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(μ) = I0μα, α > −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 F1, valid for all real α 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 |Δ F| 10⁻¹⁷. The framework encompasses the square-root law (α = 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 seven α values confirms a ∼104 × 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.