###### Abstract

Analytical models describing the dynamics of lobed radio sources are essential
for interpretation of the tens of millions of radio sources that will be
observed by the Square Kilometre Array and pathfinder instruments. We propose
that historical models can be grouped into two classes in which the forward
expansion of the radio source is driven by either the jet momentum flux or
lobe internal pressure. The most recent generation of analytical models
combines these limiting cases for a more comprehensive description. We extend
the mathematical formalism of historical models to describe source expansion
in non-uniform environments, and directly compare different model classes with
each other, and with hydrodynamic numerical simulations. We quantify
differences in predicted observable characteristics for lobed radio sources
due to the different model assumptions for their dynamics. We make our code
for the historical models analysed in this review openly available to the
community.

###### keywords:

galaxies: active; galaxies: jets; radio continuum: galaxies

1 1 0 9 June 2023 17 July 2023 https://doi.org/ Dynamics of powerful radio
galaxies Dynamics of powerful radio galaxies Ross J. Turner 1,* and Stanislav
S. Shabala 1 Ross J. Turner and Stanislav S. Shabala Turner, R. J.; Shabala,
S. S. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

The first extragalactic radio sources were identified over seven decades ago
by Bolton in the late-1940s (Bolton et al., 1949). Shortly after, in 1953, the
first resolved image was captured of Cygnus A, now known as the archetypal
“classical double” (Jennison and Gupta, 1953). This breakthrough was followed
by the pioneering efforts of radio survey groups in Australia (Mills and Slee,
1957) and the United Kingdom (Edge et al., 1959), which conducted the first
large-scale radio surveys (for a comprehensive review, refer to Norris, 2017).
The first quasar, 3C273, was discovered a few years later in 1963 (Hazard et
al., 1963; Schmidt, 1963, see Hazard et al., 2018 for a historical review).
These pivotal developments laid the foundation for observational studies of
radio galaxies. Subsequently, building upon Lynden-Bell’s (1969) proposal that
black holes are responsible for the extreme luminosities observed in quasars,
the 1970s saw the development of the first models describing the dynamical
evolution of radio galaxies.

While varying in specific details the majority of models in the literature
share a similar overarching framework. These models in general consider two
initially conical jets composed of particles that have been accelerated to
relativistic velocities. The interaction between the jets and the intracluster
medium (ICM) surrounding their host galaxy determines the subsequent
evolution. Jets which retain sufficient forward ram pressure during their
initial propagation phase (typically on galaxy scales) will be collimated by
pressure from the ambient medium, or more likely, a build up of plasma shed by
the jet in the early stages of lobe formation (Alexander, 2006; Krause et al.,
2012). Regardless, each collimated jet leads to the formation of a Mach disk
(observable as a hotspot) and, as overpressured jet material flows back
towards the equatorial plane, the inflation of a plasma lobe observable
through synchrotron radiation; such objects are generally classified to have a
Fanaroff and Riley Type-II (FR-II) lobe morphology.

On the other hand, jets which suffer substantial entrainment (e.g. from
stellar winds, Perucho et al., 2014; Wykes et al., 2015; or the interstellar
medium, Bicknell, 1995) will slow down to transsonic speeds (with respect to
the internal lobe sound speed, of order 0.1c) and be disrupted. In this
scenario, the jet momentum thrust is not important to the evolution of the
lobe, and the role of the jet is to simply supply energy to the synchrotron-
emitting lobe. The subsequent expansion of the lobes is determined by solving
a set of fluid conservation equations; typically the lobes undergo an initial
momentum-dominated supersonic phase, followed by an adiabatic-expansion driven
coasting phase, and ultimately rising buoyantly in the later phases of
evolution.

A large number of analytical and numerical models describing the evolution of
AGN jets and lobes have been published since the first models were introduced
over five decades ago. In this review, we summarise the different classes of
lobed radio galaxy models, and provide a common framework to facilitate
comparison both between the model classes and to more detailed hydrodynamic
simulations. The dynamics of jetted Fanaroff and Riley Type-I (FR-I) sources
are not considered in this work; we refer the interested reader to the
classical work of Bicknell (1995).

The review is structured as follows. Section 2 presents early models by Rees
(1971) and Scheuer (1974, Model A), in which the forward thrust of
uncollimated jets is balanced by the ram pressure from the ambient medium. We
extend the formalism of Scheuer (1974) to non-uniform environments, enabling
direct comparison with more sophisticated modern analytical models for the
first time. While these early models capture the fundamental aspects of jet
termination and lobe formation, they neglect jet collimation by sideways ram
pressure from the lobe (or ambient medium). In Section 3, we describe the
analytical models proposed by Falle (1991) and Kaiser and Alexander (1997),
which link jet collimation to subsequent lobe expansion. The past decade has
seen the advent of environment-sensitive radio galaxy models beyond the self-
similar solutions of Kaiser and Alexander (1997) and related models (e.g.
Blundell and Rawlings, 2000; Manolakou and Kirk, 2002). These models capture
the evolution of lobe morphology in realistic environments (Turner and
Shabala, 2015; Hardcastle, 2018) as well as the transition between jet- and
lobe-driven expansion (Hardcastle, 2018; Turner et al., 2023). We describe
these models in Section 4. In Section 5, we compare the consistency of
predicted radio source dynamics between the main model classes, benchmark
their evolutionary tracks against hydrodynamic simulations, and discuss their
ability to generate synthetic AGN populations for parameter inversions. We
conclude and suggest improvements to implement in next generation of
analytical models in Section 6.

##  2 Early Jet–Lobe Models

Figure 1: Schematic of the dynamical model for the Scheuer (1974) Model A. We
show a thin shocked gas shell between the contact discontinuity and bow shock
as in Figure 1 of the original paper, however, the shocked gas is not
explicitly considered in their model.

Rees (1971) proposed a model in which conical jets emanating from the central
engine of active galactic nuclei (AGNs) are pressure-balanced by the ram
pressure of the ambient medium. In this model, the jets are a beam of low-
frequency electromagnetic waves (LFEMW), the quantum field equivalent of a
pair-plasma. The radiation pressure of this beam upon absorption by the
ambient medium is $p_{rad}=Q/(\Omega R^{2}c)$ for jet kinetic power $Q$ and
beam cross-sectional area $\Omega R^{2}$ at radius $R$ from the active
nucleus. The pressure is increased if the interaction between the beam and
ambient medium results in pair production, leading to a reaction pressure up
to double that of the radiation pressure; the exact factor depends on the
angle of reflected particles. The pressure contribution from the jet is
therefore expressed as:

$p_{jet}=\frac{\kappa_{1}Q}{\Omega R^{2}c},$ (1)

where $1\leqslant\kappa_{1}<2$ is a dimensionless constant describing both the
fraction of the beam power that interacts with the ambient medium as radiation
or particles, and the angle of reflection of those particles.

The forward jet thrust is balanced by ram pressure from the ambient medium
$p_{ram}=\rho v^{2}$, where $\rho$ is the gas density of the assumed constant
density ambient medium and $v=dR/dt$ is the advance speed of the jet-head (see
Figure 1). Scheuer (1974) evaluated the resulting first-order differential
equation in $R$ assuming a constant jet half-opening angle $\theta_{j}$ (and
thus solid angle $\Omega$) in their Model A. The jet length is related to
source age $t$, and jet and environment parameters as

$R(t)=\left(\frac{\kappa_{1}Q}{\Omega\rho
c}\right)^{1/4}\left(2t\right)^{1/2}.$ (2)

The above approach assumes a constant density environment. However, the
ambient medium on scales exceeding several kiloparsecs – typical of extended
radio sources – is well represented by a symmetric power-law density profile
of the form $\rho=kr^{-\beta}$, where the density parameter
$k\equiv\rho_{0}r_{0}^{\beta}$ is a constant (e.g. Falle, 1991; Turner and
Shabala, 2015). The ram pressure applied by the ambient medium onto the
expanding jet consequently weakens with distance from the central nucleus. In
this review, we extend the Scheuer (1974) Model A for the more general case of
a power-law density profile, yielding

$R(t)=\left(\frac{\kappa_{1}Q}{\Omega
kc}\right)^{1/(4-\beta)}\left(\frac{(4-\beta)t}{2}\right)^{2/(4-\beta)},$ (3)

which converges to Scheuer’s original constant density form when $\beta=0$,
noting that $\rho=k$ in this limiting case. We use this more complete version
of the model in the remainder of this work.

The synchrotron-emitting lobes inflated by the jets are typically assumed to
have ellipsoidal morphology, with the ratio of major (aligned with the jet) to
minor axes defined by the axis ratio $A=R/R_{\perp}$; we note that this
differs (by a factor of 2) from the axial ratio $R_{T}=A/2$ of Kaiser and
Alexander (1997). Scheuer (1974) derive the volume of the ellipsoidal radio
lobe associated with their LFEMW jets by considering the work done in
inflating the cavity. The total energy in the cavity, $U$, increases over the
time interval $\delta t$ due to the input kinetic power $Q$ as

$\delta U=Q\delta t-p\delta V,$ (4)

where $\delta V$ is the differential increase in volume, and the lobe pressure
is given by (see e.g. Kaiser and Alexander, 1997, their Equation 15)

$p=\frac{U(\Gamma_{c}-1)(q+1)}{V},$ (5)

where $\Gamma_{c}$ is the polytropic index (or adiabatic index for an
adiabatic equation of state; EoS) of the lobe plasma, and $q\ll 1$ is the
ratio of energy in the magnetic field to that in the particles. This equation
assumes the energy density (and thus pressure) is approximately uniform
throughout the lobe, a reasonable assumption given the high ($\sim$$0.1c$)
sound speeds in the lobes.

