Optically-Thin Non-Thermal Gyrosynchrotron Emission (Preliminary)

Note: The procedures described in this subsection are under testing. Comments are welcome for improvement.

Based on approximation models, the relation between the physical variables of emitting region and the emission can be derived. The electron distribution function is assumed to be the power law.

$${{d NV(E)} \over {d E}} = K \left({{E}\over{E_{0}}}\right)^{-\delta}。$$

where $E$ is electron energy (keV), $NV(E)$ is number of electrons (particles) that has larger energy than $E$. This distribution is described with parameters $\delta$, $E_{0}$, and $K$. But it is usual to use $NV(E_0)=K/(\delta -1)/E_0^{\delta-1}$ instead of $K$. And we fix $E_{0} = 10\ {\rm keV}$ after Dulk (1985).

(a) From Physical Variables to Emission

Inputs; power-law index $\delta$, magnetic field strength (G), angle between magnetic field and line of sight (degree), and $NV$ -- total number (particles) of non-thermal electron (of $E> 10\ {\rm keV}$): Outputs; flux density (SFU), circular polarization degree:
By Dulk's (1985) method:
IDL> dulk_gysy,delta,bb,theta,nv,freq,fi,rc $<$CR$>$
If the size of emitting source (by solid angle in unit of sterad) is given additionally, optical depth is also obtained
IDL> dulk_gysy,delta,bb,theta,nv,freq,fi,rc,omega,tau $<$CR$>$
By Ramaty's (1969) method:
IDL> ramaty_gysy,delta,bb,theta,nv,freq,fi,rc $<$CR$>$
IDL> ramaty_gysy,delta,bb,theta,nv,freq,fi,rc,omega,tau $<$CR$>$

(b) From Emission to Physical Variables

Based on Dulk's (1985) model we may derive the physical variables from emission. After deriving $\alpha$ (see 3.5.9), power law index of non-thermal electron distribution function is
IDL> norh_alpha,indexa,fi17,indexz,fi34 ,indexal,alpha,mvda $<$CR$>$
IDL> norh_alpha2delta,alpha,delta $<$CR$>$
If V-component (R$-$L) of 17 GHz is given adding to this, physical variables can be derived to some extent. For this, we have to assume one of three variables to be obtained -- magnetic field strength (G), angle between magnetic field and line of sight (degree), and total number of non-thermal electron of $E> 10\ {\rm keV}$ (particles).
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,thetain,bb,nv,/assumetheta $<$CR$>$
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,bbin,nv,theta,/assumebb $<$CR$>$
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,nvin,theta,bb,/assumenv $<$CR$>$
Adding to this, if the size of emission source is given (as solid angle. Unit is sterad), optical thickness is derived
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,thetain,bb,nv,/assumetheta ,omega,tau $<$CR$>$
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,bbin,nv,theta,/assumebb ,omega,tau $<$CR$>$
IDL> norh_gysy_inv,fi17,fi34,delta,fv17,nvin,theta,bb,/assumenv ,omega,tau $<$CR$>$

Note: In case of images, make the size of beams same between two images (see section 3.5.8).