Power Laws of Topography and Gravity Spectra of the Solar System Bodies

Abstract

When a spacecraft visits a new planetary body, it is useful to know the properties of its shape and gravity field. This knowledge helps predict the magnitude of the perturbations in the motion of the spacecraft due to nonsphericity of a body's gravity field as well as planning for an observational campaign. It has been known for the terrestrial planets that the power spectrum of the gravity field follows a power law, also known as the Kaula rule (Kaula, 1963, https://doi.org/10.1029Rapp, 1989 Rapp, , https://doi.org/10.1111 Rapp, /j.1365 Rapp, -246X.1989. A similar rule was derived for topography (Vening Meinesz, 1951). The goal of this study is to generalize the power law dependence of the gravity and topography spectra for solid surface solar system bodies across a wide range of body sizes. Traditionally, it is assumed that the gravity and topography power spectra of planets scale as g −2 , where g is the surface gravity. This gravity scaling also works for the minor bodies to first order. However, we find that a better fit can be achieved using a more general scaling based on the body's radius and mean density. We outline a procedure on how to use this general scaling for topography to provide an a priori estimate for the gravity power spectrum. We show that for irregularly shaped bodies the gravity power spectrum is no longer a power law even if their topography spectrum is a power law. Such a generalization would be useful for observation planning in the future space missions to the minor bodies for which little is known. Plain Language SummaryWe used the available models of shape and gravity field of the solar system bodies. We study how the amplitude of the mountains and valleys in gravity and topography depends on their size. Mountains on Mars are greater than on the Earth and yet larger on asteroid Vesta compared relative to the body's size. The same is true for gravity: Relative variations in gravity are larger on smaller bodies. We quantify this dependence and develop a method to predict the variations in gravity and topography depending on the body's size and density.

Titan's gravity field and interior structure after Cassini

“…For example, if we suppose that there is a peak deviatoric stress that scales as Δρ  g  R (where Δρ is a typical density anomaly, g is gravitational acceleration, and R is radius or some comparable large length scale) then we might expect ∝ (  ) −1 ∝ −2 since g scales as R. Note however that this scaling should not be used when going from rocky bodies to icy bodies since the expected peak stresses are lower in ice and the mean density is lower, so the partial success of this scaling for terrestrial bodies does not imply it should work for icy bodies. Recently, Ermakov, et al (2018) reviewed the Kaula rule of different terrestrial bodies in the literature to achieve a more general scaling rule among them. Their result for the gravity spectra is ∝ −1.72 .…”

The gravity field and interior structure of Dione

“…Thus, the a priori uncertainties on the normalized coefficients of degree l were set using the Kaula rule K/l 2 (Kaula, 1963). This empirical law can successfully describe the gravity power spectrum of the rocky planets, the Moon, and Vesta (Ermakov et al, 2018). Moreover, a good agreement was found for Titan, even if the gravity field is available only up to degree 5 .…”

Phoebe’s differentiated interior from refined shape analysis

“…We also computed the RMS spectra for the topography, as defined in Ermakov et al (2018). Figure 2 shows the variation of the RMS spectra in blue as well as the difference with the RMS spectra deduced by Ermakov et al (2018). The two estimations are very close to each other, and the difference is below 10 −4 .…”

“…Therefore, in the following, we focus mainly on A 2,0 and use an uncertainty of 0.55 km. We also computed the RMS spectra for the topography, as defined in Ermakov et al (2018). Figure 2 shows the variation of the RMS spectra in blue as well as the difference with the RMS spectra deduced by Ermakov et al (2018).…”