from the sun, it must slow down. This applies to the Earth as well. In the northern hemisphere winter,
when the Earth is slightly closer to the sun,^1 the Earth moves faster in its orbit. (You may have noticed
that the earliest sunset in wintertime occurs about two weeks before the solstice—this is a direct
consequence of Earth’s faster orbit.)Figure 15.2 Kepler’s second law. The area “swept out” by a planet in its orbit is shaded. In equal
time intervals Δt 1 and Δt 2 , these swept areas A 1 and A 2 are the same.- A planet’s orbital period squared is proportional to its orbital radius cubed . In mathematics, we
write this as T 2 = cR 3 . Okay, how do we define the “radius” of a non-circular orbit? Well, that
would be average distance from the sun. And what is this constant c ? It’s a different value for every
system of satellites orbiting a single central body. Not worth worrying about, except that you can
easily derive it for the solar system by solving the equation above for c and plugging in data from
Earth’s orbit: c = 1 year^2 /AU^3 , where an “AU” is the distance from Earth to the sun. If you really
need to, you can convert this into more standard units, but we wouldn’t bother with this right now.Energy of Closed Orbits
When an object of mass m is in orbit around the sun, its potential energy is , where M is the
mass of the sun, and r is the distance between the centers of the two masses.
The kinetic energy of the orbiting mass, of course, is K = ½mv 2 . The total mechanical energy of the
mass in orbit is defined as U + K . When the mass is in a stable orbit, the total mechanical energy must be
less than zero. A mass with positive total mechanical energy can escape the “gravitational well” of the
sun; a mass with negative total mechanical energy is “bound” to orbit the sun.^2
All of the above applies to the planets orbiting in the solar system. It also applies to moons or
satellites orbiting planets, when (obviously) we replace the “sun” by the central planet. A useful
calculation using the fact that total mechanical energy of an object in orbit is the potential energy plus the
kinetic energy is to find the “escape speed” from the surface of a planet ... at r equal to the radius of the
planet, set kinetic plus potential energy equal to zero, and solve for v . This is the speed that, if it is
attained at the surface of the planet (neglecting air resistance), will cause an object to attain orbit.