P41C-2084
Understanding Kaula’s Rule for Small Bodies

Thursday, 17 December 2015
Poster Hall (Moscone South)
Jay McMahon1, Daniel Jay Scheeres2, Davide Farnocchia3 and Steve Chesley3, (1)University of Colorado at Boulder, Aerospace Engineering Sciences, Boulder, CO, United States, (2)University of Colorado Boulder, Boulder, CO, United States, (3)NASA Jet Propulsion Laboratory, Pasadena, CA, United States
Abstract:
Kaula’s rule gives a bound on the RMS of the gravity coefficients of each order as a power law K/n^2, where n is the degree. Kaula derived the value of K for Earth as 10-5. This rule has been used as an a priori information bound on the gravity coefficients of other planetary bodies before their gravity fields are measured by spacecraft. To apply Kaula’s rule to other bodies, the simple scaling based on the relative gravity of each body is used - (gEarth/gPlanet)2. This scaling was successfully used even for Vesta, where K = 0.011. However, if Kaula’s rule is applied to very small bodies, such as the OSIRIS-REx target asteroid Bennu, the scaling results in un-useable bounds. In this case, K ~ 105. This fact has motivated further investigation into the derivation and application of a Kaula-like power rule to bound the gravity field of small bodies. 
Our initial investigation focuses on the specific application to Bennu. This study is enabled by the fact that a fairly accurate shape model of Bennu has been derived based on three Earth-based radar apparitions along with a constrained bulk density based on astrometry and thermal measurements. Thus we investigated varying the Bennu topography within the expected accuracy of the shape model as well as the density distribution. Several interesting facts were discovered through this analysis. First, the top shape of Bennu, common to a number of near-Earth asteroids, results in the even zonal coefficients being larger than the odd zonal of one lower degree. Second, the zonals in general are significantly larger than the coefficients with order > 1, so that the zonals will dominate any fitting of K to a power law. This encourages us to have one K for the absolute value of the zonals (K=0.087), and a separate value for the RMS of the other coefficients (K=0.026). Third, variation in the topography within this uncertainty dominates the variation in the gravity field coefficients over basic inhomogenous density distribution effects. Finally, with significantly non-spherical shapes, it is not clear what value for the reference radius should be used when deriving the gravity field coefficients, and this can greatly change the exponent of the power law.
These results are investigated and compared for several other small bodies (e.g. Itokawa, 1999 FG3) in this work.
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