|Number 22: 30/10/2003
A scientific publication by SGF and NEODyS
Part 2: The Solution of the Planetary Problem
by Andrea Carusi - President SGF
New observational techniques: Tycho Brahe
A key actor in our story of the Copernican Revolution is Tycho Brahe (1546-1601). Tycho was born three years after the publication of Copernicus's De Revolutionibus Orbium Coelestium, and one can imagine that he was an enthusiastic supporter of the new theory . In fact he was not: Tycho remained for all his life an opponent of the Copernican theory but, in an ironical turn, he contributed fundamentally to its eventual victory.
I have already underlined, in my first article in this series (read part I, Astronomy at the time of Copernicus, Tumbling Stone num. 21), that the body of astronomical data available to Copernicus and his colleagues, a data set collected over more than two thousand years, was rather deficient, both in precision and accuracy. Tycho was well aware of this situation and devoted his life to build an impressive complex of instruments for naked eye observations, instruments that allowed him to collect a rich harvest of very high value and, more important, self-consistent data. To get an idea of the precision of Tycho's observations, it is sufficient to remark that he was able to determine the positions of selected stars with an approximation of a minute of arc, and that of the planets to better than four minutes of arc. By comparison, previous measurements had errors of the order of ten minutes of arc.
Moreover, Tycho planned his work in a way that we would recognise as being very modern, systematically observing the planets - Mars in particular - along the entire arc of visibility, not only in selected opportunities. In essence, Tycho revolutionized observational techniques in astronomy, this being his major merit, and stopped the practice to uncritically accepting centuries-old measurements, the veracity of which was unknown, and indeed impossible to ascertain.
Prior to examining the results of Tycho's observations, and before speaking of the so-called Tychonic system, we should consider an astronomical parameter that plays a key role in this story: stellar parallax. This idea was well known to the ancient astronomers, masters in geometry and trigonometry, and was recognised also by Renaissance astronomers. In fact, the idea of parallax was one of the main reasons for the Copernican system being generally rejected, from the following argument.
The motion of the Earth, under the Copernican system, implied that, observing the same star twice, six months apart, and taking the diameter of the supposed orbit of the Earth as a baseline, the derived stellar parallax would provide the distance of the star from the Earth (i.e. the distance of the "sphere of the fixed stars"). No measurement, however, not even those very accurate observations obtained by Tycho, had ever shown a finite value for the stellar parallax, and there were only two ways to explain this result:
1. The Earth does not move at all, and therefore there is no parallax of any sort;
2. The distance from the Earth to the stars is frighteningly huge, so that the parallax is extremely small.
The second explanation was believed impossible for a long time, because it implied that between the sphere of Saturn (the last known planet at the time) and that of the stars there was a vast extent of nothing, something that ancient belief despised as much as the infinite. Therefore, the argument said that the first explanation must be correct, and Copernicus was wrong.
We know now that neither Tycho, nor indeed his successors, could have measured any parallaxes until the XIXth century: stellar parallax (always less than one second of arc) is too small to be measured except with more accurate instruments, not available until centuries after Tycho. He was forced to reject Copernicus's hypothesis because, based on his computations, it implied a distance between Saturn and the stars about 700 times larger than the supposed distance between Saturn and the Sun, and this was not deemed acceptable. Actually, Tycho got the multiplier too small: this ratio is larger than 25,000 even for the closest star! The first stellar parallax was determined by Friedrich W. Bessel in 1838 for the star 61 Cygni, and found to be about 0.33 seconds of arc (the currently accepted value is 0.289). The stars are really very far from us and it was necessary to adopt new units of distance for them. Only sophisticated photographic technologies were able to provide precise measures of stellar parallax. This was impossible using only the naked eye.
The Tychonic system
|Figure 1. The Tychonic system
(click for an enlargement)
In the Tychonic system the Earth, as in the Ptolemaic theory, is at the centre of the universe, and the Moon, the Sun, and the stars all revolve about it. However, the other planets revolve around the Sun. Furthermore, the sphere of the stars rotates westwards, while the Moon and the Sun move eastwards. The planets move around the Sun westwards, but are trailed by the Sun that moves eastwards. Too complicated? Sure, but the Tychonic system has three properties that make it very interesting in terms of progress towards the final success of the Copernican theory.
Let us note, first, that now for the first time there are objects that do not revolve around the Earth: the planets are thought of as satellites of the Sun. This is an important step, to be compared with the discovery of the "Medicean" satellites of Jupiter by Galileo a few years later (we now call these the Galilean satellites).
