An Analytical Method for Calculating the Satellite Bow Shock/Magnetopause Interception Positions and Times

TL;DR

This paper presents an analytical method to calculate the intersection points and times of satellite trajectories with Earth's magnetosphere features like the bow shock and magnetopause, using Kepler's approximation and second-order surfaces.

Contribution

It introduces a novel analytical approach transforming the intersection problem into a solvable fourth-order equation within Kepler's orbit framework.

Findings

Provides explicit formulas for intersection points and times.

Reduces the problem to a fourth-order algebraic equation.

Enables precise prediction of satellite-magnetosphere interactions.

Abstract

This paper contains a presentation of analytical solution of the problem of calculating the places and moments of intersection of satellite trajectories with elements of the Earth's magnetosphere (bow shock and magnetopause). The satellite motion is presented in a Kepler's approximation. Magnetopause and bow shock are described by second-order surfaces- elliptic paraboloides. These surfaces are employed as situational conditions for determining the points of intersection they have (if any) with the satellite trajectory. The situational condition is herein transformed into the plane of Kepler's orbit, thereafter it is reduced to a second-order plane curve- quadric (ellipse or parabola). The solution of this system, containing the equation of this curve and Kepler's ellipse equation, allows determining the places where orbits intersect with the magnetopause or the bow shock. The solution of this system is suggested to be given by reducing the system to a fourth-order equation.