On the mechanism of radio emission in type III Solar Radiobursts

TL;DR

This paper explores how intense density fluctuations in the solar corona influence the mechanism of radio wave generation in type III solar bursts, proposing a more efficient linear energy transfer process than traditional nonlinear models.

Contribution

It demonstrates that density inhomogeneities enable a linear wave energy transformation, challenging the conventional induced scattering mechanism for type III solar radio burst emission.

Findings

Density fluctuations significantly affect beam-plasma interactions.

Reflection of Langmuir waves can partially convert electrostatic energy into electromagnetic.

Linear transformation becomes more efficient than nonlinear scattering at fluctuation levels above 1%.

Abstract

Type III solar radio bursts are generated by streams of energetic electrons accelerated at the Sun during periods of the solar activity. The generation occurs in two steps. Initially, electron beams generate electrostatic Langmuir waves and then these waves are transformed in electromagnetic emissions. It is widely accepted that the mechanism of generation of emission on fundamental frequency close to plasma frequency is due to induced scattering of Langmuir waves into electromagnetic. However this process imposes quite restrictive limit of the ratios of effective brightness temperatures of electromagnetic and Langmuir waves in the source region. Recent studies showed that the level of density fluctuations in the solar wind and in the solar corona is so high that it may significantly affect beam-plasma interaction. Here we show that the presence of intense density fluctuations not only crucially influence the process of beam plasma interaction but also changes the mechanism of energy transfer from electrostatic waves into electromagnetic. Reflection of the Langmuir waves from the density inhomogeneities may result in partial transformation of the energy of electrostatic wave into electromagnetic. We show that the linear wave energy transformation for the level of fluctuations of the order of 1\% or higher may be significantly more efficient for generation of type III solar radio bursts than conventionally considered process of nonlinear conversion due to induced scattering on ions.

Equations

$${\omega }^2={\omega }_p^2\left(1+3{\lambda }_D^2{k}^2+\frac{\delta n}{N}\right),$$

$$k_z=\sqrt{\frac{{\omega }^2-{\omega }_p^2}{(3T/m)}-k_x^2},$$

$$\frac{N+\delta n(z)}{N}=\frac{{\omega }_p^2(z)}{{\omega }_{p0}^2}=\left(1+\frac{z}{L}\right),$$

$${\omega }_p(z)=\omega \sqrt{1-\frac{3T}{m{c}^2}{\sin }^2\theta }.$$

$$\mid \eta {\mid }^2=1-\mid R{\mid }^2=\frac{(2\pi {)}^{2}q[A{i}^{\prime }(q){]}^2}{[1+{\pi }^{2}q(A{i}^{\prime }(q){)}^2{]}^2+{\pi }^4{q}^2[(A{i}^{\prime }(q){]}^2[B{i}^{\prime }(q){]}^2},$$

$$q(L,\Psi )={\left(\frac{c}{\omega L}\right)}^{2/3}\left(\frac{k^2{c}^2}{{\omega }^2}\right){\sin }^2\Psi ,$$

$$K_{\text{eff}}=\frac{W_{em}}{W_L}=P_{\text{ref}}\underset{0<q<q_{\max }}{\int }P(\Psi )P(L)|\eta (L,\Psi ){|}^2\sin \Psi d\Psi dL,$$

$$0<\sin \Psi <q_{\max }{\left(\frac{c}{\omega L}\right)}^{1/3}\left(\frac{\omega }{kc}\right),$$

$$0<k_{\perp }<q_{\max }{\left(\frac{c}{\omega L}\right)}^{1/3}\frac{\omega }{c}.$$

$$0<\frac{k_{\perp L}}{k_L}=\sin \Psi <q_{\max }{\left(\frac{c}{\omega L}\right)}^{1/3}\frac{\omega }{kc}\simeq q_{\max }{\left(\frac{c}{\omega L}\right)}^{1/3}\frac{V_b}{c}.$$

