Public Universe · Student worksheet
Kepler’s law: fit, residuals and uncertainty
How convincingly do catalogue orbital elements support a period–size relationship?
Data and starting point
Open each of the eight planet pages. Record semi-major axis a in AU and orbital period P in days, the object identifier, source, epoch and retrieval date where supplied. API users can inspect orbital.semi_major_axis_au and orbital.orbital_period_days in object detail responses. Record missing metadata as missing.
Investigation
- Build an eight-row table. Keep the original values and units, and document any exclusions. Use P_year = P_days / 365.25 as an explicit Julian-year conversion.
- Plot x = log10(a / 1 AU) against y = log10(P / 1 year). Fit y = m x + b with x on the horizontal axis. Report the slope, intercept and residuals y − (m x + b).
- Compare m with 1.5 and calculate q = (P / 1 year)^2 / (a / 1 AU)^3 for each planet. Identify the largest residual without claiming it is statistically significant.
- Check how a and P were obtained. Discuss why quantities derived from a shared dynamical model may agree by construction. A small residual alone is not evidence of independent measurements.
- Extension: for positive, independent measurements with small symmetric uncertainties, derive (σq / q)^2 ≈ (2σP / P)^2 + (3σa / a)^2. Explain why covariance or absent uncertainties prevents applying this blindly.
Submit
Submit the source table, one labelled log plot, a residual plot, and a 200-word interpretation separating model assumptions, numerical agreement and measurement uncertainty.
Record your result
Slope m and intercept b, including units/conventions:
Largest residual and a possible explanation:
Missing uncertainty or dependence that limits the claim:
Public Universe · Teaching notes
Kepler’s law: fit, residuals and uncertainty
Learning outcomes
- Fit and interpret a power law using dimensionless ratios.
- Diagnose residuals and shared assumptions in derived data.
- State what uncertainty information is missing before drawing a conclusion.
Discussion and interpretation
A slope near 1.5 is expected for approximately Keplerian orbits around a common dominant mass. q is approximately constant in these chosen units; avoid requiring exact equality to one.
The two-body relation is P² = 4π²a³ / [G(M + m)]. Planet masses, perturbations, element definitions and epoch conventions matter when interpreting differences. Do not turn this catalogue exercise into an unqualified test of gravity.
A fit that treats x as exact is a teaching simplification. Without suitable errors and covariance, reward transparent limitations rather than a numerical confidence claim. Missing uncertainties must not become zero error bars.
Assessment criteria
- Correct units, transformations and labelled plots.
- A reproducible source table and stated fitting convention.
- An explanation of correlated or derived quantities and why residuals alone do not establish significance.
Primary sources and further reading
- JPL: approximate planetary positions and orbital elementshttps://ssd.jpl.nasa.gov/planets/approx_pos.html
- NASA: Kepler’s laws of planetary motionhttps://science.nasa.gov/resource/orbits-and-keplers-laws/