Moons of Jupiter: a laboratory for gravity

Curriculum links

Quoted word for word from the official documents.

Step by step

The caption is what the class reads on screen; the talking points are in your notes drawer (N) when you present. Steps marked Pupils only appear only in the pupil lesson.

1. 01

   Jupiter's four large moons

   Presented and for pupils

   Caption

   Jupiter's four large moons at true scale, seen from above. The clock runs at six hours a second: Io goes round in about 7 seconds, Callisto in about 67.

   Talking points

   • NASA's Jovian Satellite Fact Sheet: Io orbits at 421,800 km in 1.769138 days, Europa at 671,100 km in 3.551181 days, Ganymede at 1,070,400 km in 7.154553 days and Callisto at 1,882,700 km in 16.689017 days. JPL's mean elements give the same four distances.
   • In the viewer the moons keep their real periods, to at least two decimal places of a day, and each one starts at a random point of its orbit, so their places on the screen are not their real places on today's date.
   • At true scale the moons are too small to see from this distance, so the labels mark them.

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2. 02

   Kepler's third law round Jupiter

   Presented and for pupils

   Caption

   The moons obey T² ∝ a³ just as the planets do, with a different constant. Here it is set by Jupiter's mass.

   Talking points

   • T²/a³ from NASA's figures, in s² m⁻³: Io 3.113 × 10⁻¹⁶, Europa 3.115 × 10⁻¹⁶, Ganymede 3.116 × 10⁻¹⁶ and Callisto 3.116 × 10⁻¹⁶. The four agree to within 0.08 per cent.
   • Round the Sun the same ratio is 2.975 × 10⁻¹⁹ s² m⁻³ (Earth: (3.15581 × 10⁷ s)² ÷ (1.49598 × 10¹¹ m)³), about a thousand times smaller, because T²/a³ = 4π²/GM and the Sun is about a thousand times as massive as Jupiter.

   Ask the class

   Io orbits at a = 421,800 km in 1.769 days. Use T² ∝ a³ to predict Callisto's period, given that its a = 1,882,700 km. Answer in days.

   Answer16.68 days. Answers from 16.58 to 16.78 are marked right.

   WhyT_C = T_I × (a_C ÷ a_I)^(3/2) = 1.769 × (1,882,700 ÷ 421,800)^1.5 = 1.769 × 4.4635^1.5 = 1.769 × 9.430 = 16.68 days. NASA gives 16.689 days.

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3. 03

   Weighing Jupiter

   Presented and for pupils

   Caption

   Any one moon is enough to weigh Jupiter: M = 4π²a³/GT².

   Talking points

   • GM = 4π²a³/T² from each moon: Io 1.2680 × 10¹⁷, Europa 1.2675 × 10¹⁷, Ganymede 1.2671 × 10¹⁷ and Callisto 1.2671 × 10¹⁷ m³ s⁻². NASA's Jupiter Fact Sheet gives GM = 126.687 × 10⁶ km³ s⁻², which is 1.26687 × 10¹⁷ m³ s⁻²: all four agree with it to within 0.1 per cent.
   • NASA gives Jupiter's mass as 1,898.13 × 10²⁴ kg, 1.898 × 10²⁷ kg, which is 317.83 times Earth's.
   • CODATA 2022, from NIST: G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻².

   Ask the class

   Io has a = 4.218 × 10⁸ m and T = 1.769 days. With G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻², calculate Jupiter's mass in units of 10²⁷ kg.

   Answer1.9 × 10²⁷ kg. Answers from 1.89 to 1.91 are marked right.

   WhyT = 1.769 × 86,400 = 1.528 × 10⁵ s. M = 4π²a³/GT² = 39.48 × 7.504 × 10²⁵ ÷ (6.674 × 10⁻¹¹ × 2.336 × 10¹⁰) = 1.90 × 10²⁷ kg. NASA gives 1.898 × 10²⁷ kg.

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4. 04

   The Sun against Jupiter

   Presented and for pupils

   Caption

   a³/T² is proportional to the central mass, so comparing Io's orbit with Earth's compares Jupiter with the Sun.

   Talking points

   • For Earth round the Sun, a³/T² = (1.49598 × 10¹¹)³ ÷ (3.15581 × 10⁷)² = 3.3617 × 10¹⁸ m³ s⁻². For Io round Jupiter, (4.218 × 10⁸)³ ÷ (1.52854 × 10⁵)² = 3.2120 × 10¹⁵ m³ s⁻².
   • NASA's masses give 1,988,400 ÷ 1,898.13 = 1,047.6. JPL's GM for the Sun against its GM for the Jupiter system, moons included, gives 1,047.3.

