Tycho Brahe

How to use this course

Tycho Brahe is remembered as the greatest astronomer before the telescope, the man with the metal nose and the island observatory, and the source of the data that Johannes Kepler turned into laws. He was also a sharp theorist. He used parallax to prove that a new star and a comet lay far beyond the Moon, which destroyed the solid celestial spheres of Aristotle. He then built a world system that kept the Earth still and the Copernican geometry intact. This course follows the life and explains the mathematics behind each of these steps.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. The Copernicus course explains the heliocentric geometry Tycho inherited; the Kepler course shows what Kepler made of Tycho’s observations of Mars.

Timeline

Knutstorp and Copenhagen, 1546–1562

Tyge Ottesen Brahe, known by the Latin form Tycho, was born on 14 December 1546 at Knutstorp castle in Scania, then a Danish province and now in southern Sweden. Both his parents belonged to the small group of noble families that governed Denmark with the king. His father Otte Brahe served the crown as a privy councillor and governor of fortresses. When Tycho was about two, his father’s brother Jørgen Brahe and Jørgen’s wife Inger Oxe, who were childless, took him to raise as their own heir.

On 19 April 1559 he entered the University of Copenhagen, aged twelve, to begin the humanist education expected of a future statesman. On 21 August 1560 a partial eclipse of the Sun occurred as the astronomers had predicted. The idea that the motions of the heavens could be computed in advance struck him deeply. He bought astronomical tables and a copy of Ptolemy, and started teaching himself.

Wanderjahre, 1562–1571

In 1562 his family sent him to Leipzig with a tutor, Anders Sørensen Vedel, to study law. Tycho studied the sky at night. In August 1563 he observed a close conjunction of Jupiter and Saturn and found that both the Alfonsine and the newer Copernican tables predicted it wrongly. The lesson he drew was that astronomy needed better observations before it needed better theories.

He returned to Denmark in 1565. His uncle Jørgen died that year, and Tycho left again for the German universities: Wittenberg, then Rostock. On 29 December 1566, after a quarrel, he fought a duel at Rostock with his cousin Manderup Parsberg and lost part of his nose. He wore a prosthesis for the rest of his life; analysis after his 2010 exhumation found traces suggesting it was brass. He travelled further in Germany and Switzerland, and at Augsburg had a giant quadrant built for observing altitudes.

His father died on 9 May 1571. Tycho inherited a share of Knutstorp, and with his maternal uncle Steen Bille set up a laboratory and observatory at Herrevad Abbey. Alchemy and medicine interested him all his life, but astronomy soon dominated.

The new star and the island, 1572–1576

On the evening of 11 November 1572 Tycho saw a bright star in Cassiopeia where none had been. It outshone Venus at its peak and stayed visible to the naked eye for well over a year. Aristotelian cosmology held that the heavens beyond the Moon were unchangeable, so most observers assumed the object was a sublunar phenomenon. Tycho measured it carefully against neighbouring stars and found no parallax and no motion. In De nova stella (1573) he concluded that it lay among the fixed stars. Lesson 05 explains the argument. The book gave the word nova to astronomy. Modern astronomers classify the event as a type Ia supernova, SN 1572.

In 1573 he began a lifelong union with Kirsten Jørgensdatter. As a commoner she could not marry a noble formally, and their eight children, six of whom reached adulthood, could not inherit noble estates. From September 1574 to the spring of 1575 he lectured on astronomy at the University of Copenhagen, then travelled in Germany and visited Landgrave Wilhelm IV of Hesse-Kassel, an observer whose instruments and methods impressed him.

Tycho was considering a move to Basel. To keep him in Denmark, King Frederick II offered him the island of Hven in the Øresund, with its rents, and money to build. The foundation stone of Uraniborg, the “castle of Urania”, was laid on 8 August 1576. Lesson 11 describes what he built there.

Instruments and the arithmetic of error

For the general reader. Before Tycho, astronomers recorded positions to perhaps ten minutes of arc, a third of the Moon’s width. Tycho reached about one minute for the stars, the limit of the naked eye. He did it by building large, rigid instruments with fine scales, by checking each instrument against others, and by repeating observations night after night. He was among the first astronomers to treat error as something to be measured.

