Astron. Astrophys. 362, 447-464 (2000)

3. The model

In 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 model

The 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 r is the radial distance from the source centre and is the core radius of the density distribution. X-ray observations of groups and clusters of galaxies show that the density of the hot gas in these structures is often distributed according to (e.g. Sarazin 1988)

Outside a few core radii, , the power law assumed above with provides an adequate fit to Eq. (1). Even for smaller distances r good power law approximations can be found by adjusting and (e.g. Kaiser & Alexander 1999a).

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, t is the age of the jet flow and is a dimensionless constant. The ratio, , of the pressure in the hot spot region and is constant in the model of KA.

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 f is a constant and will depend on the ratio . The strict separation of the cocoon material into the hot spot region and the rest of the cocoon with two distinctly different values of the respective pressure within these regions is of course artificial. In the cocoons of real FRII objects the transition of jet material from hot spot to cocoon will be accomplished in a continuous hydrodynamical flow along a pressure gradient much smoother than the sudden change from to described here. However, a detailed model of this flow is beyond the scope of this paper and for simplicity I will use the assumption of a strict spatial separation in the following. As the two cocoon regions are in physical contact with each other, f may become negative as the expansion of one region may influence the evolution of the other. Substituting Eqs. (2) and (3) into Eq. (6) gives

3.2. Synchrotron emission

The 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 t that was injected into the cocoon at time these losses result in an energy distribution (see KDA)

with

where is the energy density of the CMB radiation field at the source redshift z and is the rest mass of an electron. Here I have used and . The normalisation of the energy spectrum, , is given by integrating the initial power law distribution over the entire energy range

or, for ,

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, t.

3.3. Spatial distribution of the emission

So 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, r, perpendicular to the jet axis. A similar approach is used in the model of Chyzy (1997). However, in this model the dynamical evolution and the radio luminosity of the entire cocoon depend on the geometrical shape and evolution of the volume elements. In the model of KA and KDA this is not the case and the identification of the with a specific geometrical shape does not alter their results.

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, l, which I define in units of , the length of one half of the entire cocoon (see Sect. 3.1). The outer edges of the slices form the cocoon boundary or contact discontinuity. Most FRII sources have cocoons of a relatively undistorted, ellipsoidal shape (e.g. Leahy et al. 1989). I therefore parameterise the cocoon boundary as

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.

Table 1. Typical value ranges for the dimensionless constants in the dynamical model. This assumes and (e.g. Leahy & Williams 1984).

3.4. Backflow

The 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 l;

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
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