Deriving Kepler's First Law of Planetary Motion with Classical Mechanics

Introduction

Kepler's Laws of Planetary Motion represent a cornerstone in our understanding of the cosmos. Specifically, Kepler's First Law states that the orbit of a planet around the Sun is an ellipse with the Sun at one of the two foci. This article delves into how this law can be derived from the principles of classical mechanics, including Newton's laws of motion and the law of universal gravitation.

Step-by-Step Derivation

Step 1: Newton's Law of Universal Gravitation

According to Newton's law of universal gravitation, the force F between two masses m1 (the Sun) and m2 (the planet) is given by:

F G(m1m2) / r2

where:

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

Step 2: Centripetal Force and Motion

For a planet orbiting the Sun in a circular path of radius r, the gravitational force provides the necessary centripetal force:

F m2v2 / r

where v is the orbital velocity. Equating the gravitational force to the centripetal force:

G(m1m2) / r2 m2v2 / r

Simplifying this equation, we obtain:

v2 Gm1 / r

Step 3: Generalizing to Elliptical Orbits

While the above derivation shows circular orbits, Kepler's First Law applies to elliptical orbits as well. The key insight comes from the conservation of angular momentum and energy.

Angular Momentum Conservation

The angular momentum L of a planet is:

L m2r2 dθ / dt

Since there is no external torque, angular momentum is conserved.

Energy Conservation

The total mechanical energy E of the planet in orbit is:

E (1/2) m2 v2 - G(m1 m2 / r)

Step 4: The Shape of the Orbit

Using the vis-viva equation, derived from energy conservation, which relates the speed of the planet to its distance from the Sun and the semi-major axis a:

v2 Gm1(2/r - 1/a)

This equation indicates that the speed of the planet varies with its distance from the Sun, a characteristic of elliptical orbits.

Step 5: The Geometry of Ellipses

The mathematical definition of an ellipse can be described using the foci and the semi-major axis a and semi-minor axis b. The distance from any point on the ellipse to the two foci F1 and F2 satisfies:

d1 d2 2a

In the context of celestial mechanics, one focus is occupied by the Sun, and the gravitational force acting on the planet leads it to follow this elliptical path.

Conclusion

By combining Newton's laws of motion, the law of universal gravitation, and the principles of conservation of angular momentum and energy, one can derive that the orbits of planets are elliptical, thereby confirming Kepler's First Law of Planetary Motion. This derivation showcases the fundamental connection between classical mechanics and celestial motion.