Using Lyman-$\alpha$ transits to constrain models of atmospheric escape

Lyman-$\alpha$ transits provide an opportunity to test models of atmospheric escape directly. However, translating observations into constraints on the properties of the escaping atmosphere is challenging. The major reason for this is that the observable parts of the outflow often comes from material outside the planet's Hill sphere, where the interaction between the planetary outflow and circumstellar environment is important. As a result, 3D models are required to match observations. Whilst 3D hydrodynamic simulations are able to match observational features qualitatively, they are too computationally expensive to perform a statistical retrieval of properties of the outflow. Here, we develop a model that determines the trajectory, ionization state and 3D geometry of the outflow as a function of its properties and system parameters. We then couple this model to a ray tracing routine in order to produce synthetic transits. We demonstrate the validity of this approach, reproducing the trajectory of the outflows seen in 3D simulations. We illustrate the use of this model by performing a retrieval on the transit spectrum of GJ 436 b. Our model constrains the sound speed of the outflow $\gtrsim 10 \text{ km s}^{-1}$, indicating that we can rule out core-powered mass loss as the mechanism driving the outflow for this planet. The bound on planetary outflow velocity and mass loss rates are consistent with a photoevaporative wind.