Theoretical scaling law of coronal magnetic field and electron power-law index in solar microwave burst sources

It is first proposed a theoretical scaling law respectively for the coronal magnetic field strength $B$ and electron power-law index $\delta$ versus frequency and coronal height in solar microwave burst sources. Based on the non-thermal gyro-synchrotron radiation model (Ramaty in Astrophys. J. 158:753, 1969), $B$ and $\delta$ are uniquely solved by the observable optically-thin spectral index and turnover (peak) frequency, the other parameters (plasma density, temperature, view angle, low and high energy cutoffs, etc.) are relatively insensitive to the calculations, thus taken as some typical values. Both of $B$ and $\delta$ increase with increasing of radio frequency but with decreasing of coronal height above photosphere, and well satisfy a square or cubic logarithmic fitting.