Mapping stellar surfaces II: An interpretable Gaussian process model for light curves

Abstract

The use of Gaussian processes (GPs) as models for astronomical time series datasets has recently become almost ubiquitous, given their ease of use and flexibility. GPs excel in particular at marginalization over the stellar signal in cases where the variability due to starspots rotating in and out of view is treated as a nuisance, such as in exoplanet transit modeling. However, these effective models are less useful in cases where the starspot signal is of primary interest since it is not obvious how the parameters of the GP model are related to the physical properties of interest, such as the size, contrast, and latitudinal distribution of the spots. Instead, it is common practice to explicitly model the effect of individual starspots on the light curve and attempt to infer their properties via optimization or posterior inference. Unfortunately, this process is degenerate, ill-posed, and often computationally intractable when applied to stars with more than a few spots and/or to ensembles of many light curves. In this paper, we derive a closed-form expression for the mean and covariance of a Gaussian process model that describes the light curve of a rotating, evolving stellar surface conditioned on a given distribution of starspot sizes, contrasts, and latitudes. We demonstrate that this model is correctly calibrated, allowing one to robustly infer physical parameters of interest from one or more stellar light curves, including the typical radii and the mean and variance of the latitude distribution of starspots. Our GP has far-ranging implications for understanding the variability and magnetic activity of stars from both light curves and radial velocity (RV) measurements, as well as for robustly modeling correlated noise in both transiting and RV exoplanet searches. Our implementation is efficient, user-friendly, and open source, available as the Python package starry_process.

Mapping stellar surfaces I: Degeneracies in the rotational light curve problem

“…This is true even in the absence of measurement error: the mapping from a stellar surface to its rotational light curve is so degenerate that there exist an infinite number of solutions to the inverse problem. This fact has been pointed out recently in different contexts (e.g., Cowan et al 2013;Rauscher et al 2018;Sandford & Kipping 2019;Luger et al 2019;Basri & Shah 2020), but it dates back at least to Russell (1906), who demonstrated it by expanding the surface intensity of a celestial body in terms of spherical harmonics (see Figure 2). Russell (1906) showed that many of the modes comprising the intensity profile of a spherical object are in the null space, the set of surface features that have identically zero effect on the light curve.…”

“…Another important feature of S, which we hinted at above, is that it is exactly zero for all odd-degree modes above l = 1. This is a well-known fact: all odd spherical harmonics other than the dipole are in the null space regardless of inclination (e.g., Luger et al 2019). In other words, these spherical harmonics are perfectly antisymmetric in projection over the unit disk when viewed from any orientation.…”

“…To understand why, consider an expansion of the stellar surface intensity in the spherical harmonic basis out to arbitrary order 3 (see Figure 2). Assuming (for the moment) that the star rotates about an axis that points up along the page, the observed light curve may be expressed as a weighted sum of the disk-integrated intensity of each of the spherical harmonics as they rotate about that same axis (Luger et al 2019). However, not all spherical harmonics will contribute to the full light curve, as many (in fact, most) of the spherical harmonics are perfectly antisymmetric about the equator.…”

“…For rotational light curves, the vast majority of the surface modes lie in the null space. To show this, we will make use of the fact that we can express the vector of K observed fluxes f (i.e., the light curve) as a linear operation on the vector of N spherical harmonic coefficients y (Luger et al 2019):…”

Three K2 Campaigns Yield Rotation Periods for 1013 Stars in Praesepe

“…Disentangling which of these mechanisms is causing these changes in the light curves is difficult, especially if they are occurring on the same time scale (Basri & Shah 2020). Modeling these light curves with a Gaussian Process model (e.g., using starry; Luger et al 2021) may lend insight into the spot evolution that is occurring and provide a metric that can quantitatively measure such changes in the light curve (e.g., Gordon et al 2021). Investigating this is beyond the scope of this work, but we note this as a point of future exploration.…”