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\shorttitle{Evolution of Global Properties of Powerful Radio Sources II}

%%\shortauthors{}


\begin{document}

\title{Evolution of Global Properties of Powerful Radio Sources II -
Hydrodynamical Simulations in a Declining Density Atmosphere and
Source Energetics}

\author{Joel C. Carvalho\altaffilmark{1} and Christopher P. O'Dea}
\affil{Space Telescope Science Institute, 3700 San Martin Dr.,
Baltimore, MD 21218}

\altaffiltext{1}{Departamento de Fisica, UFRN, C.P. 1661, CEP 59072-970,
Natal,
RN, Brazil}
%\author{Draft: \today }


\begin{abstract}

We present 2-D numerical hydrodynamical simulations of light, supersonic
jets propagating
in atmospheres which decline in density with increasing distance in
several ways --
isothermal King-law atmospheres with the power law exponent $\beta =1$
and  $\beta
= 0.75$ and in an isobaric King-law atmosphere with power law exponent
$\beta =1$. We
explore the same very broad range of parameter space in Mach number $M$
and density
contrast $\eta$ as in Paper I in this series. We compare our results
with those for the
constant density and pressure atmosphere simulations and with the
predictions of the
self-similar models (Paper I). We also discuss the global energetics of
the sources in
these different environments. Our comparison of the constant and
declining density results
shows the following. The overall morphology of the jet, cocoon and bow
shock is similar.
However, there are some differences which start to appear when the jet
has propagated
about twice the core radius. In a declining density atmosphere, (1)
there is less
structure in the cocoon due to turbulence and (2) at a given source
size,  the cocoon
seems to be under-expanded relative to a constant density atmosphere for
a broad range of
Mach numbers $\sim 5-30$ (outside this range the cocoon may actually be
wider). In an
isobaric atmosphere with density gradient the general appearance of the
source is similar
to that of an isothermal atmosphere though the source size increases
much faster. The
overall distribution of pressure in the shocked ambient gas (SAG) region
and cocoon and
its decline with distance from the jet head is similar in both types of
atmospheres.
However, in jets in declining density atmospheres, for Case 1 (pressure
dominated by
relativistic electrons), the cocoon and SAG region remain much more
over-pressured with
respect to the ambient medium than in jets in constant density
atmospheres. The lateral
expansion speed of the bow shock and cocoon are similar in the constant
and declining
density atmospheres. In a constant density atmosphere, the jet head
velocity decelerates
slowly with time, while in the declining density atmosphere, the jet
head accelerates with
time with a rate given by exponent $z_h\propto t^m$ with $1 < m < 1.5$.
However, at high
Mach number and/or high jet density, the acceleration decreases with
time and the head
velocity approaches a nearly constant velocity -- consistent with the
predictions of the
self-similar models. In an isothermal atmosphere with a density
gradient, the volume of
the region enclosed by the bow shock tends to increase with source size
in agreement with
the predictions of the Type III models, i.e., $V \propto t^3$. The
estimated volume of the
cocoon tends to vary with time possibly due to the influence of K-H
instabilities on the
contact discontinuity, but in general $V\propto t^{2-3}$. We find that
for all
environments, the behavior of the source volume as a function of time is
closer to the
self-similar predictions than the behavior of the source size as a
function of time. This
suggests that the shape of the bow shock adjusts to keep the time
dependence of the source
volume roughly self-similar. We examine the dependence of the pressure
on the source
volume as a function of the source age for the sources in different
environments. In the
constant density atmosphere, the pressure evolution of cylindrical
slices of the SAG
region departs from that expected for adiabatic expansion both near the
head and near the
jet inlet. Near the jet head, the departure is due to energy input to
the SAG region from
the jet head (i.e., there is an effective ``source term") and near the
jet inlet, the
departure is due to the source coming into pressure balance with the
ambient medium. The
average global pressure in the SAG region and cocoon vary more slowly
with volume than
predicted by the self-similar models. These results for the evolution of
the pressure are
similar for the constant density or isothermal King atmosphere, though
for the King
atmosphere there tends to be better agreement with the predictions of
the self-similar
models. In general, the energetics of the source are similar whether the
source is
propagating in an isobaric or isothermal atmosphere, with some caveats.
For both the SAG
region and the cocoon, the pressure declines more slowly with source
size in the isobaric
atmosphere than in the isothermal atmosphere with a density gradient. In
addition, the
sources in an isobaric atmosphere show somewhat larger departure from
the predictions of
the self-similar models than do the sources in either a uniform
atmosphere or an
isothermal atmosphere with declining density gradient. The behavior of
the source and the
energetics are similar in the $\beta=1$ and $\beta=0.75$ King model
atmospheres. However,
we do note that some of the global parameters are more in agreement with
the self-similar
models for $\beta=1$. The self-similar models assume that the source is
highly
over-pressured with respect to the ambient medium and thus do not
explicitly consider a
functional form for the ambient pressure. However, we have found that
the agreement of the
simulations with the predictions of the self-similar model does depend
on the degree to
which the source is over-pressured and thus, the assumed ambient
pressure profile for the
source. The agreement between the self-similar models and the
simulations tends to improve
as the atmosphere declines more steeply, i.e., from an atmosphere with
$\beta=0$ to
$\beta=0.75$ and $\beta=1$.

\end{abstract}

\keywords{ galaxies: jets -- galaxies: intergalactic medium --
hydrodynamics -- radio continuum: galaxies}

%******************************************************************************

%******************************************************************************

\section{INTRODUCTION}

The pioneering work on numerical simulations of astrophysical jets  was
carried out for a
jet propagating in a uniform medium with constant density (Norman et al.
1982; Lind et al.
1989; Clarke, Norman \& Burns 1989; Hardee \& Norman 1990; Falle 1991;
Clarke \& Burns
1991; Cioffi \& Blondin 1992; Loken et al. 1992; Mioduszewski, Hughes \&
Duncan 1997;
Marti et al. 1997; Komissarov \& Falle 1997, 1998; Rosen et al. 1999).
However, a more
realistic environment for the jet, as it emerges from the galactic
nucleus at parsec
scales or from the galaxy itself at tens of kilo-parsec scales, is one
in which it
propagates across changes in the ambient gas density. Therefore, we
should consider a
scenario where a density gradient is present and examine the
consequences for the jet
propagation and source expansion. In fact, several {\it analytical\/}
models of source
expansion have been developed which take  into account a varying density
atmosphere
surrounding the central galaxy (e.g., Carvalho 1985, 1994, 1998;
Begelman \& Cioffi 1989;
Daly 1990; Falle 1991; Begelman 1996; Bicknell, Dopita \& O'Dea 1997;
Kaiser \& Alexander 1997; Alexander 2000).

In the last decade, several authors have used numerical simulations to
study aspects of
jet propagation in a medium where the density decreases with distance
from nucleus
(e.g., De Young 1997; Hardee et al. 1992).
De Young (1997) has conducted simulations of jets in an isobaric
atmosphere with a density
gradient to study the effect of jet energy losses on the overall source
propagation
properties as it interacts with the ambient gas. He studied compact
sources with jets
propagating on scales $< 3$ kpc. The results are in agreement with the
suggestion by
Readhead et al. (1996) that the evolution of compact symmetric and
medium symmetric
objects\footnote{Readhead and collaborators have dubbed small ($< 1
kpc$) and intermediate
size ($ 1-10$ kpc) powerful classical doubles as Compact Symmetric
Objects and Medium
Symmetric Objects, respectively. For a review of their properties see
O'Dea (1998).} is
non-self-similar.

In a more detailed study, Hardee et al. (1992) carried out 2D
simulations of slab jets
with broken symmetry propagating in a variety of environments including
pressure, density
and temperature gradients - though with fairly shallow density and
temperature gradients.
They showed that the cocoon can be diminished or eliminated in
atmospheric gradients and
the trailing part of the jet (i.e., near the origin) may interact
directly with the
ambient gas. They have also noted that propagation in the presence of a
density gradient
can result in a tail-like source morphology.

Simulations have also been performed by Loken et al. (1993) of perturbed
jets of radio
sources in dominant galaxies in a cluster cooling flow. In addition,
there have been
several studies whose main objective was to examine the effect on the
jet of crossing the
interface between the interstellar medium or extended halo and a
constant low density
intergalactic or intracluster medium. An extensive investigation of the
jet propagation
properties in such an environment has been conducted in a series of
papers by Wiita, Rosen
\& Norman (1990), Wiita \& Norman (1992), Hooda, Mangalam \& Wiita
(1994) and Hooda \& Wiita (1996, 1998).

Other authors have investigated the propagation of radio sources into
the intracluster
medium (ICM) in an attempt to explain the X-ray cavities and shells
observed around
cluster galaxies. Carilli, Perley \& Harris (1994) proposed that the
X-ray cavities
observed in X-ray surface brightness distribution around Cygnus A are a
consequence of the
radio source expansion and emission from the compressed gas. Clarke,
Harris \& Carilli
(1997) performed full 3D numerical simulations in an isothermal King
atmosphere and
calculated the X-ray brightness distribution. They showed that the
expansion of the jet
and radio lobe produces an evacuated cavity and that the compression of
the ambient gas
between the cocoon and bow shock can explain the X-ray bright feature
observed, for
example, in the {\it ROSAT} data of Cygnus A.

Heinz, Reynolds \& Begelman (1998) presented a simple analytical model
for the interaction of the expanding radio galaxy with the external
medium which predicts the X-ray appearance of radio galaxies as a
function
of the source age. Reynolds, Heinz \& Begelman (2001) studied the same
problem using 2D axisymmetric numerical simulations and showed that the
supersonically expanding cocoon produces a shell/cavity structure as
predicted by the model. They also found that even in sources where the
expansion speed is subsonic, a shell structure exists because the
cocoon has not had time to collapse and the shell is the result of the
emission from a weak shock or sonic boom driven by the supersonic motion
of the jet head.

One might also expect the expansion of the radio source through the
intracluster medium to
be asymmetric as it encounters a clumpy medium around the host galaxy.
The interaction of
the jet and expanding cocoon with clouds has been studied by several
authors both
analytically and numerically (De Young 1991; Norman 1993; Raga \&
Cant\'o 1996; Steffen et
al. 1997; Carvalho 1998; Higgins et al. 1999; Wang, Wiita \& Hooda
2000). It is found that
clouds can either slow down and/or disrupt powerful jets or, in low
power
jets, they can even halt or destroy the jet. The later can also be bent
by dense clouds.
These results could explain some observed morphological features of some
GPS and CSS
sources (e.g., O'Dea 1998). The jet/cloud interaction may also be
relevant for
the generation of optical emission lines in the narrow-line region
through the
compression of clouds.


In Carvalho \& O'Dea (2001a, hereafter Paper I) we presented the results
of 2-D
axisymmetric hydrodynamic simulations of supersonic jets propagating in
a constant density
atmosphere. We explored a very wide range of jet parameters in order to
develop a better
understanding of the evolution of the global properties of the source
and its dependence
on the jet Mach number and density contrast. Our goal is to characterize
the evolution of
global properties of the radio source and use the insight gained from
this to construct an
improved analytical model for radio source propagation. Our simulations
of  jets
propagating in a  uniform atmosphere allowed us to determine the
behavior of the jet and
cocoon expansion unaffected by any external density variations. In
particular, we examined
the evolution of the jet size, the advance speed of the jet head, the
head size, and the
lateral expansion of the cocoon and bow shock. We also compared the
results of the
simulations with the predictions of analytical self-similar models.

In this paper, we present the results of the simulations carried out for
the
same set of jet parameters used in Paper I but in a quasi-power law
density
atmosphere similar to King's law. We shall be referring to Runs A, B and
C and
the reader should refer to Table 1 of that paper. We also investigate
the
energetics of the source and how it evolves with time for the runs
presented in
Paper I (constant density) and for those studied here.

In Section $\S$~\ref{propagation} we present the results of the
numerical hydrodynamical
simulations in an atmosphere with a density gradient. We present a
description of the
structure and dynamics of the jets and compare them with the simulations
in a constant
density atmosphere. In Section $\S$~\ref{global} we compare the results
on the jet head
propagation and the evolution of the volume with the self-similar models
defined in Paper
I. In Section $\S$~\ref{isobaric} we present our results for propagation
of the jets in a
isobaric atmosphere with a density gradient and compare them to the
isothermal atmosphere
with a density gradient. The energetics of the source is studied in
Section
$\S$~\ref{energetics} for the jet propagating in both a constant density
and density
gradient atmosphere. In Section $\S$~\ref{summary} we summarize our
results and present
our conclusions.


%******************************************************************************

%******************************************************************************

\section{JET PROPAGATION IN A POWER LAW DENSITY ATMOSPHERE}
\label{propagation}

\subsection{Numerical setup}

X-ray observations have revealed the presence of extended halos of hot
gas around bright
early-type elliptical galaxies (e.g., Forman, Jones \& Tucker 1985;
Trinchieri \& Fabbiano
1985; Fabbiano 1989). These quasi-isothermal halos can extend as far as
100 kpc from the
nucleus. The gas density derived from the radiation surface brightness
distribution is
compatible with the following model
\beq
\rho_a = \rho_{ao} \left[1+{r^2\over r_{o}^2}\right]^{-\beta},
\label{king}
\eeq
where $r$ is the distance to the galactic center, $\rho_{ao}$ is the
central
densities, $r_{o}$ is the characteristic radius of the distribution
which is in
the range 1 - 10 kpc.