Size and scale

An arc of radius \(R\) graduated in units of angle \(\Delta\theta\) has marks spaced \(R\,\Delta\theta\) apart. With \(\Delta\theta=1'=2.909\times10^{-4}\) rad, a radius of 2 m gives marks about 0.58 mm apart, near the limit a reader can resolve. This is why Tycho’s best instruments were large, and why they had to be rigid: a flex of a fraction of a millimetre at the scale is a minute of arc.

Transversals

Tycho subdivided degree divisions with transversal dots: a slanting row of points between two concentric arcs, so that the sight-line’s crossing point indicates a fraction of the division. The construction is linear interpolation in geometric form.

For small \(\Delta\theta\) the arcs are nearly parallel straight lines, and the transversal crosses them like the diagonal of a parallelogram; similar triangles give the proportional division. The error terms come from the curvature and from the radial lines not being parallel. Tycho kept them small with narrow bands near large radii.

Random and systematic error

Repeating an observation reduces scatter, but a bias \(b\) shared by all observations stays. Tycho fought bias by comparing different instruments, by measuring the same quantity by different routes, and by correcting for refraction (Lesson 09). Modern analyses find that his stellar positions typically carry errors of half a minute to one minute, and that his planetary observations, which Kepler used, were reliable to about two minutes.

Parallax and the new star

For the general reader. Hold up a finger and look with one eye, then the other: the finger jumps against the background. Astronomers use the same effect, parallax, to measure distance. As the Earth turns, an observer is carried around a circle, and a nearby object like the Moon shifts against the stars. Tycho showed that the new star of 1572 did not shift, so it was far beyond the Moon.

Proof. In the triangle formed by the centre \(C\), the observer \(O\) and the body \(B\), the angle at \(O\) is \(180^\circ-z'\) and the angle at \(B\) is \(p\). The law of sines gives \(\sin p/R=\sin(180^\circ-z')/d\). For \(R\ll d\), \(p\) is small and \(z'\approx z\).

For the Moon, \(d\approx60R\), so \(\pi_h\approx57'\), nearly two lunar diameters. Parallax always makes a body appear lower than it would from the centre.

Tycho’s test

The new star was circumpolar at Hven: it never set, and it crossed the meridian twice a day, once high above the pole (upper culmination) and once below it (lower culmination). Its parallax would be small at the first and large at the second. Measured against nearby stars of Cassiopeia, which have no parallax, its position would change by the difference. Tycho found no change at the level of about a minute of arc.

The comet of 1577 and the end of the spheres

For the general reader. Comets were thought to be fires in the upper air, below the Moon. Tycho first saw the Great Comet on 13 November 1577 and followed it until January 1578. Its parallax was too small to measure, so it was far above the Moon. Worse for the old cosmology, its path carried it through the region where the planets’ crystalline spheres were supposed to be. Tycho concluded that the spheres did not exist.

This is the small-angle form of the triangle on the baseline. Tycho compared his positions of the comet with those of Thaddaeus Hagecius at Prague. Relative to nearby stars, the Moon would have appeared in noticeably different places from the two cities; the comet did not. Tycho also used the diurnal method of Lesson 05 from Hven alone.

In De mundi aetherei recentioribus phaenomenis (1588) Tycho placed the comet on a circle around the Sun, outside the orbit of Venus. A body moving through the planetary region would have to pass through the solid spheres that were thought to carry the planets. Tycho concluded that the heavens were fluid, with the planets moving freely. This conclusion freed him to draw the system of Lesson 07, in which circles intersect.

The geometry of the Tychonic system

For the general reader. Tycho admired Copernicus’s geometry but could not accept a moving Earth. It went against physics as he knew it and against Scripture, and it required the stars to be impossibly far and large. His solution, published in 1588, kept the Earth at rest with the Sun and Moon circling it, while the other planets circled the Sun. Seen from Earth, every planet moves exactly as in Copernicus’s system.

Proof. In the Copernican system the Earth is at \(-\mathbf s(t)\) relative to the Sun, so the planet relative to the Earth is \(\mathbf h_k(t)-(-\mathbf s(t))=\mathbf s+\mathbf h_k\). This is the Tychonic position by definition. The two systems differ by the time-dependent translation \(\mathbf x\mapsto\mathbf x+\mathbf s(t)\), a change of reference point from the Sun to the Earth.

