Final answer:
The Ptolemaic model is the one that depicts planets traveling around the Earth in epicycles. The heliocentric Copernican and Keplerian models later superseded this geocentric model. So the correct option is C.
Step-by-step explanation:
The model of the solar system that shows the planets travel around the Earth in epicycles is the Ptolemaic model. This model, named after the Greek philosopher Ptolemy, posits that the Earth is at the center of the universe, and celestial bodies, including the planets and stars, revolve around it in complex circular paths. To explain the observed planetary motions, like retrograde motion, Ptolemy used a system where each planet moves in a small circle called an epicycle, which in turn revolves around a larger circle called the deferent.
The other models, like the Copernican model, which is heliocentric, and the Keplerian model, which is based on ellipses, do not use the concept of epicycles. The Brahe model is a hybrid that retains Earth at the center for the orbit of the Sun while other planets orbit the Sun.