Discovering Kepler’s Third Law from Planetary Data
City Tech 20th Annual Poster Session
Boyan Kostadinov, Satyanand Singh
Mathematics Department
11/17/2022
Introduction
In this data-inspired project, we illustrate how Kepler’s Third Law of Planetary Motion can be discovered using regression modeling to fit a power model to real planetary data from NASA’s Lunar and Planetary Science Division.
NASA’s data consists of the average distances \(D\) of the planets from the Sun (in \(10^6\) km), and the orbital periods \(T\) of all planets (in Earth’s days).
We plot NASA’s planetary data on a log-log scale, which shows that all planets appear to be positioned along a straight line. This is strong evidence that the original data must satisfy a power law.
Orbital Periods and Distances on a Log-Log Scale
Kepler’s Third Law
Thus, we consider a power model for the planetary data:
\[T=\alpha D^{\beta}\]
We linearize the power model, and apply linear least squares to fit the model parameters to the planetary data. This can be done analytically, and the result is:
\[T = \alpha D^{\beta}, \quad \beta=r_{xy}\frac{s_y}{s_x}\approx 1.4988, \quad \alpha =e^{\bar{y} - \beta \bar{x}} \approx 0.2007.\]
- \(\bar{x}\) is the sample mean of the vector of the logarithms of the distances.
- \(\bar{y}\) is the sample mean of the vector of the logarithms of the periods.
- \(s_{x}, s_{y}\) are the sample standard deviations.
- \(r_{xy}\) is the sample correlation.
The Fitted Power Model
Conclusion
Fitting mathematical models to data is the backbone of modern machine learning. We hope that this project can help our students build mathematical intuition and computing skills in their effort to rediscover Kepler’s Third Law from planetary data.
In some ways, Kepler can be considered as one of the first very hard working "data scientists" since it took him about 10 years to manually fit Brahe’s Mars data to the power model, and discover Kepler’s Third Law.