In this work, we adopt the above density profile to describe the
environment of the jet
outside the central  galactic region. Most of our simulations adopt
$\beta = 1.0$  in
order to allow direct comparison with analytical models which assume
$\beta = 1.0$. The
X-ray observations indicate that the exponent $\beta\sim 0.75$ (Forman,
Jones \& Tucker
1985; Canizares, Fabbiano \& Trinchieri 1987). We have run five
simulations with $\beta =
0.75$ and our comparison (\S~\ref{beta75}) shows that the numerical
results are
consistent. In most of our simulations, we include a pressure gradient
constructed to
insure that the jet propagates in an isothermal atmosphere. Requiring a
constant
temperature implies the pressure is given by
\beq
P_a = P_{ao} \left[1+{r^2\over r_{o}^2}\right]^{-\beta},  \label{kingP}
\eeq
and to keep the atmosphere stable we introduce a suitable effective
gravitational potential.


We have also run simulations in a constant pressure atmosphere for all
corresponding
isothermal atmospheres and present some of the results. It is likely
that ``reality" is
neither perfectly isothermal nor isobaric, but somewhere in between.
Hopefully, our
simulations will serve as a guide to radio source behavior in
environments with gradients
in density, pressure and temperature.

In Paper I we performed an analysis of the classical equivalent of
relativistic jets and
showed how to calculate the corresponding classical Mach number ($M_c$)
and
jet-to-ambient-gas density ratio ($\eta_c$) for a given Lorentz factor
of the relativistic
jet. We called our set of simulations of classical equivalent jets Run A
and further
divided it into two cases: Case 1, where  the relativistic particle
content of the jet is
electrons (and positrons) and Case 2, where it is electrons plus
protons. A second set of
simulations (Run B) was run to study the effects of systematically
varying first the Mach
number at a fixed value of $\eta_c$ and, second, varying the density
ratio at a fixed
$M_c$. Finally, in a third set called Run C we varied only slightly one
of the parameters
while keeping the other fixed in order to investigate how sensitive the
results of the
simulations are to slight changes in $\eta$ and $M$.


Table 1 gives a summary of the input properties of the simulations. In
order to allow
comparison with the results of Paper I, we have run our new simulations
of jets
propagating in atmospheres declining with $\beta=1.0$ with the same
values of  Mach number
and density ratio (see Table 1 of Paper I).
Specifically,  the values of $M$ and $\eta$ in
Runs A11-A20 correspond to those in Runs A1-A10 in Paper I, and the
input values in Runs
B11-B20 correspond to those of Runs B1-B10,  while the input values in
Runs C11-C16
correspond to those of Runs C1-C6. In all these simulations, the
atmosphere is isothermal
and the pressure has a gradient proportional to the density gradient,
that is, $\nabla P/P
=\nabla\rho/\rho$. Runs X11-X20, Y11-Y20 and Z11-Z16 correspond to an
isobaric atmosphere with the same density gradient and have the same
Mach number and
density ratio of Runs A11-A20, B11-B20 and C11-C16, respectively. Our
simulations with an
isothermal atmosphere with a density gradient with $\beta=0.75$ are
presented in runs Runs
N11-N15 where  $M$ and $\eta$ are the same as those in Runs A11-A15.
%%%
Runs A11-A15, X11-X15 and N11-N15 correspond to Case 1 (see Paper I) for
which the jet
relativistic particle content is electrons (and positrons) while Runs
A16-A20, X16-X20
correspond to Case 2, that is, electrons plus protons.
%%%

We adopted a value of 5 kpc for the core radius $r_o$ (consistent with
X-ray observations,
e.g., Fabbiano 1989). The other input values are the same as in Paper I
- central ambient
medium temperature $\sim 4.8\times 10^7$ K and density $\rho_{ao}=1$
cm$^{-3}$ and a jet
radius of 500 pc. In Fig. \ref{ASNA_All1} and Fig. \ref{ASNA_All2} we
present the contours
of density for Run A at the time the bow shock front has nearly reached
the end of the
grid. The size of each panel in the figures is 80 kpc $\times$ 20 kpc.
Fig.
\ref{ASNA_View1} shows the time evolution of simulations for jets with
$\gamma_j=2$, 2.5,
5 and 7 in Case 1 (Runs A11-A14). In Fig. \ref{ASNA_View2} we show the
corresponding
simulations for Case 2 (Runs A16-A19). The time in the panels is given
in Myr.
Unfortunately, because of problems during the numerical computations,
Run B11 could only
be followed up to a distance of $\sim 40$ kpc.

%******************************************************************************

%******************************************************************************

\subsection{Structure and dynamics of the simulated jets}
\label{sec-struct}

Here we discuss the overall morphology of the sources and the
distribution of pressure and
lateral expansion speed and compare them with our results in Paper I. We
find that at a
given power, the jet advances at a higher speed in an declining density
atmosphere. Thus,
at a given source size, the ages are significantly younger when compared
with jets in a
constant density atmosphere (cf.  Fig.'s 3 and 4 of Paper I). Note that
the difference in
ages does not affect our analysis since we are primarily interested in
the evolution as a
function of source size and not age.

We also note that out to about 10 kpc (the first panel of each run in
Fig.'s
\ref{ASNA_View1} and \ref{ASNA_View2}), the appearance and age of the
source is similar in
the constant (Fig.'s 3 and 4 of Paper I) and declining density
atmospheres. Thus, the
effect of the density gradient is not seen until the jet has propagated
out to of order
twice the core radius. Beyond that scale, the differences become
increasingly apparent.

In general, {\it at a given size}, the axial ratio of the source in the
constant and
declining density atmospheres are similar -- though with some exceptions
which are a
function of Mach number. (1) At high $M$ ($\gae 30$) the bow shock is
wider than in the
constant density atmosphere. This is due to the fact that the jets and
the head are both
wider at high Mach number in the declining atmosphere (compare Fig. 2
and 3 in of Paper I
with Fig. \ref{ASNA_All1} and \ref{ASNA_View1} in this paper). (2) At
low $M$ ($\lapprox
5$) the cocoon is wider than in the constant density atmosphere. (3) At
intermediate Mach
number (5-30) the cocoon seems to be somewhat less inflated compared
with the constant
density atmosphere (compare  Fig. 4 of Paper I with Fig. \ref{ASNA_All2}
and
\ref{ASNA_View2} of this paper). In addition,  comparing sources at the
same {\it age\/}
in  Fig. 15 of Paper I and Fig. \ref{ASNA_Vc_T} in this paper, we see
that the volume of
the cocoon is larger in the declining density due to the fact that the
source has
propagated much further. Since the  width of the cocoon is also
narrower, the aspect ratio
of the radio emission (assuming it comes from the cocoon) will appear to
be smaller.

Figs. \ref{ASNA_All1} - \ref{ASNA_All2} show that the cocoon exhibits
much less structure
due to turbulence than in the constant density atmosphere. In general,
the turbulence is
relatively underdeveloped. Although we did not test for the presence of
back-flow, the
simulations carried out by Hardee et al. (1992) indicate that the
back-flow is suppressed
in an atmosphere with density gradients even at the low Mach numbers
they adopted. The
lack of back-flow would explain the relative narrowness of the cocoon in
our simulations
for Mach $5-30$ jets.

Similar to our results in Paper I, the bow shocks in Case 2 are wider
than in Case 1 in a
declining atmosphere. In addition, matching the runs in bow shock width
(compare Run A7
and A8 with Runs A1 and A2 in Fig. 4 and 3 of Paper I and Run A17 and
A18 with Runs A11
and A12 in Fig. \ref{ASNA_View2} and \ref{ASNA_View1} in this paper)
shows that
independent of the environment, the cocoons are narrower in Case 2. The
presence of
relativistic protons in the jet (Case 2) makes it heavier (see Eq. (5)
and Fig. 1 of Paper
I). In addition, in a declining atmosphere, the density ratio at the jet
head increases as
it advances making it even heavier with respect to the ambient medium.
Consequently, the
cocoon is under-inflated. Also, in a declining atmosphere, the partial
suppression of the
back-flow contributes to causing the cocoon to be under-inflated. The
consequent partial
nakedness of the jet de-focuses it (making it wider along its entire
length - contributing
to the apparent size of the cocoon) and makes the head radius larger and
thus, the bow
shock wider.

As we noted above, the radio source properties are not strongly affected
by the existence
of the gradients in the ambient medium  until the source has propagated
about twice the
length of the core radius. In particular, the jet internal shock
structure near the
nucleus seems unchanged by the existence of the density and pressure
gradient. We have
carried out the same analysis as in Paper I (\S 3.2) on the Wilson \&
Falle (1985)
suggestion that the position of the first shock $z_s$ can be used to
estimate the jet Mach
number, and we find the results essentially unchanged. We have estimated
$z_s$ by
determining the position of the first most significant maximum of the
jet density and
plotted the results against the jet response scale $\ell$ for Run B and
A respectively. We
used the input value of the jet radius, that is, $r_j=500$ pc in $\ell
\simeq 2r_jM$ and
the straight line represents $z_s = \ell$. As in the constant density
atmosphere, the
points tend to lie near the line, although the agreement is better for
small Mach number.
For high Mach number the position of the first knot is much smaller than
the predicted and
this can be attributed to the fact that the average radius of the jet
tends to decrease
with increasing Mach number. If we take the average value of $r_h$ over
a distance of the
order of the jet response scale and substitute this corrected value of
the jet radius in
the expression for $\ell$, a new value is obtained. With the exception
of the high Mach
number jets ($M>20$), the position $z_s$ is in agreement with that
predicted by the
argument that it should be of the order of the jet response scale to
changes in the
external pressure. Using this approach and assuming the shock structure
is associated with
the knots observed in extragalactic jets, in Paper I we estimated $M\sim
13$ for the jet
in Cygnus A. The present simulations demonstrate that this result still
holds when there
is an external density gradient.

%******************************************************************************

%******************************************************************************

\subsection{Pressure Profile and Expansion Speed}
\label{PV}


Fig. \ref{ASNA_Pres1} shows the pressure profile along the direction of
the jet axis for
the cocoon (thick line) and in two regions inside the bow shock: (1) the
outer SAG region
defined as $R_b/2 < R <R_b$ (thin line) and (2) the inner region defined
as $R_c<R<R_b/2$
(dashed line). Note that in an isothermal atmosphere $P_a$ is a function
of the distance
from the origin and therefore, we are plotting the ratio
$P(R,z)/P_a(R,z)$, where $R$ is
the radius of the bow shock or the cocoon at a distance $z$ along the
jet axis. In those
sections of the source where the bow shock has left the simulation grid,
we plot only the
pressure in the inner SAG region. The general behavior is similar to
that in the jets in a
constant density medium with the pressure being very high near the head
and falling
steadily towards the central object (jet inlet). As happens in the
constant density
atmosphere, the pressure in the outer SAG region is higher than in the
inner section which
is comparable to the cocoon pressure.

We find that as in Paper I, the pressure falls much faster in Case 2
than in Case 1, and
that the SAG region and cocoon remain over-pressured in Case 1 over most
of the length of
the simulated jet, whereas in Case 2 they quickly return to the ambient
pressure. This
appears to be due to the fact that in Case 2, the power supplied by the
jet is lower and
thus, the total internal energy in the cocoon is also lower. Since the
cocoon pressure
near the head is relatively high, the pressure must drop faster (in Case
2) in order to
yield a volume integral (total internal energy) which is smaller. One
major difference is
that in the jets in declining density atmospheres, the cocoon and SAG
region are much more
highly over-pressured than in the constant density atmosphere (see Fig.
6 of Paper I);
while in Case 2, the pressure declines in a similar way in both types of
atmospheres. We
also note that in both Case 1 and 2 the pressure near the head is
significantly higher in
the sources in declining density atmospheres.

The lateral expansion speed of the cocoon as well as of the bow shock
(not shown here)
follows the same general trend as in the constant density atmosphere. It
rises at the head
and then falls sharply and smoothly away from the head. Though, at later
times, the cocoon
expansion speed starts varying randomly as the result of
Kelvin-Helmholtz instabilities
that develop at the contact discontinuity. Excluding a small portion
near the head, the
average expansion speed of the cocoon is essentially zero for most of
its length giving
the cocoon its cylindrical shape with an almost constant width. In the
constant density
atmosphere the cocoon maintains a cylindrical shape but slowly expands
because the average
speed is slightly greater than zero. The bow shock lateral expansion
speed decreases very
slowly but always remains above zero (as in the constant density
atmosphere) which gives
it a round conical shape with an opening angles inversely proportional
to the jet Mach
number.

%****************************************************************************

%****************************************************************************

\section{GLOBAL PROPERTIES OF THE SIMULATIONS}
\label{global}
In this Section we describe the results from our simulations on the
behavior of the source size, head radius, advance speed, and
source volume. We compare them to the predictions of the self-similar
models. Our results are summarized in Table 2.


\subsection{The jet expansion and self-similarity}
\label{jss}

The self-similar models discussed in Paper I that apply to the
non-constant density
atmosphere are Type II and Type III with $\delta=2$. Both model types
predict that the jet
should propagate with constant speed and hence the source size $z_h$,
that is, the
distance of the head to the parent galaxy (jet inlet) is proportional to
$t$. These models
employ a pure power law to describe the density distribution whereas we
use the King law
(\ref{king}) which is not directly comparable. However, since the
characteristic radius we
have taken is 5 kpc, our distribution should be almost identical to a
power law for
distances from the origin greater than $\sim 10$ kpc.

