# Active galactic nuclei jets simulated with smoothed particle hydrodynamics

Filip Huško,1 Cedric G. Lacey1
1 Institute for Computational Cosmology, Department of Physics, University of
Durham, South Road, Durham, DH1 3LE, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Simulations of active galactic nuclei (AGN) jets have thus far been performed
almost exclusively using grid-based codes. We present the first results from
hydrodynamical tests of AGN jets, and their interaction with the intracluster
medium (ICM), using smoothed particle hydrodynamics (SPH) as implemented in
the SWIFT code. We launch these jets into a constant-density ICM, as well as
ones with a power-law density profile. We also vary the jet power, velocity,
opening angle and numerical resolution. In all cases we find broad agreement
between our jets and theoretical predictions for the lengths of the jets and
the lobes they inflate, as well as the radii of the lobes. The jets first
evolve ballistically, and then transition to a self-similar phase, during
which the lobes expand in a self-similar fashion (keeping a constant shape).
In this phase the kinetic and thermal energies in the lobes and in the shocked
ICM are constant fractions of the total injected energy. In our standard
simulation, two thirds of the initially injected energy is transferred to the
ICM by the time the jets are turned off, mainly through a bow shock. Of that,
$70\%$ is in kinetic form, indicating that the bow shock does not fully and
efficiently thermalise while the jet is active. At resolutions typical of
large cosmological simulations ($m_{\mathrm{gas}}\approx 10^{7}$
$\mathrm{M}_{\odot}$), the shape of the lobes is close to self-similar
predictions to an accuracy of $15\%$. This indicates that the basic physics of
jet-inflated lobes can be correctly simulated even at such resolutions
($\approx 500$ particles per jet).

###### keywords:

galaxies: jets – galaxies: evolution – galaxies: clusters: intracluster medium

††pubyear: 2022††pagerange: Active galactic nuclei jets simulated with
smoothed particle hydrodynamics–Active galactic nuclei jets simulated with
smoothed particle hydrodynamics

##  1 Introduction

Active galactic nuclei (AGN) feedback due to energy release by accreting
supermassive black holes (SMBHs) is an important process that contributes to
the evolution of galaxies (Bower et al. 2006, Croton et al. 2006, Henriques et
al. 2015, Lagos et al. 2018). It is thought to be the cause of quenching of
star formation in massive elliptical galaxies (Springel et al. 2005, Martig et
al. 2009, Sturm et al. 2011). AGN feedback can take the form of radiative or
wind feedback from quasars (Sanders et al. 1988, Silk & Rees 1998, Di Matteo
et al. 2005), where $5-40$ per cent of the infalling matter is converted into
energy and radiated away (Novikov & Thorne 1973, Noble et al. 2011). This
radiation interacts with the ambient gas by directly heating it, as well as
driving outflows through radiation pressure (Feruglio et al. 2010, Cicone et
al. 2014, Carniani et al. 2015).

Observations of galaxy clusters reveal the existence of an additional mode of
AGN feedback: SMBHs can launch jets of relativistic particles that may
traverse large distances (Blandford & Königl 1979, Urry & Padovani 1995), in
some cases larger than a Mpc (Dabhade et al. 2020, Mahato et al. 2021,
Andernach et al. 2021). As they travel, these jets are decelerated by and
deposit their energy into the hot halo of gas that surrounds the host galaxy
(McNamara & Nulsen 2007, Fabian 2012, McNamara & Nulsen 2012). The jets are
visible in radio frequencies due to synchrotron emission (Biermann &
Strittmatter 1987, O’Dea 1998, Markoff et al. 2001), as well as indirectly in
X-ray emission in the form of cavities in the hot gas halo (Bîrzan et al.
2004, McNamara et al. 2005, Wise et al. 2007).

AGN jets are often invoked in order to explain otherwise puzzling observations
of galaxy clusters. The cooling rates of hot gas in the central regions of
galaxy clusters, inferred from X-ray observations, are large enough that we
would generally expect large star formation rates in such environments; this
is typically not observed (Edge & Stewart 1991, Fabian 1994, McDonald et al.
2018), with some exceptions (O’Dea et al. 2009, McDonald et al. 2015).
Furthermore, observations of emission lines, which we would expect in the
presence of a cooling flow, suggest low cooling rates (Peterson et al. 2003,
Bregman et al. 2006). The observed cooling rates (and/or central X-ray
luminosities) are closely correlated to the jet powers (Rafferty et al. 2006,
Nulsen et al. 2009, Russell et al. 2013, Hlavacek-Larrondo et al. 2012),
implying the existence of a feedback loop in galaxy clusters. Jets have also
been observed in systems that are smaller than galaxy clusters, such as galaxy
groups (Sancisi et al. 1987, Baldi et al. 2009, Randall et al. 2011, Werner et
al. 2019, Eckert et al. 2021) and remnants of galaxy mergers (Heckman et al.
1986, Merritt & Ekers 2002, Ivison et al. 2012, Shabala et al. 2017). More
surprisingly, they have also been observed in disc galaxies (Ledlow et al.
2001, Singh et al. 2015, Nesvadba et al. 2021, Webster et al. 2021) and dwarf
galaxies (Pakull et al. 2010, Mezcua et al. 2019, Yang et al. 2020, Davis et
al. 2022), indicating that their effects may be widespread.

Models of galaxy formation in the form of hydrodynamical simulations have in
recent years begun to incorporate jet feedback on a cosmological scale
(Weinberger et al. 2018, Davé et al. 2019, Dubois et al. 2021). However, the
numerical resolution that can be achieved in such cosmological simulations is
often thought to be insufficient to fully capture the impact of the jets
(Bourne & Sijacki 2017, Weinberger et al. 2017). In particular, low resolution
simulations do not display instabilities in the jets, which are likely
important for their evolution and energetics (Perucho et al. 2006, Perucho et
al. 2010). Many simulations have been carried out on smaller scales (galaxy
cluster or group scales) in order to facilitate our understanding of jet
propagation, energy deposition and the overall effect of jets as a mode of
feedback (e.g. Reynolds et al. 2006, Mendygral et al. 2012, Meece et al. 2017,
Yang et al. 2019). Some of these suggest that jet-like feedback may be
reliable even at at lower resolutions, with the jet energetics (e.g. how much
energy is kinetic vs. thermal, how quickly it is transferred to the hot gas
halo) insensitive to resolution (Weinberger et al. 2017).

Kinetic, jet-like feedback is currently employed in the following large
hydrodynamical, cosmological simulations: IllustrisTNG (Weinberger et al.
2018), SIMBA (Davé et al. 2019) and HorizonAGN (Dubois et al. 2014). All of
these simulations model jet feedback by increasing the velocity of gas
particles or cells close to the SMBH by values of order $10^{4}$ km/s. This is
done in discrete injection events whose frequency is determined by some jet
power $P_{\mathrm{j}}$, which is taken to be proportional to the accretion
rate onto the black hole:
$P_{\mathrm{jet}}=\epsilon_{\mathrm{j}}\dot{M}_{\mathrm{BH}}c^{2}$. Here,
$\epsilon_{\mathrm{j}}$ is the jet efficiency, a numerical factor that
encapsulates the efficiency of energy extraction from the SMBH. The EAGLE
hydrodynamical simulation of galaxy formation currently includes only a
thermal mode of AGN feedback, implemented as isotropic heating of gas (Schaye
et al. 2015), which represents the effects of radiative (quasar) feedback. The
inclusion of a jet-like kinetic mode of feedback would likely lead to more
realistic galaxies in the simulation. The successors of the EAGLE simulation
will utilise the SWIFT code (Schaller et al. 2016), and its SPHENIX
hydrodynamical implementation based on smoothed particle hydrodynamics
(hereafter SPH; Borrow et al. 2022).

Before including a jet feedback mode in a large cosmological simulation, it is
important to verify that the hydrodynamical code it uses can realistically
simulate the effects of jets on the surrounding gas. This can be done by
simulating an individual jet episode with a constant jet power, and comparing
the behaviour of such a jet with theoretical predictions (see Komissarov &
Falle 1998 as an early example). Performing this kind of test is especially
important for cosmological simulations using SPH codes, since there has been
virtually no effort to simulate individual AGN jets with SPH. We note that
some previous SPH simulations have included AGN jets, but these simulations
were more complex (e.g. self-consistent feedback in Barai et al. 2016 or jets
as a feedback mechanism in cosmological simulations as employed by Choi et al.
2015), and thus harder to compare with theoretical expectations. In recent
years, SPH codes have been upgraded in various ways in order to better deal
with problems such as fluid mixing and conduction across shocks, that plague
traditional SPH schemes (Hopkins 2015, Menon et al. 2015, Wadsley et al. 2017,
Rosswog 2020, Borrow et al. 2022). This is particularly important for jet
simulations, since they involve extreme contrasts in fluid properties.

Theoretical studies of constant-power jets, propagating in a gaseous medium
with a power-law density profile, predict that all jets start off with a
ballistic phase (Falle 1991, Kaiser & Alexander 1997, Kaiser & Best 2007).
During this phase they easily drill through the ambient medium. After the mass
of the swept-up medium exceeds that of the jet material, the jet transitions
into a self-similar phase, which should always occur at large enough distances
(depending on the jet power and mass flux, as well as the ambient medium
density). In this phase, the jet material experiences strong shocks and begins
to inflate hot lobes of gas (also referred to as ’cocoons’, e.g. Komissarov &
Falle 1998, although we use the former term hereafter for consistency). These
lobes then collimate the jets of unshocked material. The jet-inflated lobes in
this regime expand in a self-similar fashion (with a constant aspect ratio),
hence the name. This phase of jet evolution is especially suitable for
hydrodynamical tests of jet behaviour since the dependence of jet and lobe
properties (such as length) on time, jet power and background density should
be very simple (a power law, as long as the background density profile is also
a power law). In this paper, we present results on individual jet episodes
simulated with SWIFT, using the SPHENIX hydrodynamics scheme. Our focus is on
comparing the properties of these jets with theoretical predictions in the
self-similar regime of evolution.

We use a jet power and opening angle, as well as properties of the ambient
medium, that are close to observed values so that the properties of our
simulated jets can be meaningfully compared to observations. We stress,
however, that our main aim in this paper is not to make comparisons with
observations, but rather with analytical predictions, for the purpose of
validating our numerical scheme. We thus favour simplicity over realism in
this work. The numerical resolution we achieve, of order $\sim 1$ kpc (within
the jet-inflated lobes), is on a par with many similar simulations of AGN jets
that use grid-based codes (e.g. Yang et al. 2019, Smith andJustin Donohoe
2021,Wang & Yang 2022).

The jet launching velocity, $v_{\mathrm{j}}$, is a very important parameter in
our simulations, and it has both a physical and numerical role. On the
physical side, the launching velocity determines when the jet reaches the
self-similar phase of evolution, as well as whether it is in the relativistic
regime. In addition, increasing the velocity leads to higher typical
temperatures in the jet lobes, as well as lower densities. This latter fact is
due to the kinetic energy per particle scaling as $\propto
v_{\mathrm{j}}^{2}$, so the total mass and number of particles in the jets and
lobes scale as $1/v_{\mathrm{j}}^{2}$, under the assumption that the energy
within the jets and lobes is kept fixed. The same scaling ties into the role
of the launching velocity as a numerical parameter. Less massive jets and
lobes are represented with a smaller number of particles (given a constant jet
power), so they are more poorly resolved.

We do not simulate relativistic or mildly relativistic jets with $v>0.3c$
representing some of the stronger/younger Fanaroff-Riley (FR) II sources
(Wardle & Aaron 1997, Jetha et al. 2006, O’Dea et al. 2009, Snios et al.
2018), which have been the focus of many recent simulation studies (Walg et
al. 2013, Hardcastle & Krause 2013, Tchekhovskoy & Bromberg 2016, Matsumoto &
Masada 2019, Perucho et al. 2022). This is because such jets would be poorly
resolved in our simulations. Due to this restriction, we launch jets with
subrelativistic velocities of order $0.1c$, representing FRI sources or FRII
sources that have either significantly decelerated or entrained significant
amounts of ambient material on a kpc scale (Bicknell 1995). According to
analytical models of jet and lobe evolution, the properties of jets should be
largely insensitive to the choice of launching velocity once they reach the
self-similar regime (Kaiser & Best 2007). The differences between
subrelativistic and relativistic jets, in terms of their properties such as
the lengths or shapes of the lobes, are minimal (of order tens of percent),
and they arise largely from the different adiabatic indices of the lobe
material ($4/3$ versus $5/3$).

The outline of this paper is as follows. In Section 2, we summarise existing
theoretical predictions for the evolution of jets and the lobes they inflate
in the self-similar regime, while in Section 3 we discuss our numerical
implementation of the jet launching process, as well as the different
simulations we have done. In Section 4 we discuss some general properties of
our simulated jets, including their morphology, energetics and how well they
compare to predictions of analytical models. We also compare with previous
simulations of AGN jets. Section 5 includes a parameter study, where we
compare the properties of jets simulated at different numerical resolutions,
and with different jet powers, launching velocities and half-opening angles.
In all cases, we compare our simulations with theoretical predictions. We also
show some results of AGN jets launched into power-law gaseous atmospheres. In
Section 6 we summarise and conclude.

##  2 Jet and lobe evolution in the self-similar regime

In this work, we compare our simulated jets with theoretical predictions for
the self-similar regime of jet lobe evolution (e.g. Falle 1991, Kaiser &
Alexander 1997, Komissarov & Falle (1998); see also Begelman & Cioffi 1989 and
Bromberg et al. 2011 for alternative, but similar models). In the self-similar
picture, the jet is launched from a conical region defined by a half-opening
angle $\theta_{\mathrm{j}}$. The physical quantities that determine the
evolution of the jet and its lobes are: 1) power $P_{\mathrm{j}}$, 2)
launching velocity $v_{\mathrm{j}}$ (or equivalently, mass flux
$Q_{\mathrm{j}}=2P_{\mathrm{j}}/v_{\mathrm{j}}^{2}$), 3) background density
$\rho$ and 4) background pressure $p$ (or equivalently, temperature $T$). Note
that in our notation, the jet power and mass flux refer to the total, summed
over both jets.

