A Log-Mixed Closed-Form Estimator for Escape Maneuvers of Spacecraft

We introduce a single closed-form estimator for escape Δ𝑣 that blends the low-altitude loss regime with the high-altitude vacuum/Oberth regime through a smooth, dimensionless mixing weight. The estimator evaluates in 𝒪(1) time, is fully differentiable, and calibrates from a small set of reference cases using a two-point/least-squares procedure. The resulting one-equation model avoids piecewise logic yet reproduces the expected altitude dependence and regime transition, providing design-quality initial guesses for trajectory optimization and onboard tuning. We validate the approach against standard high-fidelity references (Lambert/propagation baselines) across Earth-like scenarios and show that the calibrated form tracks the target trends closely while remaining portable across primaries via a simple seven-scalar retuning.