We have plotted $z_h$ against $t$ and performed a least squares fit to
the
function $z_h\propto t^m$
for the points with $z_h > 10$ kpc. We find that the fit has a slope $m$
that is always
larger than the value of unity which is predicted by the models.
However, as we have
pointed out in Paper I, a single fit over all ``observed" points is not
sufficient
to test
whether the jet evolves towards a self-similar regime. Thus, using the
two methods described in
Paper I (\S 5.1) we have calculated the exponent $m$ as a function of
the source size. The
results are shown in Fig. \ref{ASNA_Similar_Xh} for Runs A11-A20. The
figures confirm what
is suggested by the overall fit, i.e., the source is accelerating
($m>1$) although, over most of the parameter space, at
least within the range of the simulations, the acceleration is taking
place
at a decreasing rate ($m\lapprox 2$).

The assumption of models Type II and Type III with $\delta=2$ is that
the area of the head
increases as $t^2$ so that, if the jet luminosity is kept constant, the
head pressure
should fall as $t^{-2}$. Under ram pressure equilibrium, this
compensates the power law
decrease of the ambient gas density, resulting in a constant advance
velocity. Our
simulations provide some evidence that this is true at high Mach number.
(1) For $M=55.7$
(Run A15), the exponent $m$ is near the value 1 and for Runs A13 and
A14, $m$ approaches 1
as the source size increases. (2) For the highest value of $M$ in Case 2
(Run A20), the
exponent $m$ is also close to 1. (3)  For Runs A16 and 17, after an
excursion away from
the self-similar value, the exponent $m$ approaches this value at the
very end of the
simulation. However, it is unknown whether the exponent will remain at
this value at
larger source sizes.

We also find  that the behavior of the exponent is much more
``unsettled" than in a
constant density atmosphere, especially for Mach numbers less than $\sim
10$. Because
these sources have already propagated over 150 jet radii at the end of
the simulation, and
since they are being driven by low power jets, it is unlikely that they
will reach a
self-similar stage before their jets become disrupted in such a way as
to cease supplying
the necessary energy to push them forward (see Paper I \S 6).

Thus, it appears that in a declining density atmosphere, high Mach
number jets do evolve
towards a regime where their size is increasing roughly linearly with
time (as predicted
by the self-similar models). The jets in our simulations are nearly
always accelerating --
though the acceleration tends to drop with increasing source size at
high $M$.

%******************************************************************************

%******************************************************************************

\subsection{The head radius}
\label{sec-Rh}

The radius of the jet and cocoon at the head $r_h$ is shown in Fig.
\ref{ASNA_Rh_Xh} as a
function of the source size for Runs A11-A20. The open circles represent
the cocoon
radius, the solid circles the jet radius and the solid line the radius
of the head
calculated from the advance velocity of the simulated jet using Eq.
\ref{vh} below (which
assumes ram pressure balance). We see that the calculated head radius
closely  follows the
cocoon and jet radius, demonstrating that the hypothesis that the jet
advances in pressure
equilibrium with the external medium is a good approximation. Exceptions
are the runs
where the Mach number is very low and the ram pressure becomes of the
order of the ambient
pressure (see section 5.3 of Paper I). This can also be seen in Run A16
where the terminal
radius of the jet is not well defined (see Fig. \ref{ASNA_View2}). In
general, the
behavior of the head radius is similar to that of the constant density
atmosphere.  After
the source has propagated beyond 10 - 20 kpc, the head radius starts
increasing with
distance and develops the characteristic quasi-periodic oscillations due
to the modulation
produced by the ``vortex-shedding". Note however, that in a declining
density atmosphere,
the head radius increases faster with increasing source size than in a
constant density
atmosphere (compare Fig. 11 of Paper I with Fig. \ref{ASNA_Rh_Xh} in
this paper).


The dependence of the average value of $r_h$ upon the Mach number is
shown in Fig.
\ref{ASN_Rmed_ABC}a where the triangles and open circles represent the
average radius of
the jet head and cocoon, respectively. The solid circles represent the
head radius
calculated from the head speed in the simulations using Eq. \ref{vh}.
Here we notice some
differences compared with the constant density atmosphere (see Fig. 12a
of Paper I). In
Paper I, the head radius remained almost constant up to $M\sim 10$ and
then decreased with
increasing $M$; while, here  we see that for low Mach number, $r_h$
decreases with
increasing $M$ until $M\sim 10$, remains  approximately constant up to
$M\sim 25$ where it
rapidly drops and then begins increasing again. It should be pointed out
that, although
the observed behavior of the average value of $r_h$ seems quite complex,
the variations
occur in a consistent manner for all three radii displayed in the
figure. It is also
generally true that the average value of $r_h$ is larger than in the
constant density
atmosphere.

The rate at which the head radius increases as the jet advances is
plotted in Fig.
\ref{ASN_Rmed_ABC}b, where the vertical axis represents the exponent $d$
calculated by
fitting a curve of the type $r_h\propto z_h^d$. Again, there is a major
difference with
respect to the constant density atmosphere in that, with very few
exceptions, the exponent
$d$ is always positive (compare with Fig. 12b of Paper I) for the jet,
the cocoon and the
radius calculated from the advance speed. The growth rate is larger than
in the constant
density atmosphere. Although Fig. 12b of Paper I and Fig.
\ref{ASN_Rmed_ABC}b of this
paper show no evident trend in the growth rate as a function of $\eta$,
the dispersion in
the points is much larger for jets in a declining atmosphere. This
contributes to the
apparently complex behavior of the head radius as a function of $M$
displayed in Fig.
\ref{ASN_Rmed_ABC}a.

The larger growth rate of the jet and cocoon head in the declining
atmosphere appears to
be due to a number of factors which contribute to reduced collimation:
(1) the narrowness
of the cocoon, (2) the reduced presence of turbulent motion within the
cocoon, and, (3)
the near nakedness of the trailing part of the jet (at very high Mach
number).


%******************************************************************************

%******************************************************************************

\subsection{The advance speed of the source}
\label{sec-Vh}

Here we present our results for the advance speed of the source and its
dependence on $M$
and $\eta$. The velocity of the jet head can be calculated assuming ram
pressure balance
between the head of the source and the ambient medium (Cioffi \& Blondin
1992; Loken et
al. 1992; Hardee et al. 1992; Frank et al. 1998)
\beq
v_h=v_j \left({\mu\sqrt{\eta}\over 1+\mu\sqrt{\eta} }\right),
\label{vh}
\eeq
where $\mu=r_j/r_h$ and $\eta =\rho_j/ \rho_a$.

In Fig. \ref{ASNA_Vh_Xh} we show the advance speed of the source as a
function of its size
$z_h$ (open circles) for Run A. We  have also calculated the advance
speed by adopting for
$r_h$ in (\ref{vh}) the measured jet and cocoon radii from the
simulations shown in Fig.
\ref{ASNA_Rh_Xh}. Fig. \ref{ASNA_Vh_Xh} shows good agreement between the
calculated and
simulated advance speeds implying that the assumption of ram pressure
balance is correct.
A plot for Run B (not shown here) shows a similar result.

According to Eq. \ref{vh} we would expect that $v_h$ will increase with
$z_h$ because, as
the source propagates through a declining atmosphere, the density ratio
$\eta$ will also
increase. This is consistent with  our results at low and moderate Mach
number (e.g., Runs
A11, A12, A18, A19, B13 and B14, $20 < M$) where the advance speed
increases steadily with
increasing source size. Consequently, the source size will increase
according to
$z_h\propto t^m$, with the exponent $m$ always being  greater than 1 as
shown in Fig.
\ref{ASNA_Similar_Xh}.

On the other hand, at higher Mach number (e.g., Runs A13, A14, A15, A20
and B15), $v_h$
tends to asymptote to a roughly constant value despite the continuing
increase in $\eta$.
We attribute this to the fact that, at high $M$, the head radius
increases at such a rate
(\S~\ref{sec-Rh}) that the decrease in the ratio $\mu=r_j/r_h$
compensates  for the
increase in $\eta$.

In addition,  plotting $v_h$ for Runs B16-20 shows that, for a fixed
$M$, the
acceleration of the head clearly decreases as the jet initial density
ratio increases from
0.003 to 0.3. A plot of $r_h$ for these runs (not shown) reveals that
the head radius is
almost constant for $\eta=0.003$ and at higher $\eta$, the head radius
increases with
source size at a rate proportional to $\eta$. This supports our
hypothesis that, the
decreasing ratio $\mu=r_j/r_h$ compensates for the density variation
causing the advance
speed to decelerate.

In our study of jets in a constant density atmosphere (Paper I, \S~5.1)
we were able to
identify three ``phases" of jet head propagation whose length is
dependent on Mach number
- an initial phase of constant advance speed, a second phase of
deceleration with exponent
$m\simeq 3/5$ until the source reaches the ambient sound speed, and a
third phase of
constant advance speed. In contrast, in a declining atmosphere the jets
are generally
always accelerating at least slightly. Thus, the jets never experience
the second and
third phases seen in the constant density atmosphere.

Thus, it appears that at low and intermediate Mach number ($M < 20$) the
jets we have
simulated tend to keep accelerating in a declining density atmosphere
and are not
consistent with the self-similar (Type II and III) models. However, the
high $M$ ($\gae
20$) sources do seem to evolve towards  constant advance speed and so do
eventually become
more self-similar in their behavior.\footnote{As we noted in
\S~\ref{jss}, the propagation
is not {\it exactly\/} self-similar since the exponent $m$ in
$z_h\propto t^m$ is not
exactly equal to unity.} It is an open question whether real radio
sources have Mach
numbers higher than 20, since we estimate  that $M\sim 13$ in Cygnus A
(\S~\ref{sec-struct}).



%******************************************************************************

%******************************************************************************

\subsection{Source volume}
\label{sec-Vol}

We have calculated the volume of the cocoon and of the region enclosed
by the bow shock
(whole source) and in Fig. \ref{ASNA_Vc_T} we show the results for the
cocoon as a
function of time. The solid line represents the least squares fit to the
function
$V\propto t^n$ for points with $z_h > 10$ kpc. In order to compare this
with the
prediction of the self-similar models (see Paper I) we have also plotted
the dotted line
representing $V\propto t^2$ (Type II) and the dashed line representing
$V\propto t^3$
(Type III). For the whole source, the fit (not shown here) is generally
in agreement with
the prediction of Type III models ($n=3$) whereas for the cocoon, the
exponent of the
fit lies between $n=2$ and $n=3$.

However, in order to create a more detailed picture, we have also used
the methods
described in Paper I \S~5.1 to examine how the exponent $n$ changes as
the source expands.
The results are shown in Fig. \ref{ASNA_Similar_Vb} and Fig.
\ref{ASNA_Similar_Vc} for the
whole source and cocoon, respectively. For moderate and high $M$ we find
that the exponent
$n$ tends towards the value predicted by Type III models and this
happens relatively early
during the source expansion. Notice that, as we have shown in section
\ref{jss}, the
exponent $m$ in $z_h\propto t^m$ is not always equal to 1 as predicted
by these models.
This implies that the shape of the bow shock adjusts itself to bring
about a volume that
scales roughly as $t^3$ as required by the self-similar models.

The exponent $n$ of the cocoon does not converge and continues to vary
as the source
expands similar to the jet in the constant density atmosphere. The
variability in the
exponent is due to the K-H  instabilities that develop on the contact
discontinuity
surface causing the cocoon volume (as estimated in our simulations) to
fluctuate. Over
much of parameter space, the exponent $n$ oscillates between 3, the
value predicted by
Type III models and 2, its value in Type II models, and sometimes
becomes larger than 3.
An exception is Run A11 for which $n\sim 3$.

Thus, although the time dependence of the source volume does not agree
in detail with the
self-similar models, it does tend to evolve with time to become close to
$V\propto t^3$
(Type III). On the other hand, the cocoon volume seems to grow more
slowly with an
exponent between 2-3, though there is some uncertainly due to the effect
of K-H
instabilities on our ability to determine the cocoon volume.

%******************************************************************************

%******************************************************************************

\section{JET PROPAGATION IN AN ISOBARIC ATMOSPHERE WITH A DENSITY
GRADIENT}
\label{isobaric}

We have run all the examples shown in Table I (Runs X, Y and Z) for a
jet
propagating in an isobaric atmosphere with a density gradient. The
general
morphology of the
source is quite similar to that of the isothermal atmosphere with
declining density; i.e.,
compared to the uniform atmosphere the bow shock is over-expanded while
the cocoon seems
less inflated. Similarly, at a given age, the volume of the cocoon in
the isobaric
atmosphere is larger than in the uniform density atmosphere by a similar
factor as the
isothermal atmosphere with declining density. In addition, the cocoon in
the isobaric
atmosphere also exhibits less turbulence than in the constant density
atmosphere.

