2021ApJ...917L..30A

Accreting planets have been detected through their hydrogen-line emission, specifically Hα. To interpret this, stellar-regime empirical correlations between the Hα luminosity L_Hα and the accretion luminosity L_acc or accretion rate {dot}M have been extrapolated to planetary masses, however without validation. We present a theoretical L_acc-L_Hα relationship applicable to a shock at the surface of a planet. We consider wide ranges of accretion rates and masses and use detailed spectrally resolved, nonequilibrium models of the postshock cooling. The new relationship gives a markedly higher L_acc for a given L_Hα than fits to young stellar objects, because Lyα, which is not observable, carries a large fraction of L_acc. Specifically, an L_Hα measurement needs 10 to 100 times higher L_acc and {dot}M than previously predicted, which may explain the rarity of planetary Hα detections. We also compare the {dot}M-L_Hα relationships coming from the planet-surface shock or implied by accretion-funnel emission. Both can contribute simultaneously to an observed Hα signal, but at low (high) {dot}M the planetary-surface shock (heated funnel) dominates. Only the shock produces Gaussian line wings. Finally, we discuss accretion contexts in which different emission scenarios may apply, putting recent literature models in perspective, and also present L_acc-L_line relationships for several other hydrogen lines.