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

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 $\mathrm{\mid }\mathrm{\Delta }\mathcal{F}\mathrm{\mid }\lesssim {10}^{-17}$. The framework encompasses the square-root law ($\alpha =1\mathrm{/}2$) 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 $\sim {10}^4\times$ 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.