Plotting the average pressure inside the SAG region and cocoon as a
function of distance
along the jet axis, we find that the general behavior is similar to that
in a
constant density
atmosphere although with some differences. The pressure is high near the
head and
decreases as one approaches the jet inlet. It falls more rapidly inside
the cocoon than
inside the SAG region and the cocoon only stays over-pressured near the
jet head except for  very powerful jets (Runs
X13 and X14) and at the early stages of the
propagation. The most striking difference is that the pressure near the
head of both the
SAG region and cocoon is lower and that it falls more rapidly to levels
that are also much
lower than in the constant density atmosphere. Only for very high Mach
number is the pressure
inside the SAG region kept significantly above the ambient pressure for
most of its
length. In general, the cocoon is also under-pressured with respect to
the ambient medium
and, away from the head, the values of the cocoon pressure are
essentially the same as
that in the inner portion of the SAG region. In addition, the lateral
expansion speed
follows the same pattern described in \S~\ref{PV} for the isothermal
atmosphere.

We repeated the same analysis here as in  \S~\ref{jss} in order to study
the behavior of
the source size with time. The prediction of both Type II and III models
is that $z_h$
will scale as $t$. We used the two methods described in Paper I \S~5.1
to calculate the
exponent $m$ in $z_h\propto t^m$ as a function of the source size. We
find that, for
example, for Runs X11-X20, the source expansion in the isobaric
atmosphere is essentially
always accelerating ($m \gae 1$) and in fact is accelerating even faster
than
the isothermal atmosphere with declining density. We shall discuss this
further when we analyze the advance speed of the source head.

As in the isothermal atmosphere, the head radius also tends to increase
as the source
advances (not shown) but the behavior of its average value as a function
of the Mach
number exhibits some differences. We observe a slight tendency for $r_h$
to increase for
low Mach number ($M<6$), to decrease for moderate values of $M$, up to
$M\sim 25$ and then
to start increasing again. The variation of the average value of $r_h$
is not very large,
dropping, for example, by a factor no greater than 3 between the maximum
at $M\sim 6$ and
the minimum at $M\sim 25$. Nevertheless, it does so consistently for
both the jet and
cocoon. The overall average value of $r_h$ is larger than in the
constant
density atmosphere. We have fit the increase of the head radius with the
source size to
the function $r_h\propto z_h^d$. A plot of the exponent $d$ as a
function of the Mach
number shows that, except for $M=2$, the exponent $d$ is always positive
and is larger
than in the uniform atmosphere.

Eq. \ref{vh} implies the advance speed $v_h$ should increase with $z_h$
since the density
gradient causes the density ratio $\eta$  to become larger with
increasing distance from
the jet inlet. Examining the advance speed of the source as a function
of its
size $z_h$ for (Runs X11-20), we find that there are departures from
this behavior at
large distance, especially for higher $M$.

In the isobaric atmosphere the ambient sound speed increases linearly as
$v_s\propto z_h$.
We find that for very low values of $M$, the advance speed of the source
rapidly
intersects the sound speed. Since the source advance must remain above
the ambient sound
speed, the low $M$ sources enter a phase where $z_h$ increases
exponentially with time -
i.e., we observe a dramatic increase of the exponent $m$ in $z_h\propto
t^m$ for
Runs X16 and
X17 with $M= 3.3$ and 4.4, respectively. In a real radio source, this
exponential behavior
would end when the source stalls (Paper I, \S~6)  or when it has
propagated to the end
of the galactic halo and entered a presumably constant density
intracluster or
intergalactic  medium.

We can summarize by identifying three types of behavior for the source
expansion along the
jet axis in an isobaric atmosphere with a density gradient. As in the
constant density
atmosphere, they are characterized by the initial jet Mach number.

For low Mach number the head advance speed is also low and soon it
intersects the
increasing ambient sound speed $v_s$ as in runs X16 and X17, after which
$v_h$ increases
linearly with $z_h$. As this happens, the source starts to expand
exponentially.
For intermediate values of the Mach number (Runs X11, X12, X18, X19 and
X20), $v_h$
remains greater than $v_s$ through the entire simulation and still
increases with $z_h$
with an exponent less than 1. As a consequence, the size of the source
increases as
$z_h\propto t^m$, and the exponent $m$ being always greater than 1.

The third type of behavior occurs at high Mach number (Runs X13-X15).
Here, $v_h$ is
approximately constant and much larger than the ambient sound speed and
the source expands
almost linearly with $t$ ($m\sim 1$). This is similar to the isothermal
atmosphere
(\S~\ref{vh}) and the constant density atmosphere (Paper I) where this
behavior occurred
in phase one. As in the isothermal atmosphere, the head radius expands
at a
rate which counterbalances the negative density gradient causing the
head advance speed to
remain approximately constant.
Similar behavior is also seen in  Runs Y11-Y20.


%******************************************************************************

%******************************************************************************

\section{ENERGETICS OF THE SOURCE EXPANSION}
\label{energetics}

Here we discuss the energetics of the source in both the constant
density and declining
density atmospheres making use of our results in Paper I and this paper,
respectively. We
also compare the results of the simulations with the predictions of the
self-similar
models. Our results are summarized in Tables 2 and 3.

%******************************************************************************

%******************************************************************************

\subsection{Constant density atmosphere}
\label{energeticsU}

\subsubsection{Distribution of Pressure in the Bow Shock and Cocoon}


We have seen in \S 2.3 and  3.4 of Paper I that the pressure
distribution is not uniform
inside the cocoon and SAG region; i.e., there are pressure gradients
both parallel and
perpendicular to the jet axis. Here we examine how the pressure  evolves
as the source
expands.

The usual approach found in the literature to the problem of the
expansion of the lobe of
radio sources is to consider an adiabatic expansion and write the energy
equation as
(Begelman \& Cioffi 1989; Loken et al. 1992; Cioffi \& Blondin 1992;
Nath 1995; Begelman
1996)
\beq
{dE\over dt} = L_j,                     \label{e0}
\eeq
where $E$ is the total internal energy of the region and $L_j$ is the
energy input rate,
that is, approximately the jet luminosity. If the luminosity is
constant, the solution is
trivial and if we use a $\Gamma$ law equation of state the pressure is
given by
\beq
P = (\Gamma -1) {E\over V},             \label{e1}
\eeq
so that
\beq
P = (\Gamma -1) {L_j t\over V},         \label{e2}
\eeq
where $V$ is the volume of the lobe.

Other authors have consider the possibility of expansion losses through
the work done by
the expanding volume and write (Scheuer 1974; Eilek \& Shore 1989; Daly
1990; Kaiser \&
Alexander 1997; Bicknell, Dopita \& O'Dea 1997)
\beq
{dE\over dt} + P{dV\over dt} = L_j .    \label{e3}
\eeq
Substituting $E$ from Eq. ({\ref{e1}) into ({\ref{e3}) we obtain
\beq
{\dot{P}\over P} + \Gamma {\dot{V}\over V} = (\Gamma-1){L_j \over
PV}\,\,.
\label{e3a}
\eeq

If we put $L_j=0$ this equation describes the problem of the expansion
of a region without
energy input whose solution is the well known expression:
\beq
P \propto V^{-\Gamma}.                  \label{e4}
\eeq
If $L_j$ is constant, the solution will depend on how $V$ varies with
time. As
an example, since the volume increases with time, we adopt
\beq
V \propto t^n.                          \label{e5}
\eeq
The solution has the form
\beq
P = A V^{-\Gamma} + B V^{-1+1/n},       \label{e6}
\eeq
where $A$ and $B$ are constants. As we can see from Eq. (\ref{e2}),
(\ref{e4}) and
(\ref{e6}),  (supposing $n>1$) the pressure becomes arbitrarily small
as the volume increases.

These solutions seem quite satisfactory for studying  the early stage of
the source
expansion when its internal pressure $P$ is much larger than the ambient
pressure $P_a$.
However, at later times, when the volume is very large, we expect the
pressure to tend
asymptotically to or become comparable to $P_a$. At this stage, the bow
shock would be
only a small perturbation that propagates as a sonic wave. The same
should be true for the
internal shock (the contact discontinuity) that expands to form the
cocoon. Since the
pressure in Eq. (\ref{e6}) tends to zero, this is not realistic and it
is necessary to
incorporate the external pressure in the energy equation.

We address this in a phenomenological way. Without entering into the
detailed analysis of
the shock propagation, we consider the following {\it ansatz}: let us
introduce a
correction to the pressure $P$ by adding a term $-\Pi$ which is
proportional to $P_a$ so
that Eq. (\ref{e3}) becomes
\beq
{dE\over dt} + (P-\Pi){dV\over dt} = L_j .      \label{e7}
\eeq
The exact value of $\Pi$ can be calculated from the asymptotic condition

$P_{\infty} = P_a$ after solving the above equation. When $L_j=0$,
the solution of (\ref{e7}) is straightforward and is given by
\beq
P = \left(P_o - {\Gamma-1\over \Gamma} \Pi\right)\left({V\over
V_o}\right)^{-\Gamma} +
{\Gamma-1\over \Gamma} \Pi,                      \label{e8}
\eeq
where $P_o$ and $V_o$ are the initial pressure and volume, respectively.
Taking the limit
$V\rightarrow \infty$ we obtain the desired relation between $\Pi$ and
$P_a$, that is,
\beq
\Pi = {\Gamma\over \Gamma-1} P_a .                \label{e9}
\eeq
Therefore, in order to produce a more realistic description for the
expansion of the radio
sources when they are young as well as at the late stage of their
evolution, the
appropriate energy equation should be written in the form
\beq
{dE\over dt} + \left(P-{\Gamma\over \Gamma-1} P_a\right){dV\over dt} =
L_j
                                                \label{e10}
\eeq
which generalizes the equation usually adopted in the literature.

We decompose the SAG region into a set of slices which are initially
high-pressure discs
of gas of constant width that are left behind by the source head as it
advances along the
z-axis and will subsequently expand radially. The adiabatic expansion of
the slice in the
ambient medium whose pressure is $P_a$ is described by the energy
equation
\beq
{dE\over dr} + (P - {5\over 2}P_a) {dV\over dt} = 0,  \label{PP1}
\eeq
where $V$ and $P$ are respectively the volume and pressure, the
adiabatic index
$\Gamma=5/3$ and
$$
E = {3\over 2} P V.
$$
The solution of (\ref{PP1}) is given by
\beq
P = \left(P_o - P_a\right)\left({V\over V_o}\right)^{-5/3} + P_a
\label{PP2}
\eeq
where $P_o$ is the pressure at a fiducial volume $V_o$.

Fig. \ref{ASUA_Pb_Vs} shows the pressure in slices of the SAG as a
function of the volume
of those slices in a logarithmic plot. The curves represent averages
over 160 slices, each
of them having width equal to the grid spacing. For clarity, they have
been displaced
vertically by an arbitrary amount. The top curves represent the regions
near the jet head
while the bottom ones represent regions near the central galaxy (jet
inlet). The
latter are the
oldest set of slices while the former are the youngest slices. The
dashed curves drawn in
Fig. \ref{ASUA_Pb_Vs} represent the solution to Eq (\ref{PP2}) and were
vertically
displaced for easy comparison with the results of the simulations. We
did not attempt to
fit the theoretical curves to the points, and the values of $P_o$ and
$V_o$ are the same
in all panels.

The general trend (especially at intermediate Mach number) is that
pressure decreases with
increasing volume as expected for an adiabatic expansion, and at large
volume the curves
flatten asymptotically as the pressure approaches the ambient value.
Note also that near
the head (top curves) (1) the curves tend to flatten at small volume,
that is, at the
beginning of the expansion and (2) the pressure seems to fall more
slowly than at the jet
inlet (bottom curves).


In Fig. \ref{ASUA_Pc_Ts} we show the evolution of the pressure inside
the {\it cocoon\/}.
We plot the pressure as a function of time (instead of volume) because
the K-H
instabilities introduce uncertainty into the determination of the
location of the contact
discontinuity and thus the cocoon volume. We find that (especially at
intermediate $M$)
initially, the pressure falls sharply then asymptotes towards a constant
value. The dotted
and dashed lines correspond to $P\propto t^{-1}$ and $P\propto t^{-4/5}$
which are the
predictions of the Type I and Type III self-similar models (see Paper
I). We see that the
cocoon pressure generally does not seem to follow either of these
predictions except
possibly for brief periods of time in the transition between the initial
very sharp fall,
when the slice is young, and the later stage of the expansion when the
pressure tends to
an equilibrium pressure.

We also note that while there is only mild variation in the  behavior of
$P(V)$ in slices
of the SAG region between the jet head and inlet, the slices in the
cocoon show very
strong differences between the jet head and inlet. The pressure in
slices of the cocoon
near the head drops much faster than in slices near the jet inlet.


\subsubsection{The Evolution of the Global Pressure in the SAG region
and
Cocoon}

In order to study the evolution of the internal energy of the source
lobe as a whole we
need to take into account that the jet is continuously injecting energy
into it. Eq.
(\ref{PP1}) is appropriate for the study of a slice of the SAG region or
cocoon (assumed,
for simplicity, to be an isolated entity). However, in the entire region
we
must assume that there is a net energy input rate $Q$ into the expanding
region.
Considering adiabatic losses, the energy equation with a source term $Q$
is given by
\beq
{dE\over dt} + (P - {5\over 2}P_a) {dV\over dt} = Q.  \label{PP3}
\eeq

Here we shall assume that the time dependence of both the volume and the
source
term is a power law, that is,
$$
V = V_o \left({t\over t_o}\right)^{\alpha}
$$
and
$$
Q = Q_o \left({t\over t_o}\right)^{h}.
$$
This, of course, is a simplification since  as we have already discussed
in
\S~\ref{sec-Vol} and also in Paper I (\S~5.4), over much of parameter
space, the time
dependence of the SAG region and (especially) cocoon  volume is more
complex than a simple
power law.  However, in order to proceed and to find a simple analytical
solution for the
energy equation, we shall make this approximation.

