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

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Appendix A: diffusion of relativistic particles in the cocoon

A.1. Quasi-linear anomalous diffusion

In the model it will be assumed that the magnetic field within the cocoon is completely tangled on a scale [FORMULA] much smaller than the size of the cocoon, [FORMULA]. In this case the relativistic electrons can diffuse quickly within `patches' of coherent magnetic field of a size [FORMULA]. However, it is difficult for them to escape their patch as this would involve diffusion perpendicular to field lines. In the presence of turbulent motion within the plasma underlying the magnetic field, patches can be stretched out and in this case the diffusion of charged particles into an adjacent patch becomes more likely. Duffy et al. (1995) calculate how far a given electrons has to travel along a field line in its original patch before the patch becomes so stretched out that it can escape. Note here that this treatment is valid only in the quasi-linear regime for which the relative amplitude of irregularities of the magnetic field is much smaller than the ratio of the turbulent correlation lengths perpendicular and parallel to the local magnetic field. The efficiency of this diffusion then depends crucially on whether the electron is able to travel this distance ballistically or whether it must diffuse along the field line. The more efficient ballistic regime requires that

[EQUATION]

where [FORMULA] is the gyro-radius of the particle in the field, [FORMULA] is the diffusion coefficient perpendicular to the field and [FORMULA] is the time it takes the particle to escape the patch. For a relativistic electron moving at speed v and corresponding Lorentz factor [FORMULA] following Chuvilgin & Ptuskin (1993) [FORMULA] with [FORMULA] the Bohm diffusion coefficient and [FORMULA], where [FORMULA] is the rate of collision of the particle and [FORMULA] is its gyro-frequency. For the assumption of Blundell & Rawlings (2000) that the coherence length of the magnetic field is roughly 10 kpc in all directions, the inequality (A.1) yields [FORMULA] for a magnetic field strength of 130 µG, appropriate for the fiducial model of Sect. 4.3, and an electron with a Lorentz factor 1000. This implies that the time between collisions of this electron must be greater than 2 Myr, i.e. the mean free path of the electron is of order 600 kpc. This is clearly unphysical in the case of [FORMULA] kpc as relativistic particles would then simply escape the cocoon very quickly.

The relativistic electrons in the cocoon must therefore diffuse along the magnetic field lines in between jumps from one patch of coherent magnetic field to another. The expression of Duffy et al. (1995) for the rms diffusion length after a time t, x, in this case can be approximated by

[EQUATION]

Assuming that the mean free path of the electron is less than the coherence length of the magnetic field, i.e. less than 10 kpc, I find for the same magnetic field and Lorentz factor [FORMULA] kpc if [FORMULA] Myr. A significant mixing of relativistic particles along the cocoon due to anomalous diffusion is therefore unlikely.

A.2. Non-linear diffusion

Of course, it may be argued that as the above analysis only applies to the quasi-linear regime, the diffusion timescale in a highly turbulent flow may be much shorter. Consider such a flow to be present in the cocoons of FRII-type radio sources. In this case, the relativistic electrons may diffuse through the observed lobes within a time short compared to the age of the source. However, since diffusion is a stochastic process and the geometry of the cocoon is elongated, most of the particles will leave the cocoon sideways and will be lost to the surrounding gas before traveling large distances along the cocoon. It is likely that the diffusion time for the relativistic particles depends on their energy and so the diffusion losses, if present, will significantly change the emission spectrum which is not observed (e.g. Roland et al. 1990). Even in the case of efficient energy independent diffusion the observed radio lobes should show diffuse edges in low frequency radio maps. Again this is not observed (e.g. Roland et al. 1990, Blundell et al. 2000a,b).

From the above it is clear that in the presence of very efficient diffusion some special confinement mechanism for the relativistic particles in the cocoon preventing their escape sideways is needed. This may be provided for by the compression and shearing of the tangled magnetic field at the edges of the cocoon. This process will align the magnetic field close to the cocoon edge with this surface and therefore act as a kind of magnetic bottle. The order thus introduced in the originally tangled magnetic field due to this process leads to an enhanced polarisation of the emitted radiation in this region (e.g. Laing 1980).

Large volume compression ratios are ruled out as the sound speed in the cocoon is high (e.g. KA). However, for a conservative estimate consider a volume compression ratio of 10 in the sheet of compressed material at the edge of the cocoon of a given source. This already implies that the maximum theoretical value of polarisation of the synchrotron emission should be observed at the edge of the radio lobes (Hughes & Miller 1991). For adiabatic compression of a tangled magnetic field the strength of the field increases by a factor of roughly 4.6. The rate of collisions of a given particle is probably increased as well as the irregularities in the magnetic field are also compressed. However, the case of [FORMULA] provides for a lower limit of [FORMULA] and so with [FORMULA] it is clear that the diffusion coefficient within the compressed region perpendicular to the field lines decreases by a factor 25 at most compared to the inner cocoon. This implies that in a given time a relativistic particle diffuses a 5 times shorter distance in the compressed boundary layer compared to the inner regions of the cocoon. To significantly influence the time the particle remains in the cocoon this compressed layer must at the very least occupy 20% of the width of the cocoon which is unlikely. Note, that although the magnetic field is compressed in the boundary layer and therefore aligned with the cocoon surface some field lines may still be perpendicular to this surface. Along these field lines diffusion will be even faster allowing many electrons to escape the cocoon.

From this I conclude that diffusion, even if it is highly effective in the inner cocoon and there exists a compressed boundary layer along the edges of the cocoon, will not alter the distribution of relativistic particles within the cocoon. This allows us to use the spatial distribution of the synchrotron radio emission of FRII sources to infer their age. The model developed in the following can be viewed as an extension to the classical spectral aging methods in that it takes into account the evolution of the magnetic field in the lobe

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Online publication: October 24, 2000
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