# Jets from SANE Super-Eddington Accretion Disks: Morphology, Spectra, and
Their Potential as Targets for ngEHT

Brandon Curd1,2, Razieh Emami1, Richard Anantua3,1,2, Daniel Palumbo1,2,
Sheperd Doeleman1,2, Ramesh Narayan1,2
1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
2 Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA
02138, USA
3 Department of Physics $\&$ Astronomy, The University of Texas at San
Antonio, One UTSA Circle, San Antonio, TX 78249, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present general relativistic radiation magnetohydrodynamics (GRRMHD)
simulations of super-Eddington accretion flows around supermassive black holes
(SMBHs) which may apply to tidal disruption events (TDEs). We perform long
duration ($t\geq 81,200\,GM/c^{3}$) simulations which achieve mass accretion
rates $\gtrsim 11$ times the Eddington rate and produce thermal synchrotron
spectra and images of their jets. Gas flowing beyond the funnel wall expands
conically and drives a strong shock at the jet head while variable mass
ejection and recollimation along the jet axis results in internal shocks and
dissipation. Assuming the ion temperature ($T_{i}$) and electron temperature
($T_{e}$) in the plasma are identical, the radio/submillimeter spectra peak at
$>100$ GHz and the luminosity increases with BH spin, exceeding $\sim
10^{41}\,\rm{erg\,s^{-1}}$ in the brightest models. The emission is extremely
sensitive to $T_{i}/T_{e}$ as some models show an order of magnitude decrease
in the peak frequency and up to four orders of magnitude decline in their
radio/submillimeter luminosity as $T_{i}/T_{e}$ approaches 20. Assuming a
maximum VLBI baseline distance of $10\ {\rm{G}}\lambda$, 230 GHz images of
$T_{i}/T_{e}=1$ models shows that the jet head may be bright enough for its
motion to be captured with the EHT (ngEHT) at $D\lesssim 110$ (180) Mpc at the
$5\sigma$ significance level. Resolving emission from internal shocks requires
$D\lesssim 45$ Mpc for both the EHT or ngEHT.

###### keywords:

accretion, accretion discs - black hole physics - MHD - radiative transfer

††pubyear: 2022††pagerange: Jets from SANE Super-Eddington Accretion Disks:
Morphology, Spectra, and Their Potential as Targets for ngEHT–B

##  1 Introduction

The central black holes (BHs) of galaxies throughout the universe gain some
fraction of their mass from the tidal disruption of stars in the near vicinity
(see 2021MNRAS.500.3944P for a recent study). Random scatterings of stars
orbiting the BH occasionally place them on a chance orbit that will cross the
tidal radius ($R_{t}$, defined later in Equation 12), beyond which the star’s
self gravity is weaker than the tidal forces acting on it and it is
subsequently disrupted into an elongated stream of gas (1975Natur.254..295H;
1988Natur.333..523R; 1989IAUS..136..543P; 1989ApJ...346L..13E). The bright
transient which occurs as a result of the disruption is known as a tidal
disruption event (TDE). The most well studied TDE is that of a near solar mass
star being disrupted by a supermassive BH (SMBH). In these events, the
transient light curve roughly follows a $L\propto(t/t_{\rm{fb}})^{-5/3}$
decay. Here, the fallback time

$t_{\rm{fb}}=3.5\times
10^{6}\,{\rm{s}}\,\left(\dfrac{M_{\rm{BH}}}{10^{6}M_{\odot}}\right)^{1/2}\left(\dfrac{M_{*}}{M_{\odot}}\right)^{-1}\left(\dfrac{R_{*}}{R_{\odot}}\right)^{3/2},$
(1)

where $M_{*}$ is the mass of the disrupted star and $R_{*}$ is its radius, is
the characteristic decay time for the TDE (2013MNRAS.435.1809S). This
behaviour originates from the gas dynamics during the disruption, which causes
the star to be disrupted into an elongated stream of gas with a binding energy
distribution which dictates that mass flows back towards pericenter at the
‘fallback rate’ $\dot{M}_{\rm{fb}}\propto(t/t_{\rm{fb}})^{-5/3}$
(2013MNRAS.435.1809S). The earliest theoretical works on the subject proposed
that the radiation emitted by TDEs would originate from a small scale,
geometrically thick accretion disk located near $\sim R_{t}$ due to
circularization of the stream with gas temperatures leading to X-ray/UV
emission (e.g. see 1988Natur.333..523R).

Thanks to various surveys in the optical/UV and X-ray, the number of known
TDEs has grown substantially since their initial discovery via ROSAT (see
2015JHEAp...7..148K; 2021ARA&A..59...21G for a review). We now understand that
TDE emission is quite complicated as not all TDEs have bright X-ray emission.
Also, the optical/UV emission follows the same power-law behaviour as expected
for the X-ray. This suggests that the optical/UV component originates from
reprocessed X-rays emitted from the hot accretion disk (2018ApJ...859L..20D).
The reprocessing material may be any surrounding material such as the TDE
debris itself, clouds, outflows, or winds. This scenario is supported
observationally as several Bowen-TDEs, which show Bowen fluorescence
(requiring reprocessing of higher energy photons), have been detected
(2019ApJ...887..218L; 2021ApJ...908....4V).

Radio studies in the $5-8.4$ GHz range have only detected emission from a
handful of TDEs (2020SSRv..216...81A; 2022NatAs.tmp..252P). Four of these are
likely so-called “jetted” TDEs, which launched powerful relativistic jets with
bulk lorentz factor $\Gamma>10$ and had a peak radio luminosity of
$L_{\rm{peak,radio}}\sim 10^{40}-10^{42}\,\rm{erg\,s^{-1}}$. Jetted TDEs are
rare, making up only a small subset ($\sim 1\%$) of all TDEs
(2015ApJ...812...33S; 2022NatAs.tmp..252P; 2022arXiv221116530A). The other
events, which emitted significantly less energy in the radio with
$L_{\rm{peak,radio}}\sim 10^{37}-10^{39}\,\rm{erg\,s^{-1}}$, have been modeled
using sub-relativistic outflows, internal jet shocks, and off-axis jets. An
important difference in each of these mechanisms is the site of acceleration
for electrons which eventually produce the observed synchrotron emission. For
example, in the external shock picture the electrons are accelerated in the
shock between the outflow and the circumnuclear medium (CNM) while in the
internal acceleration picture the electrons are accelerated inside of the jet
itself.

It is worth noting that radio emission in “non-jetted” TDEs (TDEs with no
highly relativistic jet) typically appears $>30$ days after the discovery of
the TDE in the optical/UV/X-ray bands. Assuming the outflow which ultimately
results in radio emission is driven by the accretion flow itself, this may
simply be due to the initially turbulent evolution of the disk. The infall of
high density material into the BH at steep angles may suppress the formation
of a funnel region, through which the gas may be accelerated into a jet
(2021MNRAS.507.3207C). However, 2020MNRAS.495.1374B present hydrodynamical
simulations of a TDE accretion disk which appear to have formed a funnel
region promptly. It is also possible that our understanding of the disk
formation process itself is incomplete. For instance, the onset of disk
formation may not correlate to the time at which optical/UV/X-ray emission
emerges.

TDEs are of great interest for studies of accretion physics and stellar
populations close to the BH. For TDEs of Sun-like stars around BHs with mass
$M_{\rm{BH}}\lesssim 10^{7.5}\,M_{\odot}$, the peak fallback rate will exceed
the Eddington rate $\dot{M}_{\rm{Edd}}$ (2013MNRAS.435.1809S), which opens up
the possibility that the accretion rate is actually super-Eddington. For
example, a solar mass star disrupted around a $10^{6}\,M_{\odot}$ BH will have
a peak fallback rate that is $\sim 100$ times the Eddington rate, and the
fallback rate should remain above Eddington for a few fallback times.
Hydrodynamic simulations of the early stages of disk formation in a TDE
suggest that as much as 20% of the returning stream may actually cross the
horizon through an accretion flow; however, the current library of published
simulations has yet to cover a substantial range of the TDE parameter space
and the effects of magnetism have largely been ignored in long term
simulations (2009ApJ...697L..77R; 2014ApJ...783...23G; 2015ApJ...804...85S;
2016MNRAS.461.3760H; 2016MNRAS.455.2253B; 2019arXiv191010154L;
2020MNRAS.495.1374B; 2021MNRAS.504.4885B; 2021MNRAS.507.3207C;
2022MNRAS.510.1627A; 2022arXiv220610641S).

The exact nature of the accretion disk structure is still uncertain, but
previous works have applied global GRRMHD accretion disk simulations to model
the accretion flow (2018ApJ...859L..20D; 2019MNRAS.483..565C). The emission
properties of TDEs are possibly described by thick accretion disks
(2018ApJ...859L..20D; 2019MNRAS.483..565C). In particular, magnetically
arrested disks (or MADs, 2003PASJ...55L..69N) around spinning BHs produce
powerful jets and have emission properties remarkably similar to that of Swift
J1644+57 (2014MNRAS.437.2744T; 2019MNRAS.483..565C). For non-jetted TDEs, the
‘standard and normal evolution’ (or SANE) disks may be appropriate to describe
X-ray TDEs, but the optical/UV component was uncertain in 2019MNRAS.483..565C
since it originated from the outer radius of the torus, which was initialized
using an equilibrium torus model rather than from following the entire disk
formation process in a TDE. SANE models may also apply to jetted TDEs, as a
funnel region in super-Eddington accretion flows can result in highly
energetic outflows capable of explaining even jetted events
(2015MNRAS.453.3213S; 2020MNRAS.499.3158C). A numerical study of the
radio/submillimeter emission from outflows driven by SANE super-Eddington
accretion flows and its comparison with the radio emission in TDEs is thus
well motivated.

In this work, we model the radio/submillimeter emission (via thermal
synchrotron) from the outflows of super-Eddington accretion flows using GRRMHD
simulations similar to those presented in 2019MNRAS.483..565C. However, in
this work we use a grid with more resolution in the jet and run the simulation
substantially longer such that the outflow reaches radii similar to the
emission scales for known radio TDEs (2020SSRv..216...81A). We perform
simulations of SANE accretion disks around BHs of mass
$M_{\rm{BH}}=(5,10)\times 10^{6}\,M_{\odot}$ and a dimensionless BH spin of
$a_{*}=(0,0.5,0.9)$ and measure the jet power and morphology across the
parameter space. We perform general relativistic ray tracing (GRRT) of each
model to produce spectra and images of the jet emission. We examine the
viability of detecting and resolving the jet at 230 GHz, which is the
frequency employed by the Event Horizon Telescope (EHT, 2019ApJ...875L...3E).
Based on our results, we suggest that the EHT or the next generation Event
Horizon Telescope (ngEHT) could be suited for radio/submillimeter follow-up of
TDEs. A direct probe of jet launching from super-Eddington accretion disks and
radio/submillimeter emission in TDEs are both of significant interest and are
expected to increase our understanding of BH accretion and the environment of
BHs.

The paper is structured as follows. In Section 2, we motivate
radio/submillimeter observations of nearby TDEs by computing the expected
number of TDEs within $D<60$ Mpc. In Section 3, we outline the numerical
methods used in this work. In Section 4, we describe the accretion flow and
outflow properties. In Section 5, we analyse GRRT radio/submillimeter spectra
and images of each model and determine the viability of resolving the jets in
each simulation with the EHT/ngEHT. We compare our results with current TDE
radio observations in Section 6 and conclude in Section 7.

##  2 TDE Rates

Figure 1: Here we show the BHMF as calculated in the text (left panel). We
then show the volumetric TDE rate (right panel). The text above the curve
shows the TDE rate when the curve is integrated over BH mass. Figure 2: Here
we estimate the number of total TDEs expected given a limiting observable
distance per year (green line) and per 10 years (blue line). The shaded
regions denote the range of parameter space where the expected number of TDEs
in $1$ year (green) and $10$ years (blue) exceeds $1$. We use a different
shade where $D<=$ 20, 30, and 40 Mpc to illustrate the impact of increasing
the maximum distance by only 10 Mpc.

In this section, we motivate follow-up observations of TDEs in the
radio/submillimeter band by quantifying the number of TDEs in a given volume
of the nearby universe assuming conservative TDE rates. We utilize methods
described in 2016MNRAS.455..859S and the interested reader should see their
work for a detailed discussion of TDE rates. The number of possible TDEs per
year grows rapidly at higher redshift since the number of BHs in an integrated
volume of space varies as $N_{\rm{BH}}(D)\propto D^{3}$. Similarly, assuming
estimates of the TDE rates are applicable broadly within the local universe
(see 2016MNRAS.455..859S; 2020MNRAS.497.2276P), we can estimate the number of
TDEs per year within a finite distance so long as we know the mass dependent
TDE rate of a given BH ($\Gamma(M_{\rm{BH}})$) as well as the black hole mass
function (BHMF, $\widetilde{\phi}(M_{\rm{BH}}))$. The BHMF varies at larger
cosmic distance, but here we consider only the local universe ($z<0.1$), where
the BHMF is essentially redshift independent.

We make use of the BHMF adopted by 2016MNRAS.455..859S to estimate the
volumetric TDE rate. See their work for an in depth definition. To estimate
the BHMF, we first define the R band luminosity ($L_{R}$) function of the
galaxy by assuming it follows the Schechter function (1976ApJ...203..297S):

$\widetilde{\phi}(L_{R})dL_{R}=\widetilde{\phi}_{*}\left(\dfrac{L_{R}}{L_{*}}\right)^{-1.1}\exp(-L_{R}/L_{*})dL_{R},$
(2)

where $\widetilde{\phi}_{*}=4.9\times 10^{-3}h_{7}^{3}{\rm{\,Mpc^{-3}}}$ and
$L_{*}=2.9\times 10^{10}h_{7}^{-2}L_{\odot}$. Here $h_{7}=1$ is the normalized
Hubble constant. Combining the above Schechter function with the Faber-Jackson
law, $\sigma\approx 150{\rm{\,km\,s^{-1}}}(L_{R}/10^{10}L_{\odot})^{1/4}$,
with the 2013ApJ...764..184M calibration of the $M_{\rm{BH}}-\sigma$ relation,
we arrive at the BHMF (left panel in Figure 1):

$\widetilde{\phi}(M_{\rm{BH}})d\ln
M_{\rm{BH}}=2.56\widetilde{\phi}_{*}f_{\rm{occ}}\left(\dfrac{M_{\rm{BH}}}{10^{8}M_{\odot}}\right)^{-1.07}\\\
\times\exp\left(-0.647\left(\dfrac{M_{\rm{BH}}}{10^{8}M_{\odot}}\right)\right)d\ln
M_{\rm{BH}}.$ (3)

Here $f_{\rm{occ}}$ is the occupation fraction, which accounts for the
expectation that lower mass galaxies may not host a SMBH at their center.
Following 2016MNRAS.455..859S, we define the occupation fraction as:

$f_{\rm{occ}}=\begin{cases}0.5\tanh\bigg{[}\ln\left(\dfrac{M_{\rm{bulge}}}{M_{c}}\right)\times\\\
~{}~{}~{}~{}~{}~{}2.5^{8.9-\log_{10}(M_{c}/M_{\odot})}\bigg{]}+0.5,\quad
M_{\rm{bulge}}<10^{10}\,M_{\odot}\\\
1.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}M_{\rm{bulge}}>10^{10}\,M_{\odot}\end{cases}$

Here $M_{c}$ is a parameter determining the bulge mass below which the
occupation fraction turns over and begins to approach zero as illustrated in
2016MNRAS.455..859S. We relate $M_{\rm{bulge}}$ to the BH mass with the
$M_{\rm{BH}}-M_{\rm{bulge}}$ relation from 2013ApJ...764..184M. As was noted
by 2016MNRAS.455..859S, the value of $M_{c}=10^{8.5}\,M_{\odot}$ is the most
likely case, so we fix $M_{c}=10^{8.5}\,M_{\odot}$ in our analysis.

