDEs and direct capture events (Eq. 18). For MBHs with
mass above the mass separatrix, the variable of $r_{\rm consp}$ in
${\mathscr{F}^{\rm drain}}$ is $\propto M_{\rm BH}$ and does not contribute to
the dependence on $m_{\ast}$ and $R_{\ast}$ (Eq. 20). (c) ${\mathscr{F}^{\rm
drain}}$ can depend on the evolutionary ages of the different types of stars
through the factor of $T^{\frac{-2\alpha+1}{5}}$ in Equation (23).

As for the loss region draining processes, the different stellar species move
in the same triaxial galactic potential and are independent of each other, and
the ${\mathscr{F}^{\rm drain}}$ rate correlation tendencies with $M_{\rm BH}$
and $\alpha$ due to loss-region draining obtained in Section 3.2 (see Eqs. 18
and 20) can be directly applied to the different stellar species, although the
detailed fit correlation cooefficients could be affected quantitatively by
their differences in the mass separatrix of MBHs between TDEs and direct
capture events.

Based on the above analysis, we discuss the implications specifically for the
following types of stars which have different characteristic masses and radii.

  * •

Giant stars: For giant stars (with $m_{\ast}{\sim}M_{\odot}$,
$R_{\ast}{\sim}10$–$1000R_{\odot}$), the upper mass boundary of MBHs being
able to produce TDE flares increases to much larger values, i.e., $\gg
10^{8}M_{\odot}$. Based on Equation (18) and including the enlarged loss-cone
size since the transition to giant star from their main-sequence stages, the
ratio of ${\mathscr{F}^{\rm drain}}$ for giant stars to that for solar-type
stars can be estimated by the factor of
$(m_{\ast}/M_{\odot})^{\frac{2\alpha-6}{15}}(R_{\ast}/R_{\odot})^{\frac{-2\alpha+6}{5}}(T/10{\rm\,Gyr})^{\frac{-2\alpha+1}{5}}f_{\rm
g}\sim 1$, if adopting $\alpha=3/2$, $m_{*}\sim M_{\odot}$, the lifetime
$T\sim 1{\rm\,Gyr}$ for $R_{*}\sim 10R_{\odot}$ giants or $T=0.01{\rm\,Gyr}$
for $R_{*}\sim 1000R_{\odot}$ giants, and if the number ratio of the giants to
solar-type stars $f_{\rm g}$ is estimated by $\sim T/10{\rm\,Gyr}$. Note that
the estimates of ${\mathscr{F}^{\rm drain}}$ for giant stars in the above
example are significantly high, in which the TDE rates for giant stars can be
up to those for solar-type stars.

For TDEs of giant stars, we expect that the correlation between the TDE rates
and the MBH mass is similar as obtained from the single solar-type stellar
population assumption, i.e., dominated by the negative ${\mathscr{F}^{\rm
lc}}$–$M_{\rm BH}$ correlation at small $M_{\rm BH}$, and by the positive
${\mathscr{F}^{\rm drain}}$–$M_{\rm BH}$ correlation at large $M_{\rm BH}$,
The transitional MBH mass between those two correlation tendencies with
$M_{\rm BH}$ should be smaller than $\sim 10^{7}M_{\odot}$ due to the relative
increase in loss-region draining rates ${\mathscr{F}^{\rm drain}}$. At the
mass range below or above the transitional MBH mass, the analysis of the
correlation tendencies of the TDE rates with $M_{\rm BH}$ and $\alpha$ are
expected to follow those shown in Section 3.2.

  * •

Massive young stars: For massive stars, the upper mass boundary of MBHs being
able to produce TDE flares can also increase to much larger values, i.e., $\gg
10^{8}M_{\odot}$. Note that the increase of ${\mathscr{F}^{\rm drain}}$ due to
the young dynamical age has been discussed in Section 4.1. For massive stars,
the dependence of ${\mathscr{F}^{\rm drain}}$ on
$(m_{\ast}/M_{\odot})^{\frac{2\alpha-6}{15}}(R_{\ast}/R_{\odot})^{\frac{-2\alpha+6}{5}}$
is relatively not strong, for example, ${\mathscr{F}^{\rm drain}}$ can
increase by a factor of $\sim 2$–$4$ (for $m_{\ast}\sim 10$–$100M_{\odot}$ and
$R_{*}\propto m_{\ast}^{0.8}$), if adopting $\alpha=3/2$, due to the enlarged
loss-cone size.

The analysis on the correlation tendencies of ${\mathscr{F}^{\rm drain}}$ and
${\mathscr{F}^{\rm lc}}$ with $M_{\rm BH}$ and $\alpha$ and the transitional
MBH mass of the correlations for giant stars above can also be applied to the
analysis for TDE samples of massive young stars (e.g., $T\ll 1{\rm\,Gyr}$).

