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Ellipses

Gravity and orbits · level 7, lesson 3 of 6

A question to start

For two thousand years astronomers assumed planets moved in circles, and Mars never quite fitted: its position in the sky was off by up to 8 minutes of arc, a quarter of the Moon's width. In 1609 Kepler gave up circles altogether. What did he find?

The idea

Kepler's first law: each planet moves on an ellipse, with the Sun at one focus.

An ellipse is a stretched circle. Two numbers describe it:

  • the semi-major axis \(a\): half its longest width (the "average" distance);
  • the eccentricity \(e\): how stretched it is, from 0 (a circle) to nearly 1 (very long and thin).

The closest point to the Sun, perihelion, is at \(r_p = a(1-e)\); the farthest, aphelion, at \(r_a = a(1+e)\).

Sun (focus)perihelionaphelionacentreempty focus

(This ellipse is drawn much more stretched than any planet's.)

Kepler's second law: a line from the Sun to the planet sweeps out equal areas in equal times. Near the Sun the line is short, so the planet must move fast to sweep the same area; far away it moves slowly. Physically, it is conservation of angular momentum: \(r \times v_\perp\) stays the same, so at the two ends \(r_p v_p = r_a v_a\).

The orbit of Mars.

Mars Earth
Semi-major axis \(a\) 1.524 AU (227.9 million km) 1.000 AU (149.6 million km)
Eccentricity \(e\) 0.093 0.017
Perihelion 1.381 AU (206.6 million km) 0.983 AU
Aphelion 1.666 AU (249.3 million km) 1.017 AU
Speed at perihelion / aphelion 26.5 / 22.0 km/s 30.3 / 29.3 km/s
Sunlight, perihelion against aphelion 45% more 7% more

Mars's orbit is more than five times as stretched as Earth's. Its sunlight changes by 45% over the year, so its southern summer, which comes near perihelion, is warmer and shorter than its northern one, and the great dust storms usually start then.

Worked examples

1. Perihelion and aphelion. \(r_p = 1.524 \times (1 - 0.0934) = 1.382\) AU; \(r_a = 1.524 \times 1.0934 = 1.666\) AU.

2. Speeds from the second law. \(v_p / v_a = r_a / r_p = 1.666 / 1.381 = 1.206\). Measured: \(26.50/21.97 = 1.206\). ✓

3. Sunlight. Sunlight falls off as \(1/r^2\): \((1.666/1.381)^2 = 1.455\), about 45% more at perihelion.

Try it

  1. An orbit has \(a = 2\) AU and \(e = 0.5\). Find its perihelion and aphelion.
  2. A comet's perihelion is 0.6 AU and aphelion 35 AU. Find \(a\) and \(e\). (That is close to Halley's comet.)
  3. Using the second law, how much faster does that comet move at perihelion than at aphelion?
  4. Why are the Mars seasons not all the same length?
  5. Why did Kepler's discovery come from Mars and not from Venus, whose orbit he also studied?
Answers
  1. \(r_p = 2 \times 0.5 = 1\) AU, \(r_a = 2 \times 1.5 = 3\) AU.
  2. \(a = (0.6 + 35)/2 = 17.8\) AU; \(e = (35 - 0.6)/(35 + 0.6) = 0.966\).
  3. \(35 / 0.6 \approx 58\) times faster.
  4. Mars moves faster near perihelion, so the seasons around then pass more quickly.
  5. Venus's orbit is almost a circle (\(e = 0.007\)), so a circle fitted it well; Mars's larger \(e\) made the error visible in Tycho Brahe's data.

Common mistake

Putting the Sun at the centre of the ellipse. It is at a focus, off to one side. For Mars, the Sun is \(ae = 0.14\) AU, about 21 million km, from the centre.

Where it leads