- X-ray emission from expanding cocoons
- 1 Introduction
- 2 Numerical simulations
- 3 X-ray signatures: Shell and cavity
- 4 The shell temperature
- 5 The cocoon dynamics
- 6 Discussion
- 7 Summary
- Appendix A
- References

A&A 402, 949-962 (2003)

DOI: 10.1051/0004-6361:20030302

**C. Zanni ^{1,2} - G. Bodo^{2} - P. Rossi ^{2}
- S. Massaglia^{1} - A. Durbala ^{1,2,3} - A. Ferrari ^{1}**

1 - Dipartimento di Fisica Generale dell'Università,
Via Pietro Giuria 1, 10125 Torino, Italy

2 -
INAF - Osservatorio Astronomico di Torino, Strada dell'Osservatorio
20, 10025 Pino Torinese, Italy

3 -
Universitatea din Bucuresti, Facultatea de fizica atomica si nucleara,
Bucuresti-Magurele,
PO Box MG-11, Romania, Italy

Received 3 December 2002 / Accepted 21 February 2003

**Abstract**

X-ray observations of extragalactic radiosources show strong evidences
of interaction between the radio emitting plasma and the X-ray
emitting ambient gas. In this paper we perform a detailed
study
of this interaction by numerical simulations. We study the propagation
of an axisymmetric supersonic jet in an isothermal King atmosphere and
we analyze the evolution of the resulting X-ray properties and their
dependence on the jet physical parameters. We show the existence of
two distinct and observationally subsequent different regimes of interaction,
with strong and weak shocks. In the first case shells of enhanced
X-ray emission are to be expected, while in the second case we expect
deficit of X-ray emission coincident with the cocoon. By a comparison
between analytical models and the results of our numerical
simulations, we discuss the dependence of the transition between these
two regimes on the jet parameters and we find that the mean controlling
quantity results to be the jet kinetic power. We then discuss how
the observed jets can be used to constrain the jet properties.

**Key words: **X-rays: galaxies: clusters - galaxies: jets - hydrodynamics

1 Introduction

X-ray observations of extragalactic radio sources have revealed strong evidences of interaction between the radio emitting plasma and the X-ray emitting gas in the ambient medium. The observation of Cygnus A with the ROSAT HRI (High Resolution Imager) by Carilli et al. (1994) and with the Chandra X-ray Observatory (Smith et al. 2002) showed deficits of X-ray emission in the cluster gas spatially coincident with the radio lobes. Observations of the Perseus cluster by Böhringer et al. (1993), also with ROSAT, and by Fabian et al. (2000) with the Chandra X-ray Observatory showed that the cluster emitting gas was displaced by the radio lobes of the source NGC 1275. A similar behavior has been observed in the Hydra A cluster, hosting the radio source 3C 218, with the Chandra X-ray Observatory by McNamara et al. (2000) (see also Nulsen et al. 2002), and in Abell 2052 that shows regions devoid of X-ray emission coincident with the radio lobes of 3C 217 (Blanton et al. 2001). Other clear examples of the interaction between radio lobes and the surrounding cluster gas are given by A 4059 (Heinz et al. 2002, Chandra) and A 2199 (Owen & Eilek 1998, ROSAT). Disturbances by a radio source are also found in the gas halo of some giant elliptical galaxies such as M 87 (Böhringer et al. 1995, ROSAT) and M 84 (Finoguenov & Jones 2001, Chandra).

On the other hand, theoretical models predict that jets in radiogalaxies inflate overpressured cocoons that displace and compress the ambient gas and the effects of such interaction could indeed expected to be the formation of cavities and shells in the X-ray emission, as shown by observations. A simple one-dimensional model of this interaction has been presented by Begelman & Cioffi (1989). More detailed and realistic models require the use of numerical simulations. In this context, Clarke et al. (1997) carried out calculations of the jet propagation in a King atmosphere obtaining simulated X-ray images to compare with ROSAT data on Cygnus A. They demonstrated that a deficit in the X-ray brightness is indeed shown in the simulation results and found agreement between simulations and observations for moderate Mach number of the jet (). A similar scenario is depicted by the numerical simulations of Rizza et al. (2000) that showed the interaction and disruption of a jet inside a cooling flow cluster. More recently Reynolds et al. (2001) have pointed out that cocoons start being strongly overpressured, but, during their evolution, their pressure decreases, and they then become essentially in pressure equilibrium with the ambient or even underpressured. During this evolution, therefore, the shock driven in the external medium is strong at the beginning and becomes very weak at the end. Reynolds et al. (2001) call this last phase "sonic boom''. The need for weak shocks comes from the observations of cool rims surrounding some of the X-ray cavities (see A 2052, Hydra A, Perseus A) that rule out the possibility of strong shocks driven by the expanding cocoon. To explain these observations several analytical (Churazov et al. 2000; Soker et al. 2002) and numerical (Churazov et al. 2001; Brighenti & Mathews 2002; Quilis et al. 2001) models of "bubbles'' of hot plasma expanding subsonically in the ambient medium have been studied. If these bubbles are buoyant they can also explain the presence of deficits of X-ray emission far from the radio lobes as observed in Perseus A. On the other hand, Heinz et al. (1998) and Alexander (2002) have studied self-similar solutions of simplified analytical models of overpressured cocoons expanding in a stratified medium in order to explain the observed features.