We also account for the impact of different mass stars on the TDE rate by
incorporating the Kroupa initial mass functions (IMF):

$\chi_{\rm{Kro}}=\dfrac{dN_{*}}{dM_{*}}\propto\begin{cases}(M_{*}/M_{\odot})^{-1.3},\quad&0.08<(M_{*}/M_{\odot})<0.5\\\
(M_{*}/M_{\odot})^{-2.3},\quad&0.5<(M_{*}/M_{\odot})<1.\\\ \end{cases}$ (4)

Note that we have followed 2016MNRAS.455..859S and only assume stars in the
range of $0.08-1\,M_{\odot}$ are accounted for. This is reasonable
particularly if we are considering older galaxies where high mass stars would
have already evolved and died. We normalize the IMF such that
$\int\chi_{\rm{Kro}}dM_{*}=1$.

For the TDE rate, $\Gamma_{\rm{TDE}}(M_{\rm{BH}})$, we use the upper limit
estimate from 2016MNRAS.455..859S:

$\Gamma_{\rm{S16}}=2.9\times
10^{-5}{\rm{\,yr^{-1}}}\left(\dfrac{M_{\rm{BH}}}{10^{8}\,M_{\odot}}\right)^{-0.404}.$
(5)

This estimate comes from core/cusp galaxies using a sample of roughly 200
objects. We note that the TDE rate can be increased in galaxies with denser
stellar cores (see 2020MNRAS.497.2276P, for more details). However, in this
work, we restrict ourselves to conservative estimates.

We infer the volumetric TDE rate as a function of BH mass accounting for both
the distribution of stars and the fact that some stars will get swallowed and
do not contribute to the TDE rate. Therefore the volumetric TDE rate as a
function of BH mass would be:

$\dot{n}(M_{\rm{BH}})=\int_{M_{*,{\rm{min}}}}^{M_{*,{\rm{max}}}}\Gamma(M_{\rm{BH}},M_{*})\widetilde{\phi}(M_{\rm{BH}})\chi_{\rm{Kro}}dM_{*}.$
(6)

where $\Gamma(M_{\rm{BH}},M_{*})$ is the TDE rate which is given by Equation
(5). We account for cases where the star is swallowed by the BH by setting
$\Gamma(M_{\rm{BH}},M_{*})=0$ if $R_{t}<r_{H}$, where $r_{H}$ is the horizon
radius of the BH, for the star at the given BH mass. We show the volumetric
TDE rate in the right panel in Figure 1 for $M_{c}=10^{8.5}M_{\odot}$ and
$\Gamma_{\rm{S16}}$. The text above the curve shows the integral
$\dot{n}=\int\dot{n}(M_{\rm{BH}})d\ln M_{\rm{BH}}$ which is the total
volumetric TDE rate. Lower cutoff masses for the occupation fraction can
significantly increase the TDE rate, but the greatest enhancement comes from
assuming that denser stellar cores are present.

To estimate the number of TDEs occurring within a closed volume in a finite
amount of time, we use the total volumetric TDE rate, which is constant in
time, to obtain:

$N_{\rm{TDE}}=\dfrac{4}{3}\pi D^{3}\dot{n}\Delta t,$ (7)

where $D$ is the distance in Mpc and $\Delta t$ is the total time in years.
Figure 2 shows the TDE rates given a number of years (1 year and 10 years of
observation).

The vertical lines in Figure 2 show the effect on the number of observable
TDEs for a maximal observing distance. Even a modest maximum observing
distance of $D<20$ Mpc would provide 2-3 TDEs per decade while extending this
range to $D<40$ Mpc would lead to nearly 2 TDEs per year. Assuming these
flares are captured by other survey telescopes such as the Large Synoptic
Survey Telescope (LSST, 2019ApJ...873..111I), which is expected to lead to
many more TDE detections in the coming years (2020ApJ...890...73B),
radio/submillimeter follow-up of several TDEs may be possible. We note that we
have assumed full sky coverage in our analysis, which is unrealistic, but our
calculations demonstrate how the TDE rate varies with distance in order to
quantify how beneficial the detector sensitivity may be. In the following
sections, we for the first time attempt to quantify the observability of the
radio/submillimeter emission from outflows launched by a super-Eddington
accretion disks with mass accretion rates similar to the peak accretion rate
in TDEs.

##  3 Numerical Methods

The simulations presented in this work were performed using the general
relativistic radiation magnetohydrodynamical (GRRMHD) code KORAL
(2013MNRAS.429.3533S; 2014MNRAS.439..503S; 2014MNRAS.441.3177M;
2017MNRAS.466..705S) which solves the conservation equations in a fixed,
arbitrary spacetime using finite-difference method. We solve the following
conservation equations:

$\displaystyle(\rho u^{\mu})_{;\mu}$ $\displaystyle=0,$ (8)
$\displaystyle(T^{\mu}_{\nu})_{;\mu}$ $\displaystyle=G_{\nu},$ (9)
$\displaystyle(R^{\mu}_{\nu})_{;\mu}$ $\displaystyle=-G_{\nu},$ (10)

where $\rho$ is the gas density in the comoving fluid frame, $u^{\mu}$ are the
components of the gas four-velocity as measured in the “lab frame”,
$T^{\mu}_{\nu}$ is the MHD stress-energy tensor in the “lab frame”:

$T^{\mu}_{\nu}=(\rho+u_{g}+p_{g}+b^{2})u^{\mu}u_{\nu}+(p_{g}+\dfrac{1}{2}b^{2})\delta^{\mu}_{\nu}-b^{\mu}b_{\nu},$
(11)

$R^{\mu}_{\nu}$ is the stress-energy tensor of radiation, and $G_{\nu}$ is the
radiative four-force which describes the interaction between gas and radiation
(2014MNRAS.439..503S). Here $u_{g}$ and $p_{g}=(\gamma_{g}-1)u_{g}$ are the
internal energy and pressure of the gas in the comoving frame (linked by
adiabatic index $\gamma_{g}$) and $b^{\mu}$ is the magnetic field four-vector
which is evolved following the ideal MHD induction equation
(2003ApJ...589..444G). In the KORAL simulations, we assume a single
temperature plasma where the ion temperature ($T_{i}$) and the electron
temperature ($T_{e}$) are identical. This description is adequate in the
optically thick regions where collisions are common, but may not be accurate
in the extended jet where the gas density is substantially lower
(2022arXiv220103526L). We also assume that the electrons follow a thermal
distribution, which is another caveat in the initial KORAL simulations since
internal shocks in the jet are expected to accelerate electrons into a non-
thermal distribution. Magnetic reconnection events in the disk/wind may also
produce regions where a non-thermal electron population exists, but we do not
account for the possibility of these effects.

The radiative stress-energy tensor is obtained from the evolved radiative
primitives, i.e. the radiative rest-frame energy density and its four velocity
following the M1 closure scheme modified by the addition of radiative
viscosity (2013MNRAS.429.3533S; 2015MNRAS.447...49S).

The opposite signs of $G_{\nu}$ in the conservation equations for gas and
radiation stress-energy (Eqs. 9 and 10) reflect the fact that the gas-
radiation interaction is conservative, i.e. energy and momentum are
transferred between gas and radiation, (see 2017MNRAS.466..705S, for more
details). We include the effects of absorption and emission via the electron
scattering opacity ($\kappa_{\rm{es}}$), free-free absorption opacity
($\kappa_{\rm{a}}$), and bound-free absorption opacity through the Sutherland
Dopita model (1993ApJS...88..253S) and assume a solar metal abundance for the
gas. We also include the effects of thermal synchrotron and Comptonization
(2015MNRAS.454.2372S; 2017MNRAS.466..705S).

The simulations presented in this work are conducted in 2D $r-\vartheta$
coordinates and we implement the mean-field dynamo model described in
2015MNRAS.447...49S to sustain the magnetic field throughout the simulation.

###  3.1 Basic Dynamics

A star which has been captured by a SMBH will be disrupted when it can no
longer be held together by its self-gravity. This occurs at radii less than
the tidal radius:

$R_{t}/r_{g}=47\left(\dfrac{M_{\rm{BH}}}{10^{6}M_{\odot}}\right)^{-2/3}\left(\dfrac{M_{*}}{M_{\odot}}\right)^{-1/3}\left(\dfrac{R_{*}}{R_{\odot}}\right).$
(12)

It is common to describe the disruption in terms of the impact parameter,
$\beta$, which is defined as the ratio between the tidal radius and pericenter
separation such that $\beta\equiv R_{t}/R_{p}$.

The pericenter separation at which a full disruption of the star is sensitive
to the stellar composition since the compactness of the star effects how
easily it is disrupted. For Zero Age Main Sequence (ZAMS) stars, those
described by a $\gamma=5/3$ polytrope are fully disrupted if $\beta\gtrsim
0.9$ while stars described by a $\gamma=4/3$ polytrope must come within
$\beta\gtrsim 2$ (2013ApJ...767...25G; 2017A&A...600A.124M).
2019ApJ...882L..26G demonstrated that the pericenter separation required for
evolved stars is even smaller (sometimes greater than $\beta=3$ based on their
findings) as the core is no longer hydrogen dominated due to its compactness.
For our purposes, we assume a ZAMS star with a $\gamma=5/3$ polytrope was
disrupted for simplicity.

If hydrodynamical forces are neglected, then the change in the specific
binding energy of the fluid in the star as a result of the tidal interaction
can greatly exceed the internal binding energy of the star
(1988Natur.333..523R). As a result, a spread in binding energy is imparted on
the stellar material. 2013MNRAS.435.1809S find that the spread in orbital
energy $\Delta\epsilon$ is insensitive to $\beta$ since the energy is
essentially frozen in at the tidal radius. This spread is then given by:

$\Delta\epsilon\approx 4.3\times
10^{-4}\left(\dfrac{M_{\rm{BH}}}{10^{6}\,M_{\odot}}\right)^{1/3}\left(\dfrac{M_{*}}{M_{\odot}}\right)^{2/3}\left(\dfrac{R_{*}}{R_{\odot}}\right)^{-1}c^{2}.$
(13)

The orbital binding energy of the most/least bound material is given by
$\epsilon_{\rm{mb}}=\epsilon_{*}-\Delta\epsilon/2$ and
$\epsilon_{\rm{lb}}=\epsilon_{*}+\Delta\epsilon/2$. Here $\epsilon_{*}$ is the
initial orbital binding energy of the star. For parabolic orbits, which have
$\epsilon_{*}=0$, the spread in binding energy leads to half of the mass
remaining bound and the other half being ejected. In this work, we assume the
star was disrupted on a parabolic orbit since the majority of TDEs will be of
such stars.

The spread in binding energy is one of the most crucial parameters that
defines the TDE evolution. In particular, the fallback time (Equation 1) and
the peak mass fallback rate:

$\dfrac{\dot{M}_{\rm{fb,peak}}}{\dot{M}_{\rm{Edd}}}\approx
133\left(\dfrac{M_{\rm{BH}}}{10^{6}M_{\odot}}\right)^{-3/2}\left(\dfrac{M_{*}}{M_{\odot}}\right)^{2}\left(\dfrac{R_{*}}{R_{\odot}}\right)^{-3/2},$
(14)

are direct consequences of the spread in binding energy.

###  3.2 Definitions

In this section, we discuss the units adopted throughout the text and provide
brief descriptions of quantities used to study the KORAL simulation data.

Throughout this work, we use gravitational units to describe physical
parameters. For distance we use the gravitational radius $r_{g}\equiv
GM_{\rm{BH}}/c^{2}$ and for time we use the gravitational time $t_{g}\equiv
GM_{\rm{BH}}/c^{3}$, where $M_{\rm{BH}}$ is the mass of the BH. Often, we set
$G=c=1$, so the above relations would be equivalent to $r_{g}=t_{g}=M_{\rm
BH}$. 111For a BH mass of $10^{6}\,M_{\odot}$, the gravitational radius and
time in CGS units are $r_{g}=1.48\times 10^{11}$ cm and $t_{g}=4.94$ s,
respectively. Occasionally, we restore $G$ and $c$ when we feel it helps to
keep track of physical units.

We adopt the following definition for the Eddington mass accretion rate:

$\dot{M}_{\rm{Edd}}=\dfrac{L_{\rm{Edd}}}{\eta_{\rm NT}c^{2}},$ (15)

where $L_{\rm{Edd}}=1.25\times 10^{38}\,(M_{\rm{BH}}/M_{\odot})\,{\rm
erg\,s^{-1}}$ is the Eddington luminosity, $\eta_{\rm{NT}}$ is the radiative
efficiency of a thin disk around a BH with spin parameter $a_{*}$ (which is
often referred to as the Novikov-Thorne efficiency):

$\eta_{\rm{NT}}=1-\sqrt{1-\dfrac{2}{3r_{\rm{ISCO}}}},$ (16)

and $r_{\rm{ISCO}}=3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}$ is the radius of
the Innermost Stable Circular Orbit (ISCO, 1973blho.conf..343N) in the Kerr
metric, where
$Z_{1}=1+(1-a_{*}^{2})^{1/3}\left((1+a_{*})^{1/3}+(1-a_{*})^{1/3}\right)$ and
$Z_{2}=\sqrt{3a_{*}^{2}+Z_{1}^{2}}$. For $a_{*}=$ 0, 0.5, and 0.9, the
efficiency is $\eta_{\rm{NT}}=$ 5.72%, 8.21%, and 15.58%.

We compute the net mass inflow rate as:

$\dot{M}(r)=-2\pi\int_{0}^{\pi}\sqrt{-g}\rho\,u^{r}d\vartheta.$ (17)

We treat the accretion of mass onto the BH as this integral taken at the
horizon $r_{H}$.

