FFT based algorithm for efficient gravity field calculation: comparison with exact results for polyhedral shape models

In the frame of the H2020 NEO-MAPP project, we propose a Fast Fourier Transform (FFT) based algorithm for efficient computation of the gravity field created by a body with any arbitrary mass distribution defined by a mass density discretized with suitable resolution. Our algorithm starts by considering a primary three-dimensional cartesian grid that contains the considered mass distribution and has uniform point spacings \(\Delta x\), \(\Delta y\) and \(\Delta z\) in the directions parallel to the \(x\), \(y\) and \(z\) axes. The density \(\rho(r)\) is then discretized at each primary grid cell, thus characterizing the mass distribution of the body, which creates the gravity field. Next, our algorithm considers a secondary three-dimensional cartesian grid that applies an arbitrary translation to the primary one and represents the space region where the gravity field will be computed. Once the primary and secondary grids have been defined, our algorithm computes efficiently the components of the gravity vector \(\vec{g}(r)\) created by the (discretized) body within the primary grid at all secondary grid points. From the computed gravity values at such points, a suitable interpolation allows extending the calculation to any point inside the region within the secondary grid.

Our algorithm has been applied to bodies having regular and irregular shape as well as uniform and non-uniform density. Using 16 GB of RAM memory in different PCs, we set up primary and secondary grids of up to 241 points at each axis and computation times that do not exceed one minute for each secondary grid. The numerical results provided by our algorithm have been tested to be in very good agreement with exact analytical gravity fields created by homogeneous and non-homogenous mass distributions. In addition, we have applied our algorithm to compute the gravity field created by uniform polyhedral shape models like those of Didymos (65803) and compare the numerical results with the exact analytical ones for homogeneous polyhedrons. From such comparison it has been found that spatial discretization introduces a systematic error in the mass and volume that has a different impact on the gravity field in the regions inside and outside the body. Specifically, it is shown that, by adjusting the bulk density, it is possible to obtain the gravity field very accurately in the region outside the body, with a relative error whose maximum value does not exceed 1.3 % and whose average value is below 0.04 %.