$$\underset{0}{\overset{{\Psi }_{\max }}{\int }}P(\Psi )\sin \Psi d\Psi =\frac{1}{2\pi }\underset{0}{\overset{{\Psi }_{\max }}{\int }}\sin \Psi d\Psi =\frac{1}{4\pi }{q}_{\max }^2{\left(\frac{c}{\omega L}\right)}^{2/3}{\left(\frac{\omega }{kc}\right)}^2\simeq \frac{q_{\max }^2}{4\pi }{\left(\frac{c}{\omega L}\right)}^{2/3}{\left(\frac{V_b}{c}\right)}^2.$$

$$K_{\text{eff}}=0.5\frac{Q}{4\pi }{q}_{\max }^2{\left(\frac{\omega }{kc}\right)}^2{\left(\frac{c}{\omega }\right)}^{2/3}\underset{0}{\overset{\infty }{\int }}\frac{1}{L^{2/3}}P\left(L\right)dL.$$

$$P\left(\frac{\partial \ln (z)}{\partial z}\right)=P(y)=\frac{1}{\sqrt{2\pi }{A}_{sc}}\exp \left(-\frac{y^2}{2A_{sc}^2}\right).$$

$$P(L)=\frac{L_{sc}}{\sqrt{2\pi }{L}^2}\exp \left(-\frac{L_{sc}^2}{2L^2}\right)$$

$$I=\underset{0}{\overset{\infty }{\int }}{\left(\frac{1}{L}\right)}^{2/3}P(L)dL=\frac{L_{sc}}{\sqrt{\pi }}\underset{0}{\overset{\infty }{\int }}{\left(\frac{1}{L}\right)}^{8/3}\exp \left(-\frac{L_{sc}^2}{L^2}\right)dL=\frac{\Gamma (5/6)}{2\sqrt{\pi }{L}_{sc}^{2/3}},$$

$$K_{eff}=0.5Q\frac{q_{max}^2}{8\pi \sqrt{\pi }}{\left(\frac{V_b}{c}\right)}^2{\left(\frac{c}{\omega L_{sc}}\right)}^{2/3}\Gamma (5/6)\simeq \frac{Q}{10}{\left(\frac{V_b}{c}\right)}^2{\left(\frac{c}{\omega L_{sc}}\right)}^{2/3}.$$

$$U_{\omega ,\Omega }=\frac{k_t^2}{V_{gr}}\frac{{\omega }_p^2}{2(2\pi {)}^{5/2}{n}_e}{\left(\frac{T_i}{T_i+T_e}\right)}^2\int \frac{[k_t\times k_L^{{}^{\prime }}{]}^2}{k_t^2{k}_L^{2^{\prime }}}\frac{W_{kL}^{{}^{\prime }}d{k}_l^{{}^{\prime }}}{V_{ti}|k-k_L^{{}^{\prime }}|}\times \exp {\left(-\frac{1}{2}\frac{\omega -\omega (k_L)}{V_{ti}|k-k_L|}\right)}^2$$

$$U_{\omega ,\Omega }=\frac{k_t^2}{V_{gr}}\frac{{\omega }_p^2}{24(2\pi {)}^{5/2}{n}_e}\int \frac{W_{kL}^{{}^{\prime }}d{k}_l^{{}^{\prime }}}{V_{ti}{k}_L^{{}^{\prime }}}\times \exp {\left(-\frac{1}{2}\frac{\omega -\omega (k_L)}{V_{ti}{k}_L}\right)}^2$$

$$U=\frac{k_t^2}{V_{gr}}\frac{{\omega }_p^2{W}_L}{48\pi n_e}$$

$$U=\frac{V_{te}}{V_b}\frac{{\omega }_p^4}{c^3}\frac{\sqrt{3}{W}_L}{48\pi n_e}.$$

$$K_{eff}=\frac{{\pi }^3}{6\sqrt{3}}\frac{V_{te}}{V_b}\frac{V_{te}^3}{c^3}\frac{1}{{\lambda }_D^3{n}_e}$$

