Using Astrometry to Break Degeneracies in Stellar Surface Mapping

TL;DR

This work addresses starspot-induced astrometric jitter as a fundamental limit to detecting Earth-mass planets with sub-$\mu$as precision and introduces a linear forward model that maps a star's surface, expanded in spherical harmonics, to the astrometric photocenter. It proves that astrometry preferentially probes odd-degree harmonics (while photometry samples even-degree harmonics), and that inclinations mix harmonic content via Wigner-D rotations, enabling joint astrometry–photometry surface mapping to break degeneracies. A Bayesian inversion framework with a Gaussian-Markov random-field prior estimates the surface and stellar inclination from simulated data, with reconstructions showing improved spot localization and inclination constraints when combining data types. The methodology has direct relevance for forthcoming Gaia sub-$\mu$as astrometry and informs future direct-imaging strategies, offering a principled path to map stellar activity and mitigate jitter in exoplanet mass measurements via astrometry.

Abstract

Astrometric jitter noise arises when starspots on a rotating stellar surface move in and out of view, shifting the photocenter. This noise may limit our ability to detect and weigh small, sub-Neptune-sized planets around active stars. By deriving a linear forward model for the astrometric jitter signal of a rotating star in a spherical-harmonic coordinate system, we show that jitter noise can be used to reconstruct surface-brightness maps and, in principle, disentangle jitter from stellar reflex motion due to an orbiting planet. Furthermore, we show that astrometry and photometry probe complementary surface information: photometry measures even-degree spherical harmonic surfaces that are symmetric about the equator, while astrometry measures odd-degree modes. Their joint use, therefore, breaks degeneracies in surface mapping. Our model further quantifies the variation in the astrometric signal with inclination angle, which is foundational for studies of worst-case configurations of astrometric star-spot noise. For example, we show that pole-on stellar inclinations lead to poorly constrained inversions, as any stellar surface produces a purely circular astrometric jitter signal. We characterize the degeneracy in jointly identifying the stellar surface and stellar inclination, and develop a surface estimation approach. Using this approach, we present example simulations and reconstructions that demonstrate the use of astrometry data alongside light-curve data to improve stellar surface mapping and localize spot positions in latitude and longitude. With forthcoming high-precision Gaia astrometry, astrometric surface mapping provides a promising new approach to probe stellar activity.

Figures (9)

• Figure 1: Stellar rotation geometry (applied right to left). We parameterise the orientation with Euler angles $R=(\alpha,\beta,\gamma)$ in the $z-y-z$ convention, mapping the star-fixed frame to the sky via $R=R_z(\alpha)R_y(\beta)R_z(\gamma)$. Here $\beta$ is the inclination (tilt) of the spin axis, $\gamma=\omega t$ is the time-dependent rotation at angular rate $\omega$, and $\alpha$ sets the sky-plane tilt.
• Figure 2: An order $L=25$ expansion of an 8 % $R_\star$ starspot. The star surface is shown (left) viewed equatorially. The astrometric signal is shown for different inclinations ranging from $\beta = 0$ (equator on) to $\beta = \pi / 2$ (pole down). When the photo-center is at origin, the starspot is out of view. For a pole-down observer, any astrometric signal is circularized.
• Figure 3: An order $L=25$ expansion of a 10 % $R_\star$ starspot. The star surface is shown (left) at an inclination of $\beta = 30^\circ$, with an equatorial starspot. For this stellar inclination, the astrometric signal from starspots at different latitudes denoted by $\lambda$ is shown (right), ranging from $\lambda = 0^\circ$ (equatorial) to $\lambda = 60^\circ$. Grey arrows denote the egress and ingress of the starspot. As the starspot heads to egress, the photocenter moves toward the origin. While the starspot is out of view, the photocenter is fixed at zero. When the starspot moves into ingress, the photocenter begins to move again. These varying starspot latitudes $\lambda$ are shown on the stellar surface (right) as arrows pointing from the star's interior origin point. Furthermore, starspots at higher latitudes generally produce less photo-center variation.
• Figure 4: Visualizing the measurement kernels $\mathbf{k}$ for the first-order moments (astrometry) and zeroth-order moments (light curve photometry) over the spherical harmonic basis indexed by $(l,m)$. For a star viewed equatorially, where the kernel is 0 (white) for the ($l, m$) harmonic, this harmonic term produces no measurable signal. The magnitude of each kernel term is the contribution to the astrometric signal for that particular spherical harmonic, in other words how strongly that spherical harmonic component shifts the photocenter. The astrometry operator follows a pattern where even-valued $l>2$ cannot be measured, whereas the photometry operator follows the opposite pattern, odd-valued $l>2$ cannot be measured. Therefore, astrometry obtains complementary information to light curve data, and combined, can better constrain a stellar surface estimate.
• Figure 5: Visualizing the observable spherical harmonics and their weightings when the star is inclined by $\beta = 0.6$. The inclination dependent matrix in the forward model, $B_\beta$ comprises diagonal sub-matrices where the inclination rotation is applied to each of the kernel terms $\mathbf{k}_l^h$. Here we show these values over the spherical harmonic basis indexed by $(l,m)$. Where the kernel is 0 (white) for the ($l, m$) harmonic, this harmonic term does not contribute to the measured signal. For an inclined star, the kernel terms $\mathbf{k}_l^h$ have been 'mixed' over order $m$ per degree $l$, rendering unobservable terms observable and doubling the overall rank of the forward model. All odd degree $l$ harmonics now contribute to the observations in both astrometric directions, and all even degree $l$ harmonics contribute to the photometric observations. As shown, when a star's rotational axis is inclined, astrometry and photometry still generally obtain measurements from distinct collections of spherical harmonics.