# AT2022cmc: a Tidal Disruption Event with Two-component Jet in a Bondi-
profile Circumnuclear Medium

Chang Zhou Department of Astronomy, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, China  Zi-Pei Zhu Department of
Astronomy, School of Physics, Huazhong University of Science and Technology,
Wuhan, 430074, China  Key Laboratory of Space Astronomy and Technology,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100101, China  Wei-Hua Lei Department of Astronomy, School of Physics,
Huazhong University of Science and Technology, Wuhan, 430074, China  Shao-Yu
Fu Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100101, China  University
of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049,
China  Wei Xie Department of Astronomy, School of Physics and Electronic
Science, Guizhou Normal University, Guiyang, 550001, China  Guizhou Provincial
Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal
University, Guiyang, 550001, China  Dong Xu Key Laboratory of Space Astronomy
and Technology, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100101, China

###### Abstract

A supermassive black hole can launch a relativistic jet when it violently
disrupts a star that passes too close. Such jetted tidal disruption events
(TDEs) are rare and unique tool to investigate quiescent supermassive black
hole, jet physics and circumnuclear environment at high redshift. The newly
discovered TDE AT2022cmc ($z\sim 1.193$) providing rich multi-band (X-ray, UV,
optical, sub-millimeter and radio) data, has been interpreted as the fourth
on-axis jetted TDE. In this work, we constrain the circumnuclear medium (CNM)
density profile with both closure relation (CR) test and detailed forward
shock model fit with Markov chain Monte Carlo (MCMC) approach to the multi-
band (optical, sub-millimeter and radio) data of AT2022cmc. We find that the
CNM density profile of AT2022cmc is $n\propto R^{-k}$ with $k\sim 1.69$,
implying a Bondi accretion in history. Furthermore, our model fit result
suggests a two-component jet in AT2022cmc, indicating a similar jet physics to
well-studied jetted TDE Sw J1644+57.

Tidal disruption (1696)

##  1 Introduction

Tidal disruption events (TDEs) are transient events when a star comes too
close to a supermassive black hole (SMBH) to be torn apart by the black hole
(Rees, 1988; Phinney, 1989). TDEs have been observed in various wavelengths
(radio, UV/optical, X-rays, even $\gamma$-ray in some jetted TDEs), providing
a new probe to study SMBH of quiescent galaxies.

The detections of Sw J1644+57 (Bloom et al., 2011; Burrows et al., 2011; Levan
et al., 2011; Zauderer et al., 2011), Sw J2058+05 (Cenko et al., 2012) and Sw
J1112-82 (Brown et al., 2015) suggested that at least some TDEs can launch a
relativistic jet toward Earth. Recent study on IGR J12580+0134 revealed the
first TDE with an off-axis relativistic jet (Lei et al., 2016; Yuan et al.,
2016). The on-axis jetted TDEs are rare, but are raido-loud (Alexander et al.,
2020) and have special use in unveiling cosmological quiescent SMBHs, jet
physics and density profile of pre-existing CNM at $z>1$.

The CNM density profile ($n\propto R^{-k}$) provides key diagnostics of
accretion history of the SMBH. Currently, there are three special types of CNM
density profiles: 1) Sgr $\rm A^{*}$-like ($k=1$), namely, the CNM density
profile consistent with Sgr $\rm A^{*}$ whose $k=1$ (Baganoff et al., 2003),
AT2019dsg is one of the TDEs satisfying this distribution (Stein et al.,
2021); 2) Bondi-like ($k=3/2$), consistent with Bondi accretion whose $k=3/2$,
Sw J1644+57 is one of the TDEs satisfying this distribution (Eftekhari et al.,
2018); 3) ASASSN-14li-like ($k=5/2$), there are several TDEs satisfying this
distribution, such as ASASSN-14li (Alexander et al., 2016), CNSSJ0019
(Anderson et al., 2020) and Arp 299 (Mattila et al., 2018). Such a steep
profile is not expected for spherically symmetric accretion, but appears in
some models of super-Eddington accretion flows (Coughlin & Begelman, 2014,
ZEBRAs). The accretion history of a dormant SMBH especially at $z>1$ is of
great interest.

Recently, a very luminous TDE AT2022cmc has been discovered in optical at
$z=1.193$. Its unusual X-ray behavior (peak luminosity of $\geq 3\times
10^{47}\rm erg\ s^{-1}$, short variability timescales of $\sim 1000$ s, and
duration of $>30$ days) and long-lived radio/mm data (consistent with
synchrotron radiation) indicated that AT2022cmc is an on-axis jetted TDE
(Andreoni et al., 2022; Pasham et al., 2023). The rich multi-band
observational data offer a good opportunity to explore the properties of SMBH
and the CNM density distribution. For a jetted TDE, the multi-band afterglow
emissions are produced by the external shock when the jet interacts with the
CNM. Therefore, we can use the afterglow observations of TDEs to infer the CNM
density at parsec or even sub-parsec scales (Alexander et al., 2016). As the
jet expands, the afterglow emissions at different epochs can be used to
construct the CNM density profile. Such constraints are extremely valuable, as
these scales are not directly resolvable at any wavelength with current
facilities at the distance of most TDE hosts.

Based on equipartition analysis (Chevalier, 1998; Barniol Duran et al., 2013)
and afterglow modeling, Matsumoto & Metzger (2023) roughly estimated the
density profile of AT2022cmc as $k\simeq 1.5-2$, which is similar to Sw
J1644+57 and generally consistent with a Bondi-like accretion. However, a well
constraint on density profile index $k$ is expected for better understanding
the accretion history of SMBH. On the other hand, the detailed studies on the
X-ray and radio observations showed a two-component jet in of Sw J1644+57
(Wang et al., 2014; Liu et al., 2015; Mimica et al., 2015). The investigation
on the jet structure of AT2022cmc is also desired.