Full text

acceleration of particles— Sun: radio radiation— Sun: particle emission— solar wind

††journal: ApJ

1 Introduction

Solar type III radio bursts are amongst the strongest radio emissions in the heliosphere. It is wildly accepted that the high energy electrons $\sim 5-30$ keV, accelerated during reconnection of the magnetic flied lines in solar atmosphere, are responsible for generation of these radio emission (Suzuki & Dulk, 1985; Melrose, 2017). Important characteristic of the type III radio bursts is the fast frequency drift rate (Wild & McCready, 1950). Type III bursts can start at a frequency of several hundred of MHz and then go down to tens kHz within few minutes with the increasing duration at a given frequency(Alvarez & Haddock, 1973; Reid & Kontar, 2018; Krupar et al., 2018a). To explain the frequency drift, the beams should have near relativistic speeds $0.1-0.5c$ (Melrose, 1990) and generate radio emission near plasma frequency, $f_{pe}$ and double plasma frequency (harmonic) (Ginzburg & Zhelezniakov, 1958). Frequency of the emission follows local electron density along the beam path, starting at high frequencies in dense plasma close to the Sun and then decrease over time as the beam propagates in the expanding solar corona and later solar wind (Krupar et al., 2018b).

Generation of radio waves occurs in two steps. In the original study, Ginzburg & Zhelezniakov (1958) proposed that the two-stream instability of an electron beam results in the growth of electrostatic Langmuir waves (ES) that later produce electromagnetic (EM) emission at the plasma frequency due to the scattering on ions while the coalescence of two Langmuir waves can produce the harmonic emission. The theory has been subsequently refined and alternative mechanisms for the conversion of the beam-driven Langmuir waves into electromagnetic radiation have been proposed (Melrose, 1990; Malaspina et al., 2012). While exact mechanism of the ES to EM conversion is still under debate, the generation of the Langmuir waves by electron beams in the solar wind has been confirmed by in situ measurements (Lin et al., 1981; Ergun et al., 1998).

Beam-type configurations in a plasma are known to be unstable and relaxation of the beam-plasma system to the stable state results in growth of the Langmuir turbulence (Romanov & Filippov, 1961; Drummond & Rosenbluth, 1962; Vedenov, 1963). Landau resonance enables effective energy transfer from the beam electrons to the waves, as a result up to $2/3$ of the initial beam energy can be transferred to the Langmuir waves through wave-particle interaction (Vedenov & Ryutov, 1975). Recent studies showed that the level of density fluctuations in the solar wind and in the solar corona is so high that they may significantly affect beam-plasma interaction (Kontar, 2001; Zaslavsky et al., 2010; Krafft et al., 2013; Reid & Kontar, 2013; Voshchepynets et al., 2015). The effect of density fluctuations results in phase velocity variations of Langmuir waves. These variations change the wave resonance velocities of the electrons. This leads to significant decrease of the increment of instability and important increase of the relaxation length of the beam (Voshchepynets & Krasnoselskikh, 2015).

Presence of the density fluctuations has significant impact on the observed properties of the Langmuir waves (see Krasnoselskikh et al. (2007) and references therein). First, this idea has been proposed in Smith & Sime (1979) in order to explain observed clumping of Langmuir waves in type III source regions. Later, it has been proposed (Ergun et al., 2008) that the density irregularities in the solar wind can take form of cavities that might result in modulation of the the waveforms of the Langmuir waves. Analysis of the large number of individual waveforms measured by STEREO and WIND showed good agreement with theoretical predictions (Ergun et al., 2008; Malaspina & Ergun, 2008) and results of the numerical simulations (Krafft et al., 2014). To address stochastic nature of the density fluctuations, several statistical models that deduced properties of the Langmuir waves from the probability distribution function of the density fluctuations have been proposed. Stochastic growth theory proposed by Robinson (1995) predicted that distribution of the amplitudes of the Langmuir waves in type III source regions should follow log-normal distribution. While some observations (Robinson et al., 1993; Píša et al., 2015) show good agreement with these predictions, there are numerous reports of observations and simulations (Krasnoselskikh et al., 2007; Vidojevic et al., 2012; Reid & Kontar, 2017; Voshchepynets et al., 2017) that deviations of the distribution of the amplitudes of the Langmuir waves from log-normal can be rather significant.

