Kepler's first law gives the shape of an orbit. His second law tells how a planet moves along it, and the answer surprised him: not at a steady speed.
Why it works
Imagine a line from the Sun to the planet. Kepler found that this line sweeps out equal areas in equal times. Near the Sun the line is short, so to sweep a given area in a month the planet must cover a long stretch of its path: it moves fast. Far from the Sun the line is long, and a short stretch of path sweeps the same area: it moves slowly. Newton explained why. Gravity pulls straight toward the Sun, never sideways, so it never speeds up or slows down the planet's turning around the Sun (its angular momentum stays fixed). At perihelion and aphelion, where the motion is straight across the line to the Sun, this means $v_p\, r_p = v_a\, r_a$.
Key idea
A planet sweeps out equal areas in equal times, so it moves fastest at perihelion and slowest at aphelion: $v_p\, r_p = v_a\, r_a$.
Worked examples
Worked example: Earth's two speeds
Earth is $147.1$ million km from the Sun at perihelion (early January) and $152.1$ million km at aphelion (early July). How much faster does it move in January?
- At the ends of the orbit: $\frac{v_p}{v_a} = \frac{r_a}{r_p}$.
- $\frac{152.1}{147.1} = 1.034$.
- Earth moves about $3.4\%$ faster in January, which is one reason northern winter is a few days shorter than northern summer.
Answer: About $1.034$ times as fast.
Worked example: A comet slows down
A comet moves at $40$ km/s at perihelion, $0.5$ AU from the Sun. Its aphelion is $5$ AU out. How fast is it there?
- $v_a = v_p \cdot \frac{r_p}{r_a} = 40 \times \frac{0.5}{5}$.
- $v_a = 4$ km/s: ten times farther, ten times slower.
Answer: $4$ km/s.
Common mistake
It is natural to picture a planet cruising at one steady speed, but along an ellipse its speed changes all the time. Check with Earth: the March equinox to the September equinox (northern spring and summer, Earth farther out and slower) lasts about 186 days, while the other half of the year lasts about 179.