We estimate the electron scattering photosphere location for an observer at
infinity along the direction ($\vartheta$) by integrating the optical depth
radially inward from the outer boundary of the grid. Far from the BH, the
curvature of spacetime is negligible, so we simply integrate at constant
($\vartheta$) in the “lab frame”:

$\tau_{\rm{es}}(r)=\int_{r}^{R_{\rm{max}}}\dfrac{\rho\kappa_{\rm{es}}}{c}\left(u^{t}-u^{r}\right)\sqrt{g_{rr}}dr^{\prime},$
(18)

where $\kappa_{\rm{es}}=0.2(1+X)\kappa_{\rm{KN}}\,{\rm cm^{2}}$ is the
electron scattering opacity, $X$ is the Hydrogen mass-fraction which is
assumed to be the Solar abundance $X_{\odot}=0.7381$, $\kappa_{\rm{KN}}$ is
the Klein-Nishina correction factor for thermal electrons
(2017MNRAS.466..705S), and $R_{\rm{max}}$ is the radius corresponding to the
outer boundary of the grid. For the gas and radiation temperatures in the
simulations presented here, the Klein-Nishina correction is negligible and the
electron scattering opacity is essentially the Thomson opacity. In this work,
we choose the location of the photosphere as the $\tau_{\rm{es}}=1$ surface.

We define the accretion flow as three distinct regions (disk, wind, and jet)
based on the total energy via the Bernoulli parameter. We also make use of the
electron scattering opacity, $\tau_{\rm{es}}$, to determine whether the region
of the fluid under consideration is optically thin or thick. In optically
thick regions, i.e. ($\tau_{\rm{es}}\geq 1$), the radiation is advected with
the flow and can contribute to acceleration of the gas so we treat it as
contributing to the Bernoulli parameter. Meanwhile, in optically thin regions,
i.e. ($\tau_{\rm{es}}<1$), we assume that only the MHD components are relevant
to the total gas energy. We modify the Bernoulli definition used in
2019MNRAS.483..565C to include optically thin regions where the interaction
between gas and radiation can be neglected via:

$\rm{Be}=\begin{cases}-\dfrac{(T^{t}_{\ \,t}+R^{t}_{\ \,t}+\rho u^{t})}{\rho
u^{t}},\ \,\rm{(optically\,thick)}\\\ -\dfrac{(T^{t}_{\ \,t}+\rho u^{t})}{\rho
u^{t}},\ \,\rm{(optically\,thin)}\\\ \end{cases}$ (19)

The ‘disk’ is made up of bound gas with $\rm{Be}<0$. This constitutes both the
inner accretion disk and the large mass reservoir of the initial torus. Both
the ‘wind’ and ‘jet’ are unbound and are generally radially out flowing. The
wind is defined as any fluid with $0<\rm{Be}\leq 0.05$. The jet is any fluid
with $\rm{Be}>0.05$. This choice of cutoff for wind vs. jet is based on the
velocity at infinity ($v_{\infty}$), with the wind having $v_{\infty}\lesssim
0.3c$ and the jet having $v_{\infty}\gtrsim 0.3c$. It is worth noting that
this choice assumes that gas at small radii will not lose/gain energy as it
travels outward, which is not guaranteed to be the case outside of steady
state regions. A fraction of the positive Bernoulli gas in the simulation
domain could remain bound to the BH such as in 2018ApJ...863..158C; however,
the Bernoulli is still a useful approximate definition to characterize the
outflow based on energy. We make use of the Bernoulli parameter to find the
opening angle of the jet. I.e. we define $\vartheta_{\rm{jet}}(r)$ such that:

${\rm{Be}}(r,\vartheta_{\rm{jet}}(r))=0.05.$ (20)

We visualize the gas density, optical depth, and Bernoulli in simulation
m5a0.0-HR (described later in the text) in Figure 3. At small scales, the disk
is clearly separated from the funnel/outflow region based on our definition.
In addition, the wind initially constitutes a small angular region of the
funnel and is clearly a higher density gas than the jet component. At large
scales, the jet is a lower density region which is hugged by the higher
density wind.

Figure 3: Here we visualize an evolved disk/jet. We show the gas density
(colors) as well as the $\tau_{\rm{es}}=1$ (yellow line), Be$=0$ (white line),
and Be$=0.05$ (dashed white line) contours. Note that we reflect the data
about the $x$-axis for visualization purposes, but the simulation was run in
2D $r-\vartheta$ coordinates. In the top panel, we focus on the inner
accretion disk and funnel region to illustrate the separation of the disk
(region inside the solid white lines) from the wind (region between solid and
dashed white lines) and the jet (region inside of dashed lines). At large
scales, the wind (region between solid and dashed white lines) is clearly
delineated from the jet (region inside of dashed lines) and the initial bound
atmosphere (region outside of solid white lines).

The total luminosity (the net energy flux) is computed as:

$L_{\rm{net}}(r)=-2\pi\int_{0}^{\pi}\sqrt{-g}(T^{r}_{\ \,t}+R^{r}_{\ \,t}+\rho
u^{r})d\vartheta,$ (21)

where we integrate the radial flux of energy carried by gas plus magnetic
field ($T^{r}_{\ \,t}$) and radiation ($R^{r}_{\ \,t}$), and subtract out the
rest-mass energy ($\rho u^{r}$, since it does not lead to observational
consequences for an observer at infinity). When computed at the BH horizon,
Equation 21 gives the total energy extracted from the accretion flow so we
define $L_{\rm{tot}}=L_{\rm{net}}(r_{H})$. We also compute the total energy
outflowing in the wind and jet as:

$L_{\rm{MHD}}(r)=-2\pi\int_{0}^{\pi}\sqrt{-g}(T^{r}_{\ \,t}+\rho
u^{r})d\vartheta.$ (22)

We compute energy outflow in the wind at $r=1000\,r_{g}$ by using Equation 22,
but we only sum energy outflows where the gas is unbound, outflowing, and non-
relativistic (or $0<\rm{Be}<0.05$ and $u^{r}>0$). A similar choice is made to
compute the total energy outflowing in the jet at $r=1000\,r_{g}$ but we use
the criterion that energy is only summed where $\rm{Be}\geq 0.05$ and
$u^{r}>0$. Note that at $1000\,r_{g}$, the wind and jet are in an optically
thin region so the Bernoulli is computed without the radiation energy density
included.

The radiative bolometric luminosity is given by:

$L_{\rm{bol}}(r)=-2\pi\int_{0}^{\pi}\sqrt{-g}R^{r}_{\ \,t}d\vartheta,$ (23)

which gives the flux of radiation energy through a surface at a given radius.
In this work, we measure the flux through a sphere at $r=5000\,r_{g}$, which
lies beyond the outer radius of the initial torus, and define
$L_{5000}=L_{\rm{bol}}(5000\,r_{g})$. We assume rays crossing the surface
reach a distant observer.

We define the total, wind, jet, and radiation efficiencies as:

$\eta_{\rm{tot}}=\dfrac{1}{\dot{M}}L_{\rm{net}}(r=r_{H}),$ (24)
$\eta_{\rm{wind}}=\dfrac{1}{\dot{M}}L_{\rm{MHD}}(r=1000\,r_{g},0.05>{\rm{Be}}>0,u^{r}>0),$
(25)
$\eta_{\rm{jet}}=\dfrac{1}{\dot{M}}L_{\rm{MHD}}(r=1000\,r_{g},{\rm{Be}}\geq
0.05,u^{r}>0),$ (26)

and

$\eta_{\rm{rad}}=\dfrac{1}{\dot{M}}L_{\rm{bol}}(r=5000\,r_{g}),$ (27)

In each case, positive values correspond to energy being extracted from the
system.

The magnetic flux is computed as:

$\Phi(r)=-\pi\int_{0}^{\pi}\sqrt{-g}|B^{r}(r)|d\vartheta,$ (28)

where $B^{r}$ is the radial component of the magnetic field. We quantify the
magnetic field strength at the BH horizon through the normalized magnetic flux
parameter (2011MNRAS.418L..79T):

$\phi_{\rm{BH}}=\dfrac{\Phi(r_{H})}{\sqrt{\dot{M}(r_{H})}}.$ (29)

For geometrically thick disks the MAD state is achieved once
$\phi_{\rm{BH}}\sim 40-50$ (see e.g. 2011MNRAS.418L..79T;
2012JPhCS.372a2040T).

To characterize the magnetic field, we define the magnetic pressure ratio:

$\beta_{t}\equiv\dfrac{(p_{\rm{gas}}+p_{\rm{rad}})}{p_{\rm{mag}}},$ (30)

(distinguished from the impact parameter $\beta$) which is used to define the
pressure ratio in optically thick regions where the radiation is dynamically
important. Note that we also make use of the ratio between the gas and
magnetic pressure:

$\beta_{g}\equiv\dfrac{p_{\rm{gas}}}{p_{\rm{mag}}},$ (31)

which is used to characterize the optically thin jet.

We quantify the resolution of the fastest growing mode of the
magnetorotational instability (MRI, 1991ApJ...376..214B) by computing the
quantity:

$Q_{\vartheta}=\dfrac{2\pi}{\Omega
dx^{\vartheta}}\dfrac{|b^{\vartheta}|}{\sqrt{4\pi\rho}},$ (32)

where $dx^{\vartheta}$ (the grid cell in polar coordinate $\vartheta$) and
$b^{\vartheta}$ (the $\vartheta$ component of the magnetic field) are both
evaluated in the orthonormal frame, $\Omega$ is the angular velocity, and
$\rho$ is the gas density. Numerical studies of the MRI have shown that values
of $Q_{\vartheta}$ in excess of at least 10 are needed to resolve the fastest
growing mode (2011ApJ...738...84H). As we are considering 2D simulations in
$r-\vartheta$, we do not consider the $\phi$ MRI quality factor. Throughout
each simulation, we find the quality factor $Q_{\vartheta}>10$ at the mid
plane for $r<50\,r_{g}$.

| m5a0.0-HR | m5a0.5-HR | m5a0.9-HR | m10a0.0-HR | m10a0.5-HR | m10a0.9-HR
---|---|---|---|---|---|---
$M_{\rm{BH}}\,(M_{\odot})$ | $5\times 10^{6}$ | $5\times 10^{6}$ | $5\times 10^{6}$ | $10^{7}$ | $10^{7}$ | $10^{7}$
$\langle\dot{M}\rangle(\dot{M}_{\rm{Edd}})$ | $\sim 11$ | $\sim 11$ | $\sim 11$ | $\sim 12$ | $\sim 16$ | $\sim 25$
$a_{*}$ | $0$ | $0.5$ | $0.9$ | $0$ | $0.5$ | $0.9$
$\langle\phi_{\rm{BH}}\rangle$ | 4.3 | 7.6 | 5.8 | 4.5 | 4.2 | 5.8
$\eta_{\rm{tot}}$ | 4.00% | 7.19% | 14.17% | 4.29% | 6.42% | 13.23%
$\eta_{\rm{wind}}$ | 0.23% | 0.52% | 0.88% | 0.10% | 0.22% | 0.72%
$\eta_{\rm{jet}}$ | 0.38% | 0.96% | 2.75% | 0.24% | 0.63% | 1.15%
$\eta_{\rm{rad}}$ | 3.33% | 4.64% | 12.34% | 3.76% | 4.39% | 5.48%
$R_{0}(r_{g})$ | $20$ | $20$ | $20$ | $20$ | $20$ | $20$
Duration ($t_{g}$) | $83,000$ | $83,000$ | $83,000$ | $83,000$ | $83,000$ | $81,200$
$N_{r}\times N_{\vartheta}$ | $640\times 256$ | $640\times 256$ | $640\times 256$ | $640\times 256$ | $640\times 256$ | $640\times 256$
$R_{\rm{in}}\,(r_{g})/R_{\rm{out}}\,(r_{g})$ | $10/3000$ | $10/3000$ | $10/3000$ | $10/3000$ | $10/3000$ | $10/3000$
$R_{\rm{min}}\,(r_{g})/R_{\rm{max}}\,(r_{g})$ | $1.1/10^{5}$ | $1.1/10^{5}$ | $1.1/10^{5}$ | $1.1/10^{5}$ | $1.1/10^{5}$ | $1.1/10^{5}$
Table 1: Simulation parameters and properties of the three simulations
presented in this work. We specify the BH mass ($M_{\rm{BH}}$), average
accretion rate ($\langle\dot{M}\rangle$), spin of the BH ($a_{*}$), average
magnetic flux ($\langle\phi_{\rm{BH}}\rangle$), total efficiency computed at
$r_{H}$ ($\eta_{\rm{tot}}$), wind and jet efficiency computed at $1000\,r_{g}$
($\eta_{\rm{wind}}$ and $\eta_{\rm{jet}}$), radiative efficiency computed at
$5000\,r_{g}$ ($\eta_{\rm{rad}}$), simulation duration in $t_{g}$, grid
resolution, inner and outer edges of the initial torus, and the inner and
outer radial boundaries of the simulation box. Note that the accretion rate
($\langle\dot{M}\rangle$), magnetic flux ($\langle\phi_{\rm{BH}}\rangle$), and
efficiencies ($\eta_{\rm{tot}},\,\eta_{\rm{wind}},\,\eta_{\rm{jet}}$ and
$\eta_{\rm{rad}}$) are computed using time averages over the final
$50,000\,t_{g}$ of each simulation. Figure 4: Here we show the initial torus
configuration for model m5a0.0-HR in terms of the gas density (colors) and
vector potential $A_{\phi}$ (contours). The solid purple (dashed green) lines
denote positive (negative) vector potential. Figure 5: Here we show the mass
accretion rate (top), normalized magnetic flux at the BH horizon (middle), and
bolometric luminosity (bottom) for each model. Also indicated is
$\phi_{\rm{BH}}=40$ (horizontal dashed line), which is approximately the
magnetic flux at which a geometrically thick disk will enter the MAD state.

###  3.3 Numerical Models/Initial Conditions

The domain outside of the gas torus is initialized with a low density, hot gas
with a density maximum at the BH horizon of:

$\rho_{\rm{atm,max}}=4.5\times
10^{-8}\left(\dfrac{M_{\rm{BH}}}{M_{\odot}}\right)^{-1}\ [{\rm{g\,cm^{-3}}}].$
(33)

The gas temperature at the BH horizon is set to $T_{\rm{atm,max}}=10^{11}$ K.
We assume that the atmosphere follows a profile similar to a spherical
accretion flow (1952MNRAS.112..195B) and set the gas density in the atmosphere
using a radial profile $\rho_{\rm{atm}}\propto r^{-3/2}$ while the internal
energy varies as $u_{g,\rm{atm}}\propto r^{-5/2}$. The radiation in the
atmosphere is initialized with a radiation temperature of $5\times
10^{4}(M_{\rm{BH}}/M_{\odot})$ K everywhere, which ultimately defines an
atmosphere with negligible radiation energy initially.

To initialize the torus, we assume that the stream rapidly circularized and
formed an accretion flow. It is important to note that this model assumes that
the circularization process happens on a timescale that is much shorter than
the fallback time. We also assume that angular momentum and binding energy are
conserved, which allows for simple initial conditions of the initial gas torus
to be defined.

Using these inputs, we follow the method described in 2004ApJ...605..307K to
initialize the simulation domain with an equilibrium torus which has its
angular momentum vector aligned with the BH spin vector. We set the radius of
the density maximum of the torus ($R_{0}$, see Table 1) to occur at
$R_{\rm{circ}}=2R_{p}$ and the binding energy is initialized using a constant
angular momentum torus with $l=\sqrt{R_{\rm{circ}}}$ and
$\epsilon=\epsilon_{\rm{mb}}$. This results in a tenuously bound,
geometrically thick torus where the inner edge of the torus ($R_{\rm{in}}$) is
smaller than $R_{0}$ and the outer edge ($R_{\rm{out}}$) is at 1000s of
$r_{g}$ (see values in Table 1). In each model, our choice of density maximum
results in a late time accretion rate of $\gtrsim 11\,\dot{M}_{\rm{Edd}}$.

The magnetic field is initialized in terms of the vector potential
($A_{\phi}$) using multiple field loops with varying polarity across the mid-
plane and in radius (see Figure 4). This guarantees that the accretion disk
remains at a low magnetic flux throughout. Without this initialization, the
chance advection of a large poloidal field loop towards the BH horizon can
drive the disk towards a MAD state, which we cannot accurately simulate in 2D.
The magnetic field is normalized such that the maximum pressure ratio
$\beta_{t,\rm{max}}\equiv(p_{\rm{gas,max}}+p_{\rm{rad,max}})/p_{\rm{mag,max}}=33$.
This choice of magnetic pressure is somewhat arbitrary, but is sufficient to
resolve MRI in the initial torus. As the simulations in this work are
performed in spherical $r-\vartheta$ coordinates, we employ a mean field
dynamo which mimics the dynamo processes that naturally take place in full 3D
simulations and prevent the magnetic flux from weakening throughout the
simulation.

Similar to 2022MNRAS.511.3795N, we use a grid which has more cells both near
the poles to resolve the outflow and in the midplane to resolve the disk. We
use the coordinate transformation from spatial simulation “code coordinates"
$(x_{1},x_{2})$ to Kerr-Schild grid coordinates $(r,\vartheta)$ introduced in
2017MNRAS.467.3604R. The simulation grid is logarithmically spaced in radius
$r$, while the polar angle $\vartheta$ is a function of both $x_{1}$ and
$x_{2}$, designed to concentrate resolution in the jet and disk regions.

We set the following parameters for the grid, where we use the same notation
as in 2017MNRAS.467.3604R. The hyperexponential break radius is set at $r_{\rm
br}=R_{\rm{max}}$ such that the entire radial grid is spaced logarithmically.
The collimation radii are $r_{\rm coll,jet}=1000r_{\rm g}$, $r_{\rm
coll,disk}=5r_{\rm+}$, the decollimation radii are $r_{\rm decoll,jet}=r_{\rm
decoll,disk}=2r_{\rm+}$. The power-law indices are $\alpha_{\rm 1}=1$,
$\alpha_{\rm 2}=0.25$. The fraction of the angular resolution concentrated in
the jet and disk are $f_{\rm jet}=0.35$, $f_{\rm disk}=0.4$. We ’cylindrify’
cells close to the axis at small radii by expanding their size in $\vartheta$
(2011MNRAS.418L..79T; 2017MNRAS.467.3604R); the cylindrification radius
$r_{\rm cyl}=20r_{\rm g}$ and $n_{\rm cyl}=1$. This is an important step as it
increases the minimum time step of the simulation and increases the
computational efficiency by nearly 2-3 times. The polar angle code coordinate
$x_{2}$ extends from $x_{\rm 2,min}=10^{-5}$ to $x_{\rm 2,max}=1-10^{-5}$,
where $x_{2}=0$ and 1 correspond to the two polar axes.

We set the resolution $N_{r}\times N_{\vartheta}$ such that the cells are
roughly 1:1 in most of the simulation domain. We use modified Kerr-Schild
coordinates with the inner edge of the domain inside the BH horizon. At the
inner radial boundary ($R_{\rm{min}}$), we use an outflow condition while at
the outer boundary ($R_{\rm{max}}$), we use a similar boundary condition and
in addition prevent the inflow of gas and radiation. Note that our choice of
$R_{\rm{min}}$ is such that at least 6 cells in the computational domain lie
inside of the horizon. At the polar boundaries, we use a reflective boundary.
Furthermore, the inner two cells at the jet axis are not evolved and instead
we copy the primitives from the 3rd cell from the polar axis to bolster
numerical stability. We employ a periodic boundary condition in azimuth. To
maintain numerical stability, we introduce mass in highly magnetized regions
of the simulation domain using a floor condition on the magnetization
$\sigma\equiv b^{2}/\rho\leq 100$ throughout each simulation.

We identify the simulations by the BH mass in units of
$m_{6}=M_{\rm{BH}}/10^{6}\,M_{\odot}$ and the dimensionless BH spin $a_{*}$.
For instance, m5a0.5-HR corresponds to a BH mass of $5\times
10^{6}\,M_{\odot}$ with $a_{*}=0.5$. See Table 1 for descriptions of each
model.

##  4 Results

Figure 6: A zoomed in view of the accretion flow and funnel of m5a0.0-HR at
$t=83,000\,t_{g}$. The colors show the gas density (top) and radiation energy
density (bottom). The streamlines indicate the fluid velocity (top) and
radiative flux (bottom). We also indicate the photosphere ($\tau_{\rm{es}}=1$,
yellow line). Figure 7: Here we present the fraction of energy flux in the top
jet for model m5a0.0-HR (solid lines) and m5a0.9-HR (dashed lines) in the form
of radiation energy flux ($R^{r}_{\ \,t}$, green), kinetic plus gravitational
binding energy flux ($\rho u^{r}u_{t}+\rho u^{r}$, blue), and magnetic energy
flux ($b^{2}u^{r}u_{t}-b^{r}b_{t}$, orange). The data has been time averaged
over $t=33,000-83,000\,t_{g}$. For radii $r\lesssim 5000r_{g}$, we observe a
continual conversion of radiation energy to kinetic and binding energy. Beyond
roughly $5000\,r_{g}$, the fraction of energy flux in radiation steadily
increases. This is likely due to a combination of dissipation of kinetic
energy as well as the fact that the radiation is no longer forced to flow
through the funnel for $r\gtrsim 3000\,r_{g}$ and can escape laterally. The
magnetic energy flux in the jet also appears to increase with spin, which
contributes to accelerating lower density gas along the jet axis. Figure 8:
Here we show a zoomed out view of m5a0.0-HR at $t=83,000\,t_{g}$. Note that we
reflect the data about the $x$-axis for visualization purposes, but the
simulation was run in 2D $r-\vartheta$ coordinates. The colors show the gas
density (top left), radiation energy density (2nd panel, left), gas
temperature (3rd panel, left), magnetic field strength (bottom left), gas
lorentz factor $\gamma$ (top right), gas magnetization $\sigma$ (2nd panel
right), gas pressure to magnetic pressure ratio $p_{\rm{gas}}/p_{\rm{mag}}$
(3rd panel, right), and gas to radiation pressure ratio
$p_{\rm{gas}}/p_{\rm{rad}}$ (bottom right). Figure 9: The same as Figure 8,
but for model m5a0.9-HR at $t=83,000\,t_{g}$. Note that we have increased the
range of $\lambda$ to highlight the maximum jet velocity. The simple addition
of a spinning BH has a noticeable effect in many characteristics of the jet.
Namely, the jet propagates farther owing to a high velocity component near the
poles with $\gamma>2$. In addition, the jet is hotter along the jet axis and
jet head. Lastly, the magnetic field strength in the jet is larger at larger
radii ($r>30,000\,r_{g}$), which leads to more of the gas becoming strongly
magnetized with $\sigma>1$. Regions where the jet pinches towards the polar
axis (i.e. at $z\approx 25,000\,r_{g}$) signifies recollimation shocks are
also present.
Figure 10: Here we present radial profiles of the density weighted outflow
velocity in the jet for each model at the final snapshot. The behaviour is
similar in each model. The density weighted outflow velocity shows
acceleration between $10^{3}\lesssim r/r_{g}\lesssim 10^{4}$. Beyond
$10^{4}\,r_{g}$, the velocity seems to have reached a plateau of $\sim
0.2-0.35c$.

###  4.1 Accretion Flow Properties

We show the accretion rate, magnetic flux at the BH horizon, and the
luminosity of outgoing radiation computed directly from the KORAL data in
Figure 5. For each model in Figure 5, the accretion rate does not remain at
the initial accretion rate (e.g. $\sim 100$ for m10a0.0-HR) and instead goes
through phases of high and low accretion, eventually settling into a nearly
steady state accretion phase for $t\gtrsim 25,000\,t_{g}$. The magnetic flux
in each simulation is generally $\phi_{\rm{BH}}\lesssim 5$, which is well
below the limit for the MAD state ($\phi_{\rm{BH}}\gtrsim 40$).

There are, occasional periods where a loop of poloidal magnetic field is
advected towards the BH horizon and momentarily drives $\phi_{\rm{BH}}$ near
the MAD limit. As indicated by the accretion rate, this does not disrupt the
accretion flow (which typically occurs for strongly magnetized accretion flows
at the MAD limit in 2D), but the Blandford-Znajek (1977MNRAS.179..433B)
process will extract some spin energy from the BH and momentarily increase the
jet power in the case of a spinning BH. The most extreme increase in
$\eta_{\rm{tot}}$ due to the Blandford-Znajek process occurs in model
m10a0.9-HR as $\phi_{\rm{BH}}$ around $t=20,000\,t_{g}$ increases the total
efficiency to $\eta_{\rm{tot}}\sim 110$% momentarily, which is to be expected
in MAD accretion disks around rapidly rotating BHs (2019MNRAS.483..565C). The
effect of this is a brief order of magnitude increase in the escaping
radiation roughly $5000\,t_{g}$ later due to the light crossing time between
where we measure $\eta_{\rm{tot}}$ versus where we measure $L_{\rm{bol}}$. The
two peaks in $L_{\rm{bol}}$ around $t=25,000\,t_{g}$ are due to the increase
in radiation flowing through the top and bottom jets crossing the
$r=5000\,r_{g}$ surface at different times.

The other models do not have as dramatic an increase in their efficiency or
$L_{\rm{bol}}$, even during later spikes in $\phi_{\rm{BH}}$. These events are
extremely short lived (lasting at most $100\,t_{g}$, which is the cadence at
which we save snapshots of the simulations) and rare, so these deviations are
not the driving factor in the energetics and jet evolution. Aside from the
short increase in $L_{\rm{bol}}$ for model m10a0.9-HR at $t=25,000\,t_{g}$,
the escaping radiation is slightly super-Eddington at $L_{5000}\approx
3L_{\rm{Edd}}$.

Since each of the simulations has similar disk features, we focus on m5a0.0-HR
to describe the inner accretion flow properties. Visualizations of the
simulation data for m5a0.0-HR are shown in Figures 6. The accretion disk is
evidently thick and turbulent, with the turbulence being driven by MRI. The
gas accretes onto the black hole primarily along the equatorial plane. The
flow is significantly turbulent even at relatively large radii. The turbulent
structure of the velocity streamlines is in part the result of material near
the black hole gaining energy and being launched back into the disk. For such
a low binding energy disk, small perturbations can lead to the material
becoming unbound quite easily. Outflows driven by radiation and Poynting flux
are evident within $\sim 45^{\circ}$ from the pole. For an in depth
description of the accretion flow of similar SANE models, see
2019MNRAS.483..565C.

To characterize the acceleration of gas in the jet, we compute the
contributions to the energy flux in the $\rm{Be}>0.05$ region for the top jet.
Namely, we compute the kinetic plus gravitational binding energy
($\dot{E}_{\rm{kin+bind}}$), internal ($\dot{E}_{\rm{int}}$), magnetic
($\dot{E}_{\rm{mag}}$), and radiative ($\dot{E}_{\rm{rad}}$) energy fluxes.
The energy fluxes are computed as:

$\dot{E}_{\rm{kin+bind}}(r)=-2\pi\int_{0}^{\vartheta_{\rm{jet}}(r)}\sqrt{-g}\,(\rho
u^{r}u_{t}+\rho u^{r})\,d\vartheta,$ (34)
$\dot{E}_{\rm{mag}}(r)=-2\pi\int_{0}^{\vartheta_{\rm{jet}}(r)}\sqrt{-g}\,(b^{2}u^{r}u_{t}-b^{r}b_{t})\,d\vartheta,$
(35)

and

$\dot{E}_{\rm{rad}}(r)=-2\pi\int_{0}^{\vartheta_{\rm{jet}}(r)}\sqrt{-g}\,R^{r}_{\,\
t}\,d\vartheta,$ (36)

respectively. We ignore the flux of gas internal energy since the gas is
radiation pressure dominated. We illustrate the acceleration of gas by
radiation in model m5a0.0-HR in Figure 7. For radii $50\lesssim
r/r_{g}\lesssim 5000$, we see a steady conversion of radiative flux to kinetic
plus binding energy flux. However, the fraction of energy flux in the form of
radiation in the jet begins to increase for $r\gtrsim 5000\,r_{g}$. We
attribute this to two factors. First, there is some conversion of kinetic
energy into thermal and radiation energy due to internal shocks within the jet
(which we illustrate at the end of this section). In addition, there is also
the fact that the radiation is no longer laterally confined by the funnel
walls for $r\gtrsim 3000\,r_{g}$. Once the radiation can escape laterally, the
radiation force supplied to gas along the jet axis declines. The fact that the
fraction of radiation energy flux in the jet begins to climb for $r\gtrsim
5000\,r_{g}$ suggests that the optically thick gas is weakly accelerated by
radiation at larger radii.

We present a snapshot of the jet resulting from the accretion flow in Figure
8. Much of the radiation energy density is contained within the disk near the
equatorial plane. Radiation is advected in with the accretion flow and escapes
out through the funnel, driving a mildly relativistic outflow ($\gamma\lesssim
1.2$). As described in 2015MNRAS.453.3213S, this process occurs because
optically thick gas flows from the disk into the funnel region and is
subsequently accelerated by the radiation streaming through the funnel. The
non-spinning BH model m5a0.0-HR reaches similar gas velocities in the jet as
the models presented in 2015MNRAS.453.3213S, which were also of $a_{*}=0$ BHs.
The disk is radiation pressure dominated, but contributions from radiation
energy to the total pressure in the jet will be negligible due to the low
optical depth. However, the magnetic pressure can become quite large in the
jet and sometimes exceeds the gas pressure within the jet. Some regions reach
above the magnetization $\sigma$ of unity. The map of the magnetization
$\sigma$ shows that it is primarily near the pole where the gas becomes
strongly magnetized.

