Printable worksheet · Ages 16 to 18 · about 40 minutes
Kepler's third law with real satellites
A one-page worksheet: five real satellites from the live directory, a graph of T squared against r cubed, and the mass of the Earth from the gradient.
Numbers
Today's numbers come from the Satellite directory, captured when this page loaded: 18:37 UTC on 3 November 2026. Print the class set in one go so every copy matches the teacher notes. Reload for new numbers.The boxes are left blank. Pupils open the Satellite directory and copy in the numbers themselves, so answers vary a little: the teacher notes give the method and the range to accept.Live numbers are not available right now, so the boxes are blank: pupils look them up in the Satellite directory.
Orbital Radar · Space for Schools
Ages 16 to 18 · Gravitational fields
Kepler's third law with real satellites
For a circular orbit, T2 is proportional to r3, where r is measured from the centre of the Earth. Use five real satellites, with today's numbers from the Satellite directory, to find the mass of the Earth.
Name Class Date
Today's numbers from the Satellite directory, 18:37 UTC on 3 November 2026
Satellite
Height h / km
Period T / min
r / 106 m
r3 / 1020 m3
T / 103 s
T2 / 106 s2
ISS
421
93.0
NOAA 20
834
101.4
NAVSTAR 77 (GPS)
20,080
718.0
GALILEO 15
23,223
844.7
ASTRA 2E (TV)
35,784
1,436.1
In the Satellite directory, find each one by its number (25544, 43013, 43873, 41859, 39285) and copy its height and period.
Radius of the Earth = 6.371 × 106 m · G = 6.67 × 10−11 N m2 kg−2
Complete the table. r = h + the radius of the Earth.[4]
Plot T2 against r3 and draw the line of best fit. What does the shape of your graph tell you?[3]
Gravity provides the centripetal force. Show that T2 = (4π2 / GM) r3.[3]
Find the gradient of your line, and use it to work out the mass of the Earth, M.[3]
The two low orbits bunch up near the origin. Which graph would spread the points out, and what gradient would you expect?[2]
Extension The Moon orbits at r = 3.84 × 108 m with T = 27.3 days. Does it fit your line? Suggest why it is very slightly off. [2]
Show your working on the back of this sheet.
Teacher notes and answersNot on the pupil sheet. Printed only with "Print with answers".
Teacher notes: Kepler's third law with real satellites
Worked from the numbers on today's sheet (18:37 UTC on 3 November 2026).
Pupils looked up their own numbers, so answers vary a little: check the method and accept the ranges below.
Total 15 points, plus 2 for the extension.
1. The completed table
Satellite
r / 106 m
r3 / 1020 m3
T / 103 s
T2 / 106 s2
ISS
6.792
3.1
5.58
31.1
NOAA 20
7.205
3.7
6.08
37.0
NAVSTAR 77 (GPS)
26.451
185.1
43.08
1,855.9
GALILEO 15
29.594
259.2
50.68
2,568.7
ASTRA 2E (TV)
42.155
749.1
86.17
7,424.6
2 to 5
A straight line through the origin, so T2 is proportional to r3: Kepler's third law.
GMm / r2 = mω2r with ω = 2π / T, so T2 = (4π2 / GM) r3.
The gradient is 4π2 / GM, about 9.9 × 10−14 s2 m−3 (today: 9.92 in the table's units, so 9.92 × 10−14). M = 4π2 / (G × gradient) = 5.96 × 1024 kg. Accept 5.8 to 6.2 × 1024 kg.
lg T against lg r: a straight line of gradient 1.5, because T is proportional to r3/2.
Extension: T = 2.36 × 106 s, so T2 / r3 = 9.81 × 10−14 s2 m−3, about 1% below the satellites. The Moon's own mass (about 1.2% of the Earth's) adds to the pull: T2 = 4π2r3 / G(M + m).
Where the numbers come from
The directory works out each satellite's height from its measured period, using this same law. So the five points sit on the line by design: the sheet practises the method and gives back the mass of the Earth, but it does not test the law.
The Moon is the real test: its distance and its period are both measured directly.
The directory lists each satellite's lowest height. These five orbits are almost circular, so that is within a few kilometres of their average height.
Curriculum
England: A level Physics, gravitational fields and circular motion (Kepler's third law comes from exam board specifications: check yours). Wales: WJEC A level Unit 4, Orbits and the wider universe. Northern Ireland: CCEA A2 2, 5.2 Gravitational fields. Scotland: Advanced Higher Physics, Gravitation.
How these worksheets work
One page, free to print and copy for teaching. The numbers on the sheet are captured from the live tool when the page loads, and printed with the time, so a class set printed together matches the teacher notes. Reload for new numbers.
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Orbital Radar · Space for Schools