StarryStarryProcess Bayesian Surface Mapping

• StarryStarryProcess is a hierarchical Bayesian framework that models spatial surfaces using spherical harmonics and interpretable Gaussian process priors.
• It effectively combines transit mapping and rotational modulation to break degeneracies in stellar spot properties and geometric parameters.
• The method enables analytic marginalization and robust uncertainty quantification, making it adaptable for dynamic multi-system surface analyses.

StarryStarryProcess refers to a hierarchical Bayesian framework for inferring, modeling, and mapping spatial surfaces using indirect observations, with a particular emphasis on variable features such as starspots, planetary surfaces, or mass maps on spherical domains. The framework combines spherical harmonic representations, interpretable Gaussian process (GP) priors, and direct encoding of geometric and physical parameter dependencies. Applications span exoplanet transit mapping, stellar activity characterization, and weak-lensing mass-mapping on the celestial sphere.

1. Foundational Principles

StarryStarryProcess frameworks are built on analytic representations of a surface (or set of surfaces) using spherical harmonics. These coefficients encode surface brightness or relevant physical quantities. The process integrates physically interpretable GPs whose hyperparameters correspond to surface feature statistics—such as spot contrast, size, latitude distribution—or directly to underlying spatial processes. By projecting the spherical harmonic surface map into the observables (photometric, occultation, or rotational signals), the forward model remains linear in the coefficients, allowing tractable closed-form Bayesian updates.

Core to the framework is the encoding of prior information in kernels tailored to the physics: for instance, the prior covariance of starspot distributions might depend on stellar inclination, rotational period, limb-darkening, radius, and latitude statistics. The GP kernel is thus designed to be interpretable and parameter-rich, distinguishing the framework from generic black-box GP approaches (Luger et al., 2021, Luger et al., 2021, Sagynbayeva et al., 30 Apr 2025).

2. Spherical Harmonic and Transit-Based Surface Mapping

Surface maps are encoded as vectors of spherical harmonic coefficients, $y$. The expected disk-integrated flux at any point in time (or during an occultation) is then

$f_{\text{obs}} = M(\Theta) y$

where $M(\Theta)$ is a design matrix that incorporates system geometry—viewing angle, rotation, limb darkening, and, in transit applications, the instantaneous planetary transit chord.

In exoplanet transit mapping, the framework leverages spot-crossing anomalies—distinctive photometric bumps during a planetary transit—as direct constraints on the spot properties (contrast, size, latitude). When combined with rotational modulation outside transits, joint modeling dramatically reduces degeneracies inherent to light-curve-only analysis, extracting three-dimensional information on stellar orientation and spot distributions (Sagynbayeva et al., 30 Apr 2025, Sagynbayeva et al., 8 Oct 2025). The likelihood, after integrating over the GP prior on $y$, takes the form

$$p(f_{\text{obs}}\mid \Theta) \propto |B|^{-1/2} \exp\left( -\frac{1}{2} r^T B^{-1} r \right)$$

where $r=f_{\text{obs}}-\mu(\Theta)$, and $B$ encapsulates the noise and prior covariances.

The spot latitude distribution is modeled with a Beta distribution in $\cos\phi$ parameterized by physical means and variances, permitting inferences about active latitude belts or high-obliquity regions.

3. Interpretable Gaussian Process Kernels

The GP kernel is nonstationary and directly parameterized by physical hyperparameters $\theta_\bullet = (n, c, \mu_\phi, \sigma_\phi, r)$, where $n$ is the (scaling) number of spots, $c$ the fractional contrast, $r$ the characteristic spot radius, and $\mu_\phi, \sigma_\phi$ describing latitude distribution. Expectations and covariances in the spherical harmonic basis are computed via closed-form nested integrals over the assumed distributions of spot parameters, yielding analytic expressions for mean and covariance of the observable flux time series (Luger et al., 2021).

For ensemble analyses, sharing $\theta_\bullet$ across multiple stars yields robust inferences on mean spot properties and magnetic morphologies. Temporal extensions multiply spatial covariances by a time-evolution kernel (e.g., Matérn-$3/2$), supporting models of spot evolution over rotation cycles or epochs.