In this paper we analyze in detail, by using numerical simulations, the evolution of the X-ray properties of expanding cocoons and their dependence on the jet properties. The jets are characterized by their Mach number and their density ratio with the ambient medium density; the parameter plane is widely covered in order to consider a wide range of jet powers. We confirm the results presented by Reynolds et al. (2001) on the existence of two distinct and observationally different subsequent regimes of interaction, with strong or weak shocks but we are able to determine how and when the transition between these two regimes occurs, depending on the jet parameters. These results, on the other hand, show how the X-ray properties of cocoons could possibly be used as diagnostic for the jet characteristics.

The plan of the paper is the following: in Sect. 2 we describe the model, the basic equations and the initial conditions, in Sect. 3 we discuss the X-ray morphologies resulting from the simulations, in Sect. 4 we discuss the heating of the external material compressed by the expanding cocoon and the consequent changes in the X-ray emission properties, in Sect. 5 we discuss the physics of the cocoon expansion that leads to the interpretation of the different X-ray morphologies and in Sect. 6 we discuss the astrophysical relevance of our results, finally a summary is presented in Sect. 7.

2 Numerical simulations

We solve numerically the hydrodynamic equations for a supersonic
jet, in cylindrical (axial) symmetry in the coordinates (*r*,*z*)continuously injected into a gravitationally stratified (but not
self-gravitating) medium

where the fluid variables

The system of Eqs. (1) has been solved
numerically employing a PPM (Piecewise Parabolic Method) hydrocode (Woodward
& Colella 1984).
The integration domain has a size
where
,
where *a* is
the core radius, defined below, and
has been divided in
grid points. The axis of the jet is
along the left boundary of the domain (*r*=0), where we have imposed
symmetric boundary conditions for
and
antisymmetric conditions for *v*_{r}. Reflective boundary conditions are
also imposed on the boundary of injection of the jet (*z*=0) outside its
radius in order to reproduce a bipolar flow and to avoid spurious inflow
effects.
Free outflow is set on the
remaining boundaries by imposing a null gradient for each variable (
).

The undisturbed ambient medium is assumed stratified in a spherically
symmetric gravitational well, according to a classical isothermal King
model:

(2) |

with and . The resulting pressure stratification is kept in equilibrium by an appropriated external gravitational potential.

A (cylindrical) jet is injected from the bottom
boundary of the integration domain, in pressure balance with the ambient. The
initial jet velocity profile has the form

(3) |

and the corresponding density profile is

(4) |

with

Measuring lengths in units of the core radius *a*, velocities in units of the
adiabatic sound speed
in the undisturbed external medium and the density in
units of the ambient central density ,
our main parameters are the
Mach number
and the density ratio
.
Consistently the unit for the kinetic power is

and the unit of time is

(6) |

where

In Table 1 we show the parameters

M |
L_{j}/L_{k} |
||

10 | 0.1 | 10^{2} |
1.38 |

60 | 0.001 | 1.22 | |

60 | 0.01 | 0.49 | |

60 | 0.1 | 0.15 | |

120 | 0.001 | 0.76 | |

120 | 0.01 | 0.25 | |

120 | 0.1 | 0.07 |

3 X-ray signatures: Shell and cavity

The general structure of the interaction between a low density jet and
the ambient medium is well known since the first simulations of Norman
et al. (1982) (see also Massaglia et al. 1996 and Krause 2003). The flowing jet matter, slowed down by one or more
terminal shocks, inflates a cocoon that compresses and drives shocks
in the surrounding external medium. The compressed ambient material
forms a shell surrounding the cocoon: the boundary between the shell
and the cocoon is marked by a contact discontinuity, while the
boundary between the shell and the undisturbed external medium is
marked by a shock. We will call "*extended cocoon*'' the whole region
interested in the interaction between the jet and the ambient medium.
The "*extended cocoon*'' is then formed by the cocoon proper and by the
surrounding shell. The cocoon, which is formed by
the expanded jet material forms a cavity with very low density and
high temperature, in which the X-ray emissivity is strongly depressed.
On the other hand the external material in the shell has an enhanced
emission due to its compression. In addition, depending on the
external shock strength this material can be heated and its emission
properties can then change.

