Derivation of Kepler's Laws from Newton's Laws: A Comprehensive Guide

Kepler's laws of planetary motion describe the orbits of planets around the Sun. These laws can be derived from Newton’s law of universal gravitation and his laws of motion. This article outlines a step-by-step derivation, particularly focusing on the second law, the area law, and the first law, the elliptical orbits. Let's delve into the intricate yet beautiful connection between these fundamental principles of physics.

1. Newton's Law of Universal Gravitation

Newtons law states that any two masses m1 and m2 attract each other with a force F given by:

F G frac{m1 m2}{r2}

Where:

G is the gravitational constant. r is the distance between the centers of the two masses.

2. Equation of Motion

For a planet of mass m orbiting the Sun mass M, the gravitational force provides the centripetal force necessary for circular motion:

F m a m frac{v^2}{r}

Setting F equal to the gravitational force gives:

m frac{v^2}{r} G frac{m M}{r^2}

Cancelling m (assuming m ≠ 0) and rearranging gives:

v^2 frac{G M}{r}

3. Conservation of Angular Momentum

In a central force motion like planetary orbits, angular momentum L is conserved:

L m r^2 frac{dtheta}{dt} text{constant}

This implies that as a planet moves closer to the Sun, it must speed up to conserve angular momentum.

4. Deriving Kepler's Laws

Kepler's First Law: Elliptical Orbits

Using energy conservation and the effective potential approach, we can show that orbits are conic sections ellipses, parabolas, and hyperbolas. The total mechanical energy of the system is:

E K - U frac{1}{2} m v^2 - G frac{m M}{r}

The trajectory can be derived from the equations of motion in polar coordinates, leading to the conclusion that the orbits of planets are ellipses with the Sun at one focus.

Kepler's Second Law: Equal Areas in Equal Times

To show that a line segment joining a planet and the Sun sweeps out equal areas in equal times, we can analyze the angular momentum:

(frac{dL}{dt} frac{d}{dt}m r^2 frac{dtheta}{dt} 0)

This implies that r^2 frac{dtheta}{dt} is constant. The area A swept out in time t is:

A frac{1}{2} r^2 theta

Thus, if r^2 frac{dtheta}{dt} is constant, the area swept out in a given time interval is also constant, confirming Kepler's second law.

Kepler's Third Law: The Harmonic Law

To derive the third law, we can use the gravitational force and centripetal acceleration relationship. For a circular orbit, we have:

T^2 propto r^3

Where T is the period of the orbit, and r is the semi-major axis. This can be shown by equating the gravitational force to the centripetal force and rearranging to find the relationship between T and r:

T^2 frac{4 pi^2}{G M} r^3

Conclusion

Thus, from Newton's laws of motion and his law of universal gravitation, we derive Kepler's three laws of planetary motion, demonstrating the profound connection between these fundamental principles of physics.