Initial pitch-angle of narrowly beamed electrons injected into a magnetic mirror, formation of trapped and precipitating electron distribution, and asymmetry of hard X-ray and microwave footpoint emissions

A new kind of static distribution function for trapped and precipitating electrons is derived by solving a time-independent Fokker–Planck equation in a magnetic mirror, with injection of initially narrow-beamed electrons at an arbitrary initial pitch-angle. There are two independent parameters to determine the ratio of trapped and precipitating electrons, as well as their emissions, i.e., the mirror ratio and the initial pitch-angle, which is helpful for understanding some new features of asymmetrical hard X-ray and microwave footpoint emissions in solar observations.