These quantities can be combined to yield two length scales, $L_{1}<L_{2}$.
The evolutionary phase of a jet, can be determined by comparing its current
length, $L_{\mathrm{j}}$, with those length scales (Komissarov & Falle 1998).
In the initial phase of ($L_{\mathrm{j}}\ll L_{1}$), the mass in the jet is
large compared to the ambient medium being swept up by the jet. The jet is
denser than the ambient medium and it drills through it without significantly
being slowed down, due to its large inertia. The jet head moves with a
velocity equal to the launching velocity, $v_{\mathrm{j}}$, and the jet length
is thus given by $L_{\mathrm{j}}=v_{\mathrm{j}}t$. The jets have not yet
reached the self-similar regime while the above condition is true. We will
refer to jets that are in this evolutionary phase as ’ballistic’.

The length scale $L_{1}$ represents the scale at which the mass of the swept
up medium becomes comparable to the mass launched into the jet. It is given by

$L_{1}=\frac{1}{\theta_{\mathrm{j}}}\sqrt{\frac{2}{\pi\rho}\sqrt{\frac{Q_{\mathrm{j}}^{3}}{2P_{\mathrm{j}}}}}=\frac{2}{\theta_{\mathrm{j}}}\sqrt{\frac{P_{\mathrm{j}}}{\pi\rho
v_{\mathrm{j}}^{3}}}.$ (1)

The second length scale, $L_{2}$ represents the scale at which the ambient
pressure becomes important. It is given by

$L_{2}=\bigg{(}\frac{P_{\mathrm{j}}^{2}\rho}{p^{3}}\bigg{)}^{1/4}.$ (2)

Note that these length-scales represent dimensional combinations, and thus do
not necessarily include the correct numerical factors. Furthermore, previous
work implies that the transition from one regime to another, which should
occur once the jet has reached $L_{1}$ or $L_{2}$, can be fairly protracted
(Komissarov & Falle 1998).

The majority of observed FR-II sources are expected to satisfy $L_{1}\ll
L_{\mathrm{j}}\ll L_{2}$ (Komissarov & Falle 1998). In this regime, both the
mass flux and the ambient pressure are dynamically unimportant. The jet
experiences strong shocks and it is effectively slowed down. The jet head
velocity is thus expected to be much smaller than the launching velocity
$v_{\mathrm{j}}$. The jet comes into equilibrium with its own lobe (previously
shocked particles) through recollimation (reconfinement) shocks. The jet also
launches a bow shock.

Once the mass flux and external pressure are excluded, one cannot form any
length-scale from the remaining dynamical quantities, with the exception of
time. As a result, the behaviour of the jet-inflated lobes is expected to be
self-similar (Sedov 1959), and we thus refer to jets that satisfy $L_{1}\ll
L_{\mathrm{j}}\ll L_{2}$ as being in the ’self-similar’ regime. For the rest
of the analysis, we assume that the background medium follows a power law in
density:

$\rho(r)=\rho_{0}\bigg{(}\frac{r}{r_{0}}\bigg{)}^{-\beta}.$ (3)

Including time as a dynamical quantity, one can compute a length scale of the
form

$L=\bigg{(}\frac{P_{\mathrm{j}}t^{3}}{\rho_{0}r_{0}^{\beta}}\bigg{)}^{1/(5-\beta)}.$
(4)

(Falle 1991). The actual length of the jet (and the lobe) may differ from $L$
by some numerical factor, which can depend on the dimensionless parameters
that govern the jet evolution (half-opening angle $\theta_{\mathrm{j}}$ and
adiabatic index $\gamma$). The actual jet length can be computed from energy
conservation if one assumes self-similarity of the lobes and a particular type
of geometry. With a cylindrical geometry, the jet length is given by

$L_{\mathrm{j}}=c_{1}\bigg{(}\frac{P_{\mathrm{j}}t^{3}}{\rho_{0}r_{0}^{\beta}}\bigg{)}^{1/(5-\beta)},$
(5)

where $c_{1}$ is

$c_{1}=\left\\{\frac{A^{4}}{18\pi}\frac{(\gamma_{\mathrm{c}}+1)(\gamma_{\mathrm{l}}-1)(5-\beta)^{3}}{9[\gamma_{\mathrm{l}}+(\gamma_{\mathrm{l}}-1)A^{2}/2]-4-\beta}\right\\}^{1/(5-\beta)}.$
(6)

Here, $A$ is the aspect ratio of the lobe (its length divided by radius, also
equal to $1/\theta_{\mathrm{j}}$ for cylindrical jets), and
$\gamma_{\mathrm{l}}$ and $\gamma_{\mathrm{c}}$ are the adiabatic indices of
the lobe and ambient gas, respectively (Kaiser & Best 2007).

##  3 Numerical implementation

In this work, we use the open-access111https://swift.dur.ac.uk/ SWIFT
hydrodynamics and galaxy formation code (Schaller et al. 2016), and the
SPHENIX hydrodynamics scheme implemented therein (Borrow et al. 2022). SPHENIX
is an SPH method (Monaghan 1992). It includes artificial viscosity, which is
necessary in order to capture shocks since traditional SPH is dissipationless.
SPHENIX also includes artificial conductivity, which helps reduce unwanted
surface tension otherwise present in SPH simulations (Agertz et al. 2007,
Sijacki et al. 2012, Nelson et al. 2013), allowing for mixing between flows
that are in pressure equilibrium but contrasting in temperature and/or
density.

Both artificial viscosity and conductivity are crucial in our simulations:
artificial viscosity because our jets experience strong shocks (in some cases
with a Mach number, hereafter $M$, of $M\approx 100$), and artificial
conductivity since the jet-inflated lobes feature extreme density and
temperature contrasts, but are in approximate pressure equilibrium with their
surroundings. An artificial viscosity limiter is included to prevent spurious
viscosity in shear flows. An artificial conductivity limiter is also included,
to prevent spurious energy transfer in all flows.

In our simulations we do not include radiative cooling, gravity, magnetic
fields or cosmic rays, since such additional physics might cause deviations
from the simple model of self-similar jet and lobe evolution. We have,
however, performed runs with gravity and radiative cooling as a consistency
check. We found very small differences compared to purely hydrodynamical jets,
which shows that the artificial conduction limiter in SPHENIX prevents
spurious radiative losses, even with very poorly resolved jets. We do not
include relativistic effects, since we do not include very large velocities
where these effects occur (the reasoning for this choice is outlined in
Section 3.3). We therefore set the adiabatic index to $\gamma=5/3$ for all gas
in our simulations.

###  3.1 Jet launching scheme

AGN jets in an SPH code can be implemented through velocity kicks of gas
particles. Given a jet power of interest, $P_{\mathrm{j}}$, the time interval
at which particles need to be kicked is given by

$\Delta
t=\frac{2\times\frac{1}{2}m_{\mathrm{gas}}v_{\mathrm{j}}^{2}}{P_{\mathrm{j}}}.$
(7)

Here, $m_{\mathrm{gas}}$ is the mass of the gas particles in the simulation
$v_{\mathrm{j}}$ is some arbitrary launching velocity, and the factor of $2$
is present to ensure that two particles are always kicked (in opposite
directions, ensuring conservation of momentum). The total number of kicking
events can be calculated as

$N_{\mathrm{j}}=\frac{T_{\mathrm{j}}}{\Delta
t}=\frac{T_{\mathrm{j}}P_{\mathrm{j}}}{m_{\mathrm{gas}}v_{\mathrm{j}}^{2}},$
(8)

where $T_{\mathrm{j}}$ is the lifetime of the jet episode. The larger
$N_{\mathrm{j}}$, the better the jet will be resolved (as one might expect,
and as will be clear from our results).

$N_{\mathrm{j}}$ can be increased by decreasing the particle mass or launching
velocity, or by increasing the total energy launched into the jet (by
increasing either the jet power or jet duration). When attempting to simulate
the self-similar regime of jet evolution, one also has to keep in mind that
the length scale $L_{1}$ (Eqn. 1) needs to be small, while the length scale
$L_{2}$ (Eqn. 2) conversely needs to be large. These represent additional
constraints on the choice of parameters characterizing the jets and the
ambient medium.

The most natural implementation of AGN jets in SPH would involve kicking
particles from the smoothing kernel of the central black hole. The SWIFT
hydrodynamical code, which we utilise in this work, includes black holes so
this scheme is easy to implement (see Huško et al. 2022, where we use such an
implementation in the context of self-consistent accretion and feedback in
idealized galaxy groups and clusters). We have attempted this scheme in the
present work, and we find that it works in general. However, at very high
resolutions (more than $10^{4}-10^{5}$ launching events per few dozen Myrs,
the kind of resolution we are interested in when testing jet hydrodynamics),
this scheme can become computationally expensive and unreliable. In
particular, the black hole requires very small time steps between kicking
events, smaller than the typical evolutionary time step of the particles
kicked into the jet. This can result in particles being kicked more than once.

For simplicity, we instead populate the initial conditions with a reservoir of
particles that are to be used for jet launching, therefore bypassing any
issues that might arise in a setup using a BH. In general, the reservoir we
use takes the shape of two cones (defined by some half-opening angle
$\theta_{\mathrm{j}}$), placed along both directions of the $z$ axis, up to
some maximal radius (10 kpc in all our simulations). We obtain these cones by
creating a uniform cube with a grid of particles, and then cut out the desired
cones. These particles are not allowed to interact with any other particles
until they have been kicked and have cleared the region associated with the
reservoir. The particles are launched progressively from the outside-in, so
that they can immediately interact with the ambient medium, instead of
traveling through the frozen-in reservoir. The total number of particles in
the reservoir exactly matches the number to be launched into the jets we are
simulating. The density of this reservoir is $\approx 10^{-27}$
$\mathrm{g}\mathrm{cm}^{-3}$, which is $\approx 10$ times less than the
density of the ambient medium. In the case of ballistic jets with null opening
angle (only a single simulation), we instead use a spherical reservoir with a
radius of 5 kpc, from which particles are launched parallel to the $z-$axis.
We have tested a cylindrical reservoir of similar size, but found the
differences to be negligible.

###  3.2 Physical setup

The structure of realistic gaseous haloes, representing the intracluster
medium, can be represented using a density profile that is constant in the
centre, and falls off as $r^{-\alpha}$ at large distances, with $\alpha\approx
2$ (Komatsu & Seljak 2001, Croston et al. 2008), at least out to roughly the
virial radius (at larger distances the profile drops more sharply, Eckert et
al. 2012). Many jet simulation studies incorporate profiles similar to this
(e.g. English et al. 2016, Weinberger et al. 2017). While it may be more
realistic to launch jets into such a profile, we choose instead a constant
density medium ($\alpha=0$) for most of our simulations. We do this since the
jet length should scale as $t^{0.6}$ in such a setup, whereas at large
distances in a realistic profile ($\alpha=2$), the jet length scales as $t$.
This is due to the jet-inflated lobes not behaving self-similarly for
$\alpha=2$, whereas for $\alpha=0$, they are firmly in the self-similar
regime, provided an appropriate choice of parameters.

We launch most of our jets into the same background medium, with a constant
density $\rho_{0}$ (the value we choose is discussed in the next subsection).
In this case we choose periodic boxes that are slightly longer in each
dimension ($\approx 20\%$) than the predicted sizes of the jets based on the
theory outlined in Section 2. We find that this works well in all cases, and
the jets do not reach the edges of the box by the end of the simulation.

We have performed a few runs where the ambient medium instead features a
power-law density profile, such that $\rho\propto r^{-\alpha}$. We restrict
ourselves to $\alpha<2$, since the self-similar solution from the previous
section is applicable only in this regime. In these cases we use a Navarro et
al. (1996) (NFW) background gravitational potential, and we choose the gas
pressure (and therefore the temperature) in such a way that the gaseous halo
is held in hydrostatic equilibrium. For this purpose we choose NFW parameters
representing a galaxy cluster with a halo mass $M_{\mathrm{h}}=10^{15}$
$\mathrm{M}_{\odot}$ at redshift $z=0$, virial radius $R_{\mathrm{v}}\approx
2$ Mpc and concentration parameter $c=4$. Since it is impossible to implement
a power-law density profile such that the power law is valid all the way to
the centre of the halo, we use a cored $\beta$-profile

$\rho(r)=\frac{\rho_{0,\beta}}{[1+(r/r_{\mathrm{c}})^{2}]^{3\beta/2}},$ (9)

where we choose the value of $\beta$ to match our desired value of the slope
at large radii, $3\beta=\alpha$. We choose a small core, $r_{\mathrm{c}}=10$
kpc, which matches the size of our jet reservoir. The normalisation
$\rho_{0,\beta}$ is then calculated so that the total mass of the gaseous halo
is $15\%$ of the total halo mass. This choice is not necessary in this
application, since we are only interested in how our jets compare with theory
(and not how realistic they are), but we make it for simplicity.

###  3.3 Simulations

In Table 1 we summarise the parameters used for all of our simulations. In the
first row we specify the fiducial choice of parameters for our constant-
density ambient medium simulations. This choice corresponds to: 1)
$m_{\mathrm{gas}}=1.81\times 10^{5}$ M⊙, 2) $P_{\mathrm{j}}=10^{46}$ erg/s, 3)
$v_{\mathrm{j}}=15000$ km/s and 4) $\theta_{\mathrm{j}}=10\degree$. We vary
all of these parameters, but we do not vary the ambient density, which we
choose to be $\rho_{0}=1.2\times 10^{-26}$ g/cm3. We also do not vary the jet
duration, which we set to $T_{\mathrm{j}}=100$ Myr.

Our chosen ambient gas density is the typical central density of a galaxy
cluster with a halo mass $M_{\mathrm{h}}=10^{15}$ M⊙, a virial radius $2$ Mpc,
and a baryonic mass ratio $0.15$. The initial temperature of this gas is set
to $T=10^{7.2}$ K (and this is also the temperature of the gas kicked into the
jets). This value is somewhat low for the cores of realistic clusters, but we
choose it to ensure that our jets never reach the regime in which ambient
pressure is important (Eqn. 2). In any case, the aim in this work is not to
produce perfectly realistic jets, but rather to check that SWIFT can correctly
simulate jets.

Our fiducial mass resolution is 10 times better than the EAGLE simulation
(Schaye et al. 2015). At this mass resolution, the typical smoothing length
(corresponding to spatial resolution) is $\approx 1$ kpc in the ambient medium
and $\approx 7$ kpc in the jet-inflated lobes, which are about $300$ times
less dense than the ambient medium. At our highest resolution level, the
typical smoothing lengths are instead $\approx 0.3$ and $\approx 2$ kpc for
the two cases. For comparison, the lobe is roughly $400$ kpc long and $50$ kpc
in radius at the end of the simulation (see Fig. 4).

