Analytical transit light curves for arbitrary power-law limb darkening: a unified framework

Abstract

We present a unified analytical framework for computing exoplanet transit light curves under arbitrary real power-law limb darkening $I(\mu)=I_0\mu^\alpha$, $\alpha>-2$, eliminating the two-decade restriction to integer-polynomial forms. Our central result is an exact closed-form expression for the normalised stellar flux in terms of Appell's bivariate hypergeometric function $F_1$, valid for all real $\alpha$ and all geometric transit configurations. Three mutually equivalent formulations -- a geometric kernel representation, a Riemann-Liouville fractional-calculus framework, and a hypergeometric-integrand representation -- provide complementary physical insights and independent computational routes, with inter-method agreement at $|\Delta\mathcal{F}|\lesssim 10^{-17}$. The framework encompasses the square-root law ($\alpha=1/2$) essential for M-dwarf characterisation, half-integer powers required by Claret's four-parameter law, and arbitrary real values enabling empirical fitting, while recovering integer-polynomial elliptic-integral expressions as special cases. Validation at 40-digit precision across five geometric regions and eight $\alpha$ values confirms a $\sim10^4\times$ computational speedup over Monte Carlo integration and thirteen orders of magnitude accuracy advantage. Linear superposition enables exact treatment of arbitrary multi-parameter limb-darkening laws without special-case logic.