How do I derive a straightforward form of Kepler's 3rd Law if I start with Newtons's Law of Gravitation. (the opposite of what Newton did)
To derive a straightforward form of Kepler's 3rd Law starting from Newton's Law of Gravitation, we need to understand the key principles involved and the equation of motion governing the motion of planets in gravitational fields.
1. Newton's Law of Gravitation: Newton's law states that the gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Mathematically, it can be represented as:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two bodies, and r is the distance between their centers of mass.
2. Equation of Motion: To describe the motion of a planet around the Sun, we use Newton's second law of motion, which states that the net force acting on an object is proportional to its acceleration and can be expressed as:
F = m * a
where m is the mass of the planet, and a is its acceleration.
3. Centripetal Force: In circular motion, there is a force called the centripetal force, which acts towards the center of the circle and keeps the object in motion along a circular path. For an object moving in a circular orbit, this force is provided by the gravitational force.
4. Equating Gravitational and Centripetal Forces: For a planet orbiting the Sun, the gravitational force provides the necessary centripetal force for the circular motion. Therefore, we can equate the gravitational force to the centripetal force.
F = F_c
G * (m1 * m2) / r^2 = m * (v^2 / r)
where v is the orbital speed of the planet.
5. Orbital Period: The orbital period, T, is the time taken for a planet to complete one revolution around the Sun. It is the reciprocal of the orbital frequency. The frequency can be defined as the number of revolutions per unit time.
f = 1 / T
6. Speed and Distance: The orbital speed, v, of a planet can be defined as the distance traveled per unit time.
v = 2πr / T
where r is the distance between the planet and the Sun.
7. Simplification: To derive Kepler's 3rd Law, we need to eliminate the orbital speed, v, from the equation. Using the relationship between v and r from step 6, we substitute v in equation 4:
G * (m1 * m2) / r^2 = m * ((2πr / T)^2 / r)
Simplifying further, we get:
(G * (m1 * m2)) / (r^3) = 4π^2 * (m / T^2)
8. Kepler's 3rd Law: Finally, we note that the term m1 * m2 is the product of the Sun's mass (the central body) and the planet's mass, which can be denoted as M (mass of the central body) and m (mass of the planet), respectively.
Substituting M for m1 * m2, we obtain Kepler's 3rd Law:
(G * M) / (r^3) = 4π^2 * (m / T^2)
This equation demonstrates that the square of the orbital period is directly proportional to the cube of the semi-major axis of the planet's orbit, which is the statement of Kepler's 3rd Law.
By following this process, starting from Newton's Law of Gravitation and utilizing the principles of circular motion and equations of motion, we can derive a straightforward form of Kepler's 3rd Law.