Orbital Period & Kepler's Third Law | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

Orbital Period & Kepler's Third Law

We use the

T

to describe the timing of an orbit. Its unit is s.

Linking speed to period gives an equation between the period and the radius.

Definition

1 mark

What is meant by orbital period?

Model answer: The time taken for an object to complete one full orbit.

From speed to period

The distance travelled in one orbit is the circumference. Divide this distance by the period to find the speed.
$v = \dfrac{2\pi r}{T}$
Substitute this speed into the circular-orbit equation. Rearranging gives the main period equation shown next.
$\left(\dfrac{2\pi r}{T}\right)^2 = \dfrac{GM}{r}$

T^2 = \dfrac{4\pi^2 r^3}{GM}

Symbols

$T$ = orbital period (time for one orbit) (s)
$r$ = orbit radius from the centre (m)
$G$ = gravitational constant (N m² kg⁻²)
$M$ = mass of the planet (kg)

The square of the period is proportional to the cube of the radius for circular orbits around the same central mass. This is (9702/42/M/J/23 Q1(b)).

The basic T² against r³ graph

A common question asks you to show the same equation in a different arrangement (9702/42/M/J/23 Q1(b)).
$4\pi^2 r^3 = GMT^2$
If data use orbit radius $r$ for objects around the same central mass, plot $T^2$ against $r^3$ . The graph is a straight line through the origin. Use its gradient to find the central mass.
$M = \dfrac{4\pi^2}{G \times \text{gradient}}$

Change the orbit radius in the display. It shows how v and T change.
The central mass does not change. Therefore, the value of T²/r³ stays constant.

Circular Orbit Explorer

Change the orbit radius measured from the Earth's centre. Watch the speed and period change together. The model uses an Earth mass of 6.0 × 10²⁴ kg.

Orbit radius r12.0 × 10⁶ m

Swipe left or right to see the whole diagram →

orbital speed v

5.77 km s⁻¹

orbital period T

3.6 h

Kepler ratio T²/r³

9.86e-14 s² m⁻³

The ratio T²/r³ stays constant. This is Kepler's third law for circular orbits around the same central mass.