To Spark A Scientific Revolution
How Physics Was Introduced Into Astronomy For The First Time
It may seem amazing to us now, but astronomy and physics were once completely disconnected from each other. Prior to the early 17th century astronomy was considered a branch of mathematics, and physics a branch of philosophy.
It was the ancient philosophers Plato and Aristotle who between them declared that the planets, Sun and moon moved in perfect circles at uniform speed around the stationary Earth. The job of astronomy was then to account for the observed positions of these bodies in terms of uniform circular motions. The idea that there was a physical cause behind them was simply not considered due to the authority of these philosophers.
For 2000 years this was the state of affairs until one singular individual came along: Johannes Kepler, who worked out the correct shape of the orbits of the planets (immortalised in his First Law of Planetary Motion), how they moved over time (his Second Law) and how long they took to orbit the Sun (his Third Law).
In the process he married physics to astronomy. The story of how he did so is fascinating.
The motions of Mars
Something very odd happens if, night by night, you monitor the position of Mars against the background of stars at a particular time: instead of the planet moving continuously in a single direction, it periodically changes direction twice in a zig-zag motion before continuing on its path.
We now know that this is because we are looking at Mars from a moving observatory, the Earth, which is, like Mars, orbiting the Sun. So, like two cars orbiting a roundabout in different lanes, with the Earth on the inner, faster lane and Mars on the outer, slower one, as the Earth overtakes Mars, Mars appears to change direction compared to the stars, once and then again forming a loop until the overtake is complete.
This is called retrograde motion and is illustrated for Mars in the animation below. The other outer planets: Jupiter, Saturn, Uranus and Neptune, also exhibit retrograde motion, for the same reason.
Early models of the Solar System
It was the 2nd century mathematician astronomer Ptolemy, who, in order to account for retrograde motion, used geometrical devices such as off-centred (eccentric) circles, epicycles and equants, whilst always assuming that the orbits of the planets must involve perfect circles, even if in combination.
In the Ptolemaic model, each planet moves on a circle, called the epicycle. The centre of the epicycle also moves on a circular path, called the deferent. Ptolemy used this arrangement to model retrograde motion, since the planet on the epicycle spent half of its cycle moving in the opposite direction to that of the deferent and so appears from the Earth to perform retrograde motion as it does so.
In addition, it was known that the speed of a planet varied during its orbit as it moved against the background of stars (we now know this is because it moves slowest when furthest away from the Sun and fastest when closest). Ptolemy modeled this behaviour by making the centre of the deferent circle not the Earth but a point in space called the equant.
The next significant advance came with the work of Copernicus who placed the Sun, rather than the Earth, in the centre of the Solar System.
Copernicus (1473–1543)
Copernicus was dissatisfied with the Ptolemaic model chiefly because of its use of the equant which he considered to be too great a departure from uniform motion. His genius was to realise that the retrograde motion of the outer planets could be due to a single factor — the motion of the Earth around the Sun, and so he placed the Sun in the centre with the planets, including the Earth, orbiting it.
But Copernicus was still wedded to circles (specifically, crystalline spheres) and so, without the equant he needed to add more epicycles which made his system more cumbersome than the Ptolemaic model, even though it correctly placed the Sun in the centre.
Kepler and the Astronomia Nova
It was Johannes Kepler (1571–1630) who finally did away with the whole 2000-year-old machinery of circles, epicycles and equants and replaced them with his 3 laws of planetary motion:
1. The orbit of a planet is an ellipse with the Sun at one focus.
2. A line joining the Sun and a planet sweeps out equal areas in equal times.
3. The square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit.
The first two laws are contained in Kepler’s book of 1609, Astronomia Nova (The New Astronomy) and is one of the great masterpieces of science (his 3rd law was contained in Kepler’s Harmonices Mundi, published in 1619).
And it’s in the Astronomia Nova where we can pinpoint the exact place where Kepler introduces physics into astronomy, the first person ever to do so.
The Astronomia Nova is not at all written like a text book, where there is a smooth path from assumptions to results. It is instead more of Kepler’s journey of how he reached his conclusions about the movement of the planets. It is unique in the history of science, because Kepler shows his every step and misstep, every digression, every blind alley. It is quite unlike the works of Copernicus, Galileo and Newton, who presented us with only the final products of their labours.
Kepler’s personality shines through the Astronomia Nova which gives the book a warmth and charm, even if it can also be exasperating. It gives us a unique insight into the process of how a brilliant and complex mind sweeps away incorrect assumptions about the workings of the Universe that have held sway for two millennia.
However, Kepler would not have been able to achieve any of this without the observations of Mars produced by the great Danish astronomer Tycho Brahe.