Equation 4 is a first-order differential equation describing the evolution of
the total energy of the cavity. In Appendix A.1, we solve this differential
equation assuming the cavity volume expands with increasing jet length as
$V(R)=\kappa_{2}R^{\alpha}$; here $\alpha,\kappa_{2}>0$ are constants. This
yields an expression for the lobe pressure in terms of the source age,

$p(t)=\frac{Q(\Gamma_{c}-1)(q+1)}{\kappa_{2}[\alpha(\Gamma_{c}-1)(q+1)+(4-\beta)/2]}\left(\frac{\Omega
kc}{\kappa_{1}Q}\right)^{\alpha/(4-\beta)}\left(\frac{(4-\beta)t}{2}\right)^{(4-\beta-2\alpha)/(4-\beta)},$
(6)

where the constants $\alpha$ and $\kappa_{2}$ are evaluated below by
considering the lobe volume evolution.

A major limitation of the Scheuer (1974) model concerns the sideways expansion
of the lobe, which is assumed to occur at the same velocity at every point on
the lobe surface at any given time $t$. This expansion rate is derived by
equating the lobe pressure to the ram pressure presented by the ambient medium
as the lobe widens; i.e. $\rho v_{\perp}^{2}=p(t)$ where the ambient gas
density is reasonably approximated as $\rho\sim kR^{-\beta}$. This sidewards
expansion can only commence at locations already reached by the jet material.
The half-width of the lobe at some location $r$ along the jet axis is thus
given by

$R_{\perp}(r)=\int_{t(r)}^{t(R)}v_{\perp}(t^{*})dt^{*},$ (7)

where $t(r)$ is the time when the jet-head reached the location $r$ along the
jet axis, and $t(R)$ is the source age when the jet has its present length
$R$. This integral is evaluated in Appendix A.2.

The lobe volume at the source age $t\equiv t(R)$ is found by integrating over
all locations $r$ along the jet axis,

$\begin{split}V(R)\equiv\kappa_{2}R^{\alpha}&=\int_{0}^{R}\pi
R_{\perp}(r)^{2}dr\\\ &\propto R^{(14-5\beta-2\alpha)/2},\end{split}$ (8)

Dimensional analysis shows that the only possible solution is for
$\alpha=(14-5\beta)/4$. The lobe expansion is therefore not self-similar
(which would require $V\propto R^{3}$ and hence $\alpha=3$) unless
$\beta=0.4$, i.e. in a gently declining density profile representative of
cluster cores. For a uniform medium as considered by Scheuer (1974), the
exponent converges to $\alpha=\tfrac{7}{2}$, while in a steep environment
($\beta=2$) representative of the outer regions of groups or clusters, the
lobe volume scales linearly with jet length leading to a rapid increase in the
lobe axis ratio.

The constant of proportionality, $\kappa_{2}$, is similarly found by comparing
terms not involving $R$, yielding

$\kappa_{2}=\frac{16\pi^{1/2}(\Omega
c)^{3/4}k^{1/4}}{[(14-5\beta)(18-5\beta)]^{1/2}\kappa_{1}^{3/4}Q^{1/4}}\left[\frac{(\Gamma_{c}-1)(q+1)}{(14-5\beta)(\Gamma_{c}-1)(q+1)+2(4-\beta)}\right]^{1/2},$
(9)

which converges to the expression found by Scheuer (1974, their Equation 10)
in the limit of a uniform ambient medium and assuming
$\Gamma_{c}=\tfrac{4}{3}$.

This simple model neglects the sidewards ram pressure of the ambient medium
(or lobe at later times) acting on the jet, which will lead to reconfinement
shocks and ultimately the collimation of the jet. Scheuer (1974) proposed a
second model (their Model B) in which the jet is smoothly compressed into a
collimated beam by the ambient medium; however, this assumption leads to
unphysically narrow jets and consequently significantly faster jet-head
advance speeds, which scale with jet cross-section $y$ as $R(t)\propto
y^{-4/9}t^{7/9}$. Scheuer (1982) subsequently proposed that, in some sources,
the jet may precess on a timescale which is short compared to the evolutionary
timescale of the lobe. The time-averaged momentum flux of the jet is
effectively spread over a larger cross-sectional area (equivalent to a larger
jet opening angle) resulting in a slower growth rate along the jet axis (e.g.
Begelman and Cioffi, 1989). We return to this point in Section 5 when we
compare the predictions of different models.

##  3 Lobe Expansion Models

The self-similar expansion model for the growth of quasar winds by Dyson et
al. (1980) spurred a new generation of analytical models based on the
adiabatic expansion of the lobe bubble along a power-law ambient gas density
profile. In particular, Falle (1991) related the geometry and internal
pressure of the expanding lobe to the dynamics of the jet (Section 3.1),
enabling the Dyson et al. (1980) model to be modified to consider the
evolution of radio sources (Section 3.2). The Falle (1991) model forms the
basis for several models in the literature including those by Kaiser and
Alexander (1997), Blundell and Rawlings (2000), and Manolakou and Kirk (2002).

###  3.1 Jet collimation

Falle (1991) revisited the dynamical modelling of jet collimation in their
1991 work, considering an initially conical jet that reflects a strong shock
off the surrounding medium upon reaching lateral pressure equilibrium. This
reconfinement shock bounces between each side of the jet cavity preventing any
further decay of the lateral thrust against the surrounding medium; this leads
to a constant width, collimated jet with repeated cross-shaped structures of
enhanced pressure and synchrotron emissivity. Falle (1991) assumed that lobe
formation occurs prior to jet collimation, and thus that it is the lobe
pressure which opposes the sidewards component of the jet thrust, not the
ambient medium. Alexander (2006) showed that jets may in fact be collimated by
the ambient medium prior to the formation of lobes if
$\sqrt{\Gamma_{x}}\sin\theta_{j}M_{x}<1$, where $\Gamma_{x}$ is the polytropic
index of the ambient medium, $\theta_{j}$ is the half-opening angle of the
conical jet, and $M_{x}$ is the Mach number of the jet-head advance with
respect to the sound speed of the ambient medium. Jet-head advance speeds are
initially relativistic (e.g. VLBI observations of Britzen, S. et al., 2008)
and thus the external Mach number of the jet is expected to be of order
several hundred; only for very small opening angles of $<1$ degree would the
jet be expected to be collimated by the ambient medium rather than the lobe.

The location of the initial reconfinement shock, $z$, is found by applying the
Rankine-Hugoniot jump conditions for a plane-parallel shock to the lateral
component of the flow that travels along the jet edge. The pressure of the
lobe plasma is related by these conditions to the lateral component of the jet
thrust as

$p(t)=\frac{2}{\Gamma_{j}+1}\rho_{j}(z_{1})v_{j}^{2}\sin^{2}\theta_{j},$ (10)

where the jet plasma has bulk velocity $v_{j}$, polytropic index $\Gamma_{j}$,
and density $\rho_{j}(z_{1})$ at the critical radius $z_{1}$ at which the jet
begins to collimate (i.e. location where the ram pressure first matches the
lobe pressure). The density of the jet plasma will remain constant after this
point until it reaches the jet-head. We can thus derive an expression for the
pressure acting on the contact discontinuity between the jet-head and
surrounding shocked gas using the shock jump conditions. That is,

$\begin{split}p_{h}(t)&=\frac{2}{\Gamma_{j}+1}\rho_{j}(z_{1})v_{j}^{2}\\\
&=\frac{p(t)}{\sin^{2}\theta_{j}},\end{split}$ (11)

where the second equality is obtained upon substitution of Equation 10. This
expression has the same form as found by Kaiser and Alexander (1997, their
Equation 36), and yields comparable pressure ratios to the numerically
informed value obtained by Komissarov and Falle (1998).

The lobe and jet-head region are surrounded by a shell of swept-up ambient
medium that has been overrun by the bow shock generated by the expanding jet.
This shocked gas is in approximate pressure equilibrium with the proximate
lobe/jet-head plasma, but has significantly higher density and thus lower
temperature (e.g. Hardcastle and Krause, 2013; Yates-Jones et al., 2021); i.e.
$p_{s}(t)\sim p_{h}(t)$, where $p_{s}(t)$ is the pressure just inside the bow
shock along the jet axis. Together with conservation equations, these
relationships between the conditions in the lobe cavity, the bow shock and the
ambient medium are sufficient for describing the evolution of the expanding
radio source.

###  3.2 Lobe adiabatic expansion

Figure 2: Schematic of the dynamical model proposed by Falle (1991), and
subsequently refined by others including Kaiser and Alexander (1997) and
Alexander (2006). The bow shock is assumed to expand in a self-similar manner
(i.e. constant scaling to the lobe) but the energy associated with the shocked
gas is not explicitly considered.

