# Laws of Planetary Motion

### Kepler’s Laws of Planetary Motion:

One of the greatest ideas proposed in human history is the fact that the earth is a planet, among the other planets that orbit the sun.

The precise determination of these planetary orbits was carried out by Johannes Kepler, using data compiled by his teacher, the astronomer Tycho Brahe.

Johannes Kepler discovered three empirical laws by using the data on planetary motion:

(i) First law: Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse.

(ii) Second law: A line from the sun to a given planet sweeps out equal areas in equal times.

(iii) Third Law: The square of the periods of the planets are proportional to cube of their mean distance from the sun.

These laws go by the name Kepler’s laws of planetary motion. It was in order to explain the origin of these laws among other phenomena, that Newton proposed the theory of gravitation.

In our discussion we are not going to derive the complete laws of planetary motion from Newton’s law of gravitation.

Since most of the planets actually revolve in near circular orbits, we’re going to assume that the planets revolve in circular orbits. Consider a planet of mass m, rotating around the sun (mass M, M>>m) in a circular orbit of radius r with velocity v.

Then, by applying Newton’s law of gravitation and the second law of motion, we can write,

Gravitational force = mass × centripetal acceleration

$\displaystyle \frac{G M m}{r^2} = m (\frac{v^2}{r})$

$\displaystyle v^2 = \frac{G M}{r}$

$\displaystyle v = \sqrt{\frac{G M}{r}}$

As the moment of the gravitational force about S is zero, the angular momentum of the planet about the sun remains constant. This is the meaning of Kepler’s 2nd law of motion, as will be shown later.

The time period of rotation, T, of the planet around the sun is given by,

$\displaystyle T = \frac{2 \pi r}{v} = \frac{2 \pi r}{\sqrt{GM/r}}$

Squaring ,

$\displaystyle T^2 = (\frac{4 \pi^2}{G M} ) r^3$

$\displaystyle T^2 \propto r^3$

Which is Kepler’s 3rd law of motion

Note: that the constant of proportionality in the above equation depends only on the mass of the sun (M) but not on the mass of the planet.
Kepler’s Laws are also valid for the motion of satellites around the earth.

##### Kepler’s 2nd Law and its Meaning Consider a planet P that moves in an elliptical orbit around the sun, and let P and P’ be the positions of the planet at time t and t+ Δt (where Δt is a very small time interval).

If the angular displacement of the planet is Δθ, then the area swept out by the line joining the planet SP in time Δt is:

ΔA = area of the section SPP’

$\displaystyle = \frac{1}{2}r^2 \Delta\theta$ ; where r = length SP.

The areal velocity $\displaystyle v_A = \frac{\Delta A}{\Delta t}$

$\displaystyle v_A = \frac{1}{2}r^2 ( \frac{\Delta\theta}{\Delta t}) = \frac{1}{2} r^2 \omega = constant$

In other words,

m × (2vA) = m r2 ω = L = constant (m = mass of the planet)

$\displaystyle v_A = \frac{L}{2m}$

Areal velocity , $\displaystyle \frac{dA}{dt} = \frac{L}{2 m}$

This is the expression for the angular momentum of the planet,

L = Iω = m r2 ω = mr2 (dθ/dt) perpendicular to the plane of its orbit.

The gravitational force. $\displaystyle \vec{F} = -\frac{G M m}{r^2}\hat{r}$

is centripetal, and the torque on the planet is zero. as

$\displaystyle \vec{r}\times \vec{F} = \vec{r}\times (-\frac{G M m}{r^2}\hat{r}) = 0$

Hence, the angular momentum of the planet does not vary i.e the Arial velocity of the planet remains constant. At its aphelion (farthest point from the sun, r is large) the planet moves slowly, and at its perihelion (nearest point from the sun, r is small) the planet moves fastest