# Temperature dependent growth rates of the upper-hybrid waves and solar radio zebra patterns

## TL;DR

This study investigates how the growth rates of upper-hybrid waves in solar radio zebra patterns depend on temperature, revealing shifts in resonance frequencies and implications for magnetic field estimations.

## Contribution

It provides a detailed numerical analysis of temperature effects on double plasma resonance instability, enhancing understanding of zebra pattern formation.

## Key findings

- Growth-rate maxima shift to lower frequencies with increased temperature.
- Peaks in growth rate become smoother at higher hot electron temperatures.
- Wave number components deviate from analytical estimates at maximum growth rates.

## Abstract

Context. The zebra patterns observed in solar radio emission are very important for are plasma diagnostics. The most promising model of these patterns is based on double plasma resonance instability, which generates upper-hybrid waves, which can be then transformed into the zebra emission. Aims. We aim to study in detail the double plasma resonance instability of hot electrons, together with a much denser thermal background plasma. In particular, we analyse how the growth rate of the instability depends on the temperature of both the hot plasma and background plasma components. Methods. We numerically integrated the analysed model equations, using Python and Wolfram Mathematica. Results. We found that the growth-rate maxima of the upper-hybrid waves for non-zero temperatures of both the hot and background plasma are shifted towards lower frequencies comparing to the zero temperature case. This shift increases with an increase of the harmonic number s of the electron cyclotron frequency and temperatures of both hot and background plasma components. We show how this shift changes values of the magnetic eld strength estimated from observed zebras. We confirmed that for a relatively low hot electron temperature, the dependence of growth rate vs. both the ratio of the electron plasma and electron cyclotron frequencies expresse distinct peaks, and by increasing this temperature these peaks become smoothed. We found that in some cases, the values of wave number vector components for the upper-hybrid wave for the maximal growth rate strongly deviate from their analytical estimations. We confirmed the validity of the assumptions used when deriving model equations.