Analytic Light Curve for Mutual Transits of Two Bodies Across a Limb-darkened Star

Abstract

We present a solution for the light curve of two bodies mutually transiting a star with polynomial limb darkening. The term “mutual transit” in this work refers to a transit of the star during which overlap occurs between the two transiting bodies. These could be an exoplanet with an exomoon companion, two exoplanets, an eclipsing binary and a planet, or two stars eclipsing a third in a triple-star system. We include analytic derivatives of the light curve with respect to the positions and radii of both bodies. We provide code that implements a photodynamical model for a mutual transit. We include two dynamical models, one for hierarchical systems in which a secondary body orbits a larger primary (e.g., an exomoon system) and a second for confocal systems in which two bodies independently orbit a central mass (e.g., two planets in widely separated orbits). Our code is fast enough to enable inference with Markov Chain Monte Carlo algorithms, and the inclusion of derivatives allows for the use of gradient-based inference methods such as Hamiltonian Monte Carlo. While applicable to a variety of systems, this work was undertaken primarily with exomoons in mind. It is our hope that making this code publicly available will reduce barriers for the community to assess the detectability of exomoons, conduct searches for exomoons, and attempt to validate existing exomoon candidates. We also anticipate that our code will be useful for studies of planet–planet transits in exoplanetary systems, transits of circumbinary planets, and eclipses in triple-star systems.

The “Drake Equation” of Exomoons—A Cascade of Formation, Stability and Detection

“…Inspired by the success of the Mandel-Agol model [86] used in the analysis of transit light curves of exoplanets, a number of models were proposed for the analysis of a planetmoon system, including an unnamed open access GUI visualization and simulator [17], the LUNA [79], planetplanet [87,88], gefera [89], Pandora [90], the Photodynamic Agent of the Transit and Light Curve Modeler (TLCM, [91,92]), or the analytical models of [89,93] and the folding framework by [94]. Generally, these are based on two fully opaque round bodies that occult a portion of the stellar disk (Figure 2) or each other.…”

Analytic auto-differentiable ΛCDM cosmography

“…In this demonstration, I use a very simplified model for supernova cosmology, presuming to have measured the redshifts ẑs and derived standardised distance moduli μs , of N type Ia supernovae. 26 For the latter I assume Gaussian likelihoods with equal and independent uncertainties: μs |µ s ∼ N (µ s , σ 2 µ ), where σ µ is the combined uncertainty due to residual scatter after standardisation and measurement noise. For the redshifts I adopt the toy model of [63], which also has a Gaussian likelihood albeit with a redshift-dependent variance: ẑs |z s ∼ N (z s , (1 + z s ) 2 σ 2 z ), imitating a photometric estimate, and uses as prior a physically motivated and simple to calculate gamma distribution with rate parameter β.…”

“…The deep-learning revolution brought about by automatic differentiation and generalpurpose parallel computing on graphics processing units (GPUs) has motivated the development of a number of new high-performance automatically differentiable simulators (and emulators) across cosmology: e.g. for large-scale structure [31,32,38,46,47,56,57], weak lensing [4], strong lensing [17,24,29,41], gravitational waves [18], and in related fields [1,26,30,36,43,48,51,61,73,76,80]. Having access to gradients through the simulator then enables general high-dimensional likelihood-based analyses with Hamiltonian Monte Carlo (HMC) [21,34] or variational inference (VI) [35,40,66] and can be used in the context of likelihood-free simulation-based inference to speed up the training of neural networks through an additional loss term [5,15,79].…”

“…Furthermore, even though calculating the likelihood is not required for SBI, its gradient (if available) can be used to form an addition to the loss function that speeds up training[5,79] or to derive optimal summary statistics[15] 25. For overviews of the plethora of methods falling under the umbrella of SBI, see[19,50] 26. I use a hat: •, to denote an observed, measured, or otherwise affected by observational uncertainty quantity, in contrast to the intrinsic (latent) values.…”