Johannes Kepler, a german mathematician, devised some of the most important relationships we know of in astronomy: the laws of planetary motion. After analyzing observations of the motion of the planets in our solar system, he found that the planets following clear relationships and published these relationships in 1609.
These simulations let you see what Kepler did (with the benefit of some hindsight). Play around with them and test your own hypotheses about planetary motion!
Kepler's first law is a simple but powerful assertion: any orbiting body travels in an elliptical path. For things in our solar system that means that the planets travel in ellipses with the Sun at one of the foci. The planets in our solar system travel in near circular paths with the most eccentric[1] planet being Mercury.
Kepler's second law tells us about how orbiting bodies move along these ellipses. It states that for any equal area swept out by an orbiting body is traversed in the same time. Kepler drew this conclusion from noting that planets change speed as they orbit. As they get closer to the Sun they speed up and when they are far away, they slow down. This is especially clear to see with commets or other bodies that pass close to the sun.
Kepler's third law gives us a powerful relation that we can use to estimate distances and masses of orbiting bodies near and far away. It states that the time it takes for a body to complete an orbit squared is proportional to its average distance from the star cubed.
To be more specific: $$ T^2 \propto a^3 $$ Where \(T\) is the time it takes to complete one rotation (orbital period) and \(a\) is the average distance from the star (semi-major axis)
Combining this relationship with Newton's Law of Gravitation[2] allows us to estimate the masses of distant binary stars and oribitng exoplanets!
[1] Eccentricity is the ratio of the distance between the center and a focus to the semi-major axis. The higher this value, the more streched the orbit is.
[2] Newton actually showed that the realtionship between orbital period and semi-major axis contains a factor that is the sum of the two bodies' masses: $$ \frac{a^3}{T^2} = G\frac{M+m}{4\pi^2} $$ Where \(G\) is the gravitational constant, \(M\) and \(m\) are the masses of the bodies. So if you can measure semi-major axis and period, you can deduce the combined mass of the system! Read more here
If you're interested in learning more OpenStax has a great resource on this topic (with practice problems!). Check it out!
Fraknoi, A., Morrison, D. and Wolff, S. (2016). Astronomy. Houston, Texas Minneapolis: OpenStax College, Rice University,Open Textbook Library.
Weisstein, Eric W. "Eccentricity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Eccentricity.html
Williams, David R. (2019). "Planetary factsheet -- metric". Greenbelt, Maryland: NASA Goddard Space Flight Center. https://nssdc.gsfc.nasa.gov/planetary/factsheet/