STARRY: Analytic Light Curves & Surface Mapping

• STARRY is a set of analytic frameworks for modeling astrophysical light curves and mapping spherical surfaces using high-dimensional, efficiently-differentiable methods.
• It employs spherical harmonic expansions and analytic derivatives to compute transit and occultation light curves with machine-precision speed and accuracy.
• Recent extensions integrate STARRY into robotic spatial-temporal prediction frameworks, achieving high success rates through Transformer-based diffusion models.

STARRY refers to a set of distinct yet influential technical frameworks with roots in analytic modeling and high-speed inference, centered on astrophysical light curve analysis, robotic spatial-temporal prediction, and hierarchical mapping of surface features. The common unifying theme is structured, high-dimensional, efficiently-differentiable forward modeling—most widely recognized in planetary and stellar occultation analysis, but more recently extended to domains such as robotic manipulation and advanced spot-mapping in stellar photometry. Below, each major strand of the STARRY concept is detailed within its technical, mathematical, and scientific context.

1. Analytic Occultation Light Curves and the STARRY Software Framework

"STARRY" as originally defined by Luger et al. is an analytic, closed-form solution for the total flux observed from a spherical body during occultations, eclipses, and phase curves, given a surface brightness parameterized in spherical harmonics. The formalism enables rapid computation of transit and occultation light curves, bypassing the prohibitive cost and numerical artifacts of direct integration (Luger et al., 2018).

For a body with surface map $$I(\theta, \phi) = \sum_{l=0}^L \sum_{m=-l}^{+l} y_{l,m} Y_{l,m}(\theta, \phi)$$, the total flux during an occultation is computed analytically by leveraging Green’s theorem and recursion relations for integrals along the intersection of disk and occultor boundaries. This approach reduces forward-model cost to as low as $O(L^2)$ per light curve point, with machine-precision accuracy. Automatic differentiation is natively supported for all parameters, making the framework suitable for Hamiltonian Monte Carlo and other gradient-based inference routines.

The STARRY software, available as a C++/Python open-source package, yields up to six orders of magnitude speedup and far lower bias relative to direct quadrature. Applications include exoplanetary atmospheric mapping, starspot recovery, phase curve analysis, and hierarchical multi-component inference for coupled light and radial velocity datasets (Luger et al., 2018, Bartolić et al., 2021, Sagynbayeva et al., 30 Apr 2025, Murray et al., 4 Nov 2025).

2. Spherical Harmonic Modeling of Surfaces

Central to STARRY is the surface expansion in real spherical harmonics, $Y_{\ell m}$, with corresponding coefficients $y_{\ell m}$. This basis is not only mathematically complete for arbitrary smooth maps on a sphere, but also enables linear algebraic manipulations such as analytic marginalization, principal component analysis, and covariance modeling of spot distributions. The design matrix encoding projected fluxes (including through transit and occultation geometries) is separable from the map parameters, paving the way for efficient Bayesian inference (Bartolić et al., 2021, Sagynbayeva et al., 30 Apr 2025).

Surface heterogeneities—such as starspots or volcanic hotspots—are parameterized through alterations to the $y_{\ell m}$ vector, whether via direct manipulation or the imposition of Gaussian-process or hierarchical priors. For spot models, STARRY often utilizes (Gaussian-)smoothed top-hat modifications with free latitude, longitude, angular radius, contrast, and smoothing parameters, concisely encoding astrophysical priors about spatial structure (Murray et al., 4 Nov 2025).

Analytic derivatives with respect to all map and geometric parameters are available, supporting high-dimensional MCMC and HMC workflows (Luger et al., 2018).

3. Hierarchical Bayesian Surface Mapping with STARRY

Recent methodological advances, exemplified by "StarryStarryProcess" (Sagynbayeva et al., 30 Apr 2025), embed STARRY’s analytic light curve engine into a hierarchical Bayesian framework for stellar surface mapping. Here, spot distributions are modeled as draws from hyperparameterized distributions on contrast, size, and latitude; the associated $y_{\ell m}$ vector then inherits a Gaussian-process prior whose covariance $\boldsymbol\Lambda$ is derived from the spot hyperparameters.