The aim of the present study is to determine the role of the density fluctuation in the conversion process of the beam-generated ES waves into EM emissions. Reflection of the Langmuir waves from the density inhomogeneities may result in partial transformation of the energy of electrostatic wave into electromagnetic. We consider this effect of linear wave energy transformation in application to generation of type III solar radio bursts. We use the probability distribution of density fluctuations to evaluate the statistical characteristics of such process and its efficiency. We show that the mechanism of linear transformation for the relative density fluctuations of the order of $1\%$ may be significantly more efficient than widely accepted process of nonlinear conversion of Langmuir waves due to induced scattering on ions.

2 Calculation of the efficiency of energy conversion

When the density fluctuations along the waves path cannot be neglected, the propagation of the wave can be described with non-linear Bohm-Gross dispersion relation for Langmuir waves :

[TABLE]

where $\omega$ and $k$ are frequency and wave-vector of the Langmuir wave, $\omega_{p}$ is the electron plasma frequency for the electron number density $N$, $\delta n$ is a deviation of the density from the average value $N$, and $\lambda_{D}$ is the Debye length. The waves are assumed to be generated resonantly: $\omega=kV_{b}$, where $V_{b}$ is the beam velocity that significantly exceeds thermal velocity of electrons $v_{T}=\sqrt{T/m}$ ($T,m$ are electron temperature and mass). A Langmuir wave propagating in plasma with density inhomogeneities encounters density depletions and enhancements along its path. When the wave goes to the increasing density region where the local plasma frequency $\omega_{p}$ becomes equal to the wave frequency $\omega$, the wave is reflected.

Assuming that the density fluctuations are isotropic, the incident angles of Langmuir waves are distributed uniformly over a semi-sphere. For the majority of incidence angles, the reflection resembles mirror type reflection: the $k$-vector component parallel to the direction of the density gradient changes its direction, while the $k$-vector component perpendicular to the gradient remains unchanged. In the rather narrow range of incidence angles, electrostatic waves may couple with the electromagnetic and the process becomes a three-wave coupling process. In this case, the reflection results in a Langmuir wave and an electromagnetic wave, so that a part of the incident Langmuir wave energy is transformed to the electromagnetic wave. It is worth noting that initial reflection generates electromagnetic wave propagating in the direction towards the Sun. However, as an average density decreases with the distance from the Sun the secondary reflections that may be considered as mirror type will turn the wave direction towards the Earth (top panel in Figure 1).

Following the previous works (Piliya, 1966; Stenzel et al., 1974), let the Langmuir wave of frequency $\omega$ propagates obliquely to the direction of the density gradient that we choose to be along the $z$-axes (bottom panel in Figure 1). Let the perpendicular component of the $k$-vector be directed along the $x$-axes and be equal to $k_{x}=(\omega/c)\sin\theta$, so the component along the $z$-axes is

[TABLE]

The angle $\theta$ corresponds to the angle of propagation of the reflected electromagnetic wave with respect to $z$-direction. The incidence angle of electrostatic wave $\psi$ is determined by the ratio of perpendicular and parallel components of ${k}$-vector, i.e. $\tan\psi=(V_{b}/c)\sin\theta$. When the density profile is linear function of distance with characteristic scale $L$, one finds

