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Astron. Astrophys. 327, 432-440 (1997)

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4. Synchrotron radiation from anomalously transported electrons

The 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 [FORMULA] is the Lorentz factor of the particle and g a factor which depends on the magnetic field strength, and the energy density of the photon field. In units suited to the application to clusters of galaxies (see below) one has


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 [FORMULA]. Defining the differential rate [FORMULA] at which electrons are injected into the system at time t as [FORMULA], the re sulting electron density, allowing for energy losses, is


Consider electrons continuously injected with a power-law distribution in [FORMULA] at the point [FORMULA], i.e.,


where [FORMULA] 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 [FORMULA], after multiplying n by the synchrotron kernel. Furthermore, information about the magnitude and direction of the magnetic field along the line of sight is needed. For our purpose, it is sufficient to use the delta-function or 'monochromatic' approximation to the synchrotron kernel (see, for example, Mastichiadis & Kirk 1995 ) and we will simplify the discussion by using a constant, angle-averaged emissivity in the line of sight integral (see, however, Crusius & Schlickeiser 1988 ). As a result, the surface brightness at a given frequency is simply proportional to [FORMULA], where z is measured along the line of sight and where [FORMULA] is fixed, being proportional to the square root of the observing frequency.

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 x, the resulting surface brightness is directly proportional to [FORMULA] given by Eq. (26). This expression is simple to evaluate numerically. For various special cases, such as diffusion ([FORMULA]), or injection with [FORMULA] the integral can be written in closed form (see Appendix B for details). The resulting profile is shown in Fig. 2 for the three qualitatively different types of transport considered in Sect. 3.

[FIGURE] Fig. 2. The surface brightness of synchrotron radiation from a planar source of relativistic electrons, seen edge-on, as a function of distance x from the plane of injection, normalised to unity at [FORMULA]. Distance is expressed in units of the scale length [FORMULA] (Eq. 27). Electrons are injected at [FORMULA] with a spectrum [FORMULA]

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 [FORMULA], where


which is in general a function of observing frequency. Sub-diffusive transport ([FORMULA]), which tends to trap particles close to the point of injection, displays an integrable singularity at [FORMULA], whereas supra-diffusive transport, which quickly moves particles away from the source, initially increases at small x. The brightness profile for diffusive transport is monotonically decreasing with x. In Fig. 2 the surface brightness is arbitrarily normalised to unity at [FORMULA]. In a realistic situation, in which the electron source is distributed in space, it may not be possible to detect these features. In fact, the smooth almost exponential decrease of surface brightness with distance means that it would be difficult if not impossible to distinguish observationally between the different types of transport from these profiles alone.

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 [FORMULA] is simply related to the [FORMULA] -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 [FORMULA]. Defining the frequency dependence of the surface brightness by [FORMULA], we have


where [FORMULA]. The resulting spectral index as a function of x is plotted in Fig. 3 for [FORMULA].

[FIGURE] Fig. 3. The spectral index of synchrotron emission as a function of (scaled) radius. The dotted line corresponds to sub-diffusion with [FORMULA], the solid line to diffusion ([FORMULA]) and the dashed line to supra-diffusion ([FORMULA]). Electrons are injected continuously in the plane [FORMULA] with a spectrum [FORMULA]. The total emission integrated over x is thus a pure power-law of index [FORMULA].

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 [FORMULA]. This is because only recently injected particles remain close to [FORMULA] - older particles have a very small chance of return in the supra-diffusive case. As [FORMULA] 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 ([FORMULA], dashed line) has a very narrow spread of particle age at a given x. Once the distance exceeds [FORMULA], the spectrum becomes very soft, since the number of radiating particles decreases rapidly. In the case of diffusion ([FORMULA], solid line) the age distribution is gaussian and produces a more gradual softening, whereas the sub-diffusive case ([FORMULA], dotted line) not only traps some particles close to [FORMULA] but, also transports a few particles out rapidly outwards. Thus, the spectrum remains rather hard even at large radius. An observationally relevant measure of this dependence is the change in spectral index over the distance required for the brightness to drop by one order of magnitude from its value at [FORMULA]. For supra-diffusion, the spectrum softens by 1.8 from -1 at [FORMULA] to -2.9 at [FORMULA] ; for diffusion it softens by 0.8 from -1.1 at [FORMULA] to -1.9 at [FORMULA] and for sub-diffusion we find a change of only 0.3 from -1.2 at [FORMULA] to -1.5 at [FORMULA]. Thus, a relatively slow change in synchrotron spectral index indicates sub-diffusive behaviour.

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 [FORMULA]) and the outer parts (in this paper we adopt the value [FORMULA], 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 [FORMULA], 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 [FORMULA], 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 [FORMULA]. However, the model should represent the data fairly well within the range [FORMULA]. 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 in situ acceleration is negligible, diffusion is not capable of propagating the electrons outwards from the inner [FORMULA] of the cluster.

[FIGURE] Fig. 4. The one-dimensional surface brightness profiles for the three types of transport [FORMULA], 1 and 1.5 fitted to observations of the diffuse radio emission of the Coma cluster of galaxies by Hanisch (1980 ) as presented by Valtoja (1984 ). Only the outer parts of the cluster are plotted. Also shown are the predicted dependences of the spectral index on radius. The upper limit is a rough indication of the spectral softening observed by Giovannini et al. (1993 )

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 [FORMULA]) 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.

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Online publication: April 8, 1998