The hierarchical likelihood, marginalized over $y$, takes the form $\mathcal N(\mathbf{M} \mathbf{y}, \mathbf{C})$, further marginalized over the prior on $\mathbf{y}$. This approach enables simultaneous inference of spot properties, star–planet geometry, and allows for time-variable maps through linear interpolation in harmonic space across multiple epochs.

Applied to high-precision TESS and JWST light curves, this framework quantitatively resolves degeneracies between spot latitude, contrast, and radius that are fundamentally unconstrained in rotational light curves alone, thus leveraging planetary transits as deterministic surface probes (Sagynbayeva et al., 30 Apr 2025, Murray et al., 4 Nov 2025).

4. Application to Starspot-Crossing Events and Occultation Case Studies

STARRY’s analytic light curve engine has been systematically validated in the context of spot-crossing events (SCEs). Studies such as Murray & Berta-Thompson (Murray et al., 4 Nov 2025) performed extensive injection–recovery campaigns, quantifying biases and uncertainties in spot longitude, latitude, radius, contrast, and resulting planet-to-star radius ratio under the effect of the Transit Light Source Effect.

For large, high-contrast, or centrally-crossed spots, longitude errors can be $O(L^2)$0, radius $O(L^2)$1, and fractional errors on derived depths $O(L^2)$2 at moderate SNR. The analytic relations (e.g., $O(L^2)$3) allow practitioners to relate observed bump properties directly to underlying surface statistics. Strategies for breaking classic $O(L^2)$4 spot degeneracies are developed via a grid of light curve observables and used to define informative priors for MCMC, dramatically increasing efficiency and reliability of retrieval (Murray et al., 4 Nov 2025).

Occultation mapping of bodies such as Io leverages the same analytic machinery for hot spot recovery from sparse ingress–egress time series, demonstrating broad applicability to mapping thermal and albedo features across planetary, lunar, and stellar contexts (Bartolić et al., 2021).

5. Extensions to Robotic Manipulation: Spatial-Temporal Prediction with STARRY

The acronym STARRY has also been used for "Spatial-Temporal Action-Centric World Modeling," a distinct recent framework for action generation in robotic manipulation (Tian et al., 29 Apr 2026). While conceptually separate from occultation analysis, this STARRY introduces a Transformer-based policy that explicitly models the future spatial-temporal interaction structure between observations and actions. Its architecture comprises:

• Joint denoising of spatial-temporal latents and action trajectories via diffusion models.
• The Geometry-Aware Selective Attention Modulation (GASAM) mechanism, which integrates predicted depth maps and end-effector positions to modulate cross-attention with explicit 3D spatial bias.
• Direct supervision and loss terms on geometry predictions, attention weights, and diffusion denoising.

STARRY achieves up to 93.8% success rate on RoboTwin 2.0 simulations and demonstrates significant gains (up to +30 points) over state-of-the-art baselines for spatially-demanding manipulation tasks. Ablation studies highlight the necessity of structured future prediction and explicit attention modulation for high success in both simulation and real-robot (bimanual) settings, providing strong empirical evidence for the action-centric spatial-temporal modeling paradigm (Tian et al., 29 Apr 2026).

6. Code Ecosystem, Performance, and Practical Considerations

The STARRY code ecosystem encompasses a rigorously tested Python/C++ interface with integrations for automatic differentiation (Autograd, JAX, and Eigen in C++), enabling seamless embedding in modern probabilistic inference engines such as PyMC3 and NumPyro. The core Map class exposes full control over spherical harmonic expansion, occultation geometry, and synthetic/inferred surface features.

Performance benchmarks indicate throughput exceeding $O(L^2)$5 flux evaluations per second for $O(L^2)$6, and full 40k-point transit curves computed within milliseconds. Analytic gradients enable large-scale HMC and NUTS without the numerical fragility or slop of finite-difference derivatives. The code and notebooks are available as reproducible research artifacts via GitHub repositories linked in the respective publications (Luger et al., 2018, Bartolić et al., 2021, Sagynbayeva et al., 30 Apr 2025).

Ongoing applications include exoplanet mapping, stellar obliquity retrieval, volcanism inference, spot-mapping under time evolution, and robot action generation—each relying on the blend of analytic evaluation, symbolic differentiation, and hierarchical modeling that underpins the evolving STARRY paradigm.