Analytical Transit Light Curves for Arbitrary Power-Law Limb Darkening: A Unified Framework

TL;DR

This paper introduces exact analytical solutions for exoplanet transit light curves with arbitrary power-law limb darkening, overcoming limitations of previous integer-only models and enabling highly accurate, efficient analysis for modern astronomical observations.

Contribution

It provides the first exact solutions for arbitrary real power-law limb darkening in transit light curves, generalizing previous integer-based models and improving computational accuracy and speed.

Findings

Exact solutions for arbitrary power-law limb darkening derived

Achieves nine orders of magnitude accuracy improvement over Monte Carlo methods

Enables efficient Bayesian inference for upcoming astronomical missions

Abstract

We present the first exact analytical solutions for exoplanet transit light curves under arbitrary real power-law limb darkening, $I(\mu) = I_0\mu^\alpha$ with $\alpha > -2$, removing the two-decade restriction to integer-polynomial forms. This generalization addresses a fundamental barrier: physically motivated stellar atmosphere models require non-integer exponents -- most critically the square-root law ($\alpha=1/2$) essential for M-dwarf characterization and half-integer powers in Claret's four-parameter law -- yet existing frameworks support only integer powers, forcing reliance on numerical integration or polynomial approximations that introduce systematic errors at the $\sim 10^{-3}$ level. We develop three mathematically equivalent formulations: (1) a geometric kernel representation expressing blocked flux as a one-dimensional radial integral; (2) a fractional-calculus framework yielding exact hypergeometric solutions via Riemann-Liouville operators; and (3) a numerically stable hypergeometric-integrand form with guaranteed convergence. All three reproduce established elliptic-integral expressions for integer $\alpha$, confirming our framework as a true generalization. For central transits, we obtain the exact closed form $\mathcal{F} = (1-p^2)^{1+\alpha/2}$. Linear superposition enables exact treatment of multi-parameter laws without special-case logic. Validation using 45-digit precision arithmetic demonstrates inter-method agreement at $|\Delta\mathcal{F}| \lesssim 10^{-13}$ across all geometric regions, including transcendental exponents ($\pi/2$, $\sqrt{2}$, $e-2$). Compared with Monte Carlo integration, our method achieves nine orders of magnitude improvement in accuracy with $\sim 10^4\times$ computational speedup, enabling efficient Bayesian inference for JWST, TESS, and next-generation facilities.