Shape and spin distributions of asteroid populations from brightness variation estimates and large databases

Abstract

Context. Many databases on asteroid brightnesses (e.g. ALCDEF, WISE) are potential sources for extensive asteroid shape and spin modelling. Individual lightcurve inversion models require several apparitions and hundreds of data points per target. However, we can analyse the coarse shape and spin distributions over populations of at least thousands of targets even if there are only a few points and one apparition per asteroid. This is done by examining the distribution of the brightness variations observed within the chosen population. Aims. Brightness variation has been proposed as a population-scale rather than individual-target observable in two studies so far. We aim to examine this approach rigorously to establish its theoretical validity, degree of ill-posedness, and practical applicability. Methods. We model the observed brightness variation of a target population by considering its cumulative distribution function (CDF) caused by the joint distribution function of two fundamental shape and spin indicators. These are the shape elongation and the spin latitude of a simple ellipsoidal model. The main advantage of the model is that we can derive analytical basis functions that yield the observed CDF as a function of the shape and spin distribution. The inverse problem can be treated linearly. Even though the inaccuracy of the model is considerable, databases of thousands of targets should yield some information on the distribution. We employ numerical simulations to establish this and analyse photometric databases that provide sufficiently large numbers of data points for reliable brightness variation estimates. Results. We establish the theoretical soundness and the typical accuracy limits of the approach both analytically and numerically. We propose a robust brightness variation observable η based on at least five brightness points per target. We also discuss the weaker reliability and information content of the case of only two points per object. Using simulations, we derive a practical estimate of the model distribution in the (shape, spin)-plane. We show that databases such as Wide-field Infrared Survey Explorer (WISE) yield coarse but robust estimates of this distribution, and as an example compare various asteroid families with each other.

LEADER: fast estimates of asteroid shape elongation and spin latitude distributions from scarce photometry

“…We used the squared brightness L 2 for convenience; a more detailed explanation is given in Nortunen et al (2017). We were then able to compute the amplitude A from each η, and from all the amplitude values of the population, we constructed the cumulative distribution function (CDF) C(A).…”

“…In Nortunen et al (2017), we used a limit of α tol = 30 • for the WISE database. For the Pan-STARRS1 database, we used α tol = 20 • .…”

“…Even larger population-level attributes can be obtained by using all available photometric measurements from the rich survey databases also when the data are not sufficient for individual models, as shown in Nortunen et al (2017) and Cibulková et al (2017). We refer to such data as scarce photometry to distinguish it from sparse photometry.…”

“…The possibility of obtaining population-level attributes is based on the principle of using even a few data points per target to construct a set of brightness variation estimates for a given population (defined by the user). As shown in Nortunen et al (2017), such estimates are sufficient for robustly reconstructing the shape elongation and (ecliptically symmetric) spin latitude distributions of the population when there are thousands of samples available. The determination of the spin distribution requires the population to be dominated by orbits concentrated near the ecliptic plane.…”

“…The mathematical principles are presented in Nortunen et al (2017), but the computational aspects require further exposition given here. The main issues are the choice of the grid discretization level in solving the inverse problem, experimenting with various setup choices to establish the stability of the result, the "deconvolution" of the result distributions in the shape-spin plot plane, and the mandatory use of simulations (from synthetic data for the same observing geometries as in the input database) to check the reliability of the inversion.…”