The jet power we use is relatively high compared to previous similar
simulations (e.g. Weinberger et al. 2017, Bourne & Sijacki 2017). However,
observations imply that jet episodes with such powers are frequent in the most
massive galaxy clusters (Kino & Kawakatu 2005, Hlavacek-Larrondo et al. 2012).
Our choice of half-opening angle is somewhat large compared to most real jets
(Pushkarev et al. 2009), but we choose such a value to ensure that jets are
ballistic for as short a time as possible (Eqn. 1). In addition, observed
subrelativistic jets have similarly large opening angles (Pushkarev et al.
2017).

We use a non-relativistic launching velocity of order $10^{4}$ km/s for a few
reasons: 1) SWIFT does not include relativistic effects, 2) launching jets
with relativistic velocities leads to only small differences (e.g. Komissarov
& Falle 1998, English et al. 2016), 3) using velocities of order $10^{5}$ km/s
or higher would result in poorly resolved jets and 4) velocities of order
$10^{4}$ km/s are typically employed in cosmological simulations that include
jets (Weinberger et al. 2018, Davé et al. 2019). The fiducial mass resolution
we have chosen results in a total of $\approx 10^{8}$ particles in the
simulation, while the jet power and launching velocity, in combination with
the mass resolution and jet duration, yield $\approx 40000$ particles launched
per jet. The actual number of particles in the jets and lobes may be larger
due to ambient particles being swept up.

When varying any of the four parameters listed at the beginning of this
subsection, we keep other parameters fixed. The variations we have done for
our constant-density ambient medium case are given in the second row of Table
1. We have simulated jets with numerical resolutions corresponding to ten
times worse than the EAGLE simulation ($m_{\mathrm{gas}}=1.81\times 10^{7}$
M⊙), down to 3160 times better ($m_{\mathrm{gas}}=5.73\times 10^{3}$ M⊙),
differing by factors of 3.16 (logarithmic interval of 0.5). Our highest
resolution simulation has a total of $2.8\times 10^{9}$ particles, and
$1.4\times 10^{6}$ particles kicked into each jet. The lowest resolution one
has only 450 particles per jet. We vary jet powers between $10^{45}$ erg/s and
$10^{47}$ erg/s, launching velocities between $3750$ km/s and $60000$ km/s,
and half-opening angles between $0\degree$ and $25\degree$.

Finally, in the case of power-law gaseous atmospheres, we have performed three
simulations, with $\alpha=0.5$, $\alpha=1$ and $\alpha=1.5$. The parameters of
these simulations are listed in the third and final row of Table 1. These
power-law cases required a different set of parameters for two reasons. We
found that the length scales $L_{1}$ and $L_{2}$ were larger and smaller,
respectively, with our fiducial choice of jet-related parameters, than they
were in the constant-density case. This means that the jets took a longer time
to reach the self-similar phase, and would also take a shorter time to exit
the same phase due to the external pressure becoming important.

Given these restrictions, we chose to modify our fiducial parameters in the
following way. We launched the jets with: 1) a jet power of
$P_{\mathrm{j}}=10^{47}$ erg/s in order for the length scale $L_{2}$ to lie at
comfortably large distances compared to the self-similar prediction (see Eqn.
2, although note that its meaning is somewhat moot in power-law atmospheres),
2) a jet velocity of $v_{\mathrm{j}}=0.2c=60000$ km/s, bringing $L_{1}$ down
to $L_{1}=5$ kpc (given the new jet power), 3) a jet duration of
$T_{\mathrm{j}}=40$ Myr to prevent the jets from reaching large distances
(wishing to avoid both $L_{2}$ and the virial radius). With these changes to
the physical parameters, the number of particles launched into the jets is a
quarter of that in our standard constant-density simulations. For this reason
we have decreased the particle masses in the power-law simulations (i.e.
increased the resolution) by a factor of four. This ensures that the jets are
resolved with the same number of particles.

Table 1: List of all simulations and the parameters we use. In the first row we specify the parameters of our fiducial simulation with a constant-density ambient medium. In the second row we specify the parameters that are varied for this same case. In the final row we specify the parameters of the case with a power-law ambient density profile. The parameters are, in order: 1) $m_{\mathrm{gas}}$ \- mass resolution, 2) $P_{\mathrm{j}}$ \- jet power, 3) $v_{\mathrm{j}}$ \- jet velocity, 4) $\theta_{\mathrm{j}}$ \- jet half-opening angle, 5) ambient medium density $\rho$ \- constant value or power law slope and 6) $T_{\mathrm{j}}$ \- jet duration.  $m_{\mathrm{gas}}$ $[\mathrm{M}_{\odot}]$ |  $P_{\mathrm{j}}$ [$\mathrm{erg}\hskip 0.85358pt\mathrm{s}^{-1}$] |  $v_{\mathrm{j}}$ [$\mathrm{km}\hskip 0.85358pt\mathrm{s}^{-1}$] |  $\theta_{\mathrm{j}}$ [$\degree$] |  $\rho$ [$\mathrm{g}\hskip 0.85358pt\mathrm{cm}^{-3}$] |  $T_{\mathrm{j}}$ [$\mathrm{Myr}$]
---|---|---|---|---|---
$1.81\times 10^{5}$ | $10^{46}$ | $1.5\times 10^{4}$ | $10$ | $1.2\times 10^{-26}$ | 100
$5.73\times 10^{3}-1.81\times 10^{7}$ | $10^{45}-10^{47}$ | $3.75\times 10^{3}-6\times 10^{4}$ | $0-25$ | $1.2\times 10^{-26}$ | 100
$4.53\times 10^{4}$ | $10^{47}$ | $6\times 10^{4}$ | $10$ |  $\propto r^{-\alpha}$ , $\alpha=0.5-1.5$ | 40

###  3.4 Definition of jet lobe

All simulations of jets exhibit the so-called lobe, made up from hot, shocked
gas that was previously part of the jet. Jets also invariably launch a bow
shock that propagates through the ambient medium. The model of self-similar
lobe evolution predicts that their aspect ratio should be constant, and it
predicts the same for the ratio of energy in the lobes versus the energy added
to the ambient medium (as well as for how much energy is in kinetic and
thermal forms). In order to test these predictions, it is important to
numerically determine which particles belong to what we might visually call
the jet or the jet lobe, with remaining particles classified as making up the
ambient medium.

Figure 1: Visualisations of the gas temperature distribution for a ballistic
jet (effective half-opening angle $\theta_{\mathrm{j}}=0\degree$, top) and a
jet in the self-similar evolutionary phase (half-opening angle
$\theta_{\mathrm{j}}=10\degree$, bottom) at different times. All other
parameters correspond to our fiducial choice, given in the first row of Table
1. Colours represent the temperature of the gas, as given by the colour bar.
The panels show slices 10 kpc in depth.

Empirically, we find that the peak temperature $T_{\mathrm{p}}$ achieved by
gas particles serves well to define the lobe. We use a threshold value
$T_{\mathrm{p,min}}$, and all particles whose peak temperature was at some
point above this value, i.e. $T_{\mathrm{p}}>T_{\mathrm{p,min}}$, are defined
as constituting the lobe. This definition is motivated by the fact that
particles that reach extremely high temperatures are exclusively located in
the lobe, whereas the shocked ambient medium experiences temperature jumps of
a factor of several at most.

The appropriate value of $T_{\mathrm{p,min}}$ varies from simulation to
simulation, but it can easily be estimated (to better than an order of
magnitude) by assuming that all of the kinetic energy launched into the gas
becomes thermalised. From this condition, the characteristic temperature of
the lobes is $T_{\mathrm{lobe}}\approx\mu
m_{\mathrm{p}}v_{\mathrm{j}}^{2}/3k_{\mathrm{B}}$, with $\mu=0.6$ the mean
molecular weight, $m_{\mathrm{p}}$ the proton mass and $k_{\mathrm{B}}$ the
Boltzmann constant. In reality the lobes are somewhat less hot since not all
of the kinetic energy is thermalised. The appropriate value of
$T_{\mathrm{p,min}}$ can thus be expected to be a factor of several times
below $T_{\mathrm{lobe}}$. We find the particular value that we use by
plotting the total mass in the lobe $M_{\mathrm{lobe}}$ versus
$T_{\mathrm{p,min}}$, the defining peak temperature that determines how many
particles will be assigned to the lobe. We find that this dependence exhibits
a change in slope at some critical value of $T_{\mathrm{p,min}}$. Using a
larger than the critical one results in fewer and fewer particles (that are
part of the lobe) being assigned to the lobe, whereas using a lower one causes
ambient medium gas to be assigned to the lobe (usually ambient gas particles
near the jet head, where strong shocks are occurring). For our fiducial
simulation, with a jet launching velocity of $v_{\mathrm{j}}=15000$ km/s, we
use $T_{\mathrm{p,min}}=5\times 10^{8}$ K. This value is $\approx 30$ times
larger than the initial temperature of the ambient medium, and $\approx 10$
times smaller than the characteristic temperature of the lobe, which is in
this case $T_{\mathrm{lobe}}\approx 5\times 10^{9}$ K.

We also define unshocked, fast-moving, recently launched particles as
constituting the lobe, which means that we include the jets into the lobes
(for simplicity). Given a launching velocity $v_{\mathrm{j}}$, we thus define
all particles with $|\mathbf{v}|>0.5v_{\mathrm{j}}$ as also being part of the
lobe. Note that the factor $0.5$ here is fairly unimportant; using any
critical velocity value that is significantly above the sound speed of the
ambient gas leads to almost all of the unshocked particles being included in
the lobes, and none of the ambient medium. Particles not belonging to the jets
or the lobes are classified as part of the ambient medium.

###  3.5 Measuring the energetics and jet/lobe lengths and radii

Given the above definition of the lobe, measuring the thermal and kinetic
energy gains of both the lobes and the ambient medium is trivial. We calculate
the kinetic energy gain of either component by calculating the total kinetic
energy in the particles (all of the gas in the simulation is initially not
moving), while the thermal energy gain is calculated relative to the initial
temperature of all of the gas, which is $T_{0}=10^{7.2}$ K.

The length of the lobe is calculated by taking the mean distance from the
origin of $n$ farthest particles (ordered along the axis of launching), with
$n$ determined to be $3\%$ of all of the particles launched into the jet. With
this definition, $n$ is both resolution and time-dependent. For the radius of
the lobes, the procedure is similar, but we use cylindrical distances from the
launching axis. Note that this choice yields the maximal radius of the lobe,
not the average radius. For both the length and the radius of the lobe we use
a mean of the values calculated for both of the jets.

##  4 Results: general properties of simulated jets

###  4.1 Ballistic jets

In this section we will discuss properties of jets launched into a constant-
density ambient medium. Before focusing on self-similar jet and lobe
evolution, it is worth addressing some properties of ballistic jets simulated
with SWIFT. According to Eqn. (1), jets should remain ballistic at arbitrary
distances in the $\theta_{\mathrm{j}}=0\degree$ case. The top panels of Fig. 1
show visualisations of jets from a simulation with a null half-opening angle,
simulated at a mass resolution 10 times better than the EAGLE simulation
($m_{\mathrm{gas}}=1.81\times 10^{5}$ M⊙). The jet power is
$P_{\mathrm{j}}=10^{46}$ erg/s, and the launching velocity
$v_{\mathrm{j}}=15000$ km/s (i.e. these parameters, other than the opening
angle, match our fiducial choice listed in Table 1).

Figure 2: Visualisations of the gas temperature distribution in our highest
resolution jet simulation, with particle mass $m_{\mathrm{gas}}=5.73\times
10^{3}$ M⊙, at different times. All other parameters correspond to our
fiducial choice, given in the first row of Table 1. Colours represent the
temperature of the gas, as given by the colour bar. The panels are 10 kpc in
depth.

The figure shows a few visually distinct regions. The particles launched into
the jets constitute a thin, cylindrical region (often called the jet spine) of
unshocked, cold gas (with the temperature the same as the ambient medium).
This gas experiences some shocking all the way from the launching region to
the jet head. The gas shocked in this way makes up the hot lobe that can be
seen surrounding the jet itself. Finally, the action of the jet also launches
a bow shock, which transitions from being strongly supersonic near the jet
head to a sound wave in the perpendicular direction.

Visually inspecting the jet, we can surmise whether this jet is in the self-
similar or ballistic phase. It is clear that the aspect ratio of the lobes
(length vs. width) grows with time, whereas in the self-similar case it should
remain constant. Furthermore, we can see that the jet increases in length by
an approximately equal amount with each snapshot, indicating that the jet
velocity is nearly constant (as it should be in the ballistic regime). More
quantitatively, we find the power-law slope of the $L_{\mathrm{j}}-t$
dependence to be $0.9$, very near the ballistic value of $1$.

The typical velocity of the jet head is found to be $\approx 5500$ km/s.
However, a ballistic jet should drill through the medium at exactly the
launching velocity, which is $15000$ km/s in this case. Equally, one might
wonder why is there significant shocking of the jet particles along the way
from the launching region to the jet head, whereas we would expect all shocks
to happen at the latter location (for ballistic jets). We have performed other
simulations of ballistic jets, which we do not show here (since we focus on
self-similar jets), where we find that this discrepancy is due to numerical
resolution. In particular, we find that increasing the resolution leads to
less shocking occurring inside the jet spine. The jet head velocity is
consequently larger, and the jets thinner and denser. We find that simulated
jets are close to being fully ballistic only at very high resolutions
($>10^{5}$ particles per jet).

Figure 3: Top: properties of our highest resolution jet simulation after $96$
Myr (with a mass resolution $m_{\mathrm{gas}}=5.73\times 10^{3}$ M⊙, while
other parameters correspond to our fiducial choice, given in the first row of
Table 1). Bottom: properties of a simulation with a slightly lower resolution,
$m_{\mathrm{gas}}=1.81\times 10^{4}$ M⊙, a smaller opening angle
$\theta_{\mathrm{j}}=5\degree$ and a larger jet velocity
$v_{\mathrm{j}}=30000$ km/s, after $48$ Myr. Each panel is 10 kpc in depth and
shows different gas properties, as given by the titles and legends.