The equivalence applies to the planets but not to the stars. In the Copernican system the fixed stars are fixed relative to the Sun, so seen from the Earth they show an annual parallax. In Tycho’s system they are fixed relative to the Earth and show none. The absence of observed stellar parallax therefore favoured Tycho, until Friedrich Bessel measured it in 1838. Tycho also rejected the Earth’s daily rotation; his assistant Christen Sørensen Longomontanus later accepted it in a “semi-Tychonic” system. Without a dynamics, nothing in the planetary motions alone could decide.

Mars crosses the Sun’s orbit

In Tycho’s system, the Sun moves on a circle of radius \(1\) about the Earth, and Mars on a circle of radius \(a=1.52\) about the Sun. Mars’s distance from the Earth therefore ranges from \(a-1=0.52\) to \(a+1=2.52\), and its circle crosses the Sun’s. In Ptolemy’s system Mars was always beyond the Sun. Tycho saw a test: near opposition, Mars should be closer than the Sun and show a larger diurnal parallax. In the early 1580s he tried to measure it. The Sun’s true parallax is \(8.8''\), so that of Mars at a favourable opposition is under \(30''\), below his reach. Tycho, like nearly every astronomer since antiquity, believed the Sun about twenty times closer than it is, and expected a much larger effect.

The size of the stars

Tycho measured apparent stellar diameters of one to a few minutes of arc; we now know these are artefacts of the eye, since stars are point-like at that resolution. If stars showed no annual parallax above about \(1'\), their distance had to exceed about 3,400 AU, and a star of apparent diameter \(\theta\) at distance \(d\) would have true diameter \(\theta d\). Tycho concluded that in the Copernican system a typical star would be as large as the Earth’s orbit, which he found absurd.

Prosthaphaeresis: multiplying by adding

For the general reader. Reducing thousands of observations meant multiplying long numbers, mostly sines and cosines of angles. For about thirty years before logarithms, astronomers used a trick called prosthaphaeresis, from Greek words for adding and subtracting. It turns a product of two sines into a difference of two cosines, which can be read from a table.

Proof. Add or subtract the expansions \(\cos(\alpha\mp\beta)=\cos\alpha\cos\beta\pm\sin\alpha\sin\beta\) and \(\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta\).

To multiply two numbers \(x,y\in(0,1)\), find \(\alpha=\arccos x\) and \(\beta=\arccos y\) in the table, look up \(\cos(\alpha+\beta)\) and \(\cos(\alpha-\beta)\), and average them. Other numbers are first scaled by powers of ten. The method has the same structure as logarithms: an isomorphism that turns multiplication into addition, here through the angle-addition law of the circle.

The method appears in the work of Johannes Werner in the early sixteenth century. Paul Wittich, who stayed on Hven for about four months in 1580, brought it to Tycho, and the cosine form may have come through Jost Bürgi. Tycho claimed a share in the invention. Nicolaus Reimers, called Ursus, published the formulas in 1588 and credited Wittich and Bürgi; historians generally side with Ursus.

The altitude formula

The altitude \(h\) of a star of declination \(\delta\) at hour angle \(H\), seen from latitude \(\varphi\), follows from the spherical law of cosines in the triangle pole–zenith–star:

Each term is a product of trigonometric values, ideal for prosthaphaeresis.

Refraction

For the general reader. The air bends starlight, lifting everything in the sky. Near the horizon the lift is about half a degree, the width of the Sun, so the Sun can be seen after it has geometrically set. Tycho was the first astronomer to measure this effect systematically and to publish tables to correct his observations for it.

Proof. Snell’s law at each interface conserves \(n\sin z\), so \(n_0\sin z=1\cdot\sin z_0\). Writing \(z_0=z+R\) with \(R\) small, \(\sin z+R\cos z\approx n_0\sin z\), so \(R\approx(n_0-1)\tan z\). The result depends only on the index at the ground, not on the profile.

At sea level \(n_0-1\approx2.9\times10^{-4}\), so \(R\approx60''\tan z\). The formula is good to about \(75^\circ\) from the zenith. Near the horizon \(\tan z\) diverges, the curvature of the atmosphere matters, and the refraction depends on temperature and pressure; the true horizontal refraction is about \(34'\).

The Moon’s inequalities

For the general reader. The Moon’s motion is the hardest in the sky to predict, because the Sun pulls on it as well as the Earth. Ptolemy knew two irregularities. Tycho, observing at all phases and not only at eclipses, found more: the Moon runs ahead and behind between new and full Moon, a little differently in winter and summer, and its orbit tilts back and forth. Newton later explained all of these as effects of the Sun’s gravity.

