# Revisiting the Rates and Demographics of Tidal Disruption Events: Effects of
the Disk Formation Efficiency

Thomas Hong Tsun Wong Department of Physics, The University of Hong Kong
Pokfulam Road, Hong Kong, China  Hugo Pfister Department of Physics, The
University of Hong Kong
Pokfulam Road, Hong Kong, China  DARK, Niels Bohr Institute, University of
Copenhagen
Jagtvej 128, 2200 København, Denmark  Lixin Dai Department of Physics, The
University of Hong Kong
Pokfulam Road, Hong Kong, China

###### Abstract

Tidal disruption events (TDEs) are valuable probes of the demographics of
supermassive black holes as well as the dynamics and population of stars in
the centers of galaxies. In this Letter, we focus on studying how the debris
disk formation and circularization processes can impact the possibility of
observing prompt flares in TDEs. First, we investigate how the efficiency of
disk formation is determined by the key parameters, namely, the black hole
mass $M_{\rm BH}$, the stellar mass $m_{\star}$, and the orbital penetration
parameter $\beta$ that quantifies how close the disrupted star would orbit
around the black hole. Then we calculate the intrinsic differential TDE rate
as a function of these three parameters. Combining these two results, we find
that the rates of TDEs with prompt disk formation are significantly suppressed
around lighter black holes, which provides a plausible explanation for why the
observed TDE host black hole mass distribution peaks between $10^{6}$ and
$10^{7}\,M_{\odot}$. Therefore, the consideration of the disk formation
efficiency is crucial for recovering the intrinsic black hole demographics
from TDEs. Furthermore, we find that the efficiency of the disk formation
process also impacts the distributions of both stellar orbital penetration
parameter and stellar mass observed in TDEs.

accretion, accretion disks – black hole physics – galaxies: nuclei –
relativistic processes – stars: kinematics and dynamics

††journal: ApJL

††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Stellar tidal disruption events (TDEs) occur whenever stars are
gravitationally scattered into the vicinity of supermassive black holes
(SMBHs) where tidal force dominates over the self-gravity of the stars (Rees,
1988). This happens generally when the star enters the tidal disruption radius
of the black hole (BH):

$r_{T}\simeq\left(\eta^{2}\frac{M_{\rm BH}}{m_{\star}}\right)^{1/3}r_{\star}$
(1)

where $M_{\rm BH}$ is the BH mass, $m_{\star}$ and $r_{\star}$ are the mass
and radius of the star. $\eta$ is a parameter which depends on the details of
stellar structure (Law-Smith et al., 2020; Ryu et al., 2020b), and throughout
this work we take $\eta$ to be unity for simplicity. After the disruption,
approximately half of the stellar debris remains bound and forms an accretion
disk, while the remaining stellar mass leaves the SMBH. When a sufficiently
large fraction of the star is disrupted and accreted onto the BH, luminous
flares can be produced through which we can peek into the event and its host
SMBH (Guillochon & Ramirez-Ruiz, 2013; Law-Smith et al., 2019).

Figure 1: The host $M_{\rm BH}$ distribution of the observed TDE candidates
compared to the intrinsic distribution from the LC theory. The solid lines
represent our chosen observed TDE candidates, while the colors indicate the
methods used to constrain $M_{\rm BH}$ (orange: $M_{\rm BH}-\sigma$; blue:
$M_{\rm BH}-M_{\rm gal}$). Similar to the bootstrapping method, error bars of
individual TDEs are modeled as a Gaussian distribution with the scattering
from the $M_{\rm BH}$-scaling relations and the measurement errors from
$\sigma$ and $M_{\text{gal}}$. Both observed TDE $M_{\rm BH}$ distributions
center around $10^{6}-10^{7}\,M_{\odot}$, while the intrinsic TDE $M_{\rm BH}$
distribution derived from the LC dynamics (black dashed curve) peaks towards
lighter BHs. The small peak of the orange curve at $10^{8-9}\,M_{\odot}$ is
due to a single event ASASSN-15lh.

The study of TDEs has inspired great interest, as these events can allow us to
probe the vast majority of SMBHs which are dormant. Moreover, it is shown that
the observables of these events can be used to measure the BH mass, constrain
the BH spin, as well as probe the stellar mass and structure (e.g., Kesden,
2012; Mockler et al., 2019; Ryu et al., 2020a). Recently, this field has been
growing fast not only because several dozens of TDE candidates have been
detected in optical, UV, and X-ray wavebands (see reviews by Saxton et al.,
2020; van Velzen et al., 2020; Gezari, 2021), but also because thousands of
TDEs are expected to be observed in the next decade using telescopes such as
the Vera Rubin Observatory, _eROSITA_ , and _Einstein Probe_. It will
therefore be possible to use a large collection of TDEs to construct the
demographics of massive BHs.

The rates of TDEs happening in galaxies with specific stellar density profiles
can be calculated using the loss cone (LC) theory, which captures how two-body
interactions can bring stars into low angular momentum orbits with pericenter
within $r_{T}$ (e.g., Merritt, 2013). For example, Wang & Merritt (2004) apply
the LC theory upon two common galaxy density profiles and find that the TDE
rate should be around $10^{-3}-10^{-4}\,\rm{gal}^{-1}\,\rm{yr}^{-1}$, which
gradually decreases with increasing $M_{\rm BH}$. Stone & Metzger (2016)
parameterize the observed galaxy samples with further consideration of stellar
mass functions and obtain a similar TDE rate dependency on $M_{\rm BH}$. This
trend of decreasing in TDE rate with increasing $M_{\rm BH}$ is seen again
while taking into account nuclear star clusters in the center of dwarf
galaxies (Pfister et al., 2020, 2021). The TDE rates also have a close
correlation with the merger history (Pfister et al., 2019) and the structures
of the host galaxies (Law-Smith et al., 2019; French et al., 2020).

In summary, while the details depend somewhat on the stellar density profile
and galaxy structure, the TDE rate of a galaxy should negatively correlate
with $M_{\rm BH}$. Given there are more smaller galaxies hosting lighter BHs
than the opposite (Reines & Volonteri, 2015), we expect there should be more
TDEs produced around lighter BHs. However, looking at the observed population
of TDEs, one finds very few TDE candidates from BHs with $M_{\rm BH}\lesssim
10^{5}\,M_{\odot}$ (Wevers et al., 2017, 2019). Furthermore, Stone & Metzger
(2016) and Hung et al. (2017) show that the TDE host $M_{\rm BH}$ distribution
peaks between $10^{6}-10^{7}\,M_{\odot}$. This promotes the proposal that TDEs
around more massive BHs have higher chances to be observed. Here we re-examine
the observed TDE $M_{\rm BH}$ distribution given that the sample size has
grown significantly in the past few years. We refer to several recent reviews
and pick out a total of 65 likely TDE candidates (the selection criteria are
detailed in Appendix A). We also calculate their $M_{\rm BH}$ consistently
using two different methods: $M_{\rm BH}-\sigma$ (Merritt & Ferrarese, 2001)
and $M_{\rm BH}-M_{\rm gal}$ (Reines & Volonteri, 2015). The full list of the
selected TDE candidates and their $M_{\rm BH}$ are shown in Table A.1. We show
the normalized distribution of $M_{\rm BH}$ for these observed TDEs in Fig.1.
The distributions for both TDE lists still center around
$10^{6}-10^{7}\,M_{\odot}$ and drop off towards both ends of the mass
spectrum, regardless of the methods used to derive $M_{\rm BH}$. In the same
figure, we also show the theoretically computed TDE $M_{\rm BH}$ distribution
from the LC dynamics (see Appendix C for detailed calculations) for
comparison, where one sees an increasing TDE rate with decreasing $M_{\rm
BH}$.

The large discrepancy between the intrinsic and observed TDE $M_{\rm BH}$
distribution can result from many factors such as survey constraints and dusts
(van Velzen, 2018; Roth et al., 2021). Also the disk formation, accretion, and
emission processes in TDEs (Bonnerot & Stone, 2021; Dai et al., 2021; Roth et
al., 2020, respectively) can play a major role in altering the observed TDE
$M_{\rm BH}$ distribution. One possible explanation is that the TDE luminosity
is capped at the Eddington luminosity limit and therefore scales with $M_{\rm
BH}$ (Kochanek, 2016). However, the peak luminosity of the observed TDE
candidates do not show any clear linear trend with the host $M_{\rm BH}$
(Fig.A.2). Although we cannot rule out the Eddington-limit model due to
possible observational biases and the limited dynamic range of the data,
instead, we turn our focus to the disk formation process, since recent
hydrodynamics simulations show that not all TDEs can form circular disks
efficiently through stream self-crossings (see review by Bonnerot & Stone,
2021). In particular, Shiokawa et al. (2015) show that in the case of
inefficient disk circularization, the accretion onto SMBHs will happen on
timescales much longer than the fallback timescale, which can lower the
possibility of the detection of such events. Based on these simulations, Dai
et al. (2015) and Guillochon & Ramirez-Ruiz (2015) use first-order
calculations to quantify how the disk formation promptness depends on the BH
mass and stellar orbital parameters. It is found that in TDEs around more
massive BHs the stream self-crossings are closer to the BH due to stronger
general relativistic (GR) effects. The closer self-crossings give stronger
collisions, leading to faster disk formation and accretion and more prompt
flares. Therefore, the disk formation process is a natural candidate for a
mechanism which can promote TDEs around more massive SMBHs to be observed.

In this paper, we aim at mitigating the TDE $M_{\rm BH}$ distribution tension
from the perspective of disk formation efficiency. We hypothesize that
inefficient disk circularization processes significantly reduce the chance of
TDEs being detected around lighter SMBHs. Using the first-order framework of
Dai et al. (2015), we further quantify the disk formation efficiency as a
function of key parameters (Section 2). Next we obtain the differential TDE
rates from the LC theory, and check how including the correction by the disk
formation efficiency can shift the TDE $M_{\rm BH}$ distribution (Section 3).
Then we reexamine the distributions of some key parameters in TDEs (Section
4). Lastly, a summary and further discussions are given (Section 5).

##  2 Efficiency of disk formation from debris stream self-crossing

Figure 2: Comparison between the intrinsic and $\mathcal{C}$-corrected
differential TDE rates. For $M_{\rm BH}/M_{\odot}=\\{10^{4},10^{6},10^{8}\\}$
(top, middle, bottom rows), the intrinsic and corrected differential TDE rates
are shown in the left and right panels, respectively. The white regions
represent the parameter space where stars directly plunge into the BH event
horizon. Contours of disk formation efficiencies $\mathcal{C}$ are overplotted
in the left panels. $\mathcal{C}$ has a monotonically increasing trend towards
the upper-left corner, (higher $\beta$ and lower $m_{\star}$). One can see
that the $\mathcal{C}$-correction leads to a significant drop in TDE rates at
low $\beta$ for lighter BHs. For $M_{\rm BH}=10^{8}\,M_{\odot}$,
$\mathcal{C}\approx 1$ across the TDE-allowed parameter space.

