A Guided Tour of Planetary Interiors

TL;DR

This paper investigates the gravitational trajectories within planetary interiors, comparing straight and brachistochrone paths across various celestial bodies to understand their internal gravitational dynamics.

Contribution

It introduces a simplified power-law model to analyze and compare the internal gravitational fields of different planets and stars.

Findings

Larger bodies behave like point-masses with gravity decreasing with distance.

Smaller bodies exhibit uniform sphere gravity profiles.

Brachistochrone paths vary significantly across celestial bodies.

Abstract

We explore the gravitational dynamics of falling through planetary interiors. Two trajectory classes are considered: a straight cord between two surface points, and the brachistochrone path that minimizes the falling time between two points. The times taken to fall along these paths, and the shapes of the brachistochrone paths, are examined for the Moon, Mars, Earth, Saturn, and the Sun, based on models of their interiors. A toy model of the internal structure, a power-law gravitational field, characterizes the dynamics with one parameter, the exponent of the power-law, with values from -2 for a point-mass to +1 for a uniform sphere. Smaller celestial bodies behave like a uniform sphere, while larger bodies begin to approximate point-masses, consistent with an effective exponent describing their interior gravity.