= Solution
<Thermal bremsstrahlung> arises from <electron> acceleration in <ion> Coulomb fields. For a thermal <plasma> its optically thin spectral emissivity has the dependence
$$
j_\nu\propto n_e\sum_iZ_i^2n_i\,T_e^{-1/2}e^{-h\nu/(k_BT_e)}g_{\rm ff}(\nu,T_e).
$$
The exponential high-energy slope helps constrain <temperature> in X-ray spectra, with composition, absorption and other emission components included. At radio frequencies with $h\nu\ll k_BT_e$, the thin spectrum is much flatter and does not alone determine a unique <temperature>; the <Gaunt factor> supplies a weak <frequency> dependence.
For an approximately isothermal layer, <radiative transfer> gives $I_\nu=B_\nu(T_e)(1-e^{-\tau_\nu})$. Radio free-free opacity scales approximately as $\tau_\nu\propto\mathrm{EM}\,T_e^{-3/2}\nu^{-2}g_{\rm ff}$, where the hydrogen-plasma <emission measure> is $\mathrm{EM}=\int n_e^2dl$. A resolved optically thick brightness <temperature> approaches $T_e$; an unresolved filling factor lowers the observed value. The turnover between thin and thick regimes helps constrain <emission measure>.
An angular extent and distance give a physical size, and an assumed geometry supplies a line-of-sight depth. Combining that depth with <emission measure> estimates an rms <density>, $n_{e,\rm rms}\sim\sqrt{\mathrm{EM}/\ell}$. Clumping, filling factor, composition and unknown depth preserve a <density-depth degeneracy of free-free radiation>. These measurements constrain different combinations of <temperature>, <density> and size rather than independently fixing all three without assumptions.
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