Recent studies have consistently shown that the formation of TDE disks is
mainly caused by the self-crossing of the debris stream due to GR apsidal
precession near SMBHs. Some fraction of the stellar debris becomes unbound due
to shocks, while the remaining rapidly dissipates its orbital energy to form a
somewhat circular accretion disk, generally within a few orbital timescales
(Lu & Bonnerot, 2020). For stars disrupted along inclined orbits around
spinning BHs, Lense-Thirring precession can complicate the process (Kochanek,
1994; Dai et al., 2013; Guillochon & Ramirez-Ruiz, 2015), although the latest
hydrodynamical simulations show that disks still form after a short delay
(Liptai et al., 2019). Albeit these advancements, full-scale simulations of
the complex disk formation processes are still not achieved due to
computational limitations.

In this study, we follow the first-order calculations in Dai et al. (2015) for
debris dynamics and kinematics around a Schwarzschild BH. The basics are
detailed in Appendix B. Starting from there, the specific energy loss at the
self-crossing point $R_{I}$ is approximated as

$\Delta
E_{\text{first}}=\left|\frac{1}{2}v_{f}^{2}-\frac{1}{2}v_{i}^{2}\right|,$ (2)

where $v_{f}$ and $v_{i}$ are the debris speeds before and after collision
respectively. The bound debris has to lose a significant amount of orbital
energy, likely through the shocks and dissipation happening in repeated self-
crossings, in order to form a disk. In theory, if the debris could completely
circularize, the classical circularization radius can be calculated using the
conservation of angular momentum:

$r_{\text{circ}}=\frac{2r_{T}}{\beta}$ (3)

where $\beta=r_{T}/r_{p}$ is the penetration parameter quantifying how deeply
the star plunges into the BH gravitational potential well. Therefore, the
total specific orbital energy needed to be removed from the initially
elliptical orbit for complete circularization is:

$\displaystyle\Delta E_{\text{total}}\,$ $\displaystyle=$
$\displaystyle\,E_{\text{ellip}}-E_{\text{circ}}$ (4) $\displaystyle=$
$\displaystyle\,-\frac{GM_{\rm BH}}{2a_{\text{mb}}}+\frac{GM_{\rm
BH}}{2r_{\text{circ}}}$ $\displaystyle=$ $\displaystyle\,\frac{GM_{\rm
BH}}{2}\left(\frac{\beta}{2r_{T}}-\frac{1}{a_{\text{mb}}}\right)$

where $a_{\rm mb}$ is the semi-major axis of the most bound debris orbit.
(Strictly speaking, the debris falling back at the peak of the flare should be
slightly less bound than the most bound debris. Therefore, the $\Delta
E_{\text{total}}$ obtained here is a lower limit.)

Given that the subsequent collisions and interactions after the first debris
self-crossing are hard to trace analytically, we define a first-order,
dimensionless disk formation (circularization) efficiency parameter by scaling
Eq.2 with Eq.4:

$\mathcal{C}\left(M_{\rm BH},\beta,m_{\star}\right)\equiv\frac{\Delta
E_{\text{first}}}{\Delta E_{\text{total}}}$ (5)

When $\mathcal{C}\ll 1$, the debris stream self-crossing happens far away from
the BH and the disk formation is expected to be slow, which lowers the
possibility of detecting prompt flares from such events. On the other hand,
when $\mathcal{C}\sim 1$, the debris can already dissipate a large fraction of
its orbital energy through the shocks at the first self-crossing, and we
expect such TDEs will likely produce prompt, observable flares.

The tidal radius $r_{T}$ can be expressed as a function of only $M_{\rm BH}$
and $m_{\star}$ for main-sequence (MS) stars, since their masses and radii are
linked by Kippenhahn & Weigert (1990):

$\frac{r_{\star}}{\rm{R}_{\odot}}=\begin{cases}\left(\displaystyle\frac{m_{\star}}{M_{\odot}}\right)^{0.8}&\text{for}\,\,m_{\star}\leq
M_{\odot}\\\
\left(\displaystyle\frac{m_{\star}}{M_{\odot}}\right)^{0.57}&\text{for}\,\,m_{\star}\geq
M_{\odot}\end{cases}$ (6)

Therefore, $\mathcal{C}$ can be expressed as a function of three parameters:
$M_{\rm BH}$, $m_{\star}$ and $\beta$. We plot the values of $\mathcal{C}$ as
contour lines in the left column of Fig. 2 for three representative $M_{\rm
BH}$. One can see that for intermediate-mass black holes (IMBHs) with $M_{\rm
BH}\sim 10^{4}\,M_{\odot}$, extremely large $\beta>$ few $\times$ 10 will be
needed for having efficient disk formation. For $M_{\rm BH}\sim
10^{6}\,M_{\odot}$, low-mass, denser stars with $m_{\star}\sim 0.1\,M_{\odot}$
can have moderately prompt disk formation for all $\beta$, while for more
massive stars the disk formation is slow unless $\beta>$ few. For very massive
black holes with $M_{\rm BH}\sim 10^{8}\,M_{\odot}$, the disk formation is
always efficient, but only stars with $m_{\star}\gtrsim 1\,M_{\odot}$ can
still be disrupted outside the BH event horizon.

In the subsequent sections, we apply the $\mathcal{C}$-correction to the
observed TDE rates and demographics in two different ways: one way is to
directly use $\mathcal{C}$ as the probability of observing an event $P_{\rm
obs}$, the other way is that we pick a particular threshold value
$\mathcal{C}_{\text{thres}}$ to screen out TDEs with
$\mathcal{C}<\mathcal{C}_{\text{thres}}$. For the latter, reasonable choices
of $\mathcal{C}_{\text{thres}}$ can be obtained by comparing the TDE light
curve decay timescale ($\sim 1$ year for typical parameters) and the orbital
timescale of the most bound orbit ($T_{\rm mb}\sim 1$ month for typical
parameters). The timescale to circularize the elliptical orbit is
approximately $T_{\rm circ}=nT_{\rm mb}$, where $n$ is the number of stream
self-crossings (Ulmer, 1998). For the dynamical fallback pattern to be
reserved, the disk formation should in principle happen faster than the
fallback timescale, which gives $n\lesssim 10$ and $\mathcal{C}_{\rm
thres}\sim 1/n\gtrsim 0.1$.

We also note that there are ongoing discussions on whether the optical
emissions from TDEs are powered by accretion or the shocks induced in the
debris self-crossing (e.g., Piran et al., 2015; Bonnerot & Lu, 2020).
Nonetheless, the chance of observing a particular TDE flare, even if only
powered by debris self-crossing, should still have a tight correlation with
$\Delta E_{\rm first}$. Therefore, the correction between the observed TDE
rates and $\mathcal{C}$ should hold regardless of the origin of TDE optical
emissions, which awaits to be disclosed by further studies.

##  3 Revised TDE Rates

###  3.1 Intrinsic TDE rates from loss cone dynamics

We apply the LC theory and calculate the TDE rates. In this work, we only
consider bright TDE flares from full disruption scenarios, since it is hard to
distinguish whether a dim flare is produced due to slow disk formation or
partial disruption. We refer the readers to Pfister et al. (2021) for the
calculations used in this work, which we also include in Appendix C for
completeness.

Assuming the Kroupa stellar mass function (Kroupa, 2001), an isothermal
stellar density profile ($\rho\propto r^{-2}$), and the BH mass following the
$M_{\rm BH}-\sigma$ relation (Merritt & Ferrarese, 2001), the intrinsic
differential TDE rate can be analytically computed as a function of $M_{\rm
BH}$, $m_{\star}$, and $\beta$ (Pfister et al., 2021):

$\displaystyle\frac{\rm{d}^{2}\Gamma}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\,\,=$ $\displaystyle\,\,\frac{8\pi^{2}GM_{\rm
BH}r_{T}}{\beta}\phi(m_{\star})m_{\star}\times$ (7)
$\displaystyle\int^{GM_{\rm BH}/r_{T}}_{0}\mathcal{G}(E,M_{\rm
BH},\beta,m_{\star})\rm{d}E$

where $\phi(m_{\star})$ is the Kroupa stellar mass function and
$\mathcal{G}(E,M_{\rm BH},\beta,m_{\star})$ is a complex function that gathers
the essence of the LC dynamics.

Under this simplified framework, we vary $M_{\rm BH}$ in the range
$\left[10^{4}\,M_{\odot},10^{9}\,M_{\odot}\right]$ while bearing in mind the
existence of IMBHs is still under debate. We include the MS stars in range of
$\left[0.08\,M_{\odot},10\,M_{\odot}\right]$. The penetration parameter
$\beta$ varies between $1$ and $\beta_{\rm max}$, where
$\beta_{\text{max}}=r_{T}/2r_{g}$ corresponds to the orbit with $r_{p}$ at the
BH Schwarzschild radius. For a fixed $M_{\rm BH}$, $\beta_{\text{max}}$ is set
by the most massive star, so we have $\beta_{\rm
max}^{10\,M_{\odot}}=\beta_{\rm max}(M_{\rm BH},10\,M_{\odot})$, which has a
value of $\sim 10^{3}$ when $M_{\rm BH}=10^{4}\,M_{\odot}$, $\sim 40$ when
$M_{\rm BH}=10^{6}\,M_{\odot}$, and $\sim 1$ when $M_{\rm
BH}=10^{8}\,M_{\odot}$. We show the intrinsic differential TDE rate in Fig.2
(left panels). This plot depicts that most TDEs have $\beta\sim 1$, and most
disrupted stars have low masses unless for $M_{\rm BH}\gtrsim
10^{8}\,M_{\odot}$.

###  3.2 Corrected TDE rates with disk formation efficiency

Following the discussion in Sec.2 we multiply the intrinsic differential TDE
rates (Eq.7) with the disk formation efficiency parameter $\mathcal{C}$ to
obtain the predicted observed differential TDE rates:

$\frac{\rm{d}^{2}\Gamma_{\mathcal{C}}}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\,\equiv\,\,\mathcal{C}\times\frac{\rm{d}^{2}\Gamma}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}$ (8)

where $\Gamma$ and $\Gamma_{\mathcal{C}}$ stand for the intrinsic and
$\mathcal{C}$-corrected rates respectively. $\Gamma_{\mathcal{C}}$, as a
function of $\beta$ and $m_{\star}$, is shown in Fig.2 (right panels).
Comparing the two rates, the most notable difference is that the
$\beta$-distribution becomes much more even for the same $m_{\star}$ after the
$\mathcal{C}$ correction. This is because the substantial reduction of the TDE
rates mostly happens at low $\beta$ due to small $\mathcal{C}$ where the
intrinsic rate is large. Also, $\Gamma_{\mathcal{C}}$ has a more sensitive
dependence on $m_{\star}$, which results from that larger stars which are rare
initially also generally have smaller $\mathcal{C}$ values. However, for
$M_{\rm BH}\approx 10^{8}\,M_{\odot}$, $\mathcal{C}\approx 1$, so the rates
and distributions are barely altered.

