Gravitation

Lesson

Part 1 — how planets move, how masses attract, and how g changes as you climb or dig.

In this lesson you will

Equal areas in equal times: the planet speeds up near the Sun.

Concept

Kepler’s three laws

Kepler summarised the motion of the planets in three laws:

Law of orbits: every planet moves in an ellipse with the Sun at one focus.
Law of areas: the line from the Sun to the planet sweeps out equal areas in equal times.
Law of periods: the square of the period is proportional to the cube of the semi-major axis, $T^2 \propto a^3$ .
The point of the orbit nearest the Sun is perihelion, the farthest is aphelion.

Table of Kepler three laws, what each states and its physical reason
Law	Statement	Physical reason
Orbits	A planet moves in an ellipse with the Sun at one focus	Inverse-square attraction
Areas	The line from Sun to planet sweeps equal areas in equal times	Force along the line: angular momentum is constant
Periods	T² is proportional to a³ (a = semi-major axis)	Gravity falls as 1/r²

Diagram

Equal areas in equal times: the planet speeds up near the Sun.

In the picture the two wedges are swept in the same time interval.

Equal areas in equal times means the planet moves faster near perihelion and slower near aphelion.
The area swept per second is $\dfrac{dA}{dt} = \dfrac{L}{2m}$ , where $L$ is the angular momentum about the Sun.
Gravity acts along the line to the Sun, so its torque about the Sun is zero and L stays constant.
At the two extreme points $v_p r_p = v_a r_a$ , so the speed ratio is the inverse of the distance ratio.