Tycho Brahe
Tycho Brahe observations of the positions of the stars and planets were the most accurate that had ever been made.
Tycho needed Kepler to make sense of his observations of Mars, a task which had defeated his most senior assistant Longomontanus. Kepler boasted he could solve the orbit Mars in ‘a week’. In fact it took him nearly 5 years.
Although Kepler’s relationship with Tycho was difficult, he had the utmost respect for Tycho’s observations whose error in the position of Mars was never greater than 2 minutes of arc, i.e. two 60th (or 1/30th) of a degree. If Kepler discovered that a particular theory of his disagreed with Tycho’s data by more than 2 minutes then he threw out the theory, rather than ‘fudging” the theory or blaming the data.
Kepler declared that he was ‘at war’ with Mars (who was of course named after the Roman god of war).
Kepler’s battle with Mars begins
Once Kepler started by assuming that Mars orbit was an eccentric (off-centered) circle, but in a way that was different to that of Copernicus.
Unlike Copernicus, Kepler used an equant but not an epicycle as he considered an epicycle as absurd. Kepler also assumed that the distance from the centre of Mars’ orbit to the Sun and to the equant were unequal, unlike Ptolemy who had assumed they were the same. He therefore had four variables to solve for: the radius of the circular orbit, the direction of the axis containing the nearest and farthest points of Mars on the circle and the Sun-to-centre and centre-to-equant distances on that axis.
This was a complicated problem for which there is no direct formula; instead it was a process of trial and error until all the four parameters, and therefore the circular orbit, fitted the observations of Tycho that he had chosen (the position of Mars when it was opposite the Sun, known as ‘opposition’) to within a tolerable accuracy.
The labour involved beggars belief: Kepler’s draft calculations cover nine hundred folio pages in small handwriting [Koestler 1959, p325].
After this monumental effort Kepler exclaimed [Donahue 2015, p190]:
‘If this wearisome method has filled you with loathing, it should more properly fill you with more compassion for me, as I have gone through it at least seventy times at the expense of a great deal of time, and you will cease to wonder that the fifth year has now gone by since I took up Mars’.
He then tried a further 8 of Tycho observations of Mars at opposition and the circle of the orbit again fitted to within an accuracy of 2 minutes of arc.
But then disaster struck. He tried to fit his orbit to Tycho’s measurements of the distance of Mars from the Sun and alas, they did not fit.
‘Who would have thought it possible? This hypothesis, so closely in agreement with the observations, is nevertheless false’. Astronomia Nova, Chapter 19, p208.
Up to this point he was trying to fit a circular orbit to the observations. Now he realises that this isn’t possible, and he must start again and create the shape of the orbit without any pre-conceptions. But he also realises that to do this he must first understand how the Earth’s orbit behaves.
So he does something no-one else had thought of doing before — he transfers the observer from Earth to Mars, and by selecting observations of Mars that are separated by one Mars year (~687 days) he computes the distances and longitudes of Earth at these times as they would be measured from Mars itself. No wonder Einstein thought this was pure genius (see Einstein’s preface).
The result was as he expected: the orbit of the Earth was not exceptional: it moved faster when closer to the Sun and slower when further away.
The introduction of physics into astronomy
And now for the crucial moment, for it is at this point that Kepler introduces physics into astronomy and is the first person ever to do so.
For in Book III, Chapter 33 of the Astronomia Nova Kepler states his belief that the speed of a planet is inversely proportional to its distance from the Sun throughout its orbit and therefore that the Sun is somehow the cause of planetary motion.
No one before Kepler has introduced causality — physics — as an explanation for the motions of the planets [Gingerich 1972, p359]. Not surprisingly, Kepler’s physics is definitely pre-Newtonian, but it is physics nonetheless.
He has no concept that a moving body will continue to move unless a force acts upon it, instead he thinks of the Sun dragging the planets along in their orbits. He believes that the Sun is magnetic and that it is the rotation of the Sun’s magnetic field that reaches out and causes the planets to move.
This explanation of the planets’ motions is of course incorrect, though his speculation that the Sun rotates (discovered several years later) and has a magnetic field is correct.
Kepler’s 2nd Law of Planetary Motion
Now that Kepler is confident that the Sun is the cause of the motion of the planets and that a planet’s speed is inversely related to it’s distance from the Sun he can work out how long it takes to get to any particular position.
He begins by dividing up half the orbit of Mars into 180 points each one degree apart and calculates the distance of each of these to the Sun. Then, to calculate how long it takes to go from one position in its orbit to another, say from 30 to 70 degrees, he would then add up the distances from the Sun to Mars for the 40 points in between.
But even for Kepler this calculation was tedious to say the least. So he looks for an alternative.