In this paper, instead of the equipartition method, we employ a forward shock
(FS) model similar to that used in gamma-ray burst (GRB) afterglows. Different
with Matsumoto & Metzger (2023) and Yao et al. (2023), we first obtain the
closure relations (CRs) of jet-CNM interaction with an arbitrary CNM density
profile, and analyze the radio light curve and spectrum with the closure
relations. In this way, we can get a rough estimate on the CNM density profile
index $k$. Then, we perform a detailed forward shock model fit to the multi-
band (optical, sub-millimeter and radio) data of AT2022cmc with Markov chain
Monte Carlo (MCMC) method, and obtain a better constraint on the profile index
$k$. Finally, we adopt a two-component jet model as in Wang et al. (2014) but
modified with an arbitrary CNM density profile to fit the data.

Like Sw J1644+57, the early rapidly-declining X-rays are also assumed to arise
from the internal emission of the jet (Matsumoto & Metzger, 2023). Following
Andreoni et al. (2022), the blue optical plateau phase should be thermal
emission unrelated to the jet-CNM interaction. The late-time r-band ($\geq 5$
days) data is likely a thermal origin similar to other optically-selected
TDEs. Therefore, for the optical data, we only use the early r-band ($<5$
days) data in our multi-band afterglow fit, as did in Matsumoto & Metzger
(2023).

The paper is organized as follows: The analysis with closure relations are
presented in Section 2. We constrain the CNM density profile index $k$ of
AT2022cmc by using the MCMC fit to the radio, sub-millimeter and optical data
in Section 3. In Section 4, we discuss the results, e.g., the accretion
history, two-component jet origin and detectability of AT2022cmc-like events.
The main conclusions are summarised in Section 5. A standard cosmology model
is adopted with $H_{0}=67.3\ \rm{km\cdot s^{-1}Mpc^{-1}}$, $\Omega_{M}$=0.315,
$\Omega_{\Lambda}$=0.685 (Planck Collaboration et al., 2014).

##  2 Closure Relation Analysis

The afterglow (radio, sub-millimeter and early optical) of AT2022cmc can be
interpreted with synchrotron emission from the forward shock (FS) of a
relativistic jet propagating into the CNM (Matsumoto & Metzger, 2023). The
synchrotron flux can be described by a series of power-law segments
$F_{\nu}\propto t^{-\alpha}\nu^{-\beta}$ (Sari et al., 1998; Gao et al., 2013;
Zhang, 2018). In such model, the type of CNM can be tested with the closure
relations (CRs, relations between the temporal indices $\alpha$ and spectral
indices $\beta$), as did in gamma-ray bursts (GRBs) (Gao et al., 2013). Gao et
al. (2013) only presented the CRs for constant-density ($k=0$) and wind type
($k=2$). In this work, to study the CNM type of AT2022cmc, we need to obtain
the CRs for an arbitrary density profile.

To do this, we consider a relativistic thin shell with energy $E_{\rm K,iso}$,
initial Lorentz factor $\Gamma_{0}$, and opening angle $\theta_{\rm j}$
expanding into the pre-existing CNM with density $n_{1}$ (Rees & Meszaros,
1992; Mészáros & Rees, 1997; Sari et al., 1998; Zhang, 2018),

$n_{1}(R)=n_{18}\left(\frac{R}{10^{18}\rm cm}\right)^{-k}=AR^{-k},$ (1)

where $n_{18}$ is the CNM density at distance $R=10^{18}\rm cm$. We adopt an
analytical description of the main properties of the evolution and emission of
the FS from the jet-CNM interaction.

Generally, the evolution of the jet includes four phases. The first is a
coasting phase, in which we have $\Gamma(t)\simeq\Gamma_{0}$. In the second
phase, the shell starts to decelerate when the mass $m$ of the CNM swept by
the FS is about $1/\Gamma_{0}$ of the rest mass in the ejecta $M_{\rm ej}$.
The shell then approaches the Blandford & McKee (1976) self-similar evolution.
The Blandford & McKee (1976) solution for arbitrary $k$ density profile is
given by,

$\Gamma(t)\simeq\left(\frac{(17-4k)E_{\rm K,iso}}{4^{5-k}(4-k)^{3-k}\pi
Am_{\rm p}c^{5-k}t^{3-k}}\right)^{\frac{1}{2(4-k)}},\ \ \
R(t)\simeq\left(\frac{(17-4k)(4-k)E_{\rm K,iso}t}{4\pi Am_{\rm
p}c}\right)^{\frac{1}{(4-k)}},$ (2)

where $m_{\rm p}$ is proton mass. Later, as the ejecta is decelerated to the
post-jet-break phase when the $1/\Gamma$ cone becomes larger than $\theta_{\rm
j}$. Finally, the blastwave enters the Newtonian phase when it has swept up
the CNM with the total rest mass energy comparable to the energy of the
ejecta. The dynamics is described by the well-known Sedov-Taylor solution.

During the dynamical evolution of the FS, electrons are believed to be
accelerated at the shock front to a power-law distribution $N(\gamma_{\rm
e})\propto\gamma_{\rm e}^{-p}$. Assuming a fraction $\epsilon_{\rm e}$ of the
shock energy $e_{2}=4\Gamma^{2}n_{1}m_{\rm p}c^{2}$ is distributed into
electrons, this defines the minimum injected electron Lorentz factor,

$\gamma_{\rm m}=\frac{p-2}{p-1}\epsilon_{\rm e}(\Gamma-1)\frac{m_{\rm
p}}{m_{\rm e}}$ (3)

where $m_{\rm e}$ is electron mass. We also assume that a fraction
$\epsilon_{\rm B}$ of the shock energy is in the magnetic field generated
behind the shock. This gives the comoving magnetic field

$B=(32\pi m_{\rm p}\epsilon_{\rm B}n_{1})^{1/2}c.$ (4)

The synchrotron power and characteristic frequency from an electron with
Lorentz factor $\gamma_{\rm e}$ are

$P(\gamma_{\rm e})\simeq\frac{4}{3}\sigma_{\rm T}c\Gamma^{2}\gamma_{\rm
e}^{2}\frac{B^{2}}{8\pi},$ (5) $\nu(\gamma_{\rm e})\simeq\Gamma\gamma_{\rm
e}^{2}\frac{q_{\rm e}B}{2\pi m_{\rm e}c},$ (6)

where $\sigma_{\rm T}$ is the Thomson cross-section, $q_{\rm e}$ is electron
charge. The peak of the spectra power occurs at $\nu(\gamma_{\rm e})$, and