###  4.2 General properties of jets in the self-similar phase

In the bottom panels of Fig. 1 we show a jet from a simulation with an
equivalent set of parameters as the previously discussed ballistic jet
simulation, but with a half-opening angle of $10\degree$ rather than
$0\degree$. The jet power, $P_{\mathrm{j}}=10^{46}$ erg/s and the launching
velocity $v_{\mathrm{j}}=15000$ km/s, yield the $L_{1}$ length-scale of $7.7$
kpc, on order of our launching region size. This means that jets should reach
the self-similar phase almost as soon as they are launched. The $L_{2}$ length
scale is $970$ kpc, so ambient pressure should not be important.

This jet is shorter and wider than the ballistic one, as one might intuitively
expect. The central outflow of unshocked gas no longer forms a thin spine;
this jet gas instead flows conically until it is recollimated at distances of
$\approx 50$ kpc (this is referred to as recollimation since all jets are
initially collimated at small distances, e.g. Park et al. 2019, Chatterjee et
al. 2019). The recollimation is driven by previously shocked jet gas, which
constitutes a hot lobe surrounding the unshocked jet. The collimated gas then
expands and collimates again in an oscillatory fashion; these spatially
periodic recollimation shocks are expected theoretically (Falle 1991, Bamford
& Komissarov 2018), have been found in other simulations (van Putten 1996,
Mizuno et al. 2015, Hervet et al. 2017, Bodo & Tavecchio 2018, Gourgouliatos &
Komissarov 2018, Smith & Donohoe 2019) and could explain multiple hotspots
observed in some radio jets (Rees 1978, Dreher & Feigelson 1984, Hardcastle et
al. 2003).

In Fig. 2 we show a visualisation of our highest-resolution jet simulation,
with the same set of parameters but simulated at a resolution of
$m_{\mathrm{gas}}=5.73\times 10^{3}$ M⊙ (316 times better than EAGLE
resolution). The jet shown exhibits clear signs of a conical outflow of
particles, which is collimated at the point where significant shocking begins
to occur (visible by a change in temperature). Unlike the lower-resolution
simulation, we see no sign of multiple recollimation shocks. Based on
analytical expectations (Eqn. 26 from Kaiser & Alexander 1997), the initial
recollimation shock should occur at $\approx 60$ kpc along the z axis for our
jets. We find that the cone begins to lose coherence at roughly such a
distance, albeit somewhat too far ($70-100$ kpc, difficult to pinpoint exactly
since the outflow is shocked earlier in the centre of the cone than at the
edges).

Due to instabilities, we find that the jet lobe does not exhibit a simple
ellipsoidal shape. The lobe is also not very homogeneous in gas temperature
(as well as other properties; see Fig. 3), and this is likely the reason why
there are no multiple recollimation shocks (recollimation requires uniform
pressure from all sides). For the same reasons, the bow shock created by the
jet exhibits some irregularities. The deviation from a smooth ellipsoidal
shape (which we do find at lower resolutions, see Fig. 6) could be explained
in a few ways:

  1. 1.

Usage of a fairly low launching velocity means that the jet is heavy, and thus
mixes with the ambient medium less easily (Rossi et al. 2008, English et al.
2016, Donohoe & Smith 2016).

  2. 2.

Only the largest-wavelength Kelvin-Helmholtz instabilities are resolved in
this simulation. In other words, it is possible that at even higher
resolutions, a jet lobe would again be recovered, but a highly turbulent one.

  3. 3.

The recollimation shock is expected to occur at a large distance from the jet
origin (60 kpc). A lobe is not expected to exist until the jet has reached
that distance. A lobe-like feature does appear and become more prominent from
snapshot 2 to 3, and 3 to 4 in Fig. 2. If the jet was allowed to become much
larger than the distance to the recollimation shock, a clear lobe would likely
be visible. This is supported by the equivalent lower-resolution jet also
having a clear lobe only at later snapshots.

In the top panels of Fig. 3 we show several different gas properties in the
high-resolution jet simulation after $96$ Myr of evolution. The conical
outflow is constituted by cold, dense and fast-moving gas, with a high Mach
number ($M\approx 10-40$). This gas has a low entropy and is under-pressured
compared to the jet lobe. This causes a collimation of the outflowing
particles and subsequent shocking.

The jet lobe, constituted from previously shocked jet particles, is of high
temperature and low density (by a factor of up to $\approx 300$ relative to
the ambient medium, although this varies greatly within the lobes), as well as
in pressure equilibrium. Its high entropy indicates that this gas has
experienced significant shock heating. The typical velocity in the jet lobe is
of order a few $\times$ $1000$ km/s, much lower than the launching velocity.
Furthermore, although not visible on this plot, the velocity is not only in
the z-direction. In particular, near the base of the jet, the lobe particles
move in the direction opposite of the general jet direction, constituting the
so-called backflow (Lind et al. 1989, Rossi et al. 2008, Cielo et al. 2014,
Mukherjee et al. 2018).

The bow shock surrounding the jet lobes has a high density, indicating that it
is constituted from particles swept up by the jet. It is mildly supersonic
with Mach number of order $M\approx 1.5$. Such Mach numbers have been found in
deep X-ray observations of galaxy clusters, in the presence of weak shocks
(Fabian et al. 2003, Forman et al. 2007, Snios et al. 2018), and are thought
to be one of the main ways that jets can heat the intra-cluster medium in an
isotropic fashion (Reynolds et al. 2001, Brüggen et al. 2007, Li et al. 2017,
Weinberger et al. 2017). Near the base of the jets, however, we find that the
bow shock expands laterally at the sound speed. It is almost invisible on the
entropy plot, indicating that it has experienced only mild shock heating,
compared to both the lobe and the ambient medium near the jet head.

In the bottom panels of Fig. 3 we show a jet in the self-similar regime with a
narrower opening angle ($\theta_{\mathrm{j}}=5\degree$) and higher launching
velocity ($v_{\mathrm{j}}=30000$ km/s). The jet power is the same, while the
mass resolution is relatively fine at $m_{\mathrm{gas}}=1.81\times 10^{4}$
$\mathrm{M}_{\odot}$. This would generally yield a high-resolution jet, but
usage of a large launching velocity (larger by factor of two compared to our
standard choice) effectively reduces the jet resolution by a factor of four.
This jet exhibits an interesting streak of more than a dozen recollimation
shocks, owing to the narrow opening angle and large launching velocity. The
Mach number in this case reaches values up to $M\approx 100$. The large
launching velocity results in very light and hot lobes with
density/temperature contrasts of almost a factor of 1000, relative to the
ambient medium.

###  4.3 Evolution of the jet-inflated lobe length and width

We now turn to comparing the length and radius of the lobes in our simulations
to theoretical predictions of self-similar lobes discussed in Section 2. Here
we show results for our standard-resolution simulation (i.e. the one shown in
the bottom panels of Fig. 1). We do this since for that simulation we have a
relatively large number of snapshots (50), whereas we were only able to output
four snapshots for the highest-resolution simulation (due to storage
restrictions). We discuss the dependence of various properties on resolution
in Section 5.4.

Fig. 4 shows the time dependence of jet/lobe length and lobe radius (at its
widest point), as well as the bow shock radius, in our standard-resolution
simulation. We also show the prediction based on approximate theoretical
models of jets and lobes in the self-similar regime (Eqn. 5, Kaiser & Best
2007). The agreement is clearly very good, especially at late times. This is
true for both the lobe (and jet) length and the lobe radius, both of which
have the same slope ($0.6$) in the time dependence. This means that the lobe
has a constant aspect ratio, as predicted in the self-similar model, and its
value is in agreement with the predictions. We find that the bow shock radius
scales with time in the same way, which is also in agreement with the self-
similar theory. We find that the lobe is slightly too short and too wide at
$t<5$ Myr, although this is likely related to lower resolution at these times,
since only a small fraction of particles have been launched into the jets.

Figure 4: Time evolution of the jet lobe length and radius, as well as the bow
shock radius (solid and dotted lines, as marked), in our standard simulation
(first row of Table 1). These are compared to predictions from Kaiser & Best
(2007) for the lobe (dashed lines; Eqn. 5).

###  4.4 Jet, lobe and ambient medium energetics

The question of how much energy is in which component, and in what form, is an
interesting one in the context of jets as a feedback mechanism that plays a
role in the formation and evolution of galaxies. We again focus on the
energetics of our standard-resolution self-similar simulation, rather than the
high-resolution one, due to more snapshots being available. We have performed
similar analyses for the higher-resolution simulations and we find that the
energetics is well converged, down to the level of a few per cent. The
procedure for calculating the energies in the jets and lobes (here we group
the jets into the category of ’lobe’ for the purpose of simplicity), as well
as the ambient medium, is described in Section 3.5.

In the top panel of Fig. 5 we show the energetics of our standard-resolution
simulation. The top panel shows the kinetic and thermal energies of the jet-
inflated lobes and the ambient medium. The sum of all of these energies
exactly matches the total injected energy, showing that energy is conserved.
During the initial phase of jet launching, the ambient medium and the lobe are
approximately equal in total energy. At later times, the lobe carries about a
third of the energy, whereas the remainder has been transferred to the ambient
medium. At late times this ratio is constant, as one would expect for a jet in
the self-similar phase (since the volume ratios of the lobe and the region
defined by the bow shock remain constant, due to the constant shape of both
components).

For the jet lobes we find that they are dominated by kinetic energy initially,
but by the time they are turned off (at the end of the simulation), an
approximately equal amount of energy is in the thermal component. The ambient
medium initially has roughly equal amounts of added kinetic and thermal
energy, but at later times about two thirds of the added energy is in the
kinetic component. More quantitatively, at $t=100$ Myr (the end of the
simulation), the energy partition is as follows: 1) lobe kinetic - $20\%$, 2)
lobe thermal - $14\%$, 3) ambient kinetic - $47\%$ and 4) ambient thermal -
$19\%$. Across different simulations we find that these fractions can vary,
but none of the components becomes negligible.

In the bottom panel of Fig. 5 we show the energy density per unit length in
the lobe, including the jet itself (normalised by the length of the jet), at
the end of the same simulation. Note that this is the energy density per unit
length in slices perpendicular to the jet axis, and not just the energy
density per unit length exactly along that axis. The kinetic energy density is
roughly constant along the entire length, but shows signs of oscillations
around the constant component. These oscillations are a result of the multiple
recollimation shocks. Thermal energy is initially negligible, but reaches
about the same density as the kinetic component at one third the jet length.
The total energy density first rises, reaches a slight peak around half the
jet length, and then falls - this is a result of the ellipsoidal shape.
However, at the very end of the jet, both the thermal and kinetic energies
reach a peak. This is likely a feature of the terminal shock.

Figure 5: Energetics of our standard jet simulation (see the first row of
Table 1 for a list of parameters), at $t=100$ Myr. Top: Evolution of the total
energies in the jet lobe and ambient medium. Blue lines represent kinetic
energy, while red lines are the thermal energy. The sums for the lobe and
ambient medium are given by the black lines. The grey lines represent the
total injected energy and the total change in the energy of the lobes and the
ambient medium. These two lines overlap, showing that energy is numerically
conserved. Bottom: Kinetic, thermal and total energy densities per unit length
along the jet axis, normalised by the total jet length.

###  4.5 Comparison with previous simulations

Here we will compare our results with previous hydrodynamical simulations of
AGN jets, all of which were performed using grid-based codes. While there is
an extensive literature of such simulations, most of these include and study
the effects of more exotic physics that we do not include, such as magnetic
fields (e.g. Hardcastle & Krause 2014, Tchekhovskoy & Bromberg 2016, Mukherjee
et al. 2020), radiative cooling (e.g. Blondin et al. 1990, Stone et al. 1997,
Guo et al. 2018) and cosmic rays (e.g. Guo & Mathews 2011, Ehlert et al. 2018,
Yang et al. 2019). We thus do not compare with such studies. We also restrict
our comparison to studies that launch jets into a constant-density ambient
medium, since this is the focus of our study. We do not, however, restrict
ourselves to comparisons with non-relativistic jet studies. We do this since
the majority of the literature has included relativistic effects, and since
differences between classical and relativistic jets (or more accurately, the
lobes they inflate) should be minor (Kaiser & Best 2007).

The above restrictions leave only a few studies that are comparable with out
study. We begin by comparing with the results of Falle (1991), who also
presented the first analytical model of self-similar jet lobe expansion. They
performed 3D simulations of classical jets launched into an ambient medium of
fixed density. They did not use a finite (non-zero) opening angle (unlike our
study), since they argue that this requires the jet launching region to be
well resolved. Instead they used a zero opening angle and a relatively small
internal Mach number, which should be similar to using a non-zero opening
angle in combination with a large internal Mach number ($M\gtrsim 10)$. We
independently confirm this to be the case, although we do not show the results
of those simulations here.

These two different set-ups lead to a similar outcome for the following
reasons. In the finite-opening angle case, the outflowing jet gas has a
conical geometry as a direct consequence of how it was set up. The gas is
eventually recollimated once the pressure in the jet cone becomes lower than
the pressure of the lobe gas. In the case of a zero-opening angle and small
internal Mach number, the outflowing jet gas is instead hot enough that it
expands on account of thermal pressure. It recollimates for the same reason as
the finite-opening-angle case. We note, however, that the relation between an
’effective’ opening angle (or aspect ratio of the lobes) and the finite Mach
number is not obvious. The value Falle (1991) used is $M=5$, which they find
corresponds to $\theta_{\mathrm{j}}=13\degree$. They found that their jets
inflate lobes that expand self-similarly, i.e. with a constant aspect ratio
and whose length evolves with time as $L\propto t^{3/5}$. Both of these
findings are in good agreement with our results (4). For comparison, our
standard simulations have a finite opening angle of $10\degree$ and an
internal Mach number of $M\approx 20$.

