Modelling (not so) periodic signals

In this work, we studied the gravitational effects of three body dynamics.

In this case: three stars.

It turns out that most stars actually come in pairs, triples, or larger groups! Something like ~30% of stars like our Sun have a companion. But, we actually think that more massive stars always come in pairs or even in groups of several stars.

This is actually really important because pairs of stars, otherwise known as binary stars can do some really weird things, like cause each other to spin faster and even transfer mass to one another. Processes such as these are thought to be really important for producing exotic things such as black-hole binaries and their progenitors. I study these stars to understand how massive binaries can produce extreme systems such as black-hole binaries. In this published work, in particular, we studied the orbiting motion of a triple star system, which is predicted to eventually result in a black hole binary (and gravitational waves!).

The orbiting motion of multiple stars produces observable dynamic effects in the time-series brightness measurements of stars. I built a generalized probabilistic model that predicts the observable effects of this motion in multiple types of astronomical time-series data, using symbolic coding tools including pyMC. The code uses numeric sampling tools, such as Hamiltonian Monte Carlo, to derive parameter inferences of the physical stellar system by modeling the data.

Signal extraction

The open-source software that I developed, together with my collaborator Nora Eisner, is divided into two key components. The first component analyzes the stellar time-series data to accurately determine the timing of each stellar eclipse.

The process begins by creating a model of the primary eclipse (the deeper one), using a trapezoid or cosh function—both shapes that can be used to represent the shape of eclipses. This model is applied to the smoothed, phase-folded, and binned time-series data. For each primary eclipse, only two parameters are left adjustable: the timing of the eclipse and the slope of an underlying linear trend to account for any systematic variations. The same method is separately applied to the secondary eclipses. A Markov Chain Monte Carlo (MCMC) approach, implemented through the EXOPLANET software (Foreman-Mackey et al. 2021), optimizes the fits for both primary and secondary eclipses. Importantly, we keep track of the center of the fit to each eclipse, as the difference between observed and calculated eclipse times (O – C) helps reveal any dynamical effects due to a third body in the system.

Data modeling

There are two primary factors that cause deviations from perfectly periodic eclipse timings: the light travel time effect (LTTE) and the dynamical effect. LTTE arises from changes in the projected distance between the center of mass of the binary system and that of the entire triple system, while the dynamical effect is due to physical alterations in the binary's orbit caused by the gravitational pull of a third body (Borkovits et al., 2003). By modeling both the LTTE and dynamical variations in eclipse timing, we can infer properties such as the mass ratio of the tertiary (the outermost star) to the total system mass, the eccentricity of the tertiary's orbit, and the mutual inclination between the binary’s orbital plane and that of the tertiary.

The second part of the software focuses on jointly modeling eclipse timing variations and radial velocity data, using the methodology from Borkovits et al. (2016).

For model optimization, a No U-Turn Sampling (NUTS) approach is applied, which is a variant of Hamiltonian Monte Carlo (HMC) as outlined by Hoffman & Gelman (2011). HMC is a subset of MCMC methods used to numerically approximate posterior probability distributions. Unlike traditional MCMC, which uses a random walk to explore an n-dimensional parameter space, HMC leverages a Hamiltonian description of the probability distribution to more efficiently sample the posterior. While standard HMC requires manual tuning of the number of steps and step sizes—often making it inefficient—the NUTS algorithm automatically evolves the system until the proposed location begins to reverse course. It also incorporates the stochastic adopt/reject mechanism of standard MCMC, making NUTS more efficient for exploring high-dimensional parameter spaces.