The Laws of Planetary Motion
At around the same time that Galileo was observing the heavens with his newly designed telescope and conducting experiments with falling bodies (that would lay the foundations for Classical Mechanics and Gravitation) in the early 1600’s, two other scientists were advancing science with their own contributions.
These two were:
- Tycho Brahe — an observational astronomer, known both for his progressive (heretical) ideas and for running astronomical observatories which documented the positions of the celestial bodies with unrivalled precision. These records were instrumental in future mathematics relating to the heavens. Along with…
- Johannes Kepler — an immensely skilled mathematician, who applied his talents to the motions of the planets. His skill allowed him to solve problems that had existed for thousands of years, relating to planetary motion, understanding it truly, for the first time.
Together, the work of Brahe and Kepler worked in concert with one another to place the theories of the Copernican Model on an adamantine mathematical and observational foundation. This paved the way for Isaac Newton in the 1700s.
Contents
- The Tycho Brahe Observatory
- A Young Johannes Kepler
- The First Two Laws of Planetary Motion
- Kepler’s Three Laws of Planetary Motion
- Planetary Motion Problems
The Tycho Brahe Observatory
Three years after the publication of De Revolutionibus by Copernicus, Tycho Brahe was born. His family was of Danish nobility, and from an early age, he showed a penchant for the science of astronomy.
As a young man he made significant astronomical observations. One was a careful study of a supernova — the violent explosion of an old, massive star at the end of its life — which flared up in the night skies during Brahe’s early life during the 17th century.
His reputation grew, gaining the attention of King Frederick II of the Danes, who became his patron. [1] By the time Tycho Brahe was 30 years old, he had established his famous observatory on the island of Hven, in the North Sea. [1] Brahe was the last and greatest from the era of observers in Europe before the widespread use of the telescope, mapping the stars with naught but their keen eyes.
Tycho Brahe (and his team) took continuous measurements of the positions of the Sun, Moon, and planets for almost 20 years at Hven. The thorough, precise observations enabled him to note small variations of the positions of the planets from published tables (based on the work of Ptolemy). [1]
The value of these astronomical records for the development of astronomy, since precise, current, positional data is what was required for mathematical analysis. This (in hindsight) was what enabled the Laws of Planetary Motion to be discovered for the first time in our history.
(In the image above, left, Tycho Brahe can be seen in this stylized engraving using his instruments to measure the altitudes of celestial objects. That is what this large curved device was used for, measuring angles in the sky with precision. Note the grandeur of the observatory depicted here.[1])
However, Brahe was not a mathematical astronomer — certainly not a mathematician — and did not possess the skills necessary to perform this analysis himself. Thus he could not develop a model superior to Ptolemy or Copernicus himself.
He was further hampered by his own nature. He was an “extravagant and cantankerous fellow” with a skill for accumulating enemies among government officials, whom he rubbed the wrong way. When Frederick II died, the Danish King who was his Patron and supporter, he no longer enjoyed the political support he once did. So, he decided to leave Denmark for Prague.
In Prague he became the court astronomer for Emperor Rudolf of Bohemia. It was here that Brahe discovered a skilled mathematician, a young German man named Johannes Kepler — a man with the exact set of skills he needed to help him to analyze the treasure trove of planetary data he had accumulated, to develop a new model for the solar system.
A Young Johannes Kepler
Johannes Kepler was born into a family in the German province of Wurttemberg. Unlike Brahe’s upbringing (a son of a noble family) Keplers’ family was poor, and he lived much of his life amid the tumultuous Thirty Years’ War. [1]
He first studied theology at the University at Tubingen (like Copernicus and Galileo, at their respective universities). He developed his skills in mathematics here, and was both exposed — and converted — to the principles of the Copernican system and the heliocentric hypothesis.
Later on, Kepler went to Prague to serve as Brahe’s assistant. Brahe recognized the potential of Kepler to possibly be able to devise a new model of planetary motions with his mathematical skills, achieving the long-sought goal of scientists to correct the inaccuracies in the model Ptolemy had devised more than a thousand years before.
