The heating of the thermal plasma with energetic electrons in small solar flares

The energetic electrons deduced from hard X-rays in the thick target model may be responsible for heating of soft X-ray plasma in solar flares. It is shown from OSO-7 studies that if a cutoff of 10 keV is assumed, the total electron is comparable to the thermal plasma energy. However, (1) the soft X-ray emission often appears to begin before the hard X-ray burst, (2) in about one-third of flares there is no detectable hard X-ray emission, and (3) for most events the energy content (assuming constant density) of soft X-ray plasma continues to rise after the end of the hard X-ray burst. To understand these problems we have analyzed the temporal relationship between soft X-rays and hard X-rays for 20 small events observed by ISEE-3 during 1980. One example is shown. The start of soft X-ray and hard X-ray bursts is defined as the time when the counting rates of the 4.8 to 5. keV and 25.8 to 43.2 keV channels, respectively, exceed the background by one standard deviation

Momentum distribution, vibrational dynamics and the potential of mean force in ice

By analyzing the momentum distribution obtained from path integral and phonon calculations we find that the protons in hexagonal ice experience an anisotropic quasi-harmonic effective potential with three distinct principal frequencies that reflect molecular orientation. Due to the importance of anisotropy, anharmonic features of the environment cannot be extracted from existing experimental distributions that involve the spherical average. The full directional distribution is required, and we give a theoretical prediction for this quantity that could be verified in future experiments. Within the quasi-harmonic context, anharmonicity in the ground state dynamics of the proton is substantial and has quantal origin, a finding that impacts the interpretation of several spectroscopies

Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations

We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity.Comment: 29 page