$P_{\nu,{\rm max}}\simeq\frac{P(\gamma_{\rm e})}{\nu(\gamma_{\rm
e})}=\frac{m_{\rm e}c^{2}\sigma_{\rm T}}{3q_{\rm e}}\Gamma B.$ (7)

By equating the lifetime of electron to the time $t$, one can define a
critical electron Lorentz factor $\gamma_{\rm c}$

$\gamma_{\rm c}=\frac{6\pi m_{\rm e}c}{\Gamma\sigma_{\rm T}B^{2}t},$ (8)

the electron distribution shape should be modified for $\gamma_{\rm
e}>\gamma_{\rm c}$ when cooling due to synchrotron radiation becomes
significant. Accounting for the radiative cooling and the continuous injection
of new accelerated electrons at the shock front, one expects a broken power-
law energy spectrum of them, which leads to a multi-segment broken power-law
radiation spectrum separated by three characteristic frequencies at any epoch
(Gao et al., 2013; Zhang, 2018). The first two characteristic frequencies
$\nu_{\rm m}$ and $\nu_{\rm c}$ in the synchrotron spectrum are defined by the
two electron Lorentz factors $\gamma_{\rm\,m}$ and $\gamma_{\rm\,c}$,
respectively. The third characteristic frequency is the self-absorption
frequency $\nu_{\rm a}$, below which the synchrotron photons are self-absorbed
(self-absorption optical depth larger than unity).

The maximum flux density is $F_{\nu,{\rm max}}=N_{\rm e,2}P_{\nu,{\rm
max}}/4\pi D^{2}$, where $N_{\rm e,2}=\int 4\pi R^{2}n_{1}dR$ is the total
number of electrons in shocked CNM and $D$ is the distance of the source.

In this work, following Matsumoto & Metzger (2023), the early r-band data are
attributed to the synchrotron emissions of the FS. The peak of the early
optical phase ($\leq 1$ day) can be reasonably explained as the onset of the
deceleration phase. We thus focus on the second phase, i.e., self-similar
deceleration phase. The spectra is likely in $\nu_{\rm a}<\nu_{\rm m}<\nu_{\rm
c}$ regime. From Equation (2), $\Gamma\propto t^{-\frac{(3-k)}{2(4-k)}}$ and
$R\propto t^{\frac{1}{4-k}}$, one has the scalings for the FS spectra
parameters in this phase as (for $\nu_{\rm a}<\nu_{\rm m}<\nu_{\rm c}$)

$\nu_{\rm a}\propto t^{-\frac{3k}{5(4-k)}},\ \ \nu_{\rm m}\propto t^{-3/2},\ \
\nu_{\rm c}\propto t^{-\frac{4-3k}{2(4-k)}},\ \ F_{\nu,{\rm max}}\propto
t^{-\frac{k}{2(4-k)}}.$ (9)

We can then obtain,

$\displaystyle F_{\nu}=F_{\nu,{\rm
max}}\times\left\\{\begin{tabular}[]{l}$\left(\frac{\nu_{\rm a}}{\nu_{\rm
m}}\right)^{\frac{1}{3}}\left(\frac{\nu}{\nu_{\rm a}}\right)^{2}\propto
t^{\frac{2}{4-k}}\nu^{2},\ \ \ \nu<\nu_{\rm a}<\nu_{\rm m}<\nu_{\rm c}$\\\
$\left(\frac{\nu}{\nu_{\rm m}}\right)^{\frac{1}{3}}\propto
t^{\frac{2-k}{4-k}}\nu^{\frac{1}{3}},\ \ \ \nu_{\rm a}<\nu<\nu_{\rm
m}<\nu_{\rm c}$\\\ $\left(\frac{\nu}{\nu_{\rm
m}}\right)^{-\frac{p-1}{2}}\propto
t^{-\frac{12p-12+5k-3kp}{16-4k}}\nu^{-\frac{p-1}{2}},\ \ \ \nu_{\rm
a}<\nu_{\rm m}<\nu<\nu_{\rm c}$\\\ $\left(\frac{\nu_{\rm c}}{\nu_{\rm
m}}\right)^{-\frac{p-1}{2}}\left(\frac{\nu}{\nu_{\rm
c}}\right)^{-\frac{p}{2}}\propto t^{-\frac{3p-2}{4}}\nu^{-\frac{p}{2}},\ \ \
\nu_{\rm a}<\nu_{\rm m}<\nu_{\rm c}<\nu$\end{tabular}\right.$ (14)