Second, as can be seen in Figure 1, the hypothetical spheres carrying Mercury, Venus, and Mars cross that of the Sun. If the celestial bodies are carried on by "pure crystalline spheres", as the theory said, it is difficult to see how they can intersect each other. But the third and most important property of the Tychonic system is that it is directly equivalent to that of Copernicus. As a matter of fact, the modifications made by Tycho to the Ptolemaic system affect mainly the motion of the planets. The motion of the stars depends upon the diurnal rotation of the Earth: in the Tychonic system, if we let the Earth rotate (but not revolve around the Sun), then the sphere of the stars stops moving, as in the Copernican system. The Moon is really revolving around the Earth, and the daily rotation of the latter does not change the situation. Concerning the Sun, in both cases of the Earth revolving around the Sun, or vice-versa, we get a counter-clockwise rotation, and these two types of motion are equivalent, from a geometrical point of view. Finally, now the planets revolve around the Sun, like in the Copernican system.
However, the Tychonic system did not solve the irregularities of the planetary motions, especially the retrogressions. Tycho was therefore forced to include again in his system epicyclical, deferent, equant, and eccentric circles. But, even if we look at the Tychonic system as the desperate attempt of an honest scientist to settle the controversy generated by the Copernican theory, it is clear that it disproved forever numerous beliefs on the part of others, opening the way for the Copernican system to triumph.
The "nova" and the fall of some certainties
In 1572 Tycho Brahe, not yet in his thirties, observed together with many colleagues a surprising phenomenon: a nova (new) star had appeared in the constellation Cassiopeia. This star rapidly reached the brightness of Venus, but then became fainter and fainter until its disappearance in 1574. What was it? Was it really a star? A non-trivial question, because the stars enjoyed, in the minds of men, the special status of incorruptible objects: if they could change, then there was no real difference between the super-lunar and sub-lunar worlds. Then, the Earth might be considered a planet, because the heavens did not differ so much from the ever-changing terrestrial world.
Both Tycho and his colleagues concluded that this nova could only be a star, because it was impossible to measure any parallax for it. The world of the stars was no longer as perfect as previously believed.
Many flaws had now been found in the Ptolemaic system: the planets move with irregular motions; perhaps the crystalline spheres do not exist; the stars may turn on and off like lamps (and therefore may not be eternal); the comets, several examples of which were observed by Tycho himself, are not stars, nor are they close to the Earth (due to the lack of detected diurnal parallax). The astronomical world was in a situation of great embarrassment. This is the confusion that often precedes, and almost invokes, the advent of a brilliant idea capable of clarifying the matter. The road was open for Kepler.
It is interesting to note at this point that comets and novae had been observed also by earlier astronomers. Their nature could have been explained adequately in the framework of the Ptolemaic system, but this interpretation did not take place. As Kuhn remarks:
"For two millennia, before the birth of Brahe, the phenomena and the instruments necessary to observe them had been available to the astronomers; but the observations were not made or, when done, were not adequately interpreted. In the second half of the XVIth century, the meaning and importance of well known phenomena changed quickly. These changes seem incomprehensible if they are not put in the framework of the new climate of the scientific thought, a most important representative of which is Copernicus… The De Revolutionibus represented a turning point, and no return to the ancient time would have been possible."
|Figure 2. The variation of the distance of Mars from the Sun along its orbit (in AU). The horizontal line is the average distance. (click for an enlargement)|
Kepler was born in 1571, 28 years after the publication of De Revolutionibus, and was assistant of Tycho Brahe in the last years of his activity. Kepler committed himself to trying to improve the Copernican theory in order to demonstrate its validity beyond any doubt. To do that he utilised extensively the excellent data collected by Tycho, especially those related to the motion of Mars. In ten years of hard work he confronted the difficult task of computing two orbits: that of Mars around the Sun, and that of the Earth. Kepler made many attempts with many different schemes, using circles of different sizes, but none of these attempts was successful; in the best cases there were still large residual displacements from Tycho's observations, and with great honesty Kepler thought that his solutions, which would have satisfied other astronomers, could not be correct.
In the end, Kepler discarded the underlying hypothesis that planetary orbits are circles. This was another big step forward, requiring the abandonment of another firm belief. After many trials with different types of ovals, he realised that the computed residuals were not randomly distributed, but followed a regular path (see Figure 2). Based on this finding, Kepler finally reached his first law: the planets move around the Sun along ellipses of which the Sun is located at one of the foci. Moreover, Kepler was forced to demolish another certainty: not only were the orbits of the planets ellipses and not circles, but also their velocity was not constant, being greater when the planets passed closer to the Sun, and smaller at a larger distance. Kepler was able to quantify this difference in speed and discovered that in this case there was another simple law: the linear velocity is not constant, but the product of the velocity and the angle described by the Sun vector (i.e. the angular velocity) obeys a simple rule. Connecting the Sun and the planet with a segment, and computing the area of the elliptic sector that this segment described in a given time, Kepler realised that the areas of the sectors covered in equal times were always the same, that is to say that the velocity of the planet varied such that this area remains constant for equal times. The conclusion is that the planets are faster when in perihelion, and slower in aphelion.