In principle, we then expect three main features in the X-ray properties of the region of interaction between a jet and the ambient medium:

- 1.
- a region of depressed emission coincident with the cocoon;
- 2.
- a shell of enhanced emission;
- 3.
- a variation in the spectral properties of the emission by the shell.

In Figs. 1-3 we show in the first
two columns simulated X-ray flux distribution
for the seven cases described in Table 1.
The figures are symmetrical with respect to the *r* and *z* axes.
The emissivity per frequency unit is computed with a Raymond-Smith
thermal spectrum code (see HEASARC Web Page) from the electron density
and temperature distribution obtained in our calculation.
The emissivity
is then
integrated in the
band. The flux distribution
is calculated integrating the optically thin emissivity along the line of sight
that is assumed perpendicular to the jet axis:

We consider an isothermal ambient medium with a temperature

Figure 2:
Simulated X-ray fluxes,
in logarithmic scale, in the 0.1-4 keV band for the M=60 cases. The rows
refer from top to bottom to the
cases respectively.
The quantities represented are the same as in Fig. 1. |

Figure 3:
Simulated X-ray fluxes,
in logarithmic scale, in the 0.1-4 keV band for the M=10,
case.
The quantities represented are the same as in Figs. 1, 2. |

These figures show a sequence of morphologies starting from cases in
which a cavity is not present going to cases in which the cavity is
the dominant feature of the image. In particular, we see that for *M*
= 120 the brightness depression is evident only for the lighter case,
while for the *M* = 60 cases starts to be present for
during its evolution and is very
evident during the whole evolution for
.
For *M* = 10the cavity is dominant already for .

The presence or absence of the brightness
depression, as discussed by Clarke et al. (1997) depends on the
thickness of the shell. In fact, the line of sight that goes through
the cocoon region with very low emissivity crosses also the shell of enhanced
emissivity, and the observed flux is higher or lower than the undisturbed
profile depending on the interplay between the two effects. If the
shell is narrow, as a consequence of mass conservation, it will have
a high density, the emissivity will be greatly enhanced and will overcome
the decrease in the cocoon giving an observed brightness higher than the
undisturbed profile. Following Clarke et al. (1997), we can write the
ratio of the observed flux to the undisturbed one as

where we have neglected the dependence of emissivity on the temperature and is the ratio between the shell width and the cocoon radius. From Eq. (8), we see that the ratio increases as goes to 0 and becomes less than 1 for .

In Fig. 4 we show the
density distribution for the *M* = 60 cases: each row refers to a
different density ratio (
from top to bottom)
and each column is for different times corresponding to cocoon lengths 1, 1.5, 2 respectively.
We see in fact that the width of the shell
increases going from the high to the low
case. Moreover we can
see that the shell tends to widen and the bow shock becomes weaker
during the evolution of the cocoon, apart from the
case
in which the relative thickness of the shell remains constant.
Note that the localized feature appearing along the *z*=0 axis
is due to the assumed reflection boundary conditions and does not affect the
general morphological and dynamical behavior.

In order to better quantify the deficit or the enhancement in the X-ray
emission, we have computed integral measures
and
of these
quantities defined as

where the surface integrals are computed over the domains and defined as the areas over which the integrand is respectively lower (the cavity) and greater (the shell) than zero, where

Figure 5:
(Upper panels) Plot of the integrated flux deficit as a function
of cocoon length in the 0.1-4 keV band (left) and in the 4-10 keV band
( right) for the M=60 cases.
This quantity is
defined in Eq. (9).
(Lower panels) Plot of the excess emission as a function of cocoon length in the
0.1-4 keV band (left) and in the 4-10 keV band
(right) for the M=60 cases. This quantity is
defined in Eq. (10).
In the four panels the
three lines refer to the
(solid),
(dashed)
and
(dash-dotted) cases. |

