Question

Question #dea13

Answer 1

Kepler's third law states that the ratio of the square of the period of a planets orbit to the cube of its rotational distance from the sun is the same for all planets. That's a long winded way to say that;

`T^2/R^3 =` the same for all of the planets

where `T` is the time it takes a planet to orbit the sun once, and `R` is the average distance from the sun. Kepler discovered his laws empirically, meaning that he looked at his data and noticed certain patterns. To understand why it works, we can derive this law based on physics.

First off, Kepler's first law states that the planets' orbits are ellipses, with the sun at one focus. They are, however, nearly circular, so for simplification, lets assume that the orbits are circles. We'll start by combining Newton's law of gravitation and Newton's second law;

`F=G(Mm)/R^2` (law of gravity) `=ma` (2nd law)

Here, `G` is Newton's gravitational constant; `M` is the mass of the sun; `m` is the mass of the planet; `R` is the distance the planet orbits from the sun; and `a` is the acceleration of the planet toward the sun. Right away we can see that `m` is irrelevant to our calculation so we can remove it.

`GM/R^2 = a` (equation 1)

For an object moving at constant speed in a circle, the acceleration toward the center of that circle is given by;

`a = omega^2 R`

where `omega` is the rotational velocity, measured as the radial distance the planet moves over time. Since the total radial distance is `2pi`, the number or radians in a circle, and the total time is one period, `T`, we can replace `omega` with `(2pi)/T` to get;

`a = ((2pi)/T)^2R`

so that equation 1 becomes;

`GM/R^2 = ((2pi)/T)^2R`

We can now move all of the variables to one side and the constants to the other to get Kepler's third law.

`T^2/R^3 = (4(pi)^2)/(GM) =` Constant

The only thing that effects this constant is the mass of the sun. For larger stars, the ratio becomes smaller, meaning that the period decreases for a planet orbiting the same distance from the star.