Project: Kepler's third law from a table

What you will be able to do

After this lesson you can

• carry out a whole analysis of a table, from reading to a report,
• clean a table and say what you did, and
• use a fitted law to predict a missing value.

The problem

You have a table of the planets: the distance from the Sun in AU and the period in years. There is also Pluto, a dwarf planet. One value is missing: the table does not say the period of Neptune. The questions are:

1. Does Kepler's third law, $T^2 / a^3 = 1$, hold for these bodies, and how closely?
2. What is the power of the law, found from the data itself?
3. What period does the law predict for Neptune, and how good is the prediction?

This is a small example of what a scientist does with data: read it, check it, clean it, calculate from it, find a law, make a prediction, and draw a picture, and say so clearly.

The program

Predict first. Before you run it, what ratio $T^2/a^3$ do you expect for Pluto (39.5 AU, 248 years)? $248^2 = 61\,504$ and $39.5^3 = 61\,630$.

1. Read and check. pd.read_csv(io.StringIO(text)) reads the table. The program prints the number of rows, 9, and the missing values in each column. The result shows exactly one missing value: the period of Neptune.

2. Clean. p.dropna() drops the row of Neptune, so 8 rows are left. We chose to drop it, and not to fill it with a guess, because we want to use Neptune as a test later. (If we had filled it, the prediction would not be a real test.) Note what you did: "one row with a missing period was removed".

3. Derive. p.assign(ratio=...) makes a new table with a new column, the ratio. (assign returns a new table and leaves the old one as it was.) The ratios are 0.971 for Mercury, 1.030 for Venus, 1.000 for Earth, 1.006 for Mars, 1.000 for Jupiter, 1.000 for Saturn, 0.999 for Uranus and 0.998 for Pluto.

4. Summarise. The mean of the ratio is 1.0 and the spread is 0.016. The law holds to about 2 percent. Look at which ones are farthest from 1: Mercury (0.971) and Venus (1.030). The table gives their numbers with only two digits, so a rounding error of a few percent is expected. The big planets, Jupiter to Pluto, agree to about a fifth of a percent.

5. Find the power. np.polyfit on the logarithms of the distances and the periods gives the slope 1.5: the power of the law is 3/2, found from the data alone. The data say $T = a^{1.5}$, which is the same as $T^2 = a^3$.

6. Predict. Use the fitted law, $T = 10^{\text{intercept}}\,a^{\text{slope}}$, with the distance of Neptune, 30.07 AU. The prediction is 165.0 years. The real period of Neptune is 164.8 years. The prediction is correct to 0.1 percent. This is a test of the law with a body that was left out of the fit, and the law passes it. That is the main reason why the law is believed.

7. Draw. The picture shows the eight bodies on a log-log graph, with the fitted line. The points lie on the line, from Mercury at the lower left to Pluto at the upper right.

Neptune's real period is about 165 years. A prediction for a body left out of the fit is the honest test of the law.

Writing the report

A good analysis ends with a short report. For this project:

The table has 9 bodies. One period (Neptune) was missing, and the row was removed for the fit. For the other 8, the ratio $T^2/a^3$ has the mean 1.00 and the spread 0.016, so Kepler's third law holds to about 2 percent. A line fitted to the logarithms gives the power 1.50. The law predicts 165.0 years for Neptune, and the true value is 164.8 years.

Notice what is in it: the size of the data, what was cleaned, the numbers with their spreads, and the test. A reader can trust a report like this, and can repeat it.

Common mistakes

• Cleaning the data without saying so. Report every row that you drop or fill.
• Using the test body in the fit. Then it is not a test.
• Reading a ratio of 0.97 as a failure. Look at the precision of the data.
• Forgetting the units. Years and AU make the ratio 1. Other units give another constant.
• Trusting the fit without the graph. Look at the points on the line.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Project: Kepler's third law. You write the functions for the ratio, the power and the prediction.
2. Build. Open the problem Predict the moons, a project with the same method for the moons of Jupiter.
3. Stretch. Open the problem A different law.

Remember these

• An analysis has steps: read, check, clean, derive, summarise, find a law, predict, draw, report.
• Say what you cleaned. Keep a test case out of the fit.
• A fit on a log-log graph gives the power of a law. The prediction for a body that was not used is the real test.
• $T^2 / a^3 = 1$ for the planets: the power is 1.5.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why did we drop Neptune before the fit and then use its distance for a prediction?

2. What does the slope 1.5 of the log-log line tell you?

3. A planet is at 4 AU. What period does Kepler's law give?

4. What should a report say about missing data?