Exoplanet phase curves at large phase angles. Diagnostics for extended hazy atmospheres

At optical wavelengths, Titan's brightness for large Sun-Titan-observer phase angles significantly exceeds its dayside brightness. The brightening that occurs near back-illumination is due to moderately large haze particles in the moon's extended atmosphere that forward-scatter the incident sunlight. Motivated by this phenomenon, here we investigate the forward scattering from currently known exoplanets, its diagnostics possibilities, the observational requirements to resolve it, and potential implications. An analytical expression is derived for the amount of starlight forward-scattered by an exponential atmosphere that takes into account the finite angular size of the star. We use this expression to tentatively estimate how prevalent this phenomenon may be. Based on numerical calculations that consider exoplanet visibility, we identify numerous planets with predicted out-of-transit forward scattering signals of up to tens of parts-per-million provided that aerosols of $>$1 micron size form over an extended vertical region near the optical radius level. We propose that the interpretation of available optical phase curves should be revised to constrain the strength of this phenomenon that might provide insight into aerosol scale heights and particle sizes. For the relatively general atmospheres considered here, forward scattering reduces the transmission-only transit depth by typically less than the equivalent to a scale height. For short-period exoplanets the finite angular size of the star severely affects the amount of radiation scattered towards the observer at mid-transit.