The remarkable observations of Tycho Brahe preceded the invention of the telescope by a decade, the quality of telescopic measurements by a century. Brahe's measurements of cometary motion disproved the claim that comets were inside the Moon's orbit, as well as the idea that the celestial bodies moved on invisible spheres. Brahe's measurements of planetary positions provided the data for Johannes Kepler's three laws: planetary orbits are elliptical, equal orbital areas are swept out in equal times, the square of the planet's annual period is proportional to the cube of the semimajor axis (“radius” of the long dimension) of the orbit. The three laws were empirical, based entirely on Kepler's use of Brahe's data. They were not explained until Newton discovered the law of gravity.
Tycho Brahe,1 or in Danish, Tyge Brahe (1546–1601), was one of the best observational astronomers of all time. This is especially remarkable because his observations preceded the invention of the telescope. The accuracy of astronomical measurements with telescopes had to wait a century before they surpassed Tycho's. His measurements of planetary positions, especially those of Mars, were the data behind Johannes Kepler's three laws describing planetary motion.
How to draw an ellipse.
Kepler's First Law states that the planets move in elliptical orbits with the Sun in one of the two foci2 (the ellipticity is exaggerated in the drawing) (Fig. 1.2).
Kepler's First Law.
The Second Law states that a line through the Sun and the planet sweeps equal areas at equal times. Thus, the closer a planet is to the Sun, the more quickly it moves (Fig. 1.3).
Kepler's Second Law.
The Third Law states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. The semi-major axis is the distance from the center of the ellipse to the furthest point on its circumference. For the special case of the circle, it is just the radius.
Defining T as the period of a planet's orbit, viz., its “year” and a as the semi-major axis, the Third Law says
Treating the ellipse as a circle, which is approximately true for the planets,3 for the Earth, (1 year)2 = a constant times (the Earth's orbital radius)3. The value of the constant depends on what units you choose. The very simplest way is to choose T in Earth-years and a in Earth–Sun distance, called “astronomical units:” a.u. So, for the Earth, this reduces to 12 = 13, which indeed it does. For Mars, T = 1.88 years, a = 1.52 AU. T2 = 3.53; a3 = 3.51, the same, discounting the margin of error in the third significant digit, and so on, for the other planets.4
Tycho was a man with a considerable temper. At the age of 20, he had a disagreement with a fellow student that led to a sword duel. Unfortunately, for Tycho, the fight ended with a part of his nose cut off. For the rest of his life, he wore either a brass or silver prosthesis.
Tycho's fame (the Church might have called it “notoriety”) started with his observations in 1572 of a supernova5 in the constellation Cassiopeia. His careful measurements showed that the object did not change its position relative to the fixed stars. He, therefore, identified it as a new star in the heavens. This was revolutionary. According to classical Aristotelian wisdom, the planets and the moon moved on the surfaces of spheres. Anything below the Moon's sphere consisted of the four elements, earth, air, fire, and water, that could change between different forms. Above the Moon's sphere, however, everything was made of the fifth element, the “quintessence”, which was immutable. Therefore, the heavens were unchangeable. Tycho published his result in 1573 in a book, About the New Star (De Nova Stella), from whose title came the words nova and supernova. (Luckily for him, the Danish Church had become Protestant in 1536, so he was not subject to the attacks on Galileo by the Catholic Church.)
From November 1577 to January 1578, a comet was seen and meticulously observed by Tycho. He made the following very clever deduction (see Fig. 1.4).6
Comet seen by Tycho.
When the comet rose over the horizon (point D), Tycho measured the position of the comet (A) compared with the fixed stars. Just before it set (point C), he measured it again. The length DC is the diameter of the Earth. Using only the trigonometry of a triangle, he deduced the distance between the Earth and the comet. Doing this for 74 days, he could follow its orbit and prove that it moved through, not on the crystalline spheres. So, with one brilliant stroke, he disproved both Aristotle's claim that comets were atmospheric phenomena and thereby below the Moon's sphere, as well as the entire idea of crystalline spheres.
Kepler believed that Tycho's line of argument about the comet was conclusive and he was so impressed with the rest of his findings that when Tycho died, Kepler made a “midnight run” to obtain (grab) all of his data.
Another astronomer, Thadaeus Hagedius, made similar observations of the comet but drew the opposite conclusion, claiming that the comet was closer than the moon. Tycho demanded to see his data. With these in hand, he proved not only that Hagedius had miscalculated, but that instead of using the terrestrial diameter as a basis, he (Tycho) could even use the much smaller distance from Prague to Hven as the measurement baseline and get the correct answer. This provided an even stronger proof that the comet stayed outside the “lunar sphere.”
Why Prague to Hven? King Frederik II of Denmark had granted Tycho the use of the island of Hven, located in the sound between Denmark and Sweden, for his observatory. He also made Tycho the landlord of the island's farmers. Never one to waste time on matters other than the astronomical, Tycho dealt with disputes by throwing both plaintiffs and defendants into jail. He so irritated Frederik's son, King Christian IV, that the latter threw him off the island, whereupon Tycho moved to Prague, setting up a new observatory.
A nail had been driven into the coffin of the geocentric model in 1523 by the heliocentric model of Copernicus; he, however, held back publication until 1543, when he lay on his deathbed—a wise decision. Tycho himself, despite knowing of the Copernican model, propounded a modification in 1583: the Earth was still the center of the universe, with the Sun revolving around it and the other planets revolving around the Sun. This, as he admitted, was a way to save both the Copernican and the Ptolemaic systems, thus not offending anybody—another wise decision.
Tycho died of a burst bladder. The details of his death are not clear and are the subject of various anecdotes. We shall draw the curtain of charity over them.
Portrait from https://physicstoday.scitation.org/do/10.1063/PT.5.031 374/full/.
The foci of an ellipse are formally defined as follows: For two given points, the foci, an ellipse is the locus of points such that the sum of the distance to each focus is constant. In lay terms, if you attach a loose string to two fixed nails, put a pencil in the string, and then move the string with the pencil around. Keeping the string taut, you will draw an ellipse, with the two fixed points at the foci. A circle is a special case of an ellipse, where the two points become one (Fig. 1.1).
Pluto's orbital eccentricity (departure from a circle) is the largest, but then, it has been demoted.
This turned out to be a consequence of Newton's Inverse Square Law of Gravitation: the force between two objects decreases as the square of the distance between them increases. Thus, doubling the distance between two objects decreases the force between them by a factor of 22, or 4.
An exploding star.
All distances are ridiculously out of proportion. The Earth–comet distance is much larger.