   Ask the class

   Using a³/T² = 3.3617 × 10¹⁸ m³ s⁻² for Earth and 3.2120 × 10¹⁵ m³ s⁻² for Io, how many times as massive as Jupiter is the Sun?

   Answer1,047 times. Answers from 1,041 to 1,053 are marked right.

   Whya³/T² = GM/4π², so the ratio of the two is the ratio of the masses: 3.3617 × 10¹⁸ ÷ 3.2120 × 10¹⁵ = 1,047 (1,046.6 to five figures). NASA's masses give 1,047.6.

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5. 05

   The 1:2:4 resonance

   Presented and for pupils

   Caption

   Io, Europa and Ganymede are in resonance: each time Ganymede goes round once, Europa goes round twice and Io four times.

   Talking points

   • NASA: Io, Europa and Ganymede "are in what is called a resonance", and "every time Ganymede orbits Jupiter once, Europa orbits twice, and Io orbits four times."
   • From NASA's periods: Europa ÷ Io = 2.007 and Ganymede ÷ Europa = 2.015, so Ganymede ÷ Io = 4.044.
   • NASA: because the three line up with each other at the same points in their orbits over and over, the small tugs they give each other keep their orbits from becoming circular. The viewer starts the moons at random points, so it shows the periods of the resonance and leaves out where the moons line up.

   Ask the class

   Using NASA's periods, 1.769138 days for Io and 7.154553 days for Ganymede, how many times longer is Ganymede's period than Io's? Give two decimal places.

   Answer4.04. Answers from 4.03 to 4.05 are marked right.

   Why7.154553 ÷ 1.769138 = 4.044, close to 4. Rounded to two decimal places, 7.15 ÷ 1.77 = 4.04 as well.

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6. 06

   Count the laps

   Pupils only

   Caption

   Your turn. Count how many times Io goes round while Ganymede goes round once.

   Talking points

   • At six hours a second Ganymede takes about 29 seconds a lap and Io about 7.
   • Io makes 4.04 laps in one of Ganymede's, so a careful count gives 4, with Io just past where it started.

   Pupil question

   How many times did Io go round while Ganymede went round once?

   Answer4 times. Answers from 3.9 to 4.1 are marked right.

   WhyFour. NASA: every time Ganymede orbits Jupiter once, Io orbits four times. The periods give 4.04.

   Pupil task

   Pause when Ganymede is at a point that is easy to remember, such as straight to the right of Jupiter on your screen, and note where Io is. Play, count Io's laps, and pause when Ganymede is back at its starting point.

   HintIf Io is too quick to count, press Slower: at one hour a second Ganymede takes about three minutes a lap. Type your count in the box above.

   The viewer checks the task as the pupil works and says when it is done.

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7. 07

   What the orbits cannot tell us

   Presented and for pupils

   Caption

   Callisto takes 2.33 times as long as Ganymede, so it is outside the 1:2:4 pattern. All four moons give the same mass for Jupiter, and none of them gives its own.

   Talking points

   • 16.689017 ÷ 7.154553 = 2.333.
   • In GMm/r² = mv²/r the moon's mass m is on both sides and cancels, so the orbit depends on Jupiter's mass alone.
   • NASA's Jovian Satellite Fact Sheet gives the moons' masses as 893.2 (Io), 480.0 (Europa), 1,481.9 (Ganymede) and 1,075.9 (Callisto) × 10²⁰ kg. Ganymede, the most massive, has 0.008 per cent of Jupiter's mass.

   Ask the class

   In one or two sentences: why do the moons' orbits give Jupiter's mass and say nothing about the moons' own masses?

   AnswerWritten in the pupil’s own words, up to 400 characters. It is not marked.

   Sources 1 3Present from this step →

Sources

Every fact in this lesson comes from these sources, and each step lists the ones it uses.

1. Jovian Satellite Fact Sheet, NASA NSSDCA
2. Planetary Satellite Mean Elements, JPL Solar System Dynamics
3. Jupiter Fact Sheet, NASA NSSDCA
4. Europa Facts, NASA Science
5. Earth Fact Sheet, NASA NSSDCA
6. Sun Fact Sheet, NASA NSSDCA
7. Astrodynamic Parameters, JPL Solar System Dynamics
8. Newtonian constant of gravitation, CODATA 2022, NIST