Falle (1991) present their model in terms of the volume and pressure of the
lobe, however, the surrounding shocked gas shell also receives a non-
negligible fraction of the input energy from the central nucleus. We therefore
express their equations in a more complete form considering both the lobe and
shocked gas shell consistent with later work by Hardcastle (2018), Turner and
Shabala (2020) and Turner et al. (2023). These models assume that it is
primarily the thermal pressure of the lobe plasma that drives the source
expansion; this pressure is uniform throughout the lobes due to high internal
sound speed. The first law of thermodynamics relates the jet kinetic energy
input to the thermal pressure $p$ in the lobe and shell, and shocked gas
volume $V_{s}$ (see Equation 6 of Alexander, 2006):

$V_{s}\frac{dp}{dt}+\Gamma_{c}p\frac{dV_{s}}{dt}=(\Gamma_{c}-1)Q,$ (12)

where $\Gamma_{c}$ is the adiabatic index of the shocked gas and lobe plasma
($\tfrac{5}{3}$ for a non-relativistic fluid), and $Q$ is the power injected
into the shocked shell by the jet. This equation can be rewritten in terms of
the pressure at the interface of the shocked shell and ambient medium (along
the jet axis) using Equation 11. That is,

$V_{s}\frac{dp_{s}}{dt}+\Gamma_{c}p_{s}\frac{dV_{s}}{dt}=\frac{(\Gamma_{c}-1)Q}{\sin^{2}\theta_{j}}.$
(13)

Falle (1991) showed that the shocked shell expands in a self-similar manner,
leading to a constant scaling between the volume and cube of the jet length.
That is,

$V_{s}(R_{s})=\kappa_{3}R_{s}^{3}.$ (14)

where $R_{s}$ is the radius of the shocked gas shell along the jet axis, and
$\kappa_{3}$ is a constant of proportionality. Kaiser and Alexander (1997),
and some subsequent authors (e.g. Alexander, 2000), modelled the lobe as
cylindrical (see Figure 2), with the major axis of the shocked gas shell being
a factor of $A_{s}=1/\sin\theta_{j}$ longer than the minor axis. The volume of
the shocked shell is then

$V_{s}(R_{s})=\pi\sin^{2}\theta_{j}R_{s}^{3}\ \ \ \text{(cylinder)}.$ (15)

In this approach, expansion in the jet direction is driven by the ram pressure
acting on the jet-head region, $p_{h}(t)$, whilst sideways expansion is driven
by shocked shell thermal pressure, $p(t)$. The more realistic assumption of an
ellipsoid shocked shell adds a factor of $\tfrac{2}{3}$ to the above equation,
however, the sidewards ram pressure then becomes a strong function of distance
along the jet axis. The resulting average lobe and shocked shell pressures
must be calculated numerically (see Section 4.1) rather than using the simple
relation derived in Section 3.1.

The pressure of the shocked shell along the jet axis, $p_{s}(t)\sim p_{h}(t)$,
is related to the density of the ambient medium, $\rho_{x}=kR_{s}^{-\beta}$
using the Rankine-Hugoniot shock jump conditions,

$p_{s}(t)=\frac{2}{\Gamma_{x}+1}kR_{s}^{-\beta}\left(\frac{dR_{s}}{dt}\right)^{2},$
(16)

where $\Gamma_{x}$ is the adiabatic index of the ambient medium surrounding
the shocked gas shell.

Substituting Equations 15 and 16 for the shocked shell volume and pressure
along the jet axis into Equation 13 yields a second-order nonlinear
differential equation for shell radius,

$2R_{s}^{3-\beta}\frac{dR_{s}}{dt}\frac{d^{2}R_{s}}{dt^{2}}+(3\Gamma_{c}-\beta)R_{s}^{2-\beta}\left(\frac{dR_{s}}{dt}\right)^{3}=\frac{(\Gamma_{c}-1)(\Gamma_{x}+1)Q}{2k\pi\sin^{2}\theta_{j}}.$
(17)

This equation can be solved by trialling another power-law solution, with the
exponent again constrained by dimensional analysis. This yields

$R_{s}(t)=\left[\frac{(\Gamma_{c}-1)(\Gamma_{x}+1)(5-\beta)^{3}Q}{18(9\Gamma_{c}-4-\beta)k\pi\sin^{2}\theta_{j}}\right]^{1/(5-\beta)}t^{3/(5-\beta)}.$
(18)

Kaiser and Alexander (1997) include an additional correction in the
denominator of their equivalent expression due to the energy associated with
the higher pressure jet-head region; the $(9\Gamma_{c}-4-\beta)$ term in
Equation 18 becomes
$(9[\Gamma_{c}+(\Gamma_{c}-1)/\sin^{2}\theta_{j}]-4-\beta)$. We note that this
correction assumes that the differential increase in volume of the lobe and
hotspot are equal as the source grows; this assumption is not particularly
realistic for an ellipsoidal lobe geometry.

##  4 Semi-analytic Models

Improved computation has recently enabled a new generation of analytical
models with added complexity. These models typically solve systems of
differential equations which lack an analytic solution to describe the
evolutionary history of the radio source. Below, we summarise the main
developments, including atmospheres beyond power-law density profiles (Section
4.1), considering both the ram and thermal pressure contributions to the
expansion along the jet axis (Section 4.2), and modelling the relativistic jet
in a distinct expansion phase prior to the onset of lobe formation (Section
4.3).

###  4.1 RAiSE (version 2015)

Turner and Shabala (2015) developed a semi-analytic model, _Radio AGN in Semi-
analytic Environments_ (RAiSE), based on the theory of the Falle (1991) class
of models (Section 3). These authors extended existing analytic approaches by
considering piece-wise solutions to the governing differential equations in
two dimensions. The RAiSE model included three key improvements over the
earlier models: (1) ambient medium consistent with X-ray observations of
clusters and semi-analytic galaxy formation models; (2) angular dependence of
expansion velocity across the ellipsoidal contact surface; and (3) modelling
of the morphological transition from supersonic to subsonic lobe expansion by
using complete differential equations rather than limiting cases which yield
analytic expressions.

Figure 3: Schematic of the Turner and Shabala (2015) dynamical model for the
lobe and shocked shell. This framework is also used by Turner et al. (2023)
for both their jet- and lobe-dominated expansion phases, albeit the lobe
(shown in red) only forms once a critical length-scale is reached. Taken from
Figure 1 of Turner et al. (2023).

The lobe and shocked shell in their model are constructed from an ensemble of
small angular volume elements in assumed pressure equilibrium. Each element of
fixed angular width $d\theta$ is assumed to receive a constant fraction of the
jet power as the cavity expands. This assumption yields self-similar expansion
at early times when the shocked shell is expanding in the strong-shock
supersonic limit, as in the earlier models of Kaiser and Alexander (1997). The
volume of each small angular element of the shocked shell,
$[\theta-\delta\theta/2,\theta+\delta\theta/2)$, is given by

$\delta V_{s}(\theta)=\frac{2\pi R_{s}^{3}(\theta)}{3}\sin\theta\delta\theta,$
(19)

where $\theta$ is the angle between some location on the surface of the
shocked shell and the jet axis, and $R_{s}(\theta)$ is the radius of the
initially ellipsoidal shell at that location (see Figure 3). Importantly, the
shocked gas shell does not expand self-similarly as the steepness of the
ambient gas density profile encountered by the non-spherical shell will in
general differ across its surface, leading to different growth rates; this
prediction is consistent with the higher axis ratios observed in the largest
radio sources Mullin et al. (2008).

The initial radius of each volume element is related to that along the jet
axis by a geometric factor as $R_{s}(\theta,t\rightarrow
0)=\eta_{s}(\theta)R_{s}(\theta=0,t\rightarrow 0)$, where $\theta=0$ is
aligned along the jet axis and $\eta_{s}(\theta)$ is defined as

$\eta_{s}(\theta)=\frac{1}{\sqrt{(\sin^{2}\theta/\sin^{2}\theta_{j})+\cos^{2}\theta}},$
(20)

where we have assumed the same relationship between the jet half-opening
angle, $\theta_{j}$, and axis ratio of the shocked shell, $A_{s}$, as
discussed in Section 3.

Following Turner and Shabala (2015), and later work by Turner and Shabala
(2020) and Turner et al. (2023), the adiabatic expansion of each angular
volume element is related to the pressure imparted on that element at the
surface, $p_{s}(\theta)$, its volume $\delta V_{s}(\theta)$, and the fraction
of the input jet power associated with that element, $Q\delta\lambda(\theta)$.
The function $\delta\lambda(\theta)$ is defined in Equation 20 of Turner et
al. (2023). The first law of thermodynamics in Equation 13 gives

$\frac{dp_{s}(\theta)}{dt}\delta
V_{s}(\theta)+\Gamma_{c}p_{s}(\theta)\frac{d[\delta
V_{s}(\theta)]}{dt}=(\Gamma_{c}-1)Q\delta\lambda(\theta).$ (21)

Away from the contact surface, the pressure in the lobe is calculated as the
spatial average of the surface pressures $p_{s}(\theta)$.

Turner and Shabala (2015) use a similar expression for the pressure at the
contact surface to Equation 16, but additionally consider: (1) the orientation
of the expanding surface as it impacts the ambient medium; and (2) terms
describing evolution in the transonic and subsonic expansion regimes. That is,
for expansion in the supersonic and transonic phases, we have

$p_{s}(\theta)=\frac{2}{\Gamma_{x}+1}kR_{s}^{-\beta}(\theta)\left(\frac{\zeta_{s}(\theta)}{\eta_{s}(\theta)}\frac{dR_{s}(\theta)}{dt}\right)^{2}-\frac{\Gamma_{x}-1}{\Gamma_{x}+1}(kl)R_{s}^{-\beta}(\theta),$
(22)

where $\zeta_{s}(\theta)$ is a further geometric factor defined in Equation 13
of Turner et al. (2023). The radial temperature profile of the ambient medium
is defined by Turner and Shabala (2015) as $T=(\bar{m}/k_{b})lr^{-\varsigma}$
for Boltzmann constant $k_{B}$, and average particle mass $\bar{m}\sim
0.6m_{p}$ (where $m_{p}$ is the proton mass). Below, we present their results
assuming an isothermal medium for consistency with other authors, i.e. adopt
$\varsigma=0$. The pressure in the subsonic regime is equal to the ambient
pressure (see Equation 4 of Turner and Shabala, 2015).