Key formulae include:

$$\mu = 1 + M \cdot \mathbb{E}[y \mid \theta_\bullet], \quad \Sigma = M \cdot \operatorname{Cov}[y \mid \theta_\bullet] \cdot M^T$$

$$\ln{\mathcal{L}} = -\frac{1}{2} r^T (\Sigma + C)^{-1} r - \frac{1}{2} \ln|\Sigma + C| - \frac{K}{2}\ln(2\pi)$$

where $C$ is the observational noise covariance.

4. Extensions: Evolving Surfaces and Dynamic Mapping

StarryStarryProcess supports dynamic models where the surface may evolve over time. The time-dependent geography is vectorized, and the full map is endowed with a spatio-temporal GP prior, typically constructed as a Kronecker product of spatial and temporal kernels:

$K = \alpha\, K_S \otimes K_T$

For large datasets, analytic marginalization over the high-dimensional map parameters and use of matrix identities (e.g., Woodbury) enable computational efficiency even with $>10^6$ unknowns. This facilitates inference on evolving geography, retrieval of axial tilt, rotation period, or seasonal atmospheric cycles for exo-Earths or variable stars (Kawahara et al., 2020).

Key innovations include efficient sampling strategies for the marginalized likelihood:

$$p(d \mid \vartheta) = \mathcal{N}(d \mid 0, \Sigma_d + W K_S W^T)$$

where $\vartheta$ includes nonlinear parameters (spin geometry, GP hyperparameters).

5. Joint Modeling, Degeneracy Breaking, and Stellar Geometry Recovery

The combined analysis of transit light curves and rotational modulation permits disambiguation of surface features and stellar orientation parameters. For instance, spot features in the transit chord—otherwise invisible in unresolved photometry—directly fix latitude distributions and break degeneracies in inclination and obliquity. The framework recovers characteristic spot radii ($\sim$10° angular scale for Kepler-63 and Kepler-17), measures bimodal latitude belts, and delivers obliquity estimates consistent with previous independent measurements, as in

• Kepler-63: high-obliquity, bimodal latitude distribution at $\pm30^\circ$
• Kepler-17: low-obliquity, equatorially concentrated spots at $\pm15^\circ$ (Sagynbayeva et al., 8 Oct 2025)

Observable spot-induced flux decrements are encoded as:

$$\mathcal{d} = n c r^2 / R_\star^2,\quad \mathrm{RMS_{spot}} = \mathcal{d} / \sqrt{n}$$

which link inferred spot properties with surface feature statistics.

6. Comparison to Traditional Methods and Broader Implications

Unlike parametric spot models or generic GPs, StarryStarryProcess provides a full Bayesian quantification of uncertainties, interpretable physical parameter mapping, and analytic marginalization over nuisance variables such as inclination. This approach avoids strong prior assumptions (e.g., fixed number of spots), reduces bias in spot property recovery, and handles the null space problem in rotational light-curve mapping.

Applications extend to exoplanet characterization (radius and orbital parameter estimation robust to stellar activity), inference of magnetic morphologies, and multi-system comparative analysis. The method’s generality permits adaptations to planetary mapping, solar system bodies (e.g., Io), and functional data clustering when combined with mixture or spline regression models (Bartolić et al., 2021, Chamroukhi, 2015).

7. Future Directions and Research Extensions

The framework’s modularity allows extensions to higher spherical harmonic resolution (small-scale spot mapping), time-dependent GP priors for long-term evolution, ensemble approaches for statistical population studies, and multi-wavelength inference for disentangling chromatic activity signatures. Releases emphasize reproducibility, with all code and data publicly available for the community.

A plausible implication is that as large time-domain surveys continue, this approach may reveal the diversity of stellar magnetic activity, clarify the impact of spot distributions on exoplanet inferences, and inform dynamo theory via statistically robust comparative spot mapping.

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StarryStarryProcess thus serves as a comprehensive, hierarchical Bayesian surface-mapping methodology, integrating interpretable GPs, spherical harmonics, and transit/event geometry to extract detailed physical information on stellar, planetary, and functional surfaces from indirect measurements (Sagynbayeva et al., 30 Apr 2025, Sagynbayeva et al., 8 Oct 2025, Luger et al., 2021).