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Going back to basics: accelerating exoplanet transit modelling using Taylor-series expansion of the orbital motion
Monthly Notices of the Royal Astronomical Society ( IF 4.7 ) Pub Date : 2020-09-25 , DOI: 10.1093/mnras/staa2953
H Parviainen 1, 2 , J Korth 3
Affiliation  

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 $\sim1\,\mu$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.

中文翻译:

回归基础:使用轨道运动的泰勒级数展开加速系外行星凌日建模

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主恒星之间的预计距离。对于圆形轨道,这是一种相对较快的操作,但对于偏心轨道则较慢。但是,由于行星的位置及其时间导数对于轨道相位中的任何特定点都是恒定的,因此可以通过扩展行星在天空中的 $x$ 和 $y$ 位置来快速准确地计算凌日附近的投影距离飞机在中途进入泰勒级数。使用位置向量的四个一阶导数(速度、加速度、加加速度和捕捉)计算椭圆轨道的投影距离比使用牛顿方法计算要快 $\sim100$ 倍,并且比计算圆形轨道的 $z$ 快得多,因为该方法不使用数值昂贵的三角函数。投影距离计算中的速度增益导致过境模型评估速度提高 2-25 倍,具体取决于过境模型的复杂性和轨道偏心率。使用数值微分计算四个位置导数需要 $\sim1\,\mu$s 使用现代笔记本电脑并且只需要对给定轨道进行一次,并且近似引入过光曲线的最大误差低于 1 ~ppm 表示物理上合理的轨道参数空间的主要部分。投影距离计算中的速度增益导致过境模型评估速度提高 2-25 倍,具体取决于过境模型的复杂性和轨道偏心率。使用数值微分计算四个位置导数需要 $\sim1\,\mu$s 使用现代笔记本电脑并且只需要对给定轨道进行一次,并且近似引入过光曲线的最大误差低于 1 ~ppm 表示物理上合理的轨道参数空间的主要部分。投影距离计算中的速度增益导致过境模型评估速度提高 2-25 倍,具体取决于过境模型的复杂性和轨道偏心率。使用数值微分计算四个位置导数需要 $\sim1\,\mu$s 使用现代笔记本电脑并且只需要对给定轨道进行一次,并且近似引入过光曲线的最大误差低于 1 ~ppm 表示物理上合理的轨道参数空间的主要部分。
更新日期:2020-09-25
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