Modeling the Gravitational Field by Using CFD Techniques
The Laplace equation represents harmonic (i.e., both source-free and curl-free) fields. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge
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Abstract
The Laplace equation represents harmonic (i.e., both source-free and curl-free) fields. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, known as the Complex-boundary Value Problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a harmonic vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of comet 67P/Churyumov-Gerasimenko with a complex shape. The method is validated in a closedloop simulation by comparing the result with a direct integration of Newton’s formula. It shows a good consistency between them, with a relative magnitude discrepancy at percentage level and with a maximum directional difference of 5ı . Moreover, the numerical scheme adopted in our method is able to overcome the divergence problem and hence has a good potential for solving the CBVPs. Keywords
CFD techniques Comet 67P/Churyumov-Gerasimenko Finite Volume Method Gravitational field modeling Potential flow
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Introduction
A gravitational field is the influence that a mass body extends into the space around itself, producing a force on another mass body. The gravitational field outside the masses has two important properties: source free and curl free (i.e., Z. Yin () N. Sneeuw Institute of Geodesy, University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected]
harmonicity). The main problem of physical geodesy is to establish a gravitational model, on the Earth’s surface and in the outer space, to the extent made possible by existing data, mathematical analytical tools and numerical computational tools (Sansò et al. 2012). The analytical solution of the Laplace equation (e.g., spherical harmonics) plays an important role in building advanced geopotential models. The convergence of spherical harmonics is guaranteed outside a mass-enclosing reference sphere, also referred to as the Brillouin sphere (Hobson 2012). However, when it comes to the gravitational field modeling of complex shaped asteroids
International Association of Geodesy Symposia, https://doi.org/10.1007/1345_2019_72, © Springer Nature Switzerland AG 2019
Z. Yin and N. Sneeuw
or comets, the series at points
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