Conical Coordinates for Elliptical Orbits

Abstract

Two-body orbital motion is analyzed within a three-dimensional conical reference frame in which the center of the gravitational field is within the cone volume and the apex angle is fixed at 90 degrees. In this frame the satellite position is characterized by slant height R along the cone surface in addition to the classical radial distance r. The constraint of the cone half-angle being 45 degrees gives a line along the angular momentum vector to the cone apex with the magnitude of √(ap), where a is the semi-major axis and p is the semi parameter. Three results are developed. First, the tilt angle Φₕ of the orbital plane relative to the cone axis satisfies sin(2Φₕ) = e, providing a direct geometric encoding of eccentricity that is intrinsic to the frame and requires no reference to the orbit. Second, applying Gauss's law for gravitation to the conical surface yields a logarithmic flux integral that is shown to be equivalent pointwise to Kepler's equation through the eccentric anomaly E, establishing a three-dimensional geometric interpretation of that classical result. Third, the logarithmic structure allows for a Newton-Raphson positioning method with dual convergence tolerances for both the Kepler residual and the position variable, offering geometric consistency and balanced numerical resolution throughout the orbit. This is especially useful for high-eccentricity cases.