Which of the following is Kepler’s Third Law?
a
W = F*d
b
T^2 ⍺ s^3
c
F = G*(m1*m2)/r^2
d
U = M - KE
b. T^2 ⍺ s^3
The correct option for Kepler's Third Law is option (b) T^2 ⍺ s^3.
The correct answer is option b: T^2 ⍺ s^3.
Kepler's Third Law is a fundamental law of planetary motion discovered by the astronomer Johannes Kepler. It relates the period (T, time taken for a planet to complete one orbit around the sun) and the semi-major axis of the planet's orbit (s, average distance from the planet to the sun).
To understand how to derive Kepler’s Third Law, we need to know some basic concepts:
1. Newton's Law of Universal Gravitation: F = G * (m1 * m2) / r^2
This equation describes the gravitational force (F) between two objects with masses m1 and m2 separated by a distance r. G is the gravitational constant.
2. Centripetal Force: F = (m * v^2) / r
This equation represents the centripetal force (F) required to keep an object with mass m moving in a circle of radius r at velocity v.
Now, let's derive Kepler's Third Law step by step:
1. Equate the gravitational force equation with the centripetal force equation:
G * (m1 * m2) / r^2 = (m * v^2) / r
Note: We assume that the mass of the planet (m) cancels out, as it does not affect the relationship between T and s.
2. Rearrange the equation to eliminate velocity:
G * (m1 * m2) / r^2 = (m * (2 * π * r / T)^2) / r
We can simplify this equation by canceling out one r:
G * (m1 * m2) / r = (4 * π^2 * r^2) / T^2
3. Rearrange the equation to express T^2 in terms of r^3:
(T^2) * (G * (m1 * m2)) = (4 * π^2 * (r^3))
Thus, we can conclude that T^2 is proportional to r^3.
Therefore, Kepler’s Third Law states that the square of the orbital period (T^2) of a planet is directly proportional to the cube of its average distance from the sun (s^3). This law applies to all objects in our solar system, not just planets.