Analytical transit light curves for arbitrary power-law limb darkening: a unified framework
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Chishtie, Farrukh A. Saeed, Mohammad I. Goderya, Shaukat N. |
| author_facet | Chishtie, Farrukh A. Saeed, Mohammad I. Goderya, Shaukat N. |
| contents | We present a unified analytical framework for computing exoplanet transit light curves under arbitrary real power-law limb darkening $I(μ)=I_0μ^α$, $α>-2$, 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 $F_1$, 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 $|Δ\mathcal{F}|\lesssim 10^{-17}$. 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 eight $α$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytical transit light curves for arbitrary power-law limb darkening: a unified framework Chishtie, Farrukh A. Saeed, Mohammad I. Goderya, Shaukat N. Earth and Planetary Astrophysics Instrumentation and Methods for Astrophysics Solar and Stellar Astrophysics We present a unified analytical framework for computing exoplanet transit light curves under arbitrary real power-law limb darkening $I(μ)=I_0μ^α$, $α>-2$, 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 $F_1$, 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 $|Δ\mathcal{F}|\lesssim 10^{-17}$. 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 eight $α$ 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. |
| title | Analytical transit light curves for arbitrary power-law limb darkening: a unified framework |
| topic | Earth and Planetary Astrophysics Instrumentation and Methods for Astrophysics Solar and Stellar Astrophysics |
| url | https://arxiv.org/abs/2506.18860 |