Radio synchrotron spectra for a leaky box approximation

The synchrotron emission observed from many astrophysical objects is commonly modelled as arising from relativistic electrons in an emission region occupied by a spatially homogeneous, but tangled, magnetic field. However, we know that magnetic fields embedded in ionized gases tend to form flux ropes interspersed with regions of much lower magnetic field strengths. Here we develop a full description of the evolution of the energy distribution of relativistic electrons in a plasma divided into two distinct regions with different strengths of the magnetic field. Electrons are able to leak continuously from the low-field region into the high-field region. The model becomes fully analytic for physically reasonable assumptions. We show that such a leaky box model produces two distinct breaks in the electron energy distributions which give rise to three breaks in the resulting synchrotron spectrum. The spectral slopes in between the breaks are in general not constant and thus allow for significant curvature of the spectrum. These spectra are consistent with spatially resolved observations of the radio spectra of the lobes of radio galaxies. The exact form of the spectra depends on the adopted diffusion rate. The leaky box model significantly extends the time over which synchrotron emission can be detected at a given frequency compared to the usually assumed case of homogeneous magnetic fields. The spectral ages inferred for the electron population from standard techniques for the leaky box model are considerably younger than their real age.