  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.