The second-order non-linear differential equation that results from
substituting Equations 19 and 22 into the Equation 21 cannot in general be
solved to yield an analytic solution. Turner and Shabala (2015) instead
rewrite the resulting equation as a system of two coupled first-order ordinary
differential equations. These differential equations describe the velocity and
acceleration at the contact surface of a given volume element $\delta
V_{s}(\theta)$,

$\begin{split}\dot{R}_{s}(\theta)&=v_{s}\\\
\dot{v}_{s}(\theta)&=\frac{3(\Gamma_{x}+1)(\Gamma_{c}-1)QR_{s}^{\beta-3}\delta\lambda}{8\pi
v_{s}(\zeta_{s}/\eta_{s})^{2}k\sin\theta\delta\theta}+\frac{(\beta-3\Gamma_{c})v_{s}^{2}}{2R_{s}}\\\
&\quad\quad+\frac{(\Gamma_{x}-1)(3\Gamma_{c}-\beta)l}{4R_{s}(\zeta_{s}/\eta_{s})^{2}},\end{split}$
(23)

where $v_{s}$, $R_{s}$, $\delta\lambda$, $\zeta_{s}$ and $\eta_{s}$ are
explicit functions of $\theta$, whilst the properties of the ambient medium,
$k$, $l$ and $\beta$, are implicit functions of $\theta$ as different sections
of the contact surface reach a given distance into the spherically symmetric
environment at different times. Turner and Shabala (2015) use a standard
fourth-order Runge-Kutta method to solve this system of equations, providing
the analytic solution for the strong-shock limit as an initial condition.

###  4.2 Hardcastle model

Figure 4: Schematic of the dynamical model proposed by Hardcastle (2018). We
depict a conical jet, noting that in this model the cross-sectional area at
the jet-head is related to the lobe volume/radius by a constant scaling factor
$\kappa_{1}$, and hence the dynamics of the jet are not critical to model
behaviour.

Hardcastle (2018) improves on the lobe-dominated expansion model of Turner and
Shabala (2015) by explicitly considering the momentum flux of the jet plasma.
This model considers expansion for two angles across the surface of the
shocked shell, $\theta=0$ and $\tfrac{\pi}{2}$ (see Figure 4). The choice of
these two angles is sufficient to model changes to the axis ratio of the lobe
and shocked shell in lobed FR-IIs, with the larger number of angles considered
by Turner and Shabala (2015) only important in the transonic and subsonic
expansion phases when the lobe deforms from its initial ellipsoidal shape.

The shocked shell pressures are derived from the ram pressure component along
the jet axis (Equation 1), and a component due to the internal energy of the
relativistic lobe plasma acting along both axes (Equation 5, with $U=Qt$ and
$q\ll 1$). The lobe and shocked gas are assumed to be in pressure equilibrium,
as discussed in Section 3. The shocked shell pressures along the major and
minor axes are then given by

$\begin{split}p_{s}(\theta=0)&=\frac{\varepsilon
QR_{s}(\theta=0)}{2cV}+\frac{(\Gamma_{c}-1)\xi Qt}{V}\\\
p_{s}(\theta=\tfrac{\pi}{2})&=\frac{(\Gamma_{c}-1)\xi Qt}{V}.\end{split}$ (24)

where $V/R_{s}(\theta=0)$ is the cross-sectional area of the jet-head region
for a cylindrical lobe of volume $V$, $\varepsilon\sim 4$ here acts as a
geometric correction factor reflecting more realistic lobe shapes, and
$\xi\sim\tfrac{1}{2}$ is the fraction of the input jet kinetic power found in
hydrodynamic simulations to be stored as internal energy of the relativistic
lobe plasma; the remainder is stored as thermal and kinetic energy in the
shocked gas shell.

Hardcastle (2018) relates the volume of the lobe, $V$, to that of the shocked
shell, $V_{s}$ (including the interior lobe), by considering the ratio of
total internal energies,

$\frac{V}{V_{s}}=\frac{(\Gamma_{c}-1)\xi
Qt}{[\xi\Gamma_{c}+(1-\xi)\Gamma_{s}-1]Qt+f(N,T,v_{s})},$ (25)

where $f(N,T,v_{s})$ is a function describing the internal energy of the
ambient medium swept-up by the bow shock. This function depends on the total
number of swept-up particles, $N$, their temperature, $T$, and their bulk
velocity due to the expansion of the shocked shell, $v_{s}=dR_{s}/dt$
(Hardcastle, 2018, their Equations 5 and 7). For young sources, when the
thermal energy of these particles is lower than the energy supplied by the jet
to the forming shocked gas shell, the ratio of lobe to shocked shell volumes
tends to $\xi$ for $\Gamma_{c}=\Gamma_{s}=\tfrac{5}{3}$, or $\xi/(2-\xi)$ if
the lobe is assumed to have a relativistic plasma ($\Gamma_{c}=\tfrac{4}{3}$).
The volume of the shocked gas shell is of course also directly related to the
lengths of the major and minor axes; for an ellipsoidal geometry this gives

$V_{s}=\frac{2\pi R_{s}(\theta=0)R_{s}^{2}(\theta=\tfrac{\pi}{2})}{3}.$ (26)

We can therefore express the pressure along the major and minor axes of the
shocked shell (Equation 24) as a function of both axis lengths upon
substitution of Equations 25 and 26.

Hardcastle (2018) rewrites the Rankine-Hugoniot shock jump conditions
(Equation 22) in terms of the velocities along the major and minor axes of the
lobe. However, unlike the previously discussed models, Hardcastle (2018)
expresses the jump conditions in terms of the sound speed, $c_{s}$, and
adiabatic index, $\Gamma_{s}$, of the shocked gas shell surrounding the lobe.
The expansion rates along the major and minor axes are then given by

$\begin{split}\frac{dR_{s}(\theta=0)}{dt}&=c_{s}\sqrt{\frac{(\Gamma_{s}+1)[p_{s}(\theta=0)/p_{x}(R_{s}(\theta=0))]-(\Gamma_{s}-1)}{2\Gamma_{s}}}\\\
\frac{dR_{s}(\theta=\tfrac{\pi}{2})}{dt}&=c_{s}\sqrt{\frac{(\Gamma_{s}+1)[p_{s}(\theta=\tfrac{\pi}{2})/p_{x}(R_{s}(\theta=\tfrac{\pi}{2}))]-(\Gamma_{s}-1)}{2\Gamma_{s}}},\end{split}$
(27)

where $p_{x}(r)$ is the spherically symmetric ambient gas pressure profile.
This coupled system of non-linear ordinary differential equations can be
solved using a standard fourth-order Runge-Kutta method, with initial
conditions of $R_{s}(\theta=0)=ct_{0}$ and
$R_{s}(\theta=\tfrac{\pi}{2})=ct_{0}$ for some small time $t_{0}$.

###  4.3 RAiSE (version 2023)

The RAiSE (Turner and Shabala, 2015) model discussed in Section 4.1 was first
extended to make predictions for the spatial distribution of emission at radio
(Turner et al., 2018) and X-ray wavelengths (Turner and Shabala, 2020); and
most recently to incorporate important changes to jet and lobe dynamics. The
Turner et al. (2023) model includes: (1) a relativistic jet expansion phase
modelled prior to the formation of a lobe; (2) formation of lobes within a
surrounding bow shock; and (3) a separation of the ram and thermal components
of the jet and lobe pressure.

The Turner et al. (2023) model is implemented using the same computational
framework as the original RAiSE model discussed in Section 4.1, specifically,
using coupled differential equations which are solved for small angular volume
elements of the lobe and shocked shell. Below, we present a concise derivation
of their solution for the expansion of the relativistic jet (Section 4.3.1),
summarise their methodology to model the subsequent lobe formation/inflation
(Section 4.3.2), and finally present their method to separate the ram and
thermal components of the jet and lobe pressure (Section 4.3.3).

####  4.3.1 Relativistic jet expansion

The relativistic hydrodynamic conservation equations relate the properties of
fluids upstream and downstream of a shock discontinuity via the stress-energy
tensor. The conservation equations for a relativistic fluid are expressed in
terms of comoving quantities including gas density $\rho$, gas pressure $p$,
dimensionless specific enthalpy $h$ (i.e. enthalpy divided by $c^{2}$), and
the non-zero spatial component of the four-velocity $u=\gamma v/c$ (hereafter
shortened to four-velocity) relative to the shock front. The conservation
equations for a relativistic fluid are (e.g. Turner et al., 2023, and
references therein):

$\displaystyle\rho\gamma v=\rho_{1}\gamma_{1}v_{1}\ \ \ \text{(continuity)}$
(28a) $\displaystyle\rho
h\gamma^{2}v^{2}+p=\rho_{1}h_{1}\gamma_{1}^{2}v_{1}^{2}+p_{1}\ \ \
\text{(momentum)}$ (28b) $\displaystyle\rho(h\gamma-1)\gamma
v=\rho_{1}(h_{1}\gamma_{1}-1)\gamma_{1}v_{1}\ \ \ \text{(energy)}.$ (28c)

The fluid downstream of the shock is represented by the subscript ‘1’; no
subscript refers to the upstream fluid.

The conservation of energy expression in Equation 28c is related to the rate
of energy input by the jet, $Q$, by multiplying by the cross-sectional area of
the jet (cf. Walg et al., 2013, their Equation 26). That is,

$Q=\rho_{j}(h_{j}\gamma_{j}-1)\gamma_{j}v_{j}c^{2}\Omega r^{2},$ (29)

where the factor of $c^{2}$ is added to convert the dimensionless enthalpy,
$h_{j}$, to the specific enthalpy; $v_{j}$ is the bulk velocity of the jet
plasma and $\gamma_{j}$ is the corresponding Lorentz factor. We can therefore
obtain an expression for the density of the jet plasma some distance $r$ along
the jet in terms of the dimensions and energetics of the jet as follows:

${\rho}_{j}(r)=\frac{Q}{\gamma_{j}{v}_{j}c^{2}(h_{j}{\gamma}_{j}-1)\Omega
r^{2}},$ (30)

where the density at the jet-head, $\rho_{j}\equiv\rho_{j}(R_{s})$, is of
particular interest for the radio source dynamics.