Under this assumption, the solution is given by
\beq
P = \left[(1-\delta)P_o - P_a\right]\left({V\over V_o}\right)^{-5/3} +
\delta
P_o\left({V\over V_o}\right)^{{h-\alpha+1\over \alpha}} + P_a
\label{PP4}
\eeq
where $\delta$ is defined as
\beq
\delta = {2\over 3h+2\alpha+3}\left({Q_o t_o\over V_o}\right)
P_o^{-1}.      \label{PP5}
\eeq
We can estimate $\delta$ by taking an approximate power law solution for
the pressure that
is valid when $P\gg P_a$. In this limit, we put $P_a=0$ in Eq.
(\ref{PP3}) and solve the
modified equation. If we take $P\propto t^m$ the solution is
\beq
P = P_o\left({V\over V_o}\right)^{{h-\alpha+1\over \alpha}}. \label{PP6}
\eeq
Therefore, here $m=h-\alpha+1$, which corresponds to the second term in
the right hand side of (\ref{PP4}). We also obtain
$$
P_o = {2\over 3h+2\alpha+3}\left({Q_o t_o\over V_o}\right)
$$
which, when compared with (\ref{PP5}), gives $\delta=1$. In general, we
might expect
$\delta = O(1)$, with $\delta=0$ corresponding to zero energy input
($Q=0$).

The overall pressure of a given region in the simulation is calculated
by multiplying the
pressure in each individual slice by its volume, summing over all slices
and then dividing
by the total volume. In Fig. \ref{ASUA_Pt_Vt} we present a logarithmic
plot of the overall
pressure inside the cocoon and SAG region as a function of the total
volume of each
region. The open and filled circles represent the SAG region and cocoon,
respectively. We
show the curves $P(V)$ given by Eq. (\ref{PP4}) for three values of
$\delta$, namely
$\delta=0$ (solid line), $\delta=0.5$ (dashed line) and $\delta=1$
(dash-dotted line). The
curves were calculated taking a constant source term ($h=0$) and
calculating $\alpha$ from
the fit to the $\log {V} \times \log {t}$ plot.

The pressure in the self-similar models Type I and III described in
Paper I is the
solution of Eq. (\ref{PP3}) with $P_a=0$ and a constant source term.
This corresponds to
Eq. (\ref{PP6}) for $h=0$ and, if we substitute $\alpha=2$ as given by
the Type I model
and $\alpha=9/5$ given by Type III, we find $P\propto V^{-1/2}$ and
$P\propto V^{-4/9}$
respectively. These curves are represented by dotted lines in Fig.
\ref{ASUA_Pt_Vt}.

In the model developed by Hardee et al. (1992) to analyze simulations of
asymmetric jets,
the cocoon pressure is calculated assuming an adiabatic expansion
without a source term,
which is equivalent to the first term of Eq. (\ref{PP4}) with $\delta=0$
and
$P_a=0$. This
solution, but with $P_a\ne 0$ is represented in the figure by the solid
line. We see that
it does not fit the observed variation of the pressure because it
initially falls too fast
as the volume increases. On the other hand, for very high Mach number
(Runs A4
and A5 in Fig. \ref{ASUA_Pt_Vt}), the pressure is so
high that the simple power law given by (\ref{PP6}) and suggested by the
self-similar
models seems quite appropriate to describe the pressure evolution.
However, for moderate and low Mach number ($M\lae
30$), the full solution of the energy equation given by (\ref{PP4}) is
required and the
results of the simulations are in good agreement with Eq.~\ref{PP4}
with  $\delta=1$.

We conclude that, despite the oversimplification made by taking
$V\propto t^{\alpha}$, the
overall behavior of the pressure inside the SAG region can be described
assuming an
adiabatic expansion of a volume having an almost uniform energy density
distribution and
which is continuously supplied with energy near the jet head. This, in
part, explains the
similarity between most of the curves in Fig \ref{ASUA_Pb_Vs} (mostly
near the jet inlet)
and also why, near the head (upper curves), where energy is being fed,
the pressure falls
somewhat more slowly. The behavior of the cocoon pressure is far less
clear and this may
be due to the strong variations in the way the pressure evolves at
different locations
along the source (see Fig. \ref{ASUA_Pc_Ts}). This makes it much more
difficult to produce
a simple meaningful analytical model for the whole cocoon.

We also note that the fact that the SAG pressure changes with volume
approximately
according to Eq. (\ref{PP4}) suggests its evolution is not strictly
self-similar. Another
way to look at the issue of self-similarity of the pressure evolution is
as follows. In a
constant density atmosphere, the pressure scales as $z_h^{-1}$ and
$z_h^{-4/3}$, according
to  Type I and III models, respectively. Fig. \ref{ASUA_Pt_Xh} shows the
average pressure
inside the cocoon and SAG region as a function of $z_h$. The curves show
the expectations
of the self-similar models -  dotted lines have slope $-1$ (Type I) and
the dashed lines
have slope $-4/3$ (Type III). We note that over much of parameter space,
the pressure
falls more slowly with source size than the expectations of the
self-similar models. In
order to examine the behavior of the exponent of $P\propto z_h^b$ in
more detail, we have
employed the two methods of estimation described in Paper I (\S~5.1). We
show the results
in Fig. \ref{ASUA_Similar_Pb} and Fig. \ref{ASUA_Similar_Pc} for the SAG
region and cocoon
respectively. We see that, at moderate values of $M$, the exponent $b$
is of order $-0.5$,
which is higher than predicted by the self-similar models. However, when
the SAG region is
highly over-pressured (i.e., for high Mach number - Runs A3, A4 and A5),
the pressure
evolution inside the SAG region is consistent with Type I models, that
is, $P\propto
z_h^{-1}$.  On the other hand, even at high Mach number, the simulations
are never in
agreement with the Type III models.

The behavior of the pressure evolution of the cocoon is  broadly similar
to that of the
SAG region, though for the cocoon, the exponent $b$ is much less stable
as the source size
increases. In general, the cocoon pressure decreases with source size
more slowly than
predicted by the self-similar models. There is also some indication that
at moderate Mach
number (Runs A2, A3, A9 and A10) the exponent is close to that of the
SAG pressure.
However, for high Mach number (Runs A4 and A5), the pressure appears
initially
to decrease slowly with distance as $b\simeq -0.5$ and later more
rapidly
as $b\simeq -1$. This is consistent with the time evolution of the
pressure
profiles shown in Fig.'s 6 and 7 of Paper I where
the SAG region and cocoon profiles seem to fall approximately together.

Cioffi \& Blondin (1992) also noticed that the SAG pressure in a
simulated jet falls with
time more slowly than predicted by the Begelman \& Cioffi (1989)
self-similar Type I model
($t^{-1}$). They attributed this  to the increase of the head area and
consequent
deceleration of the source. This in turn, would cause the volume to
increase at a slower
rate. However, in Fig. \ref{ASUA_Pt_Vt} we have plotted pressure against
the volume and
there still is a discrepancy with the self-similar model. Thus, we do
not believe this to
be the complete explanation. We shall return to this question in Paper
III.

Another possible reason given by Cioffi \& Blondin for the discrepancy
is the increase of
the ratio between the internal energy and total energy $E_i/E_t$ in the
source with time.
This would be due to the fact that a fraction of the energy injected by
the jet is
initially in the form of kinetic energy of large scale turbulent motion.
As the large
scale kinetic energy cascades to smaller scales, reaching the scale of
the numerical grid,
it is dissipated and transformed into thermal energy. Therefore, the
internal energy
fraction increases.

If this effect is significant, it would play the role of an extra source
term in the
energy equation. A plot of $E_i/E_t$ as a function of the volume $V$
over the whole volume
for both the cocoon and SAG region shows that $E_i/E_t$ increases by a
factor no larger
than 1.7 as $V$ increases by a factor of $\sim 400$ in the SAG region.
For the cocoon, in
 low Mach number jets, the largest increase in $E_i/E_t$ is by a factor
$\sim 2$ as
$V$ increases by a factor of $\sim 600$. Therefore, we conclude that in
general, the
dissipation of turbulent kinetic energy into internal energy should have
only a marginal
effect on the evolution of the pressure with volume.

At this point, we note that, the behavior of turbulent motion is
different in two and
three dimensions (see also \S 3.1 of Paper I). Thus, contrary to what
happens in
three dimensions, where energy in
large scale eddies cascades to smaller scales, in 2D the energy cascades
to larger scales.
Hence, the expected effect (dissipation of kinetic energy into thermal
energy) should not
be observable in the present axisymmetric two-dimensional simulations.

Although the behavior of the ratio $E_i/E_t$ in the cocoon is in
agreement with
that of the SAG
region at low Mach number, it becomes more complex at high $M$. At high
$M$, the ratio
$E_i/E_t$ exhibits a sharp drop during the early stages of the expansion
(Runs A4, A5, A9
and A10) and then starts increasing slowly again. We attribute this to
the fact that, at
around $z_h \sim 10$ kpc, the bow shock suddenly narrows (see Fig. 3 and
4 of Paper I) and
the jet becomes more focused, and the jet and cocoon radius decrease
(Fig. 11 of Paper I).
When this happens, nearly all of the cocoon energy is in the form of
kinetic energy, most
of it due to the large-scale forward motion of the source and only a
small amount is in
turbulent motion (Fig. 2 in Paper I) which can be transformed into
internal energy.
Consistent with this, we  observe small glitches in the cocoon pressure
in Fig.
\ref{ASUA_Pt_Vt} (Runs A4 and A5) which are due to this phenomenon - the
pressure falls
and then increases slightly.

Later, in \S~5.3, we shall suggest that the slow decrease of the
internal energy with time
observed mainly for low Mach number jets, can be partially attributed to
the fact that the
pressure in these sources rapidly reaches the external pressure.

Finally, it is worth mentioning another aspect of some of the
self-similar models. For
example, the Begelman (1996) model (Type III) is based on the assumption
that the ratio
between the average pressure in the head $P_h$ and the cocoon pressure
is a constant $\sim
2$. On the other hand, in Type I models, the pressure ratio scales as
$t$ (see Table 2 of
Paper I). We have estimated $P_h$ in the following way: first, we
calculate the maximum
pressure $P_{max}$ on the simulation grid. This occurs in the maximum
compression region
inside the head. We then define the region around this point where the
pressure is in the
range $0.8 P_{max} < P < P_{max}$ and calculate the average pressure
$P_h$ inside this
region. The ratio between $P_h$ calculated in this way and the average
pressure as a
function of the source size (solid line) is shown for the cocoon in Fig.
\ref{ASUA_PhPc_Xh}. The figure also shows the ratio calculated using the
ram pressure
$\sim \rho v_h^2$ (open circles) and the results are similar. We see
that, for the cocoon,
at low and moderate $M$  the ratio is almost constant or decreases
slowly; while at high
Mach number ($M \gae 30$) the ratio initially increases and then levels
off.  In the
extreme examples, the ratio may vary for more than a factor of 10 during
the period the
simulation has been carried out. This contradicts the prediction of Type
III models. In
addition, even where the ratio increases, it does not increase as much
as
required by the Type I model. Thus, the behavior of the cocoon is
inconsistent with both
models.

The behavior of the pressure ratio in the SAG region (not shown here) is
even worse as far
as Type III is concerned since the pressure ratio may vary in some runs
by almost a
factor 100. On the other hand, at very high Mach number (Runs A4 and
A5), the ratio scales
pretty much as predicted by the Type I model ($\propto t$). In addition,
as we have seen
(Fig. \ref{ASUA_Similar_Pb}), these are the regimes where the exponent
$b$ of the SAG
pressure in $P\propto z_h^b$ is also almost constant and  $b \simeq -1$
in accordance
with the
model. Thus, except for a limited range of parameter space at high Mach
number, the
behavior of the ratio of the peak to average pressure is not consistent
with the
expectations of the self-similar models.



%******************************************************************************

%******************************************************************************

\subsection{Atmosphere with density gradient}


%******************************************************************************

%******************************************************************************

\subsubsection{The isothermal atmosphere}
\label{energeticsT}

In this section, we briefly discuss the time evolution of the pressure
for a source
propagating in an isothermal atmosphere with a density gradient and
compare the results
with the predictions of the self-similar models. The appropriate models
are Type II and
Type III with $\delta=2$ (see Table 2, Paper I). They predict that the
overall pressure
should scale with the source size as $z_h^{-1}$ and $z_h^{-2}$,
respectively.

We have calculated the exponent of average pressure as a function of
source size $P\propto
z_h^b$ as before and show the results in Fig. \ref{ASNA_Similar_Pb} and
Fig.
\ref{ASNA_Similar_Pc} for the SAG region and cocoon, respectively. For
the SAG region the
exponent starts out between -1 and -2 and approaches the value -2 in
accordance with Type
III models. This is seen more clearly for large $M$ as in Runs A13, A14,
A15 and A20, and
is in contrast to the behavior in  a constant density atmosphere where
the pressure
evolves according to Type I models with $b \simeq -1$ (see Fig.
\ref{ASUA_Similar_Pb}).