[TABLE]

so the incident Langmuir wave reflects when the local plasma frequency is

[TABLE]

where $c$ is the speed of light. A fraction of incident wave energy is reflected as Langmuir wave, while the other part is converted to an electromagnetic wave in this mode conversion point $\omega_{p}(z)=\omega$. The electromagnetic wave propagates outward into the direction of the density decrease beyond its cutoff frequency at $\omega_{p}(z)=\omega\cos\theta$. The problem has been studied by many authors starting with the pioneering work by Denisov (1957). Several methods have been developed to evaluate the conversion coefficient, (e.g. review by Piliya, 1966). Recently, Hinkel-Lipsker et al. (1989, 1991) have performed analytical study and obtained an analytical expression for the conversion coefficient:

[TABLE]

where $R$ is the reflection coefficient defined as the ratio of the reflected Langmuir wave amplitude to the amplitude of the incident wave, $\eta$ is the ratio of the electromagnetic and the incident Langmuir wave amplitudes, $q=[c/(\omega L)]^{2/3}\sin^{2}\theta$, $Ai(q)$ and $Bi(q)$ are Airy functions, and $Ai^{\prime}(q),$ and $Bi^{\prime}(q)$ their derivatives. The dependence of the energy conversion coefficient on parameter $q$ is shown in Figure 2.

It is convenient to re-write the reflection coefficient in terms of incidence angle of Langmuir wave $\Psi$. The parameter $q$ can be written

[TABLE]

where $L$ is the characteristic density inhomogeneity scale. Here we take into account that the conversion coefficient has non-zero values only in the range of $q,$ $0<q<q_{\max}$. In our probabilistic model, the efficiency of the beam-generated Langmuir waves conversion into EM emission is evaluated averaging over angles and the density fluctuation scales. To evaluate the ensemble averaged values taking into account the probability distribution of the density fluctuations, we choose hereafter the reference frame, where $z$ -axes is directed along the wave vector of the propagating Langmuir wave. The average probability can be written as the product of probability distributions in angle $P(\Psi)$ and in scale $P(L)$:

[TABLE]

where $K_{\text{eff}}$ is the energy conversion coefficient from ES Langmuir waves to EM, $W_{em}$ is energy of reflected EM wave, $W_{L}$ is the energy of incident Langmuir wave, and $P_{\text{ref}}$ is the probability of ES wave reflection. As it was shown in Voshchepynets et al. (2015) and Voshchepynets & Krasnoselskikh (2015), in the plasma with random density fluctuations $P_{\text{ref}}$ can be calculated by making use of a probability distribution function of the amplitudes of the density fluctuations $P(\delta n/N)$.

Since the conversion occurs only when the parameter $q$ is in the range from about [math] to $q_{\max}$, this leads to the limited angular range of reflected electromagnetic waves given by:

[TABLE]

that corresponds to values of the perpendicular component of the $k$-vector of incident Langmuir wave

[TABLE]

Taking into account that the $k$-vector of the Langmuir wave is approximately equal to $k_{L}\simeq\omega/V_{b}$, the conversion may occur only when the angle $\Psi$ given by

[TABLE]

To simplify the evaluation of the integrals, we shall take the conversion coefficient to be approximately constant (corresponding to its average value $\mid\eta(L,\Psi)\mid^{2}=Q\simeq 0.16$ in the range of $q$ between zero and $q_{\max}$. Under this assumption the integration over angles and scales $L$ may be carried out independently step by step. Assuming $P(\Psi)=(2\pi)^{-1}\sin\Psi$, the integration over angles results in

[TABLE]

Then the conversion coefficient may be re-written:

