Kepler’s Orrery

A working model of the solar system for teaching gravitation and orbital motion — positions computed from J2000 orbital elements by solving Kepler’s equation.

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Guided investigation — ages 16–18 physics

Three laws, measured rather than told

Work through the investigations in order using the orrery above. Each one asks you to predict first, then measure. Record your prediction before you touch the controls — a prediction written after the fact teaches nothing.

IEllipses

Where exactly is the Sun?

Mode: Ellipses  ·  15 min

Predict

The Earth’s orbital eccentricity is 0.0167. If you drew the Earth’s orbit accurately on this page, would you be able to see that it is not a circle? Justify your answer with a number before checking.

  1. Set e = 0.600. Switch on Focal radii and watch r1 and r2 as the planet moves. Each one changes constantly. State what their sum does.
  2. Change e and a freely. Deduce the relationship between the constant sum and the semi-major axis a.
  3. Read off c = ae, the distance from the centre of the ellipse to the Sun. Explain why e is best understood as a measure of off-centredness, not of squashedness.
  4. Press the Earth 0.017 preset and read b/a. Compare with your prediction. Then read c in AU and convert it to kilometres.
  5. Press Halley 0.967. Compare the perihelion and aphelion distances. Halley's comet spends most of its 75-year period in the outer part of that orbit — suggest why, then confirm your reasoning in Investigation II.
IIEqual areas

The wedges that give the game away

Mode: Equal Areas  ·  20 min

Predict

A comet at perihelion is 5× closer to the Sun than at aphelion. Predict the ratio of its speeds at those two points, and state the physical principle you used.

  1. Set e = 0.700 and Δt = 10% of T. Let four wedges build. Describe the shape of a wedge near perihelion against one near aphelion — short and fat, or long and thin?
  2. Read the measured areas. They are computed from the pixels on screen, not assumed. Comment on the spread between the largest and smallest.
  3. Now watch the angular momentum panel. r changes by a factor of nearly 6; v changes too. Explain why rv sin θ does not.
  4. Find the two points in the orbit where θ = 90°. Identify them and explain why they are the only two places where h is simply rv.
  5. Using h = rv sin θ = constant, derive the perihelion-to-aphelion speed ratio in terms of e. Check it against the vis–viva values in the Orrery panel for Mercury.
IIIHarmonies

One straight line, five decades wide

Mode: Harmonies  ·  25 min  ·  complete the table below

Predict

Sketch the gradient you expect for a graph of log(T /yr) against log(a /AU), and state what the intercept should be if T is measured in years and a in AU. Explain both.

  1. Complete the table. Enter a and T from the Orrery panel; the last three columns compute themselves so you can see the pattern form.
  2. State the value and unit of the constant T 2/a 3. Explain why choosing years and AU makes it come out at 1.
  3. In the Harmonies plot, determine the gradient of the line of best fit. Compare with 3/2 and comment on the agreement across five orders of magnitude in a.
  4. Set your own body somewhere between 0.15 and 80 AU and press Measure its period numerically. This integrates Newton’s law directly. Outline why agreement with the line is evidence that Kepler’s third law is a consequence of Newtonian gravitation, not an independent law.
  5. Halley’s comet has e = 0.967 — wildly non-circular. It still sits on the line. Explain what this tells you about which orbital parameter the period actually depends on.
Body a  / AU T  / yr a 3  / AU3 T 2  / yr2 T 2/a 3
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IVExtension

Beyond the syllabus

For students who finish early, or for extension discussion

Weighing Jupiter

Jupiter’s T 2/a 3 is 0.99904, not 1.00000. Using T 2 = 4π2a3/G(M+m), determine m/M from that residual. Compare with the accepted 1/1047.

The closing of the orbit

In The Force, set n = 2.10 and watch the ellipse precess. Mercury’s orbit precesses by 43″ per century more than Newton predicts. Estimate what deviation in n would mimic that — then read about why the answer was general relativity instead.

The knife-edge

In the Sandbox, try to launch a body with exactly ε = 0. Record your closest attempt. Discuss why a parabolic orbit is a mathematical idealisation that no real body occupies.

Why the speed limit at aphelion

Combine vis–viva with h = rv sin θ to show that a body cannot be at aphelion and moving fastest. Then explain, in energy terms, where the kinetic energy went.

Orbital elementsSemi-major axes, eccentricities, longitudes of perihelion and mean longitudes at J2000 follow the standard low-precision series for the major planets. Sidereal periods are observed values.
How positions are foundMean anomaly is advanced linearly in time, Kepler’s equation M = E − e sin E is solved by Newton–Raphson to 10−12, and the true anomaly follows. No small-eccentricity approximation is used, so Halley-class orbits are exact too.
What is not to scalePlanet discs are drawn on a logarithmic size scale so all eight are visible at once; only distances are true, and only in True distance mode. Orbits are projected into a common plane — real inclinations are ignored.
UnitsAU and years throughout, with GM⊙ = 4π2 AU3 yr−2. Speeds are converted at 1 AU yr−1 = 4.74057 km s−1.