Figure 6:
Temperature plots
for the M=120 cases (
from left to right respectively)
at a time corresponding to a cocoon length of 2 core radii. (Upper panels)
Plot of the quantity E(T) as defined in Eq. (11).
It represents the energy emitted in the 0.1-10 keV band by the gas at the temperature T.
The vertical line marks the temperature
of the ambient isothermal medium while the peak correponds to the compressed shell emission.
The emission is normalized to the total emission from the unperturbed atmosphere.
( Lower panels)
Plot of the shell average temperature as a function of the angle
with origin in r=0, z=0. The
angle corresponds
to the direction of the jet axis. The ambient temperature is taken 2.3 keV. |

4 The shell temperature

Figure 7:
Temperature plots
for the M=60 cases (
from left to right respectively)
at a time corresponding to a cocoon length of 2 core radii. The quantities plotted are the same
as in Fig. 6. |

Figure 8:
Temperature plots
for the M=10,
case at a time corresponding to a cocoon length of 2 core radii.
The quantities plotted are the same
as in Figs. 6 and 7. |

As we discussed, the shock driven in the external medium can heat it
and its effects can be more or less evident depending on its
strength. The heating of the shell can change the typical
emission energies, an example of the consequences of this effect has
been shown in the bottom panel of the first column of
Fig. 5 where we have seen that the case
with
seemed to present an anomalous behavior. We can now
compare the first column of Fig. 5 with the second column where we
show the same quantities but in the range 4-10 keV. Looking at the
bottom panels we see that the case ,
which in the softer
band presented an excess emission below that of the case
,
shows in the harder band an excess emission larger than the other cases.
The gas in the compressed shell becomes in fact hotter as we increase
the value of
and the emission is then shifted towards higher
emission energies. To quantify in more detail this temperature change, we
have plotted in the first row of Figs. 6-8
the quantity

where is the temperature at a given position, is the Dirac delta function and are the radiative losses per unit volume in the 0.1-10 keV band as calculated with the Raymond-Smith code. This quantity measures the energy emitted by the gas at the temperature

From the figures we see that, for the cases with
and *M*=60,120, the
shell temperature is everywhere larger than 8 keV and reaches
temperatures up to 850 keV at the jet head of the *M*=120 case. The lower density cases,
instead, present an increase in temperature mainly concentrated in the
forward part of the cocoon. This temperature distribution will have
consequences in the shell morphology as seen in different X-ray
bands. In fact, we expect to see emission from the forward part of the
cocoon at high energies, while the backward part is expected to be
more dominant at lower energies.

5 The cocoon dynamics

5.1 Expansion in a uniform medium

The first attempt to build an analytical model for the cocoon dynamics
is due to Begelman & Cioffi (1989) and Cioffi & Blondin (1992). They
consider only the case of strongly overpressured cocoons, for which
they consider only strong shocks driven in the external medium and
they essentially neglect the external pressure. For our purposes, we
have to extend the description to a more general situation, in which
the external shock can be of any strength and the external pressure is
taken into account. We describe in detail the model in the Appendix,
here we give only the resulting behavior of the extended cocoon
radius and cocoon pressure versus time: the cocoon radius is
given by

and the cocoon pressure is

Equations (12), (13) describe two different phases in the evolution: initially the cocoon is strongly overpressured and the solutions behave in the same way as described by the Begelman & Cioffi (1989) model, with the radius proportional to

Therefore, for a given Mach number and for a given length of the cocoon, we expect the jets with lower density ratios to be less overpressured and therefore to form wider and less dense shells.

In a similar way we can also find a solution assuming a spherical symmetry
for the cocoon, that is solving the system of Eqs. (A.6-A.8) taking
.
Taking into account separately the
strongly overpressured (
)
and the weakly overpressured
(
)
phases we can find two distinct regimes of expansion:
a supersonic one during which the external radius behaves like

and the pressure decreases as

and a weakly overpressured phase during which the cocoon pressure tends to a constant value and the radius expands

In the spherical symmetry model the cocoon properties scale with the jet power since it is the only jet parameter that enters the system of equation solved in the Appendix. It is worthwhile noticing that our solution for the spherical strongly overpressured cocoon (Eqs. (15) and (16)) is the same, apart from numerical constants, of Heinz et al. (1998) who solved a system of equations similar to ours but limited their solution to the strongly overpressured regime.

Looking at the geometry and at the detailed structure of the shell and the cocoon, these analytical models are too simplified to describe in detail the radiative properties of the cocoon but still they show clearly how the relative thickness of the shell can increase only in a weakly overpressured regime.

5.2 Expansion in a stratified medium

The cocoon expansion in a stratified medium is, of course, much more
complicated, since the external pressure is not constant, and the
development of an analytical model becomes more problematic. However
the above discussion can provide a framework for understanding the
results of numerical simulation.