Table 1:  Closure relations (CRs) for different CNM profile in the $\nu_{a}<\nu_{m}<\nu_{c}$ spectral regime. |  | $\beta$ | $\alpha$ | $\alpha(\beta)$
---|---|---|---|---
k=0, slow cooling | $\nu<\nu_{a}$ | -2 | $-\frac{1}{2}$ | $\frac{1}{4}\beta$
$\nu_{a}<\nu<\nu_{m}$ | $-\frac{1}{3}$ | $\frac{1}{2}$ | $-\frac{3}{2}\beta$
$\nu_{m}<\nu<\nu_{c}$ | $\frac{p-1}{2}$ | $\frac{3(p-1)}{4}$ | $\frac{3}{2}\beta$
$\nu_{c}<\nu$ | $\frac{p}{2}$ | $\frac{3p-2}{4}$ | $\frac{3\beta-1}{2}$
k=1, slow cooling | $\nu<\nu_{a}$ | -2 | $-\frac{2}{3}$ | $\frac{1}{3}\beta$
$\nu_{a}<\nu<\nu_{m}$ | $-\frac{1}{3}$ | $\frac{1}{3}$ | -$\beta$
$\nu_{m}<\nu<\nu_{c}$ | $\frac{p-1}{2}$ | $\frac{9p-7}{12}$ | $\frac{9\beta+1}{6\beta}$
$\nu_{c}<\nu$ | $\frac{p}{2}$ | $\frac{3p-2}{4}$ | $\frac{3\beta-1}{2}$
k=1.5, slow cooling | $\nu<\nu_{a}$ | -2 | $-\frac{4}{5}$ | $\frac{2}{5}\beta$
$\nu_{a}<\nu<\nu_{m}$ | $-\frac{1}{3}$ | $\frac{1}{5}$ | $-\frac{3}{5}\beta$
$\nu_{m}<\nu<\nu_{c}$ | $\frac{p-1}{2}$ | $\frac{15p-9}{20}$ | $\frac{15\beta+3}{10}$
$\nu_{c}<\nu$ | $\frac{p}{2}$ | $\frac{3p-2}{4}$ | $\frac{3\beta-1}{2}$
k=2.0, slow cooling | $\nu<\nu_{a}$ | -2 | -1 | $\frac{1}{2}\beta$
$\nu_{a}<\nu<\nu_{m}$ | $-\frac{1}{3}$ | 0 | 0
$\nu_{m}<\nu<\nu_{c}$ | $\frac{p-1}{2}$ | $\frac{3p-1}{4}$ | $\frac{3\beta+1}{2}$
$\nu_{c}<\nu$ | $\frac{p}{2}$ | $\frac{3p-2}{4}$ | $\frac{3\beta-1}{2}$
k=2.5, slow cooling | $\nu<\nu_{a}$ | -2 | $-\frac{2}{3}$ | $\frac{1}{3}\beta$
$\nu_{a}<\nu<\nu_{m}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$ | $\beta$
$\nu_{m}<\nu<\nu_{c}$ | $\frac{p-1}{2}$ | $\frac{9p+1}{12}$ | $\frac{9\beta+5}{6\beta}$
$\nu_{c}<\nu$ | $\frac{p}{2}$ | $\frac{3p-2}{4}$ | $\frac{3\beta-1}{2}$

The CRs for different CNM profile in the $\nu_{a}<\nu_{m}<\nu_{c}$ spectral
regime are listed in Table 1. Inspecting the radio spectrum as given by
Andreoni et al. (2022), the low frequency bands $<20$ GHz should be in the
$\nu<\nu_{\rm a}$ regime with spectral index $\beta=-2$ and temporal index
$\alpha=-2/(4-k)$. The temporal index for 15.5 GHz is $\alpha\simeq-0.79$
obtained by Pasham et al. (2023), which is close to the $k=1.5$ case
($\alpha=-4/5$) by inspecting Table 1. Applying the closure relation, i.e.,
equating $2/(4-k)=0.79$, one can find $k\simeq 1.47$, preferring a bondi-like
profile. This result is also consistent with the equipartition analysis result
of Matsumoto & Metzger (2023), and is similar to that implied by modeling the
early radio emission from the first jetted TDE Sw J1644+57 (Metzger et al.,
2012; Berger et al., 2012).

##  3 Constrain the CNM Profile with FS Model Fit

To get a better constraint on the CNM profile index $k$, we employ a numerical
code PyFS111https://github.com/leiwh/PyFS for FS model described in Wang et
al. (2014), Lei et al. (2016) and Zhu et al. (2023). The dynamical evolution
of the shell are calculated numerically using a set of hydrodynamical
equations (Huang et al., 2000)

$\frac{dR}{dt}=\beta c\Gamma(\Gamma+\sqrt{\Gamma^{2}-1}),$ (15)
$\frac{dm}{dR}=2\pi R^{2}(1-\cos\theta_{\rm j})n_{1}m_{\rm p},$ (16)
$\frac{d\Gamma}{dm}=-\frac{\Gamma^{2}-1}{M_{\rm ej}+2\Gamma m},$ (17)

where $R$ and $t$ are the radius and time of the event in the burster frame,
$m$ is the swept-up mass, $M_{\rm ej}=E_{\rm K,iso}(1-\cos\theta_{\rm
j})/2(\Gamma_{0}-1)c^{2}$ is the ejecta mass, and
$\beta=\sqrt{\Gamma^{2}-1}/\Gamma$. The density profile is described by a
power-law in radius $R$ as Equation (1). The synchrotron spectra of the jet
are calculated following the standard broken-power-law spectral model
separated by the three characteristic frequencies ($\nu_{\rm a},\nu_{\rm
m},\nu_{\rm c}$) as described in Section 2 (Gao et al., 2013; Zhang, 2018, for
a detailed review).

Now, we can numerically calculate the synchrotron light curve and spectrum
from the forward shock for different CNM density profile with our PyFS code.
Markov chain Monte Carlo (MCMC) with PyFS is adopted for multi-band fitting to
place constraints on the model parameters. The MCMC fitting is done using the
python package emcee (Foreman-Mackey et al., 2013), which utilizes a group of
parallel-tempered affine invariant walkers to explore the parameter space.

Eight free parameters are considered in the fitting, i.e., the isotropic
kinetic energy $E_{\rm K,iso}$, the initial lorentz factor $\Gamma_{0}$ of the
jet, the jet opening angle $\theta_{\rm j}$, the CNM profile parameter $k$,
the number density of CNM medium $n_{18}$ at $R=10^{18}\rm cm$, the electron
distribution power-law index $p$, the energy fraction in electrons
$\epsilon_{\rm e}$ and in magnetic field $\epsilon_{\rm B}$. The viewing angle
$\theta_{\rm obs}$ is fixed to 0 in the fit due to the on-axis jet as
suggested by Andreoni et al. (2022) and Pasham et al. (2023), and also by the
nearly zero observed polarization degrees (Cikota et al., 2023).

The prior range of these parameters are given based on the estimations from
observations. First, the CR studies in Section 2 suggest the CNM density
profile index of $k\simeq 1.5$, favoring a Bondi-like profile. However, other
CNM density profile types ($k=1$ or $5/2$) can not be completely ruled out by
such CR study. We therefore set the prior range of $k$ as $1-3$.