Instead of the sum of the distances to the points on the orbit he considers whether it’s the area subtended by the chosen points and the Sun. Strangely, he knew that this was mathematically objectionable and explains at length why this is so, does it anyway, and is if “by a miracle” it worked, and so we have Kepler’s 2nd Law: A line joining the Sun and a planet sweeps out equal areas in equal times.
Kepler’s 1st Law of Planetary Motion
Kepler’s path to realising that the shape of the orbit of Mars is an ellipse, and not a circle, was a tortuous one and took him nearly two years.
At this point he has two circular orbits for Mars; one that fits Tycho’s positional data for Mars, the other that fits the distances. In order to come up with an orbit that satisfied both the positional and the distance measurement he adds, horror of horrors, an epicycle (though admittedly a small one)!
And although it looks like an ellipse (which would be the correct shape), it is in fact an oval, i.e. egg shaped, more flattened at perihelion and more pointed at aphelion.
Kepler’s preoccupation with this egg went on for a year. He added up 180 distances from the Sun to Mars, a calculation which he repeats 40 times. Ironically on 4th July 1603 he writes to his friend David Fabricius that he is unable to solve the geometric problems with his egg but:
‘if only the shape were an ellipse all the answers would be found in Archimedes’. [TS, p335].
So close, yet so far. But he doesn’t realise the correct shape is an ellipse, and instead the egg remains his idée fixe.
Eventually he realised that he could not get his egg to fit the data. So he once again calculates the location of Mars at different points on the orbit. It again looked like a flattened circle.
Kepler then draws a diagram in which his curve is inscribed within a circle (see illustration below) with a sickle-shaped gap in between:
He found that the maximum thickness of each sickle was 0.00429 of the radius. For some reason he then became interested in the angle at M and found that at its maximum value of 5⁰ 18’, and that its secant (hypotenuse (MC) divided by the adjacent (MS)) was 1.00429. Kepler says[Donahue 2015, p407]:
‘And when I saw that this was 1.00429, it was as if I were awakened from sleep to see a new light…’
It led to a formula, but Kepler didn’t realise at this point that it represented an ellipse.
He then entered into one last goose-chase — he tried to draw the curve that corresponded to the formula, didn’t know how, made a mistake and arrived at the wrong curve. So he threw out the formula, tried a new idea — the ellipse — and found that this was the correct curve after all [Koestler 1959, p338].
It was a triumph. After 7 years of intense work, he had at last conquered Mars and discovered that the shape of its orbit is an ellipse with the Sun at one focus, his First Law of Planetary Motion:
‘At last, when he [Mars] saw that I held fast to my goal, while there was no place in the circuit of his kingdom where he was safe or secure, the enemy turned his attention to plans for peace: sending off his parent Nature, he offered to allow me the victory; and, having bargained for liberty within limits subject to arbitration, he shortly thereafter moved over most eagerly into my camp, with Arithmetic and Geometry pressing closely at his sides.’ Astronomia Nova, Dedication to Rudolph II.
Kepler: The first astrophysicist
Before Kepler, physics was considered the domain of philosophy, with astronomy relegated to ‘saving the phenomena’ using a fantastical mixture of circles, equants, epicycles and deferents, something that seems barely credible today.
After Kepler, physics was forever married to astronomy, the fruits of which have produced an ever-accelerating cascade of developments in our understanding of the Universe that continue to this day: Newton’s universal theory of gravity, Einstein’s Special and General Theories of Relativity, Quantum Mechanics, the Standard Model of Particle Physics, String Theory, the list goes on.
We are so fortunate that Kepler wrote down all the steps, labyrinths and blind alleys he took before he finally rid the Solar System of combinations of eccentric circular motions, to replace them with elliptical orbits and in so doing clear away the 2000 year old fixation with uniform circular motion and the rest.
It allows us to point to precisely when the very first human-being realises that physical causes — physics — and not sky-geometry, is the key to understanding the Universe.
So, as you were no doubt taught at school, always show your working. Just in case you spark a Scientific Revolution.
Donahue, W. (2015) Johannes Kepler, Astronomia Nova (translated from Latin), Green Lion Press, US.
Love, D (2015) Kepler and the Universe, Prometheus Books, New York.
Stephenson, B (1987) Kepler’s Physical Astronomy, Springer-Verlag, New York.
Gingerich, O (1972) Johannes Kepler and the New Astronomy, (The George Darwin Lecture, 1971) Quarterly Journal of the Royal Astronomical Society (1972), Vol. 13, p346–373.
Koestler, A (1959, reprinted in Arkana 1989) The Sleepwalkers, Penguin Books Ltd, England.