Komissarov & Falle (1998) performed simulations that are more directly
comparable to ours, since they used finite opening angles
($\theta_{\mathrm{j}}=5-20\degree$) and very large internal Mach numbers.
These simulations were also 3D, and included both the classical and
relativistic variety of jets. They found that the latter are very similar to
the former in qualitative behaviour, so we do not differentiate between the
two for the purpose of this comparison. With their set-up, they also found
jets whose lobes reach the self-similar regime at late times, with $L\propto
t^{3/5}$. However, they found that this regime is not reached as soon as the
jet is much larger in length (e.g. several times) than the transition length-
scale $L_{1}$ that separates the ballistic and self-similar evolutionary
period (Eqn. 1). Instead, the transition is complete once the width of the
jet-inflated lobe is a few times larger than $L_{1}$. They also found that the
transition is very prolonged, and there is no obvious break in the velocity,
aspect ratio or jet length (i.e. the jet length does not suddenly change from
scaling linearly with time to a $\propto t^{3/5}$ scaling around $L_{1}$). Our
findings are also in good agreement with these results. From Fig. 4 we see
that there is no sudden transition at around $L_{1}$, which is $\approx 8$ kpc
in our case. Instead, the jet is too short but also too wide (and therefore
too stumpy) compared to the self-similar prediction. By the time the width of
the lobe is $\approx 15$ kpc $=(2-3)\times L_{1}$, the jet has, however,
reached the self-similar regime. Both the length and the radius of the lobe
begin to agree with the self-similar prediction at the same time. This is in
good agreement with the finding of Komissarov & Falle (1998), who argue that
this transition is complete once the aspect ratio has increased enough to
reach the value it should have in the self-similar regime.

Finally, we briefly compare with some of the newer work on hydrodynamical AGN
jets by Krause et al. (2012). They used finite opening angles in a 2.5D
simulation with a constant-density ambient medium. Their jets are very similar
in appearance to those of Komissarov & Falle (1998), both of which show more
mixing than our jets. It is hard to compare our results quantitatively (since
Krause et al. 2012 do not show results on the evolution of the lobe length and
radius), but we note that Krause et al. (2012) also find a distinct transition
that occurs once the jets reach the $L_{1}$ length-scale.

##  5 Results: comparison of jets with varying parameters

In this section we compare the properties of jets simulated with different
parameters. As in the previous section, all of these results concern our
constant-density ambient medium case, with the exception of the results
discussed in Section 5.6.

The parameters we vary here are jet power, half-opening angle, launching
velocity and mass resolution. The standard choice and ranges of variations are
given in Table 1, and we also repeat the former here:
$m_{\mathrm{gas}}=1.81\times 10^{5}$ M⊙, $P_{\mathrm{j}}=10^{46}$ erg/s,
$\theta_{\mathrm{j}}=10\degree$ and $v_{\mathrm{j}}=15000$ km/s. When varying
any one of the parameters, we keep all other parameters fixed. For each
variation, we have performed 5 simulations with different parameter choices
(with the exception of mass resolution, where we have performed 8 simulations
in total, in order to probe the full range form very low to very high-
resolution jets).

We varied the jet power by factors of $\sqrt{10}\approx 3.16$ between
$P_{\mathrm{j}}=10^{45}$ erg/s and $P_{\mathrm{j}}=10^{47}$ erg/s, the half-
opening angle in increments of $5\degree$ from $\theta_{\mathrm{j}}=5\degree$
to $\theta_{\mathrm{j}}=25\degree$, the jet launching velocity by factors of 2
from $v_{\mathrm{j}}=3750$ km/s to $v_{\mathrm{j}}=60000$ km/s, and finally
the mass resolution by factors of $\sqrt{10}\approx 3.16$ from
$m_{\mathrm{gas}}=1.81\times 10^{7}$ $\mathrm{M}_{\odot}$ to
$m_{\mathrm{gas}}=5.73\times 10^{3}$ $\mathrm{M}_{\odot}$. Fig. 6 shows the
visualisation of jets from all of these simulations after $100$ Myr of
evolution. In Fig. 7 we show the time evolution of jet/lobe lengths for all
four cases of parameter variations.

Figure 6: Comparison of jets simulated with different parameters (see Table 1
for the range of variations), after $100$ Myr of evolution. The standard set
of parameters is: $P_{\mathrm{j}}=10^{46}$ erg/s,
$\theta_{\mathrm{j}}=10\degree$, $v_{\mathrm{j}}=15000$ km/s and
$m_{\mathrm{gas}}=1.81\times 10^{5}$ M⊙. When varying one parameter, all
others remain fixed. From top to bottom, each row contains variations of: jet
power, half-opening angle, launching velocity and mass resolution. The colours
show gas temperature, according to the colour bars. The panels are 10 kpc in
depth.

###  5.1 Varying the jet power

The main effect of increasing the jet power, as is visible in the first row of
Fig. 6, is the lengthening of the jets. The typical temperature of the jet-
inflated lobes remains the same. This is because the launching velocity is
kept constant ($v_{\mathrm{j}}=15000$ km/s). At lower jet powers, we find a
roughly ellipsoidal shape for jet lobes, and multiple recollimation shocks. At
higher jet powers, instabilities destroy this ellipsoidal lobe shape. This is
possibly a result of these higher-power jets being better resolved due to more
particles being injected into the jets. The lobe disruption results in a much
more complex structure, similar to our high-resolution simulation with
standard jet power (Fig. 2). In our highest jet power simulation, the lobe
(and therefore the bow shock) takes on a horned shape at the jet head, which
is also similar to the high-resolution simulation shown in Fig. 2. This is a
result of the transition length scale $L_{1}$ (that determines when the jet
goes from the ballistic to the self-similar regime) being larger at higher jet
powers (Eqn. 1). As a result, this jet is ballistic for a longer portion of
its lifetime shown here.

From the top left panel of Fig. 7, we can see that the jet/lobe length
evolution is self-similar for the three lower jet powers, and in fairly good
agreement with theoretical predictions. At high jet powers, however, the slope
of the time dependence is closer to $0.7$ than the self-similar value of
$0.6$. This is possibly due to the already-mentioned instabilities causing
some deviation from the self-similar model. Alternatively, the time dependence
in these two highest power simulations could be interpreted as showing the
gradual transition from a ballistic phase with $L_{\mathrm{j}}\propto t$ at
early times to a self-similar phase with $L_{\mathrm{j}}\propto t^{0.6}$ at
late times.

###  5.2 Varying the half-opening angle

From the second row of Fig. 6, we see that jets become shorter as their half-
opening angle increases, also accompanied by an increase in the aspect ratio
of the lobes. This is followed by the disappearance of multiple recollimation
shocks. Instead, the outflow takes the shape of a simple cone with a single
recollimation shock. The ballistic phase of the jet can be seen in the shocked
jet gas near the bow shock. In the two lower-angle simulations, this gas takes
the form of a thin strip, whereas at larger opening angles we see evidence of
a horned feature. From the top right panel of Fig. 7 we can see that all jets,
except that for $\theta_{\mathrm{j}}=5\degree$, agree well with the self-
similar prediction of jet and lobe length evolution. This is because the
$\theta_{\mathrm{j}}=5\degree$ jet is ballistic for a fairly long time.

###  5.3 Varying the launching velocity

Figure 7: Jet and lobe length as a function of time for simulations with
various parameters. The standard set of parameters is:
$P_{\mathrm{j}}=10^{46}$ erg/s, $\theta_{\mathrm{j}}=10\degree$,
$v_{\mathrm{j}}=15000$ km/s $m_{\mathrm{gas}}=1.81\times 10^{5}$ M⊙, and the
range of variations is given in Table 1. Each panel represents a different
variation of parameters, as visible in the legends (top left - jet power, top
right - half-opening angle, bottom left - launching velocity, bottom right -
mass resolution). Orange lines represent theoretical predictions from Kaiser &
Best (2007). Where we show only one line, there should be no change of jet
length with the parameter being varied.

The third row of Fig. 6 shows that the jet-inflated lobes change shape
significantly, as well as becoming hotter, as we increase their launching
velocity. The latter finding is expected, since there is more kinetic energy
per particle available to be thermalised (due to constant total energy and a
smaller number of particles). According to the self-similar model, changing
the launching velocity of a jet should not result in any change in the
jet/lobe length (see Eqn. 5). However, the launching velocity can affect when
the evolution changes from ballistic to self-similar. We find that jets become
shorter, relative to the $v_{\mathrm{j}}=15000$ km/s case, as we either
increase or decrease the launching velocity.

The shortening at higher $v_{\mathrm{j}}$ is unexpected since the ballistic
phase should in this case be even shorter. This effect is possibly due to
lower resolution from having a smaller number of particles in the jet
(preventing a coherent central jet spine from traveling to larger distances).
However, based on our results with varying mass resolution, down to very low
resolutions of $\approx 500$ particles per jet (see next subsection), these
high-velocity jets should still reach the same size as our standard velocity
case. According to the simulations by English et al. (2016) and Li et al.
(2018), jet-inflated lobes in simulations with higher launching velocities
should be lighter and wider at the base, due to a stronger backflow of gas
near the terminal shock. We find that the two lobes are indeed wider at the
base, to the point of the top and bottom lobes merging. This is possibly
either due to a poorly-resolved backflow, or thermal expansion. Since the
self-similar model of jet evolution takes the latter into account, but not the
former, we conclude that these stubbier jets at very high velocities are
likely due to a backflow, in agreement with previous simulations. We have
performed other simulations with similarly large velocities, including 2D
simulations with very high resolutions and near-relativistic velocities. We
find the presence of strong backflows in all cases, causing the merging of the
two lobes and a relatively stunted lobe.

We now move on to the lower-velocity cases. In the $v_{\mathrm{j}}=7500$ km/s
simulation, the jet is slightly shorter than the $v_{\mathrm{j}}=15000$ km/s
case, again due to instabilities being resolved better, which is itself a
consequence of a larger number of particles being launched into the jets with
lower velocities. However, at these velocities we are also approaching the
ballistic limit, where the jet head velocity cannot be larger than the
launching velocity. The ballistic nature is visible in the central jet
outflow, which shows no sign of multiple recollimation shocks, but instead
takes on a conical shape. Between this cone and the bow shock is a significant
amount of shocked gas, indicating that this jet is not fully ballistic. The
shocked gas, and therefore the bow shock, takes on a horned morphology. This
can be traced to the fact that gas in the centre of the conical outflow is
shocked at earlier times/smaller distances than that at the edges of the cone,
resulting in a horned feature (if viewed as a slice; in 3D this feature is a
ring). Similar features have been found in other simulations (Omma et al.
2004, Matsumoto & Masada 2019, Talbot et al. 2020). In the
$v_{\mathrm{j}}=3750$ km/s case, the jet is fully ballistic, which is visible
by the conical outflow extending all the way to the bow shock. This jet is
shorter because its launching velocity is smaller than the self-similar jet
head velocity we expect in this case ($\approx 5000$ km/s). While this is a
small difference, there is invariably some shocking affecting the ballistic
jet, causing its effective velocity to be less than $3750$ km/s.

From the bottom left panel of Fig. 7 we see the evolution of jet lengths with
time for all five cases. The $v_{\mathrm{j}}=7500$ km/s jet length agrees very
well with our fiducial choice $v_{\mathrm{j}}=15000$ km/s, but only at late
times (once it has reached the self-similar phase). The higher velocity jet
with $v_{\mathrm{j}}=30000$ km/s is similar to our fiducial case at early
times, but slightly deviates from the theoretical prediction later on,
possibly when a backflow begins to operate. Our highest velocity jet is in the
self-similar regime at all times shown here, but with a lower normalisation,
probably due to a backflow.

###  5.4 Varying the mass resolution

As is visible in the last row of Fig. 6, the mass resolution does not affect
the lobe lengths and widths significantly, which is encouraging. If anything,
the jet lengths decrease slightly with resolution, showing that high
resolutions are not necessary in order for jets to be able to deposit their
energy at large distances$-$a conclusion relevant for cosmological
simulations. The jets are somewhat shorter at high resolutions since the
Kelvin-Helmholtz instability is better resolved, and this instability
increases the effective inertia of the lobes by mixing them with the ambient
medium. We find that the lobes take the expected ellipsoidal shape at lower
resolutions, whereas such a shape is only beginning to form at high
resolutions. At the same time, due to the lack of a coherent lobe, there is
only a single recollimation shock at high resolution, at the end of the
conical outflow, whereas we see multiple such shocks at low resolutions. We
posit that this is due to the lobe no longer being as uniform in pressure (due
to instabilities), which prevents the outflowing jet gas from being uniformly
collimated at regular intervals. We find that the jets and lobes have not
fully converged in structure by our highest resolution level. At even higher
resolutions, smaller-scale vortices caused by the Kelvin-Helmholtz instability
would possibly help to recover a more ellipsoidal lobe.

At all resolutions, we can see that the main ellipsoidal lobe does not extend
all the way to the bow shock, but rather out to roughly three quarters of the
distance to it. The lobe is instead connected to the bow shock by a thin strip
of shocked jet particles. These are most easily visible in the three lower
resolution snapshots, and especially at $m_{\mathrm{gas}}=1.81\times 10^{5}$
$\mathrm{M}_{\odot}$. These particles are among the first that were launched
into the jet, and they are the relic from its ballistic phase, with the self-
similar lobe being built up in its wake. At higher resolutions, these
particles are not easily visible because they were likely mixed with the
ambient medium. From the bottom right panel of Fig. 7, we can see that despite
the qualitative differences in jet morphology, the evolution of jet lengths is
in good agreement with the self-similar theory at all resolution levels. Our
simulated jets show a slightly steeper evolution with time than expected from
self-similar models, more consistent with $L_{\mathrm{j}}\propto t^{0.7}$ than
$L_{\mathrm{j}}\propto t^{0.6}$.

###  5.5 Varying the mass resolution - implications for cosmological
simulations

Figure 8: Comparison of jets simulated with varying numerical resolutions (see
top of each panel, these are typical resolutions employed in cosmological
simulations of galaxy clusters or large-volume cosmological simulations). The
number of particles in each jet varies from $\approx 450$ to $\approx 14000$.
Other jet parameters used are the same as the fiducal case, see Table 1. The
panels show the jets after $100$ Myr of evolution. The colours show gas
temperature, according to the colour bar. The panels are 10 kpc in depth.

In the bottom row of of Fig. 6 we showed how our jets vary visually across 5
resolution levels, separated by factors of 3.16. However, the lowest-
resolution case shown ($m_{\mathrm{gas}}=5.73\times 10^{5}$
$\mathrm{M}_{\odot}$) is itself better than the EAGLE cosmological simulation
(Schaye et al. 2015) by a factor of 3.16, and it has a relatively high number
of particles per jet, $\approx 14000$. As a result, the comparisons shown in
those figures, while interesting, are not necessarily relevant for
cosmological simulations.