We recall that the presence of a brightness depression depends substantially on the relative thickness of the shell. We then must first study the evolution of the extended cocoon radius and the cocoon proper one. We first analyze the behavior of the extended cocoon radius by comparing the results directly obtained by the numerical simulations with an estimate obtained, using the average cocoon pressure as the driver for the expansion. More precisely, we obtain this estimate by integrating numerically Eq. (A.8), where for we use the average value obtained at every time step from the simulations. For defining the extended cocoon radius in the simulations, we concentrate on the base portion of the cocoon and we take an average value of the bow shock radius between 1/8 and 1/4 of the total length of the cocoon. The results of the comparison are represented in Fig. 9, where we plot and as a function of time for all the cases we have considered. In all the panels the solid curves correspond to the results obtained in the simulations, while the dashed curves correspond to the estimates obtained through the average pressure. The figures show that this estimate reproduces very well the actual behavior of the extended cocoon radius: this result tells us that the average cocoon pressure is a good estimate of the local pressure driving the cocoon expansion. In a similar way we can proceed for the cocoon radius , for which we can use Eq. (A.5) where with we intend the fraction of jet power that is thermalized and goes into internal energy of the cocoon. This fraction can be estimated from Eq. (A.6). Equation (A.4) can be integrated giving

where again for we can use the average cocoon pressure obtained above. This equation gives us an estimated behavior of the cocoon volume and from this we can estimate the behavior of the cocoon radius as where

Looking in more detail at the behavior of the extended cocoon radius,
we observe that, as expected from the discussion of the uniform case,
its expansion velocity decreases faster initially, when the cocoon is
strongly overpressured, and then decreases more slowly, when the shock
becomes weaker. Fitting a power law
for the
quantities ,
,
in the initial and
terminal part of the evolution for the cases with *M* = 60, we get the
exponents
reported in Table 2. We see
that, at later times, the exponents of
become systematically larger than those
obtained at the initial times. Moreover we see that the differences
between the two exponents are larger for the lower density cases and
for
are minimal. In this last case, in fact the cocoon
stays strongly overpressured during the whole evolution.

= 0.1 | < 1.0 | > 1.5 |

0.85 | 0.85 | |

0.54 | 0.67 | |

0.43 | 0.56 | |

= 0.01 | < 1.0 | > 1.5 |

0.75 | 0.95 | |

0.56 | 0.71 | |

0.45 | 0.53 | |

= 0.001 | < 1.0 | > 1.5 |

0.77 | 0.89 | |

0.69 | 0.81 | |

0.46 | 0.07 |

A similar analysis can be done for the cocoon radius . We see that in the high and intermediate density cases the exponent for also increases but less than the exponent for . In the low density case, instead, becomes almost constant in time. This behavior tells us that, when the cocoon is not any more strongly overpressured, the relative shell thickness start to increase as it happened in the homogeneous case. The behavior of the relative thickness of the shell can be in fact derived from Table 2, since it is related to the ratio . We then see that it has in general a slower increase at the beginning of the evolution, and accelerates in the following phases. As discussed above, the effects of not being strongly overpressured are more evident by decreasing the value of and, in fact, the increase of the relative cocoon width is larger for smaller . If we compare the cocoon widths for equal distances of jet propagation we have a further effect that amplify the consequences of the behavior discussed above, namely we have to take into account that the advance velocity of the head of the jet decreases when we decrease the value of .

In this section we have seen that the transition from a strongly overpressured cocoon to a cocoon which is essentially in pressure equilibrium with the surroundings leads to different properties of the shell of the compressed external material. In Sect. 5.3 we will try to see whether it is possible to determine in a more quantitative way how this transition depends on the jet physical parameters.

5.3 The transition from strong to weak shocks

From the above discussion we have seen that the average cocoon
pressure can be used for interpreting the cocoon dynamics and therefore
determining the transition between the strong and weak shock regimes.
In the case of uniform external medium, the cocoon
pressure, after a decrease proportional to *t*^{-1}, tends to a
constant. In the present case, with a decreasing external
pressure, we expect that the cocoon pressure does not tend to a
constant but that it will tend to follow the behavior of *P*^{*},
defined in Eq. (A.7).

