Kepler's laws and planetary motion

G3.5 G3.6 G3.7 G4.3 G4.4 G5.5

By the end of this lesson you should be able to

State the three laws of planetary motion. G3.5
Describe the relationship between the Law of Universal Gravitation and uniform circular motion and recognise this as the third law of planetary motion. G3.6
Solve problems involving the third law of planetary motion using T_a^2/r_a3 = T_b^2/r_b3 = 4π^2/(GM). G3.7
Appreciate how the accepted model of the solar system slowly shifted under the influence of carefully collected and analysed data. G4.3
Explore the difficulties experienced by scientists who supported a heliocentric model of the solar system and the hindrances to the acceptance of their discoveries by society. G4.4
Consider the difference between the heliocentric and geocentric models of the solar system. G5.5

Key ideas

Kepler's 1st law: planets move in ellipses with the Sun at one focus.
2nd law: a line from the Sun to a planet sweeps out equal areas in equal times, so planets move fastest when closest to the Sun.
3rd law: T² is proportional to r³ for every object orbiting the same central mass.
Setting gravity equal to the centripetal force (\(GMm/r^2 = mv^2/r\), with \(v = 2\pi r/T\)) gives the 3rd law. The orbiting mass cancels.
The move from geocentric (Ptolemy) to heliocentric (Copernicus, Kepler) models was driven by better data (Tycho Brahe) and observations (Galileo).

Key formulas (QCAA formula book notation)

\[ \frac{T^2}{r^3} = \frac{4\pi^2}{G M} \]

\[ \frac{T_a^2}{r_a^3} = \frac{T_b^2}{r_b^3} \]

Worked example

Mars orbits at 1.52 times Earth's orbital radius. Find its period in Earth years.

Common mistakes