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rodluger / starry

Tools for mapping stars and planets.
https://starry.readthedocs.io
MIT License
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Higher order limb darkening #36

Closed rodluger closed 6 years ago

rodluger commented 6 years ago

This is easy. Figure out the general expression for the nth order limb darkening coefficient in terms of the spherical harmonic coefficients and allow users to provide an array of mu coefficients when instantiating a Star object.

ericagol commented 6 years ago

As I mentioned, perhaps this should be a separate paper?

rodluger commented 6 years ago

Yes! I'm thinking we can submit two papers at the same time that cite each other. The Mandel & Agol improvements and the limb darkening stuff will go into the second paper, but we might as well code it all up into starry. I'll start a repo for this second paper -- we have most of the math, and we just need to organize it!

ericagol commented 6 years ago

Yes, I've been meaning to write some code to see which of the various expression/expansions gives the best accuracy and/or speed for different values of b & r. It might also be interesting to include a limb-brightening term given in Schlawin et al. And, it would be interesting to fit some stellar atmosphere limb-darkening models to see how many coefficients are needed to represent these accurately. Another thing I've been thinking about is what to do about the derivatives at the contact points. I think it is mainly the non-limb-darkened term that is the problem. In practice, maybe this isn't a concern since it is a set of measure zero. The final thing I would like to think about is whether there is a better way to time-integrate. I looked back at the Gimenez (2006, A&A, 450, 1231; DOI: 10.1051/0004-6361:20054445) to paper to compare your solution to his. He computes the limb-darkened light curve up to \mu^N, and for each \mu^n for 0 < n <= N, he gives the solution in terms of an infinite series where each term in the series involves polynomials and Gamma functions. It's cool that your solution does this all analytically in terms of three elliptic integrals and a polynomial! It may, however, be worth comparing with his code for accuracy and/or speed. Other papers to compare to are Kjurkchieva et al. (2013) & Abukekerov et al. (2013). Also, 2016MNRAS.459.2078A computes second derivatives of a binary model.

rodluger commented 6 years ago

@ericagol Arbitrary order limb darkening is now implemented in starry! Here's a concocted example of fitting a 10th order limb darkening model to an 11-point stellar intensity grid to get an "exact" limb darkening model. image I'll write up all the math in the limb darkening paper this week.

ericagol commented 6 years ago

Excellent! Do the higher order coefficients require a Taylor expansion in this case?

rodluger commented 6 years ago

Only when $b$ is very close to zero -- that's really the only unstable region for small occultors.

ericagol commented 6 years ago

That could possibly be fixed with using the generalized complete elliptic integrals.

rodluger commented 6 years ago

True! I'll play around with that.

On Mon, May 21, 2018 at 9:42 AM Eric Agol notifications@github.com wrote:

That could possibly be fixed with using the generalized complete elliptic integrals.

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