Tycho set this task for Kepler. However, he guarded his trove of astronomical knowledge like a jealous dragon, only reluctantly doling out small quantities of material at a time. He was afraid that Kepler would discover the secrets of universal motion without him, taking all of the glory for himself. [1]
When Brahe died in 1601, Kepler finally obtained unlimited access, getting full possession of these priceless records. [1]
Analysing this extensive astronomical data, Kepler developed principles which changed astronomy (and science) forever: now known as Kepler’s Three Laws. These laws described precisely the behaviour of planets, and their motion through space, based entirely on their observed (apparent) 2D motion through the fixed stars. The first two laws of planetary motion were published in 1609 in The New Astronomy — around the same year that Galileo gave us a new perception of the heaven’s by developing his powerful telescope.
The First Two Laws of Planetary Motion
The path of a celestial object through space is called its orbit. Like all of those before him going back to the Greeks, Kepler initially assumed the orbit of the planets are circular. However, again, like the results of all those before him also indicated, circular orbits clashed with observation. (This discrepancy would have been especially apparent with the extremely precise, detailed, and extensive data accumulated by Brahe.)
Focusing on the orbit of Mars (an interesting choice since it is the next closest planet to Earth, along with Venus, but with a slightly longer orbital period than Earth) pouring over the data associated with Mars, Kepler discovered that it had the orbit of a slightly flattened circle: an ellipse.
Next to the circle, an ellipse is the simples kind of closed curve, from the family of curves known as conic sections.
In a circle, the center is a special point, called the focus of a circle. A circle is unique because the distance from the focus to any point on the circumference of the circle is the exact same (which is what makes the circle unique). This distance is called the radius.
In an ellipse, there is not one significant point within its perimeter, but two. These points are called foci (singular: focus) — a word invented by Kepler for this purpose. In an ellipse, the sum of the distance from the two foci is always the same.
This property of ellipses suggest an interesting way to draw ellipses. If we create a loop out of string (tying the two ends together) and place the loop around two tacks pushed through a piece of paper, if we take our pencil and pull on the string until the string is taut (around both tacks and the pencil) as we draw our ellipse with the pencil, the string guides it into a curve — this special curve called an ellipse, but commonly known as an oval — based on two factors: the circumference of the string loop, and the distance of the tacks from one another.
The tacks act as the foci of this ellipse. At any point where the pencil-tip may be, the sum of the distance from the pencil to each tack will always be the same. (Measure the distance from each tack to a point on the perimeter of the ellipse, add them together, and that sum will always be the same.)
This gives us two features of an ellipse: the semi-major axis and the major axis. The widest diameter of an ellipse is called its major axis. Half of this distance — the distance of the major axis, but from the perimeter on one end to the center of the ellipse between the foci — is called the semimajor axis.
The semimajor axis is usually used to specify the size of an ellipse (just like the radius is used to note the size of a circle, and the semimajor axis is, in essence, the largest radius of an ellipse). For example, the semimajor axis of the orbit of Mars is 228 million kilometers, which is also the average distance between Mars and the Sun.
The shape (roundness) of an ellipse, how close it is to a perfect circle, depends on how close the foci are to one another, compared with the major axis. The distance between the foci is one value, which is a fraction of the major axis. The ratio of the distance between the foci, to the length of the major axis, is called the eccentricity.
If the two foci are in the same location, there is no distance between them, which gives them an eccentricity equal to zero. This creates a circle, at which point the semimajor axis becomes the radius (and the major axis, the diameter). This is one extreme of an ellipse: a circle.
On the other extreme, we can extend the foci further and further from one another. This creates an increasingly greater eccentricity, and a more elongated ellipse, up to a maximum eccentricity of 1.0. This is when the ellipse becomes “flat” forming what is essentially a line. This is the opposite extreme of an ellipse from the circle.
We can completely specify the size and shape of an ellipse with two values: the semimajor axis and the eccentricity. This tells us everything we need to know about an ellipse. Kepler, through his studies of Brahe’s astronomical data, discovered that Mars has an elliptical orbit. The Sun is at one foci, and the other is empty, with an eccentricity of about 0.1.
To give you a frame of reference for this eccentricity, if we drew the orbit of Mars to scale, you would barely be able to differentiate the ellipse from a circle. However, while “practically indistinguishable”, this minute differentiation had profound implications for our understanding of planetary motions.
The First Law Of Planetary Motion
In his fist law of planetary motion, Kepler applied this discovery about the orbit of Mars to all the planets:
- The First Law of Planetary Motion — the orbits of all planets are ellipses.
