# Analytic Planetary Transit Light Curves and Derivatives for Stars with Polynomial Limb Darkening

## TL;DR

This paper presents new analytic solutions for modeling planetary transit light curves with polynomial limb darkening, including derivatives, which are faster and more stable than previous methods.

## Contribution

It introduces closed-form expressions for transit light curves with polynomial limb darkening of arbitrary order, along with their derivatives, improving speed and numerical stability over existing models.

## Key findings

- Analytic formulas for polynomial limb darkening of any order.
- Fast and stable computation of transit light curves and derivatives.
- Code implementations in multiple programming languages.

## Abstract

We derive analytic, closed-form solutions for the light curve of a planet transiting a star with a limb darkening profile which is a polynomial function of the stellar elevation, up to arbitrary integer order. We provide improved analytic expressions for the uniform, linear, and quadratic limb-darkened cases, as well as novel expressions for higher order integer powers of limb darkening. The formulae are crafted to be numerically stable over the expected range of usage. We additionally present analytic formulae for the partial derivatives of instantaneous flux with respect to the radius ratio, impact parameter, and limb darkening coefficients. These expressions are rapid to evaluate, and compare quite favorably in speed and accuracy to existing transit light curve codes. We also use these expressions to numerically compute the first partial derivatives of exposure-time averaged transit light curves with respect to all model parameters. An additional application is modeling eclipsing binary or eclipsing multiple star systems in cases where the stars may be treated as spherically symmetric. We provide code which implements these formulae in C++, Python, IDL, and Julia, with tests and examples of usage.