Deriving Kepler’s Third Law Using Newton’s Law of Gravitation: A Comprehensive Guide

Introduction

Kepler's laws of planetary motion, derived in the early 17th century, have been foundational in our understanding of celestial mechanics. Among these, Kepler's Third Law is particularly intriguing, stating that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This law can be mathematically derived using Newton's Law of Gravitation. In this article, we will explore the step-by-step derivation of Kepler's Third Law from Newton's law of gravitation, providing a deep dive into the underlying physics and mathematical principles.

Newton's Law of Gravitation

Statement of Newton's Law of Gravitation: According to Sir Isaac Newton, the gravitational force [{F}] between two masses [{m_1}] and [{m_2}], separated by a distance [{r}], is given by:

[{{F} frac{{G m_1 m_2}}{{r^2}}}

where [{G}] is the gravitational constant.

Centripetal Force

Centripetal Force in Circular Motion: For a planet of mass [{m_1}] in a circular orbit around the sun, the gravitational force provides the necessary centripetal force to keep it in that orbit. The centripetal force [{F_c}] required for circular motion is given by:

[{{F_c} frac{{m_1 v^2}}{{r}}}

where [{v}] is the orbital speed of the planet.

Setting Forces Equal

Equating Gravitational and Centripetal Forces: Since the gravitational force provides the centripetal force, we can set the two equations equal to each other:

[frac{{G m_1 m_2}}{{r^2}} frac{{m_1 v^2}}{{r}}

By canceling [{m_1}] from both sides, we obtain:

[frac{{G m_2}}{{r^2}} frac{{v^2}}{{r}}

Multiplying both sides by [{r}] gives:

[v^2 frac{{G m_2}}{r}

Relating Orbital Speed to Period

Expression for Orbital Speed: The orbital speed [{v}] can also be expressed in terms of the orbital period [{T}]:

[v frac{{2pi r}}{T}

Squaring both sides, we get:

[v^2 frac{{4pi^2 r^2}}{T^2}

Substituting for [{v^2}]

Substitution into Gravitational Force Equation: Substituting this expression for [{v^2}] back into the equation we derived from the gravitational force:

[frac{{4pi^2 r^2}}{T^2} frac{{G m_2}}{r}

Rearranging this equation to solve for [{T^2}] gives:

[4pi^2 r^2 frac{{G m_2}}{r} T^2] [T^2 frac{{4pi^2 r^3}}{G m_2}]

Thus, showing that:

[T^2 propto r^3]

For elliptical orbits, where the semi-major axis [{a}] is relevant, we can replace [{r}] with [{a}], and state that:

[T^2 propto a^3]

Conclusion

The derivation of Kepler's Third Law from Newton's law of gravitation demonstrates that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. This confirms Kepler's findings through the lens of Newtonian mechanics, providing a profound understanding of planetary motion and its mathematical underpinnings.