With the establishment of these two laws, the problem of the motion of planets suddenly disappeared. The observations agreed perfectly with the computed data, and this exceptional result had been obtained simply by introducing a curve different from a circle and a new law of motion. A puzzle lasting thousands of years was finally solved.
There is, however, an aspect of this achievement that needs to be noted. Kepler was successful where many had failed for a series of reasons: first, because he was a Copernican, and therefore very much tied to the idea of a central Sun. Second, because he had available the data collected by Tycho, a data set more precise than any previous data set. But he succeeded also because he was a Neoplatonic, convinced that natural phenomena are based on simple mathematical laws, and that the Sun was the origin of all celestial motions.
Kepler's physical theory
We must now mention the physics that stretches beyond the discoveries of Kepler. He asked himself: what moves the planets along their orbits? Obviously the Sun. The Sun emits a anima motrix (driving soul), a driving force that keeps the planets bound. It is a bizarre idea that is worth mentioning: the Sun emits rays in all directions on the ecliptic plane; a planet intercepts a number of these rays, and its velocity is proportional to this number. If the planet is far, the number of intercepted rays is smaller, and this is why it moves more slowly. This was a curious but effective explanation; unfortunately it is wrong. As a matter of fact, if the planet is put at double the distance, this reasoning would predict a velocity reduced to a half, something that does not agree with the observations. By admitting this initial error, Kepler finally reached his second law, which is correct and is not a simple proportion.
The first two laws were published in 1609, while the third did not appear until ten years later. The origin of the third law must again be searched for in Kepler's trust in an harmonic physical world based on mathematics. This law, for the first time in history, established a relationship between the motions of two planets. Nobody before Kepler had ever considered the possibility that the motions of the planets are connected together, but Kepler was occasionally affected by mystic beliefs, and for all his life he looked for harmonies hidden in the heavens. Now, the third law states that the motions of two planets are not random, but depend upon the average distances from the Sun. In particular, the ratio of the squares of the orbital periods (which obviously depend upon the average velocities) is equal to the ratio of the cubes of the semi-major axes of the ellipses.
It would be necessary to wait for Newton in order to get a mathematical explanation of this relationship. Kepler, however, took another important step forward: for the first time it was possible to compute the period (and therefore the average velocity) of a planet simply knowing its distance from the Sun or, conversely and even more interesting, it was possible to compute the distance once the period was known. Obviously, since the law is expressed by ratios, one has to know at least a one planet's pair of period and distance in order to compute all the others.
Closing this important chapter of our story, I would like to make a few comments. Kepler formulated three laws that Newton demonstrated mathematically consistent, the three laws that even now are used to compute orbits, but did not formulate a dynamical theory. That is, Kepler did not provide a sound and demonstrable explanation of why did the planets follow those laws and not others. This is not a minor consideration, because it is just this lack of a solid conception of the physics of the problem that makes Kepler an "innovator", but not a "founder". This honour is reserved to Galileo Galilei and, even more, to Isaac Newton.
On the other hand, to say that Kepler did not try to imagine how the Sun could act on the planets is not completely true. On the contrary, he was the first scientist to provide a non-animistic picture of the concept that later was called "force". The history of this concept, as well as of that of "mass", intimately connected to the former, is fascinating in itself, but would lead us too far away from the main theme of this article. However, it is worth spending a few words on it, because it is on these concepts that the fathers of physics founded the science.
Force (dunamiV, in Greek, hence "dynamics") has always been associated with the capacity to act, but it assumed a mechanical - and then quantifiable - significance only in the XVIIth century. Kepler played an important role in this change of paradigm. Certainly, Kepler did not have available any precedent; in fact he was even lacking a word to express this thought. As we have seen, he first defined the capacity of the Sun to keep and move the planets as an anima motrix, as a somehow spiritual quality; but later also his terminology became more precise. In the annotations to the second edition of his Mysterium Cosmographicum (1621), he wrote:
"If you substitute the word "anima" [soul] with the word "force" you obtain the true principle on which the celestial physics is based... Some time ago I believed that the cause of the planetary motion was a soul,...But when I realised that these driving causes became weaker at greater distances from the Sun, I came to the conclusion that this force is something physical, at least in a certain sense, if not in rigorous terms."