Introduction

For most of human history, the motion of the planets, the Moon and the Sun across the sky was described only in terms of how they appeared to move, without any single physical principle explaining why they moved that way. Johannes Kepler, working from decades of careful naked-eye observations, was the first to describe the planets' actual paths accurately -- as ellipses, not circles -- and to find three precise empirical laws that their motion obeyed, without yet knowing WHY those laws should hold. It was Isaac Newton who supplied the missing physical cause: a single force of attraction, acting between every pair of masses in the universe, whose strength falls off as the inverse square of the distance between them. This one law of universal gravitation was strong enough to derive all three of Kepler's laws as direct mathematical consequences, and, remarkably, to explain something as everyday as an apple falling to the ground using exactly the same force that keeps the Moon in its orbit around the Earth. This chapter builds up this account in order: Kepler's laws first (Section 7.2), then Newton's law of gravitation and the gravitational constant $G$ (Sections 7.3-7.4), the acceleration due to gravity and how it varies with altitude, depth and the Earth's own rotation (Sections 7.5-7.7), gravitational potential energy and potential (Sections 7.8-7.9), and finally escape velocity, orbital velocity, the energy of an orbiting satellite, and the special case of a geostationary satellite (Sections 7.10-7.13).