回归基础：使用轨道运动的泰勒级数展开加速系外行星transit建模
Going back to basics: accelerating exoplanet transit modelling using Taylor-series expansion of the orbital motion
摘要：
一个系外行星凌日模型评估时间的很大一部分用于计算行星与其宿主恒星之间的投影距离。对于圆形轨道，这是一个相对较快的操作，但对于偏心轨道则较慢。然而，由于行星的位置和它的时间导数对于轨道相位中的任何特定点都是恒定的，因此可以通过将行星在天空平面中的$x$和$y$位置展开成中星时的泰勒级数来快速准确地计算出中星附近的投影距离。使用位置矢量的四个一阶导数（速度、加速度、加加速度和瞬动）计算椭圆轨道的投影距离比使用牛顿法计算它快$\sim100$倍，并且也比计算圆形轨道的$z$快得多，因为该方法不使用数值上昂贵的三角函数。根据凌日模型的复杂性和轨道偏心率，预计距离计算中的速度增益导致凌日模型评估速度快2-25倍。用数值微分法计算四个位置导数需要$\sim1\,μ$ s的现代笔记本电脑，并且对于给定的轨道只需要做一次，并且对于物理上合理的轨道参数空间的大部分，近似引入的凌日光变曲线的最大误差小于1~ppm。
英文摘要：
A significant fraction of an exoplanet transit model evaluation time is spent calculating projected distances between the planet and its host star. This is a relatively fast operation for a circular orbit, but slower for an eccentric one. However, because the planet's position and its time derivatives are constant for any specific point in orbital phase, the projected distance can be calculated rapidly and accurately in the vicinity of the transit by expanding the planet's $x$ and $y$ positions in the sky plane into a Taylor series at mid-transit. Calculating the projected distance for an elliptical orbit using the four first time derivatives of the position vector (velocity, acceleration, jerk, and snap) is $\sim100$ times faster than calculating it using the Newton's method, and also significantly faster than calculating $z$ for a circular orbit because the approach does not use numerically expensive trigonometric functions. The speed gain in the projected distance calculation leads to 2-25 times faster transit model evaluation speed, depending on the transit model complexity and orbital eccentricity. Calculation of the four position derivatives using numerical differentiation takes $\sim1\,μ$s with a modern laptop and needs to be done only once for a given orbit, and the maximum error the approximation introduces to a transit light curve is below 1~ppm for the major part of the physically plausible orbital parameter space.
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