Does a planet move more rapidly at perihelion or aphelion?
Answers
There are a few ways of looking at this problem.
The first comes from Kepler. Kepler's second law states:
As the planet moves in its orbit, a line from the sun to the planet sweeps out equal areas in equal times.
Gravity is a central force – no matter where a planet is, in its orbit, gravity pulls it towards the sun. The torque for this would be the cross product of the gravitational force and the radius vector between the two bodies (sun and planet). Since these two vectors are in the same direction, their cross product is zero. This means that angular momentum is constant.
There is a concept called areal velocity. Areal velocity is the rate at which area is swept out by an object as it moves along a curve. If you really want to see the math for that it is easily found by searching for areal velocity, but that math shows that the areal velocity equals the radius vector crossed with the velocity vector divided by two (DA/dt = (rxv)/2). We know that the angular momentum equals rxmv (r cross mv). Substitution then shows that the angular momentum equals 2m(DA/dt).
So, if angular momentum is conserved, then the area swept out over time must also be constant. Looking at the above diagram, we can see that for the area swept out to be constant, the planet must move faster when it is close to the sun and slower when it is far from the sun.
2. The second comes from Newton. Although Newton also did a mathematical proof of Kepler's way of looking at the problem, he provided another way to look at it.
Conservation of Energy.
The total energy of an orbiting planet is the sum of its kinetic and potential energies.
KE = 1/2(mv^2)
PE = -G(Mm/r)
We can see from the two equations that as the distance between the planet and the sun ( r ) decreases, the potential energy decreases (gets farther from zero) and as the distance between the planet and the sun increases, the potential energy approaches zero. Similarly, the kinetic energy must also change proportionately.
At perihelion, kinetic energy is at maximum and potential energy is at minimum. At aphelion, the reverse is true.
Since the mass of the planet (m) is a constant, the only way to change the magnitude of the kinetic energy is to alter the velocity of the planet. So, if kinetic energy is at maximum at perihelion, then the planet must be traveling faster there, and if it is at minimum at aphelion, then the planet must be traveling slower, there.
3. A more base way to imagine the problem is to consider the vectors involved.
As we stated above, gravity is a central force. That means there will always be a force (and thus an acceleration) towards the sun, from the planet. Because the planet is moving about the sun, the relationship between the direction of that acceleration and the velocity of the planet will continually change. As the planet moves away from the sun, the acceleration will decelerate it and as the planet moves towards the sun, the acceleration will accelerate it. So, we can see that at aphelion, the planet will be traveling its slowest and at perihelion it will be traveling its fastest.