## 3. The modelIn this section I briefly summarise the dynamical and radiative model that form the basis for the extended treatment presented in this paper. Following this the prescription for the spatial distribution of the synchrotron emission within the radio lobes is developed. ## 3.1. The dynamical modelThe large scale structure of radio galaxies and radio-loud quasars of type FRII is formed by twin jets emanating from the central AGN buried inside the nucleus of the host galaxy. The jets propagate into opposite directions from the core of the source. They end in strong jet shocks and, after passing through these shocks, the jet material inflates the cocoon surrounding the jets. The cocoon is overpressured with respect to the ambient medium and therefore drives strong bow shocks into this material. Falle (1991) and KA showed that the expansion of the bow shock and
the cocoon should be self-similar which is supported by observations
(e.g. Leahy & Williams 1984, Leahy et al. 1989, Black 1992). In
these models the density distribution of the material the radio source
is expanding into is approximated by a power law,
, where Outside a few core radii, , the
power law assumed above with
provides an adequate fit to Eq. (1). Even for smaller distances
In the model of KA it is also assumed that the rate at which energy is transported along each jet, , is constant and that the jets are in pressure equilibrium with their own cocoon. The very high sound speed within the cocoon results in a practically uniform pressure within this region apart from the tip of the cocoon. The pressure in this `hot spot' region, named for the very strong radio emission originating in the shock at the end of the jets, is somewhat higher as the cocoon material injected by the jets at these points is not yet in pressure equilibrium with the rest of the cocoon. In the following I will concentrate on only one jet and the half of the cocoon it is contained in. From KA I take the expressions for the evolution of the uniform cocoon pressure, and that of the physical length of the jet, Here, is the ratio of specific
heats of the gas surrounding the radio source, KDA extend the model of KA to include the synchrotron emission of the cocoon. This is done by splitting up the cocoon into small volume elements, , the evolution of which is then followed individually. By assuming that these elements are injected by the jet into the hot spot region at time during a short time interval and then become part of the cocoon, KDA find where is the ratio of specific heats for the cocoon material and . Because the expansion of the cocoon is self-similar we can set for the volume of the total volume of the cocoon, following the notation of KA, , where is a dimensionless constant and depends on the geometric shape of the cocoon. In order to ensure self-consistency the integration of Eq. (4) over the injection time from to must be equal to the total cocoon volume . Substituting Eq. (3) for then yields Note, that this expression for is different from the one given by KA. In their analysis they used the expression for conservation of energy for the entire cocoon where is the volume of the hot spot region with pressure . KA then used the simplifying assumption that the cocoon has a cylindrical geometry, the expansion of which is governed along the jet axis by while its growth perpendicular to this direction is driven by . This implies , where is the ratio of the length of one jet and the full width of the associated lobe halfway down the jet. Kaiser & Alexander (1999b) subsequently derived empirical fitting formulae for as functions of and from an analysis of the flow of shocked gas between the bow shock and the cocoon. Their results showed that the original approximation tends to overestimate the value of . In the following I use a generalised empirical fitting formula based on their result and further calculations with additional values of , In order to satisfy Eqs. (5) and (6) I now generalise the
approach of KA by setting . Because
of the self-similar expansion of the cocoon ## 3.2. Synchrotron emissionThe cocoon volume elements are filled with a magnetised plasma and a population of relativistic electrons accelerated at the shock terminating the jet flow at the hot spot. They therefore emit synchrotron radio radiation. In optically thin conditions the monochromatic luminosity due to this process can be calculated by folding the emissivity of single electrons with their energy distribution (e.g. Shu 1991). Following KDA I assume that the initial energy distribution of the
relativistic electrons as they leave the acceleration region of the
hot spot follows a power law with exponent
between
and
. The electrons are subject to energy
losses due to the adiabatic expansion of
, the emission of synchrotron
radiation and inverse Compton scattering of the CMB. For a given
volume element at time where is the energy density of
the CMB radiation field at the source redshift where is the total energy density
of the relativistic particle distribution. At time
I set
and for simplicity
. In the following I assume that the
minimum energy condition (e.g. Miley 1980) is initially fulfilled in
each volume element and therefore
. From Eqs. (2) and (3) it
follows that and with the assumption
of completely tangled magnetic fields I find
(see also KA). With this the set of
equations describing the radio synchrotron emissivity,
, of a given volume element injected
into the cocoon at time only depends
on the present value of the pressure in the cocoon,
, and the age of the radio source,
## 3.3. Spatial distribution of the emissionSo far the cocoon volume elements were only characterised by their injection time into the cocoon, . From the analysis above it is not possible to decide where they are located spatially in the cocoon. From the above analysis it is clear that the radio spectrum emitted
by a given cocoon volume element depends on the `energy loss history'
of this part of the cocoon. Adiabatic losses only change the
normalisation of the emitted spectrum while its slope at a given
frequency is governed by the radiative loss processes of the
relativistic electrons. In the model described above the strength of
the magnetic field which determines the magnitude of synchrotron
losses is tied to the pressure in the cocoon,
. The value of
in turn depends on the energy
transport rate of the jet, , and a
combination of parameters describing the density distribution of the
gas the cocoon is expanding into, .
