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Lyman-$\alpha$ transits provide an opportunity to test models of atmosphericescape directly. However, translating observations into constraints on theproperties of the escaping atmosphere is challenging. The major reason for thisis that the observable parts of the outflow often comes from material outsidethe planet's Hill sphere, where the interaction between the planetary outflowand circumstellar environment is important. As a result, 3D models are requiredto match observations. Whilst 3D hydrodynamic simulations are able to matchobservational features qualitatively, they are too computationally expensive toperform a statistical retrieval of properties of the outflow. Here, we developa model that determines the trajectory, ionization state and 3D geometry of theoutflow as a function of its properties and system parameters. We then couplethis model to a ray tracing routine in order to produce synthetic transits. Wedemonstrate the validity of this approach, reproducing the trajectory of theoutflows seen in 3D simulations. We illustrate the use of this model byperforming a retrieval on the transit spectrum of GJ 436 b. Our modelconstrains 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 drivingthe outflow for this planet. The bound on planetary outflow velocity and massloss rates are consistent with a photoevaporative wind.

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