  * •

Stellar compact remnants (white dwarfs, neutron stars, and stellar-mass BHs):
Stellar compact remnants can generally be swallowed directly by MBHs when they
move sufficiently close to the MBH, with bursts of gravitational waves, except
that white dwarfs can be tidally disrupted by MBHs if $M_{\rm BH}\lesssim
10^{5}M_{\odot}$ (at the lower boundary of or beyond the mass range considered
in this paper).

The TDE rates of white dwarfs should be dominated by the loss-region refill
rate due to two-body relaxation, and the rate estimation is subject to the
uncertainties in the statistics of the MBH population at the low-$M_{\rm BH}$
end and their stellar environments.

For $M_{\rm BH}>10^{5}M_{\odot}$, if the mass segregation effect of compact
objects at galactic centers is ignored, the direct capture rates of compact
objects and their correlation tendencies follow the similar analysis on
${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm drain}}$ shown in Section 3,
with including the number fraction of the different types of compact objects.
Note that the loss-region draining rates of the different types of compact
objects are irrelevant with their detailed masses and radii, as their $r_{\rm
consp}=2r_{\rm swl}$ are the same given the same $M_{\rm BH}$ (see also Eq.
20).

##  5 Conclusions

In this work, we study the correlations of the stellar consumption rates by
the central MBHs of galaxies with both the MBH mass $M_{\rm BH}$ and the inner
slope of the host galaxy stellar number/mass density distribution $\alpha$.
The rates of stellar consumption due to two-body relaxation ${\mathscr{F}^{\rm
lc}}$ and stellar orbital precession in nonspherical potentials
${\mathscr{F}^{\rm drain}}$ are considered. By exploiting a simplified power-
law model, i.e., considering a single power-law stellar number/mass density
distribution under the Keplerian potential of the central MBH, we derive
approximated expressions for the stellar consumption rates due to both the
mechanisms. Then by inspecting the relative contributions from different terms
in the approximated rate expressions to the correlations, we identify the
dominant factor(s) responsible for both the slopes and scatters of the
correlations. We summarize the main conclusions of this study below.

  * •

In both cases of the two-body relaxation in spherical galaxy potentials and
the loss-region draining in nonspherical potentials, the stellar consumption
rates estimated based on the power-law model ${\mathscr{F}^{\rm lc,PL}}$ and
${\mathscr{F}^{\rm drain,PL}}$ are consistent with the rates estimated based
on the full model, i.e., ${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm
drain}}$, respectively. This not only verifies the effectiveness of using the
simplified power-law model to inspect the correlations, but also provides an
efficient and simple way to estimate the stellar consumption/flaring rates due
to both the mechanisms.

  * •

${\mathscr{F}^{\rm lc}}$ correlates negatively with $M_{\rm BH}$ while
positively with $\alpha$. As for the ${\mathscr{F}^{\rm lc}}$–$M_{\rm BH}$
correlation, the best-fit linear relation has a slope of $-0.28$. Both the
slope and the scatter of the correlation are dominated by the term
$\sigma_{\rm h}^{3}M_{\rm BH}^{-1}$ in the approximated expression of the
stellar consumption rate ${\mathscr{F}^{\rm lc,PL}}$, where $\sigma_{\rm
h}^{2}\equiv GM_{\rm BH}/r_{\rm h}$ and the influential radius of the central
MBH $r_{\rm h}$ is defined as the radius within which the stellar mass equals
the MBH mass. As for the ${\mathscr{F}^{\rm lc}}$–$\alpha$ correlation, the
best-fit linear relation has a slope of $1.37$. Again, both the slope and the
scatter of the correlation are dominated by $\sigma_{\rm h}^{3}M_{\rm
BH}^{-1}$.

  * •

${\mathscr{F}^{\rm drain}}$ correlates positively with $M_{\rm BH}$ while
negatively with $\alpha$. The latter correlation appears to be mild due to the
large scatter in the relation between ${\mathscr{F}^{\rm drain}}$ and
$\alpha$. As for the ${\mathscr{F}^{\rm drain}}$–$M_{\rm BH}$ correlation, the
best-fit linear relation has a slope of $0.90$. Both the slope and scatter of
the correlation are dominated by the term ${\kappa(M_{\rm BH},\sigma_{\rm
h})}$ (Eq. 22) in the approximated expression of ${\mathscr{F}^{\rm
drain,PL}}$, which scales as $\sim\sigma_{\rm h}^{3}$ and $\sim\sigma_{\rm
h}^{3}M_{\rm BH}^{2/5}$ when the $M_{\rm BH}$ is below and above the
separatrix, respectively, if set $\alpha=3/2$. As for the ${\mathscr{F}^{\rm
drain}}$–$\alpha$ correlation, the best-fit linear relation has a slope of
$-0.64$. The term $\chi(\alpha)$ (Eq. 21) in ${\mathscr{F}^{\rm drain,PL}}$
dominates the slope while ${\kappa(M_{\rm BH},\sigma_{\rm h})}$ dominates the
scatter of the correlation.