We now derive the Rankine-Hugoniot jump conditions relating the density and
velocity of both the jet plasma and the ambient medium. The bulk velocity of
the ambient medium in the observer frame is zero at all times for random
particle motions. As a result, the bulk velocity of these particles in the
frame of the shock front, $v_{1}$, is exactly equal to the expansion rate of
the shock in the observer frame, $v_{s}$; i.e. $v_{1}\equiv-v_{s}$. By
contrast, the bulk velocity of the upstream fluid particles in the jet is non-
zero, defined as ${v}_{j}$ in the observer frame. Following (Turner et al.,
2023), the conservation of momentum flux equation can therefore be rewritten
as

${\rho}_{j}h_{j}{\gamma}_{j}^{2}\gamma_{s}^{2}({v}_{j}-v_{s})^{2}=\rho_{x}h_{x}\gamma_{s}^{2}v_{s}^{2},$
(31)

where $h_{j}$ is the dimensionless specific enthalpy of the jet, and
$\rho_{x}$ and $h_{x}$ are the density and dimensionless specific enthalpy of
the (external) ambient medium respectively. Rearranging yields a relationship
between the jet-head advance speed and the bulk velocity of the jet (cf. Martí
et al., 1994, 1997; Rosen et al., 1999, and subsequent authors):

$v_{s}\equiv\frac{dR_{s}}{dt}=\frac{{v}_{j}}{1+[{\rho}_{j}h_{j}{\gamma}_{j}^{2}/(\rho_{x}h_{x})]^{-1/2}},$
(32)

where the dimensionless quantity
$\eta_{R}={\rho}_{j}h_{j}{\gamma}_{j}^{2}/(\rho_{x}h_{x})$ is a function of
properties of the jet and ambient medium. That is,

$\eta_{R}(r)=\frac{Qh_{j}{\gamma}_{j}}{kh_{x}{v}_{j}c^{2}(h_{j}{\gamma}_{j}-1)\Omega
r^{2-\beta}}.$ (33)

where we have made use of the power-law approximation for the local density of
the ambient medium, $\rho=kr^{-\beta}$, and Equation 30 for the jet plasma
density.

The jet length is found by integrating Equation 32 with respect to time,
however, an analytical solution is only possible in the limits
$\eta_{R}\rightarrow 0$ and $\eta_{R}\rightarrow\infty$ (e.g. Matzner, 2003;
Bromberg et al., 2011). Turner et al. (2023) solve this integral numerically
using a fourth-order Runge-Kutta method on the following system of three
ordinary differential equations:

$\begin{split}\dot{R}_{s}&=v_{s}\\\
\dot{v}_{s}&=\frac{(\beta-2){v}_{j}v_{s}}{2R_{s}\eta_{R}^{1/2}[1+\eta_{R}^{-1/2}]^{2}}\\\
\dot{\gamma}_{s}&=\frac{\gamma_{s}^{3}v_{s}\dot{v}_{s}}{c^{2}}.\end{split}$
(34)

We note that in the interests of clarity we have omitted the transverse
density and velocity structures of the flow along the jet from the above
derivation; we refer the reader to Section 2.2.2 of Turner et al. (2023) for a
complete description.

####  4.3.2 Lobe formation

The energy supplied by the central nucleus is initially focussed over a small
range of angles within the half-opening angle of the jet. Beyond some lobe
formation length-scale the energy must be distributed across the $2\pi$
steradians of the shocked shell. The source expansion in these two phases is
described by the differential equations for the relativistic jet (Section
4.3.1) and lobe and shocked shell (Section 4.1). Turner et al. (2023) combine
these frameworks by modelling the expansion of the radio source as a two-phase
fluid, where each angular volume element is assumed to comprise a fraction
$\Lambda(t)$ of lobe plasma at any given time $t$. Turner et al. (2023) relate
the acceleration of the ellipsoidal bow shock surrounding the lobe to the
acceleration in the jet- and lobe-dominated expansion phases,
$\dot{v}_{s,j\\!\;\\!e\\!\;\\!t}$ (Equation 34) and
$\dot{v}_{s,l\\!\;\\!o\\!\;\\!b\\!\;\\!e}$ (Equation 23) respectively, as
follows:

$\dot{v}_{s}(\theta)=[1-\Lambda]\dot{v}_{s,j\\!\;\\!e\\!\;\\!t}\eta(\theta)+\Lambda\dot{v}_{s,l\\!\;\\!o\\!\;\\!b\\!\;\\!e}(\theta),$
(35)

where $\Lambda$ is the fractional contribution of the lobe plasma to the
acceleration of the bow shock at a given time. The other two coupled ordinary
differential equations (for velocity and derivative of the Lorentz factor) are
identical for both fluids and thus do not require any modification.

Turner et al. (2023) define the transition from a jet-dominated to a lobe-
dominated flow based on the length-scale at which lobe formation commences.
This length-scale is calculated by equating the densities of the jet plasma
and ambient medium (e.g. Alexander, 2006; Krause et al., 2012). Turner et al.
(2023) parametrise the transition from jet- to lobe-dominated expansion by
using the ratio of these densities,

$\mathcal{L}(t)=\frac{{\rho}_{j}}{\rho_{x}}=\ \
\frac{\eta_{R}(R_{s}(\theta=0,t))}{{\gamma}_{j}^{2}}$ (36)

where $\eta_{R}(r)$ is defined in Equation 33 and is evaluated for the length
of the jet at time $t$. Turner et al. (2023) use this ratio to calculate the
fractional contribution of the lobe to source expansion,

$\Lambda(t)=e^{-\mathcal{L}^{2}(t)/(2\log 2)},$ (37)

where $\Lambda(t)\rightarrow 0$ in the jet-dominated expansion phase and
$\Lambda(t)\rightarrow 1$ in the lobe-dominated phase.

This two-phase fluid model describes the evolution of the bow shock across the
transition from a jet-dominated to lobe-dominated flow. A more complete
description requires consideration of lobe formation inside the shock front.
We refer the interested reader to Section 2.4.2 of Turner et al. (2023) for
these details.

####  4.3.3 Thermal pressure

The Turner et al. (2023) relativistic jet model (Section 4.3.1) and their
earlier lobe-dominated expansion model (Section 4.1) derive the jet and lobe
length evolution by considering conservation of momentum flux (Equation 31),
however, the relative magnitudes of the ram and thermal pressure components
after the interaction are not explicitly calculated. These pressure components
are difficult to separate directly using the conservation equations, however,
we know the lobe evolution is driven entirely by the thermal component in the
limit $t\rightarrow\infty$. Turner et al. (2023) therefore find the thermal
pressure at earlier times by iteratively solving (backwards in time) the
relevant differential equations with the initial condition at
$t\rightarrow\infty$. We refer the interested reader to Section 2.3.4 of
Turner et al. (2023) for a complete description of the separation of the ram
and thermal components of the lobe internal pressure.

##  5 Discussion

In preceding sections, we have presented the theory underpinning the key
classes of analytical models describing the dynamics of kiloparsec-scale radio
AGN jets and lobes. The same physical principles are considered in each of
these models, notably ram pressure against the ambient medium and an adiabatic
equation of state, however their implementation between model classes differs
greatly – as we discuss in Section 5.1. We compare the accuracy of predictions
for each model type relative to the outputs of a three-dimensional
relativistic hydrodynamic simulation in Section 5.2. We then assess for which
regions of parameter space the different model classes yield comparable
results, and conversely those where large differences are expected, in Section
5.3.

###  5.1 Similarity of key model classes

The four key classes of analytical models examined in this review share common
physical principles to explain the dynamics of kiloparsec-scale radio sources
(see Table 1). Scheuer (1974) models the forward expansion of the source based
on the momentum flux of the jet and invokes internal energy to calculate the
sidewards expansion of the lobe. Falle (1991) instead models the forwards
expansion by considering the adiabatic expansion of the lobe due to an
increase in internal energy while using the jet momentum flux to relate the
shape of the lobe to the opening angle of the jet. Meanwhile, the Hardcastle
(2018) and Turner et al. (2023) models smoothly transition their dynamics
between the jet- and lobe-dominated expansion phases predicted by these
earlier models.

Table 1: Summary of the model assumptions for each of the four key classes of
analytical models considered in this review. First column: analytical model.
Second column: type of solution (analytical or semi-analytic), together with a
note on the assumed ambient gas density profile. The third and fourth column:
physical process driving the jet and lobe length expansion at early and late
times, respectively. The fundamental equations used to derive the source
evolution, and the time dependence of the length expansion, are also listed.

Model |  Type |  Early-time evolution |  Late-time evolution
---|---|---|---
Scheuer (1974; Model A) |  analytical (constant density) |  momentum flux (Bernoulli equation) $R(t)\propto t^{2/(4-\beta)}$ |  –
Falle (1991) |  analytical (power-law density profile) |  – |  internal pressure (first law of thermodynamics) $R(t)\propto t^{3/(5-\beta)}$
Hardcastle (2018) |  semi-analytic (spherically symmetric density profile) |  momentum flux (non-relativistic shock-jump conditions) $R(t)\propto t^{2/(4-\beta)}$ |  internal pressure (non-relativistic shock-jump conditions) $R(t)\propto t^{3/(5-\beta)}$
Turner et al. (2023; RAiSE) |  semi-analytic (spherically symmetric density profile) |  momentum flux (relativistic hydrodynamics) $R(t)\propto t$† |  internal pressure (first law of thermodynamics) $R(t)\propto t^{3/(5-\beta)}$

$\dagger$ Solution for jet-head length expansion prior to lobe formation
length-scale.