For the cocoon, the exponent $b$ also tends to asymptote to the value $b
\simeq
-2$ given by Type III models. However, the behavior of the cocoon is
more unstable
than that of the SAG region, and by the end of the simulation, the
exponent
has varied greatly. As
before, for $M=3.3$ where the cocoon is under-pressured, the pressure
decreases more
slowly than at higher $M$ and the exponent is near -1. In general, for
Runs A11-A15, there
is only a mild tendency for the exponent to decrease with increasing
Mach number for the
SAG region and cocoon. This could be due to the fact that the SAG region
and cocoon are
both significantly over-pressured with respect to the ambient medium.

The pressure as a function of volume is shown in Fig. \ref{ASNA_Pt_Vt}.
We find that at
high Mach number (Runs A13, A14 and A15), the cocoon pressure falls
roughly as $\sim
V^{-5/3}$ which is much faster than seen in the constant atmosphere
(except at high Mach
number - see Runs A4 and A5 in Fig. \ref{ASUA_Pt_Vt}). This behavior is
characteristic of
an adiabatic expansion without energy replenishment
(\S~\ref{energeticsU}) and suggests
that  very little thermal energy is being provided to the cocoon, i.e.,
most of the jet
energy is not being promptly thermalized and remains in the form of
kinetic energy inside
the cocoon. This effect can be seen when we plot $E_i/E_t$ as a function
of the volume $V$
at high Mach number in the constant density atmosphere and is even
stronger in the
declining density atmosphere.

In an atmosphere with a density gradient, the self-similar models
predict that the ratio
between the head pressure and pressure inside the cocoon and SAG region
should be either
constant (Type III) or scale as $t^{-1}$ (Type II). In both the cocoon
and SAG region, we
find a tendency for the average ratio to increase by over an order of
magnitude as the
Mach number increases from $\sim 3$ to $\sim 50$. Generally speaking,
the pressure ratio
remains approximately constant with source size as predicted by Type III
models, except at
high $M$ where there is a slight trend for the pressure ratio to
decrease as the source
size grows.

The results in the section are in general consistent with the results
for the constant
density atmosphere (\S~\ref{energeticsU}). We do find that sources
propagating in an
atmosphere with a  density gradient behave in only very rough agreement
with the
predictions of the self-similar models, though the discrepancies are
less than in the
constant density atmosphere.


%******************************************************************************

%******************************************************************************

\subsubsection{The isobaric atmosphere }
\label{energeticsP}

Here we present the main results regarding the energetics of a source
propagating in an
isobaric atmosphere with a density gradient. Note that the temperature
in the external
medium increases as the density drops. The behavior of the average
pressure as a
function of the source size (the exponent $b$ in the relation $P\propto
z_h^b$) is shown
in Fig. \ref{ASNX_Similar_Pb} and Fig. \ref{ASNX_Similar_Pc} for the SAG
region and
cocoon, respectively.

For runs X11-X15, at lower Mach number the exponent is close to -1,
whereas at higher $M$,
the exponent is  between -1 and -2, i.e., the SAG pressure declines more
slowly with
source size then predicted by the Type III models ($b=-2$) but slightly
faster than
predicted by the Type II models ($b=-1$). At higher $M$ the SAG region
is over-pressured
along its entire length and for the whole duration of the simulation. We
note that the
comparable runs in a constant density atmosphere A1-A5 were in
relatively good agreement
with the predictions of the Type I models. In addition, for Runs
X16-X20, the exponent
tends to be larger than -1, i.e. the pressure declines more slowly with
source size then
predicted by both the Type II and III models. The behavior of the cocoon
is similar to
that in the SAG region. In addition, the exponent $b$ shows large
variations (even larger
than in the runs in a constant density atmosphere). For both the SAG
region and the
cocoon, the pressure declines more slowly with source size than in the
source in an
isothermal atmosphere with a density gradient. In general, the sources
in an isobaric
atmosphere show somewhat larger departure from the predictions of the
self-similar models
than do the sources in either a uniform atmosphere or an isothermal
atmosphere with
declining density gradient.

In an isothermal atmosphere, in jets with high Mach number, the SAG
region and
cocoon  pressure falls with volume as approximately $V^{-5/3}$ (not
shown).
Again we suggest this is because of the relatively high speed
propagation of
the source, and the reduced amount of thermal energy being fed into the
cocoon
- thus it expands essentially adiabatically (without an effective source
input term).

We have calculated the ratio between the pressure in the head  and
cocoon and SAG region
as a function of the source size. On the whole, for Runs X11-X15 the
pressure ratio is approximately constant (though with some departures)
as predicted by
Type III models for all Mach number. However, for Runs X16-X20 the ratio
tends to
eventually decrease with source size at a rate that becomes closer to
that predicted by
the Type II model as the Mach number increases. The apparent
contradiction is due to the
different density ratios $\eta$ in the two sets of simulations, i.e.,
the jets in Runs
X11-X15 have density ratios that are relatively smaller than in Runs
X16-X20. The results
of Runs Y11-Y15 with fixed $\eta$ and  Runs Y16-Y20 with fixed Mach
number (not shown)
indicate that the pressure ratio decreases faster with source size for
both increasing $M$
and $\eta$. This result is also in contrast to the results for a source
in an isothermal
atmosphere with a declining density where the pressure ratio has no
systematic trend with
source size.

Thus, in general, the energetics of the source are similar whether the
source is
propagating in an isobaric or isothermal atmosphere, with some caveats.
For both the SAG
region and the cocoon, the pressure declines more slowly with source
size in the isobaric
atmosphere than in the isothermal atmosphere with a density gradient. In
addition, the
sources in an isobaric atmosphere show somewhat larger departure from
the predictions of
the self-similar models than do the sources in either a uniform
atmosphere or an
isothermal atmosphere with declining density gradient. The ratio of
pressure in the head
to the cocoon or SAG region has no systematic trend with source size in
the isothermal
atmosphere, but tends to eventually decrease with source size in the
isobaric atmosphere.



%******************************************************************************

%******************************************************************************

\subsubsection{The isothermal atmospheres with King model $\beta=0.75$ }

\label{beta75}

Here we discuss our results for Case 1 (pressure dominated by
relativistic electrons) for
the source propagating in an isothermal atmosphere with a density
gradient using a value
$\beta = 0.75$ in Eq. (\ref{king}). The Type III self-similar model
adopting $\delta=1.5$
(see Table 2 of Paper I) is the appropriate comparison. The general
appearance of the
source (not shown) is almost identical to that for $\beta=1$ (Fig.
\ref{ASNA_View1}). The
SAG region and cocoon pressure distributions along the jet axis are also
very similar;
while the ages for $\beta = 0.75$ are only slightly larger due to the
smaller density
gradient.

The relation given by Type III models for the time dependence of the
source size is
$z_h\propto t^{6/7}$ and we have compared it with the exponent
calculated from the
simulations. The overall behavior is similar to that of the atmosphere
with $\beta=1$ (see
Fig.~\ref{ASNA_Similar_Xh}), i.e., the jets accelerate with time (with
the acceleration
dropping with increasing size at high $M$). However, the acceleration is
larger at high
$M$ for $\beta = 0.75$, i.e., there is a larger disagreement with the
predictions of the
Type III model.

The behavior of both the head radius and the head advance speed with
source size (not
shown) are similar to that in the atmosphere with $\beta=1$, i.e., the
head radius
increases  with increasing source size steadily at low $M$ and after a
initially constant
regime ($\sim 20 r_j$) at high $M$. Again, the head radius increases
faster at high values
of the Mach number and compensates for the density gradient causing the
advance speed to
remain nearly constant.

According to the Type III self-similar model, the source volume should
increase with time
as $t^{18/7}$ and we have compared with the exponent derived from the
simulations for the
SAG region and cocoon. The agreement with the model prediction is not as
good as for
$\beta=1$, the exponent being slightly over the predicted value 18/7.
Nevertheless, we
again find that the behavior of the source volume as a function of time
is closer to the
self-similar predictions than the behavior of the source size as a
function of time.  This
suggests that the shape of the bow shock adjusts to keep the time
dependence of the source
volume roughly self-similar. The exponent for the cocoon is in agreement
with the
predicted value for $M=9.7$ and $M=12.8$ (Runs N11 and N12); while at
higher $M$ it shows
large variations as a function of source size.

The Type III self-similar model predicts that in the atmosphere with
$\beta = 0.75$, the
average lobe pressure should scale as $P\propto z_h^{-11/6}$. Fig.
\ref{ASNN_Similar_Pc}
and Fig. \ref{ASNN_Similar_Pb} show the exponent $b$ for the cocoon and
SAG region,
respectively. The behavior of the SAG region and cocoon pressure with
source size are
similar for both $\beta = 0.75$ and $\beta=1$. For $\beta = 0.75$ and
high $M$, the
exponent $b$ asymptotes to the predicted value at large source size.

In summary, the results are similar for the source expanding in an
isothermal atmosphere
where the density decreases asymptotically like $\rho\propto r^{-1.5}$
and $\rho\propto
r^{-2}$. However, we find a tendency of some global properties to show a
greater departure
from the predictions of the self-similar models when $\beta = 0.75$ than
for
$\beta = 1$.

%******************************************************************************

%******************************************************************************

\subsection{Pressure variation and self-similar models}

The self-similar models assume that the source is highly over-pressured
with respect to
the ambient medium and thus do not explicitly consider a functional form
for the ambient
pressure. However, we have found that the agreement of the simulations
with the
predictions of the self-similar model does depend on the degree to which
the source is
over-pressured and thus, the assumed ambient pressure profile for the
source.

In sections \ref{energeticsU} and \ref{energeticsT} we have discussed
that one of the main
departures from the simulations results from the self-similar models is
that the pressure
seems to fall much slowly than predicted. In a constant density
atmosphere, the pressure
inside the SAG and cocoon rapidly approaches the external pressure,
except for high Mach
number jets. We suggest that this may be the main reason for the
``apparent"
slow decrease of the pressure with time in these sources. Indeed, Fig.'s
6 and 7
of Paper I show that, because the pressure, e.g., in the low Mach number
cocoons,
has reached the external pressure, it then remains almost constant
throughout
the rest of the source evolution,
although the volume continues increasing. As a consequence, the exponent
$b$ is much
higher that that predicted by the self-similar models. In an isothermal
atmosphere with
a density gradient, the regions are kept over-pressured for a longer
period of time and
the exponent tends to agree more closely with the predicted value. In
support of this is
the fact that, as we move from an atmosphere with $\beta=0$ to
$\beta=0.75$
and $\beta=1$, the agreement between model and simulation seems to
improve.

%******************************************************************************

%******************************************************************************

\section{SUMMARY}
\label{summary}

We present 2-D numerical hydrodynamical simulations of jets propagating
in atmospheres
which decline in density with increasing distance in several ways --
isothermal King-law
atmospheres with the power law exponent $\beta =1$ and  $\beta = 0.75$
and in an isobaric
King-law atmosphere with power law exponent $\beta =1$. We explore the
same very broad
range of parameter space in Mach number $M$ and density contrast $\eta$
as in Paper I in
this series. We compare our results with those for the constant density
and pressure
atmosphere simulations and with the predictions of the self-similar
models (Paper I). We
also discuss the global energetics of the sources in these different
environments.

\subsection{Morphology}

Our comparison of the constant and declining density results shows the
following. The
overall morphology of the jet, cocoon and bow shock is similar. However,
there are some
differences which start to appear when the jet has propagated about
twice the core radius.
In the declining density atmosphere,  (1) there is less structure in the
cocoon due to
turbulence and (2) at a given source size,  the cocoon seems to be
under-expanded relative
to the constant density atmosphere for a broad range of Mach numbers
$\sim 5-30$ (outside this
range the cocoon may actually be wider). These may both be caused by the
fact that there
appears to be  reduced back-flow in the sources in a declining density
atmosphere.

In an isobaric atmosphere with density gradient the general appearance
of the source is
similar to that of an isothermal atmosphere. The bow shock is
over-expanded while
the cocoon seems less inflated and the turbulence is weaker and
underdeveloped.
The source size increases much faster than in the isothermal atmosphere
and this
is observed for all values of $M$.

We find that the location of the first shock in the jet provides a
useful estimate of the
jet Mach number (see Wilson \& Falle 1985) similar to a jet in a
constant
density atmosphere (see sec 3.2 Paper I). Thus, this method is not
sensitive to the
presence or absence of large scale pressure gradients in the jet
environment (e.g., a
pressure gradient with a core radius of 5 kpc).

\subsection{Pressure profiles and Lateral Expansion Velocity}

The overall distribution of pressure in the shocked ambient gas (SAG)
region
and cocoon and its decline with
distance from the jet head is similar in both types of atmospheres.
However, in jets in
declining density atmospheres, for Case 1 (pressure dominated by
relativistic electrons),
the cocoon and SAG region remain much more over-pressured with respect
to the ambient
medium than in jets in constant density atmospheres.

In the isobaric atmosphere, the pressure falls more rapidly inside the
cocoon than inside
the SAG region and the cocoon only stays over-pressured near the jet
head except in
very powerful jets. The most striking difference is that the pressure
near the
head of both the SAG region and cocoon is lower and that it falls more
rapidly to levels
that are also much lower than in the constant density atmosphere.