[TABLE]

where $P(L)$ is a probability distribution of the density gradients (scales). In order to calculate $P(L)$ one should use spatial profiles of the density fluctuations. In the present study we use synthetic density data calculated from published density power spectra. Kellogg et al. (1999) proposed a procedure based on the inverse Fourier transform that allows to reconstruct density profiles $n(t)$ from the power spectrum assuming the phases of waves to be random. It is known from in situ spacecraft measurements in the solar wind (Celnikier et al., 1987; Chen et al., 2012) that the density spectrum can be considered as a broken power-low, with different spectral indices, about $5/3$ for low frequency part and about $1$ for higher frequency part with the transition at about $0.6Hz$. In order to transform these profile to the spatial profiles $n(z)$, one can use Taylor hypothesis assuming that fluctuations are convected with the characteristic velocity of the solar wind, $V_{SW}\sim 400km/s$. We used power spectrum in the range of frequencies between $10^{-2}Hz$ and $530Hz$. Lower frequency limit defines maximal length of the synthetic density profile ($100s$ or $4\cdot 10^{4}km$). Highest frequency limit defines the smallest scale of the density fluctuation presented in this profile. In the present study this scale is set to be of approximately $750m$ that for the plasma conditions relevant for 1AU is about $50\lambda_{D}$. After the profiles were generated, normalization procedure was applied to ensure that $\left\langle n(z)\right\rangle_{z}=N_{0}$ and $\sqrt{\left\langle(n(z)-N_{0})^{2}\right\rangle_{z}}=\delta n$ for each of the density profiles (here brackets denote averaging). We consider different levels of density fluctuations, $\delta n/N_{0}$, between $0.01$ and $0.1$. For more details on the procedure we refer to Voshchepynets & Krasnoselskikh (2015).

Locally the density profiles may be approximated by linear function of $n(z)$, thus the probability distribution of the characteristic scales could be retrieved from the distribution of density gradients, $(1/n)\nabla n=\partial\ln{n(z)}/\partial z=1/L$. Left panel in Figure 3 shows normalized probability distribution $P(\partial\ln{n(z)}/\partial z)$ obtained from the large number (about 200) of the density profiles $n(z)$ with level of density fluctuations $\delta n/N_{0}=0.01$. It is found that the distribution $P(\partial\ln{n(z)}/\partial z)$ is very close to Gaussian with characteristic scale $A_{sc}$:

[TABLE]

Taking into account that $y=1/L$ and $P(L)=P(y^{-1})$ one can get $P(L)$ as follows:

[TABLE]

where $L_{sc}=1/A_{sc}$. The functions $P(y^{-1})$ and $P(L)$ are shown at left panel in Figure 3. By making use of $P(L)$ one can integrate last part in equation for energy conversion coefficient as follows:

[TABLE]

where $\Gamma$ is a Gamma function. Substituting the integral $I$ one can find $K_{\text{eff}}$:

[TABLE]

The characteristic scales of the density gradients obtained from synthetic density profiles are shown in Figure 4. We start with $\delta n/N_{0}=0.001$ that results in $L_{sc}>10^{3}km$. As one can see $L_{sc}$ drops significantly with increasing level of the density fluctuations, that in its turn results in increase of the efficiency of energy transformation from ES to EM waves. Thus for $\delta n/N_{0}=0.1$, that was measured onboard Helios satellite closer to the Sun (Bavassano & Bruno, 1995) characteristic scale may be less than $15km$.

It is worth noting that the power spectrum used in this study is relevant for solar wind density fluctuations around 1AU. Closer to the Sun the spectrum characteristics may be different and as a result density fluctuations can be described by the different statistics. In order to avoid speculations (though the method developped here is applicable), we consider emissions in the frequency range typical for solar type III radio burst around 1 AU: $10kHz$ to $100kHz$ (Mann et al., 1999).

