= Planet-star radius estimate in the Rayleigh-Jeans limit
For <blackbody> star and planet observed at one common <wavelength> sufficiently long for the <Rayleigh-Jeans law> to hold for both,
$$
f_\lambda=\frac{F_{p,\lambda}}{F_{*,\lambda}}=\left(\frac{R_p}{R_*}\right)^2\frac{T_p}{T_*},\qquad R_p=R_*\sqrt{f_\lambda T_*/T_p}.
$$
This follows by substituting $B_\lambda=2ck_BT/\lambda^4$ in the <thermal eclipse depth>. A <radiative flux> ratio alone does not determine the <radius> without a planetary <brightness temperature> and a stellar <temperature>. Spectral peaks interpreted with <Wien's displacement law> must refer to $F_\lambda$, not $F_\nu$ or $\lambda F_\lambda$.
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