####  5.1.1 Early-time evolution

The source length evolution in the Scheuer (1974) model, and the (jet-
dominated) early-time expansion phases of the Hardcastle (2018) and Turner et
al. (2023) models, are derived considering the relative amplitudes of the
momentum flux of the jet and the thermal pressure of the ambient medium
(Equation 28b). The relativistic hydrodynamic equations used in the theory of
Turner et al. (2023) can be simplified to obtain the expressions found by the
Scheuer (1974) class of models. In particular, the jet-head advance speed
derived in Equation 32 is integrated to yield the source length as follows:

$\begin{split}R(t)&=\int_{0}^{t}\frac{dR_{s}}{dt}dt\\\
&\approx\left(\frac{Qh_{j}{\gamma}_{j}v_{j}}{\Omega
kh_{x}c^{2}(h_{j}{\gamma}_{j}-1)}\right)^{1/(4-\beta)}\left(\frac{(4-\beta)t}{2}\right)^{2/(4-\beta)},\end{split}$
(38)

where the second equality is valid for $\eta_{R}(r)\ll 1$, which coincides
with the formation of lobes on a length-scale of order 1 kpc (Turner et al.,
2023). This equation converges to that proposed by Rees (1971) and Scheuer
(1974, their Model A) by taking their limit of massless particles with
velocity $c$ (i.e. $h_{j,x}\rightarrow 1$ and $\gamma_{j}\rightarrow\infty$),

$R_{s}(t)=\left(\frac{Q}{\Omega
kc}\right)^{1/(4-\beta)}\left(\frac{(4-\beta)t}{2}\right)^{2/(4-\beta)},$ (39)

where for their assumption of a uniform ambient medium we set $\beta=0$ and
$k=\rho$.

The similarity of the early-time evolution predicted by the Hardcastle (2018)
model and Scheuer (1974) class of models is immediately apparent by comparing
their expression for the jet-head pressure (Equation 24). In the limit
$t\rightarrow 0$, their expression converges to the ram pressure component as
follows:

$\begin{split}p_{s}(\theta=0)&=\frac{\varepsilon QR_{s}(\theta=0)}{2cV}\ \ \
\text{(\cite[citeauthor]{\@@bibref{Authors}{Hardcastle+2018}{}{}})}\\\
&=\frac{\kappa_{1}Q}{\Omega R_{s}^{2}(\theta=0)c}\ \ \
\text{(\cite[citeauthor]{\@@bibref{Authors}{Scheuer+1974}{}{}})},\end{split}$
(40)

where the geometric factor assumed by Hardcastle (2018) is identified as
$\varepsilon=2\kappa_{1}/\Omega$ using the terminology of Scheuer (1974); in
other words, this geometric factor largely corresponds to the solid angle of
the jet and the jet-head region. Hardcastle (2018) derives the jet-head
advance speed using the non-relativistic Rankine-Hugoniot shock jump
conditions (Equation 27), largely equivalent to the ram pressure argument
employed by Scheuer (1974). Because of this, the source length evolution
predicted by the Hardcastle (2018) model early in the source lifetime matches
that of the Scheuer (1974) class of models for the same input parameters; i.e.
$R\propto t^{1/2}$ for a constant density ambient medium. By contrast, the
jet-dominated expansion phase of the Turner et al. (2023) model yields a very
different evolutionary history to these two model types, with $R\propto t$
prior to lobe formation, as those authors do not take the non-relativistic
limit of the hydrodynamic equations.

####  5.1.2 Late-time evolution

The lobe length expansion in the Falle (1991) class of models, and the (lobe-
dominated) late-time expansion phases of the Hardcastle (2018) and Turner et
al. (2023) models, are calculated by considering the internal energy of the
relativistic lobe plasma evolving under an adiabatic equation of state. The
first law of thermodynamics (Equation 12) applied to a self-similar
ellipsoidal shell yields lobe length growth of the form $R\propto t^{3/5}$
(Equation 18, constant gas density form is stated here for simplicity),
consistent with findings for supernova remnants. The Falle (1991) class of
models use this dependence directly, while the lobe-dominated expansion phase
of the Turner et al. (2023) model also permits an evolving lobe axis ratio due
to a non-power law ambient medium. The late-time evolution of the Turner and
Shabala (2015) model (and subsequent versions of RAiSE) ultimately transitions
from the supersonic to subsonic regime as the jet-head advance speed slows
($R\propto t^{1/3}$). This is not critical when modelling powerful lobed radio
sources which drive strong shocks – the focus of this review – but is
essential when considering the coasting (inactive) phase of remnant radio
sources.

Hardcastle (2018) does not make any explicit assumptions about the lobe axis
ratio, however, in the limit $t\rightarrow\infty$ their expressions for the
expansion rate along both the major and minor axes converge, leading to a
spherical lobe (at least in the special case of a constant density ambient
medium). The expression for their growth rate at late times (Equation 27)
becomes:

$\frac{dR_{s}}{dt}=\sqrt{\frac{(\Gamma_{s}+1)([\xi\Gamma_{c}+(1-\xi)\Gamma_{s}-1]Qt+f(N,T,v_{s}))}{2\rho_{x}V_{s}}},$
(41)

where we have assumed that the pressure along both axes is dominated by the
internal energy component (Equation 24), that the sound speed in the shocked
gas is comparable to that of the ambient medium (i.e.
$c_{s}\sim\sqrt{\Gamma_{s}p_{x}/\rho_{x}}$), and applied the relationship for
the ratio of the lobe and shocked shell volumes (Equation 25). This
differential equation again yields a solution of the form $R\propto t^{3/5}$
for a constant density ambient medium, if the internal energy contribution
from the shocked gas, $f(N,T,v_{s})$, is ignored. Such an assumption is
reasonable for high jet powers or moderate ambient densities, however, the
late-time evolution diverges significantly from this prediction for weak jets
or high density environments; we examine this point further in Section 5.2.2.

####  5.1.3 Source morphology

The lobe volume, and hence axis ratio, are calculated using the remaining
equations not previously invoked in the calculation of the source length
expansion. For example, in the Scheuer (1974) model, source expansion along
the jet axis is modelled based on ram pressure arguments, while the sidewards
expansion is derived considering the internal energy of the injected lobe
plasma; Hardcastle (2018) makes similar arguments, and Turner et al. (2023)
invoke internal energy to describe the formation and inflation of their lobe
within the confines of a surrounding shocked gas shell. By contrast, the self-
similar expansion assumed in the Falle (1991) class of models, and the lobe-
dominated expansion phase of the Turner et al. (2023) model, implicitly sets
the lobe volume based on its length, which is calculated from internal energy.
These models use ram pressure arguments to relate the internal conditions of
the lobe to those of the ambient medium (Equation 11), and to relate the jet
half-opening angle to the source axis ratio.

###  5.2 Comparison to hydrodynamic simulations

To test the analytical models, we compare the dynamics of the four key model
classes introduced above to the results of hydrodynamic simulations run using
the PLUTO code (Mignone et al., 2007). Below, we describe the existing
simulations run by Yates-Jones et al. (2022) for powerful FR-II radio
galaxies, and test the analytical models by comparing the predicted evolution
of the source length, lobe axis ratio, and jet-head pressure throughout the
source lifetime to the simulation results.

####  5.2.1 Hydrodynamic simulation dynamics

The hydrodynamic simulations of Yates-Jones et al. (2022) consider an
initially conical jet of half-opening angle $\theta_{j}=10$ degrees and bulk
flow with Lorentz factor $\gamma_{j}=5$. Their high-powered jet ($Q=3\times
10^{38}$ W) expands into a spherically symmetric King profile with core
density of $\rho_{c}=2.41\times 10^{-24}$ kg m-3, core radius of $r_{c}=144$
kpc, and slope described by the coefficient $\beta^{\prime}=0.38$ (for
details, see Yates-Jones et al., 2021). Those authors consider both jets
located at the centre of the gas distribution, as well as offset jets. In this
review, we will only consider their cluster-centred jet simulation as the
theory underpinning the analytical models assumes a spherically symmetric
environment. Their simulations result in lobed Fanaroff and Riley (1974) Type-
II sources forming on a timescale of $\sim 1$ Myr, and consider the late-time
evolution up to an age of 35.1 Myr.

We extract time series for the evolution of the source length, the lobe axis
ratio, and the jet-head pressure from the hydrodynamic simulation outputs (for
details, see Turner et al., 2023). These are critical dynamical quantities in
calculation of both the source evolution and the radio-frequency synchrotron
emission, and thus should be considered to assess the accuracy of the
analytical models. The potentially more informative lobe volume and volume-
weighted pressure are poorly constrained prior to lobe formation as large
regions near the core remain partially occupied by ambient gas; the
calculation of these quantities in the hydrodynamical simulation is highly
dependent on the threshold used to separate ambient gas from the jet plasma.

####  5.2.2 Accuracy of analytical models

The critical intrinsic parameters characterising radio source evolution in
analytical models are the jet kinetic power, source age and properties of the
ambient medium. The spherically symmetric King profile used by Yates-Jones et
al. (2021) for the ambient medium in their simulations is readily modelled by
both the Hardcastle (2018) and Turner et al. (2023) models, but not the older
models. The original form of the Scheuer (1974) model assumes a constant
density ambient medium, while the Falle (1991) model employs a slightly more
general power-law of the form $\rho=kr^{-\beta}$; in Section 2 we similarly
derived the Scheuer (1974) model for a power-law gas density profile. To
facilitate a meaningful comparison between all four model classes, we modify
the Falle (1991) and Scheuer (1974) model following the approach of Turner and
Shabala (2015), by approximating the gas density profile as a series of
contiguous power-laws with piecewise solutions. The implementation of these
models for a general ambient medium is available on our GitHub online
repository111https://github.com/rossjturner/analytical_models.

