Analytic Approximations for Transit Light‐Curve Observables, Uncertainties, and Covariances

Abstract

The light curve of an exoplanetary transit can be used to estimate the planetary radius and other parameters of interest. Because accurate parameter estimation is a nonanalytic and computationally intensive problem, it is often useful to have analytic approximations for the parameters as well as their uncertainties and covariances. Here, we give such formulae, for the case of an exoplanet transiting a star with a uniform brightness distribution. We also assess the advantages of some relatively uncorrelated parameter sets for fitting actual data. When limb darkening is significant, our parameter sets are still useful, although our analytic formulae underpredict the covariances and uncertainties.

Ground-based near-UV observations of 15 transiting exoplanets: constraints on their atmospheres and no evidence for asymmetrical transits

“…Limb darkening and red noise cause the error estimation in equations (7) and (8) to increase (Seager 2011). The error estimations we find using EXOMOP have the same behaviour as the analytic estimates by Carter et al (2008) exactly for both σ δ and σ t . For example, if the noise doubles in our white noise tests then the error estimates on R p /R * also double (Table 4).…”

“…We calculate the transit duration, τ t , of each of our transit model fits with the following equation (Carter et al 2008):…”

“…During the analysis for each model, the time of mid-transit (T c ) and R p /R * are allowed to float. We only model these two parameters for the comparison tests because the errors on them are analytically tractable (see below; Carter et al 2008). The i, e, ω, μ 1 , μ 2 , a/R * , and the P p of the planet are fixed.…”

“…Carter et al (2008) derive an analytic estimate for the 1σ uncertainty in δ (σ δ ) to be a The red and white noise are calculated using the wavelet likelihood technique (Carter & Winn 2009) described in Section 3.1.1. b The red and white noise are calculated using the time-averaging method (Pont et al 2006) described in Section 3.1.1. where σ g are the Gaussian errors in the relative flux (in our case the noise added) and n is the number of data points. Additionally, the analytic estimate of the 1σ uncertainty (σ t ) in the mid-transit time is (Carter et al 2008)…”

The WASP transit surveys

“…A simultaneous MCMC analysis of the light curve and radial-velocity curve establishes the stellar density directly from the transit profile via Kepler's 3rd law (Carter et al 2008) . By comparing the star's location in the (T eff , R/M 1/3 ) plane against isochrones and evolutionary tracks derived from the stellar evolution models of Girardi et al (2000), we refine our estimates of the stellar mass and age.…”