Figure 10:
Plot of the pressure vs. cocoon length. The rows refer to the
different Mach numbers while columns to the different values. The solid line shows the
evolution of the average cocoon pressure with time in the different cases. The
dashed one represents the external pressure averaged on the cocoon volume as defined in
Eq. (A.7). |

In Fig. 10 we have plotted the behavior of
(solid curve) and *P*^{*} (dashed curve) against the cocoon length, where
is the
average cocoon pressure and *P*^{*} is the external pressure averaged
over the cocoon volume. The ratio between the two quantities gives a
measure of the average strength of the shock driven in the external
medium.
Looking at the behavior of the cocoon evolution in the different cases we
have an always strongly overpressured expansion for the cases *M*=120 and
and for *M*=60 and ,
in these three cases the transition length
would be larger than the actual longitudinal size of our domain; two cases showing
a transition from strongly overpressured to weakly overpressured cocoons for *M*=120,
and
*M*=60, ;
finally we have two cases where the cocoon is
always weakly overpressured,
i.e. for *M*=60,
and *M*=10, .
This is in agreement with
the discussion on the X-ray flux distributions following Figs. 1-3.

In order to determine the length of the cocoon at which the transition occurs, we look for
a scaling law for the quantity
as a function of the parameters ,
*M* and
during the initial strongly overpressured phase.
Taking into account only the cases that are stronlgy overpressured at the beginning of the evolution,
we notice that the initial decrease of the average cocoon pressure follows a similar
behavior for all the cases that we have considered:
with a power law fit
to the
initial evolution of the different cases we find a mean value
with
an uncertainness of 10%.
In Table 3 we show the
values of the quantity
at the beginning of the evolution,
when the cocoon length is one third the core radius and the cocoon is
typically strongly overpressured (notice that the cases
*M*=10,
and *M*=60,
make an exception). These values can be
represented, with good approximation by the scaling
.
The general scaling law thus becomes:

The scaling relation (19) shows that the initial pressure of the cocoon is not proportional to an arbitrary combination of the jet parameters

Figure 11:
Plot of the values of
at a cocoon length
(see Table 3) against jet kinetic luminosity
.
The solid line is a
,
that is also the best fit, excluding the value of the case M=10,
that is
in a weakly overpressured regime already. |

Of course the transition from a strongly
to a weakly overpressured regime is not sudden, but gradually happens when
becomes of the order of unity. Setting a constant value for
of this order
in Eq. (19)
we derive the scaling law for transition length
(as we have done
for the uniform case, see Eq. (14)).

or in terms of the jet kinetic power

Equation (20) represents a family of curves in the plane (), and, for a given cocoon length one of the curves of this family will mark the separation between cocoons that drive strong shocks in the ambient medium (strongly overpressured regime) and cocoons that drive only weak shocks (weakly overpressured regime). In Sect. 6 we will discuss the astrophysical relevance of these results.

6 Discussion

In Sects. 3-5 we have seen that strongly
and weakly overpressured cocoons present different X-ray
morphologies. The former ones will show no deficit of emission
accompanied by a strong
emission from a shell marking the shock, driven by the cocoon
expansion. The material in the shell will be much hotter than the
ambient medium, the shell emission will then be shifted to higher
frequencies and therefore more visible at higher
X-ray energies, especially near the jet head. Weakly
overpressured cocoons will be instead characterized by the presence of a
deficit of emission in the cocoon, while the emission from the shell will be much
less visible than in the previous case; in addition, the material in the shell will be
essentially at the same temperature as the ambient medium and there
will be no change in the emission spectrum. These two regimes depend
on the jet physical parameters and on the age of the cocoon.
High Mach
number jets with high densities will be strongly overpressured for a
longer fraction of their life, decreasing the density and the Mach
number the transition to weakly overpressured cocoon will occur at an
earlier stage. We have found that the scaling of the transition
length from one regime to the other is given to a good approximation by
Eq. (20). Fixing a length of the cocoon, from this
equation we can then obtain a relation between Mach number and density
ratio such as
.
This scaling law has been
represented in Fig. 12 with the dotted, dashed and
dash-dotted lines for cocoon lengths equal to 0.5, 1 and 2 core
radii respectively. The proportionality constant
has been set
to fulfill the transition conditions in the cases where the
transition between the two regimes is observed, i.e.
and
.
In the same figure we can
also notice the presence of another line, that individuates a region in
which the jet becomes subsonic with respect to its internal sound
speed.