The lateral expansion speed of the bow shock and cocoon are similar in
the constant and
declining density atmospheres.

\subsection{Advance Speed and Head Radius}

In the constant density atmosphere, the jet head velocity decelerates
slowly
with time, while in the declining density atmosphere, the jet head
accelerates
with time with a rate given by the exponent of $z_h \propto t^m$ with
$1 < m < 1.5$ (see Table 2).
However, at high Mach number
and/or high jet density the acceleration decreases with time and the
head velocity
approaches a nearly constant velocity consistent with the predictions of
the self-similar
models. The sources do not go through the three phases seen in the
constant density
atmospheres; instead they tend to remain either at constant velocity or
slightly
accelerating.

There are three types of behavior for the source expansion along the jet
axis in an
isobaric atmosphere. For very low values of $M$, the advance speed
rapidly approaches the
sound speed limit which increases linearly with $z_h$ (because the
temperature
increases as the density drops) and the source size increases
exponentially with time. For intermediate values of the Mach number
$v_h$ is greater than
the sound speed, increases with $z_h$ and the size of the source
increases as $z_h\propto
t^m$ with $m>1$. The third type of behavior occurs at high Mach number
where $v_h$ is
approximately constant and much larger than the ambient sound speed and
the source expands
almost linearly with $t$.

In all environments, the radius of the jet head varies with source size
as expected
-- consistent with ram pressure balance. For the declining density
atmosphere,
(1) the jet head radius is larger at a given source size (i.e., it
expands
more rapidly in the declining density); and (2)  the behavior of the
radius
of the jet and cocoon head as a function of Mach number is more complex,
though there is a general trend for the jet head to be narrower at
higher Mach number.

\subsection{Time Dependence of the Source Volume}

In an isothermal atmosphere with a density gradient, the volume of the
SAG region tends to
increase with source size in agreement with the predictions of the Type
III models, i.e.,
$V \propto t^3$. The estimated volume of the cocoon tends to vary with
time possibly due
to the influence of  K-H instabilities on the contact discontinuity, but
in general
we find $V \propto t^{2-3}$.

We find that for all environments, the behavior of the source volume as
a function of time
is closer to the self-similar predictions than the behavior of the
source size as a
function of time. This suggests that the shape of the bow shock adjusts
to keep the time
dependence of the source volume roughly self-similar.

\subsection{Source Energetics}

We examine the dependence of the pressure on the source volume as a
function of the source
age for the sources in different environments. In the constant density
atmosphere, the
pressure evolution of cylindrical slices of the SAG departs from that
expected for
adiabatic expansion both near the head and near the jet inlet. Near the
jet head, the
departure is due to energy input to the SAG region from the jet head
(i.e., there is an
effective ``source term") and near the jet inlet, the departure is due
to the source
coming into pressure balance with the ambient medium. The dependence is
also a function of
the jet Mach number, with lower Mach number jets showing much less
variation of SAG region
pressure with volume. In addition, at a given Mach number, cylindrical
slices of the
cocoon tend to show much steeper dependence of pressure with volume than
the SAG region.

In a constant density atmosphere, the average global pressure in the SAG
region and cocoon
vary more slowly with volume than predicted by the self-similar models.
This indicates
that there is an additional source term which is not present in the
self-similar models.

In a constant density atmosphere, the ratio between the head pressure
and the average
cocoon pressure is relatively constant with source size for $M <10$ but
increases with
source size for $M>10$ by up to an order of magnitude -- i.e., the head
pressure becomes
more dominant as the source expands. This is in contrast to the
assumption in the Type III
models that this ratio is constant (see Table 3).

These results for the evolution of the pressure are similar for the
isothermal
King atmosphere, though here there tends to be better agreement with the
predictions of
the self-similar models. The variation of SAG region pressure with
source size tends to
start out flatter than predicted by the Type III models, but evolves to
become  consistent
with this prediction ($P \propto z_h^{-2}$), with higher Mach number
jets evolving to this
state more rapidly. The cocoon pressure behaves in a similar manner,
though it exhibits
more variation, possibly due to the effect of K-H instabilities on the
derived cocoon
size. The overall average cocoon pressure falls as expected due to
adiabatic expansion
with no energy replenishment. This is probably due to the diminished
back flow in the
cocoon which removes the effective source term present in the constant
density
atmosphere.

In general, the energetics of the source are similar whether the source
is propagating in
an isobaric or isothermal atmosphere, with some caveats. For both the
SAG region and the
cocoon, the pressure declines more slowly with source size in the
isobaric atmosphere than
in the isothermal atmosphere with a density gradient. In addition, the
sources in an
isobaric atmosphere show somewhat larger departure from the predictions
of the
self-similar models than do the sources in either a uniform atmosphere
or an isothermal
atmosphere with declining density gradient. The ratio of pressure in the
head to the
cocoon or SAG region has no systematic trend with source size in the
isothermal
atmosphere, but tends to eventually decrease with source size in the
isobaric atmosphere.

The behavior of the source and the energetics are similar in the
$\beta=1$ and
$\beta=0.75$ King model atmospheres. However, we do note that some of
the global
parameters are more in agreement with the self-similar models for
$\beta=1$.

\subsection{Pressure variation and self-similar models}


We note that, at low Mach number, the pressure exhibits a slower
decrease  with time than
predicted by the self-similar models, which results in a large value for
the exponent $b$. We suggest that this is due to the fact that the 
pressure inside the SAG and cocoon rapidly asymptotes to the
ambient pressure and then does not decrease by a significant amount
as the source volume increases.

The self-similar models assume that the source is highly over-pressured
with respect to
the ambient medium and thus do not explicitly consider a functional form
for the ambient
pressure. However, we have found that the agreement of the simulations
with the
predictions of the self-similar model does depend on the degree to which
the source is
over-pressured and thus, the assumed ambient pressure profile for the
source. The
agreement between the self-similar models and the simulations tends to
improve as we move
from an atmosphere with $\beta=0$ to $\beta=0.75$ and $\beta=1$.


%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

%@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

\acknowledgements

JCC acknowledges the financial support of PRONEX/Finep and CNPq and the
hospitality of STScI where this work was carried out. CPO received
partial
support from NASA through grant GO-07855.01-96A from the Space Telescope
Science
Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS 5-26555.
We thank Paul Wiita for helpful discussions and the referee for detailed
and helpful comments.

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\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{deluxetable}{ccccc}
\tablecolumns{5}
\tablewidth{0pc}
\tablecaption{Simulation Parameters}
\tablehead{
\colhead{}    &  \multicolumn{4}{c}{Atmosphere} \\
%%\multicolumn{3}{c}{Shell Stars} \\
\cline{2-5} \\
\colhead{Run} & \colhead{density}   & \colhead{pressure} &
\colhead{temperature}
&
\colhead{$\beta$} }

\startdata
A11 - A20 & $\nabla\rho$ & $\nabla p$ & constant   &   1  \\
B11 - B20 & $\nabla\rho$ & $\nabla p$ & constant   &   1  \\
C11 - C16 & $\nabla\rho$ & $\nabla p$ & constant   &   1  \\
\cline{1-5}
X11 - X20 & $\nabla\rho$ & constant   & $\nabla T$ &   1  \\
Y11 - Y20 & $\nabla\rho$ & constant   & $\nabla T$ &   1  \\
Z11 - Z16 & $\nabla\rho$ & constant   & $\nabla T$ &   1  \\
\cline{1-5}
N11 - N15 & $\nabla\rho$ & $\nabla p$ & constant   & 0.75 \\
%%%\cline{1-5}

\tablecomments{
The values of $M$ and $\eta$ in
Runs A11-A20 correspond to those in Runs A1-A10 in Paper I, and the
input values in Runs
B11-B20 correspond to those of Runs B1-B10,  while the input values in
Runs C11-C16
correspond to those of Runs C1-C6. In all these simulations, the
atmosphere is isothermal
and the pressure has a gradient proportional to the density gradient,
that is, $\nabla P/P
=\nabla\rho/\rho$. Runs X11-X20, Y11-Y20 and Z11-Z16 correspond to an
isobaric atmosphere with the same density gradient and have the same
Mach number and
density ratio of Runs A11-A20, B11-B20 and C11-C16, respectively. Our
simulations with an
isothermal atmosphere with a density gradient with $\beta=0.75$ are
presented in runs Runs
N11-N15 where  $M$ and $\eta$ are the same as those in Runs A11-A15.
%%%
Runs A11-A15, X11-X15 and N11-N15 correspond to Case 1 (see Paper I, \S
2.2,7)
for which the jet relativistic particle content is electrons
(and positrons) while Runs A16-A20, X16-X20 correspond to Case 2,
that is, electrons plus protons.}
\enddata
\end{deluxetable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    PAPER  II
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{deluxetable}{cccccccccccc}
\tabletypesize{\small}
\rotate
\tablecolumns{11}
\tablewidth{0pc}
\tablecaption{Comparison of the Simulations with Self-Similar Models }
\tablehead{
\colhead{} &
\colhead{} & \multicolumn{2}{c}{Self-Similar Model} &
\colhead{} & \multicolumn{3}{c}{Simulation}
           & \multicolumn{3}{c}{Simulation} &
\colhead{} \\
\colhead{} & \colhead{} & \colhead{} & \colhead{} & \colhead{} &
 \multicolumn{3}{c}{case 1} & \multicolumn{3}{c}{case 2} & \colhead{}\\
%%\multicolumn{3}{c}{Shell Stars} \\
%\cline{2-3} \cline{5-7}\\
\colhead{Relationship} & \colhead{exponent} & \colhead{} & \colhead{}
 & \colhead{}
 & \multicolumn{2}{c}{exponent}  & \colhead{note}
 & \multicolumn{2}{c}{exponent}  & \colhead{note} & }
\startdata
\multicolumn{2}{l}{Declining Density} &Type II&Type III&&&&&&&&\\

$z_h\propto t^m$ &m& 1 & 1 &&$1.0$ to $1.7$& all $\gamma$ &  a,c  &$1.1$
to
$1.3$ & all
$\gamma$ & b,c & Sec. 3.1\\

\cline{1-12}
%%%%bow shock &&&&&&&&&&\\

$V\propto t^n$ &n& 2 & 3 &SAG region     &$\sim 3.0$    & all $\gamma$ &
d
&$\sim 3.0$    & all
$\gamma$& d& Sec. 3.4\\
%%%%cocoon &&&&&&&&&&\\
               & &   &   &cocoon   &$\sim 3.0$    &low  $\gamma$ & e
&$2.0$ to
$3.0$&low

$\gamma$& f& Sec. 3.4\\
               & &   &   &         &$2.0$ to $3.0$&high $\gamma$ & g &$>
2.0$
   &high
$\gamma$& g&         \\
\cline{1-12}
\multicolumn{2}{l}{Constant Density} &Type I&Type III&&&&&&&&\\

$P\propto z_h^b$& b &$-1$&$-4/3$&SAG region     &$\sim -1.0$&high
$\gamma$ & h,i
&$\sim -1.0$ to
$-0.5$&high  $\gamma$& h,i & Sec. 5.1.2\\

                &   &    &      &cocoon   &$\sim -1.0$ to $-1.5$&low
$\gamma$ &
j &$\sim
-0.5$
to $-2.0$&low  $\gamma$& j,k & \\
                &   &    &      &         &$\sim -1.0$ to $-0.5$&high
$\gamma$&
j &$\sim
-0.5$
        &high $\gamma$& j   & \\
\multicolumn{2}{l}{Declining Density} &Type II&Type III&&&&&&&&\\

$P\propto z_h^b$& b &$-1$&$-2$&SAG region      &$-1.0$ to $-2.0$&all
$\gamma$ &
l,m &$-1.0$ to
$-2.0$&all $\gamma$& l,m & Sec. 5.2.1\\
                &   &    &    &cocoon    &$-1.0$ to $-2.0$&all $\gamma$
& l,m
&$-1.0$ to
$-2.0$&all $\gamma$& l,n &           \\

\tablecomments{
a - $m$ decreases with increasing source size and asymptotes to final
value
$\sim 1$ for
high
$\gamma$\\
b - $m$ is relatively constant with source size\\
c - $m$ increases with increasing source size for low $\gamma$\\
d - $n$ increases slightly with increasing source size and asymptotes to
final
value $\sim
3$\\
e - $n$ asymptotes to final value $\sim 3$\\
f - $n$ decreases with increasing $\gamma$\\
g - $n$ increases with increasing source size\\
h - $b$ tends to increase (become less negative) as the source size
increases\\
i - $b$ is not well defined for low $\gamma$\\
j - $b$ shows large oscillations as a function of source size\\
k - $b$ tends to decrease (become more negative) as the source size
increases\\
l - $b$ shows little dependence on $\gamma$ \\
m - $b$ decreases with increasing source size and asymptotes to final
value
$\sim -2$\\
n - $b$ is relatively constant for low $\gamma$ and decreases below
$-2.0$ for
high
$\gamma$}

\enddata
\end{deluxetable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{deluxetable}{ccccc}
\tablecolumns{11}
\tablewidth{0pc}
\tablecaption{Comparison of the Ratio Head to Bow Shock and Head to
Cocoon
Pressure in the
Simulations with Self-Similar Models }
\tablehead{
\multicolumn{2}{c}{Self-Similar Model} &\multicolumn{3}{c}{Simulation}
\\

