This is a series of the three men that have made physics and astronomy what they are today.

Since the days of Aristotle, it was assumed through rigorous philosophical reasoning that the planets moved in perfect concentric circles. The universe surely could exist in no other fashion! The geometric idea of a circle gave philosophers a rigorous explanation for the perfection of our universe.

We see circles everywhere, from the rings on a tree, ripples in a pond, and even soap bubbles (following from the laws of least action). In the words of the great Steven Strogatz, “As we gaze at [circles], they gaze back at us, literally, There they are in the eyes of our loved ones, in the circular outlines of their pupils and irises… Their eternal return suggests the cycle of the seasons, reincarnation, eternal life, and never-ending love. No wonder circles have commanded attention for as long as humanity has studied shapes”. It allowed a path so precise that no point was too close or too far from the origin. The orbiting body would return to the same point in space ad infinitum, infinite symmetries, and an explanation for why the sun sets every night and rises the following morning.

Johannes Kepler was born in what is now the south German state of Baden Württemberg. He was captivated by astronomy from a young age, observing both the Great Comet in 1577 and a lunar eclipse. These events were impactful on young Kepler, and he began to research why these events took place. Unlike Galileo’s direct opposition of the church, the sheer wonder of these events led him to believe in the mystical and philosophical perfection of the Aristotelian circle. His dreams of observing the universe and making calculations were soon lost, as his vision was too poor to do the work required of an astronomer. He did however show exceptional promise in mathematics, so he was recommended for a teaching position in Graz and took it graciously. While a professor there, he developed his theories to describe the heavens that so fascinated him as a young man.

While researching at Graz, he happened upon the problem of coincidence of Jupiter, Mars and Saturn in the same spot in the sky every 800 years, while Saturn and Jupiter coincided every 20 years. Attracted by the theological aspect that could provide answers about the Star of Bethlehem, he set forth to find explanations that resonated with the newly developed heliocentric theories of the Solar System. There were complex methods at the time to explain the abnormalities in orbits of the planets, now we know that there are a plethora of reasons for these perturbations, and we still have yet to find an analytical expression to take all the universe’s perturbing forces into account. Kepler turned to a method inspired by Platonic solids, these are shapes that have simple structures but were thought to be fundamental to the geometry of our universe.

He was taken under the study of Tycho Brahe, a wealthy astronomer who saw the mathematical talent in Kepler. Brahe and Kepler had a tenuous relationship from the start, having a dispute where Kepler ended up walking out and not returning until Brahe settled a stipend and living quarters.

Brahe had been a keen observer of the heavens, however not as great of a mathematician, he had collected a vast wealth of data for Kepler to sift through. His ultimate goal was to determine why Mars appeared to move backwards in the sky each night sometimes, what we call now retrograde motion and know that it happens as the Earth passes the slower-moving Mars in their respective orbits about the Sun.

Brahe was jealous of Kepler’s mathematical talent, his patience and persistent. Brahe believed in a geocentric model of the solar system, and actually withheld the bulk of his observational data so that Kepler could not find the actual truth, as Brahe was sure that he would.

Trying to make the circular and geocentric orbits that Brahe believed in work, Kepler eventually gave up. He used numerical methods of approximation and curve fitting, however there was always an error that was too high for Kepler to be satisfied with. He began studying a particularly challenging work by an old Greek polymath Apollonius of Perga. The work itself explained fairly simple concepts that are well known to mathematicians today, and taught to first year calculus students, however it was incredibly convoluted, and very dense reading.

While working through the dense book and trying to fit many shapes to the orbit of Mars, he noticed that an imaginary line drawn from a planet to the Sun swept out an equal area of space in equal times, regardless of where the planet was in its orbit. He came to realize this was a result of angular momentum being a conserved quantity for a central force. This central force, according to Copernicus and Kepler’s own theory, was the Sun, it was not until Newton came along that we understood exactly why this was the case.

If you draw a triangle out from the Sun to a planet’s position at one point in time and its position at a fixed time later—say, 5 hours, or 2 days—the area of that triangle is always the same, anywhere in the orbit. For all these triangles to have the same area, the planet must move more quickly when it is near the Sun, but more slowly when it is farthest from the Sun. Mathematically, we can describe the forces that act on the planet to show that gravity accelerates the planet towards the sun after perihelion, until gravitational potential is 0. Then decelerates while it’s momentum carries it away from the gravitational source until it is 0. Because angular momentum, and gravity are conserved forces, the total energy of the system says the same!

= \(\frac{\mathrm{dA} }{\mathrm{dt}}=\frac{\mathrm{r^2} }{\mathrm{2}} \frac{\mathrm{d\theta} }{\mathrm{dt}}\)

The conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater’s arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, and the precession of gyroscopes. We can define a simple law for this conservation, by saying that the external torque on the object is 0, which we can prove quite simply.

Angular momentum is conserved because around a point C, centered at the sun, gravity is a constant force, in the perfectly opposite direction of the position vector. Our torque vector, \(\tau\), can be described as \(\tau =rf \sin(\theta )\). Since the angle is by definition perfectly opposite to the position vector, it is always 180, and the term \(\sin(\theta) =0\), we see now that we have no torque from the central force. Now that we have this fact suitably defined, we can move on to why the swept angle is a constant!

Given an infinitesimal triangle with angle \(\theta\), \(A\) representing the area, and \(r\) the infinitesimally small base. \(A = \frac{1}{2}r(r\theta)\), imagine that this distance \(r\) is covered in some small time \(t\), which then gives us the equation for the change in area with respect to time as \(\frac{dA}{dt}=\frac{1}{2}r(r \frac{d{\theta}}{dt})\). Simplifying, we arrive at \(\frac{dA}{dt}=\frac{1}{2}rv\), (using the fact that the term \(r\frac{d\theta}{dt}\) denotes the velocity). Because \(\vec r \times \vec v\) evaluates to \(0\), our total torque goes to 0. Since we have this definition of angular momentum, we can apply this to our previous equation to get \(\frac{dA}{dt}=\frac{1}{2}\frac{L}{m}\). This is now equal to a constant!

http://adsabs.harvard.edu/full/1937JRASC..31..417B