Shape determination

Orbital parameters are currently known for more than 500 000 asteroids. On the other hand, rotational states and shapes were determined only for a small fraction of them. In the Minor Planet Lightcurve Database, periods for ∼11 000 asteroids are stored. Realistic shapes and spin vectors have been determined only for ∼400 asteroids. While there are various methods how to derive shape models of asteroids (with a very different quality), the leading method is the lightcurve inversion.

Space probes

Most detailed shapes can be determined by the space probe flybys. So far (January 2013), 13 asteroids (including dwarf planets Ceres and Pluto) and 8 comets have been visited and photographed by several space probes, such as near-Earth asteroid (433) Eros by NEAR Shoemaker or main-belt asteroid (243) Ida with its moon Dactyl by Galileo (Figure below). The typical asteroid shape is non-spherical with non-convex features, the surface is covered by impact craters of diverse sizes (the largest recorded craters have diameters of a third of the whole asteroid sizes) and boulders, we can also see long and deep valleys and sharp edges. The largest asteroids are not far from a spherical or tri-axial ellipsoidal shape (some of them should be even differentiated), on the other hand, smaller asteroids are more irregular.

Figure. Asteroid (243) Ida with its moon Dactyl observed by Galileo, NASA.

Radar (Delay-Doppler Images)

The shape can be successfully derived also from radar observations. Shape models that describe global features and contain no information about low-scale structures such as craters are usually presented, see e.g., Ostro et al. (2000, 2002); Chapman (2002); Cheng (2002). However, radar observations of some NEAs taken during their close encounters (e.g., Toutatis in 2012) revealed structures reminding craters and boulders. The interpretation of such features is object of many current studies and theoretical models. In Figure below, we show radar observations of asteroid (216) Kleopatra and its shape model.

Figure. Radar observations of asteroid (216) Kleopatra and its reconstructed ”bone- like” shape model. Each quadrant shows Arecibo delay-Doppler images (top), correspond- ing images calculated from the shape model (middle), and corresponding plane-of-sky views of the model (bottom), Ostro et al. (2000).

Ellispoidal models

Lightcurve inversion

The brightness in the optical and the very near infrared part of the light spectra (i.e., reflected light) of an asteroid exhibits temporal variations for several reasons:

• The usually non-spherical asteroid is rotating around its rotational axis, and so the illuminated part of the body with respect to the observer is changing. For single objects observed near the opposition (geometry when the Sun, the Earth and the asteroid lie on a straight line or close to this configuration), we have a typical double sinusoidal pattern.
• The geometry of observation affects the brightness of the asteroid. The closer to the opposition the asteroid is, the brighter it gets, because the part of the surface both illuminated by the Sun and observable from the Earth has its maximum there (the geometry of observation can be described by the solar phase angle, which is the Sun-asteroid-Earth angle). The solar phase angle is close to zero near the opposition. Typical photometric observations of main-belt asteroids are taken with solar phase angles <30 degrees. Lightcurves observed at larger phase angles are more affected by the shape irregularities and thus are more far from the double sinusoidal pattern.
• Albedo influences the absolute brightness of the asteroids, but has the same effect as the dimension of the object: increasing albedo is equivalent to expanding the body and vice versa. This is the reason why it is impossible to derive the size only from the optical data itself.
• The scattering parameters describe how the surface reflects the incident light. Several scattering laws can be found in the literature (ref). The opposition effect is an example of a light scatter where the lack of shadowing and the coherent backscattering close to the opposition cause a significant growth (usually described by an exponential function) of the asteroid brightness.

Multi-data approaches

non-convex inversion: KOALA, genetic algorithms, ADAM.

References

-- JosefHanus - 27 Aug 2015