  * •

The above correlations of ${\mathscr{F}^{\rm lc}}$ and ${\mathscr{F}^{\rm
drain}}$ serve as the backbones of the correlation tendencies of the stellar
consumption rates at the low-mass ($M_{\rm BH}\lesssim 10^{7}{M_{\odot}}$) and
the high-mass ($M_{\rm BH}\gtrsim 10^{7}{M_{\odot}}$) ranges of MBHs,
respectively.

  * •

We use the ${\mathscr{F}^{\rm lc}}$–$\alpha$ correlation to explain the
overrepresentation of TDEs in those rare E+A/poststarburst galaxies found by
some recent observational studies (e.g., Arcavi et al. 2014; French et al.
2016, 2017; Law-Smith et al. 2017; Graur et al. 2018; Hammerstein et al.
2021). According to the correlation, the expectation value of
${\mathscr{F}^{\rm lc}}$ is increased by a factor of ${\sim}24$ when $\alpha$
is increased from $1$ to $2$, or equivalently, when $\gamma$ is increased from
$0$ (core-like) to $1$ (cuspy). This factor is broadly consistent with the
overrepresentation factor found by Law-Smith et al. (2017), i.e.,
${\sim}25$–$48$, indicating that the preference of TDEs in these rare subclass
of galaxies can be largely explained by the ${\mathscr{F}^{\rm lc}}$–$\alpha$
correlation. Besides the ${\mathscr{F}^{\rm lc}}$–$\alpha$ correlation, loss-
region draining in these dynamically young systems can also enhance the
observed TDE rate by a factor of a few, depending on the dynamical age and the
MBH mass. Future observations of TDEs and their host galaxy properties are
expected to test those different contribution origins.

  * •

The stellar consumption rates and their correlation tendencies are discussed
for different types of stars, including giant stars, massive young stars, and
stellar compact remnants. We find that the estimates of the TDE rates of giant
stars can be high enough to be up to or those of solar-type stars, due to the
large tidal disruption radii of giant stars. How to distinguish the different
TDEs of giant stars and main-sequence stars in observations deserves further
investigations.

With the increasing power of the time domain surveys, the TDE samples are
expanding rapidly in recent years (Gezari, 2021), which enables the study of
the MBH demographics from a brand new perspective (e.g., Ramsden et al. 2022).
For example, TDEs illuminate those dormant MBHs, which provides a powerful
tool to study the mass function and occupation fraction of MBHs in quiescent
galaxies, especially at the low-mass end where most, if not all, TDEs occur
(Stone & Metzger, 2016; Fialkov & Loeb, 2017). In addition, the upper mass
limit of a MBH that can tidally disrupt a star depends sensitively on the MBH
spin (Beloborodov et al., 1992; Kesden, 2012; Leloudas et al., 2016; Mummery &
Balbus, 2020). Therefore, the observed MBH mass distribution near
${\sim}10^{8}$–$10^{9}{M_{\odot}}$ from a large sample of TDEs can set strong
constraints on the spin distributions of MBHs inside the galaxy centers.
Moreover, a stellar system containing a binary MBH or a recoiled MBH could
undergo a short period of time during which TDEs promptly happen (Chen et al.,
2011; Stone & Loeb, 2011; Wegg & Nate Bode, 2011; Stone & Loeb, 2012). Despite
these merits, the results from TDE observations should be interpreted with
cautions when constraining the MBH demographics, since our study reveals that
some bias may be induced by the different galaxy properties. Therefore, we
conclude that a thorough understanding of the dependence of the TDE rate on
both properties of MBHs and their host galaxies is the prerequisite of the MBH
demographic study with TDE observations.

## Acknowledgments

This work is partly supported by the National Natural Science Foundation of
China (grant Nos. 12173001, 11721303, 11873056, 11690024, 11991052), the
National SKA Program of China (grant No. 2020SKA0120101), National Key Program
for Science and Technology Research and Development (grant Nos.
2020YFC2201400, 2016YFA0400703/4), and the Strategic Priority Program of the
Chinese Academy of Sciences (grant No. XDB 23040100).

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