Despite maintaining $\phi_{\rm{BH}}\sim 4-7$ in each model, the spin $a_{*}>0$
models each show increased jet power and relativistic ($\gamma>2$) but low
density outflows near the poles. We illustrate this for m5a0.9-HR in Figure 9.
We also note that the magnetic field strength is greater along the extended
jet and, as a consequence, the jet is more magnetized in general as indicated
by the extent of the $\sigma>1$ region. The fractional energy flux profiles in
model m5a0.9-HR (Figure 7) illustrate that the simulations show an increase in
Poynting flux in the jet as the spin increases. The high velocity component
with $\gamma>2$ is likely due to Poynting acceleration, which is more
efficient in more magnetized jets (2019MNRAS.490.2200C). Also of note is the
fact that the funnel is optically thin down to $r\approx r_{H}$ near the poles
(Figure 6), which suggests radiative acceleration will be weak near the poles
and cannot accelerate the $\gamma>2$ component. In addition, the trend of
increasing jet velocity as spin increases was demonstrated in GRMHD SANE
models by 2013MNRAS.436.3741P.

Although we performed similar simulations in 2019MNRAS.483..565C but with
$M_{\rm{BH}}=10^{6}\,M_{\odot}$, the SANE $a_{*}=0.9$ model in that work
(model s09) did not produce a relativistic jet (2019MNRAS.483..565C, Figure
C5), contrary to the models with $a_{*}\geq 0.5$ in this work. The results for
the outflow in s09 were assumed to be robust since the angular resolution of
$N_{\vartheta}=320$ is large compared to many previous simulations in the
literature. For instance, SANE models presented by 2013MNRAS.436.3741P
launched a relativistic jet with only $N_{\vartheta}=128$. As we demonstrate
in Appendix A, it is necessary to employ adequate resolution in the polar
region for the jet launching and structure to be captured. This is possibly
due to the large scale initial torus that we employ since the flow properties
can be effected by the scale of the initial torus as was illustrated by
2019MNRAS.490.2200C.

The increased jet power and magnetization in the jet as the BH spin increases
suggests the Blandford-Znajek mechanism is a driving factor. Although, the
total jet efficiency will be very weak ($\lesssim 1$%) assuming the magnetic
flux and BH spin values in Table 1. Given the appearance of a relativistic jet
in our models, which are a more conservative assumption for the accretion disk
state than a MAD scenario, one wonders why the vast majority of TDEs appear to
not have a jet and if the SANE scenario can still apply to non-jetted TDEs.

The fact that TDEs likely form misaligned disks provides a plausible
explanation, especially in the case of weak jets, since a weak jet launched
along the BH spin axis will have a hard time penetrating the surrounding TDE
debris. In addition, the flux of poloidal magnetic field threading the BH,
which is the result of our initial conditions and the magnetic dynamo that we
employ, may be even lower than in our simulations. Assuming a purely
Blandford-Znajek powered jet, even a magnetic flux of unity would reduce the
jet power by more than factor of 10.

Since the accretion rate is only marginally super-Eddington, radiation can
more efficiently escape and the radiative efficiency $\eta_{\rm{rad}}$ is only
a few percent smaller than the Novikov-Thorne (NT) efficiency as a result. The
accretion flow generates a jet and wind with a total energy flux that is a
fraction of the accretion power. We find
$(\eta_{\rm{wind}}+\eta_{\rm{jet}})\approx 0.6-3.6\%$. We also observe a clear
trend of increasing energy outflowing in the jet and wind as the spin
increases for a given BH mass, reflecting the overall increase in energy
extracted from the system ($\eta_{\rm{tot}}$) as the BH spin increases.

Although the jet power increases and the jet near the polar axis becomes
significantly more relativistic as the spin increases, the density weighted
outflow velocity is surprisingly similar across BH mass and spin. As we show
in Figure 10, each model only achieves an outflow velocity of $\sim 0.2-0.35c$
in a density weighted sense. The acceleration of the gas between
$10^{3}\lesssim r/r_{g}\lesssim 10^{4}$ is also clearly illustrated. This is
in agreement with the terminal density weighted jet velocity of $\approx 0.3c$
reported by 2015MNRAS.453.3213S, but we have demonstrated that the spin and
jet magnetization does not appear to play a role in the velocity of the higher
density component of the jet in SANE, super-Eddington accretion disks with
$\dot{M}\gtrsim 11\dot{M}_{\rm{Edd}}$.

Figure 11: Here we show the gas (right) and radiation (left) entropy in the
top panel. The bottom panel shows a trace of the radial velocity (red) and
both gas and radiation entropy (blue) taken at $\vartheta=3^{\circ}$
(indicated as the pink line in the top panel). Dissipation leads to an
increasing entropy in both gas and radiation overall. The sharpest jump occurs
at the jet head ($z\approx 40,000\,r_{g}$). There are also several jumps in
entropy down the jet axis (beginning at $r\lesssim 30,000\,r_{g}$ in this
snapshot) which are internal shocks driven by fast moving gas shocking with
slow moving gas within the jet. Similar features are seen in each simulation
presented in this work.

To search for dissipation in the jet, we examine the gas entropy per unit
mass:

$s_{\rm{gas}}=\dfrac{1}{\gamma_{g}-1}\ln{\left(\dfrac{p_{\rm{gas}}}{\rho^{\gamma_{g}}}\right)},$
(37)

and the log scaled radiation entropy per unit mass:

$s_{\rm{rad}}=\log_{10}\left(\dfrac{4aT^{3}}{3\rho}\right).$ (38)

The variable velocity of gas flowing along the jet is expected to lead to
shocks, wherein dissipated kinetic energy leads to heating of the gas and thus
an increase in internal energy. In addition, we observe multiple recollimation
features, which are also sites of shocks and dissipation, along the jet axis
which appear more significant in the models with $a_{*}>0$ (see the top left
panel in Figure 9 for instance). Recollimation features have been seen in
various numerical studies of jets (2012MNRAS.422.2282K; 2012ApJ...750...68L;
2015ApJ...809...38M; 2016MNRAS.456.1739B; 2017A&A...606A.103H) and it has been
demonstrated that recollimation shocks appear due to a pressure mismatch
between the jet and the surrounding medium. We consider the radial velocity,
gas entropy, and radiation entropy in m5a0.5-HR in Figure 11. The trace of the
entropy in both gas and radiation at $\vartheta\approx 3^{\circ}$ shows quite
clearly that (i) a strong shock exists at the jet head, (ii) internal shocks
are present within the jet, (iii) the entropy shows similar oscillations to
the velocity, and (iv) the entropy generally increases between $z\sim
5000-40,000\,r_{g}$. Taken together, this suggests that the jet head is where
dissipation is largest and we are indeed observing dissipation of kinetic
energy via internal shocks within the jet at radii smaller than the jet head.

Figure 12: Here we show the half opening angle of the jet (solid lines) for
each model. Note that we have symmetrized the data by taking the mean of the
half opening angle measurement for the bottom and top jet. We find that the
jet expands roughly conically from $z\approx 1000\,r_{g}$ until nearly
$z\approx 40,000\,r_{g}$ in each model.

###  4.2 Jet Structure

We define the jet boundary using the Bernoulli parameter via Equation 19 and
measure the half opening angle $\vartheta_{\rm{jet}}$ for both the top and
bottom jet as a function of distance along the jet ($z$, see Figure 12). We
find nearly identical opening angles for both the top and bottom jet in each
model. We find that at distances $r_{H}/r_{g}<z/r_{g}\lesssim 1000$ the jet
undergoes very nearly parabolic expansion with a power-law of roughly
$\omega\propto z^{0.7}$, where $\omega$ is jet width. The jet appears to
maintain a roughly conical structure at jet lengths $1000\lesssim
z/r_{g}\lesssim 30,000$ with $\vartheta_{\rm{jet}}$ maintaining a nearly flat
profile. For $z/r_{g}\gtrsim 40,000$, the jet angle shrinks rapidly at the jet
head.

An important caveat that we must point out is that the break between the
parabolic and conical region is a byproduct of the choice of initial torus. In
each model, the break occurs very close to the radial extent of the initial
torus model ($\sim 3000\,r_{g}$, see Table 1). This point marks the transition
from higher density gas, which can provide lateral pressure support, to the
substantially lower density atmosphere, which provides negligible pressure
support in comparison.

Conically expanding jets were successfully applied to ASASSN-14li in the
models of 2018ApJ...856....1P, who modeled the radio synchrotron emission as a
superposition of synchrotron emitting regions in a conically expanding jet.
Figure 12 demonstrates that SANE super-Eddington disks produce conically
expanding jets, at least with our chosen initial torus configuration. The
half-opening angles that we find are all larger than the best fit models of
2018ApJ...856....1P. We note that their model assumed a previously cleared
funnel was present while we have assumed a low density atmosphere which does
not provide substantial lateral confinement. Future work on TDE jets should
explore the effects of the atmosphere and initial torus in greater detail.

Figure 13: Here we show the synchrotron emissivity (top panel) and the
bremsstrahlung emissivity (bottom panel) at 1 GHz for m5a0.0-HR. We note that
the spectra as presented in Sections 5.1 and 5.2 peak at $>10$ GHz; however,
the choice to show the 1 GHz emissivities is to better visualize the spatial
difference between bremsstrahlung and synchrotron emission in the jet. The
synchrotron emissivity is dominant outside of the optically thick disk. The
bremsstrahlung emissivity shown here is representative of the low frequency
emissivity as the bremsstrahlung cutoff occurs at $\nu>10^{16}$ Hz, far beyond
the frequencies we consider in this work.

##  5 Synchrotron Emission Imaging Analysis

In Figure 13, we compare the emissivity of synchrotron and bremsstrahlung
processes in m5a0.0-HR. We compute the emissivities using functions defined in
2017MNRAS.466..705S since these are the frequency dependent functions
currently used by KORAL. Note that the equations presented in
2017MNRAS.466..705S describe a two temperature plasma while we assume a single
temperature plasma with $T_{i}=T_{e}=T_{\rm{gas}}$ to compare the
contributions of synchrotron and bremsstrahlung processes. We use the fitting
function for the ultrarelativistic synchrotron emissivity in Gaussian-CGS
units:

$j_{\rm{Synch}}=4.43\times
10^{-30}\nu_{M}n_{e}\dfrac{x_{M}I^{\prime}(x_{M})}{2\theta_{e}^{2}}\,\left[\dfrac{{\rm{erg}}}{{\rm{cm^{3}\,s\,Hz}}}\right],$
(39)

where $n_{e}\equiv\rho/(\mu m_{p})$ is the electron number density in a fully
ionized gas given a mean molecular weight $\mu$ and proton mass $m_{p}$,
$\theta_{e}\equiv kT_{\rm{gas}}/m_{e}c^{2}$ is the dimensionless electron
temperature, $x_{M}=\nu/\nu_{M}$, $\nu_{M}=(3/2)eB\theta_{e}^{2}/2\pi m_{e}c$
is the characteristic synchrotron frequency, and $I^{\prime}(x_{M})$ is a
fitting function provided by 1996ApJ...465..327M:

$I^{\prime}(x_{M})=\dfrac{4.0505}{x_{M}^{1/6}}\left(1+\dfrac{0.40}{x_{M}^{1/4}}+\dfrac{0.5316}{x_{M}^{1/2}}\right)\exp(-1.8899x_{M}^{1/3}).$
(40)

The bremsstrahlung emissivity is computed as:

$\begin{split}j_{\rm{Br}}=&\dfrac{6.8\times
10^{-38}}{4\pi}T_{\rm{gas}}^{-1/2}n_{e}^{2}\overline{g}R(T_{\rm{gas}})\times\\\
&\exp(-h\nu/kT_{\rm{gas}})\left[\dfrac{{\rm{erg}}}{{\rm{cm^{3}\,s\,Hz}}}\right],\end{split}$
(41)

where we assume a Gaunt factor $\overline{g}=1.2$ and $R(T_{\rm{gas}})\equiv
1+4.4\times 10^{-10}(T_{\rm{gas}}/\rm{1\,K})$ is a relativistic correction
adopted from 1979rpa..book.....R. The frequency $\nu_{M}$ sets the peak of the
synchrotron emission in this formulation.

We find that the synchrotron emission dominates the jet while bremsstrahlung
dominates in the optically thick disk, thus we expect the jet emission will
largely be from synchrotron processes. The periodic outflows appear as
discrete ‘bubbles’ of high synchrotron emission along the jet. Here the gas is
hotter and high frequency synchrotron emission ($\nu\gtrsim 10^{11}$) is
expected as a result. We note however that these computations were done
directly from the KORAL data and do not represent the spatial intensity
information as we have not accounted for opacity effects, viewing angle, and
resolution, all of which may change which features can be observed.
Nevertheless, this analysis demonstrates that the jet emission is not
continuous. The previous description is true of each model considered in this
work.

We also estimate the Compton-$y$ parameter
($y=\tau_{\rm{es}}kT_{\rm{gas}}/m_{\rm{e}}c^{2}$, 1979rpa..book.....R) in the
extended jet to test whether Compton effects are important. We find that $y\ll
1$ throughout the bulk of the jet, so Compton effects will also be minimal due
to the low opacity. As such, the synchrotron emission alone can provide a
reasonable model of the jet emission.

We make use of the GRRT code ipole (2018MNRAS.475...43M) to produce images of
the jet that include ray-tracing and radiative transfer effects. Before post-
processing the data, we set the gas density to zero, $\rho=0$, in regions
where $\sigma>1$. GRMHD codes inject mass and energy in these regions to keep
the simulation stable. As such, the accuracy of the radiation field from these
regions is less certain than where $\sigma<1$. This method, which is the most
conservative choice, has been employed in other studies of ray-traced GRMHD
simulations (2019MNRAS.486.2873C). Because ipole does not presently include
bremsstrahlung or Compton effects, we also cut out data for $r<5000\,r_{g}$
prior to ray-tracing.