###  3.3 TDE host black hole mass distribution

Figure 3: The intrinsic and $\mathcal{C}$-corrected volumetric TDE rates and
$M_{\rm BH}$ distributions. Panel (a) shows the TDE volumetric rates as a
function of $M_{\rm BH}$, with different ways of implementing the disk
formation efficiency correction. Panel (b) shows how the value of
$\mathcal{C}_{\rm thres}$ affects the TDE $M_{\rm BH}$ distribution. The
intrinsic and observed distributions (orange dotted: $M_{\rm BH}-\sigma$) same
as in Fig.1 are included for comparison. The distribution with
$\mathcal{C}_{\text{thres}}=0.1$ (red solid) best replicates the median of the
observed TDE samples. Overall, the rate suppression around light SMBHs due to
slow disk formation is evident in both plots.

The volumetric TDE rates as a function of the host BH mass can be obtained
with the results above and the information of the black hole mass function
(BHMF). The BHMF $\Phi_{\rm BH}$, defined as the number of SMBHs per co-moving
volume between masses $M_{\rm BH}$ and $M_{\rm BH}+dM_{\rm BH}$, provides
knowledge on the growth and evolution of SMBHs and galaxies. Currently, the
observationally constrained BHMF still has a large uncertainty (e.g., Shankar
et al., 2016; Gallo & Sesana, 2019). In particular, the uncertainty in BH
occupation fraction increases significantly towards lighter BHs of $M_{\rm
BH}\lesssim 10^{5}\,M_{\odot}$, which also results in uncertainty of TDE rates
around such BHs. We have shown in Appendix D that the uncertainty in the
moderately constrained BHMF at the low-mass end is insufficient to explain the
TDE BH mass distribution as observed.

In this work, we adopt the BHMF $\Phi_{\rm BH}$ derived by Gallo & Sesana
(2019)111The equation is the corrected version of Gallo & Sesana (2019),
reported in Pfister et al. (2021).:

$\begin{split}\log_{10}\left(\frac{\Phi_{\rm
BH}}{\text{Mpc}^{-3}\,M_{\odot}^{-1}}\right)=&-9.82-1.10\times\log_{10}\left(\frac{M_{\rm
BH}}{10^{7}\,M_{\odot}}\right)\\\ &-\left(\frac{M_{\rm BH}}{128\times
10^{7}\,M_{\odot}}\right)^{1/\ln(10)}\end{split}$ (9)

which constrains the local SMBH occupation fraction and incorporates the
latest galaxy stellar mass function with _Chandra_ X-ray data. Most
importantly, this BHMF takes into account the non-unity SMBH occupation
fraction in galaxies towards the near-IMBH mass spectrum. Furthermore, since
all TDE candidates discovered so far are at low redshifts $z\sim 0.01-0.4$,
this particular BHMF derived from the local SMBHs is applicable for this
redshift range.

The predicted volumetric intrinsic and $\mathcal{C}$-corrected TDE rates are
respectively:

$\displaystyle\dot{n}(M_{\rm BH})$ $\displaystyle=\Phi_{{\rm
BH}}\times\int\left(\frac{\rm{d}^{2}\Gamma}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln\beta\,\rm{d}\ln m_{\star}$ (10a)
$\displaystyle\dot{n}_{\mathcal{C}}(M_{\rm BH})$ $\displaystyle=\Phi_{{\rm
BH}}\times\int\left(\frac{\rm{d}^{2}\Gamma_{\mathcal{C}}}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln\beta\,\rm{d}\ln m_{\star}$ (10b)

which are plotted in Fig.3(a). The integrated volumetric rate of all $M_{\rm
BH}$ is $\lesssim 10^{-5}\,\rm{Mpc}^{-3}\,\rm{yr}^{-1}$ for $\mathcal{C}_{\rm
thres}\gtrsim 0.1$, which lies a little beyond the TDE rate inferred from
current observations $\dot{n}\approx 4\times 10^{-8.4}-5.4\times
10^{-6}\,\rm{Mpc}^{-3}\,\rm{yr}^{-1}$ (Stone et al. 2020 and references
therein). This could result from the isothermal stellar density profile used
in our calculation, the BHMF uncertainty at low $M_{\rm BH}$, and
observational limitations. Therefore, we check how the normalized TDE $M_{\rm
BH}$ distribution changes after considering the disk formation efficiency in
Fig.3(b), which depends less on various assumptions. Similar to the rates,
after applying the $\mathcal{C}$ corrections, the $M_{\rm BH}$ distribution
shifts towards the more massive side of the spectrum and peaks around
$10^{6-7}\,M_{\odot}$.

Suppose that the disk formation efficiency is the sole mechanism affecting the
observed $M_{\rm BH}$ distribution, we find $\mathcal{C}_{\rm thres}\approx
0.1$ would give the best-fit distribution according to the centroid values at
$M_{\rm BH}\approx 10^{6.5}\,M_{\odot}$. This suggests that current transient
surveys favor the detection of TDEs with moderately prompt disk formation.

##  4 Distributions of TDE: Penetration Parameter and Stellar Mass

###  4.1 Distribution of the penetration parameter $\beta$

Figure 4: Numerical and empirically-fitted $\beta$-distributions derived from
intrinsic and $\mathcal{C}$-corrected TDE rates. Different colors indicate
different $M_{\rm
BH}=10^{4}\,M_{\odot}\,(\text{blue}),\,10^{6}\,M_{\odot}\,(\text{green}),\,10^{8}\,M_{\odot}\,(\text{red})$.
In panels (a) and (b), empirical fitting formulae of Eq.12a and 12b are
plotted in dashed lines with their corresponding numerical results in solid
lines. In panel (c), two different
$\mathcal{C}_{\text{thres}}=0.1\,(\text{dashed})$ and $0.5\,(\text{dotted})$
are included.

The intrinsic $\mathcal{P}(\beta)$ and $\mathcal{C}$-corrected
$\mathcal{P}_{\mathcal{C}}(\beta)$ $\beta$ probability density functions can
be numerically computed respectively as:

$\displaystyle\frac{\rm{d}\mathcal{P}}{\rm{d}\beta}$
$\displaystyle=\frac{1}{\beta\Gamma}\int\left(\frac{\rm{d}^{2}\Gamma}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln m_{\star}\,$ (11a)
$\displaystyle\frac{\rm{d}\mathcal{P}_{\mathcal{C}}}{\rm{d}\beta}$
$\displaystyle=\frac{1}{\beta\Gamma_{\mathcal{C}}}\int\left(\frac{\rm{d}^{2}\Gamma_{\mathcal{C}}}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln m_{\star}\,$ (11b)

We show the intrinsic $\beta$-distributions (Fig.4(a)) for three
representative $M_{\rm BH}$. For $1\ll\beta\ll\beta_{\rm
max}^{10\,M_{\odot}}$, we find $d\mathcal{P}/d\beta\propto\beta^{-2}$, which
is consistent with the rate dominated by the full LC regime (see Stone &
Metzger 2016). After $\beta$ reaches around $\beta_{\rm
max}^{0.08\,M_{\odot}}$ (maximal possible $\beta$ for TDEs of
$m_{\star}=0.08\,M_{\odot}$), low-mass stars start to plunge into the horizon,
resulting in a drop in the slope. The $\beta$-distributions is cut off at
$\beta_{\rm max}^{10\,M_{\odot}}$ when all stars undergo direct plunging. On
the other side, as $\beta\rightarrow 1$, $d\mathcal{P}/d\beta$ is expected to
peak owing to the additional contribution from TDEs in the diffusive regime.

The $\beta$-distributions with $\mathcal{C}$ correction implemented are shown
in Fig.4(b) and Fig.4(c). As expected, low-$\beta$ TDEs are removed, and
larger $\mathcal{C}_{\text{thres}}$ values generally yield higher low-$\beta$
cutoffs. Adopting the nominal value of $\mathcal{C}_{\rm thres}=0.1$, we can
see that for $M_{\rm BH}\lesssim 10^{6}\,M_{\odot}$, low-$\beta$ TDEs are
significantly reduced. Furthermore, for IMBHs with $M_{\rm BH}\lesssim
10^{4}\,M_{\odot}\,(10^{5}\,M_{\odot})$, only TDEs with $\beta\gtrsim 20\,(5)$
can have prompt disk formation.

The numerically obtained $\beta$-distributions in Fig.4(a,b) could be
accurately fitted in log-log space with the following empirical formulae:

$\displaystyle\frac{\rm{d}\mathcal{P}}{\rm{d}\beta}\bigg{|}_{\text{fit}}\propto\,\,$
$\displaystyle\left[\left(\frac{\beta}{\beta-1}\right)^{0.1}+\left(\frac{\beta}{\beta-1}\right)^{-0.1}-2\right]\times$
$\displaystyle\exp\left[-2\left(\frac{\beta}{\beta_{\text{max}}^{10\,M_{\odot}}}\right)^{2.7}\right]\,\,\text{for}\,\,1\lesssim\beta<\beta_{\text{max}}^{10\,M_{\odot}},$
(12a)
$\displaystyle\frac{\rm{d}\mathcal{P}_{\mathcal{C}}}{\rm{d}\beta}\bigg{|}_{\text{fit}}\propto\,\,$
$\displaystyle\left[\left(\frac{\beta}{\beta-1}\right)^{0.6}+\left(\frac{\beta}{\beta-1}\right)^{-0.6}\right]\times$
$\displaystyle\exp\left[-9\left(\frac{\beta}{\beta_{\text{max}}^{10\,M_{\odot}}}\right)^{1.1}\right]\,\,\text{for}\,\,1\lesssim\beta<\beta_{\text{max}}^{10\,M_{\odot}},$
(12b)

Figure 5: The stellar mass probability density functions in TDEs before and
after the correction with $\mathcal{C}$. For four distinct $M_{\rm
BH}=10^{4}\,M_{\odot}\,(\text{blue}),\,10^{6}\,M_{\odot}\,(\text{green}),\,10^{7}\,M_{\odot}\,(\text{orange}),\,10^{8}\,M_{\odot}\,(\text{red})$,
we show the intrinsic (solid), $\mathcal{C}$-corrected (dashed), and
$\mathcal{C}_{\rm thres}=0.1$ case (dotted) stellar mass distributions in
TDEs. The Kroupa initial mass function is plotted for comparison (grey solid).
Low-mass stars have a lower intrinsic presentation in TDEs due to a
combination of the LC physics as well as their direct plunge when $M_{\rm BH}$
is large. For TDEs with $M_{\rm BH}\gtrsim 10^{8}\,M_{\odot}$, there exists a
cutoff of low-mass stars. For TDEs with $M_{\rm BH}\lesssim
10^{6}\,M_{\odot}$, the $\mathcal{C}$-correction can over-compensate the
suppression of low-mass stars.

which are overplotted on the numerical results. The generic form of the
formulae captures the entire behaviour across the full $\beta$ range: the
rising towards $\beta\approx 1$, the asymptotic $\beta^{-2}$ proportionality
($\beta$-independent) in the $1\ll\beta\ll\beta_{\rm max}^{10\,M_{\odot}}$
regime, and the fast exponential decay while approaching $\beta_{\rm
max}^{10\,M_{\odot}}$, with a maximum error of $0.39$ dex.

