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A significant fraction of an exoplanet transit model evaluation time is spentcalculating projected distances between the planet and its host star. This is arelatively fast operation for a circular orbit, but slower for an eccentricone. However, because the planet's position and its time derivatives areconstant for any specific point in orbital phase, the projected distance can becalculated rapidly and accurately in the vicinity of the transit by expandingthe planet's $x$ and $y$ positions in the sky plane into a Taylor series atmid-transit. Calculating the projected distance for an elliptical orbit usingthe four first time derivatives of the position vector (velocity, acceleration,jerk, and snap) is $\sim100$ times faster than calculating it using theNewton's method, and also significantly faster than calculating $z$ for acircular orbit because the approach does not use numerically expensivetrigonometric functions. The speed gain in the projected distance calculationleads to 2-25 times faster transit model evaluation speed, depending on thetransit model complexity and orbital eccentricity. Calculation of the fourposition derivatives using numerical differentiation takes $\sim1\,\mu$s with amodern laptop and needs to be done only once for a given orbit, and the maximumerror the approximation introduces to a transit light curve is below 1~ppm forthe major part of the physically plausible orbital parameter space.

Going back to basics: accelerating exoplanet transit modelling using Taylor-series expansion of the orbital motion