AT2022cmc is bright in X-ray with peak luminosity $\geq 3\times 10^{47}\rm
erg\ s^{-1}$, which is comparable with the well-studied relativistic jetted
TDE Sw J1644+57 (Andreoni et al., 2022; Matsumoto & Metzger, 2023). The X-ray
light curve shows rapid decay as $L_{\rm X,iso}\simeq 3\times 10^{47}\rm erg\
s^{-1}(t/5{\rm days})^{-2}$, indicating an isotropic equivalent energy of
$E_{\rm X,iso}>10^{53}{\rm erg}$. The kinetic energy $E_{\rm K,iso}$ can be
larger or smaller than this X-ray energy $E_{\rm X,iso}$ (depending on the
efficiency of converting jet power to X-ray radiation), we take the prior
range of isotropic kinetic jet energy $E_{\rm K,iso}$ as $10^{52}-10^{54}{\rm
erg}$.

From the day-timescale variability of the radio data, Rhodes et al. (2023)
infer that the bulk Lorentz factor of the jet is $\,\gtrsim 8$. By
extrapolation of the equipartition analysis results, Matsumoto & Metzger
(2023) obtained $\Gamma_{0}\simeq 5$ for narrow jet model or $2.5$ for wide
jet model. Using the synchrotron afterglow model of the relativistic jet, a
Lorentz factor of $12$ was obtained by Andreoni et al. (2022). In Pasham et
al. (2023), the bulk Lorentz factor of the jet is $86$ for the emission model
with only synchrotron and synchrotron self-Compton (SSC), and $5$ for the
model with external Compton (EC) included. The prior range $1.1-50$ is used
for $\Gamma_{0}$ in the MCMC fit.

The jet opening angle is poorly constrained from observations. It is
reasonable to assume $\theta_{\rm j}\sim 1/\Gamma_{0}\sim 0.2$ (0.4) if
$\Gamma_{0}\simeq 5$ (2.5) is adopted. Another choice $\theta_{\rm j}\sim 0.1$
is motivated by the modeling of Sw J1644+57 (Metzger et al., 2012). We use
prior range $0.052-0.35$ for $\theta_{\rm j}$.

Taking the peak ($\sim 1$ day) of the early optical phase (coincides with the
peak of X-ray) as the onset of deceleration phase, one finds the deceleration
time in the engine’s rest frame as $t_{\rm dec}\simeq 1\ {\rm day}/(1+z)\sim
0.5\ {\rm day}$ (Matsumoto & Metzger, 2023). The deceleration radius is given
by $R_{\rm dec}\simeq 2\Gamma_{0}^{2}ct_{\rm dec}\sim 6.5\times 10^{16}{\rm
cm}\ (\Gamma_{0}/5)^{2}(t_{\rm dec}/0.5\ {\rm day})$. We thus can estimate the
jet kinetic energy $E_{\rm K,iso}\simeq\frac{4\pi}{3-k}m_{\rm
p}c^{2}\Gamma_{0}^{2}R_{\rm dec}^{3}n_{1}(R_{\rm dec})$, where $n_{1}(R_{\rm
dec})$ is the density at $R_{\rm dec}$. The CNM density at $R=10^{18}\rm cm$
can thus be given by

$n_{18}\simeq 19\times{\rm cm^{-3}}\left(\frac{E_{\rm K,iso}}{10^{53}{\rm
erg}}\right)\left(\frac{\Gamma_{0}}{5}\right)^{-5}\left(\frac{t_{\rm
dec}}{0.5\ {\rm day}}\right)^{-1.5},$ (18)

where $k=1.5$ is taken as suggested by our CR studies in Section 2. We use a
relatively wide prior range $10^{-3}-200\ \rm cm^{-3}$ for $n_{18}$ by
considering the uncertainty in the above estimations.

The multi-band fit to early optical non-thermal component finds a spectral
index $\beta=-1.32\pm 0.18$ (Andreoni et al., 2022), leading to an electron
energy index of $p=2.64\pm 0.36$ (Matsumoto & Metzger, 2023). We thus fix $p$
to 2.64 in the fit.

For the two energy fractions $\epsilon_{\rm e}$ and $\epsilon_{\rm B}$, we
adopt the typical value ranges. However, the low polarization (consistent with
zero) favor the jet with low magnetic field energy density (Cikota et al.,
2023). We therefore use a relatively smaller value of the lower-limit for
$\epsilon_{\rm B}$. The descriptions, prior type and prior range for each
model parameter are presented in Table 2.

The X-ray light curves are highly variable (on timescales of 1000 s), which
are assumed to arise from the internal emission of the jet as in Sw J1644+57
(Matsumoto & Metzger, 2023). The optical and UV observations revealed a fast-
fading red “flare” ($<5$ days) with a peak luminosity $\nu L_{\nu}\simeq
10^{46}\rm{erg\ s^{-1}}$ that transitioned quickly to a slow blue “plateau” of
luminosity $\sim 10^{45}\rm{erg\ s^{-1}}$ lasting at least a couple of months,
suggesting a two component emission model (Andreoni et al., 2022): jet
component (early r-band) and accretion disk thermal component (blue optical
plateau with a blackbody temperature of $\simeq(2-4)\times 10^{4}$ K). The
late-time r-band ($\geq 5$ days) data is likely a thermal origin similar to
other optically-selected TDEs. The radio counterpart was identified in Karl G.
Jansky Very Large Array (VLA) observations on 15 February 2022 (four days
after the ZTF trigger). The radio/sub-millimeter observations display a
typical synchrotron self-absorption spectrum (Andreoni et al., 2022; Pasham et
al., 2023). As studied in Andreoni et al. (2022) and Matsumoto & Metzger
(2023), the radio, sub-millimeter and early r-band optical emissions should
originate from the jet-CNM interactions. Therefore, for our MCMC fitting with
PyFS, we use the sub-millimeter/radio (7.0 GHz, 8.5 GHz, 10.5 GHz, 11.5GHz,
15.5 GHz, 17.4GHz, 33.5 GHz, 86 GHz, 102 GHz, 225 GHz and 350 GHz), and the
early r-band optical ($<5$ days) data.