###  4.2 Distribution of the disrupted stellar mass $m_{\star}$

In a similar fashion, we can calculate the intrinsic $\mathcal{P}(m_{\star})$
and $\mathcal{C}$-corrected $\mathcal{P}_{\mathcal{C}}(m_{\star})$ stellar
probability density functions as:

$\displaystyle\frac{\rm{d}\mathcal{P}}{\rm{d}m_{\star}}$
$\displaystyle=\frac{1}{m_{\star}\Gamma}\int\left(\frac{\rm{d}^{2}\Gamma}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln\beta\,,$ (13a)
$\displaystyle\frac{\rm{d}\mathcal{P}_{\mathcal{C}}}{\rm{d}m_{\star}}$
$\displaystyle=\frac{1}{m_{\star}\Gamma_{\mathcal{C}}}\int\left(\frac{\rm{d}^{2}\Gamma_{\mathcal{C}}}{\rm{d}\ln\beta\,\rm{d}\ln
m_{\star}}\right)\rm{d}\ln\beta\,.$ (13b)

Fig.5 shows the predicted stellar mass distribution for a range of black hole
masses. One can see that the intrinsic LC process somewhat suppresses the
representation of low-mass stars in TDEs (see Pfister et al. 2021 for the
explanation). The suppression is not strong unless when $M_{\rm
BH}>10^{7}\,M_{\odot}$ and low-mass stars on high-$\beta$ orbits start to
directly plunge. When $M_{\rm BH}\gtrsim 10^{8}\,M_{\odot}$, one can see all
low-mass stars are removed.

The disk formation efficiency, however, is higher for low-mass stars and
therefore leads to an enhancement of their presence in the observed TDE
population. When putting all the factors together, for the majority TDE
population ($M_{\rm BH}\lesssim 10^{6}M_{\odot}$), we should still expect to
observe more TDEs from low-mass stars than the initial stellar mass
distribution.

##  5 Discussion and Summary

In this paper, we consider how the physics of TDE disk formation can affect
their observed rates and demographics. We summarize our findings as below:

  * •

We have systematically recalculated the host BH masses for an updated list of
observed TDEs using two methods, $M_{\rm BH}-\sigma$ and $M_{\rm BH}-M_{\rm
gal}$, based on the observed values of $\sigma$ and $M_{\rm gal}$ from
previous literature. We find that the observed TDE $M_{\rm BH}$ distribution
peaks between $10^{6}-10^{7}\,M_{\odot}$, which shows a drastic discrepancy
when compared to the intrinsic distribution of TDE $M_{\rm BH}$ obtained using
the LC dynamics and the BHMF (Fig.1).

  * •

We address this discrepancy by considering the efficiency of TDE disk
formation ($\mathcal{C}$), which is quantified to the first order as the ratio
of the energy loss in the first stream self-crossing over the total energy
needed to be reduced for complete circularization (Eq.5). We find that for BHs
with $M_{\rm BH}\lesssim 10^{6}\,M_{\odot}$, $\mathcal{C}$ is typically very
low ($\lesssim 0.1$) for typical TDEs with low $\beta$ (Fig.2).

  * •

While applying $\mathcal{C}$ to correct the TDE rates, we naturally obtain a
significant suppression of TDE rates around lighter BHs. The $M_{\rm BH}$
distribution peak can be shifted to fit the observed one when we only keep the
TDEs with $\mathcal{C}\gtrsim 0.1$ (Fig.3).

  * •

There are a few interesting consequences induced by including the
$\mathcal{C}$ correction. First, high-$\beta$ TDEs are favored since they have
more prompt disk formation. Therefore, the $\beta$-distribution of the TDEs
with prompt disk formation is much flatter than the intrinsic one (Fig.4). We
provide the fitted formulae for both the intrinsic and corrected
$\beta$-distributions (Eq.12a and 12b). Second, in most TDEs the low-mass star
population is expected to be enhanced as compared to their intrinsic weight,
unless when $M_{\rm BH}\gtrsim 10^{7}\,M_{\odot}$ for which low-mass stars
directly plunge (Fig.5).

With the next-generation all-sky transient surveys such as the Vera Rubin
Observatory, the TDE sample size will increase by tens of times, making it
possible to use TDEs as probes of the demographics of massive black holes and
stars in the centers of galaxies. In this work, using first principles, we
have demonstrated that it is important to understand how physical processes in
TDEs, such as how promptly the debris forms a disk, can significantly impact
the distributions of $M_{\rm BH}$ and $m_{\star}$ obtained from TDE
observations.

It is important to stress that our study aims at providing one important link
narrowing the gap between the intrinsic and the observed TDE rates and
demographics. There can be other factors are affecting the observed TDE
$M_{\rm BH}$ distribution, such as dust obscuration/reddening and the
sensitivity and bias of the instruments. Also, the chance of detecting a TDE
necessarily depends on its luminosity (in monochromatic bands), and more
studies of TDE emission processes are required for understanding how the
luminosity links to $M_{\rm BH}$ and other parameters. Moreover, we
acknowledge that the low-mass end of the BHMF still has large uncertainties,
which directly links to the rates of TDEs around low-mass SMBHs. Eventually,
comprehensive forward modeling including all these factors will be needed to
bring the theory and observation together.

We thank Alex Dittmann, Tiara Hung, Cole Miller, Brenna Mockler, and Sjoert
van Velzen for useful comments and discussions. The authors acknowledge the
support from the Hong Kong Research Grants Council (HKU27305119, HKU17304821)
and the National Natural Science Foundation of China (HKU12122309).

## References

  * Auchettl et al. (2017) Auchettl, K., Guillochon, J., & Ramirez-Ruiz, E. 2017, ApJ, 838, 149
  * Bonnerot & Lu (2020) Bonnerot, C., & Lu, W. 2020, MNRAS, 495, 1374
  * Bonnerot & Stone (2021) Bonnerot, C., & Stone, N. C. 2021, Space Sci. Rev., 217, 16
  * Dai et al. (2021) Dai, J. L., Lodato, G., & Cheng, R. 2021, Space Sci. Rev., 217, 12
  * Dai et al. (2013) Dai, L., Escala, A., & Coppi, P. 2013, ApJ, 775, L9
  * Dai et al. (2015) Dai, L., McKinney, J. C., & Miller, M. C. 2015, ApJ, 812, L39
  * French et al. (2020) French, K. D., Wevers, T., Law-Smith, J., Graur, O., & Zabludoff, A. I. 2020, Space Sci. Rev., 216, 32
  * Gallo & Sesana (2019) Gallo, E., & Sesana, A. 2019, ApJ, 883, L18
  * Gezari (2021) Gezari, S. 2021, arXiv e-prints, arXiv:2104.14580
  * Graur et al. (2018) Graur, O., French, K. D., Zahid, H. J., et al. 2018, ApJ, 853, 39
  * Guillochon & Ramirez-Ruiz (2013) Guillochon, J., & Ramirez-Ruiz, E. 2013, ApJ, 767, 25
  * Guillochon & Ramirez-Ruiz (2015) —. 2015, ApJ, 809, 166
  * Hung et al. (2017) Hung, T., Gezari, S., Blagorodnova, N., et al. 2017, ApJ, 842, 29
  * Kesden (2012) Kesden, M. 2012, Phys. Rev. D, 85, 024037
  * Kippenhahn & Weigert (1990) Kippenhahn, R., & Weigert, A. 1990, Stellar Structure and Evolution
  * Kochanek (1994) Kochanek, C. S. 1994, ApJ, 422, 508
  * Kochanek (2016) —. 2016, MNRAS, 461, 371
  * Komossa et al. (2008) Komossa, S., Zhou, H., Wang, T., et al. 2008, ApJ, 678, L13
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231
  * Law-Smith et al. (2019) Law-Smith, J., Guillochon, J., & Ramirez-Ruiz, E. 2019, ApJ, 882, L25
  * Law-Smith et al. (2020) Law-Smith, J. A. P., Coulter, D. A., Guillochon, J., Mockler, B., & Ramirez-Ruiz, E. 2020, ApJ, 905, 141
  * Liptai et al. (2019) Liptai, D., Price, D. J., Mandel, I., & Lodato, G. 2019, arXiv e-prints, arXiv:1910.10154
  * Lu & Bonnerot (2020) Lu, W., & Bonnerot, C. 2020, MNRAS, 492, 686
  * Magorrian & Tremaine (1999) Magorrian, J., & Tremaine, S. 1999, MNRAS, 309, 447
  * Merritt (2013) Merritt, D. 2013, Classical and Quantum Gravity, 30, 244005
  * Merritt & Ferrarese (2001) Merritt, D., & Ferrarese, L. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 249, The Central Kiloparsec of Starbursts and AGN: The La Palma Connection, ed. J. H. Knapen, J. E. Beckman, I. Shlosman, & T. J. Mahoney, 335
  * Mockler et al. (2019) Mockler, B., Guillochon, J., & Ramirez-Ruiz, E. 2019, ApJ, 872, 151
  * Pfister et al. (2019) Pfister, H., Bar-Or, B., Volonteri, M., Dubois, Y., & Capelo, P. R. 2019, MNRAS, 488, L29
  * Pfister et al. (2021) Pfister, H., Toscani, M., Wong, T. H. T., et al. 2021, arXiv e-prints, arXiv:2103.05883
  * Pfister et al. (2020) Pfister, H., Volonteri, M., Dai, J. L., & Colpi, M. 2020, MNRAS, 497, 2276
  * Piran et al. (2015) Piran, T., Svirski, G., Krolik, J., Cheng, R. M., & Shiokawa, H. 2015, ApJ, 806, 164
  * Rees (1988) Rees, M. J. 1988, Nature, 333, 523
  * Reines & Volonteri (2015) Reines, A. E., & Volonteri, M. 2015, ApJ, 813, 82
  * Roth et al. (2018) Roth, N., Mushotzky, R., Gezari, S., & van Velzen, S. 2018, in American Astronomical Society Meeting Abstracts, Vol. 231, American Astronomical Society Meeting Abstracts #231, 405.03
  * Roth et al. (2020) Roth, N., Rossi, E. M., Krolik, J., et al. 2020, Space Sci. Rev., 216, 114
  * Roth et al. (2021) Roth, N., van Velzen, S., Cenko, S. B., & Mushotzky, R. F. 2021, ApJ, 910, 93
  * Ryu et al. (2020a) Ryu, T., Krolik, J., & Piran, T. 2020a, ApJ, 904, 73
  * Ryu et al. (2020b) Ryu, T., Krolik, J., Piran, T., & Noble, S. C. 2020b, ApJ, 904, 98
  * Saxton et al. (2020) Saxton, R., Komossa, S., Auchettl, K., & Jonker, P. G. 2020, Space Sci. Rev., 216, 85
  * Shankar et al. (2016) Shankar, F., Bernardi, M., Sheth, R. K., et al. 2016, MNRAS, 460, 3119
  * Shiokawa et al. (2015) Shiokawa, H., Krolik, J. H., Cheng, R. M., Piran, T., & Noble, S. C. 2015, ApJ, 804, 85
  * Stone & Metzger (2016) Stone, N. C., & Metzger, B. D. 2016, MNRAS, 455, 859
  * Stone et al. (2020) Stone, N. C., Vasiliev, E., Kesden, M., et al. 2020, Space Sci. Rev., 216, 35
  * Strubbe (2011) Strubbe, L. E. 2011, PhD thesis, University of California, Berkeley
  * Ulmer (1998) Ulmer, A. 1998, in American Institute of Physics Conference Series, Vol. 431, Accretion processes in Astrophysical Systems: Some like it hot! - eigth AstroPhysics Conference, ed. S. S. Holt & T. R. Kallman, 141–144
  * van Velzen (2018) van Velzen, S. 2018, ApJ, 852, 72
  * van Velzen et al. (2020) van Velzen, S., Holoien, T. W. S., Onori, F., Hung, T., & Arcavi, I. 2020, Space Sci. Rev., 216, 124
  * van Velzen et al. (2021) van Velzen, S., Gezari, S., Hammerstein, E., et al. 2021, ApJ, 908, 4
  * Wang & Merritt (2004) Wang, J., & Merritt, D. 2004, ApJ, 600, 149
  * Wevers et al. (2017) Wevers, T., van Velzen, S., Jonker, P. G., et al. 2017, MNRAS, 471, 1694
  * Wevers et al. (2019) Wevers, T., Stone, N. C., van Velzen, S., et al. 2019, MNRAS, 487, 4136