Conversion coefficient as a function of beam velocity, $V_{b}$ and Langmuir wave frequency $f$ is shown in Figure 5. Left panel shows $K_{eff}$ for $\delta n/N_{0}=0,1$. As one can see for $V_{b}>0.15c$ conversion coefficient is above $10^{-5}$ in the whole range of frequencies. An increase of the level of the density fluctuations results in a decrease of the characteristic scale of the density gradient. As a result, reflection of the Langmuir waves will occur more often and $K_{eff}$ will increase. Right panel in Figure 5 shows $K_{eff}$ for $\delta n/N_{0}=0.04$. We found that the conversion coefficient is above $10^{-5}$ in the whole frequency range for $V_{b}>0.1c$. For faster beams with $V_{b}>0.2c$ conversion coefficient is above $10^{-4}$ for frequencies below $50kHz$.

3 Induced scattering

It is well known and widely accepted from the very early articles by Ginzburg & Zhelezniakov (1958), Kaplan & Tsytovich (1969), Melrose (1974) and Melrose (1987) that under condition $T_{e}\sim T_{i}$ (typically in the solar wind $T_{e}\simeq 2T_{i})$ the damping of ion sound waves is quite strong and the major nonlinear process of transformation of electrostatic waves onto electromagnetic (ES-EM) is induced scattering on ions. Generated electromagnetic waves have frequencies lower than the frequency of the primary Langmuir waves. To describe this process one should introduce emission coefficient $U_{\omega,\Omega}$ for electromagnetic waves that is defined as the energy per unit frequency interval generated in a unit volume and in a unit solid angle. This coefficient is expressed making use a power, $Q$, radiated by current $j$ excited by Langmuir waves.

For the conversion of plasma waves into electromagnetic due to induced scattering on ions, $U_{\omega,\Omega}$ can be found as follows (Kaplan & Tsytovich, 1969):

[TABLE]

For the sake of simplicity we assume $T_{i}=T_{e}$. One can also assume with the accuracy up to coefficient of the order of unity that radiation is isotropic (implies that $[k_{t}k_{L}]^{2}=k_{t}^{2}k_{L}^{2}/3$) and $k_{L}\gg k_{t}$. This results in:

[TABLE]

Suggesting that for the beam plasma interaction the maximum of the energy density $W_{kL}$ of the plasma turbulence occurs at a definite value of the phase velocity $V=V_{b}$, and neglecting the width of the beam choosing $W_{kl}=W_{l}\delta(k_{\parallel}-k_{L})$ (here $k_{L}=\omega_{p}/V_{b}$, and $k_{\parallel}$ is along the direction of the propagation of the beam). This results in total emission coefficient $U$ that is found integrating over $\omega$ and $\Omega$:

[TABLE]

The resonance conditions for the wave conversion impose the following relation between wave vectors of electromagnetic and Langmuir waves: $k_{t}=\sqrt{3}k_{L}V_{te}/c$ and $V_{gr}=\sqrt{3}k_{L}V_{te}c/\omega_{p}$. Thus the one can find:

[TABLE]

One can use total emission coefficient to calculate averaged over period wave energy density of the EM emission, $W_{t}$. For the comparison with the linear transformation coefficient determined as $K_{eff}=W_{t}/W_{l},$ for ES-EM transformation due to induced scattering on thermal ions it is found to be equal to:

[TABLE]

One can see that for typical parameters of the solar wind it is many orders of magnitude smaller than the efficiency coefficient for linear transformation. For the conditions at 1 AU $T_{e}\sim 10eV,n_{e}=10cm^{-3}$, $E_{b}\simeq 10keV$, $K_{eff}\sim 10^{-18}$.

3.1 Conclusions

We show that the process of linear conversion of Langmuir waves onto electromagnetic on density fluctuations can be dominant for the generation of the type III radio emissions and is significantly more efficient than conventionally accepted nonlinear process of induced scattering of Langmuir waves on ions.

As we show the efficiency of linear conversion is strongly dependent upon statistical properties of density fluctuations and their gradients. These characteristics may significantly vary with the distance from the Sun. The study of these dependencies comes beyond the scope of this paper and will be addressed in future publications.

V.K. acknowledges financial support by CNES through grant ”Stereo-Waves invited scientist”.