We performed a parameter search with 64 walkers over 20000 iterations,
discarding the first 10000 as burn-in steps. The contour plot of the model
parameters are shown in Fig. 1. The best fit of each parameter is given in
Table 2 as: $k=1.69$, $n_{18}=6.02\ {\rm cm^{-3}}$, $E_{\rm K,iso}=3.47\times
10^{53}$ erg, $\Gamma_{0}=4.76$, $\theta_{\rm j}=5.20\times 10^{-2}$ rad,
$\epsilon_{\rm e}=3.30\times 10^{-1}$ and $\epsilon_{\rm B}=2.48\times
10^{-2}$. In Fig. 2, we have shown the optical, sub-millimeter and radio
afterglow light curves of AT2022cmc along with the best fit model. The CNM
type constrained with our fit is consistent with the CR analysis in Section 2,
both indicating a Bondi-like profile.

Table 2: The input parameters, prior type, prior range, best-fit value of multi-band modelling of AT2022cmc. Parameters | Prior Type | Prior Range | Best fit
---|---|---|---
$k$ | log flat | 1-3 | 1.69
$n_{18}\rm\ (cm^{-3})$ | log flat | $10^{-3}-200$ | 6.02
$E_{\rm K,iso}$ (erg) | log flat | $10^{52}-10^{54}$ | $3.47\times 10^{53}$
$\Gamma_{0}$ | log flat | 1.1-50 | 4.76
$\theta_{\rm j}$ (rad) | flat | $0.052-0.35$ | $5.20\times 10^{-2}$
$\epsilon_{\rm e}$ | log flat | $10^{-3}-0.33$ | $3.30\times 10^{-1}$
$\epsilon_{\rm B}$ | log flat | $10^{-4}-0.33$ | $2.48\times 10^{-2}$
Figure 1: Posterior distribution and parameter constraints obtained using FS
modelling of AT2022cmc with MCMC. The median values with the $1\sigma$ error
regions are also shown in the one-dimensional probability distribution. Figure
2: Optical, sub-millimeter and radio afterglow data of AT2022cmc along with
the best fit with FS code. For convenience, the offset is added when drawing
the light curve.

##  4 Discussion

###  4.1 Accretion History of the SMBH in AT2022cmc

The pre-existing CNM density profile $n_{1}=n_{18}(R/10^{18}{\rm cm})^{-k}$
probed by the FS can provide clues to star formation activity locally in the
star cluster or accretion history of central SMBH.

First, an upper limit on SMBH mass of AT2022cmc $M_{\bullet}<4.7\times
10^{8}M_{\odot}$ is obtained by using the galaxy bulge - black hole mass
relation and the upper limit on the AT2022cmc galaxy mass
($<10^{11.2}M_{\odot}$) (Andreoni et al., 2022). The X-ray light curve shows a
variability timescale of $\delta t\simeq 1000/(1+z){\rm s}\simeq 456\rm s$.
Assuming the variability timescale is defined by the marginally stable orbit
of the accretion disk $R_{\rm ms}/c$, the SMBH mass can be estimated as

$M_{\bullet}\simeq 18\times 10^{6}M_{\odot}\left(\frac{R_{\rm ms}}{5R_{\rm
g}}\right)^{-1}\left(\frac{\delta t}{456\rm s}\right),$ (19)

where $R_{\rm g}=GM_{\bullet}/c^{2}$, and $R_{\rm ms}\simeq 6R_{\rm g}$
($1R_{\rm g}$) for SMBH spin $a_{\bullet}=0$ (1). Under assumption that the
jet is powered by Blandford-Znajek mechanism(Blandford & Żnajek, 1977), one
obtains $a_{\bullet}\geq 0.3$ for the beaming-corrected peak jet power $L_{\rm
jet}=3\times 10^{46}{\rm erg\ s^{-1}}$(Andreoni et al., 2022). Thus,
$a_{\bullet}=0.3$ is adopted in the estimations ($R_{\rm
ms}(a_{\bullet}=0.3)\simeq 5R_{\rm g}$), and we have $M_{\bullet}<18\times
10^{6}M_{\odot}$. On the other hand, given that the TDE thermal optical
emission originates from a quasi-spherical hydrostatic envelope radiating near
the SMBH Eddington limit, the observed optical plateau luminosity $\sim
10^{45}\rm{erg\ s^{-1}}$ of AT2022cmc would require a SMBH mass
$M_{\bullet}\sim 10^{7}M_{\odot}$.

Our best fit result $k=1.69$ (see Section 3) is close to the Bondi accretion
prediction ($k=1.5$) (Quataert et al., 1999). Using the best fit result
$n_{18}\simeq 6.02\ \rm cm^{-3}$, we obtain the mass accretion rate of
$\dot{M}_{\rm acc}\simeq 1.6\times 10^{-3}\dot{M}_{\rm
Edd}m_{\bullet,6}^{-1/2}$, where $\dot{M}_{\rm Edd}$ is the Eddington
accretion rate, and $m_{\bullet,6}\equiv M_{\bullet}/10^{6}M_{\odot}$ is the
mass of SMBH in units of $10^{6}M_{\odot}$. This would support the nucleus of
AT2022cmc being a low-luminosity active galactic nucleus (AGN). The total disk
luminosity of this AGN will be (assuming a thin disk model)

$L_{\rm disk}=(1-E_{\rm ms})\dot{M}_{\rm acc}c^{2}\sim 1.4\times 10^{41}{\rm
erg\ s^{-1}}m_{\bullet,6}^{1/2},$ (20)

where $E_{\rm ms}$ is the specific energy corresponding to the radius of the
marginally stable orbit $R_{\rm ms}$ (Bardeen et al., 1972), and $1-E_{\rm
ms}\simeq 0.06$ (0.42) for SMBH spin $a_{\bullet}=0$ (1). In Equation (16),
$a_{\bullet}=0.3$ is adopted in the estimation of AGN disk luminosity. This
AGN luminosity $L_{\rm disk}\sim 4.4\times 10^{41}{\rm erg\ s^{-1}}$ (if
$M_{\bullet}\sim 10^{7}M_{\odot}$) is much weaker than the observed long-
lasting optical “plateau” luminosity ($\sim 10^{45}\rm{erg\ s^{-1}}$) of
AT2022cmc.

