Efficient, uninformative sampling of limb darkening coefficients for two-parameter laws

By David Kipping. 2013

507 citations recorded by the source corpus.

Stellar limb darkening affects a wide range of astronomical measurements and is frequently modelled with a parametric model using polynomials in the cosine of the angle between the line of sight and the emergent intensity. Two-parameter laws are particularly popular for cases where one wishes to fit freely for the limb darkening coefficients (i.e. an uninformative prior) due to the compact prior volume and the fact that more complex models rarely obtain unique solutions with present data. In such cases, we show that the two limb darkening coefficients are constrained by three physical boundary conditions, describing a triangular region in the two-dimensional parameter space. We show that uniformly distributed samples may be drawn from this region with optimal efficiency by a technique developed by computer graphical programming: triangular sampling. Alternatively, one can make draws using a uniform, bivariate Dirichlet distribution. We provide simple expressions for these parametrizations for both techniques applied to the case of quadratic, square-root and logarithmic limb darkening laws. For example, in the case of the popular quadratic law, we advocate fitting for q1 ≡ (u1 + u2) 2 and q2 ≡ 0.5u1(u1 + u2) −1 with uniform priors in the interval [0, 1] to implement triangular sampling easily. Employing these parametrizations allows one to derive model parameters which fully account for our ignorance about the intensity profile, yet never explore unphysical solutions, yielding robust and realistic uncertainty estimates. Furthermore, in the case of triangular sampling with the quadratic law, our parametrization leads to significantly reduced mutual correlations and provides an alternative geometric explanation as to why naively fitting the quadratic limb darkening coefficients precipitates strong correlations in the first place.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/04a2e482d0295d6735cef07f24731689f84bdbbc

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In Astronomy and Astrophysics

In Atomic and Molecular Physics, and Optics

Citations in the corpus

The JWST Early Release Observations (2022, methodology): “The limb-darkening coefficients were fitted at each wavelength using the sampling scheme of Kipping (2013) and a square-root law, but using as priors limb-darkening coefficients obtained through the limb-darkening software combined with the SPAM algorithm, as described in Espinoza & Jordán (2015);…”