M \ |
0.1 | 0.01 | 0.001 |

120 | 192.03 | 31.30 | 6.99 |

60 | 34.35 | 8.13 | 2.17 |

10 | 0.99 |

Since Mach number and density ratio are physically very meaningful parameters, but cannot be determined for actual jets, it is favorable to express the scaling relation in terms of the jet kinetic power. In order to do so, we make use of Eq. (21) where is a quantity that depends on the properties of the environment. From this relation we see that jets with the same kinetic power have the same transition length between the two phases. The curves in Fig. 12 separating the strongly and weakly overpressured regimes, which are given by , will therefore correspond also to a constant value of . These considerations tell us that the separation between the two regimes is essentially determined by the jet kinetic power: the transition will occur at higher values of for high kinetic power jets and at lower for low kinetic power jets. For instance the dashed curve shown in Fig. 12 representing the separation between the two regimes for a cocoon length equal to 2 core radii corresponds to . The properties of the environment enter in the parameter that fixes a measure for the jet kinetic power. For a typical cluster environment with a central density , a core radius 100 kpc and a temperature 3 keV we have and the dividing kinetic power for a cocoon length equal to 1.5 core radii will be . It is important to notice that for jets that are slightly supersonic with respect to their internal sound speed the jet enthalpy flux becomes comparable to the kinetic one. In these cases our estimates for the jet kinetic power should be corrected at most by a factor two in order to obtain an approximation of the total power of the jet.

Considering the particular case of Cygnus A, its cluster
environment is characterized by a central density
,
a core radius 35 kpc and a temperature 3.4 keV
(Carilli et al. 1994), giving
.
The radio lobes of Cygnus A
show an extent of 70 kpc, twice the core radius
(
). For this
length the dividing power is given by
,
as it is possible to see in Fig. 12. Since Cygnus A clearly
shows deficit of X-ray emission and therefore is in a weakly overpressured
regime, these estimates give an upper limit to the kinetic power of its jet
.
Nevertheless the pressure of the
expanding cocoon must be higher than the ambient one since observations
by Chandra (Smith et al. 2002) show clearly that the shell is slightly
hotter than the surrounding medium. Then the cocoon is still expanding
as a (weak) shock wave more than a sound wave.

Figure 12:
Representation of the different regimes of the cocoon expansion in the
plane. Inside the dashed area the jet becomes subsonic with respect to
its internal sound speed. The solid lines correspond to a constant kinetic jet power
given in unit of the power
defined in Eq. (5). The
fragmented curves separate strongly and weakly overpressured regimes for different
lengths of the cocoon given in core radii units. It can be seen that these lines correspond to
lines of constant kinetic jet power. The diamonds correspond to the simulated cases. |

Another well known example of the interaction between a radio source and
the thermal gas of a cluster is the FR I type radio galaxy 3C 84 inside the
Perseus cluster. The cluster core can be modeled with a central electron
density
,
a core radius kpc and a central
temperature
keV (Schmidt et al. 2002) yielding
.
The radio lobes show an average extent of 22 kpc, giving a dividing power
.
These
estimates give an upper limit on the kinetic luminosity of the jet
.
This upper limit can be lowered
considering that the shell of enhanced emission surrounding the cavities
contains clearly the X-ray coolest gas in the cluster (Fabian et al. 2001).
This fact rules out
the presence of strong shocks driven by the expanding cocoon. In our scheme
this means that the cocoon must be in a weakly overpressured regime in the
first phase of expansion yet as for example in the case *M*=10, .
As it can be seen in Fig. 8 the average temperature of the shell for this
simulation is equal to the ambient one and in the shell there is also emission from gas cooler
than the ambient one. This cooling is due to the adiabatic expansion
of the shell after being compressed by a weak shock, in a way similar to that
described in the simulations of expanding hot bubbles
by Brighenti & Mathews (2002). Taking this
case (
)
to determine a limit on the kinetic power of the jet
we obtain
.
For deriving the total jet power this estimate should be corrected by the enthalpy
term, that for the above values of *M* and
is approximately equal to the 30% of the kinetic power.
This estimate is similar to the
one by Fabian et al. (2001) who used the analytical bubble model by Churazov et al. (2000) to
determine their limits. This substantial agreement can be understood considering that
the Churazov et al. (2000) model solves exactly Eq. (A.4) with spherical symmetry
assuming that the bubble pressure is equal to the external one.
The limits of Fabian et al. (2001) are then determined requiring
that the bubble has the observed dimensions, that it expands subsonically
and that it is not buoyant. This situation is similar to what we refer to as a weakly
or not overpressured regime.
The great difference between our low power cases
and a hot underdense bubble is the presence of the highly collimated jet that forms
a thermal hot spot and is likely not effective in uplifting
the cooling flow gas as proposed for example by
Soker et al. (2002) to explain the presence of cool gas in the rims around the cavities.
A collimated jet tends to go through the gas of the cluster displacing it aside and
without uplifting it. Once the jet has terminated its active phase is rapidly
destroyed (Reynolds et al. 2002) and the lobes can evolve like
buoyant bubbles (see for example Churazov et al. 2001 or Quilis et al. 2001).
An example similar to Perseus A is the Hydra A cluster containing the powerful FR I radio
source 3C 218. Given the properties of the X-Ray gas ( kpc,
,
keV, David et al. 2001) and a cocoon length of 49 kpc
we obtain an upper limit on the jet power
.
Since
there is no indication that the gas surrounding the radio lobes is hotter than the ambient
cluster gas this limit can be lowered at least by an order of magnitude.
On the other hand it is possible to find some examples of radio sources inside galaxy clusters
that do not show the presence of X-ray cavities. In our scheme these sources could correspond
to radio lobes still in a strongly overpressured phase. For example the FR II radio source 3C 295
is found inside a cluster whose core is characterized by a radius
kpc, a central
density
and a temperature
keV
(Allen et al. 2001). This cluster does not show any cavity in its X-ray emission.
Given the longitudinal dimension of the radio source 17.6 kpc we can estimate
a lower limit of the jet power
.

