A theory of transmission spectroscopy of hydrodynamic outflows from planetary atmospheres: Spectral-line saturation and limits on mass-loss constraints

Transmission spectroscopy is a key technique in the characterization of exoplanet atmospheres and has been widely applied to planets undergoing hydrodynamic escape. While a robust analytic theory exists for transmission spectra of hydrostatic atmospheres, the corresponding interpretation for escaping atmospheres has so far relied on numerical modeling, despite the growing number of observations of planetary winds. In this work, a theory of transmission spectroscopy in hydrodynamically escaping atmospheres is developed by coupling the standard transmission geometry to a steady-state, spherically symmetric, isothermal outflow. This approach yields closed-form expressions for the chord optical depth and effective transit radius of a planetary wind and allows the optical depth inversion problem to be examined. The analytic solution reveals that transmission spectroscopy of planetary winds naturally separates into two regimes. In an opacity-limited regime, transmission depths retain sensitivity to the atmospheric mass-loss rate $\dot{M}$. Beyond a critical threshold, however, spectral-line cores become saturated and no longer provide a unique constraint on the escape rate. This transition is marked by a sharp analytic boundary of the form $\sigma(\lambda)\,\dot{M} \le C_{\rm sat}$, where $\sigma(\lambda)$ is the line absorption cross-section and $C_{\rm sat}$ is a constant set by the thermodynamic and geometric properties of the wind. This condition specifies when the inversion between transmission depth and mass-loss rate admits a real solution. Once it is violated, the effective transit radius is no longer controlled by opacity or mass loss, but by the geometric extent of the absorbing wind.