In order to probe more poorly-resolved jets, such as those likely to occur in
large-scale cosmological simulations, we extend our analysis down to even
lower resolutions. In Fig. 8 we show the equivalent visualisations of jets as
in the bottom row of Fig. 6, but for resolutions down to ten times worse than
EAGLE, i.e. $m_{\mathrm{gas}}=1.81\times 10^{7}$ $\mathrm{M}_{\odot}$. The
highest-resolution simulation shown is the one with
$m_{\mathrm{gas}}=5.73\times 10^{5}$ $\mathrm{M}_{\odot}$. As we can see,
reducing numerical resolution results in a shortening of the jets and lobes
they inflate. This is likely due to spherical averaging from SPH effects, as
the total number of the jet particles approaches the number of particles
expected in a single SPH smoothing kernel ($\approx 50$). Despite the
shortening, the jet (lobe) length is still within $15$% of the self-similar
prediction. This is true even for the lowest-resolution case shown here, which
has only $\approx 450$ particles launched per jet. We have simulated cases
down to $100$ particles - these jets appear very spherical due to SPH
averaging, with their jet and lobe components blended together. They are also
significantly (factor of $\approx 2$) shorter than the self-similar
prediction.

Our worst-resolution simulation, with $m_{\mathrm{gas}}=1.81\times 10^{7}$
$\mathrm{M}_{\odot}$, matches the resolution of some simulations of galaxy
clusters (e.g. Dubois et al. 2010, Martizzi et al. 2014, Richardson et al.
2016, Hahn et al. 2017) and of cosmological simulations that are large enough
in volume to contain many galaxy clusters (e.g. Bocquet et al. 2016, Kaviraj
et al. 2017, Pillepich et al. 2018). This means that jet episodes with our
fiducial power ($P_{\mathrm{j}}=10^{46}$ erg/s) and duration
($T_{\mathrm{j}}=100$ Myr) are likely to be well-converged in such
simulations, at least in terms of basic properties.

State-of-the-art zoom-in simulations of galaxy clusters often have resolutions
significantly better than $m_{\mathrm{gas}}\approx 10^{7}$
$\mathrm{M}_{\odot}$ (e.g. $m_{\mathrm{gas}}\approx 10^{6}$
$\mathrm{M}_{\odot}$ in Barnes et al. 2017, Bahé et al. 2017, or
$m_{\mathrm{gas}}\approx 10^{5}$ $\mathrm{M}_{\odot}$ for a
$M_{\mathrm{h}}=10^{14}$ $\mathrm{M}_{\odot}$ halo in Tremmel et al. 2019), so
even weaker jet episodes are likely to be resolved at a basic level in these
simulations ($P_{\mathrm{j}}=10^{44}$ erg/s in a galaxy cluster with
$M_{\mathrm{h}}=10^{15}$ $\mathrm{M}_{\odot}$ or $P_{\mathrm{j}}=10^{43}$
erg/s in one with $M_{\mathrm{h}}=10^{14}$ $\mathrm{M}_{\odot}$). Modern
cosmological simulations with box sizes of $L\approx 100$ Mpc have a
resolution of $m_{\mathrm{gas}}\approx 10^{6}$ $\mathrm{M}_{\odot}$
(Vogelsberger et al. 2014, Schaye et al. 2015, Pillepich et al. 2018), and
thus our numerical scheme can correctly model jets in such simulations at
least down to systems with $M_{\mathrm{h}}=10^{14}$ $\mathrm{M}_{\odot}$, and
possibly $M_{\mathrm{h}}=10^{13}$ $\mathrm{M}_{\odot}$.

Small zoom-in cosmological simulations often aim to reproduce individual
Milky-Way type systems (Kim et al. 2014, Hopkins et al. 2014, Sawala et al.
2015, Fattahi et al. 2016,Grand et al. 2016), or host multiple such systems in
group environments within a $L=25$ Mpc box (Crain et al. 2015, Tremmel et al.
2017, Dubois et al. 2021). There is some evidence that jet feedback could be
important even in such lower-mass systems (Kaviraj et al. 2015, Singh et al.
2015, Olguín-Iglesias et al. 2020). These simulations do not host any galaxy
clusters, and can thus be simulated at much higher resolutions of
$m_{\mathrm{gas}}\approx 10^{3}-10^{5}$ $\mathrm{M}_{\odot}$, depending on the
type of simulation. While this is very low, the halo masses in these systems
are lower by similar factors (relative to a resolution of
$m_{\mathrm{gas}}\approx 10^{6}-10^{7}$ $\mathrm{M}_{\odot}$). The black holes
hosted by these systems are less massive by even larger factors, due to the
approximate $M_{\mathrm{BH}}\propto M_{\mathrm{h}}^{1.5}$ scaling (Croton et
al. 2006, Bandara et al. 2009). In combination with shorter jet episodes of
$1-10$ Myr (Guo & Mathews 2012, Garofalo et al. 2018, Davis et al. 2022),
instead of few 10s or up to 100 Myr (Konar et al. 2006, Machalski et al. 2007,
Mahatma et al. 2020), it is possible that jet episodes in these systems could
be less energetic by factors of up to $10^{6}$. It is unclear if simulations
with $m_{\mathrm{gas}}\approx 10^{3}-10^{5}$ $\mathrm{M}_{\odot}$ can
correctly include such jet episodes, in the context of our numerical scheme.

As an example, we take a jet episode with $P_{\mathrm{j}}=10^{42}$ erg/s,
lasting $5$ Myr. This jet power and duration correspond to the medians for
jets in radio AGN (Mezcua et al. 2019, Davis et al. 2022), and the total
energy matches the episode that likely recently occurred in the Milky Way (Guo
& Mathews 2012, Predehl et al. 2020). Such an episode is $2\times 10^{5}$
times less energetic than the standard one we simulate. Assuming the same
launching velocity of $v_{\mathrm{j}}=15000$ km/s, the mass resolution
required to resolve each of these jets with $450$ particles (the resolution we
have shown to be sufficient) would be $m_{\mathrm{gas}}\approx 10^{2}$
$\mathrm{M}_{\odot}$. This is about an order of magnitude smaller than
achievable with the highest-resolution cosmological simulations performed thus
far (Hopkins et al. 2018). A convenient workaround, but still a physically
relevant one, is to launch jets in lower-mass systems with lower velocities.
As an example, $v_{\mathrm{j}}=1500$ km/s in a Milky-Way type system would
produce a jet that is similarly underdense and hot compared to its ambient
medium as in our simulations (factor of $\approx 100$), due to a lower sound
speed in the gas halo of the Milky Way ($c_{\mathrm{s}}\approx 150$ km/s with
$T_{\mathrm{vir}}=10^{6}$ K). This would imply that our example jet episode
with $P_{\mathrm{j}}=10^{42}$ erg/s and $T_{\mathrm{j}}=5$ Myr could
realistically be simulated with $m_{\mathrm{gas}}\approx 10^{4}$
$\mathrm{M}_{\odot}$ or lower. This is in reach for modern cosmological
simulations.

Overall, for our standard case, we find jets of very similar lengths and
shapes (and therefore energetics, as the two are closely connected in self-
similar models of jet lobes) across a very large range of numerical
resolution: from $m_{\mathrm{gas}}=1.81\times 10^{7}$ $\mathrm{M}_{\odot}$
down to $m_{\mathrm{gas}}=5.73\times 10^{3}$ $\mathrm{M}_{\odot}$, a factor of
3160 difference. The implications of these findings, as discussed above, is
that jets are likely to be correctly simulated at a basic level in SPH codes
in all systems where they may be relevant. A caveat to this is that the
highest resolutions achievable may need to be employed. Otherwise, low-power
jet episodes will be poorly resolved. However, this is not necessarily a large
issue, since it is likely that stronger episodes are more relevant in terms of
feedback on galaxy formation.

An additional caveat is that one may need to carefully vary the jet launching
velocity from system to system to ensure that the jets are sufficiently
resolved, but also that sufficiently strong shocks occur in these systems (as
well as sufficiently large density and temperature contrasts). For this
reason, constant-velocity schemes with jet launching velocities of $\approx
10^{4}$ km/s (e.g. Dubois et al. 2012) will likely lead to very poorly
resolved jets in any system with a dark matter halo mass below
$M_{\mathrm{h}}\approx 10^{14}$ $\mathrm{M}_{\odot}$, at resolutions
comparable to EAGLE ($m_{\mathrm{gas}}\approx 10^{6}$ $\mathrm{M}_{\odot}$).
We instead suggest that scalings such as
$v_{\mathrm{j}}=AM_{\mathrm{h}}^{\gamma}$ or
$v_{\mathrm{j}}=AM_{\mathrm{BH}}^{\gamma}$ may be more appropriate. The
normalization and slope can be chosen in such a way that the ratio between
typical temperature of shocked jet gas (i.e. the lobe temperature) and that of
the ambient medium is roughly constant regardless of the mass of the system
(and of order e.g. 100, corresponding to jets that are supersonic by a factor
of 10).

###  5.6 Power-law ambient medium density profiles

Here we will discuss some results of simulations where jets were launched into
gaseous atmospheres with power-law density profiles ($\rho\propto
r^{-\alpha}$), rather than into a constant-density ambient medium. The set-up
for these simulations is described towards the end of Section 3.2. We test
cases with $\alpha=0.5$, $\alpha=1$ and $\alpha=1.5$, all of which should, in
principle, feature the self-similar phase of jet evolution (if various
parameters are chosen correctly). Given the $\beta$ profile we have chosen
(Eqn. 9), outside the small core, the density profile is
$\rho(r)=\rho_{0,\beta}r^{-\alpha}$. The normalization $\rho_{0,\beta}$
depends on $\alpha$ in our set-up, since we have chosen to keep the mass of
the gaseous halo constant within the virial radius of the external NFW
potential. This means that an increase in $\alpha$ is also followed by an
increase in $\rho_{0,\beta}$. An alternative would have been to keep the
normalization constant, but to instead track the evolution of the jets over
larger distances. We chose against this, and instead chose to keep the jets
confined within the extent of gaseous haloes of the same size. We did this in
order to be able to compare the jets at a similar size scale.

In Fig. 9 we show a visual comparison of all three power-law simulations we
have done, at different times up to $t=40$ Myr. All three simulations feature
very similar jets and jet-inflated lobes. However, there are differences
between these simulations and the constant-density ones featured in the rest
of the paper. Among these is that the jet-inflated lobes appear to have a
larger aspect ratio (i.e. they are thinner), despite having the same half-
opening angle $\theta_{\mathrm{j}}=10\degree$. In addition, the unshocked,
colder jet gas appears to extend all the way to the jet head in all three
cases. This is a feature also shared by ballistic jets (see top row of Fig.
1). However, these jets also clearly feature multiple recollimation shocks,
which only appear in the self-similar regime. Inspecting the unshocked jet gas
more closely, one can see that the recollimation shocks appear at distances
closer to the origin, while at large distances (close to the jet head), they
blend in into a single stream (spine) of unshocked gas, appearing very similar
to the unshocked jet gas in ballistic jets (see top panels of Fig. 1).

Figure 9: Comparison of the evolution of jets launched into power-law ambient
density profiles (see top-right-hand corner of the first panel in each row).
Other jet-related parameters used are listed in the last row of Table 1. The
colours show gas temperature, according to the colour bar. The panels are 25
kpc in depth.

In Fig. 10 we show the time evolution of the lobe lengths and radii for the
three simulations compared to predictions from Kaiser & Best (2007). In all
cases, the lobes start off as being shorter and wider than predicted by the
self-similar theory. As explained in Section 4.5, this is potentially a result
of the rather long transition from the initial ballistic phase to the self-
similar phase. It could also be a result of the initial phase of the jet
evolution being relatively poorly resolved (simply due to a small number of
particles having been launched into the jets), which tends to make the poorly-
resolved lobes more spherical due to SPH averaging effects (see Fig. 8).

At late times, in all three cases the lobes transition to behaviour that is in
better agreement with the theoretical predictions. However, the lobes appear
to be somewhat too long at late times, while also too thin (both disagreements
are at a level of $20-30\%$). These findings confirm the visual appearance of
the jets shown in Fig. 9. We posit here that this may be a result of the self-
similar theoretical predictions relying on assumptions that may not hold in
these simulations. One of these is that the lobe is cylindrical, whereas we
find a more ellipsoidal shape. The second assumption is that the lobes have a
constant pressure throughout. However, the lobes are more likely to be in
pressure equilibrium with the ambient medium locally (i.e. at a given radius).
The pressure profile of the ambient medium varies significantly with radius,
which means that the lobe has a higher pressure at smaller radii than near the
jet head. This could help the jet propagate or the lobes to expand more easily
in the radial direction rather than laterally. It could also explain why the
recollimation shocks (which occur by means of the lobe pressure acting on the
unshocked jet gas) are stronger at smaller radii.

Figure 10: Comparison of jet-inflated lobe lengths and radii in simulations
with power-law ambient density profiles (see top-left corner of each panel).
Other jet-related parameters used are listed in the last row of Table 1. The
solid lines show measured lengths and radii from the simulations, while the
dashed lines give equivalent predictions for self-similar jet-inflated lobe
expansion from Kaiser & Best (2007).

##  6 Summary and conclusions

Using the SWIFT code, and its SPHENIX smoothed particle hydrodynamics scheme,
we have simulated the interaction of hydrodynamical jets from active galactic
nuclei with the ambient medium up to very high resolutions (more than a
million particles per jet, unprecedented for an SPH simulation of jets). These
simulations intentionally did not include additional physics, such as gravity,
radiative cooling, magnetic fields or cosmic rays, in order for our results to
be comparable with simple analytical models of jet evolution.

We launch jets with a constant power and launching velocity, and with finite
opening angles, by kicking particles from initially placed reservoirs. These
jets propagate through ambient media with constant gas densities as well as
power-law density profiles. We find that jets initially evolve ballistically,
during which time they drill through the medium with ease, while also
launching a bow shock. Once the mass of the swept up ambient medium becomes
comparable to that injected into the jet, the jets transition to a self-
similar regime. This transition is fairly prolonged, and evidence of the
ballistic phase can be seen in the jet morphology even at late times.