##  Appendix A TDE Host Black Hole Masses

In this appendix, we present the criteria used to select the likely TDEs among
the many other TDE candidates as our observed sample, and further elaborate on
the properties of both the host black hole mass and luminosity distributions
we find.

###  A.1 TDE Selection and Black Hole Mass Calculation

Among a total of a hundred or so detected TDE candidates, we select 65 TDEs to
include in our analysis. The chosen TDE candidates are listed in Table A.1
with some of their host galaxy properties. Detailed criteria for the selection
of TDE candidates are as follows.

  * •

Categorized as a possible/likely/confirmed TDE in at least _two_ of the
following literature (Auchettl et al., 2017; Wevers et al., 2017, 2019; French
et al., 2020; van Velzen et al., 2021)

  * •

All TDEs in Table 1 of Gezari (2021)

  * •

All ZTF TDEs (van Velzen et al., 2021)

Also, we intentionally exclude all likely partial TDEs, e.g., iPTF-16fnl and
PS1-11af, from our sample, since our calculation only applies to fully
disrupted stars.

We also obtain the TDE host black hole mass with two different methods. We
either derive their $M_{\rm BH}$ all using the $M_{\rm BH}-\sigma$ relation
(Merritt & Ferrarese, 2001):

$\log_{10}\left(\frac{M_{{\rm
BH},\sigma}}{M_{\odot}}\right)=\log_{10}\left[10^{8}\cdot\left(1.48\pm
0.24\right)\left(\frac{\sigma}{200\,\text{km s}^{-1}}\right)^{4.65\pm
0.48}\right]$ (A1)

or using the $M_{\rm BH}-M_{\text{gal}}$ relation (Reines & Volonteri, 2015):

$\log_{10}\left(\frac{M_{{\rm BH},\text{gal}}}{M_{\odot}}\right)=\left(7.45\pm
0.08\right)+\left(1.05\pm
0.11\right)\log_{10}\left(\frac{M_{\text{gal}}}{10^{11}\,M_{\odot}}\right)$
(A2)

The derived black hole masses as well as other quantities such as the peak
luminosity (assuming blackbody emission) are listed in Table A.1. The error
bar of each derived $M_{\rm BH}$ is computed by modeling a Gaussian
distribution using the measurement errors of the velocity dispersion $\sigma$
and the total galaxy mass $M_{\rm gal}$, with the curve-fitting $1\sigma$
error of Eq.A1 and Eq.A2. The Gaussian distributions peak at the mean value
$\log_{10}\bar{M}_{\rm BH}$ with standard deviation $s=\left(\log_{10}M_{\rm
BH}^{\text{up}}-\log_{10}M_{\rm BH}^{\text{low}}\right)/2$.

We plot the $M_{\rm BH}$ distribution of all TDE samples obtained from either
method in Fig.1. The overall trends are similar, but the detailed shapes of
the two distributions are somewhat different from each other. The main reason
for this the that for many TDEs we lack measurements of $\sigma$. We further
separate between the optical/UV strong (magenta) and X-ray strong (green) TDE
samples in Fig.A.1. It can be clearly seen that the X-ray strong TDEs have a
bimodal distribution in $M_{\rm BH}$, but we believe that this effect is
primarily due to the limited sample size of this TDE category. Thus, shifting
of the distribution, perhaps reducing to a single peak, is highly likely when
more are observed. Another possibility is related to the TDE emission
mechanism in different wavebands which deserves extensive study.

