The fundamentals of Lyman-alpha exoplanet transits.

Lyman-$\alpha$ transits have been detected from a handful of nearby exoplanets and are one of our best insights into the atmospheric escape process. However, the fact interstellar absorption often renders the line-core unusable means we typically only observe the transit signature in the blue-wing, and they have been challenging to interpret. This has been recently highlighted by non-detections from planets thought to be undergoing vigorous escape. Pioneering 3D simulations have shown that escaping hydrogen is shaped into a cometary tail receding from the planet by tidal forces and interactions with the circumstellar environment. Motivated by this work, we develop the fundamental physical framework in which to interpret Lyman-$\alpha$ transits. We consider how this tail of gas is photoionized, and radially accelerated to high velocities. Using this framework, we show that the transit depth is often controlled by the properties of the stellar tidal field rather than details of the escape process. Instead, it is the transit duration that encodes details of the escape processes. Somewhat counterintuitively, we show that higher irradiation levels, which are expected to drive more powerful outflows, produce weaker, shorter Lyman-$\alpha$ transits. This result arises because the fundamental controlling physics is not the mass-loss rate but the distance a neutral hydrogen atom can travel before it's photoionized. Thus, Lyman-$\alpha$ transits do not primarily probe the mass-loss rates, but instead, they inform us about the velocity at which the escape mechanism is ejecting material from the planet, providing a clean test of predictions from atmospheric escape models. Ultimately, a detectable Lyman-$\alpha$ transit requires the escaping planetary gas to be radially accelerated to velocities of $\sim 100$ km~s$^{-1}$ before it becomes too ionized.