The self-similar evolutionary phase is characterised by significant gas
shocking, which results in the formation of an ellipsoidal lobe of low-density
and high-temperature gas. This shocked gas recollimates and shocks recently
launched jet particles. We find that the jet and lobe lengths and lobe radii
are in agreement with self-similar predictions where this is expected (for
finite half-opening angles and large enough launching velocities). In the
self-similar phase, the aspect ratio of the lobes is constant, as is the
fraction of initially injected energy contained in the lobes (including the
unshocked jets) versus that in the ambient medium. This is also true for the
kinetic and thermal components of both the lobes and ambient medium. In our
standard constant-density simulation ($m_{\mathrm{gas}}=1.81\times 10^{5}$
$\mathrm{M}_{\odot}$, $P_{\mathrm{j}}=10^{46}$ $\mathrm{erg}\mathrm{s}^{-1}$,
$v_{\mathrm{j}}=15000$ km/s and $\theta_{\mathrm{j}}=10\degree$), these
fractions are: 1) lobe kinetic - $20\%$, 2) lobe thermal - $14\%$, 3) ambient
kinetic - $47\%$ and 4) ambient thermal - $19\%$. This shows that energy is
effectively transferred from the jet to the ambient gas, although it is not
immediately and fully thermalised while the jet is active. We find that these
fractions depend on the particular simulation, but none of the components is
ever negligible.

We have performed a series of simulations in the constant-density case, with
varying parameters such as the jet power, half-opening angle, launching
velocity and mass resolution. Increasing the jet power results in longer jets,
but also better resolution in the jets due to more particles being launched
(at a fixed launching velocity). Increasing the half-opening angle results in
shorter jets. We find that jet (lobe) lengths match expectations from
analytical models of self-similar evolution for all values of jet power and
half-opening angle that we have simulated.

We find that varying the jet launching velocity changes both the physical
properties of jets (i.e. temperature and mass of the lobes), as well as the
degree to which the jet and lobe are resolved. By varying numerical resolution
through the mass of gas particles, we find that our high-resolution
($m_{\mathrm{gas}}\approx 10^{4}$ $\mathrm{M}_{\odot}$) jet lobes deviate
somewhat from the picture expected theoretically. In this case, Kelvin-
Helmholtz instabilities prevent the formation of a classical ellipsoidal lobe.
For the same reason, the jet spine takes the shape of a conical outflow, and
not a series of multiple recollimation shocks, as expected and as found in the
lower-resolution cases. We speculate that at even higher resolutions, smaller-
scale Kelvin-Helmholtz instabilities would disrupt the large vortices and
again give rise to an ellipsoidal lobe.

We find that high-resolution simulations are not necessary in order for the
jets to deposit their energy where they should. On the contrary, we find that
very poor resolution ($\approx 500$ particles launched per jet) is sufficient
for an accuracy of $15$% (for the lengths and radii of the jet-inflated
lobes). We find that the level at which jets are resolved depends not only on
numerical resolution directly (through the gas particle mass), but also
indirectly on physical parameters such as jet power and launching velocity.
This means that different jet episodes will be resolved at different levels
depending on the details of the launching scheme and on the energetics of a
particular jet episode. We find that resolving a jet with $\approx 500$
particles is sufficient to reproduce basic properties such as jet lengths and
energetics. The implication of this finding is that jet launching velocities
of $\approx 10^{4}$ km/s are appropriate for typical modern cosmological
simulations of galaxy clusters. However, we suggest that the jet launching
velocity be varied from system to system in order to achieve a balance between
resolution and sufficiently strong shock heating of the ambient medium (e.g.
through a scaling with the halo mass $v_{\mathrm{j}}\propto
M_{\mathrm{h}}^{\gamma}$, or with the black hole mass $v_{\mathrm{j}}\propto
M_{\mathrm{BH}}^{\gamma}$).

We find that the self-similar regime is also reached in simulations of jets
launched into power-law density profiles of the ambient medium ($\rho\propto
r^{-\alpha}$, with $\alpha<2$). However, the lobes inflated by these jets
appear to be up to $20-30\%$ too long and too thin compared to self-similar
predictions. This may be a result of some of the theoretical assumptions
breaking down in the power-law case (e.g. pressure equilibrium throughout the
lobes), rather than our simulations being inaccurate.

In future, we plan to extend our analysis to more realistic systems (e.g.
galaxy cluster in hydrostatic equilibrium) with additional physics included,
and to Gyr time-scales (after the jets have been turned off and formed
bubbles). We will also study jets as a self-consistent feedback mechanism,
where the jet power is calculated based on the spin and environment of a black
hole.

## Acknowledgements

We thank Josh Borrow for pointing out several errors in the initial
manuscript. We also thank the reviewer for a comprehensive report that helped
to substantially improve the quality of the paper. The research in this paper
made use of the SWIFT open-source simulation code (http://www.swiftsim.com,
Schaller 2018) version 0.9.0. The swiftsimio Python library was used to
analyze and visualize the data from the simulations (Borrow & Borrisov 2020,
Borrow & Kelly 2021). This work was supported by the Science Technology
Facilities Council through a CDT studentship (ST/P006744/1), and the STFC
consolidated grant ST/T000244/1. This work used the DiRAC@Durham facility
managed by the Institute for Computational Cosmology on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and
ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC
is part of the National e-Infrastructure.

## Data availability

The data underlying this article are available upon reasonable request to the
corresponding author.

