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

Abstract

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 q 1 ≡ (u 1 + u 2 ) 2 and q 2 ≡ 0.5u 1 (u 1 + u 2 ) −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.

ACCESS I. AN OPTICAL TRANSMISSION SPECTRUM OF GJ 1214b REVEALS A HETEROGENEOUS STELLAR PHOTOSPHERE

“…We used a uniform prior on w 1 , with boundary values set by the triangular sampling method of Kipping (2013).…”

“…In the first approach we fixed the limb darkening coefficients w 1 and w 2 during the MCMC analysis to the values from the PHOENIX atmospheric model (Husser et al 2013). In the second we utilized the triangular sampling method proposed by Kipping (2013 3 ). Both methods enlarged the s 1 uncertainties on the transmission spectrum by 0.2%.…”

The Kepler-454 System: A Small, Not-Rocky Inner Planet, a Jovian World, and a Distant Companion

“…We assumed a circular orbit model and fit for quadratic limb darkening coefficients using the parameterization suggested by Kipping (2013) We constrained the transit parameters by performing a MCMC analysis with a Metropolis-Hastings acceptance criterion (Metropolis et al 1953), starting with the initial parameters provided by Batalha et al (2013). We initialized the chains with starting positions set by perturbing the solution of the preliminary fit by up to 5σ for each parameter.…”

Characterization of the planetary system Kepler-101 with HARPS-N

“…The free parameters of our global model are the transit epoch T 0 , the orbital period P, two systemic radial velocities for HARPS-N data obtained with both the original (V r,o ) and the replaced chip (V r,r ), the radial-velocity semi-amplitude K, √ e cos ω and √ e sin ω, where e is the eccentricity and ω the argument of periastron, the transit duration T 14 , the ratio of the planet to stellar radii R p /R * , the inclination i between the orbital plane and the plane of the sky, and the two limb-darkening coefficients (LDC) q 1 = (u a + u b ) 2 and q 2 = 0.5u a /(u a + u b ) (Kipping 2013), where u a and u b are the coefficients of the quadratic limb-darkening law (e.g., Claret 2000). A DE-MCMC analysis with a number of chains equal to twice the number of free parameters was then carried out.…”