Gravity field of a tesseroid by variable-order Gauss–Legendre quadrature

Abstract

The Gauss–Legendre quadrature (GLQ) integration is a numerical method to calculate the gravitational field of a tesseroid and the approximation error of the GLQ integration increases as the tesseroid gets closer to the computation point. There are two ways to counterbalance this effect. One way is the subdivision of a tesseroid into smaller units and the other way is to increase the quadrature order. In this paper, we develop variable-order GLQ to model the gravitational field of a tesseroid based on the relation between the quadrature order and the node spacing. This algorithm uses a scalar, referred to distance-to-spacing ratio ($R_{\text{LS}}$), to control the order of the GLQ integration and the calculation accuracy without the subdivision strategy. We perform the approximation error analysis for the gravitational potential, attraction, and Marussi tensor components to examine the applicability of the variable-order GLQ. Numerical experiments show that the variable-order GLQ is suitable for the case that the horizontal dimensions of the tesseroids are larger than $0.5^\circ$ and the computation height is over 50 km. We obtain the optimal values for the ratio $R_{\text{LS}}$ based on the relation between the approximation errors and the ratio $R_{\text{LS}}$ for the tesseroid with the horizontal dimensions $0.5^\circ$ to $5^\circ$. Numerical results show that if the maximum tolerate approximation error is $10^{ - 3} \%$, the values of the ratio $R_{\text{LS}} = 2.5, 5.5$ and 10.5 are required for the gravitational potential $V$, the vertical gravitational attraction $V_{z}$, and the vertical gravitational gradient $M_{zz}$, respectively.

Appendix A: Analytical gravitational potential and its radial derivatives of a homogenous shell

The analytical gravitational potential $V^{\text{shell}}$, the radial component $V_{z}^{\text{shell}}$ of the gravitational attraction and the component $M_{zz}^{\text{shell}}$ of Marussi tensor of a homogenous spherical shell at an arbitrary point with a radial distance $r$ above the shell can be computed by (Heck and Seitz 2007; Grombein et al. 2013)

$$ V^{\text{shell}} \left( r \right) = \frac{4\pi G\rho }{3r}\left( {R_{2}^{3} - R_{1}^{3} } \right), r \ge R_{2} $$

(A1)

$$ V_{z}^{\text{shell}} \left( r \right) = \frac{{V^{\text{shell}} }}{r}, r \ge R_{2} $$

(A2)

and

$$ M_{zz}^{\text{shell}} \left( r \right) = \frac{{2V^{\text{shell}} }}{{r^{2} }}, r \ge R_{2} $$

(A3)

here $R_{1}$ and $R_{2}$ are inner and outer radii of the shell, respectively.