It should be noted that the medium density around SMBH might be asymmetric.
The radio emission from outflow may provide additional information to study
the complex structure, e.g, the cloud (Mou et al., 2022; Perlman et al., 2022;
Bu et al., 2023) and torus (Mou & Wang, 2021). The surrounding dust
distribution at comparable distances can also be indirectly probed via the
detection of infrared dust echoes (Jiang et al., 2016).

The instability developed in the jet-CNM interactions may influence the
afterglow emissions, which should be explored in future MHD simulations.

###  4.2 Two-component Jet Model Fit

Inspecting Fig. 2, we find that the late-time data of AT2022cmc are generally
consistent with the FS model predictions. The early light curves ($t<8$ days)
are, however, poorly reproduced with this one-component jet model. Modeling
the multi-wavelength radio light curves of the well-studied jetted TDE Sw
J1644+57 implied a two-component jet structure (Wang et al., 2014; Liu et al.,
2015; Mimica et al., 2015). The radio spectrum study of AT2022cmc by Matsumoto
& Metzger (2023) suggested an extra jet component or energy injection.
Motivated by these studies, we the try to fit the multi-band light curves of
AT2022cmc with a two-component jet model developed in Wang et al. (2014) but
modified to adapt to an arbitrary CNM density profile. The calculation for the
FS is still based on the PyFS code.

As in Section 3, the MCMC fit with two-component jet model are also performed
with 64 walkers over 20000 iterations. The contour plot of the model
parameters are shown in Fig. 3. We find a fast-component jet with initial
Lorentz factor of $\Gamma_{0,\rm f}=36.61$, $E_{\rm K,iso,f}=5.21\times
10^{53}$ erg, $\theta_{\rm j,f}=4.97\times 10^{-2}$ rad, $p_{\rm f}=2.21$,
$\epsilon_{\rm e,f}=1.08\times 10^{-3}$ and $\epsilon_{\rm B,f}=6.25\times
10^{-3}$, and a slow-component with initial Lorentz factor of $\Gamma_{0,\rm
s}=4.55$, $E_{\rm K,iso,s}=3.97\times 10^{53}$ erg, $\theta_{\rm
j,s}=5.23\times 10^{-2}$ rad, $p_{\rm s}=3.00$, $\epsilon_{\rm e,s}=3.08\times
10^{-1}$ and $\epsilon_{\rm B,s}=1.39\times 10^{-2}$. We get the CNM
environment of AT2022cmc are $k=1.84$ and $n_{18}=7.14\ {\rm cm^{-3}}$. The
best-fit values of the parameters are given in Table 3. The subscripts “f ”
and “s” denote the fast and slow components, respectively.

The best fit results for the optical, sub-millimeter and radio afterglow light
curves are presented in Fig. 4. The fast and slow components are plotted with
dotted and dashed lines, respectively. As one can see, the two-component jet
model can well interpret both the early (by the combination of the fast and
slow components) and late time data (dominated by the slow component see the
dashed lines in Fig. 4) of AT2022cmc. The result in Section 3 may correspond
to the slow component in Fig. 4. The constraint CNM density profile index
$k\sim 1.8$ is also consistent with the one-component jet model ($k\sim 1.7$)
in Section 3. Therefore, our result support that AT2022cmc may contain a two-
component jet as in Sw J1644+57. These two jetted TDEs (AT2022cmc and Sw
J1644+57) may share similar jet physics.

Figure 3: Posterior distribution and parameter constraints obtained using a two-component jet model. The median values with the $1\sigma$ error regions are also shown in the one-dimensional probability distribution. Table 3: The input parameters, prior type, prior range, best-fit value of two-component jet model of AT2022cmc. Parameters | Prior Type | Prior Range | Best fit
---|---|---|---
$k$ | log flat | 1-3 | 1.84
$n_{18}\rm\ (cm^{-3})$ | log flat | $10^{-5}-10^{3}$ | 7.14
$E_{\rm K,iso,f}$ (erg) | log flat | $10^{49}-10^{54}$ | $5.21\times 10^{53}$
$\Gamma_{0,\rm f}$ | log flat | 1.0-50 | 36.61
$\theta_{\rm j,f}$ (rad) | flat | $0.052-0.35$ | $4.97\times 10^{-2}$
$p_{\rm f}$ | flat | $2.0-3.0$ | 2.21
$\epsilon_{\rm e,f}$ | log flat | $10^{-3}-0.33$ | $1.08\times 10^{-3}$
$\epsilon_{\rm B,f}$ | log flat | $10^{-4}-0.33$ | $6.25\times 10^{-3}$
$E_{\rm K,iso,s}$ (erg) | log flat | $10^{49}-10^{54}$ | $3.97\times 10^{53}$
$\Gamma_{0,\rm s}$ | log flat | 1.1-50 | 4.55
$\theta_{\rm j,s}$ (rad) | flat | $0.052-0.35$ | $5.23\times 10^{-2}$
$p_{\rm s}$ | flat | $2.0-3.0$ | 3.00
$\epsilon_{\rm e,s}$ | log flat | $10^{-3}-0.33$ | $3.08\times 10^{-1}$
$\epsilon_{\rm B,s}$ | log flat | $10^{-4}-0.33$ | $1.39\times 10^{-2}$
Figure 4: Optical, sub-millimeter and radio afterglow data of AT2022cmc along
with the best fit with a two-component jet (a fast component with dotted
lines, and a slow component with dashed lines) model. For convenience, the
offset is added when drawing the light curve.

###  4.3 Detectability of AT2022cmc-like Events with Einstein Probe

AT2022cmc is unique, and is the first on-axis jetted TDE discovered in
optical. It is also the furthest jetted TDE discovered to date. Future optical
surveys, e.g., Large Synoptic Survey Telescope (LSST) and Wide Field Survey
Telescope (WFST), have great potential to discover more AT2022cmc-like events
(Bricman & Gomboc, 2020; WFST Collaboration et al., 2023). However, X-ray
observations might still be the key to the confirmation of a jetted TDE from
the optical selected candidates.

