Kepler's laws derived from Newton

From observation to explanation

Johannes Kepler discovered three laws of planetary motion from careful astronomical data. Newton later showed that these laws follow from his laws of motion plus universal gravitation. This was one of the first great unifications in physics: the same force that makes an apple fall also governs the Moon and planets.

Newton's law of gravitation states that two masses attract with force

F=Grac{Mm}{r^2}

where $M$ and $m$ are the masses, $r$ is their separation, and $G$ is the gravitational constant. The force points along the line connecting the bodies.

Central forces and angular momentum

Gravity is a central force: it always points toward the attracting mass. For a planet orbiting the Sun, the torque about the Sun is zero because

ec{ au}=ec{r} imesec{F}=0

when ec{F} is parallel or antiparallel to ec{r}. Therefore angular momentum is conserved:

ec{L}=mec{r} imesec{v}=constant

This conservation law leads directly to Kepler's second law: a line from the Sun to a planet sweeps out equal areas in equal times.

The areal velocity is

rac{dA}{dt}=rac{L}{2m}

Since $L$ is constant, $dA/dt$ is constant.

Elliptical orbits

Solving Newton's equations for an inverse-square central force gives conic sections: ellipses, parabolas, and hyperbolas. Bound orbits are ellipses. This gives Kepler's first law: planets move in ellipses with the Sun at one focus.

For an ellipse, the closest point is perihelion and the farthest point is aphelion. The semi-major axis $a$ sets the overall size of the orbit. The eccentricity $e$ describes how stretched it is. A circle has $e=0$, while a very elongated ellipse has $e$ closer to 1.

Period and semi-major axis

For a small body orbiting a much larger mass $M$, Newtonian gravity gives Kepler's third law:

T^2=rac{4pi^2}{GM}a^3

This says the square of the orbital period is proportional to the cube of the semi-major axis. Planets farther from the Sun take longer to orbit.

For two comparable masses, replace $M$ by the total mass:

T^2=rac{4pi^2}{G(M+m)}a^3

where $a$ is the semi-major axis of the relative orbit.

Circular orbit as a special case

For a circular orbit of radius $r$, gravity provides centripetal force:

Grac{Mm}{r^2}=mrac{v^2}{r}

so

v=sqrt{rac{GM}{r}}

The period is circumference divided by speed:

T=rac{2pi r}{v}

which leads to the circular version of Kepler's third law.

What Newton added

Kepler's laws described planetary motion, but Newton explained why those patterns occur. Conservation of angular momentum gives equal areas. Inverse-square attraction gives conic-section orbits. The strength of gravity gives the period-size relationship.

The big idea

Kepler's laws are not separate rules pasted onto astronomy. They are natural consequences of Newton's mechanics and inverse-square gravity. Celestial mechanics begins when geometric patterns in the sky become dynamical predictions from forces, energy, and angular momentum.