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Identifiability of Rotating Stellar Surfaces from Astrometric Jitter

Jamila Taaki, Lia Corrales, Alfred Hero

Abstract

Astrometry measures shifts in a star's photocentre and can be used to detect reflex motion due to orbiting exoplanets. Brightness asymmetries (e.g. starspots) rotating in and out of view can also cause apparent motion of the photocenter, termed astrometric jitter, that has previously been considered a source of noise. Here, we explore whether it can be used to map stellar surfaces. We derive a Cramer-Rao bound on the minimum variance for which a stellar surface can theoretically be estimated, quantifying the information content in rotational astrometric jitter. To regularize and break singularities in the Fisher information, we impose a spatial-smoothness Gaussian-Markov random field prior. A key challenge in mapping surfaces arises for stars with unknown rotational axis inclinations, requiring joint estimation of the inclination and the stellar surface. We characterize the coupling between them and quantify the precision gain when inclination is known versus unknown.

Identifiability of Rotating Stellar Surfaces from Astrometric Jitter

Abstract

Astrometry measures shifts in a star's photocentre and can be used to detect reflex motion due to orbiting exoplanets. Brightness asymmetries (e.g. starspots) rotating in and out of view can also cause apparent motion of the photocenter, termed astrometric jitter, that has previously been considered a source of noise. Here, we explore whether it can be used to map stellar surfaces. We derive a Cramer-Rao bound on the minimum variance for which a stellar surface can theoretically be estimated, quantifying the information content in rotational astrometric jitter. To regularize and break singularities in the Fisher information, we impose a spatial-smoothness Gaussian-Markov random field prior. A key challenge in mapping surfaces arises for stars with unknown rotational axis inclinations, requiring joint estimation of the inclination and the stellar surface. We characterize the coupling between them and quantify the precision gain when inclination is known versus unknown.

Paper Structure

This paper contains 12 sections, 17 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. FORWARD MODEL
  3. Measurement model
  4. Signal Model
  5. Cramér Rao Bound
  6. Bayesian Likelihood
  7. Hybrid Fisher Information
  8. Stellar Surface CRB
  9. Inclination Angle CRB
  10. GMRF prior
  11. Experiment
  12. Conclusion

Figures (3)

  • Figure 1: Simulated star (left), recovered inclination and stellar surface (right) as the MAP estimate for our measurement model. $L = 9$ for both. The simulated noise level $\sigma^2$ is at $5 \%$ of the signal standard deviation.
  • Figure 2: Simulated star and astrometric moments at two inclinations (face-on $\beta = 0$ and polar $\beta = \pi/2$). The first row shows a simulated stellar surface as a GMRF, with a starspot from the observers perspective. The surface is represented by $L=20$ spherical harmonics. The second row shows the noiseless astrometric signal observed.
  • Figure 3: The gain in optimal precision in estimating $\mathbf{s}$ from prior knowledge of inclination $\beta$. (a) Gain vs. $\beta$ for $\sigma^2/N\in\{0.1,1,10\}$ at $L=5$; (b) $\operatorname{tr}(\mathrm{CRB})$ vs. $L$ at $\beta=\pi/4$ for the prior exponent factor $\alpha\in\{0.5,1,2,3\}$, for known and unknown $\beta$. Overall, the results underscore that stellar surface mapping from first moments is strongly resolution-limited.