%\multicolumn{2}{c}{} & \multicolumn{1}{c}{case 1}
&\multicolumn{1}{c}{case 2}
\\

\colhead{} & \colhead{} & \colhead{region} & \colhead{}& \colhead{note}}

\startdata
\multicolumn{5}{c}{Constant Density} \\
Type I       & Type III &         &             & \\
$\propto z_h$& constant &SAG region     &low  $\gamma$&approximately
constant
\\
             &          &         &high $\gamma$&increases with source
size
($\propto z_h$)
\\
             &          &cocoon   &low  $\gamma$&approximately constant
or
decreases slowly
\\
             &          &         &high $\gamma$&increases initially and
levels
off
 \\
\cline{1-5}
\multicolumn{5}{c}{Declining Density} \\
Type II           & Type III &         &             & \\
$\propto z_h^{-1}$& constant &SAG region     &all $\gamma$
&approximately
constant, increasing
with
$\gamma$\\
                  &          &cocoon   &all  $\gamma$&approximately
constant,
increasing
with
$\gamma$ \\

\enddata
\end{deluxetable}

%%%%%%%%%
%\end{document}

\clearpage


\begin{figure}[ht]
\figurenum{1}
\epsscale{0.9}
\plotone{f1.eps}
\figcaption{
Contours of density showing the simulation of a light jet in Case 1 in
an atmosphere with declining density with $\beta=1.0$
 (Runs A11-A15) with the following values of $\eta_c$ and $M_c$: (a)
0.0044,
9.7; (b)
0.0069, 12.8; (c) 0.0276, 27.4; (d) 0.0542, 38.8 and (e) 0.1106, 55.7.
\label{ASNA_All1}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{2}
\plotone{f2.eps}
\figcaption{
Contours of density  showing the simulation of a light jet in Case 2
in an atmosphere with declining density with $\beta=1.0$ (Runs
A16-A20) with the following values of $\eta_c$ and $M_c$: (a) 0.0210,
3.3; (b)
0.0328, 4.4; (c) 0.1310, 9.4; (d) 0.2568, 13.3 and (e) 0.5241, 19.2.
\label{ASNA_All2}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{3}
\epsscale{0.5}
\plotone{f3.eps}
\figcaption{
Time evolution of the jet simulations for Case 1 (Runs A11-A14) in an
atmosphere with density gradient and $\beta=1.0$. The curves represent
the
outline of the bow shock, cocoon and jet. The jet Mach number and
density
ratio are indicated and the time in each panel is given in Myr.
\label{ASNA_View1}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{4}
\plotone{f4.eps}
\figcaption{
As in Fig \ref{ASNA_View1} but for Case 2 (Runs A16-A19).
\label{ASNA_View2}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{5}
\epsscale{0.8}
\plotone{f5.eps}
\figcaption{
Pressure along the direction of the jet axis in Case 1 inside the cocoon
(thick
line), in the outer SAG region $R_b/2 < R <R_b$ (thin line) and in the
inner region $R_c<R<R_b/2$ (dashed line). The horizontal axis is the
distance to
the central object (jet inlet) and the vertical axis is the logarithm of
$P$
relative to the ambient pressure $P_a$. The lower value on the vertical
axis
corresponds to $P/P_a = 0.1$ and the upper value to $P/P_a = 10^3$. The
horizontal dotted line represents $P/P_a=1$ and the time is given in
Myr.
Note that in an isothermal atmosphere, $P_a$ is a function of the
distance
from the origin and therefore, we are plotting the ratio
$P(R,z)/P_a(R,z)$, where $R$ is the radius of the bow shock or the
cocoon at a distance $z$ along the jet axis.
\label{ASNA_Pres1}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{6}
\plotone{f6.eps}
\figcaption{
Exponent of the curve $z_h\propto t^m$ (source size as a function of
time) for Runs A11-A20 (a jet propagating in an isothermal 
atmosphere with a density
gradient) estimated using the two methods described in Paper I \S 5.1.
The open circles are the values derived using the first method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative of $z_h$ (derived using
the second method) with respect to $t$ and plotted as a function of
$z_h$.
The dotted line is the value predicted by the Type II and Type
III models, i.e., $m=1$.
\label{ASNA_Similar_Xh}
} \end{figure} \clearpage



\begin{figure}[ht]
\figurenum{7}
\plotone{f7.eps}
\figcaption{
Average radius of the jet (solid circles) and cocoon (open circles) at
the source
head as a function of the size of the source. The solid line is
described in \S~\ref{sec-Vh}.
\label{ASNA_Rh_Xh}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{8}
\epsscale{0.5}
\plotone{f8.eps}
\figcaption{
(a) Average radius of the jet head (triangles) and cocoon (open circles)
and
(b) the growth rate of the head radius as a function of the Mach number
for all
26 runs (A, B and C). The solid circles represent the head radius
calculated from the head speed of the simulations using Eq. \ref{vh} 
which assumes ram
pressure balance. The calculated head radius
tends to be in good agreement with the actual cocoon radius (open
circles).
\label{ASN_Rmed_ABC}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{9}
\epsscale{0.8}
\plotone{f9.eps}
\figcaption{
Advance speed of the source head (open circles) for Runs A11-A20 --
a jet propagating in an isothermal atmosphere with a density gradient.
The solid line is the advance speed calculated using Eq. 
(\ref{vh}) and using
the radius of the {\it cocoon\/} head determined from the simulations
and shown in Fig. \ref{ASNA_Rh_Xh}.
The dotted line represents the advance speed calculated using the radius
of the {\it jet\/} head also from Fig. \ref{ASNA_Rh_Xh}. The dashed line
represents the ambient sound speed.
\label{ASNA_Vh_Xh}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{10}
\plotone{f10.eps}
\figcaption{
Volume of the cocoon as a function of time for a jet propagating in an
isothermal atmosphere with a density gradient. The solid line represents
the least squares fit, the dotted line  is $V\propto t^{2}$ and 
the dashed line $V\propto t^{3}$.
\label{ASNA_Vc_T}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{11}
\plotone{f11.eps}
\figcaption{
Exponent of $V_b\propto t^n$ (source volume as a function of time)
estimated using the two methods described in the Paper I \S 5.1
for Runs A11-A20 (a jet propagating in an
isothermal atmosphere with a density gradient.)
The open circles are the values derived using the first method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative $d(\log{V_b})/d(\log{t})$
(derived using the second method) and plotted as a function of $z_h$.
The dotted lines are the values predicted by Type II (2) and Type III
(3) models.
\label{ASNA_Similar_Vb}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{12}
\plotone{f12.eps}
\figcaption{
As in Fig. \ref{ASNA_Similar_Vb} but for the cocoon volume.
\label{ASNA_Similar_Vc}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{13}
\plotone{f13.eps}
\figcaption{
The pressure in slices of the SAG as a function of the volume of the
slices for a jet
propagating in a constant density atmosphere. The curves represent
averages over 160
slices, with width equal to the grid spacing and which have been
displaced vertically by
an arbitrary amount. The  top curves represent the regions near the jet
head, and the
bottom curves represent  the regions near the central galaxy (jet
inlet). At low Mach
number (Runs A6 and A7), only a small range of volume is sampled because
the bow shock
leaves the grid laterally very quickly. The dashed curves are the
expected behavior for adiabatic expansion in a constant density 
ambient medium (eq.~\ref{PP2}).
\label{ASUA_Pb_Vs}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{14}
\plotone{f14.eps}
\figcaption{
The pressure in slices of the cocoon as a function of time for a jet
propagating in a
constant density atmosphere. The curves represent averages over 160
slices with width
equal to the grid spacing and which have been displaced vertically by an
arbitrary amount.
Top curves represent the regions near the jet head, and the bottom
curves represent the
regions near the central galaxy (jet inlet). The dotted lines correspond
to  $P\propto
t^{-1}$ (Type I models) and the dashed lines to $P\propto t^{-4/5}$
(Type III models).
\label{ASUA_Pc_Ts}} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{15}
\plotone{f15.eps}
\figcaption{
Average overall pressure of SAG region (open circles) and cocoon (solid
circles)
as a function of volume in a constant density atmosphere. The curves are
the
function $P(V)$ given by Eq. (\ref{PP4}) for $\delta=0$ (solid line),
$\delta=0.5$ (dashed line) and $\delta=1$ (dash-doted line).
The curves have not been fit to the data, but have been normalized to
the initial value of the SAG region pressure.
The dotted lines are the predictions of Type I ($P\propto V^{-1/2}$) and
Type III ($P\propto V^{-4/9}$) models.
\label{ASUA_Pt_Vt}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{16}
\plotone{f16.eps}
\figcaption{
Average overall pressure of the SAG region (open circles) and cocoon
(solid circles) as a
function of source size in a jet propagating in a constant density
atmosphere. The dotted
and dashed lines represent the Type I model ($P\propto z_h^{-1}$) and
the Type III model ($P\propto z_h^{-4/3}$), respectively.
\label{ASUA_Pt_Xh}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{17}
\plotone{f17.eps}
\figcaption{
Exponent of average pressure in the SAG region $P_b\propto z_h^b$ for
Runs A1-A10 (in a
uniform atmosphere) estimated using the two methods described in Paper I
\S~5.1.
The open circles are the values derived using the first method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative
$d((\log{P_b})/d(\log{z_h})$
(derived using the second method) and plotted as a function of $z_h$.
The dotted lines represent the predictions of the Type
I model (-1) and Type III models (-4/3).
\label{ASUA_Similar_Pb}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{18}
\plotone{f18.eps}
\figcaption{
As in Fig. \ref{ASUA_Similar_Pb} but for the average cocoon pressure.
\label{ASUA_Similar_Pc}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{19}
\plotone{f19.eps}
\figcaption{
Ratio between average pressure in the head (in the region defined by
80\% of maximum
pressure) and the cocoon as a function of source size (solid line) in a
jet propagating in
a constant density atmosphere. The open circles represent the ratio
between the ram pressure and cocoon pressure.
\label{ASUA_PhPc_Xh}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{20}
\plotone{f20.eps}
\figcaption{
Exponent of average pressure in the SAG region as a function of source
size $P_b\propto
z_h^b$ for Runs A11-A20 (a jet propagating in an isothermal atmosphere
with a density
gradient) estimated using the two methods described in the Paper I
\S~5.1.  The open circles are the values derived using the first 
method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative $d(\log{P_b})/d(\log{z_h})$
(derived using the second method) and
plotted as a function of $z_h$. The dotted lines are the values of $b$
predicted by Type II (-1) and Type III (-2) models.
\label{ASNA_Similar_Pb}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{21}
\plotone{f21.eps}
\figcaption{
As in Fig. \ref{ASNA_Similar_Pb} but for the cocoon pressure.
\label{ASNA_Similar_Pc}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{22}
\plotone{f22.eps}
\figcaption{
Average overall pressure of the SAG region (open circles) and cocoon
(solid circles) as a
function of volume in an isothermal atmosphere with a density gradient.
The curves are the function $P(V)$ given by Eq. (\ref{PP4}) 
for $\delta=0$ (solid line), $\delta=0.5$ (dashed
line) and $\delta=1$ (dash-doted line). The curves have not been fit to
the data, but have been normalized to the initial value of the 
SAG region pressure. The dotted lines are the
predictions of Type II ($P\propto V^{-1/2}$) and Type III ($P\propto
V^{-2/3}$) models.
\label{ASNA_Pt_Vt}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{23}
\plotone{f23.eps}
\figcaption{
Exponent of average pressure in the SAG region as a function of source
size $P_b\propto
z_h^b$ for Runs A11-A20 (a jet propagating in an isobaric atmosphere
with a density
gradient) estimated using the two methods described in the Paper I
\S~5.1.
The open circles are the values derived using the first method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative $d(\log{P_b})/d(\log{z_h})$
(derived using the second method) and plotted as a function of $z_h$.
The dotted lines are the values of $b$ predicted by Type
II (-1) and Type III (-2) models.
\label{ASNX_Similar_Pb}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{24}
\plotone{f24.eps}
\figcaption{
As in Fig. \ref{ASNX_Similar_Pb} but for the cocoon pressure.
\label{ASNX_Similar_Pc}
} \end{figure} \clearpage


\begin{figure}[ht]
\figurenum{25}
\plotone{f25.eps}
\figcaption{
Exponent of average pressure in the SAG region as a function of source
size $P_b\propto
z_h^b$ for Runs N11-N15 (a jet propagating in an isothermal atmosphere
with a density
gradient with $\beta=0.75$) estimated using the two methods described in
Paper I \S~5.1.
The open circles are the values derived using the first method, and
the horizontal axis represents the position of the first fitted point.
The solid line is the logarithmic derivative
$d((\log{P_b})/d(\log{z_h})$ (derived using the second method)
 and plotted as a function of $z_h$.
The dotted line is the value of $b$ predicted by Type III model (-11/6).
\label{ASNN_Similar_Pb}
} \end{figure} \clearpage

\begin{figure}[ht]
\figurenum{26}
\plotone{f26.eps}
\figcaption{
As in Fig. \ref{ASNN_Similar_Pb} but for the cocoon pressure.
\label{ASNN_Similar_Pc}
} \end{figure} \clearpage

\end{document}

