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Appendix B: Scaling law for free-free brightness depression

From the EUV observations is well known, that solar atmosphere has significant emission measure in the temperature range, intermediate between coronal-chromospheric values. It should lead to significant contribution to microwave emission, contrary to observations (see Zirin et al. [1991] for review).

As was proposed by Grebinskij ([1987]), this dilemma may be solved, ifone takes into account the spatial inhomogeneity of atmosphere. Following that paper, we clarify this conclusion with simple scaling laws for model example of isothermal plasma volume with some electron temperature $%
T_{\rm o}$ = const, and some fixed value of emission measure $EM_{\rm o}$ = const. We show, that observed microwave f-f emission brightness $T_{\rm eff}$ of such plasma is strongly dependent on spatial inhomogeneity and becomes negligible for strong inhomogeneity.

We compare two spatial configurations, in the same projected area, say $%
S_{\rm o}=L_{\rm o}\times L_{\rm o}$, with depth $h_{\rm o}$. At homogeneous case, with $%
N_{\rm e}(\vec{r})={\rm const}=N_{\rm o}$, we have for total emission measure $%
EM_{\rm tot}=N_{\rm o}^{2}L_{\rm o}^{2}h_{\rm o}$. As the model inhomogeneity, we assume en ensemble of some number K of horizontal slabs with dimensions $%
L_{\rm o}\times d\times d$ and constant number density $N_{\rm e}(\vec{r}%
)={\rm const}=N_{*}$ inside and zero outside each slab, with the same $%
EM_{\rm tot}^{*}=KN_{*}^{2}L_{\rm o}d^{2}=EM_{\rm tot}$, and total projected area $%
S_{*}=KL_{\rm o}\times d$. From the EM balance, one would have

\begin{displaymath}N_{*}^{2}d=\frac{L_{\rm o}^{2}}{KL_{\rm o}d}\equiv \frac{1}{\varepsilon }%
N_{\rm o}^{2}h_{\rm o}
\end{displaymath} (B1)

where $\varepsilon \equiv S_{*}/S_{\rm o}$ is a surface filling factor. Now, we consider the normalized optical depth $t\equiv \tau (\lambda =1$ cm) along the line of sight for both configurations, as

\begin{displaymath}t_{\rm o}\equiv k^{\prime }N_{\rm o}^{2}h_{\rm o},\ t_{*}\equiv k^{\prime
}N_{*}^{2}d,\ k^{\prime }={\rm const}
\end{displaymath} (B2)

and, taking into account Eq. (B1), we find a scaling law for optical depth

\begin{displaymath}t_{*}=\frac{1}{\varepsilon }t_{\rm o},\ \varepsilon \leq 1.
\end{displaymath} (B3)

Here it is crucial, that we have only one slab at the line of sight, if condition $\varepsilon \leq 1$ fulfilled (from contrary: if total surface $%
S_{*}>S_{\rm o}$, then slabs should overlap, because they do not match projected area). Now, we find a scaling law for the horizontally smoothed observed microwave brightness $<T_{\rm b}>$ as:

\begin{displaymath}<T_{\rm b}>_{\rm o}\equiv T_{\rm b,o}=T_{\rm o}(1-{\rm e}^{-t_{\rm o}\lambda ^{2}}),
\end{displaymath} (B4)

\begin{displaymath}<T_{\rm b}>_{*}\equiv \varepsilon T_{b,\varepsilon }=\varepsilon
T_{\rm o}(1-{\rm e}^{-t_{\rm o}\lambda ^{2}/\epsilon })
\end{displaymath} (B5)

which are correct for $\varepsilon \leq 1$ condition and becomes same at $%
\varepsilon =1$.

Equations (B1)-(B5) reveal a scaling nature of brightness depression, irrespective of detailed spacing of microscale structures: if the filling factor is sufficiently small (i.e. $\varepsilon <<1$, and $t_{\rm o}\lambda
_{*}^{2}/\epsilon >1$), then at some wave band $\lambda \geq \lambda _{*}$ we have: $<T_{\rm b}>_{*}\ =\ \varepsilon T_{\rm o}<<T_{\rm o}$. At the shorter wave band $\lambda \leq \lambda _{*}$with $t_{\rm o}\lambda^{2}/\epsilon <1$, the emission is optically thin and does not depend on inhomogeneity, but remains reduced as $<T_{\rm b}>_{*}\ =\ <T_{\rm b}>_{\rm o}\ =\ $ $%
t_{\rm o}\lambda ^{2}T_{\rm o}<<T_{\rm o}$.

These scalings, together with the observed small EUV-filling factors $%
\varepsilon <<1$, solves the problem of consistency results of optical and microwave observations: at EUV band we have optically thin emission, proportional to emission measure, but we never detect it at microwaves, with depressed brightness at the optically-thick regime (with $t_{\rm o}\lambda
^{2}<<1$, but $t_{\rm o}\lambda ^{2}/\epsilon $ >>1 at the cm waveband).

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