Table A.1: Chosen TDE Candidates with Their Host Galaxy Properties

Name | Redshift |  $\sigma$ [km s-1] | $\log_{10}\left(\displaystyle\frac{M_{{\rm BH},\sigma}}{M_{\odot}}\right)$ | $\log_{10}\left(\displaystyle\frac{M_{\text{gal}}}{M_{\odot}}\right)$ | $\log_{10}\left(\displaystyle\frac{M_{{\rm BH},\text{gal}}}{M_{\odot}}\right)$ | $\log_{10}\left(L_{\text{bb},\text{peak}}\left[\text{erg s}^{-1}\right]\right)$
---|---|---|---|---|---|---
X-ray TDEs |  |  |  |  |  |
2MASX J0249 | 0.0186 |  $43^{+4}_{-4}$ ${}^{\text{f}}$ | $5.07^{+0.55}_{-0.62}$ |  $9.1^{+...}_{-...}$ ${}^{\text{e}}$ | $5.46^{+...}_{-...}$ |  $\cdots$
2XMMi J1847-63 | 0.0353 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $42.82^{+...}_{-...}$ ${}^{\text{h}}$
3XMM J1500 | 0.1454 |  $59^{+3}_{-3}$ ${}^{\text{e}}$ | $5.71^{+0.41}_{-0.45}$ |  $9.3^{+...}_{-...}$ ${}^{\text{e}}$ | $5.67^{+...}_{-...}$ |  $43.08^{+...}_{-...}$ ${}^{\text{h}}$
3XMM J1521+0749 | 0.179 |  $58^{+2}_{-2}$ ${}^{\text{f}}$ | $5.67^{+0.39}_{-0.41}$ |  $10.17^{+0.11}_{-0.20}$ ${}^{\text{c}}$ | $6.58^{+0.27}_{-0.40}$ |  $43.51^{+...}_{-...}$ ${}^{\text{h}}$
ASASSN-14li | 0.0206 |  $81^{+2}_{-2}$ ${}^{\text{b}}$ | $6.35^{+0.30}_{-0.32}$ |  $9.71^{+0.05}_{-0.10}$ ${}^{\text{g}}$ | $6.10^{+0.26}_{-0.34}$ |  $43.66^{+0.02}_{-0.02}$ ${}^{\text{g}}$
LEDA 095953 | 0.0366 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
NGC 5905 | 0.011 |  $97^{+5}_{-5}$ ${}^{\text{f}}$ | $6.71^{+0.31}_{-0.35}$ |  $10.83^{+0.22}_{-0.06}$ ${}^{\text{c}}$ | $7.27^{+0.31}_{-0.17}$ |  $40.94^{+...}_{-...}$ ${}^{\text{h}}$
RBS 1032 | 0.026 |  $49^{+7}_{-7}$ ${}^{\text{f}}$ | $5.33^{+0.60}_{-0.71}$ |  $9.19^{+0.15}_{-0.16}$ ${}^{\text{c}}$ | $5.55^{+0.42}_{-0.47}$ |  $41.70^{+...}_{-...}$ ${}^{\text{h}}$
RX J1242-1119-A | 0.05 |  $\cdots$ |  $\cdots$ |  $10.3^{+...}_{-...}$ ${}^{\text{e}}$ | $6.72^{+...}_{-...}$ |  $42.60^{+...}_{-...}$ ${}^{\text{h}}$
RX J1420+5334-A | 0.147 |  $131^{+13}_{-13}$ ${}^{\text{f}}$ | $7.32^{+0.33}_{-0.40}$ |  $10.53^{+0.07}_{-0.07}$ ${}^{\text{c}}$ | $6.96^{+0.19}_{-0.22}$ |  $43.38^{+...}_{-...}$ ${}^{\text{h}}$
RX J1624+7554 | 0.0636 |  $155^{+9}_{-9}$ ${}^{\text{f}}$ | $7.66^{+0.22}_{-0.26}$ |  $10.4^{+...}_{-...}$ ${}^{\text{e}}$ | $6.82^{+...}_{-...}$ |  $43.38^{+...}_{-...}$ ${}^{\text{h}}$
SDSS J0159 | 0.3117 |  $124^{+10}_{-10}$ ${}^{\text{f}}$ | $7.21^{+0.31}_{-0.36}$ |  $10.37^{+0.11}_{-0.06}$ ${}^{\text{c}}$ | $6.79^{+0.25}_{-0.22}$ |  $\cdots$
SDSS J1201+3003 | 0.146 |  $122^{+4}_{-4}$ ${}^{\text{f}}$ | $7.17^{+0.23}_{-0.25}$ |  $10.61^{+0.08}_{-0.16}$ ${}^{\text{c}}$ | $7.04^{+0.20}_{-0.31}$ |  $45.00^{+...}_{-...}$ ${}^{\text{h}}$
SDSS J1311-0123 | 0.195 |  $\cdots$ |  $\cdots$ |  $8.7^{+...}_{-...}$ ${}^{\text{f}}$ | $5.04^{+...}_{-...}$ |  $41.74^{+...}_{-...}$ ${}^{\text{h}}$
SDSS J1323+4827 | 0.0875 |  $75^{+4}_{-4}$ ${}^{\text{f}}$ | $6.19^{+0.36}_{-0.40}$ |  $10.38^{+0.06}_{-0.07}$ ${}^{\text{c}}$ | $6.80^{+0.20}_{-0.23}$ |  $44.30^{+...}_{-...}$ ${}^{\text{h}}$
Swift J1112 | 0.89 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Swift J1644+57 | 0.3543 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Swift J2058 | 1.186 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
TDXF J1347-32 | 0.0366 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $42.73^{+...}_{-...}$ ${}^{\text{h}}$
WINGS J1348+26 | 0.0651 |  $\cdots$ |  $\cdots$ |  $8.48^{+...}_{-...}$ ${}^{\text{s}}$ | $4.80^{+...}_{-...}$ |  $41.79^{+...}_{-...}$ ${}^{\text{h}}$
XMM J0740 | 0.0173 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $42.61^{+...}_{-...}$ ${}^{\text{h}}$
Optical/UV TDEs |  |  |  |  |  |
ASASSN-14ae | 0.044 |  $53^{+2}_{-2}$ ${}^{\text{f}}$ | $5.49^{+0.41}_{-0.44}$ |  $9.73^{+0.13}_{-0.13}$ ${}^{\text{c}}$ | $6.12^{+0.34}_{-0.37}$ |  $43.87^{+0.01}_{-0.01}$ ${}^{\text{g}}$
ASASSN-14li | 0.0206 |  $81^{+2}_{-2}$ ${}^{\text{b}}$ | $6.35^{+0.30}_{-0.32}$ |  $9.71^{+0.05}_{-0.10}$ ${}^{\text{g}}$ | $6.10^{+0.26}_{-0.34}$ |  $43.66^{+0.02}_{-0.02}$ ${}^{\text{g}}$
ASASSN-15lh | 0.233 |  $225^{+15}_{-15}$ ${}^{\text{d}}$ | $8.41^{+0.16}_{-0.21}$ |  $10.8^{+...}_{-...}$ ${}^{\text{d}}$ | $7.24^{+...}_{-...}$ |  $45.6^{+...}_{-...}$ ${}^{\text{d}}$
ASASSN-15oi | 0.048 |  $61^{+7}_{-7}$ ${}^{\text{f}}$ | $5.77^{+0.51}_{-0.60}$ |  $10.05^{+0.04}_{-0.04}$ ${}^{\text{g}}$ | $6.45^{+0.22}_{-0.23}$ |  $44.45^{+0.01}_{-0.01}$ ${}^{\text{g}}$
ASASSN-19dj | 0.022 |  $\cdots$ |  $\cdots$ |  $9.82^{+0.16}_{-0.13}$ ${}^{\text{i}}$ | $6.21^{+0.36}_{-0.36}$ |  $44.50^{+0.02}_{-0.02}$ ${}^{\text{g}}$
AT2018bsi | 0.051 |  $\cdots$ |  $\cdots$ |  $10.63^{+0.05}_{-0.05}$ ${}^{\text{i}}$ | $7.06^{+0.17}_{-0.18}$ |  $43.87^{+0.08}_{-0.08}$ ${}^{\text{g}}$
AT2018dyb | 0.018 |  $96^{+1}_{-1}$ ${}^{\text{s}}$ | $6.69^{+0.24}_{-0.25}$ |  $9.86^{+0.08}_{-0.15}$ ${}^{\text{g}}$ | $6.25^{+0.28}_{-0.38}$ |  $44.08^{+...}_{-...}$ ${}^{\text{h}}$
AT2018dyk | 0.037 |  $112^{+4}_{-4}$ ${}^{\text{f}}$ | $7.00^{+0.25}_{-0.28}$ |  $10.6^{+...}_{-...}$ ${}^{\text{f}}$ | $7.03^{+...}_{-...}$ |  $\cdots$
AT2018hco | 0.088 |  $\cdots$ |  $\cdots$ |  $9.95^{+0.12}_{-0.16}$ ${}^{\text{i}}$ | $6.35^{+0.31}_{-0.38}$ |  $44.25^{+0.04}_{-0.04}$ ${}^{\text{g}}$
AT2018hyz | 0.0458 |  $60^{+5}_{-5}$ ${}^{\text{f}}$ | $5.74^{+0.46}_{-0.52}$ |  $9.84^{+0.09}_{-0.14}$ ${}^{\text{i}}$ | $6.23^{+0.29}_{-0.37}$ |  $44.10^{+0.01}_{-0.01}$ ${}^{\text{g}}$
AT2018fyk | 0.059 |  $\cdots$ |  $\cdots$ |  $10.58^{+0.12}_{-0.21}$ ${}^{\text{g}}$ | $7.01^{+0.24}_{-0.37}$ |  $44.48^{+...}_{-...}$ ${}^{\text{h}}$
AT2018iih | 0.212 |  $\cdots$ |  $\cdots$ |  $10.63^{+0.18}_{-0.14}$ ${}^{\text{i}}$ | $7.06^{+0.29}_{-0.28}$ |  $44.62^{+0.04}_{-0.04}$ ${}^{\text{g}}$
AT2018lna | 0.091 |  $\cdots$ |  $\cdots$ |  $9.49^{+0.11}_{-0.12}$ ${}^{\text{i}}$ | $5.86^{+0.35}_{-0.38}$ |  $44.56^{+0.06}_{-0.06}$ ${}^{\text{g}}$
AT2018lni | 0.138 |  $\cdots$ |  $\cdots$ |  $10.00^{+0.09}_{-0.14}$ ${}^{\text{i}}$ | $6.40^{+0.27}_{-0.35}$ |  $44.21^{+0.29}_{-0.17}$ ${}^{\text{g}}$
AT2019ahk | 0.0262 |  $\cdots$ |  $\cdots$ |  $9.72^{+0.09}_{-0.10}$ ${}^{\text{g}}$ | $6.11^{+0.30}_{-0.34}$ |  $44.08^{+...}_{-...}$ ${}^{\text{h}}$
AT2019bhf | 0.1206 |  $\cdots$ |  $\cdots$ |  $10.25^{+0.14}_{-0.12}$ ${}^{\text{i}}$ | $6.66^{+0.30}_{-0.30}$ |  $43.91^{+0.04}_{-0.05}$ ${}^{\text{g}}$
AT2019cho | 0.193 |  $\cdots$ |  $\cdots$ |  $10.20^{+0.11}_{-0.14}$ ${}^{\text{i}}$ | $6.61^{+0.27}_{-0.33}$ |  $43.98^{+0.01}_{-0.01}$ ${}^{\text{g}}$
AT2019dsg | 0.0512 |  $\cdots$ |  $\cdots$ |  $10.46^{+0.11}_{-0.19}$ ${}^{\text{i}}$ | $6.88^{+0.25}_{-0.36}$ |  $44.26^{+0.04}_{-0.05}$ ${}^{\text{g}}$
AT2019ehz | 0.074 |  $\cdots$ |  $\cdots$ |  $9.74^{+0.08}_{-0.09}$ ${}^{\text{i}}$ | $6.13^{+0.29}_{-0.33}$ |  $44.03^{+0.01}_{-0.02}$ ${}^{\text{g}}$
AT2019eve | 0.064 |  $\cdots$ |  $\cdots$ |  $9.31^{+0.10}_{-0.15}$ ${}^{\text{i}}$ | $5.68^{+0.36}_{-0.44}$ |  $43.14^{+0.02}_{-0.03}$ ${}^{\text{g}}$
AT2019lwu | 0.117 |  $\cdots$ |  $\cdots$ |  $9.86^{+0.09}_{-0.13}$ ${}^{\text{i}}$ | $6.25^{+0.29}_{-0.35}$ |  $43.60^{+0.03}_{-0.04}$ ${}^{\text{g}}$
AT2019meg | 0.152 |  $\cdots$ |  $\cdots$ |  $9.70^{+0.15}_{-0.08}$ ${}^{\text{i}}$ | $6.09^{+0.36}_{-0.32}$ |  $44.36^{+0.03}_{-0.04}$ ${}^{\text{g}}$
AT2019mha | 0.148 |  $\cdots$ |  $\cdots$ |  $10.07^{+0.10}_{-0.18}$ ${}^{\text{i}}$ | $6.47^{+0.28}_{-0.39}$ |  $44.05^{+0.06}_{-0.05}$ ${}^{\text{g}}$
AT2019qiz | 0.0151 |  $\cdots$ |  $\cdots$ |  $10.01^{+0.09}_{-0.13}$ ${}^{\text{i}}$ | $6.41^{+0.27}_{-0.34}$ |  $43.44^{+0.01}_{-0.01}$ ${}^{\text{g}}$
F01004 | 0.1178 |  $132^{+29}_{-29}$ ${}^{\text{f}}$ | $7.33^{+0.51}_{-0.72}$ |  $9.8^{+...}_{-...}$ ${}^{\text{f}}$ | $6.19^{+...}_{-...}$ |  $\cdots$
GALEX-D1-9 | 0.326 |  $89^{+4}_{-4}$ ${}^{\text{e}}$ | $6.54^{+0.31}_{-0.35}$ |  $10.3^{+...}_{-...}$ ${}^{\text{d}}$ | $6.72^{+...}_{-...}$ |  $44.1^{+...}_{-...}$ ${}^{\text{d}}$
GALEX-D23-H1 | 0.1855 |  $84^{+4}_{-4}$ ${}^{\text{f}}$ | $6.42^{+0.33}_{-0.37}$ |  $10.08^{+0.15}_{-0.07}$ ${}^{\text{c}}$ | $6.48^{+0.33}_{-0.26}$ |  $43.9^{+...}_{-...}$ ${}^{\text{d}}$
GALEX-D3-13 | 0.3698 |  $133^{+6}_{-6}$ ${}^{\text{f}}$ | $7.35^{+0.23}_{-0.27}$ |  $10.7^{+...}_{-...}$ ${}^{\text{c}}$ | $7.14^{+...}_{-...}$ |  $44.3^{+...}_{-...}$ ${}^{\text{d}}$
iPTF-15af | 0.079 |  $106^{+2}_{-2}$ ${}^{\text{f}}$ | $6.89^{+0.23}_{-0.25}$ |  $10.31^{+0.08}_{-0.10}$ ${}^{\text{c}}$ | $6.73^{+0.23}_{-0.28}$ |  $44.10^{+0.10}_{-0.08}$ ${}^{\text{g}}$
iPTF-16axa | 0.108 |  $82^{+3}_{-3}$ ${}^{\text{f}}$ | $6.37^{+0.32}_{-0.35}$ |  $10.25^{+0.05}_{-0.08}$ ${}^{\text{c}}$ | $6.66^{+0.21}_{-0.25}$ |  $43.82^{+0.03}_{-0.02}$ ${}^{\text{g}}$
OGLE16aaa | 0.1655 |  $\cdots$ |  $\cdots$ |  $10.43^{+0.09}_{-0.11}$ ${}^{\text{g}}$ | $6.85^{+0.23}_{-0.27}$ |  $44.22^{+...}_{-...}$ ${}^{\text{h}}$
PS1-10jh | 0.1696 |  $65^{+3}_{-3}$ ${}^{\text{f}}$ | $5.90^{+0.38}_{-0.42}$ |  $9.63^{+0.10}_{-0.13}$ ${}^{\text{g}}$ | $6.01^{+0.33}_{-0.38}$ |  $44.47^{+0.07}_{-0.07}$ ${}^{\text{g}}$
PS16dtm | 0.0804 |  $45^{+13}_{-13}$ ${}^{\text{f}}$ | $5.16^{+0.84}_{-1.15}$ |  $9.77^{+0.11}_{-0.13}$ ${}^{\text{c}}$ | $6.16^{+0.32}_{-0.37}$ |  $\cdots$
PS17dhz | 0.1089 |  $\cdots$ |  $\cdots$ |  $9.47^{+0.11}_{-0.13}$ ${}^{\text{g}}$ | $5.84^{+0.36}_{-0.40}$ |  $43.82^{+...}_{-...}$ ${}^{\text{h}}$
PS18kh | 0.075 |  $\cdots$ |  $\cdots$ |  $9.95^{+0.12}_{-0.24}$ ${}^{\text{i}}$ | $6.35^{+0.31}_{-0.48}$ |  $43.78^{+0.02}_{-0.02}$ ${}^{\text{g}}$
PTF-09axc | 0.115 |  $60^{+4}_{-4}$ ${}^{\text{f}}$ | $5.74^{+0.43}_{-0.48}$ |  $9.84^{+0.06}_{-0.09}$ ${}^{\text{c}}$ | $6.23^{+0.27}_{-0.31}$ |  $43.46^{+0.03}_{-0.02}$ ${}^{\text{g}}$
PTF-09djl | 0.184 |  $64^{+7}_{-7}$ ${}^{\text{f}}$ | $5.87^{+0.49}_{-0.57}$ |  $9.91^{+0.13}_{-0.17}$ ${}^{\text{c}}$ | $6.31^{+0.32}_{-0.40}$ |  $44.42^{+0.04}_{-0.04}$ ${}^{\text{g}}$
PTF-09ge | 0.064 |  $82^{+2}_{-2}$ ${}^{\text{f}}$ | $6.37^{+0.30}_{-0.32}$ |  $9.87^{+0.13}_{-0.17}$ ${}^{\text{c}}$ | $6.26^{+0.33}_{-0.40}$ |  $44.04^{+0.01}_{-0.01}$ ${}^{\text{g}}$
PTF-10iya | 0.224 |  $\cdots$ |  $\cdots$ |  $9.3^{+...}_{-...}$ ${}^{\text{f}}$ | $5.67^{+...}_{-...}$ |  $\cdots$
SDSS-TDE1 | 0.136 |  $126^{+7}_{-7}$ ${}^{\text{b}}$ | $7.24^{+0.26}_{-0.30}$ |  $10.08^{+0.08}_{-0.12}$ ${}^{\text{c}}$ | $6.48^{+0.26}_{-0.32}$ |  $43.5^{+...}_{-...}$ ${}^{\text{d}}$
SDSS-TDE2 | 0.2515 |  $\cdots$ |  $\cdots$ |  $10.59^{+0.17}_{-0.10}$ ${}^{\text{g}}$ | $7.02^{+0.28}_{-0.24}$ |  $44.54^{+0.08}_{-0.06}$ ${}^{\text{g}}$
SDSS J0748+4712 | 0.0615 |  $126^{+7}_{-7}$ ${}^{\text{f}}$ | $7.24^{+0.26}_{-0.30}$ |  $10.18^{+0.06}_{-0.09}$ ${}^{\text{c}}$ | $6.59^{+0.23}_{-0.28}$ |  $\cdots$
SDSS J0952+2143 | 0.0789 |  $95^{+...}_{-...}$ ${}^{\text{a}}$ | $6.67^{+...}_{-...}$ |  $10.37^{+0.06}_{-0.07}$ ${}^{\text{c}}$ | $6.79^{+0.20}_{-0.23}$ |  $\cdots$
SDSS J1342+0530 | 0.0366 |  $72^{+6}_{-6}$ ${}^{\text{f}}$ | $6.11^{+0.42}_{-0.48}$ |  $9.64^{+0.23}_{-0.07}$ ${}^{\text{c}}$ | $6.02^{+0.45}_{-0.31}$ |  $\cdots$
SDSS J1350+2916 | 0.0777 |  $\cdots$ |  $\cdots$ |  $9.94^{+0.17}_{-0.20}$ ${}^{\text{c}}$ | $6.34^{+0.35}_{-0.43}$ |  $\cdots$

