Cosmos Notebook

Library · Experiment

Weigh Jupiter

Track Jupiter's moons through binoculars over a few weeks, then use Kepler's third law to work out the mass of the largest planet.

Astrophysics
When
Mid Dec – Jan · Into space
Time
Several clear nights over 2–4 weeks
Cost
£0 with binoculars you own
Difficulty
3 of 3

Part of The Road Map (Into space) · Measuring the Solar System (Weigh a planet)

The question

Can I work out how heavy Jupiter is, using nothing but binoculars, a notebook and Kepler's laws?

Learn just enough first

  • Read about Kepler's third law: how a moon's orbit size and period depend on the planet's mass
  • Use Stellarium to find Jupiter and learn which moon is which

Kit

  • Binoculars (10x50 is ideal), steadied on a tripod, fence post or car roof
  • A notebook with a pre-drawn line for each night
  • Stellarium, to check identifications on the same night

My prediction

The accepted mass is about \(1.9 \times 10^{27}\) kg — over 300 Earths. How close will a hobbyist get?

Method

  1. On each clear night, sketch Jupiter as a dot and mark each moon's position. Estimate distances in Jupiter-widths.
  2. Note the date and time every time. Aim for as many nights as possible over 2–4 weeks — weekend nights are fine.
  3. Concentrate on Ganymede (orbits in about 7 days) or Callisto (about 17 days).
  4. Plot each moon's distance from Jupiter against time. It traces out a wave: the peak is the orbit radius \(a\), and the time for one full wave is the period \(T\).
  5. Convert Jupiter-widths to metres using Jupiter's diameter, about 143,000 km.

Tip: UK winters are cloudy. If gaps appear, use Stellarium to fill missing nights — note in the data which points are observed and which are simulated.

Data

DateTimeMoonSide (E/W)Distance (Jupiter-widths)

Result

Kepler's third law, rearranged for the planet's mass:

$$ M = \frac{4\pi^2 a^3}{G\,T^2} $$

with \(G = 6.674 \times 10^{-11}\) (SI units), \(a\) in metres and \(T\) in seconds. As a check, Ganymede's real orbit (\(a \approx 1.07 \times 10^{9}\) m, \(T \approx 7.15\) days) gives about \(1.9 \times 10^{27}\) kg.

Explain it to a friend

A heavier planet pulls its moons round faster. Measure how far out a moon orbits and how long it takes, and gravity tells you the planet's mass. It's the same method used to weigh the Sun, black holes and whole galaxies.

New questions

  • Do two different moons give the same mass?
  • How does an error in the orbit size affect the answer? (Hint: it's cubed.)