Orbital mechanics and Kepler's laws

Orbits as falling motion

An orbit is continuous free fall around a central body. A satellite in orbit is always falling toward Earth, but it also has enough sideways velocity that Earth's surface curves away beneath it. Gravity provides the centripetal acceleration needed to bend the path.

For a circular orbit of radius $r$ around mass $M$, set gravitational force equal to the required centripetal force:

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

Solving gives

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

The orbiting mass $m$ cancels.

Orbital period

For a circular orbit, speed is circumference divided by period:

v=rac{2pi r}{T}

Combining with $v=sqrt{GM/r}$ gives

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

This is the circular-orbit version of Kepler's third law.

Kepler's first law

Kepler's first law says planets move in ellipses with the Sun at one focus. A circle is a special case of an ellipse. Newtonian gravity explains why inverse-square central forces produce conic-section orbits: ellipses, parabolas, or hyperbolas depending on energy.

Bound orbits are ellipses. Escape trajectories are parabolic or hyperbolic.

Kepler's second law

Kepler's second law says a line from the Sun to a planet sweeps out equal areas in equal times. This is a statement of angular momentum conservation.

Gravity is a central force, meaning it points along ec{r}. Therefore the torque about the central mass is zero:

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

So angular momentum is conserved:

ec{L}=ec{r} imesec{p}=constant

When the planet is closer to the Sun, it moves faster; when farther away, it moves slower.

Kepler's third law

Kepler's third law relates orbital period and size. For an elliptical orbit with semi-major axis $a$,

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

This says larger orbits have longer periods. The relationship is not linear: period squared is proportional to semi-major axis cubed.

Orbital energy

The total mechanical energy of a circular orbit is

$E=K+U$

where

K=rac{1}{2}mv^2

and

U=-Grac{Mm}{r}

Using $v^2=GM/r$,

K=Grac{Mm}{2r}

so

E=-Grac{Mm}{2r}

The negative total energy indicates a bound orbit.

Changing orbits

To move to a higher circular orbit, a spacecraft needs energy added. Higher orbits have larger total energy, meaning less negative energy. However, the orbital speed in a higher circular orbit is lower because $v=sqrt{GM/r}$ decreases with $r$.

This can feel counterintuitive: raising an orbit requires energy, but the final circular speed is smaller.

The big idea

Orbital mechanics is Newtonian gravity plus conservation laws. Circular orbit speed comes from balancing gravity with centripetal acceleration. Kepler's laws emerge from inverse-square central forces and angular momentum conservation. Orbital energy determines whether motion is bound or unbound.