Figure 14: Here we show spectra for m5a0.0-HR (top), m5a0.5-HR (middle), and
m5a0.9-HR (bottom) computed for viewing angles of $10^{\circ}$ (left) and
$90^{\circ}$ (right) at snapshots ranging from $t=28,000-78,000\,t_{g}$. In
each case, the jet is generally brightening across all frequencies; however,
the jet becomes more energetic and the peak frequency becomes larger as the
spin increases. Figure 15: The same as Figure 14 but for models m10a0.0-HR
(top), m10a0.5-HR (middle), and m10a0.9-HR (bottom). In all cases, the
spectrum is generally shifting to the left and the peak frequency also
decreases as the jet expands and some of the gas cools. However, m10a0.0-HR
begins to dim substantially and the 230 GHz luminosity drops from $\sim
5\times 10^{39}\,\rm{erg\ s^{-1}}$ at $t\leq 58,000\,t_{g}$ to $\sim
10^{38}\,\rm{erg\ s^{-1}}$ after $t=68,000\,t_{g}$. Model m10a0.9-HR on the
other hand is still brightening at 230 GHz by the final snapshot. Figure 16:
Here we present model spectra of each model produced by varying
$T_{i}/T_{e}=\mathcal{R}$. Here we use a camera angle of $\theta=90^{\circ}$
(viewing the disk/jet edge on). Increasing $\mathcal{R}$ has the effect of
decreasing both the peak frequency and luminosity in each model. Even a modest
temperature ratio of $\mathcal{R}=5$ can diminish the jet luminosity by 1-2
orders of magnitude. A temperature ratio of $\mathcal{R}\geq 5$ may also
substantially diminish the 230 GHz flux for colder jets which has important
consequences for detecting and resolving potential sources.

###  5.1 Time Evolution and Viewing Angle Dependence of Emission for
$T_{e}=T_{i}$

We present viewing angle dependent spectra for the $M_{\rm{BH}}=5\times
10^{6}\,M_{\odot}$ models computed over $\nu=10^{9}-10^{12}$ Hz for snapshots
ranging from $t=28,000-78,000\,t_{g}$ in Figure 14. Here we assume a single-
temperature gas with $T_{e}=T_{i}=T_{\rm{gas}}$, where $T_{\rm{gas}}$ is taken
directly from the KORAL simulation.

Focusing on the spectrum over time for each BH spin, we find that the jet is
generally brightening across all frequencies. The emission becomes
significantly brighter as the spin increases, reflecting the substantial
increase in jet power as the spin increases (see Table 1). We observe weak
beaming comparing the $10^{\circ}$ and $90^{\circ}$ spectra, as indicated by
the shift to the left of each spectrum as the viewing angle increases.

For the $M_{\rm{BH}}=10^{7}\,M_{\odot}$ models (Figure 15), we again observe
only weak beaming and the jet becomes brighter as the BH spin increases, but
there are several notable differences. Firstly, the jet in m10a0.0-HR and
m10a0.5-HR is actually dimming over time. For instance, at $t=28,000\,t_{g}$,
the jet in m10a0.0-HR has a peak luminosity of $\sim 5\times 10^{39}\,\rm{erg\
s^{-1}}$ and a peak frequency near 700 GHz, but by $t=78,000\,t_{g}$ the jet
has become less bright with a peak luminosity of $\sim 5\times
10^{38}\,\rm{erg\ s^{-1}}$ and a peak frequency near 100 GHz. The jet in
m10a0.9-HR on the other hand is still brightening by the time we stop the
simulation.

We note that the behaviour of m5a0.0-HR is atypical amongst the simulations
presented in this work in that the jet first brightens, dims, and then
rebrightens before the end of the simulation. After quickly brightening
following the onset of accretion, the jet initially becomes less luminous and
the peak frequency decreases until it reaches only $\sim 40$ Hz at
$t=58,000t_{g}$. For $t\lesssim 58,000\,t_{g}$, the jet emission is dominated
by material near the head of the jet and not very much emission is present
within the jet itself. As material at the jet head expands and cools, the
brightness declines and the spectrum shifts to lower frequencies overall.

However, at $\sim 40,000\,t_{g}$, a high velocity (density weighted velocity
of $\sim 0.1c$) component appears at the base of the jet, passing through the
slower moving outflow and decelerating as it dissipates its kinetic energy. By
$\sim 68,000\,t_{g}$, this component has caught up to material downstream and
drives internal shocks within the jet between $z\sim 5000-20,000\,r_{g}$.
These shocks cause a significant amount of heating, driving the gas
temperature where shocks occur to increase from $\sim 10^{9}$ K to $\sim
10^{10}$ K. The internal shocks lead to high frequency emission which is
substantially brighter than that from the jet head alone. For instance, by
$t=78,000\,t_{g}$ the jet has brightened to a luminosity of nearly $5\times
10^{39}\,\rm{erg\ s^{-1}}$ at the peak frequency versus the initial $\sim
10^{38}\,\rm{erg\ s^{-1}}$ at $t=28,000\,t_{g}$. This demonstrates clearly how
significant the contribution of emission due to internal shocks is in the
context of these models.

With the exception of m5a0.0-HR and m10a0.0-HR, the difference between the
$M_{\rm{BH}}=5\times 10^{6}\,M_{\odot}$ and $M_{\rm{BH}}=10^{7}\,M_{\odot}$
models at late times ($t\gtrsim 58,000\,t_{g}$) is in part because the jets
have propagated a shorter physical distance due to the shorter time in
physical units. Although each model was run $81,200-83,000\,t_{g}$ this only
corresponds to $\sim 24$ days for $M_{\rm{BH}}=5\times 10^{6}\,M_{\odot}$ and
$\sim 48$ days for $M_{\rm{BH}}=10^{7}\,M_{\odot}$. Consequently, the jets for
$5\times 10^{6}\,M_{\odot}$ have propagated roughly half the distance of the
jets for $M_{\rm{BH}}=10^{7}\,M_{\odot}$. The jets for the
$M_{\rm{BH}}=5\times 10^{6}\,M_{\odot}$ models may also show dimming given a
longer run time; however, m5a0.0-HR suggests that rebrightening events can
also occur.

It is interesting to note the evolution of m10a0.0-HR, which is only bright at
230 GHz until $t=58,000\,t_{g}$. This suggests that high frequency
radio/submillimeter emission (i.e. $\nu>100$ GHz) from TDE jets may only last
for several weeks depending on the BH spin. However, we have not taken into
account the effects of varying the external medium nor the possible
misalignment between the BH spin and accretion disk in this work. These two
factors will have a non-negligible effect on the jet power and the jet
evolution. Future studies should explore these factors to better understand
TDE jets.

###  5.2 Effects of Two-Temperature Plasma

Internal shocks in the jet are expected to produce non-thermal electrons,
which we do not model in this analysis. In addition, these shocks are expected
to lead to differential heating of electrons and ions. The plasma will retain
memory of this because electrons and ions and cannot efficiently thermalize in
low density plasmas. This effect has been treated in studies of hot accretion
flows using a simple two-temperature prescription. For instance, it is common
to define an electron temperature which depends on $\beta_{g}$ to
differentiate the electron temperature in the jet and disk separately
(2016A&A...586A..38M). Since we only model the jet emission, we adopt a simple
approach and define the electron temperature via:

$\dfrac{T_{i}}{T_{e}}=\mathcal{R},$ (42)

where $T_{i}$ is set to $T_{\rm{gas}}$ from the KORAL simulation. This
simplified prescription smooths over the microphysics, which depend on
magnetic reconnection and shock properties; however, it does provide some
handle on how the electron and ion populations must differ within a particular
model to produce a specific emission property, albeit in a parameterized
fashion. A recent study of electron heating in jets in active galactic nuclei
(2019Galax...7...14O; 2020MNRAS.493.5761O) has demonstrated that $T_{i}/T_{e}$
can become as large as $10-100$ depending on the heating physics. In these
simulations, shocks led to ion heating while weak Coulomb coupling prevented
the ion and electron populations from equilibrating. Our simulation results
indicate that internal shocks due to variable ejections and recollimation take
place. Furthermore, the time between Coulomb collisions is $>10^{5}$ years for
typical densities and temperatures in the jet, thus it is reasonable to model
the electrons in the jet with $T_{i}/T_{e}>1$.

We present full spectra at selected times for each model in Figure 16. In all
models, we find that increasing $\mathcal{R}$ decreases the peak frequency and
overall luminosity. In model m10a0.5-HR at $t=83,000\,t_{g}$ for instance, as
$\mathcal{R}$ is varied from $1$ to $20$, the peak frequency shifts from $\sim
200$ GHz to $\sim 20$ GHz and the luminosity declines by nearly four orders of
magnitude from $\sim 10^{41}\,\rm{erg\ s^{-1}}$ to $\sim 10^{37}\,\rm{erg\
s^{-1}}$. As models m5a0.0-HR, m10a0.0-HR and m10a0.5-HR illustrate, the
electron temperature is extremely important for the high frequency emission as
each of these models show greatly diminished emission at 230 GHz for
$\mathcal{R}>2-5$. For models m5a0.5-HR, m5a0.0-HR and m10a0.5-HR, the jet
luminosity at $\nu\gtrsim 230$ GHz is not as strongly diminished as
$\mathcal{R}$ increases and $\mathcal{R}>20$ is required to drop the peak
frequency below $230$ GHz. We explore the effects of $\mathcal{R}$ on the
detectability of these models in the following section.

Studies of electron and ion heating suggest that the electron temperature
might depend on $\beta_{g}$ (e.g. 2010MNRAS.409L.104H; 2017ApJ...850...29R)
which would introduce spatial variation in $\mathcal{R}$. This would
significantly change the behaviour in the spectra if regions that emit the
most also have large values of $\mathcal{R}$. Without a heating prescription
implemented during the simulation, however, we do not think it is justified to
use a model which varies with $\beta_{g}$. That being said, a more accurate
prescription may also need to account for the radiation pressure and use a
two-temperature model which scales based on $\beta_{t}$. We leave a precise
analysis of the spatial electron temperature ratio to a future analysis.

Figure 17: Here we present GRRT images of the thermal synchrotron emission
(mapped via the brightness temperature $T_{b}$) at $\nu=230$ GHz with a
viewing angle of $\theta=90^{\circ}$ for model m10a0.9-HR. Here we have simply
assumed $T_{e}=T_{i}$. We plot a white line of length $25,000\,r_{g}$ in the
left column for scale. We show snapshots at $28,000\,t_{g}$ (top row),
$58,000\,t_{g}$ (middle row), and $81,200\,t_{g}$ (bottom row). We show the
infinite resolution images in the 1st column from the left. In addition, we
apply Gaussian smoothing with a FWHM of $\Delta R/r_{g}=$ 10000 (2nd column
from the left), 25000 (3rd column from the left), and 50000 (right column) to
illustrate the effects of blurring due to distance and resolution. The jet
head is the brightest feature and in the infinitely resolved case appears to
show bow shock features. For well resolved images, bright ‘bubbles’ along the
jet axis can be seen, similar to what we presented in the emissivity maps in
Figure 13. The jet head features are best resolved when $\Delta R\lesssim
25,000\,r_{g}$. Observations with similar resolution scales may allow for
direct measurement of the ejecta velocity.

Figure 18: Similar to Figure 17, but here each row shows m10a0.9-HR at
$t=81,200\,t_{g}$ but with a viewing angle of $\theta=10^{\circ}$ (top row),
$\theta=45^{\circ}$ (middle row), and $\theta=90^{\circ}$ (bottom row). The
images at $\theta=45^{\circ}$ illustrate that internal shock features cannot
be distinguished even for fairly well resolved sources as the viewing angle
decreases from edge-on. For jets viewed nearly down the axis (top row) only
the jet heads are seen and the jet appears as two blurred lobes. For poorly
resolved sources, a nearly face-on jet begins to look like a compact source.

Figure 19: Similar to Figure 18, but here we show m10a0.0-HR at
$t=58,000\,t_{g}$ (top row) and $t=83,000\,t_{g}$ (bottom row) to illustrate
the dimming of the jet. The jet in m10a0.0-HR did not show significant
internal shock heating and as a result the jet head is responsible for most of
the emission. As the jet head expands and cools, it eventually becomes dim,
and possibly undetectable.

Figure 20: Similar to Figure 19, but here we show m5a0.0-HR at
$t=58,000\,t_{g}$ (top row) and $t=83,000\,t_{g}$ (bottom row) to illustrate
the dimming of the jet. Similar to m10a0.0-HR, m5a0.0-HR also did not
initially show significant internal shock heating but the initial jet was also
quite weak and not significantly bright. By $t=68,000\,t_{g}$, the jet had
brightened substantially due to internal shocks and shocks near the jet head
where fast moving gas caught the slower moving jet head and energized it.

###  5.3 230 GHz Images

In this section, we analyze viewing angle dependent thermal synchrotron images
at 230 GHz which were calculated assuming $T_{e}=T_{i}$ and employ the same
methods as in Section 5.1. In addition, we apply a Gaussian smoothing function
with a FWHM of $\Delta R$ to each model to simulate the effects of the
resolved angular scale ($\Delta\theta$) and distance to the source ($D$). The
resolved scale in $r_{g}$ can be related to both quantities by:

$\dfrac{\Delta R}{r_{g}}\approx
2022\left(\dfrac{M_{\rm{BH}}}{10^{6}\,M_{\odot}}\right)^{-1}\left(\dfrac{D}{1{\rm{\,Mpc}}}\right)\left(\dfrac{\Delta\theta}{20{\rm{\mu
as}}}\right).$ (43)

For a given resolution scale, the image could represent a jet that is both
distant and well resolved or nearby and poorly resolved. We blur each image
with a Gaussian using $\Delta R/r_{g}=(10000,25000,50000)$. We first focus on
m10a0.9-HR to describe key emission features as it is the brightest model with
the largest physical scale.

In Figure 17, we show the time evolution of the jet emission for a viewing
angle of $90^{\circ}$ in the leftmost column. The jet in this model is
extremely bright with $T_{b}>10^{10}$ K. The jet emission is primarily from
the jet head but there is also significant emission along the jet axis, where
relativistic magnetized gas resides. Discrete features within the jet due to
internal shocks which are spatially separated are also apparent in the
emission features. In the next three columns, we apply a Gaussian smoothing
function of size $\Delta R/r_{g}=10000,25000,$ and $50000$, respectively. Even
for the resolution scale $\Delta R/r_{g}=25000$, the jet is incredibly bright
with $T_{b}>10^{9}$ K and the motion of the expanding jet head could be
tracked if the source were resolved with $\Delta R\lesssim 25,000\,r_{g}$.
Tracking the jet position may allow for the jet velocity to be constrained
with a method independent of emission model assumptions.

In Figure 18, we show the viewing angle dependence of the jet emission in
m10a0.9-HR for a viewing angle of $10^{\circ}$ (top row), $45^{\circ}$ (middle
row) and $90^{\circ}$ (bottom row). The distinguishing features of the jet
(the jet head and discrete internal shock emission) cannot be distinguished at
$45^{\circ}$ even with $\Delta R/r_{g}=10000$. At $10^{\circ}$, the jet
appears as a double lobed structure due to the top and bottom jet, but this
would be seen as a compact source unless the jet is well resolved.

