Fast estimation of orbital parameters in Milky-Way-like potentials

Orbital parameters, such as eccentricity and maximum vertical excursion, of stars in the Milky Way are an important tool for understanding its dynamics and evolution, but calculation of such parameters usually relies on computationally-expensive numerical orbit integration. We present and test a fast method for estimating these parameters using an application of the St\"ackel fudge, used previously for the estimation of action-angle variables. We show that the method is highly accurate, to a level of $<1\%$ in eccentricity, over a large range of relevant orbits and in different Milky Way-like potentials, and demonstrate its validity by estimating the eccentricity distribution of the RAVE-TGAS data set and comparing it to that from orbit integration. Using the method, the orbital characteristics of the $\sim 7$ million $\textit{Gaia}$ DR2 stars with radial velocity measurements are computed with Monte Carlo sampled errors in $\sim 116$ hours of parallelised cpu time, at a speed that we estimate to be $\sim 3$ to $4$ orders of magnitude faster than using numerical orbit integration. We demonstrate using this catalogue that $\textit{Gaia}$ DR2 samples a large range of orbits in the solar vicinity, down to those with $r_\mathrm{peri} \lesssim 2.5$ kpc, and out to $r_\mathrm{ap} \gtrsim 13$ kpc. We also show that many of the features present in orbital parameter space have a low mean $z_\mathrm{max}$, suggesting that they likely result from disk dynamical effects.