Figure 5: Comparison of the four classes of analytical models to the
hydrodynamic simulation (grey shading) used by Turner et al. (2023, their
Figure 3) to assess the success of their model near the commencement of lobe
formation. Top panel: source length evolution. Middle panel: lobe axis ratio.
Bottom panel: jet head pressure. The Scheuer (1974) and Falle (1991) class
models are shown for a range of jet half-opening angles $\theta_{j}$, while
the Hardcastle (2018) model is shown for a range of jet-head cross-sections
$\varepsilon$. The RAiSE model (Turner et al., 2023) is shown for its
optimised set of parameters.

##### Source length

The source length evolution for the four analytical models is shown in the top
panel of Figure 5. The jet power, source age and ambient gas density profile
are in all cases identical to the inputs to the hydrodynamic simulation of
Yates-Jones et al. (2021), however, some of the more minor model parameters
are varied to obtain the best representation of each model class.
Specifically, the Scheuer (1974) and Falle (1991) model evolutionary tracks
are shown for three plausible values of the jet half-opening angle
$\theta_{j}$, while the Hardcastle (2018) model is shown for three values of
their equivalent geometric factor $\varepsilon$. The free parameters in the
Turner et al. (2023) model have previously been calibrated based on this
hydrodynamic simulation, and thus results for only a single set of parameters
for this model are shown. The resulting evolutionary tracks are consistent
with the discussion in Section 5.1: the Scheuer (1974) and (jet-dominated)
early-time Hardcastle (2018) models follow an approximately $R\propto t^{1/2}$
growth rate (the dependence expected for a flat atmosphere), while the Falle
(1991) and (lobe-dominated) late-time Hardcastle (2018) models follow an
$R\propto t^{3/5}$ expansion rate (also see Figure 6). By contrast, the
relativistic hydrodynamic equations used by Turner et al. (2023) yield quite
different evolutionary tracks in the jet-dominated expansion phase, and are
more consistent with the hydrodynamic simulation. At late times ($>10$ Myr),
the Turner et al. (2023) model predicts slower expansion close to an $R\propto
t^{1/2}$ relationship, converging towards the same limit as the other models
at later times. We explore this in more detail in Section 5.3.

##### Lobe axis ratio

The evolution of the lobe axis ratio is shown in the middle panel of Figure 5.
The Turner et al. (2023) model captures axis ratio evolution prior to lobe
formation as it considers a two-phase fluid with an initially low, but non-
zero, fraction of jet plasma in the region between the jet and bow shock. By
contrast, the other analytical models initially disagree with the hydrodynamic
simulation as they assume a plasma-filled lobe structure throughout the source
lifetime. Upon lobe formation, the Hardcastle (2018) model agrees well with
the hydrodynamic simulation assuming a compact jet-head region with a
geometric factor $\varepsilon\gtrsim 40$ (corresponding to a jet-head of
radius 12.6 kpc for a typical 100 kpc jet). The Falle (1991) class of models
of course yields a constant lobe axis ratio (which is why these models are
often referred to as “self-similar”) throughout the source evolutionary
history, while the Scheuer (1974) model predicts a rapidly increasing axis
ratio as the lobe encounters the steeper sections of the ambient gas density
profile ($A\propto t^{(5\beta-2)/(16-4\beta)}$; see Section 2). These last two
models do not explicitly separate the lobe and shell material, so for a fairer
comparison, we are guided by the numerical results of Turner and Shabala
(2020) in assuming that the lobe axis ratio scales to that of the shell as
$A=A_{s}^{1.7}$.

##### Jet-head pressure

We finally compare the predicted evolution of the jet-head pressure to that
measured from the hydrodynamic simulation (Figure 5, bottom panel). The
majority of the evolutionary history probed by the hydrodynamic simulation
($\lesssim 10$ Myr) is associated with a significant ram pressure component at
the jet-head, in addition to a thermal component that scales approximately in
proportion to the ram pressure component. The Scheuer (1974) and Hardcastle
(2018) models directly consider the ram pressure component, and hence predict
a power-law evolution of jet head pressure ($p\propto t^{-1}$ for a constant
gas density ambient medium; see Sections 2 and 4.2) which is broadly similar
to to the thermal jet head pressure component measured by the hydrodynamic
simulation. The self-similar expansion model of Falle (1991) yields a flatter
relationship of the form $p\propto t^{-4/5}$. Turner et al. (2023) instead
explicitly model both the thermal and ram pressure components throughout the
source evolutionary history; this model accurately captures the steepening in
the rate of change of jet-head pressure during the transition between these
two limiting cases.

###  5.3 Parameter space exploration

In this section, we compare the consistency of observable predictions between
the four model classes. We select three parameters for our comparison. Source
length and axis ratio are directly measurable model predictions, while
synchrotron luminosity integrated over the entire lobe is a quantity which can
be approximated from source dynamics – noting that detailed consideration of
particle acceleration and loss processes is required for a full calculation
(e.g. Kaiser et al., 1997). These radio source attributes are critical for
estimating the jet energy budget through a parameter inversion of observables
(Shabala et al., 2008; Turner and Shabala, 2015).

We investigate the behaviour of the different model classes for a range of
input parameters, specifically the single-jet kinetic power $Q$, core density
$\rho_{0}$, and scale radius $r_{c}$ of the ambient gas density profile. The
base-case set of parameters is chosen to match the hydrodynamic simulation of
Yates-Jones et al. (2021), but with $\rho_{0}=10^{-23}$ kg m-3 and $r_{c}=100$
kpc. The jet half-opening angles for the Scheuer (1974) and Falle (1991)
models are set to $\theta_{j}=10^{\circ}$ and $\theta_{j}=20^{\circ}$
respectively, as these closely match the hydrodynamic simulation evolution for
the three key dynamical parameters of source length, lobe axis ratio and jet-
head pressure (cf. Figure 5). For the same reasons, we set the geometric
factor to $\varepsilon=40$ in the Hardcastle (2018) model. The evolutionary
tracks for each model class are considered both for the base set of parameters
and for a factor of ten variation to one of either the jet power ($Q=3\times
10^{37}$, $3\times 10^{38}$ or $3\times 10^{39}$ W), core density
($\rho_{0}=10^{-24}$, $10^{-23}$ or $10^{-22}$ kg m-3), or scale radius
($r_{c}=10$, 100 or 1000 kpc).

##### Source length

Figure 6: Source length evolution predicted by the four classes of analytical
models for a range of input intrinsic parameters. The evolutionary tracks for
the base set of parameters, i.e. $Q=3\times 10^{38}$ W, $\rho_{0}=10^{-23}$ kg
m-3 and $r_{c}=100$ kpc, are shown as solid lines. The source evolution for
either a lower ($Q=3\times 10^{37}$ W) or higher ($Q=3\times 10^{39}$ W) jet
power is shown in the top panel with dashed and dot-dashed lines respectively.
The middle panel shows evolutionary tracks with a lower ($\rho_{0}=10^{-24}$
kg m-3) or higher ($\rho_{0}=10^{-24}$ kg m-3) core densities. The bottom
panel plots the evolution for either a lower ($r_{c}=10$ kpc) or higher
($r_{c}=1000$ kpc) scale radius of the ambient gas density profile.

The source length evolution for the four model classes are shown in Figure 6.
Changes in jet power and core density largely result in a constant offset (in
logarithmic space) to the source length, consistent with the expected scalings
in flat atmospheres of $R\propto Q^{1/4}$ and $R\propto\rho_{0}^{-1/4}$ for
the jet-dominated (Scheuer, 1974) model, and $R\propto Q^{1/5}$ and
$R\propto\rho_{0}^{-1/5}$ for the lobe-dominated (Falle, 1991) model class.
The more sophisticated models of Hardcastle (2018) and Turner et al. (2023)
capture the transition between these limiting cases. For the same input
parameters, the Hardcastle (2018) and Turner et al. (2023) models predict
similar dynamical evolution for the majority of simulated sources. However,
the jet-dominant phase lasts longer for high-powered jets and in denser
environments in the Turner et al. (2023) model, making this model more
sensitive to changes in these parameters than the other models. Variations to
the scale radius of the ambient gas density profile yield qualitatively
similar behaviour between the model classes, with faster expansion occurring
for steeper atmospheres.

##### Lobe axis ratio

Figure 7: Lobe axis ratio evolution predicted by the four classes of
analytical models for a range of input intrinsic parameters. Panels and line
styles are as in Figure 6.

The evolution in the lobe axis ratio is shown in Figure 7. The Hardcastle
(2018) and Turner et al. (2023) models both predict the lobe axis ratio
evolutionary tracks shift horizontally along the source age axis in response
to variations in the jet power and core density, with lobe formation occurring
earlier for less powerful jets and/or denser ambient media (e.g. see Turner et
al., 2023). The Scheuer (1974) model predicts a similar response to changes in
the jet power and core density for the approximately constant ambient gas
density section of the lobe axis ratio evolutionary tracks (i.e. $\lesssim 1$
Myr), but is inconsistent at later times once the ambient density profile
begins to steepen (see Section 5.2). By contrast, variations to the scale
radius produce lobe axis ratio evolution that is largely inconsistent between
the different model classes. The self-similar model of Falle (1991) yields a
constant lobe axis ratio for all input parameters by construction.

