Radio Emission in a Random Magnetic Field

The observed emission at frequency nu from a point with magnetic field B at angle theta to the line of sight is proportional to

epsilon = (B sintheta)^{n+1}nu^{-n}, (1)

where the spectral index n corresponds to a power law electron energy distribution with power law index gamma = 2n+1. As the field strength and direction vary from cell to cell of the random field, so will the emitted radio intensity. I assume throughout that the electron pitch angle distribution is isotropic, that the electron energy spectrum is independent of position, and that there is no spectral curvature.

I assume the magnetic field to be turbulent, with the turbulence being homogeneous and isotropic. No mean field is present. I further assume that the magnetic field is a Gaussian random field. This makes it easy to construct examples from a power spectrum. Also, the angles are isotropic and B is drawn from a Maxwellian distribution. The mean emissivity averaging over all angles and field strengths is

<epsilon> = [2(3/2)^{-(n+1)/2} / (n+3)] {Gamma[(n+4)/2] / Gamma(3/2)}nu^-n (2)

and the mean square emissivity is

<epsilon^2> = [(3/2)^{-(n+1)} / (n+2)] {Gamma[(2n+4)/2] / Gamma(3/2)}nu^-2n, (3)

where I have set the rms field strength equal to unity. I define the contrast Delta as the dispersion divided by the mean,

Delta = sqrt[(<epsilon^2>/<epsilon>^2)-1]. (4)

This is of order unity showing that the possible emissivities of a coherent magnetic cell cover a wide range, the width of the distribution being larger for steeper spectra.

If an emitting region is made up of a number of cells along the line of sight each with weight W_i then the total contrast is easily shown to be

Delta_T^2 = Sigma_i W_i^2 Delta^2 /(Sigma_i W_i)^2, (5)

with Delta the contrast for each cell. For N equal cells the contrast is reduced by a factor 1/sqrtN, as expected. For a Gaussian emissivity profile of equal FWHM the contrast is further reduced by a factor [(2 ln2)/pi]^1/4 = 0.81. If one could measure the contrast in a radio halo then the cell size could be estimated. One complicating factor is that radio halos are observed with only finite resolution so that if the cells are small the contrast is reduced by averaging within the observing beam.

For such a structured magnetic field the emissivity differs from that expected if the field strength is everywhere the same. The ratio is simply the average of the appropriate power of the field,

<B^(n+1)> = (3/2)^{-(n+1)/2} Gamma[(n+4)/2]/Gamma(3/2). (6)

Differences from the uniform field strength case are rather small, so that the mean emissivity gives a good estimate of the rms field strength.

The relativistic electrons lose energy by inverse Compton scattering and synchrotron emission. The inverse Compton scattering is equivalent to synchrotron losses in a 3.2microGauss field and will dominate if the magnetic field is weaker than this. This paper considers radio halos with field strengths of order 1microGauss, so that electrons lose energy at a rate approximately independent of the local field strength. I therefore ignore changes in energy losses from cell to cell. If the field is stronger then electrons in the most luminous regions lose energy faster and the emission fades relative to other cells, reducing the intensity contrast. I only consider situations where the field is sufficiently weak that this effect can be neglected, which would not be true for powerful radio sources.

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Peter Tribble,