## References

  * Agertz et al. (2007) Agertz O., et al., 2007, MNRAS, 380, 963
  * Andernach et al. (2021) Andernach H., Jiménez-Andrade E. F., Willis A. G., 2021, Galaxies, 9, 99
  * Bahé et al. (2017) Bahé Y. M., et al., 2017, MNRAS, 470, 4186
  * Baldi et al. (2009) Baldi A., Forman W., Jones C., Kraft R., Nulsen P., Churazov E., David L., Giacintucci S., 2009, ApJ, 707, 1034
  * Bamford & Komissarov (2018) Bamford J. T., Komissarov S. S., 2018, arXiv e-prints, p. arXiv:1811.06764
  * Bandara et al. (2009) Bandara K., Crampton D., Simard L., 2009, ApJ, 704, 1135
  * Barai et al. (2016) Barai P., Murante G., Borgani S., Gaspari M., Granato G. L., Monaco P., Ragone-Figueroa C., 2016, MNRAS, 461, 1548
  * Barnes et al. (2017) Barnes D. J., et al., 2017, MNRAS, 471, 1088
  * Begelman & Cioffi (1989) Begelman M. C., Cioffi D. F., 1989, ApJ, 345, L21
  * Bicknell (1995) Bicknell G. V., 1995, ApJS, 101, 29
  * Biermann & Strittmatter (1987) Biermann P. L., Strittmatter P. A., 1987, ApJ, 322, 643
  * Bîrzan et al. (2004) Bîrzan L., Rafferty D. A., McNamara B. R., Wise M. W., Nulsen P. E. J., 2004, ApJ, 607, 800
  * Blandford & Königl (1979) Blandford R. D., Königl A., 1979, ApJ, 232, 34
  * Blondin et al. (1990) Blondin J. M., Fryxell B. A., Konigl A., 1990, ApJ, 360, 370
  * Bocquet et al. (2016) Bocquet S., Saro A., Dolag K., Mohr J. J., 2016, MNRAS, 456, 2361
  * Bodo & Tavecchio (2018) Bodo G., Tavecchio F., 2018, A&A, 609, A122
  * Borrow & Borrisov (2020) Borrow J., Borrisov A., 2020, The Journal of Open Source Software, 5, 2430
  * Borrow & Kelly (2021) Borrow J., Kelly A. J., 2021, arXiv e-prints, p. arXiv:2106.05281
  * Borrow et al. (2022) Borrow J., Schaller M., Bower R. G., Schaye J., 2022, MNRAS, 511, 2367
  * Bourne & Sijacki (2017) Bourne M. A., Sijacki D., 2017, MNRAS, 472, 4707
  * Bower et al. (2006) Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS, 370, 645
  * Bregman et al. (2006) Bregman J. N., Fabian A. C., Miller E. D., Irwin J. A., 2006, ApJ, 642, 746
  * Bromberg et al. (2011) Bromberg O., Nakar E., Piran T., Sari R., 2011, ApJ, 740, 100
  * Brüggen et al. (2007) Brüggen M., Heinz S., Roediger E., Ruszkowski M., Simionescu A., 2007, MNRAS, 380, L67
  * Carniani et al. (2015) Carniani S., et al., 2015, A&A, 580, A102
  * Chatterjee et al. (2019) Chatterjee K., Liska M., Tchekhovskoy A., Markoff S. B., 2019, MNRAS, 490, 2200
  * Choi et al. (2015) Choi E., Ostriker J. P., Naab T., Oser L., Moster B. P., 2015, MNRAS, 449, 4105
  * Cicone et al. (2014) Cicone C., et al., 2014, A&A, 562, A21
  * Cielo et al. (2014) Cielo S., Antonuccio-Delogu V., Macciò A. V., Romeo A. D., Silk J., 2014, MNRAS, 439, 2903
  * Crain et al. (2015) Crain R. A., et al., 2015, MNRAS, 450, 1937
  * Croston et al. (2008) Croston J. H., et al., 2008, A&A, 487, 431
  * Croton et al. (2006) Croton D. J., et al., 2006, MNRAS, 365, 11
  * Dabhade et al. (2020) Dabhade P., et al., 2020, A&A, 635, A5
  * Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
  * Davis et al. (2022) Davis F., et al., 2022, arXiv e-prints, p. arXiv:2201.09903
  * Di Matteo et al. (2005) Di Matteo T., Springel V., Hernquist L., 2005, Nature, 433, 604
  * Donohoe & Smith (2016) Donohoe J., Smith M. D., 2016, MNRAS, 458, 558
  * Dreher & Feigelson (1984) Dreher J. W., Feigelson E. D., 1984, Nature, 308, 43
  * Dubois et al. (2010) Dubois Y., Devriendt J., Slyz A., Teyssier R., 2010, MNRAS, 409, 985
  * Dubois et al. (2012) Dubois Y., Devriendt J., Slyz A., Teyssier R., 2012, MNRAS, 420, 2662
  * Dubois et al. (2014) Dubois Y., et al., 2014, MNRAS, 444, 1453
  * Dubois et al. (2021) Dubois Y., et al., 2021, A&A, 651, A109
  * Eckert et al. (2012) Eckert D., et al., 2012, A&A, 541, A57
  * Eckert et al. (2021) Eckert D., Gaspari M., Gastaldello F., Le Brun A. M. C., O’Sullivan E., 2021, Universe, 7, 142
  * Edge & Stewart (1991) Edge A. C., Stewart G. C., 1991, MNRAS, 252, 414
  * Ehlert et al. (2018) Ehlert K., Weinberger R., Pfrommer C., Pakmor R., Springel V., 2018, MNRAS, 481, 2878
  * English et al. (2016) English W., Hardcastle M. J., Krause M. G. H., 2016, MNRAS, 461, 2025
  * Fabian (1994) Fabian A. C., 1994, ARA&A, 32, 277
  * Fabian (2012) Fabian A. C., 2012, ARA&A, 50, 455
  * Fabian et al. (2003) Fabian A. C., Sanders J. S., Allen S. W., Crawford C. S., Iwasawa K., Johnstone R. M., Schmidt R. W., Taylor G. B., 2003, MNRAS, 344, L43
  * Falle (1991) Falle S. A. E. G., 1991, MNRAS, 250, 581
  * Fattahi et al. (2016) Fattahi A., et al., 2016, MNRAS, 457, 844
  * Feruglio et al. (2010) Feruglio C., Maiolino R., Piconcelli E., Menci N., Aussel H., Lamastra A., Fiore F., 2010, A&A, 518, L155
  * Forman et al. (2007) Forman W., et al., 2007, ApJ, 665, 1057
  * Garofalo et al. (2018) Garofalo D., Singh C. B., Zack A., 2018, Scientific Reports, 8, 15097
  * Gourgouliatos & Komissarov (2018) Gourgouliatos K. N., Komissarov S. S., 2018, Nature Astronomy, 2, 167
  * Grand et al. (2016) Grand R. J. J., Springel V., Gómez F. A., Marinacci F., Pakmor R., Campbell D. J. R., Jenkins A., 2016, MNRAS, 459, 199
  * Guo & Mathews (2011) Guo F., Mathews W. G., 2011, ApJ, 728, 121
  * Guo & Mathews (2012) Guo F., Mathews W. G., 2012, ApJ, 756, 181
  * Guo et al. (2018) Guo F., Duan X., Yuan Y.-F., 2018, MNRAS, 473, 1332
  * Hahn et al. (2017) Hahn O., Martizzi D., Wu H.-Y., Evrard A. E., Teyssier R., Wechsler R. H., 2017, MNRAS, 470, 166
  * Hardcastle & Krause (2013) Hardcastle M. J., Krause M. G. H., 2013, MNRAS, 430, 174
  * Hardcastle & Krause (2014) Hardcastle M. J., Krause M. G. H., 2014, MNRAS, 443, 1482
  * Hardcastle et al. (2003) Hardcastle M. J., Worrall D. M., Kraft R. P., Forman W. R., Jones C., Murray S. S., 2003, ApJ, 593, 169
  * Heckman et al. (1986) Heckman T. M., Smith E. P., Baum S. A., van Breugel W. J. M., Miley G. K., Illingworth G. D., Bothun G. D., Balick B., 1986, ApJ, 311, 526
  * Henriques et al. (2015) Henriques B. M. B., White S. D. M., Thomas P. A., Angulo R., Guo Q., Lemson G., Springel V., Overzier R., 2015, MNRAS, 451, 2663
  * Hervet et al. (2017) Hervet O., Meliani Z., Zech A., Boisson C., Cayatte V., Sauty C., Sol H., 2017, A&A, 606, A103
  * Hlavacek-Larrondo et al. (2012) Hlavacek-Larrondo J., Fabian A. C., Edge A. C., Ebeling H., Sanders J. S., Hogan M. T., Taylor G. B., 2012, MNRAS, 421, 1360
  * Hopkins (2015) Hopkins P. F., 2015, MNRAS, 450, 53
  * Hopkins et al. (2014) Hopkins P. F., Kereš D., Oñorbe J., Faucher-Giguère C.-A., Quataert E., Murray N., Bullock J. S., 2014, MNRAS, 445, 581
  * Hopkins et al. (2018) Hopkins P. F., et al., 2018, MNRAS, 480, 800
  * Huško et al. (2022) Huško F., Lacey C. G., Schaye J., Schaller M., Nobels F. S. J., 2022, MNRAS, 516, 3750
  * Ivison et al. (2012) Ivison R. J., et al., 2012, MNRAS, 425, 1320
  * Jetha et al. (2006) Jetha N. N., Hardcastle M. J., Sakelliou I., 2006, MNRAS, 368, 609
  * Kaiser & Alexander (1997) Kaiser C. R., Alexander P., 1997, MNRAS, 286, 215
  * Kaiser & Best (2007) Kaiser C. R., Best P. N., 2007, MNRAS, 381, 1548
  * Kaviraj et al. (2015) Kaviraj S., Shabala S. S., Deller A. T., Middelberg E., 2015, MNRAS, 454, 1595
  * Kaviraj et al. (2017) Kaviraj S., et al., 2017, MNRAS, 467, 4739
  * Kim et al. (2014) Kim J.-h., et al., 2014, ApJS, 210, 14
  * Kino & Kawakatu (2005) Kino M., Kawakatu N., 2005, MNRAS, 364, 659
  * Komatsu & Seljak (2001) Komatsu E., Seljak U., 2001, MNRAS, 327, 1353
  * Komissarov & Falle (1998) Komissarov S. S., Falle S. A. E. G., 1998, MNRAS, 297, 1087
  * Konar et al. (2006) Konar C., Saikia D. J., Jamrozy M., Machalski J., 2006, MNRAS, 372, 693
  * Krause et al. (2012) Krause M., Alexander P., Riley J., Hopton D., 2012, MNRAS, 427, 3196
  * Lagos et al. (2018) Lagos C. d. P., Tobar R. J., Robotham A. S. G., Obreschkow D., Mitchell P. D., Power C., Elahi P. J., 2018, MNRAS, 481, 3573
  * Ledlow et al. (2001) Ledlow M. J., Owen F. N., Yun M. S., Hill J. M., 2001, ApJ, 552, 120
  * Li et al. (2017) Li Y., Ruszkowski M., Bryan G. L., 2017, ApJ, 847, 106
  * Li et al. (2018) Li Y., Wiita P. J., Schuh T., Elghossain G., Hu S., 2018, ApJ, 869, 32
  * Lind et al. (1989) Lind K. R., Payne D. G., Meier D. L., Blandford R. D., 1989, ApJ, 344, 89
  * Machalski et al. (2007) Machalski J., Chyży K. T., Stawarz Ł., Kozieł D., 2007, A&A, 462, 43
  * Mahatma et al. (2020) Mahatma V. H., Hardcastle M. J., Croston J. H., Harwood J., Ineson J., Moldon J., 2020, MNRAS, 491, 5015
  * Mahato et al. (2021) Mahato M., Dabhade P., Saikia D. J., Combes F., Bagchi J., Ho L. C., Raychaudhury S., 2021, arXiv e-prints, p. arXiv:2111.11905
  * Markoff et al. (2001) Markoff S., Falcke H., Fender R., 2001, A&A, 372, L25
  * Martig et al. (2009) Martig M., Bournaud F., Teyssier R., Dekel A., 2009, ApJ, 707, 250
  * Martizzi et al. (2014) Martizzi D., Jimmy Teyssier R., Moore B., 2014, MNRAS, 443, 1500
  * Matsumoto & Masada (2019) Matsumoto J., Masada Y., 2019, MNRAS, 490, 4271
  * McDonald et al. (2015) McDonald M., et al., 2015, ApJ, 811, 111
  * McDonald et al. (2018) McDonald M., Gaspari M., McNamara B. R., Tremblay G. R., 2018, ApJ, 858, 45
  * McNamara & Nulsen (2007) McNamara B. R., Nulsen P. E. J., 2007, ARA&A, 45, 117
  * McNamara & Nulsen (2012) McNamara B. R., Nulsen P. E. J., 2012, New Journal of Physics, 14, 055023
  * McNamara et al. (2005) McNamara B. R., Nulsen P. E. J., Wise M. W., Rafferty D. A., Carilli C., Sarazin C. L., Blanton E. L., 2005, Nature, 433, 45
  * Meece et al. (2017) Meece G. R., Voit G. M., O’Shea B. W., 2017, ApJ, 841, 133
  * Mendygral et al. (2012) Mendygral P. J., Jones T. W., Dolag K., 2012, ApJ, 750, 166
  * Menon et al. (2015) Menon H., Wesolowski L., Zheng G., Jetley P., Kale L., Quinn T., Governato F., 2015, Computational Astrophysics and Cosmology, 2, 1
  * Merritt & Ekers (2002) Merritt D., Ekers R. D., 2002, Science, 297, 1310
  * Mezcua et al. (2019) Mezcua M., Suh H., Civano F., 2019, MNRAS, 488, 685
  * Mizuno et al. (2015) Mizuno Y., Gómez J. L., Nishikawa K.-I., Meli A., Hardee P. E., Rezzolla L., 2015, ApJ, 809, 38
  * Monaghan (1992) Monaghan J. J., 1992, ARA&A, 30, 543
  * Mukherjee et al. (2018) Mukherjee D., Bicknell G. V., Wagner A. Y., Sutherland R. S., Silk J., 2018, MNRAS, 479, 5544
  * Mukherjee et al. (2020) Mukherjee D., Bodo G., Mignone A., Rossi P., Vaidya B., 2020, MNRAS, 499, 681
  * Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
  * Nelson et al. (2013) Nelson D., Vogelsberger M., Genel S., Sijacki D., Kereš D., Springel V., Hernquist L., 2013, MNRAS, 429, 3353
  * Nesvadba et al. (2021) Nesvadba N. P. H., et al., 2021, A&A, 654, A8
  * Noble et al. (2011) Noble S. C., Krolik J. H., Schnittman J. D., Hawley J. F., 2011, ApJ, 743, 115
  * Novikov & Thorne (1973) Novikov I. D., Thorne K. S., 1973, in Black Holes (Les Astres Occlus). pp 343–450
  * Nulsen et al. (2009) Nulsen P., Jones C., Forman W., Churazov E., McNamara B., David L., Murray S., 2009, in Heinz S., Wilcots E., eds, American Institute of Physics Conference Series Vol. 1201, The Monster’s Fiery Breath: Feedback in Galaxies, Groups, and Clusters. pp 198–201 (arXiv:0909.1809), doi:10.1063/1.3293033
  * O’Dea (1998) O’Dea C. P., 1998, PASP, 110, 493
  * O’Dea et al. (2009) O’Dea C. P., Daly R. A., Kharb P., Freeman K. A., Baum S. A., 2009, A&A, 494, 471
  * Olguín-Iglesias et al. (2020) Olguín-Iglesias A., Kotilainen J., Chavushyan V., 2020, MNRAS, 492, 1450
  * Omma et al. (2004) Omma H., Binney J., Bryan G., Slyz A., 2004, MNRAS, 348, 1105
  * Pakull et al. (2010) Pakull M. W., Soria R., Motch C., 2010, Nature, 466, 209
  * Park et al. (2019) Park J., et al., 2019, ApJ, 887, 147
  * Perucho et al. (2006) Perucho M., Lobanov A. P., Martí J. M., Hardee P. E., 2006, A&A, 456, 493
  * Perucho et al. (2010) Perucho M., Martí J. M., Cela J. M., Hanasz M., de La Cruz R., Rubio F., 2010, A&A, 519, A41
  * Perucho et al. (2022) Perucho M., Martí J.-M., Quilis V., 2022, MNRAS, 510, 2084
  * Peterson et al. (2003) Peterson J. R., Kahn S. M., Paerels F. B. S., Kaastra J. S., Tamura T., Bleeker J. A. M., Ferrigno C., Jernigan J. G., 2003, ApJ, 590, 207
  * Pillepich et al. (2018) Pillepich A., et al., 2018, MNRAS, 475, 648
  * Predehl et al. (2020) Predehl P., et al., 2020, Nature, 588, 227
  * Pushkarev et al. (2009) Pushkarev A. B., Kovalev Y. Y., Lister M. L., Savolainen T., 2009, A&A, 507, L33
  * Pushkarev et al. (2017) Pushkarev A. B., Kovalev Y. Y., Lister M. L., Savolainen T., 2017, MNRAS, 468, 4992
  * Rafferty et al. (2006) Rafferty D. A., McNamara B. R., Nulsen P. E. J., Wise M. W., 2006, ApJ, 652, 216
  * Randall et al. (2011) Randall S. W., et al., 2011, ApJ, 726, 86
  * Rees (1978) Rees M. J., 1978, MNRAS, 184, 61P
  * Reynolds et al. (2001) Reynolds C. S., Heinz S., Begelman M. C., 2001, ApJ, 549, L179
  * Reynolds et al. (2006) Reynolds C. S., Garofalo D., Begelman M. C., 2006, ApJ, 651, 1023
  * Richardson et al. (2016) Richardson M. L. A., Scannapieco E., Devriendt J., Slyz A., Thacker R. J., Dubois Y., Wurster J., Silk J., 2016, ApJ, 825, 83
  * Rossi et al. (2008) Rossi P., Mignone A., Bodo G., Massaglia S., Ferrari A., 2008, A&A, 488, 795
  * Rosswog (2020) Rosswog S., 2020, MNRAS, 498, 4230
  * Russell et al. (2013) Russell H. R., McNamara B. R., Edge A. C., Hogan M. T., Main R. A., Vantyghem A. N., 2013, MNRAS, 432, 530
  * Sancisi et al. (1987) Sancisi R., Thonnard N., Ekers R. D., 1987, ApJ, 315, L39
  * Sanders et al. (1988) Sanders D. B., Soifer B. T., Elias J. H., Madore B. F., Matthews K., Neugebauer G., Scoville N. Z., 1988, ApJ, 325, 74
  * Sawala et al. (2015) Sawala T., et al., 2015, MNRAS, 448, 2941
  * Schaller (2018) Schaller M. e. a., 2018, SWIFT: SPH With Inter-dependent Fine-grained Tasking, Astrophysics Source Code Library (ascl:1805.020)
  * Schaller et al. (2016) Schaller M., Gonnet P., Chalk A. B. G., Draper P. W., 2016, arXiv e-prints, p. arXiv:1606.02738
  * Schaye et al. (2015) Schaye J., et al., 2015, MNRAS, 446, 521
  * Sedov (1959) Sedov L. I., 1959, Similarity and Dimensional Methods in Mechanics
  * Shabala et al. (2017) Shabala S. S., Deller A., Kaviraj S., Middelberg E., Turner R. J., Ting Y. S., Allison J. R., Davis T. A., 2017, MNRAS, 464, 4706
  * Sijacki et al. (2012) Sijacki D., Vogelsberger M., Kereš D., Springel V., Hernquist L., 2012, MNRAS, 424, 2999
  * Silk & Rees (1998) Silk J., Rees M. J., 1998, A&A, 331, L1
  * Singh et al. (2015) Singh V., Ishwara-Chandra C. H., Sievers J., Wadadekar Y., Hilton M., Beelen A., 2015, MNRAS, 454, 1556
  * Smith & Donohoe (2019) Smith M. D., Donohoe J., 2019, MNRAS, 490, 1363
  * Smith andJustin Donohoe (2021) Smith andJustin Donohoe M. D., 2021, arXiv e-prints, p. arXiv:2101.03517
  * Snios et al. (2018) Snios B., et al., 2018, ApJ, 855, 71
  * Springel et al. (2005) Springel V., Di Matteo T., Hernquist L., 2005, ApJ, 620, L79
  * Stone et al. (1997) Stone J. M., Xu J., Hardee P., 1997, ApJ, 483, 136
  * Sturm et al. (2011) Sturm E., et al., 2011, ApJ, 733, L16
  * Talbot et al. (2020) Talbot R. Y., Bourne M. A., Sijacki D., 2020, arXiv e-prints, p. arXiv:2011.10580
  * Tchekhovskoy & Bromberg (2016) Tchekhovskoy A., Bromberg O., 2016, MNRAS, 461, L46
  * Tremmel et al. (2017) Tremmel M., Karcher M., Governato F., Volonteri M., Quinn T. R., Pontzen A., Anderson L., Bellovary J., 2017, MNRAS, 470, 1121
  * Tremmel et al. (2019) Tremmel M., et al., 2019, MNRAS, 483, 3336
  * Urry & Padovani (1995) Urry C. M., Padovani P., 1995, PASP, 107, 803
  * Vogelsberger et al. (2014) Vogelsberger M., et al., 2014, Nature, 509, 177
  * Wadsley et al. (2017) Wadsley J. W., Keller B. W., Quinn T. R., 2017, MNRAS, 471, 2357
  * Walg et al. (2013) Walg S., Achterberg A., Markoff S., Keppens R., Meliani Z., 2013, MNRAS, 433, 1453
  * Wang & Yang (2022) Wang S.-C., Yang H. Y. K., 2022, arXiv e-prints, p. arXiv:2201.05298
  * Wardle & Aaron (1997) Wardle J. F. C., Aaron S. E., 1997, MNRAS, 286, 425
  * Webster et al. (2021) Webster B., Croston J. H., Harwood J. J., Baldi R. D., Hardcastle M. J., Mingo B., Röttgering H. J. A., 2021, MNRAS, 508, 5972
  * Weinberger et al. (2017) Weinberger R., Ehlert K., Pfrommer C., Pakmor R., Springel V., 2017, MNRAS, 470, 4530
  * Weinberger et al. (2018) Weinberger R., et al., 2018, MNRAS, 479, 4056
  * Werner et al. (2019) Werner N., McNamara B. R., Churazov E., Scannapieco E., 2019, Space Sci. Rev., 215, 5
  * Wise et al. (2007) Wise M. W., McNamara B. R., Nulsen P. E. J., Houck J. C., David L. P., 2007, ApJ, 659, 1153
  * Yang et al. (2019) Yang H. Y. K., Gaspari M., Marlow C., 2019, ApJ, 871, 6
  * Yang et al. (2020) Yang J., Gurvits L. I., Paragi Z., Frey S., Conway J. E., Liu X., Cui L., 2020, MNRAS, 495, L71
  * van Putten (1996) van Putten M. H. P. M., 1996, ApJ, 467, L57