In the analytical scenario presented here the volume elements are the
building blocks of the cocoon and the variation of the radio surface
brightness of the cocoon of an FRII along its major axis can therefore
potentially provide information on the properties of the source
environment. For this I now identify the cocoon volume elements
with infinitesimally thin
cylindrical slices with their radius, The volume of a thin cylindrical slice of the cocoon is given by
, where
is the very small thickness of the
slice along the jet axis. The radius of the slices,
, depends on their position along the
jet axis, where , and can be determined from radio observations of the cocoon. Following the observational results of Leahy & Williams (1984) and Leahy et al. (1989) KA and KDA used the aspect ratio, , to characterise the geometrical shape of the cocoons of FRII sources. This ratio is defined as the length of one side of the cocoon measured from the radio core to the cocoon tip divided by its width measured half-way along this line. Using this definition it is straightforward to express in terms of as The dimensionless volume constant now becomes where is the complete Beta-function. Table 1 shows typical values for the dimensionless constants in the model.
## 3.4. BackflowThe cylindrical slices are injected into the cocoon at a time . For simplicity I assume that the slices remain and thus move within the cocoon as entities afterwards. In other words, I neglect any mixing of material between slices and I also assume that the geometrical shape of the slices does not deviate from the initial thin cylinders. Numerical simulations (e.g. Falle 1994) show that the gas flow in the cocoon is rather turbulent, at least close to the hot spot region. It is likely that large-scale turbulent mixing in the cocoon leads to large distortions of the projected cocoon shape as seen in radio observations. In this case, the regular cocoon shape described by Eq. (12) will be a poor representation of the `true' cocoon shape. For such distorted sources it is unlikely that the model presented here will provide a good description. However, the simple picture of cylindrical slices may still represent the `average' behaviour of the gas flow in more regularly shaped cocoons rather well. The model is designed to constrain source and environment parameters using mainly the gradient of the radio surface brightness along the cocoon. Problems with this simplified model will therefore arise if the relativistic electrons in the cocoon are distributed efficiently by diffusion. In Sect. 2 I show that the diffusion of relativistic particles is unlikely to change their distribution on large scales. It is therefore reasonable to assume that the relativistic particles are effectively tied to the cocoon slice they were originally injected into. Numerical simulations of the large scale structure of FRII sources strongly suggest that a backflow of material along the jet axis is established within the cocoon (e.g. Norman et al. 1982). The model describing the source dynamics predicts the growth of the cocoon to be self-similar and therefore the backflow within the cocoon should be self-similar as well. This suggests that the position of a slice of cocoon material injected into the cocoon at time is given by , where governs the velocity of the backflow at a given position along the cocoon. In order for the model to be self-consistent, all have to add up to the total volume of the cocoon, . Using Eqs. (15), (4), (5), (13) and (14) and replacing by an implicit Eq. for can be found from this integration; Note that this expression requires and gives . For the values of the shape parameters used in Sect. 5 ( and ) and I find . Note that for . The backflow velocity within the cocoon is given by where a dot denotes a time derivative. In the rest frame of the host galaxy the backflow is observed to flow in the direction of the source core for . For the cocoon material is stationary in this frame and for positive values the backflow is strictly speaking not a `backflow' but trailing after the advancing hot spot. The backflow is fastest just behind the hot spot and decelerates along the cocoon. The deceleration implies a pressure gradient along the cocoon which may seriously violate the assumption made for the dynamical model of a constant pressure throughout the cocoon away from the hot spots. To estimate the magnitude of the pressure gradient I use Euler's equation The density of the cocoon material, , is given by where I assumed that the entire energy in the jet is transported in
the form of kinetic energy of the flow with a bulk velocity
corresponding to the Lorentz factor
. With this, it is straightforward
with the help of Eq. (16) to integrate Eq. (17) which yields
the pressure along the cocoon as a function of where and I have used the condition . The mean advance speed of the cocoon of an FRII source, , is inferred from observations of lobe asymmetries to be in the range from 0.05 c (Scheuer 1995) to 0.1 c (Arshakian & Longair 2000). This is in good agreement with the predictions of the dynamical model used here (see KA). The ratio is usually of order 5 (see Eq. 7) and so even for only mildly relativistic bulk flow in the jet () and large gradients in the backflow velocity, e.g. , Eq. (19) predicts . Note that for the lower limit of this ratio increases to 0.82. From this I conclude that the existence of a pressure gradient along the jet within the cocoon is required to decelerate the backflow of the cocoon material. However, the pressure varies only by a factor of a few at most along the entire length of the cocoon. This implies that within this limit the model is self-consistent. © European Southern Observatory (ESO) 2000 Online publication: October 24, 2000 |