Analytic Planetary Transit Light Curves and Derivatives for Stars with Polynomial Limb Darkening
Abstract
We derive analytic, closedform solutions for the light curve of a planet transiting a star with a limbdarkening profile that is a polynomial function of the stellar elevation, up to an arbitrary integer order. We provide improved analytic expressions for the uniform, linear, and quadratic limbdarkened cases, as well as novel expressions for higherorder integer powers of limb darkening. The formulae are crafted to be numerically stable over the expected range of usage. We additionally present analytic formulae for the partial derivatives of instantaneous flux with respect to the radius ratio, impact parameter, and limbdarkening coefficients. These expressions are rapid to evaluate and compare quite favorably in speed and accuracy to existing transit lightcurve codes. We also use these expressions to numerically compute the first partial derivatives of exposuretimeaveraged transit light curves with respect to all model parameters. An additional application is modeling eclipsing binary or eclipsing multiple star systems in cases where the stars may be treated as spherically symmetric. We provide code which implements these formulae in C++, Python, IDL, and Julia, with tests and examples of usage (https://github.com/rodluger/Limbdark.jl).
 Publication:

The Astronomical Journal
 Pub Date:
 March 2020
 DOI:
 10.3847/15383881/ab4fee
 arXiv:
 arXiv:1908.03222
 Bibcode:
 2020AJ....159..123A
 Keywords:

 Exoplanet astronomy;
 Exoplanet detection methods;
 Transit photometry;
 Computational astronomy;
 Astronomy software;
 Analytical mathematics;
 Transits;
 Light curves;
 Eclipses;
 Limb darkening;
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 489;
 1709;
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 38;
 1711;
 918;
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 Astrophysics  Earth and Planetary Astrophysics;
 Astrophysics  Instrumentation and Methods for Astrophysics
 EPrint:
 57 pages, 18 figures. Accepted to AJ