“This was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.” [1]
Kepler’s Second Law
The first law defines the shape of a planetary orbit. The second law, deals with the speed the planet travels over the course of its elliptical orbit. This is known as orbital speed.
As a planet orbits the Sun, it has a point its orbit where it is at its closest position to the star, and a point in its orbit where it is furthest way. Working with the data for Mars, Kepler discovered that the planet speeds up when it is closer to Mars, and slows down when it is further away.
Kepler described this phenomenon by imagining Mars and the Sun as being connected by a straight, elastic cord. When Mars was at its furthest point from the Sun, the cord was stretched a great deal, and the planet moves slowly. When Mars was closer to the Sun, the elastic line was not stretched significantly, and the planet moves more rapidly.
[Hint: don’t concern yourself with trying to understand why the elastic cord being stretched or slack would effect the motion of the planet (as I did) because that is not the point at all!]
The significance of the elastic line is that it sweeps out areas within the ellipse of the planets orbit. Specifically, Kepler found that given equal intervals of time (t), the areas swept out by this imaginary line are also equal to one another — even when they might look totally different in shape, like they do below.
The significance of this is the relationship between orbital speed and distance travelled in a unit of time, which varies as the planet orbits the star. The area A and area B are equal to one another, as are the time intervals $t_1$ to $t_2$, and $t_3$ to $t_4$. The planets cover more distance during an equal interval of time (when they are closer relative to when they are father from the star) because they are moving faster. Yet the area remains the same.
If the orbit of a planet was circular, the planet would be moving at a constant speed — and the elastic line would be stretched the same amount consistently. What Kepler discovered, is that the speed of planets orbiting their star (Moon’s orbiting their planet, or asteroids orbiting whatever they are orbiting) changes because the orbit is elliptical.
Kepler’s Third Law
Kepler’s first two laws describe the elliptical shape of planetary orbits, as well as the relationship between the ellipse and orbital speed (allowing us to calculate the speed of motion at any point on the ellipse).
While Kepler was pleased with these discoveries, he was not content, since his quest to understand planetary motions completely was not yet finished. [1]
He desired to understand a deeper question — he called it the “harmony of spheres” — which is an explanation of why orbits were nested as they were. (A question which he did not solve, and which has not been solved to this day.) For years he sought a mathematical relationship for the spacing of the planets, and in so doing, made another tangential discovery relating the time it takes a planet to go around the Sun with their distance from our local star.
In 1619 Kepler discovered a basic relationship between the orbits of the planets and their relative distance from the Sun. A planet’s orbital period (P) is the time it takes a planet to travel once around the Sun. We also define the semimajor axis of a planet as $a$, which is equal to its average distance from the Sun.
The relationship that Kepler discovered, which we call Kepler’s Third Law, relates the orbital period to the semimajor axis. Specifically, the orbital period squared is proportional to the semimajor axis cubed:
\[ P^2 \propto a^3 \]
When $P$ is measured in Earth-years and $a$ is measured in $AU$, astronomical units (the average Earth-Sun distance) the two sides of the formula are not just proportional, but equal. Therefore in units of Earth-years and AU, the formula becomes
\[ P^2 = a^3 \]
The reason why this works is because both units are specific measurements to Earth. I.e. the Earth’s orbital period and Earth’s semimajor axis specifically. Kepler’s third applies to all of the celestial bodies orbiting the Sun — all of the planets — and allows us to calculate their relative distances from the Sun if we can measure their orbital period. (It is much easier to calculate in this direction, historically speaking, since it was easier to observe a planets orbital period than its distance from the Sun.)
Kepler’s Three Laws of Planetary Motion
In summary, Kepler’s Three Laws of Planetary Motion are as follows:
- Kepler’s First Law — Each planet moves around the Sun in an elliptical orbit, with the Sun at one focus of the ellipse.
- Kepler’s Second Law — A line drawn between the Sun and planet sweeps out equal areas in space in equal intervals of time.
- Kepler’s Third Law — The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.
Kepler’s three laws provide the first ever precise geometrical and mathematical description of planetary motion. This achievement was an historical scientific breakthrough. It was the first time that we ever understood our local environment in space — the actual dynamics of our solar system.
Moreover, it proved mathematically and geometrically the Copernican Model, and more generally, The Heliocentric Model, of the solar system.
Planetary Motion Problems
Problem 3.1 — Applying Kepler’s Laws To The Orbit Of Mars
Mars orbits the Sun in about 1.88 Earth years (so the length of the year on Mars, the time the sun takes to return to the same position among the stars as viewed from the Martian surface, is almost twice that of the Earths).
Using this value of Mar’s orbital period (1.88 Earth years) calculate the semimajor axis of Mars.
Using Kepler’s Third Law
\[ P^2 = a^3 \]
we can see how the orbital period relates to the semimajor axis. We manipulate the equation and solve for the length of the semimajor axis in AU as follows:
\begin{equation} a & = \sqrt[3]{P^2} = \sqrt[3]{1.88^2} \\ & = \sqrt[3]{3.5344} = 1.52325 \\ & \simeq 1.52 \end{equation}
Therefore we find that the semimajor axis, $a$, of Mars is about 1.52 AU — just over 150% of the Earth-Sun distance. In kilometers, since the Earth’s semimajor axis is about 150,000,000 km, Mars is about 228.5 million km from the Sun.
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Problem 3.2 — Calculating Periods
Imagine that an object is orbiting around the Sun. If we measure the semimajor axis of its orbit at 50 AU, what can we expect its orbital period to be?
We solve this with Kepler’s Third Law as well, but rearranging for the orbital period, $P$, as
\begin{equation} P & = \sqrt{a^3} = \sqrt{50^3} \\ & = \sqrt{125,000} = 353.55339 \\ & \simeq 353.6 Earth-years \end{equation}
An object at 50 AU (which is beyond the orbit of Pluto) would have an orbital period of about 353.6 Earth-years.
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Problem 3.3 — Another Orbital Period Calculation
What would be the orbital period of an asteroid (a chunk of space rock travelling in our solar system) located between Mars and Jupiter at 3 AU?
\[ P = \sqrt{3^3} = \sqrt{27} = 5.1962 \]
The orbital period of an asteroid located between Mars and Jupiter is about 5.2 years.
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Example 4 — Orbital Periods and Semimajor Axes of Earth and Venus
Venus has an orbital period $P=0.62$ and a semimajor axis of $a=0.72$, while Earth has $P=1.00$ and $a=1.00$. Calculate $P^2$ and $a^3$ for both, and confirm they obey Kepler’s third law.
Using the equation for Kepler’s third law, $P^2 \propto a^3$ we can first Verify the proportionality for Venus
\begin{equation} \begin{align} P^2 & \propto a^3 \\ 0.62^2 & \propto 0.72^3 \\ 0.3844 & \propto 0.3732 \end{align} \end{equation}
We see that these numbers are proportional to one another. The discrepency here, can be chalked up to the fact that we rounded both $P$ and $a$ for Venus, which is why their square and cube, respectively, are slightly off.
For the Earth we do the same, and find that
\begin{equation} \begin{align} P^2 & \propto a^3 \\ 1.00^2 & \propto 1.00^3 \\ 1.00 & = 1.00 \end{align} \end{equation}
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Example 5 — Orbital Periods and Semimajor Axes of Saturn and Jupiter
Saturn has an orbital period $P=29.46 years$ and $a=9.54 AU$. Jupiter has $P=11.86$ and $a=5.20$. Use Kepler’s third law and verify that they
Using the equation for Kepler’s third law, $P^2 \propto a^3$ we can first verify the proportionality for Jupiter
\begin{equation} \begin{align} P^2 & \propto a^3 \\ 11.86^2 & \propto 5.20^3 \\ 140.6596 & \propto 140.608 \end{align} \end{equation}
We see that Kepler’s third law is valid for Jupiter. Next we apply it to Saturn and discover that
\begin{equation} P^2 & \propto a^3 \\ 29.46^2 & \propto 9.54^3 \\ 867.8916 & \propto 868.2507 \\ 868 & = 868 \end{equation}
it is also valid for Saturn.
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Resources
- Andrew Fraknoi, David Morrison, Sidney Wolff. “Astronomy 2e”. OpenStax. Mar 9, 2022. Book URL: https://openstax.org/books/astronomy-2e/pages/1-introduction. Section URL: https://openstax.org/books/astronomy-2e/pages/3-1-the-laws-of-planetary-motion. Accessed 19 May 2026.