= Solution
<thermal bremsstrahlung> is continuum radiation from <electrons> accelerated in ion electric fields. For <Maxwellian> <electrons>,
$$
j_\nu\propto n_e\sum_i Z_i^2n_i\,T_e^{-1/2}e^{-h\nu/(k_BT_e)}g_{\rm ff}.
$$
At high <frequency> the exponential cutoff constrains $T_e$; X-ray fitting must also model foreground absorption, line emission, detector response and possible multitemperature or nonthermal components. At radio <frequencies>, $h\nu\ll k_BT_e$, an <optically thin> free-free spectrum is almost flat, with the weak <frequency> dependence set mainly by the <Gaunt factor>.
For pure ionized hydrogen, radio <free-free opacity> gives
$$
\tau_\nu\propto T_e^{-3/2}\nu^{-2}g_{\rm ff}\,\mathrm{EM},\qquad
\mathrm{EM}=\int n_e^2dl,\qquad
I_\nu=B_\nu(T_e)(1-e^{-\tau_\nu}).
$$
The turnover constrains <emission measure>; a resolved, <optically thick> source has <brightness temperature> approaching $T_e$. An unresolved source has an unknown beam <filling factor>, so its observed brightness need not equal its <electron> <temperature>. Angular size and an independent distance give a geometric depth $L$; a uniform model then yields \b[$n_e=\sqrt{\mathrm{EM}/L}$]. Clumping instead measures a density-squared average, not the mean <density>.<Essential Radio Astronomy by James J. Condon and Scott M. Ransom>[chapter 4, equations (4.57)–(4.60); emission measure and thick/thin spectra]{ref}
\b[A free-free spectrum alone does not uniquely determine <density> and size.] The <density-depth degeneracy of free-free radiation> is explicit: for a resolved uniform column, $n_e\mapsto qn_e$, $L\mapsto L/q^2$ leaves both emissivity-integrated brightness and <optical depth> unchanged at fixed <temperature>. With independent SZ information and known $T_e$, the <electron> column is $N_e=ym_ec^2/(\sigma_Tk_BT_e)$; then
$$
\boxed{n_e=\mathrm{EM}/N_e,\qquad L=N_e^2/\mathrm{EM}}.
$$
These equalities require consistent composition and geometry, and uniform <density> and <temperature>. Relating $L$ to angular extent can give distance; asphericity and the <gas clumping factor> are important limitations of this combined cluster method.
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