What it shows
The square of a planet’s orbital period is proportional to the cube of its average distance from the Sun. Plotting real data for the six naked-eye planets gives a straight line of slope 1 when the units are years and astronomical units.
The science
Kepler found that for all planets T squared is proportional to a cubed, where T is the orbital period and a is half the longest diameter of the orbit. Newton later showed that this follows from gravity weakening as the inverse square of distance. In years and astronomical units, with one AU the Earth’s mean distance, the constant is 1, so T squared = a cubed. For Earth, T = 1 and a = 1. For Mars, a is 1.52 and T is 1.88 years, and 1.88 squared is 3.53 while 1.52 cubed is 3.51.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A table of planetary data from a textbook or a science website
- Graph paper, preferably log-log
- A calculator
- A ruler
- Coloured pens
- A spreadsheet if available
- A strip of paper to make a scale drawing
Safety
For this project: No special safety. Use only reliable data from a book or an observatory website and write the source.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Collect these values from a book: Mercury 0.387 AU, 0.241 years; Venus 0.723, 0.615; Earth 1.000, 1.000; Mars 1.524, 1.881; Jupiter 5.203, 11.86; Saturn 9.537, 29.46.
Make a table with columns a, T, a cubed and T squared and fill them using the calculator.
Plot T squared on the vertical axis against a cubed on the horizontal axis on graph paper. The points for Jupiter and Saturn need a large scale, so the inner planets may bunch near the origin.
Draw the best straight line through the points and the origin and find its slope.
Plot the same data on log-log paper or take logs of both columns. The slope of log T against log a should be 1.5.
Use the law to predict the period of a planet or asteroid at 2.77 AU, the dwarf planet Ceres, and compare with its known period of 4.6 years.
Make a scale drawing of the orbits with 1 AU = 2 cm on a long strip of paper and mark the planets.
Find the percentage difference of each point from the line and write why the match is so close.
What you should see
A straight line of slope close to 1 for T squared against a cubed, and a log-log slope of 1.5. The predicted period of Ceres comes out near 4.6 years.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1State Kepler’s third law.
The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit.
2Why is the constant equal to 1 in years and AU?
The Earth has T = 1 year and a = 1 AU, and the law holds for all planets around the Sun.
3What does the law tell us about the outer planets?
They are slower and take much longer to orbit.
4Who explained why the law holds?
Newton, by his law of gravitation.
Take it further
Use the Moon’s distance and period or the data of Jupiter’s four large moons to find the mass of the planet from the law.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
Free ebooks for this project
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