Double plasma-resonance surfaces in flare loops and radio zebra emission

Aims. The zebra structures observed in radio waves during solar flares are some of the most important structures used as diagnostics of solar flare plasmas. We here not only analyze the so-called double plasma-resonance (DPR) surfaces, but also estimate the effects of their form on the size of the zebra sources and brightness temperature. Methods. To compute the DPR surfaces, we used numerical and analytical methods. Results. We found that except for the case of a constant magnetic field across the loop, the DPR surfaces deviate from the constant plasma density surfaces. We found that the regime with a finite height scale has three forms of resonance surfaces depending on the magnetic field variation across the loop. This magnetic field variation also determines if in the generated zebra structure, an increase in gyro-harmonic number leads to an increase or decrease of the zebra stripe frequency. In the case with an infinite height scale, the resonance surfaces are parallel to the loop axis. Furthermore, we found that for highly polarized zebra structures that are generated at DPR surfaces close to the plasma frequency, the zebra emission is limited to the narrow escaping cone and the emitting source area increases with increasing viewing angle compared to the loop axis. Moreover, with increasing deviation of the DPR surfaces from those of constant density surfaces, the frequency bandwidth of the DPR emission increases and can cause the zebra stripes to overlap, which limits the zebra generation. For the zebra structures observed on 14 February 1999, 6 June 2000, and 1 August 2010 and the observed view perpendicular to the loop axis, we estimated that the brightness temperature is 3.67 × 1014 K, 6.58 × 1013 K, and 7.35 × 1015 K, respectively. These brightness temperatures are much lower than those derived for the view along the loop axis (up to 1017 K), and thus are more realistic. The area of the emitting source for coronal loops in the view perpendicular to the loop axis can be larger by several orders of magnitude than that in the view along the loop axis.