The Gravitational Field of a Homogeneous Polygon

Abstract We derive closed-form expressions for the Newtonian potential, gravitational acceleration, and gravitational gradient tensor generated by a homogeneous, planar, zero-thickness polygon embedded in three-dimensional space. The analysis is formulated for a general simple polygon with constant areal mass density $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> and is implemented by decomposing the polygon into oriented triangular elements. For each triangle, the formulation expresses the potential and acceleration using edge log-factors and the signed solid angle, and provides an explicit dyadic representation for the tensor in terms of gradients of these quantities. For the important special case of a rectangle, compact corner-sum formulas provide efficient expressions for the potential, acceleration, and tensor. These formulas are numerically stable except at analytically singular boundary configurations, such as edges and corners of the ideal zero-thickness surface-mass model. Four independent benchmarks validate the theory: (i) a disk benchmark that demonstrates rapid convergence of the triangulated solution to the analytical axis field; (ii) the thin-cuboid limit, confirming that the rectangle model is the correct zero-thickness limit of a finite cuboid (or plate) with matching masses; (iii) reconstruction of the cuboid potential by stacking rectangles, confirming correct volume integration behavior; and (iv) agreement of rectangle and two-triangle decompositions at both acceleration and tensor levels. These results establish the correctness and consistency of the proposed polygonal gravitational primitives for geodetic and geophysical forward modeling.