7 Summary

In this paper we performed numerical simulations of axisymmetric supersonic jets propagating in an isothermal background atmosphere. In agreement with the results presented by Reynolds et al. (2001), we find two distinct and subsequent regimes of interaction between the cocoon and the external medium. In the first phase of evolution, the overpressured cocoon drives a strong shock in the ambient medium, forming a thin, hot and compressed shell of shocked material, in the second phase the shock becomes very weak and the shell widens, decreasing its density and temperature. The resulting X-ray morphology in the two phases is different: in the strongly overpressured phase, we expect a shell of enhanced X-ray emission surrounding the radio emitting material, while, in the weak shock phase, we expect a deficit of X-ray emission coincident with the radio lobes. We have studied the dependence of the transition between these two phases on the physical jet parameters, by a wide coverage of the parameter space and by a comparison of the results of numerical simulations with analytical models. We find that the transition length between the two regimes depends essentially only on the jet power scaled over a value dependent on the properties of the ambient medium.

The authors acknowledge the italian MIUR for financial support, grant No. 2001-028773. The numerical calculations have been performed at CINECA in Bologna, Italy, thanks to the support of INAF.

As we have discussed, the jet replenishes the cocoon
of matter and energy, therefore the cocoon expands compressing the
ambient medium and if the expansion speed of the cocoon is highly
supersonic, it will drive a strong shock in the external medium.
The first attempts to give an analytic description of theses processes
was done by Begelman & Cioffi (1989) and Cioffi &
Blondin (1992). To describe the energy input by the jet, they write
the following energy equation for the extended cocoon

where is the total energy in the extended cocoon and is the jet power. Assuming a constant energy input by the jet and that all the energy is converted in thermal energy, we can then write the average (extended) cocoon pressure as

where is the volume of the (cylindrical) cocoon equal to , is the extended cocoon radius and the cocoon length assumed to increase as , with given by the one-dimensional estimate , where . The lateral expansion of the extended cocoon can then be obtained by assuming a strong lateral shock, for which we can write

where is the external density. Equation (A.3) can be integrated giving for the time behavior of a dependence on

where is now the total energy of the cocoon proper and its volume. This equation can be written in an integral form, neglecting the initial value of the volume:

Inserting in this equation the expression for the pressure derived above and assuming that the volume is proportional to and the cocoon length behaves in the same way as that of the extended cocoon we can derive that the cocoon radius behaves as

where the second term on the right hand side represents the contribution described above and is the pressure of the external medium. The equation for the pressure, instead of Eq. (A.2) becomes

where

In the case of uniform external pressure, we have, of course, that . A second extension to the Begelman & Cioffi (1989) model is done by considering a lateral shock of arbitrary strength and writing the expansion speed of the cocoon in a more general way as (see e.g. Landau & Lifshitz 1959)

The system of Eqs. (A.6-A.8) can be solved analytically in the case of uniform external pressure assuming a cocoon length and , i.e. the kinetic jet power. This approximation is strictly valid for jets that are greatly supersonic with respect to their internal sound speed, so as to neglect the enthalpy flux term. Neglecting its initial value (jet radius), the cocoon radius then is given by

and the cocoon pressure is

Substituting the solution of the pressure in the weakly overpressured regime in Eq. (A.5), we obtain that the internal radius begins to expand subsonically and tends asymptotically to a constant value.

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