12.20 - Summary
Newton's law of gravitation states that the force of gravity between two particles is
proportional to the product of their masses and inversely proportional to the square
of the distance between them.
The gravitational constantG appears in Newton's law of gravitation and is the same
everywhere in the universe. It should not be confused with g, the acceleration due
to gravity near the Earth's surface. The value of gvaries slightly according to the
location on the Earth. This is due to local variations in altitude and to the Earth's
bulging shape, nonuniform density and rotation.
A circular orbit is the simplest type of orbit. The speed of an object in a circular orbit
is inversely proportional to the square root of the orbital radius. The speed is also
proportional to the square root of the mass of the object being orbited, so orbiting a
more massive object requires a greater speed to maintain the same radius.
Johannes Kepler set forth three laws that describe the orbital motion of planets.
Kepler's first law says that the planets move in elliptical orbits around the Sun, which
is located at one focus of the ellipse. Most of the planets' orbits in the solar system
are only slightly elliptical.
Kepler's second law, the law of areas, says that an orbiting body such as a planet
sweeps out equal areas in equal amounts of time. This means that the planet’s
speed will be greater when it is closer to the Sun.
Kepler's third law, the law of periods, states that the square of the period of an orbit
is proportional to the cube of the semimajor axis a, which is equal to one half the
width of the orbit at its widest. For a circular orbit, a equals the radius of the orbit.
The orbital energy of a satellite is the sum of its gravitational potential energy (which
is negative) and its kinetic energy. The total energy is constant, though the PE and
KE change continuously if the satellite moves in an elliptical orbit.
The escape speed is the minimum speed necessary to escape a planet's
gravitational attraction. It depends on the mass and radius of the planet, but not on
the mass of the escaping object.
Newton’s law of gravitationGravitational accelerationCircular orbitKepler’s second lawKepler’s third lawEnergy in circular orbitsEnergy in elliptical orbitsEscape speed