Note. — The columns are: TDE name, redshift, velocity dispersion $\sigma$,
black hole mass $M_{\rm{BH},\sigma}$ derived from the $M_{\rm BH}-\sigma$
relation, galaxy stellar mass $M_{\rm gal}$, black hole mass $M_{\rm{BH},gal}$
derived from $M_{{\rm BH}}-M_{\text{gal}}$ relation, and peak luminosity
assuming blackbody emission $L_{\rm bb,peak}$. The chosen TDEs are separated
into either the optical/UV strong or X-ray strong categories, except for
ASASSN-14li which is in both categories. References for the values of
$\sigma$, $M_{\text{gal}}$ and $L_{\text{bb,peak}}$: ${}^{\text{a}}$Komossa et
al. (2008), ${}^{\text{b}}$Wevers et al. (2017), ${}^{\text{c}}$Graur et al.
(2018), ${}^{\text{d}}$van Velzen (2018), ${}^{\text{e}}$Wevers et al. (2019),
${}^{\text{f}}$French et al. (2020), ${}^{\text{g}}$van Velzen et al. (2020),
${}^{\text{h}}$Gezari (2021), ${}^{\text{i}}$van Velzen et al. (2021), and
${}^{\text{s}}$source discovery papers. Their 1$\sigma$ error bars of $M_{\rm
BH}$ are calculated by modeling each TDE as a Gaussian distribution with the
scattering from the $M_{\rm BH}$-scaling relations and the measurement errors
from $\sigma$ and $M_{\rm gal}$.

Figure A.1: TDE host $M_{\rm BH}$ distribution for X-ray strong or optical/UV
strong TDEs. The two coloured curves in Fig.1 are further decomposed into two
groups: optical/UV strong (magenta) or X-ray strong (green). The optical/UV
TDE $M_{\rm BH}$ distribution still peaks somewhere between
$10^{6}-10^{7}M_{\odot}$, while that of the X-ray TDEs exhibits a double peak
behavior.

###  A.2 TDE Luminosity Distribution

In Fig.A.2 we plot the TDE peak luminosity $L_{\rm bb,peak}$ distribution
against $M_{\rm BH}$. One might expect that the luminosity should generally
increase somewhat linearly with $M_{\rm BH}$ if the Eddington limit is the
primary factor determining $L_{\rm bb,peak}$ (Kochanek, 2016), however, here
one could barely see any trend of $L_{\rm bb,peak}$ as a function of $M_{\rm
BH}$. We note that the currently sample size is limited, and the range of
observed TDE $M_{\rm BH}$ is narrow, which forbid us from drawing any
conclusion on whether or how much the Eddington limit is playing a role in
determining the luminosity of TDEs. We also recognize that there are further
layer of complications like the flux limits of different instruments (van
Velzen, 2018) and dust obscuration from host galaxies (Roth et al., 2018). The
number of detected TDEs of a given transient survey has a nonlinear dependency
on its effective flux limit, and their limits could be used to derive the
maximum redshifts each survey are capable of reaching given a peak luminosity.
To stimulate a detection, not only should the event peak luminosity surpasses
the survey sensitivity in specific wavebands, but it also has to have
significant contrast compared with the flux of its host galaxy. Even worse,
the effects of dust (neutral gas) at optical/UV (X-ray) wavelengths in the
host galaxies would obstruct our observation and selection capability. Hence
the observed TDE samples unavoidably suffer from further external
observational bias.

Figure A.2: TDE peak luminosity $L_{\rm bb,peak}$ v.s. $M_{\rm BH}$ from Table
A.1 (modified from Fig.16 Hung et al. 2017). The optical/UV strong TDEs are
marked by magenta dots and the X-ray strong by green crosses, with the
horizontal line showing the error bar of $M_{\rm BH}$. The dashed lines
correspond to four different Eddington ratios, while the dotted line shows the
$\dot{M}_{\text{peak}}\propto M_{\rm BH}^{-1/2}$ proportionality relation
derived in Guillochon & Ramirez-Ruiz (2013) Eq.A1 with $m_{\star}=1M_{\odot}$,
$\gamma=4/3$, $\beta=1$, and radiative efficiency $\eta=0.1$. We find here
that some TDEs have peak luminosity exceeding the Eddington limits of their
host SMBHs: ASASSN-14ae, ASASSN-15lh, ASASSN-15oi, ASASSN-19dj, AT2018hyz,
AT2018lna, AT2019meg, PS1-10jh, and SDSS J1323-4827. Overall, no apparent
correlation is found between $L_{\rm bb,peak}$ and $M_{\rm BH}$.

##  Appendix B Derivations of Energy Dissipation in Stream-stream Collision

We summarize the results in Dai et al. (2015) on debris stream self-crossing
and disk formation in TDEs. The most bound debris has a specific binding
energy of

$E_{\text{mb}}=\frac{1}{2}v_{T}^{2}-\frac{GM_{\rm
BH}}{r_{T}-r_{\star}}\approx-\frac{GM_{\rm
BH}r_{\star}}{r_{T}^{2}}\,\,\text{for}\,\,r_{T}\gg r_{\star}$ (B1)

where $v_{T}$ is the orbital velocity at the tidal radius $r_{T}$ and
$r_{\star}$ is the stellar radius. Furthermore, the semi-major axis and
eccentricity of this orbit are $a_{\text{mb}}=r_{T}^{2}/2r_{\star}$ and
$e_{\text{mb}}=1-r_{p}/a_{\text{mb}}\approx 1-2(m_{\star}/M_{\rm
BH})^{1/3}/\beta$ respectively, where $r_{p}$ is the pericenter distance and
$\beta$ is the penetration parameter. The stellar debris orbits around a
Schwarzschild SMBH goes through apsidal precession. We approximate the process
using an instantaneous shift at $r_{p}$ with an angle of:

$\phi=\frac{6\pi GM_{\rm BH}}{c^{2}a(1-e^{2})}$ (B2)

Using these, the intersection radius and the angle of the incoming and
outgoing debris stream are calculated to be

$R_{I}(M_{\rm
BH},\beta,m_{\star})=\frac{(1+e_{\text{mb}})r_{T}}{\beta\left[1-e_{\text{mb}}\cos(\phi/2)\right]}$
(B3)
$\cos\Theta=\frac{1-2e_{\text{mb}}\cos(\phi/2)+e_{\text{mb}}^{2}\cos\phi}{1-2e_{\text{mb}}\cos(\phi/2)+e_{\text{mb}}^{2}}.$
(B4)

Assuming totally inelastic collision of the debris streams and using the
conservation of momentum, the debris speeds before collision $v_{i}$ and after
collision $v_{f}$ are related as

$v_{f}=v_{i}\cos\left(\Theta/2\right)$ (B5)

where $v_{i}$ can be calculated using the conservation of energy:

$-\frac{GM_{\rm BH}}{2a_{\text{mb}}}=-\frac{GM_{\rm
BH}}{R_{I}}+\frac{1}{2}v_{i}^{2}.$ (B6)

We can then obtain the specific energy loss in the first self-crossing for the
calculation of disk formation efficiency

$\Delta E_{\text{first}}(M_{\rm
BH},\beta,m_{\star})=\left|\frac{1}{2}v_{f}^{2}-\frac{1}{2}v_{i}^{2}\right|=\frac{1}{2}v_{i}^{2}\sin^{2}\left(\Theta/2\right).$
(B7)

