## 4. Synchrotron radiation from anomalously transported electronsThe synchrotron radiation emitted by relativistic electrons is the most important diagnostic available for the study of the transport of these particles in astrophysical environments. In this section we compute the spatial dependence of the surface bright ness and of the spectral index which is to be expected when the different kinds of particle transport discussed above dominate. The energy losses suffered by a relativistic particle emitting synchrotron radiation, or undergoing inverse compton scattering whilst propagating through a homogeneous magnetic field, or a homogeneous soft radiation field can be described by the equation where is the Lorentz factor of the particle
and where the photon field is assumed to be that of the cosmic
microwave background, which is for this purpose equivalent to a
magnetic field strength of . Defining the
differential rate at which electrons are
injected into the system at time Consider electrons continuously injected with a power-law distribution in at the point , i.e., where is a constant. At all points and at all energies of interest the distribution has had sufficient time to reach a stationary state, which is given by the integral In order to compute the surface brightness on the sky of the
synchrotron radiation from these electrons, two more integrations are
required - firstly over the line of sight through the source and
secondly over , after multiplying In the one-dimensional case in which electrons are injected in a
plane, and the line of sight is parallel to this plane, displaced by a
distance
It is important to note that the shapes of these profiles are characteristic of the type of transport involved and are independent of the frequency of observation. The assumptions entering the calculation are that the magnetic field strength is constant, that the source has the stated geometry, and that the electron injection has been steady over a sufficiently long period of time. The abscissa in Fig. 2 is the scaled radius , where which is in general a function of observing frequency.
Sub-diffusive transport (), which tends to trap
particles close to the point of injection, displays an integrable
singularity at , whereas supra-diffusive
transport, which quickly moves particles away from the source,
initially increases at small A more promising approach lies in the measurement of synchrotron spectra as a function of position. In the delta-function approximation, the synchrotron spectral index is simply related to the -dependence of the electron density integrated along the line of sight. To compute this, one must take account of the dependence of the scaling radius on . Defining the frequency dependence of the surface brightness by , we have where . The resulting spectral index as a
function of
This figure shows that the different types of transport produce
very different spectra as the particles propagate away from the
source. Close to the plane of injection, supra-diffusion gives
approximately the "uncooled" value of the spectral index
. This is because only recently injected
particles remain close to - older particles
have a very small chance of return in the supra-diffusive case. As
decreases, more and more older particles are
confined close to the point of injection, so that the spectrum for
diffusive and sub-diffusive transport is softer. In general, the age
distribution of particles produced by each transport process
determines the spectrum. Thus, supra-diffusion
(, dashed line) has a very narrow spread of
particle age at a given As an example, we consider the diffuse radio emission observed from the Coma cluster of galaxies. There is active debate as to the origin of the relativistic electrons responsible for this emission. It appears that simple diffusion from one of the galaxies near the centre is not, by itself, a viable explanation (e.g., Kirk et al. 1996b ) and that some kind of distributed injection and/or acceleration is necessary (Schlickeiser et al. 1987 ). However, observations by Giovannini et al. (1993 ) have shown that there is a distinction between the central parts of the cluster (say within a radius of ) and the outer parts (in this paper we adopt the value , so that 1 sec of arc is equivalent to 700 pc). If we assume that the processes of injection and acceleration occur within a sphere of radius , we can apply the above propagators to model the surface brightness and spectral index of the emission in the outer parts. Fig. 4 shows the observed emission at ,
as presented by Valtaoja (1984 ), from the data of Hanisch (1980 ).
The horizontal "error bars" in this representation are an expression
of spread in size of the emission when measured along the E-W and N-S
directions. Superimposed on the data are three models computed using
the one-dimensional propagators. Clearly, the (almost) spherical
geometry of the cluster is important for .
However, the model should represent the data fairly well within the
range . It is apparent from this figure that
each of the three types of transport can produce the required fall off
with distance, but that the associated predicted softening in the
spectral index, which is also shown in the figure, is quite different
in each case. The spectral index maps of Giovannini et al. (1993 ),
which were computed from the ratio of the fluxes at 326 and 1380 MHz,
show that within a few hundred kpc of the edge (at 500 kpc) the
spectrum softens rapidly - by more than 0.6 in 200 kpc. This is
indicated by an upper limit in the figure. Of the curves presented in
Fig. 4, only that corresponding to supra-diffusion produces a
comparable effect. Thus, in the stationary state, and assuming
The conclusion that anomalous transport is necessary is based on assuming that both the transport coefficients and the magnetic field (and also the type of transport as determined by ) are not functions of position, and further, that the transport is independent of particle energy. Although it is not straightforward to include such effects in a transport theory, they do not provide a simple explanation of the observations shown in Fig. 4 in terms of standard diffusive transport. For example, a magnetic field which decreases with distance from the core, could explain the fall-off in the intensity, but could not account for the observed softening of the spectrum, since energy losses would then be less important at larger radius than assumed in the figure. Similarly, if the spatial diffusion coefficient increases in proportion to particle energy - as expected in the case of gyro-Bohm diffusion - then particles responsible for higher frequency radiation are more mobile than those radiating at lower frequency. In this case, the intensity should fall off more slowly at higher frequency, which would tend to produce a harder spectrum at larger radius, contrary to the observations. © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |