From Epicycles to Ellipses: Kepler's Laws of Planetary Motion, Part 1 by Lanz Lagman


This is the story of how an apprentice meticulously gathered and analyzed data for decades to lead us closer to the truth on how the heavens moved.

His groundbreaking work helped lead to the Scientific Revolution, allowing us to break free from the confines of our home planet and reach the once unreachable heavens.

"I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; 
that it is insignificantly small, and is borne through the stars"
Johannes Kepler. Image credit: ESA
That apprentice was Johannes Kepler (1571-1630).

He established the laws of planetary motion after years of hard work analyzing the records of his superior, Tycho Brahe, then struggling to fit them into the available models of the Solar System. 

There are three laws: the first states that a planet moves in an elliptical orbit around a Sun, with the Sun located at one focus.

An elliptical orbit.
The second law states that the area covered by the line connecting a planet to the Sun sweeps equal areas in equal intervals of time.

This implies that the planet will move faster as it approaches the Sun, and will move slower when it is much farther from it. Planets with circular orbits should only have a constant orbital speed.

The area covered from p1 to p2 is equal to the area covered from p3 to p4.
The third law states that the cube of the mean distances is approximately equal to the square of the period of the planet's revolution. This means that the greater the distance of the planet from the Sun, the slower it revolves. 

In this article, we will discuss how the first law was established by Kepler based on the knowledge and technology available during his time. Afterward, we'll tackle how later scientific advancements, some long after his death, provided the grounds for justifying why orbits are elliptical. 

Renaissance Astronomy: A Background

Kepler lived during a time when three models described the motions of planets: the widely accepted but increasingly questioned Ptolemaic system, the obscure but more realistic Copernican system, and finally the forced combination of these two models, the Tychonic system.

From left to right: Ptolemaic, Copernican, and Tychonic System
These models tackle the position of the Sun, Mercury, Venus, Earth, Moon, Mars, Jupiter, Saturn, and the stars. The main conflict between these two models were: 

1. What's at the center of the universe, the Earth or the Sun?
2. What planet is orbited by the others? 

According to Ptolemy (100-168), the immovable Earth was at the apparent center of the universe, with the others surrounding and orbiting it. Nicolaus Copernicus (1473-1543) disagreed with Earth being the center, and replaced it with the Sun, the other planets revolving around it.

Tycho Brahe (1546-1601), the astronomer that Kepler assisted, saw the benefits of the Copernican system and combined it with the more philosophical and religiously aesthetic Ptolemaic system, proposing that the Sun orbits the Earth, and the other planets orbit around the Sun. 

Despite their fundamental differences, they were made for the same purpose: to predict how the planets moved throughout the sky. Each of these models were proposed and modified from data continuously accumulated from night sky observations that were conducted thousands of years before. 

Astronomers obtain these data by recording the positions of planets at a fixed time, every day (as much as possible). Remember that the Renaissance was an era where instead of cars, we had horse carriages; exclusive libraries instead of Google, and 'selfies' only being possible if you were a skilled male artist.

All of these models were made at the vantage point of the three astronomers, and they had to observe for decades while working on these, while using the data that the others have collected before. 

Astronomy was then fused with astrology, not physics. This means that the movement of stars and planets were only tracked, and their main concern was to make a model that recognized the pattern of planetary movements - not what made them move in that pattern.

Leonardo da Vinci's #selfie
Image credit: leonardodavinci.net
It's also important to mention how Ptolemy's work held so much influence; despite how the heliocentric model of Copernicus diverged from it, a lot of the Ptolemaic model's elements were still retained. These included the concept of nested celestial spheres, where the planets are embedded in rotating hollow spheres made of Aether.

The rotation of these celestial spheres was believed to cause the movement of the planets, causing them to move around in circles with unchanging speed. However, the stars are all embedded in the outermost celestial sphere. 

Contrary to popular belief, these models were actually similar in accuracy. The accuracy is dictated not by how they define what is at the center, and what orbits which, but how the models employ epicycles. 

Established Epicycles: What Are They For?

The Earth revolves around its axis as an example of uniform circular motion. The stars appear to stay in their place at all times, which explains why a picture taken of the night sky with long exposure (to reveal star trails), will look like this:

Star trails. Image credit: Jerry Lodriguss
The stars seem to revolve around the star at the axis of rotation, which is the North Star, Polaris. Since the models state that the planets move in a uniform circular motion, we would expect that tracing their paths would reveal curved streaks - expected of a segment of a circle - and when they are connected, will form a perfect circle.

But, what's this? 

Movement of Mars in 2003. Image credit: NASA
Soon, the notion of uniform circular motion and geocentrism started accumulating holes. Mercury and Venus are both unseen at midnight, and are only clearly visible when the Sun is either starting to rise or set.

Planets appear to speed up then slow down, and their brightness during night subtly varies over time. The illustration above is an example of retrograde motion. 

 Mars, Jupiter, and Saturn have been observed to suddenly move backward, then move forward again. Segments of perfect circles aren't supposed to look like that, right? Perhaps the motion is not circular, right? Nope. Enter the epicycle.

Ptolemy's epicycle.
Epicycles literally mean circles within circles. In this diagram of Ptolemy's model of an orbit, the red circle is called the deferent, and the blue circle is the epicycle.

The planet revolves around the center of the epicycle, then the center moves through the deferent. Now this is the confusing part, because these seemingly contradict the notion of circular motion. 

The center of the epicycle revolves around the center of the deferent called the eccentric, at a constant distance; that's why the deferent is circular. However, with respect to the point called equant, the center of the epicycle revolves at a constant rate. 

In this way, the planets seem to move faster when they're approaching our Earth, then slowing down when they're moving away.

This was Ptolemy's clever way of devising his epicycles not only to explain why the planets sometimes appear to speed up, but also why retrograde motion occurs. However, when the path of the planets are traced, they'll look like this:

Orbits of the Ptolemaic System. Image credit: James Ferguson
The battle between the Ptolemaic and Copernican models are interesting in how they used epicycles.

Ptolemy altered the size of the epicycles of Mercury and Venus so that his model would explain why they are invisible at night. Ptolemy's epicycles are more complicated, however, because of the equants. 

Copernicus sought a simpler model, leading to the heliocentric model. Mercury and Venus now revolved around the Sun, a simpler explanation for why they can't be seen at night. Smaller epicycles then replace the idea of equants.

The Copernican model might have simpler epicycles, but it also employs more epicycles than the Ptolemaic system. 

In conclusion, both systems have their own complexities, but are almost the same in terms of accuracy. Inevitably, these models proved to be inadequate over time upon comparison to thousands of years of compiled data; but everything changed when Kepler attacked. 

Enter Kepler


Tycho Brahe at Uraniborg.
 Image credit: Astronomiae instauratae mechanica

Johannes Kepler was employed in 1600 by Tycho Brahe as an assistant in his observatory in Castle Uraniborg, in Sweden. Of all the tasks assigned to him, the most important one came when Brahe unexpectedly contracted illness.

On his deathbed, he instructed his assistant to complete their work on the Rudolphine Tables, and use his formerly restricted, but accurate data to prove his Tychonic system.

Kepler was earlier assigned to analyze Brahe's observational data on Mars since there were a lot of errors discovered. Kepler was a known adherent of the Copernican system, and sought to analyze Brahe's data in order to prove and modify it.

He began his study of the data in 1601, and fortunately concentrated on studying Mars's orbit. Years passed, and he realized he had to bring back Ptolemy's equant and modify the Copernican model. This led him to come up with the idea of an ovoid-shaped orbit (resembling an egg, or a teardrop).

He then set out to calculate the ovoid orbit using trial-and-error. After many mistakes, he realized that perhaps, the orbit was elliptical instead!

After repeating his calculations with this new idea in mind and discovering that the orbit was indeed elliptical, Kepler finally declared that orbits in general are elliptical in shape.

An egg-shaped orbit versus an elliptical orbit.

Kepler's second law was actually discovered before the first. The result of his hard work was the publication of Astronomia Nova (New Astronomy) in 1609.

Additionally, geocentrism and the epicycles slowly died in Europe, partially due to Kepler's works, but also because of the discoveries made by Galileo Galilei through his telescopes of the phases of Venus and the moons of Jupiter.

What he observed of the phases of Venus could not be justified by any geocentric model or the modified epicycles, and the moons orbiting around Jupiter should be orbiting around the Earth.

The appearance of comets during the Renaissance and subsequent studies also revealed that the heavenly bodies have highly elliptical orbits, refuting the notion of uniform circular motion.

While Kepler could show that planetary orbits are ellipses, he couldn't exactly determine why - like learning from a friend that a new flavor of pizza tastes great, but neither of you could figure out why.

This would be proven decades later by none other than Sir Isaac Newton (1642-1727), who was so badass that he almost single-handedly invented his own math (calculus) and his own physics (Newtonian mechanics).

Now why would an orbit be an ellipse instead of a circle? 

Sir Isaac Newton.
Image credit: Royal Society Print Shop
It was Kepler's laws of planetary motion that served as one of Newton's inspirations in searching for a rational basis on why and how things move in general.

In his most famous work, the Principia, Newton established his well-known laws of motion, theory of gravitation, and the beginnings of every math-hater's worst nightmare, calculus.

Together, these three were used to justify Kepler's laws and in his section, we'll discuss how these concepts lead to the elliptical shape of planetary orbits.

The three could be summarized as follows. Newton's three laws of motion state the following in order: things will stay at rest unless they are moved (law of inertia); moving objects would have greater acceleration if more force is applied (law of acceleration); and for every action, there is an equal and opposite reaction (law of action and reaction).

The second law could be summarized as the force applied being equal to the mass multiplied by its acceleration:
The gravitational theory states that everything with mass gets attracted to each other by the gravitational force in an inverse-square distance relationship. In simpler terms, the nearer two objects are to one another, the stronger they get attracted to each other.

Objects with greater masses are more attractive, but that may not be a good way to describe how attractive your crush is. The equation for the gravitational force is expressed as:


In this equation, Fg is the gravitational force, G is the gravitational constant, M and m are bigger and smaller masses of the two objects respectively, and r is the distance between them.

It's now easier to understand the concept of inverse square law; since the divisor is the square of r, that means doubling the distance of these two objects would mean the gravitational force between them would be reduced to a quarter of its former value.

Combining the two equations, we produce an important mathematical expression: acceleration due to gravity.

For now, do not mind the weird markings, for they are the result of deeper applications of Newtonian mechanics that we will not discuss.

Calculus is the mechanical study of change. Newton and Gottfried Leibniz (1646 - 1716) both independently invented it. This new form of math is perfect for accurately describing how slopes become steep and flat at some sections, and in this case, how the variations in the speed of planetary orbits influence the shape of their orbits.

The full use of these principles leads to the following expression:


This dictates the distance of the orbiting body from its parent body (r) with respect to time (t). It turns out, if you draw a graph of the expression above, what you get is a special family of shapes called conic sections. Conic sections are the shapes you get when you slice a cone, hence the name.

There are several kinds of conic sections. The kind will depend on the numerical value of the letter ε. It's the eccentricity of the conic section and this tells us how much the orbit deviates from a circle!

 For a circle, ε = 0; for an ellipse, 1 > ε > 0 or in other words, an ellipse has an eccentricity greater than zero but less than one, while a parabola has ε = 1, and a hyperbola has ε > 1.

Possible shape of orbits as dictated by their eccentricities.

In fact, a circle is a special type of ellipse; this means that perfectly circular planetary orbits are impossible in nature. For example, the orbits of the planets Mercury, Venus, Earth and Mars are the following: 0.205, 0.007, 0.017, and 0.094.

Despite Mercury being the most eccentric, astronomers failed to notice this at all, because Mercury is less-studied. After all, it moves quickly and could not be seen at night when planetary movements are easily analyzed.

The Orbits of the Inner Planets.
Image credit: NASA's Eyes

Notice that all planetary orbits are closed; this means they are not parabolic or hyperbolic. This means a huge amount of force must be exerted upon them to have an open orbit, since as per the laws of motion, they are massive and harder to move because of more inertia.

Spacecraft and comets however, have significantly less mass and are therefore easier to accelerate, making it easier for them to achieve an open orbit.

We already knew what happened next: Newtonian mechanics dethroned the method of thinking that resulted in the geocentric models, and became the most authoritative tool for hundreds of years for scientists in explaining how everything moves and interacts with one another.

Soon, Uranus was discovered, a planet too dim for the eyes of our ancestors. Then followed Neptune, whose movements were predicted by Newtonian mechanics, and thus solidifying its authority.

Who would have thought that what we discovered about the motions of these planets would be used as a noble attempt to understand one of the most basic building blocks of everything?

The atoms, once considered the smallest unit of matter, were once thought to be miniature solar systems themselves: with the nucleus as the Sun and the electrons as planets.

We as humans must marvel and appreciate the courage, curiosity, diligence and passion in the search for truth displayed by Kepler, Newton and all other scientists that brought us the Scientific Revolution.

In just a several hundred years, we may now reach the heavens through our rockets and spacecraft. We may now finally fulfill what our ancestors thousands of years ago could only dream about and marvel for the rest of their lives.

For the advanced readers who can stomach some math, we will derive Kepler's first law using the mathematical expressions of Newton's principles.


For the Advanced Reader

In order to derive Kepler's first law using Newtonian dynamics, we have to use vectors, and it is expected that the readers of this section would be familiar with basic vector calculus. They are entities with a magnitude and direction.

In the case of planetary orbits, the central mass providing the central force is the Sun and the planet with a much smaller mass moves around it because of the Sun's gravitational force. We will study this system in three dimensions and therefore, we have to observe the system at a viewpoint perpendicular to the plane where the Sun and the planet's initial positions are located.

That viewpoint would be the z-axis. The mathematical expression using vectors would be:



In the following diagram and equations, vector L, the orbital angular momentum, is the cross product of the radius vector r and the linear momentum p. The masses of the Sun and the planet are denoted by M and m respectively.

We could think of p as something that causes the planet to go straight forward with respect to the Sun but r keeps it from going further away, just like a string attached to a yo-yo. The orbital angular momentum dictates how hard it would be for the planet to keep orbiting the Sun.

Since the system is moving, the position of mass m changes over time, and it leaves its  initial position. To describe this mathematically, we take the time derivative of equation (2).






Vectors that are parallel and antiparallel to each other cancel out. Therefore, equation (3) is reduced to:



This means that vector L is constant and the orbital angular momentum is conserved. Any force whose direction is toward the center or away from the center has a zero cross product with r. 

As long as the system is conserved and no moving stars or black holes go near the system, the movement of the planet is confined on the plane (gray area) perpendicular to L, as if it only glides through it and therefore does not jump up and down with respect to it.



We have to get back to equation (2) and expand it; we will see later how practical these steps would be. Let's expand the vectors r and v as products of their magnitudes and direction vectors. Since the time derivative of the position r is velocity v, we get:



Through product rule, equation (5) now expands to:



Let's leave first equation (7) as it is and introduce the child of Newton's second law and the law of gravity, the acceleration due to the gravity of the planet. Since the Sun would be the point of reference and we're looking at how the planet is attracted toward it, the vector form would be negative. This is expressed as:



We can clearly see that the acceleration toward the center is an inverse square. This means that the acceleration becomes greater as the planet approaches the Sun.

We will now get to the cross product of acceleration due to gravity and angular momentum. This may seem weird but again will pay off later because of vector identities.



Notice that at equation (11), the common factors are the time identities. Remember that the time derivative of vector L is 0, just as we've shown in equation (4).



By integrating both sides, we get:



This may seem a dead end, but the BAC-CAB rule saves the day for us. Thus, we transform the left-hand side of equation (14) to:



Notice that the left-hand side is the dot product of vector L and the cross product of vectors r and v. Remember that the dot product of the same vector is equivalent to its magnitude, by using equation (2), we could now transform equation (15) to:



This equation seems familiar already. Let's isolate r to the left side so that we would know the present distance of the planet from the Sun at a particular time.



Equation (17) looks very similar to some well-known mathematical equation. Maybe making the following substitutions would do the trick.



We now get the equation for a conic section, with the eccentricity dictating its type. This is a much more general statement for all orbiting bodies.

Spacecraft that orbit the Sun have so little mass compared to planets, and therefore orbital angular momenta. This means that spacecraft orbits are easier to alter, and why we don't see planets with parabolic and hyperbolic orbits.



It's just almost impossible to find a perfectly circular orbit, but also impossible for planets to perform epicycles.

REFERENCES: 

1. Caroll, B.W., & Ostlie, D.A. (2009). An introduction to modern astrophysics. San Francisco, Calif; Munich: Pearson/Addison Wesley.
2. Caspar, M. (1959). Kepler. London: Abelard-Schuman.
3. Riebeek, H. (2009, June 7). Planetary Motion: The History of an Idea That Launched the Scientific Revolution. Retrieved from http://earthobservatory.nasa.gov/Features/OrbitsHistory/
4. Johannes Kepler: His Life, His Laws and Times. (n.d.). Retrieved from http://earthobservatory.nasa.gov/Features/OrbitsHistory/
5. Williams, D.R. (n.d.). Planetary Fact Sheet - Metric. Retrieved from http://nssdc.gsfc.nasa.gov/planetary/factsheet

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