##  Appendix C Supplementary Information on Loss Cone Dynamics

In classical LC calculations, the commonly used variables are defined as: the
specific energy $E$, the specific angular momentum $J$, the scaled
dimensionless angular momentum $\mathcal{R}=J^{2}/J_{C}^{2}$, where $J_{C}$ is
the specific angular momentum of a circular orbit. The orbital period $P(E)$
and the stellar distribution function $f(E,\mathcal{R})$ can be derived using
these variables (e.g., Merritt, 2013; Stone & Metzger, 2016). Considering $f$
outside of the disruption zone $(\mathcal{R}>\mathcal{R}_{\text{LC}})$, where
$\mathcal{R}_{\text{LC}}$ is defined as the critical angular momentum at which
stars would inevitably by tidally disrupted if their
$\mathcal{R}<\mathcal{R}_{\text{LC}}$:

$\lim_{E\ll GM_{\rm BH}/r_{T}}\mathcal{R}_{\text{LC}}(E,M_{\rm
BH},m_{\star})\approx\frac{4Er_{T}}{GM_{\rm BH}}\ll 1$ (C1)

it could be deduced that the mean $\mathcal{R}$-integrated distribution
function has the limiting behaviour of $\mathcal{R}\approx 1$:

$\bar{f}(E)=\frac{\int^{1}_{\mathcal{R}_{\text{LC}}}f(E,\mathcal{R})\rm{d}\mathcal{R}}{\int^{1}_{\mathcal{R}_{\text{LC}}}\rm{d}\mathcal{R}}\approx
f(E,1)$ (C2)

The limit $E\ll GM_{\rm BH}/r_{T}$ is used here since the majority of stars
approach in elliptical orbits with semi-major axes $r_{a}\gg r_{T}$.

The rate calculation in this work follows closely that in Pfister et al.
(2021), which have shown in detail the derivations of TDE rate by assuming:

  * •

A stellar population described by the Kroupa stellar mass function the MS
stars (Kroupa, 2001):

$\frac{\phi(m_{\star})}{M_{\odot}^{-1}}=\phi_{0}\begin{cases}\left(\displaystyle\frac{m_{\star}}{0.5\,M_{\odot}}\right)^{-1.3}&\text{for}\,\,0.08\,M_{\odot}\leq
m_{\star}\leq 0.5\,M_{\odot}\\\
\left(\displaystyle\frac{m_{\star}}{0.5\,M_{\odot}}\right)^{-2.3}&\text{for}\,\,10\,M_{\odot}\leq
m_{\star}\leq 0.5\,M_{\odot}\\\ 0&\text{else}\end{cases}$ (C3)

The normalization constant $\phi_{0}$ ensures that
$\displaystyle\int^{\infty}_{0}\phi(m_{\star})\rm{d}m_{\star}=1$.

  * •

A power-law stellar density profile near the SMBH:

$\rho(r)=\rho_{0}\left(\frac{r}{r_{\text{inf}}}\right)^{-\alpha}$ (C4)

where $r_{\rm inf}$ is the SMBH radius of influence, defined as the radius
where $2\pi\displaystyle\int^{r_{\rm inf}}_{0}u^{2}\rho(u)\rm{d}u=M_{\rm BH}$.
We assume an isothermal profile $(\alpha=2)$, which leads to

$r_{\text{inf}}=\frac{GM_{\rm BH}}{2\sigma^{2}}$ (C5)

where $\sigma$, the velocity dispersion of the galaxy, is constant in the case
of an isothermal sphere.

  * •

An $M_{\rm BH}-\sigma$ black hole mass scaling relation by Merritt & Ferrarese
(2001):

$\sigma=200\left(\frac{M_{\rm BH}}{(1.48\pm 0.24)\times
10^{8}\,M_{\odot}}\right)^{1/(4.65\pm 0.48)}\,\text{km s}^{-1}$ (C6)

Based on the above assumptions, the differential TDE rate as a function of
$(M_{\rm BH},\beta,m_{\star})$ after careful consideration of stellar
diffusion through two-body gravitational scattering is given by Pfister et al.
(2021) and Strubbe (2011):

$\displaystyle\mathcal{G}(E,M_{\rm BH},\beta,m_{\star})\,\,=\,\,$
$\displaystyle\frac{f(E)}{1+q^{-1}\xi\ln\left(\mathcal{R}^{-1}_{\text{LC}}\right)}\times$
(C7)
$\displaystyle\left[1-2\sum^{\infty}_{m=1}\frac{e^{-\alpha_{m}^{2}q/4}}{\alpha_{m}}\frac{J_{0}\left(\alpha_{m}\beta^{-1/2}\right)}{J_{1}(\alpha_{m})}\right],$
$\displaystyle\xi\,\,=\,\,$ $\displaystyle
1-4\sum^{\infty}_{m=1}\frac{e^{-\alpha_{m}^{2}q/4}}{\alpha_{m}^{2}},$ (C8)
$\displaystyle q(E,M_{\rm BH},m_{\star})\,\,=\,\,$
$\displaystyle\frac{P(E)\bar{\mu}(E,\langle
m_{\star}^{2}\rangle^{1/2})}{\mathcal{R}_{\text{LC}}},$ (C9)

where $q$ and $\bar{\mu}$ are the LC filling factor and orbit-averaged
diffusion coefficient respectively. $J_{0}$ and $J_{1}$ are the Bessel
functions of the first kind of order zero and one respectively, and
$\alpha_{m}$ is the $m$-th zero of $J_{0}$.

By assuming a power-law stellar density profile around the SMBH, the stellar
distribution function $f$ and the LC re-filling factor $q$ could be computed
analytically rather than numerically in the usually complicated scenarios. The
expressions are as follows (Magorrian & Tremaine, 1999; Strubbe, 2011;
Merritt, 2013; Stone & Metzger, 2016):

$\displaystyle f(E)$
$\displaystyle=\left(2\pi\sigma_{\text{inf}}^{2}\right)^{-3/2}\frac{\rho_{0}}{\langle
m_{\star}\rangle}\frac{\gamma(\alpha+1)}{\gamma(\alpha-1/2)}\left(\frac{E}{\sigma_{\text{inf}}^{2}}\right)^{\alpha-3/2},$
(C10) $\displaystyle q(E,m_{\star})$
$\displaystyle=\nu\left(\frac{E}{\sigma_{\text{inf}}^{2}}\right)^{\alpha-4},$
(C11) $\displaystyle\nu(m_{\star})$
$\displaystyle=\frac{8\sqrt{\pi}}{3}(3-\alpha)\frac{\gamma(\alpha+1)}{\gamma(\alpha-1/2)}\left[\frac{5}{32(\alpha-1/2)}+\frac{3I_{B}(1/2,\alpha)-I_{B}(3/2,\alpha)}{4\pi}\right]\left(\frac{G\langle
m_{\star}^{2}\rangle}{\sigma_{\text{inf}}^{2}r_{T}}\right)\ln\Lambda,$ (C12)

where $\sigma_{\text{inf}}=\sqrt{GM_{\rm BH}/r_{\text{inf}}}$ is the velocity
dispersion at the radius of influence of BH with mass $M_{\rm BH}$,
$\ln\Lambda=\ln(0.4M_{\rm BH}/\langle m_{\star}\rangle)$ is the Coulomb
logarithm, $\gamma$ is the Euler-Gamma function, and $I_{B}$ is an expression
defined in Strubbe (2011):

$I_{B}\left(\frac{n}{2},\alpha\right)=\int^{1}_{0}t^{-\frac{n+1}{2}}(1-t)^{3-\alpha}B\left(t,\frac{n}{2},\alpha-\frac{1}{2}\right)\rm{d}t,$
(C13)

where
$B\left(t,\displaystyle\frac{n}{2},\alpha-\displaystyle\frac{1}{2}\right)$ is
the incomplete Euler-Beta function.

Initially, the orbit-averaged diffusion coefficient $\bar{\mu}$ in Eq.C9
should be a function of the specific energy of the disrupted star $E$, the
mass of the star $m_{\star}$ and the mass of a background star
$m_{\text{bg}}$, assuming a monochromatic distribution of stellar mass.
Magorrian & Tremaine (1999) show that the derived rate using a stellar
population is the same as a monochromatic distribution of stellar masses of
$m_{\text{bg}}=\langle m_{\star}^{2}\rangle^{1/2}$, where $\langle
m_{\star}^{2}\rangle=\int m_{\star}^{2}\phi(m_{\star})\rm{d}m_{\star}$.
Pfister et al. (2021) ignores the effect of mass segregation in which this
approximation is no longer valid when stellar distribution becomes a function
of position.

In order to apply a semi-analytical calculation of TDE rate based on loss cone
dynamics onto a general galaxy system, numerous assumptions and approximations
have to be made. For a more accurate estimate of the TDE rate in a specific
galaxy, characteristics of the galaxy, such as the stellar density profile and
the host black hole mass, shall not simply be assumed to follow Eq.C3, C4, and
C6.

##  Appendix D Effect from Uncertainty in Black Hole Mass Function

One might suspect that the suppression of the TDE host $M_{\rm BH}$
demographics at low-mass end could result from the large uncertainty of BHMF
at $M_{\rm BH}\lesssim 10^{6}\,M_{\odot}$. For example, the $2\sigma$
uncertainty regions in Fig.2 of Gallo & Sesana (2019) indicate that the
volumetric density of SMBHs might increase or decrease with decreasing $M_{\rm
BH}$ in this mass range. In the lower-mass end ($M_{\rm BH}\sim
10^{4-5}\,M_{\odot}$), the maximum spread is about
${}^{+0.5\rm{dex}}_{-0.8\rm{dex}}$.

We calculate the corresponding TDE volumetric rates and their host $M_{\rm
BH}$ distributions using the upper and lower 95% confidence regions in Gallo &
Sesana (2019) (Fig.D.1). The overall shapes of the intrinsic and the
$\mathcal{C}$-corrected distributions still remain somewhat similar to that of
Fig.3, despite some boost (upper $2\sigma$) or further suppression (lower
$2\sigma$) at the low-mass ends. It can be seen that even with the lower
$2\sigma$ BHMF, the intrinsic TDE black hole mass distribution still has too
much weight towards the lower side. In order to shift the theoretical
distributions to peak at that of the observed, $\mathcal{C}_{\rm thres}\approx
0.1$ is still needed in both cases.

Figure D.1: Regenerated Fig.3 using the 95% confidence regions of BHMF Fig.2
in Gallo & Sesana (2019). The upper (lower) panel corresponds to using the
BHMF upper (lower) margin of the light yellow areas of Gallo & Sesana (2019)
Fig.2. The distributions with $\mathcal{C}_{\text{thres}}=0.1$ (red solid)
still best replicate the median of the observed TDE samples. The rate
suppression around light SMBHs due to slow disk formation is more dominant
than that by the lower $2\sigma$ of the unconstrained BHMF.