Asteroid phase curve modeling with empirical correction for shape and viewing geometry

TL;DR

This paper develops an empirical pipeline to correct asteroid phase curves for rotational and viewing geometry by normalizing sparse photometry to pole-on geometry using precomputed spin- and shape-models, enabling consistent phase-curve fitting across apparitions. It implements two phase-function families, $H,G_1,G_2$ and $H,G_{12}$, and derives physically motivated bounds on their parameters by enforcing monotonicity and slope limits on the phase curve; derivative-free optimization is recommended for the $H,G_{12}$ family to avoid convergence artifacts near $G_{12}=0.2$. The method is demonstrated on ATLAS photometry for thousands of asteroids, producing phase-curve parameters for tens of thousands of objects and generally improving fit RMS residuals. Model comparison via Bayesian Information Criterion identifies the preferred spin–shape solution per object, enabling large-scale, cross-apparition phase-curve characterizations without full spin–shape–phase inversions.

Abstract

We present a novel empirical method for correcting asteroid phase curves for rotational and geometrical effects using precomputed spin-and-shape models. Our approach normalizes sparse photometric data to a pole-on geometry, enabling consistent phase-curve fitting across apparitions. We fit both the H,G1,G2 and H,G12 phase functions to the normalized data. We also numerically derive new constraints on parameter ranges that ensure physically meaningful solutions. These constraints are based on the requirement that the reduced magnitude must monotonically decrease with phase angle and remain within plausible slope bounds. Compared to earlier bounds, our new constraints are more permissive. We also compare derivative-based and derivative-free optimization methods, highlighting convergence issues with the HG12 function and offering mitigation strategies. We applied our method to over 25,000 asteroids observed by the ATLAS survey, demonstrating its usability. The new method enables the selection of the preferred spin-and-shape solution based on either statistical phase-curve model selection criteria and/or physically motivated constraints on the phase-curve shape.

Figures (11)

• Figure 1: Data normalization for asteroid (1182) Ilona. Raw and geometry-corrected photometric data are denoted in colors. The left panels correspond to the ATLAS c filter and the right panels to the o filter. Rows show results for the following shape models: Cellinoid (top), DAMIT model 1 [DAMIT No. 2259] (middle), and DAMIT model 2 [DAMIT No. 2260] (bottom).
• Figure 2: Data normalization for asteroid (1998) Titius. Black points represent the raw photometric data, while red points show the model- and geometry-corrected photometry reduced to pole-on geometry. The left panels correspond to the ATLAS c filter and the right panels to the o filter. Rows show results for the following shape models: Cellinoid (top), DAMIT model 1 [DAMIT No. 4777] (middle), and DAMIT model 2 [DAMIT No. 4778] (bottom).
• Figure 3: Histogram of fitted $G_{12}$ values using the Nelder–Mead (NM) and least-squares (LS) methods. The dashed vertical line marks $G_{12} = 0.2$, a common convergence point in least-squares optimizations. Comparison of fitted $G_{12}$ values using the Nelder–Mead (NM) and least-squares (LS) methods. (a) Histograms of the two methods for a sample of 12$\,$114 asteroids; the dashed vertical line marks $G_{12} = 0.2$, a common convergence point in LS optimizations. (b) Difference between the LS and NM distributions (LS $-$ NM), highlighting the small residuals between the methods and confirming that the LS convergence artifact near $G_{12} = 0.2$ is significantly mitigated by the NM method.
• Figure 4: Uncertainty envelopes for the $H,\!G_{1},\!G_{2}$ and $H,\!G_{12}$ phase curves of asteroid (5647) Sarojinianidu.
• Figure 5: Left: Phase curves for physical $(G_1, G_2)$ combinations, but violating the penttila2016h conditions. Right: Derived constraints visualized in the $(G_1, G_2)$ space.