Additional 230 GHz images of the jets of each model near the brightest point
in their evolution for a viewing angle of $90^{\circ}$ are shown in Figure 27.
In general, the jets become brighter as the spin increases owing to the
greater overall jet power. In addition, the jet head and internal shocks are
also apparent for fairly well resolved jets viewed edge on for models with
$M_{\rm{BH}}=5\times 10^{6}\,M_{\odot}$.

In Figure 19, we illustrate the dimming in m10a0.0-HR. The emission in this
model was dominated by the jet head with only weak internal shocks. As a
result, the jet appears as simply two lobes separated spatially. As the jet
head expands and cools, the high frequency emission declines rapidly as
described in Section 5.1. If detected as a radio/submillimeter source,
m10a0.0-HR would appear as a bright radio/submillimeter source for several
weeks before likely dropping below detection limits.

On the other hand, the model m5a0.0-HR (Figure 20) initially did not show
internal shocks and was incredibly dim. Instead, the jet would have likely
appeared to be dormant for several weeks before brightening if this model were
observed immediately following the launch of the outflow. Although the
timescale before the jet brightens ($\sim 20$ days) is shorter than the delays
seen in many radio quiet TDEs ($>30$ days, 2020SSRv..216...81A) and we are
imaging in a higher frequency than the 5-8.4 GHz most detections have been
made at, this behaviour shows that delayed brightening can occur in super-
Eddington accretion disk jets.

Figure 21: Here we show the maximum distance at which the jet in each model is
detectable and resolved given a limiting baseline distance and a
$5\sigma_{\rm{230\,GHz}}$ (with $\sigma_{\rm{230\,GHz}}=10$ mJy for the EHT
and $\sigma_{\rm{230\,GHz}}=5$ mJy for the ngEHT) detection threshold. It is
assumed that $T_{e}=T_{i}$ ($\mathcal{R}=1$). Note that in the top panels, the
curves for m5a0.5-HR and m5a0.9-HR (as well as m10a0.5-HR and m10a0.9-HR)
largely coincide since the maximum distance is resolution limited instead of
flux limited as the baseline distance increases. Focusing first on the results
assuming observations with the EHT (left), the estimated maximum distance at
which each model can be detected increases with BH spin due to the increase in
jet power and brightness. The non-spinning models require $D\lesssim 6-9$ Mpc
regardless of resolution scale and are thus unlikely targets. The spin
$a_{*}=0.5$ models are substantially brighter and may be suitable targets at
up to $D\lesssim 35$ Mpc. In the brightest model (m10a0.9-HR), resolving the
internal shocks ($\Delta R=10,000\,r_{g}$) requires that the TDE occur within
$D\lesssim 45$ Mpc while resolving only the jet head ($\Delta
R=50,000\,r_{g}$) only requires $D\lesssim 110$ Mpc. The increased detector
sensitivity expected in the ngEHT (right) compared to the EHT (left) increases
the maximum distance at which the jets can be detected and resolved by a
factor of $\sim 1.4-1.8$. The greatest improvement is for large baseline
distances and $a_{*}=0.9$. For instance, the maximum distance increases from
$\sim 110$ Mpc to $\sim 180$ Mpc at a baseline distance of
$10\,\rm{G\lambda}$. Figure 22: Here we show the maximum distance at which the
jet in each model is detectable and resolved as $\mathcal{R}$ increases for
$\Delta R=50,000\,r_{g}$ at a limiting baseline distance of $10\
\rm{G\lambda}$. We compare EHT ($\sigma_{\rm{230\,GHz}}=10$ mJy, dashed lines
with open circles) and ngEHT ($\sigma_{\rm{230\,GHz}}=5$ mJy, solid lines with
filled circles) limits for the correlated flux density and a
$5\sigma_{\rm{230\,GHz}}$ detection threshold. The curve is cut off at
$\mathcal{R}$ where the jet becomes unresolved.

To estimate the viability of detection and resolution of each model, we
estimate the visibility amplitude as a function of baseline distance assuming
a detector limit of $\sigma_{\rm{230\,GHz}}=10$ mJy for the EHT
(2019ApJ...875L...3E) and $\sigma_{\rm{230\,GHz}}=5$ mJy for the ngEHT
(2019BAAS...51g.256D) and assuming that Earth baselines at 230 GHz will not
exceed 10 G$\lambda$. Visibility amplitudes were calculated using the eht-
imaging library (2018zndo...1173414C). We consider a jet model detectable and
resolvable if (i) the total image (zero fringe spacing baseline) flux is above
5$\sigma_{\rm{230\,GHz}}$, (ii) the difference between the total image flux
and any other, longer baseline (up to 10 G$\lambda$) flux is above
5$\sigma_{\rm{230\,GHz}}$, and (iii) the nominal resolution of the baseline
($1/u$ for baseline length $u$) is less than $\Delta R$. This definition
assumes that the baseline length is the maximum baseline separation available
for the observations. Given this assumption, it additionally requires that the
baseline can resolve the jet at the desired length scale. We compute the
maximum distance for viewing angles of $10^{\circ}$, $45^{\circ}$, and
$90^{\circ}$ and then average over viewing angle to obtain a representative
maximum distance. We also smooth the maximum distance profiles using a moving
average with a window of $\sim 2.7\ {\rm{G}}\lambda$ in order to reduce
scatter introduced by the variable behaviour in the visibility amplitude.

We present the estimated maximum distance for imaging a resolved jet near the
brightest period in each simulation assuming $\mathcal{R}=1$ with
$5\sigma_{\rm{230\,GHz}}$ significance at length scales $\Delta
R=10,000\,r_{g}$ and $\Delta R=50,000\,r_{g}$ in Figure 21. As expected, we
find that the maximum distance at which the jet can be detected and resolved
increases with BH spin due to the increased jet power. Assuming only the jet
head is resolved, the brightest jets may be detected with the EHT (ngEHT) at
up to $D\sim 110$ (180) Mpc depending on the baseline coverage available. The
non-spinning models on the other hand are only detectable within $D\lesssim
6-9$ ($8-13$) Mpc. Assuming a middling BH spin of $a_{*}=0.5$, detection may
be possible at $D\lesssim 35$ (60) Mpc. Resolving finer structure requires
that the TDE occurs nearby. For instance, m10a0.9-HR is only detectable and
resolved within $D\lesssim 45$ Mpc if $\Delta R=10,000\,r_{g}$. The same
maximum distance is found for the ngEHT assuming a resolution length scale of
$\Delta R=10,000\,r_{g}$. A similar effect on the maximum distance is seen in
the other models as the resolved length scale decreases.

We quantify the effects of a two-temperature plasma by determining the maximum
detectable and resolved distance for each model as $\mathcal{R}$ is varied
assuming a baseline separation of $10\ \rm{G\lambda}$ and a length scale
$\Delta R=50,000\,r_{g}$ in Figure 22. Due to the decline in overall
luminosity and the shift in peak frequency, the maximum distance can decrease
by up to an order of magnitude. For instance, the maximum distance for
m10a0.9-HR, assuming EHT detector limits, declines from $\sim 110$ Mpc for
$\mathcal{R}=1$ to $\sim 8$ Mpc for $\mathcal{R}=20$. In models m5a0.0-HR,
m10a0.0-HR, and m10a0.5-HR, even a small increase in $\mathcal{R}$ shifts the
peak frequency to less than 230 GHz which leads to unresolved 230 GHz jet
emission for values of $\mathcal{R}>1-2$ (assuming a lower threshold on the
distance to the TDE of 1 Mpc). A similar result is found assuming the expected
detector sensitivity of ngEHT, but the maximum distance increases by a factor
of $\sim 1.4-1.8$. In some cases, the maximum temperature ratio $\mathcal{R}$
that the jet can have and still be detectable increases (i.e. model
m5a0.9-HR).

Our results suggest that the number of sources that can be detected and
resolved will depend on the spin distribution of SMBHs across cosmic distance
due to the increase in jet power with BH spin. Current simulations and
observations which have estimated the spins of BHs at redshift $z\approx 0$
indicate that SMBHs in the mass range of $\sim 10^{6.5}-10^{7.5}\,M_{\odot}$
will tend towards $a_{*}\sim 1$ (2013CQGra..30x4004R; 2013ApJ...762...68D;
2014MNRAS.440.1590D; 2019MNRAS.490.4133B). This is encouraging since, with the
EHT (ngEHT), at least $\sim 45$ (200) TDEs are expected per year within
$D\lesssim 110$ (180) Mpc assuming $\mathcal{R}=1$. This suggests that the
increased detector sensitivity expected in the ngEHT could increase the number
of possible TDE targets by a factor of nearly 4 compared to the EHT. On the
other hand, if $\mathcal{R}$ significantly exceeds unity, less than one
resolvable source may occur per year, even with the improved sensitivity of
the ngEHT. In addition, if accretion occurs in the MAD accretion state for a
significant fraction of a BH’s lifetime, this could skew the BH spin
distribution towards lower spin values due to spin down (2022MNRAS.511.3795N)
which would certainly reduce the number of detectable TDE jet sources at
larger distances based on our models.

##  6 Discussion

Figure 23: Here we compare our model atmospheres (gray hatched region) with
CNM profiles of ASASSN-14li (2016ApJ...819L..25A), CNSS J0019+00
(2020ApJ...903..116A), Swift J1644+57 (Sw J1644+57, 2018ApJ...854...86E),
AT2019dsg (2021NatAs...5..510S), and Sagittarius A* (Sgr A*,
2003ApJ...591..891B; 2019ApJ...871..126G). The scaling in our models is such
that the density of the CNM at large radii is similar to that inferred in
other TDEs and Sgr A*.

###  6.1 Initial Atmosphere vs. Inferred Profiles in TDEs

As noted in Section 3.3, we initialized each simulation with a low density
atmosphere where $\rho\approx(0.5-1\times
10^{-14}\,\rm{g\,cm^{-3}})(r/r_{0})^{-3/2}$, where $r_{0}=r_{H}$. The
$r^{-3/2}$ profile that we have chosen is shallower than the inferred profile
of roughly $r^{-5/2}$ for ASASSN-14li (2016ApJ...819L..25A). We note however
that estimates of the density profile of ASASSN-14li were only obtained up to
$\gtrsim 10^{5}r_{g}$ (or $\gtrsim 10^{4}\,r_{g}$ if the outflow geometry is
conical). Assuming that such a profile does indeed continue to the BH, this
implies that the atmosphere in ASASSN-14li may reach a density near the
horizon of $\sim 10^{-11}\,\rm{g\ cm^{-3}}$, nearly three orders of magnitude
denser than we have implemented. On the other hand, the profile of the TDE
AT2019dsg (2021NatAs...5..510S) appears to be turning over to a shallower
profile of nearly $r^{-1}$ at $r\lesssim 10^{3}\,r_{g}$, which is similar to
the inferred properties of the CNM in Sagittarius A∗ (2003ApJ...591..891B;
2019ApJ...871..126G). Extrapolating the profile of AT2019dsg down to the
horizon implies a maximum density of $\sim 10^{-17}\,\rm{g\ cm^{-3}}$, which
is orders of magnitude less than what we have implemented.

Despite these differences in the behaviour at small radii, our model
atmospheres are similar to that of previous radio TDEs over $r\sim
10^{4}-10^{5}\,r_{g}$ (see Figure 23), the region over which we present our
spectra and images. The density profiles of TDEs have only been estimated for
a small handful of radio TDEs. As of yet, the behaviour of the density profile
down to $r\sim 1000r_{g}$ has only been measured in AT2019dsg while the other
known TDEs have profile measurements down to $r\gtrsim 10^{4}r_{g}$.
Additionally, the overall scale of the gas density (i.e. the value of
$\rho_{\rm{atm,max}}$ as defined in Section 3.3) appears to vary between
systems. For instance, the CNM in the radio TDE CNSS J0019+00 at $r\approx
10^{6}\,r_{g}$ has the same estimated density as ASASSN-14li at $r\approx
10^{5}\,r_{g}$. Simulations with more dense atmospheres should be explored, as
well as more or less steep gas density profiles, as the jet may experience
strong deceleration and/or collimation as it encounters the CNM (e.g. see
2017MNRAS.469.4957B).

###  6.2 Comparison with Radio TDEs

The density weighted outflow velocity of each model is $\sim 0.2-0.35c$, which
is mildly relativistic but substantially faster than the $\lesssim 0.1c$ found
in most radio quiet TDEs so the outflow properties cannot explain the inferred
velocities in these transients. Of note however is the production of
$\gamma>2$ gas in the jet for the $a_{*}\geq 0.5$ models. 2015MNRAS.453.3213S
found that the isotropic equivalent luminosity for highly super-Eddington
($\dot{M}>10^{3}\dot{M}_{\rm{Edd}}$) accretion flows can exceed
$10^{47}\,\rm{erg\,s^{-1}}$, which is sufficient to explain the $\sim 10^{48}\
\rm{erg\,s^{-1}}$ X-ray equivalent luminosity seen in Swift J1644+57
(2011Sci...333..203B; 2011Natur.476..421B). However, they performed
simulations with $a_{*}=0$ models and suggested that a $\gamma\gtrsim 2$
component may be untenable with radiative acceleration. Our simulation results
suggest that for mass accretion rates $\sim 11-25\dot{M}_{\rm{Edd}}$ (compared
to the $\sim 45-4800\dot{M}_{\rm{Edd}}$ in 2015MNRAS.453.3213S), the Poynting
flux may result in a relativistic component. Not only does the jet power
increase substantially as the BH spin increases in our simulations, but a
relativistic jet component with $\gamma>2$ is produced. This is due to
increased magnetic energy flux in the jet, which leads to Poynting
acceleration of gas where $\sigma>1$.

Assuming $T_{e}=T_{i}$, the thermal synchrotron spectra presented in this work
suggest that super-Eddington accretion disks can produce highly energetic jets
which primarily emit at $\nu>100$ GHz and are bright with $L\sim
10^{41}\,\rm{erg\ s^{-1}}$ for $>48$ days. This is in staunch contrast with
the radio properties of non-jetted TDEs. Non-jetted TDEs all appear to have a
peak frequency of the order $\nu_{\rm{peak}}\sim 10$ GHz and have a luminosity
of $L_{\rm{radio}}\approx 10^{37}-10^{39}\,\rm{erg\ s^{-1}}$ at the time of
detection.

The jet emission models which are brightest at 230 GHz, and thus the most
favorable for detection in terms of ngEHT observations, are $\mathcal{R}=1$.
However, these models are in conflict with TDE observations in terms of the
peak frequency. The thermal synchrotron models which peak at smaller
frequencies, and are thus more similar to known radio TDEs in terms of the
peak frequency at least, have $\mathcal{R}>10$. Nevertheless, none of our
models are able to remain bright enough ($L_{\rm{radio}}\gtrsim 10^{37}\
\rm{erg\,s^{-1}}$) and simultaneously peak at $\lesssim 10$ GHz.

