Theory of type III radio bursts in the interplanetary space. II. Study of instability of the electron beam-solar wind plasma system in a linear approximation

Frequency characteristics of disturbances of a one-dimensionally inhomogeneous electron beam-solar wind plasma system are studied in the geometrical optics approximation on the basis of the Maxwell equations closed by the material equation obtained earlier. The corresponding dispersion equation is derived and solved. It is found that resonance interaction of a wave with an electron beam can occur only at two spatial points. Perhaps, such a short-time (point-like) mechanism of the resonance clarifies one of the main problems of physics of electron beams generated by solar flares: their time of existence is much longer than the time following from the previous theoretical estimates of the beam energy loss rate due to radiation.     

Theory of type III radio bursts in the interplanetary space: I. Derivation of permittivity tensor for the system consisting of electron beam and solar wind plasma

The tensor of permittivity for the system “electron beam - plasma of the interplanetary space” is derived in the approximation of geometrical optics. The problem is one-dimensional; all parameters such as density of the beam and of the solar wind plasma, and the induction of the interplanetary magnetic field are assumed to be dependent only on the distance to the Sun. The beam is generated by an active region during a solar flare, and it is a source of radio bursts of type III in the interplanetary space. The tensor of permittivity was obtained to close field equations by a material equation. On the basis of these equations it becomes possible to study theoretically the amplitude-frequency characteristics of the radio bursts as disturbances of the above-described beam-plasma system.     

无穷维情形下的Rolle定理

设Ω是自反的实Banach空间X中的有界开凸集,Y为一实赋范线性空间。证明了一个无穷维情形下的Rolle定理：如果算子A∶Ω→Y在Ω上强连续,在Ω内Frèchet可微,并且存在Y上的非0连续线性泛函f,使得f（Ax）=0对一切x∈Ω成立,则至少存在一点∈Ω,使对一切u∈X,都成立f（A′（x）u）=0。