Kepler's Laws and Universal Gravitation: Quick Revision

A brief note on Kepler's Laws and Universal Gravitation. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Kepler's first law: each planet moves on an ellipse with the Sun at one focus.
• Kepler's second law: the line to the Sun sweeps equal areas in equal times, so a planet is fastest when nearest. At the nearest and farthest points, $v_1r_1 = v_2r_2$.
• Kepler's third law: $T^2 \propto a^3$. For bodies round the Sun, $T^2 = a^3$ with $T$ in years and $a$ in AU.
• Newton's law of gravitation: $F = \dfrac{Gm_1m_2}{r^2}$, $G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$. Pulls from several bodies add as vectors.
• Gravity at a planet's surface: $g = \dfrac{GM}{R^2}$. It falls with height and with depth.

The facts at a glance

A worked example

An asteroid comes within $1.8\ \text{AU}$ of the Sun and goes out to $3.0\ \text{AU}$.

1. Semi-major axis. Half of nearest plus farthest: $a = \dfrac{1.8\ \text{AU} + 3.0\ \text{AU}}{2} = 2.4\ \text{AU}$.
2. Its year. $T = a^{3/2} = 2.4^{1.5} = 3.72$ years.
3. Speeds. $\dfrac{v_{\text{near}}}{v_{\text{far}}} = \dfrac{3.0\ \text{AU}}{1.8\ \text{AU}} = 1.67$: it moves $1.67$ times faster when nearest.

Practice with new numbers

A program to try

The program works out the asteroid of the worked example.

Mistakes to avoid

• $r$ measured from the surface. In $\dfrac{Gm_1m_2}{r^2}$, $r$ runs between the centres.
• Adding pulls as plain numbers. They are vectors; mind their directions.
• $1 - \dfrac{2h}{R}$ for a big height. It works only when $h$ is much smaller than $R$.
• The same rule above and below the ground. Above, $g$ falls as $\dfrac{1}{r^2}$; below, in step with $r$.
• The semi-major axis as the farthest distance. It is the average of nearest and farthest.

Check yourself

1. Two $5.0\ \text{kg}$ balls are $0.20\ \text{m}$ apart. What is the pull between them?

2. The distance between two bodies is tripled. What happens to the pull?

3. A body circles the Sun at an average of $4\ \text{AU}$. How long is its year?

4. What is $g$ at a height equal to the Earth's radius?

5. Why is $g$ a little smaller at the equator than at the poles?