Before Galileo and the telescope, Kepler used naked-eye measurements to make three major discoveries:

• The planets move in elliptical orbits.

• Planets “sweep out” equal areas in equal times.

• For circular orbits, the square of a planet’s period is proportional to the cube of the orbit’s radius.

Kepler’s discoveries were the first solid proof of Copernicus’s assertion that earth and the other planets orbit the sun.

Newton proposed that any two masses M and m are attracted toward each other by a gravitational force of magnitude

where r is the distance between the masses and G is the gravitational constant.

• Newton’s law is an inverse-square law.

• Newton’s law predicts the value of g.

In addition to a specific force law for gravity, Newton’s three laws of motion apply to all objects in the universe.

The gravitational potential energy of two masses is

Gravitational potential energy is negative, with a zero point at infinity. The $U_{\text{G}}=mgy$ that you learned in Chapter 10 is a special case for objects very near the surface of a planet. Gravitationally interacting stars, planets, and satellites are always isolated systems, so mechanical energy is conserved.

《 LOOKING BACK Chapter 10 Potential energy and energy conservation

Kepler’s laws can be derived from Newton’s theory of gravity:

• Orbits can be circular or elliptical.

• Orbits conserve energy and angular momentum.

• Geosynchronous orbits have the same period as the rotating planet.

《 LOOKING BACK Sections 8.2-8.3 Circular motion

Satellites, space stations, the GPS system, and future missions to planets all depend on Newton’s theory of gravity. Our modern understanding of the cosmos—from stars and galaxies to the Big Bang—is based on understanding gravity. Newton’s theories of motion and gravity were the beginnings of modern science.