Abstract
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 $100$
km~s$^-1$ before it becomes too ionized.
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