##### Synchrotron luminosity

Figure 8: Lossless synchrotron luminosity at 151 MHz (see text for details).
Panels and line styles are as in Figure 6.

Radio galaxies are detectable through their synchrotron emission. In
analytical radio source models, this emission is typically calculated by
assuming a scaling between the lobe pressure and magnetic field, acceleration
of emitting particles at the jet termination shock, and subsequent losses due
to adiabatic, synchrotron and inverse-Compton losses due to upscattering of
cosmic microwave background photons.

A full calculation of the radio-frequency luminosity of each lobe is beyond
the scope of this review (see e.g. Kaiser et al., 1997; Shabala et al., 2008;
Turner and Shabala, 2015; Hardcastle, 2018). However, a useful estimate can be
obtained by considering only adiabatic losses, which are directly related to
the evolution of lobe pressure. The “lossless” luminosity calculated in this
approach is related to the lobe volume and pressure as $L_{\nu}\propto
p^{(\alpha+3)/2}V$, where $\alpha\sim 0.7$ is the spectral index of the non-
aged radio spectrum. The total internal energy of the lobe, $U\sim pV$,
represents the fraction of the jet energy that is transferred to the lobe, and
hence is very similar in all models for a fixed time and jet power. In this
case, the synchrotron luminosity scales with pressure as $L_{\nu}\propto
p^{(\alpha+1)/2}U$.

The evolution in the lossless synchrotron luminosity for each of the model
classes is shown in Figure 8 at a rest-frame frequency of 151 MHz. The
predictions of the Scheuer (1974) model are once again inconsistent with the
later models, with radio luminosities up to a factor of 100 greater than the
other models at late times due to their higher lobe axis ratios, and
subsequently smaller volumes and higher pressures. The remaining three models
are consistent at late times – when their lobe pressures are largely derived
based on changes in lobe internal energy – for all sets of input jet powers,
core densities and scale radii. The Falle (1991) and Hardcastle (2018) model
classes also agree at early times, in contrast to the Turner et al. (2023)
model which predicts higher luminosities as a result of significantly higher
pressures prior to lobe formation when the system is dominated by the momentum
flux of the relativistic jet. At these early times, the luminosities are a
factor of 100-1000 higher than in the lobe-only models of Falle (1991) and
Hardcastle (2018).

##  6 Concluding remarks

We have summarised and compared the main classes of analytical models
describing the dynamics of kiloparsec-scale lobed radio galaxies. These models
can be separated into two main classes, depending on the whether the expansion
of the radio source is driven by the momentum flux from the jet, or by the
internal lobe pressure. We presented the Scheuer (1974) and Falle (1991)
models, respectively, to describe the general characteristics of other
literature models in either of these two classes. We also examined separately
the more recent models proposed by Hardcastle (2018) and Turner et al. (2023),
which combine aspects of both jet momentum flux and lobe pressure.

We compared the different model classes against each other, and with high-
resolution hydrodynamic simulations, for a range of realistic input
parameters. Our key findings are as follows:

  * •

Jet momentum flux and lobe internal pressure dominate the early- and late-time
radio source evolution, respectively. Both must be considered for a complete
radio source model describing source dynamics after the lobe formation phase
($\sim$1 Myr; Section 5.2.2).

  * •

Realistic ambient gas density profiles (i.e. not constant or power-law)
produce radio sources which are inconsistent with self-similar lobe evolution
predicted by the Falle (1991) class of models (Sections 4.1 and 4.2). This
naturally explains the large axis ratios seen in giant radio galaxies.

  * •

Relativistic jet dynamics is important for an accurate description of early
source evolution, before the lobe formation phase (Section 5.2.2).

We make three of the four models considered in this review openly available.
The code extending the Scheuer (1974) and Falle (1991) models to general
atmospheres, and RAiSE (version 2023), are available in our GitHub online
repository222https://github.com/rossjturner/analytic_models${}^{\\!\;\\!,}$333https://github.com/rossjturner/RAiSEHD.
Hardcastle (2018) has also made their code available; we refer the interested
reader to their paper.

We conclude with a brief reflection on the next generation of analytical
models. Existing analytical models neglect the interaction between the jet and
the multi-phase interstellar medium of the host galaxy. Hydrodynamic
simulations that model this interaction predict that the jet can spend
$\gtrsim 1$ Myr in the galaxy (e.g. Bicknell et al., 2018). The majority of
observed radio sources are compact and short-lived (Hardcastle et al., 2019;
Shabala et al., 2020), and hence these processes are likely to be relevant to
the bulk of the radio source populations. At the other end of the radio galaxy
evolution scale, Hardcastle (2018) pointed out that for extreme losses, such
as expected in large sources at high redshift, it is possible for the majority
or even all of the jet energy to be radiated away. This mechanism can
potentially limit the maximum size to which a radio source can grow. Existing
analytical models decouple source dynamics from the synchrotron and inverse
Compton radiative loss mechanisms, and hence are not currently capable of
tackling this issue.

Conceptualisation, R.T. and S.S.; methodology, R.T.; software, R.T.;
validation, R.T.; formal analysis, n/a; investigation, R.T.; resources, R.T.;
data curation, n/a; writing—original draft preparation, R.T. and S.S.;
writing—review and editing, R.T. and S.S.; visualisation, R.T.; supervision,
n/a; project administration, n/a; funding acquisition, n/a. All authors have
read and agreed to the published version of the manuscript.

This research received no external funding.

The authors declare no conflict of interest.

no

##  Appendix A Early Jet–Lobe Models

We present a complete derivation for the lobe pressure and volume evolution of
the Scheuer (1974) model assuming a power-law ambient density profile (rather
than their assumed constant density medium), as outlined in Section 2.

###  A.1 Lobe pressure

The increase in the total energy of the cavity, $U$, over the time interval
$\delta t$ due to the input kinetic energy $Q$ is given in Equation 4. Scheuer
(1974) rewrite this first-order differential equation in terms of the jet
length by defining a constant scaling between the lobe volume and this length
of the form $V(R)=\kappa_{2}R^{\alpha}$; here $\alpha,\kappa_{2}>0$ are
constants. That is,

$\delta
U=\left[Q\frac{dt}{dR}-\alpha\frac{U(\Gamma_{c}-1)(q+1)}{R}\right]\delta R,$
(42)

where the time derivative of Equation 3 gives the jet-head advance speed,

$\begin{split}\frac{dR}{dt}&=\left(\frac{\kappa_{1}Q}{\Omega
kc}\right)^{1/(4-\beta)}\left(\frac{(4-\beta)t}{2}\right)^{(\beta-2)/(4-\beta)}\\\
&=\left(\frac{\kappa_{1}Q}{\Omega kc}\right)^{1/2}R^{(\beta-2)/2}.\end{split}$
(43)

The solution to Equation 42, upon substituting the second expression above for
the jet-head advance speed and assuming the initial condition $U(0)=0$ (i.e.
initially zero energy in the lobe), is

$U(R)=\frac{(Q\Omega
kc)^{1/2}}{[\alpha(\Gamma_{c}-1)(q+1)+(4-\beta)/2]\kappa_{1}^{1/2}}R^{(4-\beta)/2}.$
(44)

The average lobe pressure is meanwhile related to total energy in the lobe
cavity and its volume (i.e. Equation 5, or e.g. Equation 15 of Kaiser et al.
(1997)). We can therefore rewrite the above expression for the total energy in
terms of the lobe pressure, recalling $V(R)=\kappa_{2}R^{\alpha}$, as

$p(R)=\frac{(Q\Omega
kc)^{1/2}(\Gamma_{c}-1)(q+1)}{[\alpha(\Gamma_{c}-1)(q+1)+(4-\beta)/2]\kappa_{1}^{1/2}\kappa_{2}}R^{(4-\beta-2\alpha)/2}.$
(45)

This relationship is presented in Equation 6 of Section 2 as a function of the
source age upon the further substitution of Equation 3.

###  A.2 Lobe volume

The sidewards expansion rate of the lobe is derived by equating the internal
pressure to the ram pressure presented by the ambient medium as the lobe
widens; i.e. $\rho v_{\perp}^{2}=p(t)$ where the ambient gas density is
reasonably approximated as $\rho\sim kR^{-\beta}$ for somewhat spherical
lobes. As discussed in Section 2, the width of the lobe at some location $r$
along the jet axis is

$\begin{split}R_{\perp}(r)&=\int_{t(r)}^{t(R)}v_{\perp}(t^{*})dt^{*}\\\
&=\int_{r}^{R}v_{\perp}(R^{*})\frac{dt}{dR^{*}}dR^{*},\end{split}$ (46)

where $t(r)$ is the time when the jet-head reached the location $r$ along the
jet axis, and $t(R)$ is the current time (i.e. jet-head has length $R$). The
change of variables in the second equality allows the width of the lobe to be
evaluated in terms of known limits and the pressure in Equation 45. The
integral is evaluated upon substitution of Equations 43 and 45,

$\begin{split}R_{\perp}(r)&=\frac{4(\Omega
c)^{3/4}k^{1/4}}{(12-5\beta-2\alpha)\kappa_{1}^{3/4}\kappa_{2}^{1/2}Q^{1/4}}\left[\frac{(\Gamma_{c}-1)(q+1)}{\alpha(\Gamma_{c}-1)(q+1)+(4-\beta)/2}\right]^{1/2}\\\
&\quad\quad\quad\times(R^{(12-5\beta-2\alpha)/4}-r^{(12-5\beta-2\alpha)/4}).\end{split}$
(47)

The lobe volume is derived in Equation 8 of Section 2 by integrating this
expression over all locations $r$ along the jet axis.

References

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