tokens: α and are the polar and azimuth angles of the wave vector of a transverse electromagnetic wave. In this case, the angle α is the observation angle. The spectral power of radiation at the emission frequency Q ω can be obtained in accordance with the expression for the total radiation power: Comparing expressions ( 4 ) and ( 5 ) we obtain: In our work, we will not apply head-on approximations, which, as noted in the work of Willes et al. ( 1996 ), can be violated at small wavenumbers of Langmuir waves and their narrow spectra. We will also not use the narrow-spectrum approximation proposed in Willes et al. ( 1996 ). The cited work also contains a scheme with the spatial orientation of the wave vectors of the merger of Langmuir waves and the generated transverse wave. In order not to limit ourselves to particular cases, we present the results of calculations according to direct expressions ( 2 ), ( 3 ), without using the simplifications usually inherent in these models, which are described in detail in Willes et al. ( 1996 ). Let us now consider radio emission at different spectra of Langmuir waves. Taking into account the work of Willes et al. ( 1996 ), we consider the case of axial symmetry of the radiating region and the spectrum of Langmuir waves in the form: where B is the normalizing coefficient in accordance with the normalization condition: W is the total energy density of Langmuir waves; and φ are the polar and azimuth angles of the Langmuir wave, where the angle is calculated relative to the axis of symmetry of the radiating region, δ is an even positive number. Let us first consider the case when , i.e. when does not depend on the Langmuir wave vector. Fig. 1(a) shows the distribution of radiation ( Q k ) over the wavenumber for different viewing angles α at δ = 2, k max = ω pe /(3 V T e ), k min = ω pe /(15 V T e ), and T e = 3 × 10 6 K, n = 10 16 m −3 . These values of k max and k min are chosen for the following reasons: Langmuir waves with a value greater than k max are effectively absorbed by thermal electrons as a result of Landau damping, and therefore we will not consider them. Langmuir waves in plasma are usually generated by beams of superthermal electrons, so we consider the region of phase velocities up to 15 V T e and assume that there are no Langmuir waves with large phase velocities (smaller wavenumbers). The observation angle α is the angle between the symmetry axis of the radiating region and the vector k . In this case, we assume that k 0 = ( k min + k max )/2, and σ is less than the value of k 0. Fig. 1(b) shows the distribution of radiation ( Q ω ) as a function of the frequency of the transverse electromagnetic wave. The calculation results for this case are shown in Figs 1 and 2 for different σ. At a small standard deviation (e.g. σ = 0.1 k 0 , Figs 1a , b ) the radiation is concentrated in a narrow frequency range. A similar type of Langmuir spectra can be generated, for example, by a fast electron beam. With
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