Exoplanet secondary eclipse

Secondary eclipses are one of the methods of detecting exoplanets and also probe planetary emission.

During secondary eclipse the planet passes behind its star. The lost system light measures the planet-star flux ratio from reflected starlight and planetary thermal emission.

Geometric albedo compares a body's full-phase brightness with that of a flat, perfectly diffusing reference disc of the same cross-section.

The orbital variation of a planet's reflected stellar light as the visible illuminated area changes. In a simple spherical model, $F_p/F_{\ast }=A_g(R_p/a{)}^2\Phi (\alpha )$, where $\Phi (0)=1$ and $\alpha$ is the star-planet-observer angle. Exoplanet clouds can alter the geometric albedo and the angular scattering law; this observable differs from an exoplanet thermal phase curve.

The phase integral converts a monochromatic geometric albedo into the corresponding reflected power fraction. The bolometric Bond albedo is the incident-spectrum-weighted average of $q_{\lambda }{A}_{g,\lambda }$.

Bond albedo is the fraction of total incident power reflected in all directions and over all wavelengths.

This idealized surface follows Lambertian reflectance, with projected-area dependence governed by Lambert's cosine law.

A Lambertian surface has direction-independent radiance. A perfectly reflecting Lambertian patch receiving normal flux $F$ emits specific intensity $I=F/\pi$ into its outward hemisphere.

For blackbody star and planet, define the planet-star contrastA complete exoplanet secondary eclipse removes the planet's light from the combined out-of-eclipse light. Its exact normalized depth is $d_{\lambda }=r_{\lambda }/(1+r_{\lambda })$. The common approximation $d_{\lambda }\simeq r_{\lambda }$ requires $r_{\lambda }\ll 1$. This thermal model omits reflected light and spatial variations.

For blackbodies with $T_p<T_s$, the contrast $r_{\lambda }=(R_p/R_s{)}^2(e^{x_s}-1)/(e^{x_p}-1)$ increases with wavelength, where $x_i=hc/(\lambda k_B{T}_i)$. Indeed $d\log r/d\log \lambda =G(x_p)-G(x_s)>0$, with $G(x)=x/(1-e^{-x})$ increasing for $x>0$ because $1-(1+x)e^{-x}>0$. The contrast approaches $(R_p/R_s{)}^2{T}_p/T_s$ in the Rayleigh-Jeans law limit. This behavior differs from the peaked absolute blackbody flux.

For blackbody star and planet observed at one common wavelength sufficiently long for the Rayleigh-Jeans law to hold for both,This follows by substituting $B_{\lambda }=2c{k}_{B}T/{\lambda }^4$ 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 $\lambda F_{\lambda }$.