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As shown by
Melrose & Brown in 1976, the trapped electrons can be expressed as a
convolution of the injection function with some kernel representing the trapp
processes. We could find a simple analytic solution for the trapped electrons
under an assumption that the loss mechanism in the corona is only the escape-like term. Use the solution to
calculate microwave emissivity, and applied here to fit the gradual part. The
left frame: the observed flux at three fre vs time. The smooth curves are the
model fit to the observed. The rest, spiky part are regarded as due to
primary precipitation not trapped and therefore leave them unfit. Used only the optically thin flux, I.e. the
emissivity. Flux at different frequency changes and this depends on the
energy-dependent term in the escape term. We found \nu\propto velocity is
good enough. Meaning that bounce frequency,i.e.,strong diffusion case.One may
think if strong diffusion the trap may loose particles so rapidly, but it was
not, because of the small loss cone angle, only the energy dependence tells
that it is strong diffusion. Matching this relavtive fluxes at different
frequencies is equilvalent to matching the spectral index. The right panel
shows the spectral index a good coincidence, mainly determined by the energy
dependence of the trap kernel, the strong diffusion in this case. \propto
\beta The spectral slope in the right = mainly beta dependence. The
proportional constant is related to the loop property as:
\nu_0=0.5*alpha0^2(c/L). Alpha0 is a small value, meaning a very good trap in
other words.
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