Einstein Probe (EP) is an ideal instrument to search systematically for and
study TDEs (Yuan et al., 2015). EP will be able to detect more relativistic
TDEs with relativistic jet out to redshift $z>1$. In Fig. 5, we present the
NICER (black points) and Swift (red points) X-ray data of AT2022cmc and the FS
model predicted X-ray afterglow (solid lines), as well as the EP/WXT and
EP/FXT sensitivity (dotted-dashed lines). It is found that AT2022cmc-like TDEs
can be well monitored by EP/FXT, but are slightly below the EP/WXT
sensitivity. Considering that the peak X-ray luminosity of AT2022cmc is larger
than $3\times 10^{47}\rm erg\ s^{-1}$, such events still have good
opportunities to be triggered in X-ray by EP/WXT.

As pointed out by Lei et al. (2016) and Yuan et al. (2016), a good fraction of
jetted TDEs might be viewed off-axis. The observed flux density is further
subject to a correction factor due to the viewing angle for an off-axis
observer (Granot et al., 2002)

$F_{\nu}(\psi,t)=a_{\rm off}^{3}F_{\nu/a_{\rm off}}(0,a_{\rm off}t),$ (21)

where $\psi=\rm{max}(0,\theta_{obs}-\theta_{j})$ is the angle between the
near-edge of the jet and the observer, and

$a_{\rm off}=\frac{{\mathcal{D}}_{\rm off}}{{\mathcal{D}}_{\rm
on}}=\frac{1-\beta}{1-\beta\cos\psi},$ (22)

is the ratio of the on-beam Doppler factor to the off-beam Doppler factor,
with $\beta=\sqrt{1-1/\Gamma^{2}}$.

We take AT2022cmc as a prototype and investigate the detectability by EP when
assuming different view angles $\theta_{\rm obs}$, see the gray points in Fig.
5. One can find that AT2022cmc-like events can be followed by EP/FXT when
$\theta_{\rm obs}\leq 36.3^{\circ}$. Therefore, the combination observations
by WFST and EP have large chance to reveal AT2022cmc-like events in future.

Figure 5: X-ray light curves of AT2022cmc. The black (NICER) and red (Swift)
points are the X-ray observations of AT2022cmc adopted from Pasham et al.
(2023). The blue solid line is the X-ray emission from the forward shock (FS)
with the best-fit result of the one-component jet model (see Table 2). The
X-ray emission from the FS of the best-fit two-component jet model (see Table
3) is represented with yellow lines, in which the fast component, the slow
component and combiantion of the two components are plotted with yellow dotted
line, yellow dashed line and yellow solid line, respectively. The sensitivity
of the WXT and FXT of EP are also shown with purple and green dotted-dashed
lines, respectively. The gray points are the observational X-ray data viewed
off-axis with $\theta_{\rm obs}=36.3^{\circ}$, which can be marginally
detected by EP/FXT.

##  5 Summary

AT2022cmc was discovered as a luminous, rapidly evolving transient by the
Zwicky Transient Facility (Andreoni et al., 2022; Pasham et al., 2023). We
employed the CR and MCMC analysis to constrain the CNM density profile of
AT2022cmc under the framework of the FS model. Our conclusions are summarized
as follows:

1\. The r-band optical light curve as well as the radio/sub-millimeter
observations of AT2022cmc can be explained by the synchrotron emission from
the forward shock of the jet-CNM interactions. The best-fit parameter values
of the one-comopnent jet model are given in Table 2. The initial Lorentz
factor of the jet is 4.76. The isotropic jet kinetic energy $E_{\rm
K,iso}=3.47\times 10^{53}\rm erg$ indicates a prompt radiative efficiency of
the jet $\epsilon_{\rm X}\equiv E_{\rm X,iso}/(E_{\rm X,iso}+E_{\rm
K,iso})\simeq 0.22$ if the observed X-ray energy $E_{\rm X,iso}=10^{53}\rm
erg$ is used.

2\. The closure relation analysis suggests the CNM density profile of
AT2022cmc as $n\,\propto\,R^{-1.47}$. The detailed study with FS model fit
finds $n\,\propto\,R^{-1.69}$. The results are similar to that of Sw J1644+57,
and are consistent with a Bondi-like accretion in history. The nucleus of
AT2022cmc may be a low-luminosity AGN, but its luminosity is much weaker than
the observed long-lasting optical “plateau” luminosity.

3\. Compared with the one-component jet predictions, the excess in the early
sub-millimeter and radio observed data support a two-component jet model (a
narrow-fast component and a wide-slow component). The best-fit parameter
values of the two-component jet model are given in Table 3. Such two-component
jet structure is also revealed in the first jetted TDE candidate Sw J1644+57,
indicating similar jet physics in these two events.

4\. AT2022cmc provides a good prototype for unveiling jetted TDEs from optical
and X-ray surveys. Such kind of events can be well monitored by EP/FXP even if
it is viewed off-axis with view angle $\theta_{\rm obs}\leq 36.3^{\circ}$.

## acknowledgments

We are very grateful to Yanan Wang, Lixin Dai, Hui Li, Bing Zhang, Yuan-Chuan
Zou, He Gao and Jumpei Takata for helpful discussions. This work is supported
by the National Key R&D Program of China (Nos. 2020YFC2201400), the National
Natural Science Foundation of China under grants U2038107, U1931203 and
12021003. D.X. acknowledges support by the science research grants from the
China Manned Space Project with NO. CMS-CSST-2021-A13 and CMS-CSST-2021-B11.
W. H. Lei acknowledge support by the science research grants from the China
Manned Space Project with NO.CMS-CSST-2021-B11. W. Xie acknowledge support by
the Science and Technology Foundation of Guizhou Province (grant No.
QianKeHeJiChu ZK[2021]027). The authors acknowledge Beijing PARATERA Tech CO.,
Ltd. for providing HPC resources that have contributed to the research results
reported within this paper.

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