Spectrum Analysis

By coordinating multiple frequency observation, we may dervive the spectrum of the emission. The obtained variable is $\alpha$ whose definition is

$$F_{\nu} \propto {\nu}^{\alpha}.$$

In real, $\alpha$ is not uniform over the frequencies. But the spectrum in microwave range is approximately fitted by the fucntion in the form :

$$F_{\nu} = \widehat{F_{\nu}} \left(\frac{\nu}{\widehat{\nu}}\... ... tn}} & \mbox{for $\nu \gg \widehat{\nu}$} \end{array}\right.$$ (1)

Here are 4 fitting parameters: $\widehat{\nu}$ is turn-over frequency, $\widehat{F_{\nu}}$ is turn-over flux density, $\alpha_{\rm tk}$ is power index of low frequency side (optically thick side) and $\alpha_{\rm tn}$ is power index of high frequency side (optically thin side), respectively. The procedures are as follows:
IDL> day='2002-07-18'
IDL> timerange=['2002-07-18 3:30','2002-07-18 4:00'] $<$CR$>$
IDL> norp_rd_dat,day,mvd,tim,fi,fv,freq,timerange=timerange $<$CR$>$
IDL> norp_rd_avg,day,timavg,fiavg,fvavg $<$CR$>$
IDL> for m=0,6 do fi(m,*)=fi(m,*)-fiavg(m) $<$CR$>$
IDL> norp_alpha,freq,fi,mvd,mvdfit,alpha_tk,alpha_tn,freqpk,fluxpk $<$CR$>$
After subtracting the pre-flare flux levels (here daily averages are used), the flare fluxes are given to the IDL procedure norp_alpha. mvdfit is an array including the values of unity (1) for valid data or zero (0) for non-valid data.

Note: The fitting will usually fail in the simple manner shown here. We need (1) to take longer intergation time for getting better S/N ratio and (2) to remove the inadequate data for fitting (because, say, the emission in that frequency is not due to gyrosynchrotron). See 4.8.3 for better procedures.