Brightness Temperature of Radio Zebras and Wave Energy Densities in Their Sources

We estimated the brightness temperature of radio zebras (zebra pattern – ZP), considering that ZPs are generated in loops having an exponential density profile in their cross section. We took into account that when in a plasma there is a source emitting in all directions, then in the escape process from the plasma the emission has a directional character nearly perpendicular to the constant-density profile. Owing to the high directivity of the plasma emission (for emission at frequencies close to the plasma frequency), the region from which the emission escapes can be very small. We estimated the brightness temperature of three observed ZPs for two values of the density scale height (1 and 0.21 Mm) and two values of the loop width (1 and 2 arcsec). In all cases, high brightness temperatures were obtained. For the higher value of the density scale height, the brightness temperature was estimated to be 1.1 × 10 15 – 1.3 × 10 17 K , and for the lower value, it was 4.7 × 10 13 – 5.6 × 10 15 K . These temperatures show that the observational probability of a burst with a ZP, which is generated in the transition region with a steep gradient of the plasma density, is significantly higher than for a burst generated in a region with smoother changes of the plasma density. We also computed the saturation energy density of the upper-hybrid waves (according to the double plasma resonance model, they are generated in the zebra source) using a 3D particle-in-cell model with a loss-cone type of distribution of hot electrons. We found that this saturated energy is proportional to the ratio of hot electron and background plasma densities. Thus, comparing the growth rate and collisional damping of the upper-hybrid waves, we estimated minimum densities of hot electrons as well as the minimum value of the saturation energy density of the upper-hybrid waves. Finally, we compared the computed energy density of the upper-hybrid waves with the energy density of the electromagnetic waves in the zebra source and thus estimated the efficiency of the wave transformation.