Another possibility is that jets in these models will continue evolving
towards lower frequencies and luminosity and we are simply observing a higher
energy stage of the jet. For example, the low frequency luminosity in each of
our models at $\nu\approx 5-8.4$ GHz (where most radio TDEs have been
observed) is dimmer than non-jetted TDEs. However, our models showed evolution
towards lower frequencies, and in the case of m10a0.0-HR the luminosity at the
peak frequency is also decreasing over time. This suggests that our models may
evolve towards spectra more similar to radio quiet TDEs given enough time.

This is an observationally interesting possibility since the majority of
follow-up observations of TDEs in the radio have been at $5-8.4$ GHz. Our
models have a luminosity of $\lesssim 10^{36}\,\rm{erg\ s^{-1}}$ at 5-8.4 GHz,
placing them below upper limits in the majority of TDEs where no radio
emission was detected (mostly $>10^{37}\,\rm{erg\ s^{-1}}$, see Figure 1 in
2020SSRv..216...81A). If there is indeed a radio/submillimeter component at
$\nu>100$ GHz that is bright for the first few weeks of non-jetted TDEs, low
frequency searches are likely to miss this emission. Our results suggest
earlier follow-up observations at high frequencies may be necessary to capture
the full radio/submillimeter activity of some TDEs.

##  7 Conclusions

We have analyzed outflows from SANE super-Eddington accretion disks across
mass and spin parameter space with mass accretion rates $\dot{M}\gtrsim
11\dot{M}_{\rm{Edd}}$. We confirm that dissipation takes place at the jet head
as well as along the jet due to internal shocks. These internal shocks are due
to variable ejections of high velocity gas, which propagate along the jet axis
and shock with slower moving gas within the jet, as well as recollimation
shocks. The jet power and maximum velocity of gas near the pole increases with
BH spin and models with $a_{*}\geq 0.5$ launch $\gamma>2$ gas that is low
density. However, we find that the density weighted outflow velocity is
roughly $\sim 0.2-0.35c$ for each model.

Through GRRT post-processing with ipole, we produce time dependent thermal
synchrotron spectra assuming a single temperature plasma. The spectra peak at
$>100$ GHz in each model and the overall radio/submillimeter luminosity
increases with BH spin. In general, the jet head brightens as it expands into
the CNM. However, models m10a0.0-HR and m10a0.5-HR appear to be dimming as
they evolve. In addition, model m5a0.0-HR shows a delayed jet brightening with
its jet remaining initially dim with $L\sim 10^{38}\,\rm{erg\ s^{-1}}$ and
increasing to $L\sim 5\times 10^{39}\,\rm{erg\ s^{-1}}$ at $t>58,000\,t_{g}$.

We also test a simple electron temperature model. We find that increasing
$T_{i}/T_{e}=\mathcal{R}$ has the effect of reducing the peak frequency and
the overall luminosity. For instance, we found that $\mathcal{R}=20$ reduced
the peak frequency from $\sim 200$ GHz to $\sim 20$ GHz and the luminosity
from $\sim 10^{41}\,\rm{erg\ s^{-1}}$ to $\sim 10^{37}\,\rm{erg\ s^{-1}}$ in
model m10a0.5-HR. We assumed the temperature ratio was independent of
$\beta_{g}=p_{\rm{gas}}/p_{\rm{mag}}$, but there may be spatial variation in
$\mathcal{R}$. In addition, a more accurate prescription may require some
accounting for the radiation pressure as well since our simulations are in
GRRMHD.

The 230 GHz images show that the brightest feature in each jet is the jet
head, which shocks on the CNM. Internal shocks also produce bright ‘bubbles’
of emission at smaller radii in the jet. For sources resolved on scales
$\Delta R\lesssim 10000\,r_{g}$, we predict that the jet head and the internal
shocks can be distinguished so long as the jet is not viewed at steep viewing
angles. At poorer resolution ($\Delta R\lesssim 50000r_{g}$), the jet head is
the most distinguishable feature.

We tested the viability of detecting and resolving the jets in each model at
230 GHz assuming flux limits appropriate to the EHT (ngEHT). The spin
$a_{*}=0.9$ models are bright enough to resolve the jet head within $\lesssim
110$ (180) Mpc if $\mathcal{R}=1$. At this distance, $\sim 45$ (200) TDEs are
expected per year which suggests that several TDEs could be potential targets
for radio/submillimeter follow up during future observing missions. Our
calculations suggest that at most a factor of 4 increase in the number of TDE
targets is possible if the detector sensitivity is improved to $5$ mJy in the
ngEHT. If the electrons are significantly cooler than the ions, the maximum
distance where resolved jets may be detected is significantly reduced. For
example, we find a maximum resolved distance of $\sim 8$ (18) Mpc for
$\mathcal{R}=20$ which would reduce the number of potential targets per year
to less than one even for the ngEHT. Our simulations may apply to the super-
Eddington phase of a TDE, roughly $\sim 2t_{\rm{fb}}$ ($\approx 1-$several
months). We suggest that high frequency radio/submillimeter follow-up of
nearby TDEs during this early period be conducted to search for
radio/submillimeter jets.

We must stress that much is unknown about jet launching in TDEs, thus while
the number of TDEs increases with distance, only a fraction of these may ever
appear in the radio/submillimeter. In this work, we have merely provided an
analysis of a single mechanism by which radio/submillimeter emission can be
produced, that is jets from SANE, super-Eddington accretion disks which
undergo conical expansion, and determine the viability of detection of such
jets with the EHT/ngEHT. As of this writing, the majority of known radio TDEs
appear to have produced their radio emission through sub-relativistic gas
shocking with the CNM. However, if the accretion flow is indeed super-
Eddington, some fraction of future radio TDEs may launch jets similar to those
described in this work. Our model spectra were below detection limits at 5-8.4
GHz, where most radio follow-ups to date have been conducted, which suggests
that such jets may have escaped detection previously. Radio/submillimeter
follow-up of TDEs at $>100$ GHz may reveal a higher energy component
associated with a bipolar jet produced by a super-Eddington accretion disk.

Before we conclude, we must point out several caveats in this work. Firstly,
our initial conditions assume that the TDE debris stream efficiently and
rapidly circularizes into an accretion disk. This may only apply to a subset
of TDEs. For instance, 2015ApJ...804...85S found that circularization during
disk formation was quite poor and the disk remained highly eccentric for
several times the fallback time (see also 2021MNRAS.507.3207C;
2022MNRAS.510.1627A). The case for the existence of poorly circularized TDE
disks existing is supported observationally as well (2022A&A...666A...6W).
Nevertheless, a recent long duration ($>60$ days in physical time), high
resolution radiative hydrodynamics simulation by 2022arXiv220610641S has
demonstrated that efficient disk circularization is possible. They report disk
circularization near the peak of the TDE emission (at nearly 40 days in the
simulation) for the fiducial case of a $10^{6}\,M_{\odot}$ BH disrupting a
$1\,M_{\odot}$ star on a parabolic orbit with $R_{p}=R_{t}$; ergo, a
circularized accretion disk near the peak emission of the TDE may be expected
in some cases. 2020MNRAS.495.1374B; 2021MNRAS.504.4885B also demonstrate
substantial circularization of the TDE debris in their work.

Our simulation results demonstrate that in super-Eddington accretion disks,
the jet magnetization is an important factor since it can produce a
relativistic component driven by Poynting acceleration. Simulations of TDE
accretion disk formation have not yet demonstrated the presence of significant
magnetic flux in the inner accretion flow nor the onset of jet launching so
the highly magnetized, bright jets produced in the $a_{*}\geq 0.5$ simulations
are somewhat of an idealization given the assumed initial torus configuration.

As we are limited to studying 2D simulations due to computational limitations,
the explicit assumption that the jet is axisymmetric does not allow 3D
instabilities to arise. The jet may kink and twist due to various MHD
instabilities as it propagates (e.g. 2017MNRAS.469.4957B). The kink
instability in particular may give rise to additional dissipation of magnetic
energy into thermal energy within the jet.

The mass accretion rates that our models achieve is not precisely at the peak
accretion rate. This will change the jet power and potentially the
radio/submillimeter brightness since $L_{\rm{net}}\propto\dot{M}$
(2015MNRAS.453.3213S). In other words, the spectra we provide here may
under/over estimate the jet brightness by a factor of a few. This may
increase/decrease the maximum resolvable distance in some models, but the
overall finding that some nearby TDEs may produce jets which are detectable
and resolvable is robust.

The choice to cut out the $\sigma>1$ region of the jet before ray tracing is
conservative and well motivated, but the effects on the ray traced images has
not yet been well studied. Given that much of the emission in the brightest
jets originates from near the poles, our spectra provide a lower limit on the
luminosity of the jet. In addition, as we have emphasized throughout this
section, jets with internal shocks are likely to produce a non-thermal
electron population and may also consist of a two-temperature plasma. Neither
of these were employed during the simulations, and they could modify the jet
properties since radiation is dynamically important.

## Acknowledgements

We thank Ben Prather and George Wong for their assistance with the ipole code
in this work. We thank Angelo Ricarte, Dominic Pesce, and Lindy Blackburn for
helpful comments and discussions. We also thank the anonymous EHT publication
committee referee who provided useful feedback. Lastly, we thank the anonymous
referee who provided useful comments and critiques which improved the quality
and clarity of this work. This work was made possible thanks to the NSBP/SAO
EHT Scholars program and was supported in part by NSF grant AST-1816420, and
made use of computational support from NSF via XSEDE resources (grant TG-
AST080026N). This work was carried out at the Black Hole Initiative at Harvard
University, which is supported by grants from the John Templeton Foundation
and the Gordon and Betty Moore Foundation.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

##  Appendix A Effects of Jet Resolution on Dynamics

To test the impact of the chosen grid resolution on the jet dynamics, we
conducted a suite of reruns using the same initial conditions as model
m10a0.9-HR but for various grid resolutions. We describe each simulation in
Table 2 and compare the resulting jet at the same simulation time in Figure
24. Each rerun was run to $t=25,000\,t_{g}$, which is ample time for the
accretion flow to evolve and produce a jet. Note that we use abbreviated names
for the models for simplicity in the following paragraphs.

Model LRa employs the same grid resolution as 2019MNRAS.483..565C, which had
an increased resolution around $\vartheta=\pi/2$, but sacrificed resolution in
the jet (giving a five times smaller angular resolution near the pole compared
to model HR). The angular coordinate $\vartheta$ is related to a code
coordinate $x_{2}$. The precise relationship is given by

$\vartheta=\dfrac{\pi}{2}\left[1+\cot\left(\dfrac{\pi}{2}H_{0}\right)\tan\left(H_{0}\pi(x_{2}-\dfrac{1}{2})\right)\right],$
(44)

where $H_{0}$ is a parameter which increases the resolution around
$\vartheta=\pi/2$ as it approaches unity. For 2019MNRAS.483..565C and the
simulations in this appendix, we choose $H_{0}=0.7$. As with the simulations
presented in the main text, we copy the primitives from the 3rd cell from the
polar axis to bolster numerical stability. The jet in LRa evolves in a similar
way as s09 in 2019MNRAS.483..565C. The maximum $\gamma$ is less than 2,
indicating the resolution is too poor to capture the evolution of the jet near
the polar axis. When the angular resolution is doubled in LRb, we begin to see
greater acceleration of the jet but only marginally so.

For models IR and HR, we employ the same grid choices as described in Section
3.3 but with differing resolution choices. Model IR employs an identical grid
to HR but with half the resolution in both $r$ and $\vartheta$. Although the
total number of cells in both $r$ and $\vartheta$ is lower, the added
resolution and the cylindrification of the grid near the polar axis allows us
to capture the acceleration of relativistic material. However, comparing IR to
HR, we find that yet higher resolution is necessary to capture recollimation
shocks, which contribute to the structure and evolution of the jet.

We must point out that the jets in these models impact the radio-submillimeter
emission. However, the higher energy emission (optical to X-ray) is dominated
by the torus and inner accretion disk since comptonization in the jet is weak.
Hence, the qualitative viewing angle effects in optical/UV/X-ray spectra
reported in 2019MNRAS.483..565C for s09 will be unchanged by resolving the jet
properly.

Model Name | Abbreviation | Resolution | Added Resolution
---|---|---|---
|  | $N_{r}\times N_{\vartheta}$ |
m10a0.9-LRa | LRa | $512\times 320$ | disk
m10a0.9-LRb | LRb | $512\times 640$ | disk
m10a0.9-IR | IR | $320\times 128$ | disk+jet
m10a0.9-HR | HR | $640\times 256$ | disk+jet
Table 2: Here we indicate the resolution and grid type of each rerun of model
m10a0.9-HR. Each rerun was evolved until $t=25,000\,t_{g}$. Note that the grid
with added resolution only in the disk has the same grid settings as in
2019MNRAS.483..565C while the other models have the same grid setup as in this
paper (as described in Section 3.3).

Figure 24: Here we show the gas density (top panels) and lorentz factor
(bottom panels) at $t=25,000\,t_{g}$ for each model in Table 2. Note that the
abbreviated model name is given above each panel. The resolution within the
jet increases from left to right. As the resolution increases, we generally
find that the acceleration of gas near the pole is better captured. Of note is
the need for cylindrification and high resolution near the pole to capture
both the acceleration and recollimation shocks properly, as can be seen by
comparing IR and HR.

##  Appendix B Additional Figures

Here we show additional figures for comparison with fiducial models described
in the text. We show the jet properties of m5a0.5-HR in Figure 25. Compared to
m5a0.0-HR, the jet has travelled nearly the same distance ($\sim
40,000\,r_{g}$), but the jet core is much faster and the gas temperature in
the core is hotter. In addition, the magnetic field strength along the jet
axis is larger.

We show the jet properties of m10a0.9-HR in Figure 26. Compared to m5a0.9-HR,
the jet features are generally similar. This is in well agreement with the
general result that we find throughout this work that the BH mass does not
have a noticeable effect on the jet. The most important factor is instead the
BH spin.

In Figure 27, we show the 230 GHz jet emission viewed at $90^{\circ}$ near its
brightest point in each simulation.

Figure 25: The same as Figure 8 but for m5a0.5-HR. Figure 26: The same as
Figure 8 but for m10a0.9-HR.

Figure 27: Here we compare the jets for each model viewed at $90^{\circ}$. The
simulation time ray traced in each image is $t=83,000\,t_{g}$ for models
m5a0.0-HR, m5a0.5-HR, m5a0.9-HR and m10a0.5-HR, $t=58,000\,t_{g}$ for model
m10a0.0-HR, and $t=81,200\,t_{g}$ for model m10a0.9-HR. The time is also
specified at the bottom left of the leftmost panel for each model.