Exoplanet secondary eclipse

An exoplanet secondary eclipse occurs when a planet passes behind its host star. Immediately outside the event the detector receives light from both bodies; during complete occultation it receives only stellar light. Their difference isolates planetary light, consisting of reflected stellar radiation, planetary thermal radiation, or both. Its fraction of the combined out-of-eclipse light is the eclipse depth.

Measuring that depth across wavelengths gives occultation spectroscopy of the visible dayside. The thermal eclipse depth connects a simple blackbody model to the radius ratio and brightness temperature. At visible wavelengths, reflected light can instead constrain geometric albedo. Changes in the host star or detector response must be modelled because the planetary signal is a small difference between two much brighter measurements.

The difference between star-plus-planet light outside secondary eclipse and star-only light during eclipse isolates the planet's dayside spectrum. Thermal radiation probes its temperature-pressure structure and opacity, while optical reflected light probes scattering and albedo. Contrast and the subtraction of stellar photons limit sensitivity.

For a fractional feature amplitude $d$, equal independent in- and out-of-eclipse samples with $N_{\gamma }$ stellar photons each have fractional difference noise approximately $\sqrt{2/N_{\gamma }}$. The ideal significance is $d\sqrt{N_{\gamma }/2}$. Correlated systematics, stellar variability and unequal exposure budgets change this result.

The geometric albedo measures a body's brightness at full phase relative to a perfectly reflecting, diffusely scattering flat disc of the same projected area, illuminated identically. It is therefore a directional brightness comparison. For an unresolved spherical planet at separation $a$, the full-phase reflected contrast is $F_p/F_{\star }=A_g(R_p/a{)}^2$ under the usual distant-source approximation.

An optical exoplanet secondary eclipse can constrain this contrast after accounting for planetary thermal radiation. The geometric albedo differs from Bond albedo, which measures reflected power over every direction and wavelength. Strong backscattering can even make a directional geometric albedo exceed one without violating conservation of energy; total reflected power remains bounded by the incident power.

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 $q_{\lambda }{A}_{g,\lambda }$.

The Bond albedo is the fraction of total incident radiant power that a body reflects, summed over outgoing directions and over the incident spectrum. The absorbed fraction is $1-A_B$. It is a global energy-budget quantity, so it appears directly in the planetary equilibrium temperature and must lie between zero and one for a passive reflecting body.

A bright appearance from one direction does not determine Bond albedo. Geometric albedo measures full-phase brightness, and the planetary phase integral supplies the angular weighting needed to connect that brightness to reflected power at a given wavelength. The spectral weighting also depends on the illuminating star: the same reflecting material need not absorb the same fraction of light from hot and cool stars.

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 $\lambda F_{\lambda }$.