Optimal computation of brightness integrals parametrized on the unit sphere

TL;DR

This paper introduces a fast, accurate method based on Lebedev quadratures for computing brightness integrals of irregularly shaped bodies like asteroids, improving efficiency over traditional polyhedral methods.

Contribution

The paper presents a novel Lebedev quadrature-based approach for brightness integral computation that is faster and simpler than existing polyhedral methods, without surface tessellation.

Findings

Up to ten times faster at 0.01 mag accuracy compared to polyhedral sums.

Applicable to irregular shapes without surface tessellation.

Easily integrated into existing lightcurve inversion algorithms.

Abstract

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software.