Context. The zebra patterns observed in solar radio emission are very important for flare 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 field 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 expresses 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.

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 plasma there is a source emitting in all directions, then in the escape process from the plasma the emission obtains 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 as 1.1 × 10 15 -1.3 × 10 17 K, and for the lower value as 4.7 × 10 13 -5.6 × 10 15 K. These temperatures show that the observation probability of a burst with 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 (which according to the double plasma resonance model are generated in the zebra source) using a 3D particle-in-cell model with the losscone 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 minimal densities of hot electrons as well as the minimal 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.

Context. Several important mechanisms that explain coherent pulsar radio emission rely on streaming (or beam) instabilities of the relativistic pair plasma in a pulsar magnetosphere. However, it is still not clear whether the streaming instability by itself is sufficient to explain the observed coherent radio emission. Due to the relativistic conditions that are present in the pulsar magnetosphere, kinetic instabilities could be quenched. Moreover, uncertainties regarding specific model-dependent parameters impede conclusions concerning this question. Aims. We aim to constrain the possible parameter range for which a streaming instability could lead to pulsar radio emission, focusing on the transition between strong and weak beam models, beam drift speed, and temperature dependence of the beam and background plasma components. Methods. We solve a linear relativistic kinetic dispersion relation appropriate for pulsar conditions in a more general way than in previous studies, considering a wider parameter range. In doing so, we provide a theoretical prediction of maximum and integrated growth rates as well as of the fractional bandwidth of the most unstable waves for the investigated parameter ranges. The analytical results are validated by comparison with relativistic kinetic particle-in-cell (PIC) numerical simulations. Results. We obtain growth rates as a function of background and beam densities, temperatures, and streaming velocities while finding a remarkable agreement of the linear dispersion predictions and numerical simulation results in a wide parameter range. Monotonous growth is found when increasing the beam-to-background density ratio. With growing beam velocity, the growth rates firstly increase, reach a maximum and decrease again for higher beam velocities. A monotonous dependence on the plasma temperatures is found, manifesting in an asymptotic behaviour when reaching colder temperatures. A simultaneous change of both temperatures proves not to be a mere linear superposition of both individual temperature dependences. We show that the generated waves are phase-coherent by calculating the fractional bandwidth. Conclusions. Plasma streaming instabilities of the pulsar pair plasma can efficiently generate coherent radio signals if the streaming velocity is ultra-relativistic with Lorentz factors in the range 13 <  γ <  300, if the background and beam temperatures are small enough (inverse temperatures ρ0; ρ1 ≥ 1, i.e., T0; T1 ≤ 6 × 109), and if the beam-to-background plasma density ratio n1/(γbn0) exceeds 10−3, which means that n1/n0 has to be between 1.3 and 20% (depending on the streaming velocity).

A number of possible pulsar radio emission mechanisms are based on streaming instabilities in relativistically hot electron–positron pair plasmas. At saturation, the unstable waves can, in principle, form stable solitary waves, which could emit the observed intense radio signals. We searched for the proper plasma parameters that would lead to the formation of solitons, and investigated their properties and dynamics as well as the resulting oscillations of electrons and positrons that possibly lead to radio wave emission. We utilized a one-dimensional version of the relativistic particle-in-cell code ACRONYM initialized with an appropriately parameterized one-dimensional Maxwell–Jüttner particle distribution in velocity space to study the evolution of the resulting streaming instability in a pulsar pair plasma. We found that strong electrostatic superluminal L-mode solitons are formed for plasmas with normalized inverse temperatures ρ ≥ 1.66 or relative beam drift speeds with Lorentz factors γ > 40. The parameters of the solitons fulfill the conditions for wave emission. For appropriate pulsar parameters the resulting energy densities of superluminal solitons can reach 1.1 × 105 erg cm−3, while those of subluminal solitons reach only 1.2 × 104 erg cm−3. Estimated energy densities of up to 7 × 1012 erg cm−3 suffice to explain pulsar nanoshots.

Context. The double plasma resonance (DPR) instability plays a basic role in the generation of solar radio zebras. In the plasma, consisting of the loss-cone type distribution of hot electrons and much denser and colder background plasma, this instability generates the upper-hybrid waves, which are then transformed into the electromagnetic waves and observed as radio zebras. Aims. In the present paper we numerically study the double plasma resonance instability from the point of view of the zebra interpretation. Methods. We use a 3-dimensional electromagnetic particle-in-cell (3-D PIC) relativistic model. We use this model in two versions: a) a spatially extended "multi-mode" model and b) a spatially limited "specific-mode" model. While the multi-mode model is used for detailed computations and verifications of the results obtained by the "specific-mode" model, the specific-mode model is used for computations in a broad range of model parameters, which considerably save computational time. For an analysis of the computational results, we developed software tools in Python. Results. First using the multi-mode model, we study details of the double plasma resonance instability. We show how the distribution function of hot electrons changes during this instability. Then we show that there is a very good agreement between results obtained by the multi-mode and specific-mode models, which is caused by a dominance of the wave with the maximal growth rate. Therefore, for computations in a broad range of model parameters, we use the specific-mode model. We compute the maximal growth rates of the double plasma resonance instability with a dependence on the ratio between the upper-hybrid ωUH and electron-cyclotron ωce frequency. We vary temperatures of both the hot and background plasma components and study their effects on the resulting growth rates. The results are compared with the analytical ones. We find a very good agreement between numerical and analytical growth rates. We also compute saturation energies of the upper-hybrid waves in a very broad range of parameters. We find that the saturation energies of the upper-hybrid waves show maxima and minima at almost the same values of ωUH/ωce as the growth rates, but with a higher contrast between them than the growth rate maxima and minima. The contrast between saturation energy maxima and minima increases when the temperature of hot electrons increases. Furthermore, we find that the saturation energy of the upper-hybrid waves is proportional to the density of hot electrons. The maximum saturated energy can be up to one percent of the kinetic energy of hot electrons. Finally we find that the saturation energy maxima in the interval of ωUH/ωce = 3-18 decrease according to the exponential function. All these findings can be used in the interpretation of solar radio zebras.