  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.