Why eclipses hide the variation. At new and full Moon \(D=0^\circ\) or \(180^\circ\), so \(\sin2D=0\). Ancient lunar theory was fitted chiefly to eclipses, so it never saw a term in \(\sin2D\). The term is largest at the octants, \(D=45^\circ,135^\circ,\dots\), halfway between syzygies and quarters. Tycho’s observations at all phases revealed an error of about \(40'\) there.

Why the Sun matters. In the Earth–Moon frame, the Sun exerts a tidal acceleration of order \(GM_\odot r/a_\odot^3\) on the Moon, compared with the Earth’s pull \(GM_\oplus/r^2\). The ratio is

using Kepler’s third law for each orbit. Perturbation theory gives the variation amplitude as a series beginning \(\tfrac{11}{8}m^2\) radians, about \(26'\); higher-order terms bring it to the observed \(39.5'\). The annual equation arises because the tidal force varies as \(a_\odot^{-3}\), and the Earth–Sun distance changes over the year. Tycho also found that the inclination of the Moon’s orbit oscillates around \(5^\circ\) and that the nodes regress unevenly. His lunar theory was completed with Longomontanus and published by Kepler in the Progymnasmata (1602).

Uraniborg, 1576–1597

Uraniborg was built from 1576 to about 1580 on a hill in the middle of Hven: a Renaissance house with observing platforms, a library, a chemical laboratory in the basement and formal gardens. Its cost has been estimated at about one per cent of the Danish crown’s revenues during construction. Tycho added a printing press in the gatehouse and a paper mill powered by fishponds, so that he could publish his own books. In 1584 he built Stjerneborg, a second observatory nearby, with instruments set in pits below ground level to protect them from the wind.

The island was a research institute in all but name. Assistants and students, among them Longomontanus from 1589, worked in teams on the instruments, recorded observations in log books and reduced them with prosthaphaeresis. Visitors included James VI of Scotland in 1590. Tycho was also lord of the island, with rights over its tenant farmers, and his relations with them were hard. Over twenty years the institute produced the most extensive and accurate observational record before the telescope, and a manuscript star catalogue of about 1,000 stars.

Frederick II died in 1588. Under the regency for the young Christian IV, and then under Christian himself, Tycho lost royal favour and parts of his income. The causes were partly financial and partly his neglect of duties attached to his fiefs. His last observation on Hven was made on 15 March 1597. Soon afterwards he left Denmark with his family, assistants and portable instruments. Uraniborg was destroyed after his death.

Exile, Prague and Kepler, 1597–1601

Tycho spent the winter of 1597–98 at Wandsbek, near Hamburg, as the guest of Heinrich Rantzau. There he printed Astronomiae instauratae mechanica (1598), an illustrated account of his instruments, and presented a manuscript of his star catalogue to potential patrons. In 1599 Emperor Rudolf II appointed him Imperial Mathematician, and gave him Benátky castle north-east of Prague as an observatory. He had replaced Ursus, the man he accused of stealing his world system; Ursus died in Prague in October 1600.

Tycho needed a theorist. Johannes Kepler, whose Mysterium cosmographicum he had read, arrived at Benátky on 4 February 1600, as religious pressure on Lutherans in Graz was growing. Their relations were tense: Tycho guarded his data, and Kepler needed both the data and a salary. Tycho set him to work on the orbit of Mars. He also asked Kepler to write a defence of his priority against Ursus, which Kepler drafted but never published. Longomontanus, who had rejoined Tycho in Prague in January 1600, finished the lunar theory and returned to Denmark in August.

On 13 October 1601 Tycho fell ill after a banquet in Prague at the house of Peter Vok of Rožmberk. According to Kepler’s account he would not leave the table before the end, out of politeness, and afterwards could not pass urine. He died on 24 October 1601, and was buried in the Týn Church in Prague. Rumours of poisoning were investigated by exhumations in 1901 and 2010. The 2010 study found no lethal level of mercury, and murder has been ruled out. Two days after his death Kepler was appointed his successor as Imperial Mathematician, and he kept the observations despite disputes with Tycho’s heirs.

Legacy

• Data: Kepler’s three laws rest on Tycho’s observations of Mars, good to about two minutes, which ruled out the best circular model by eight minutes (see the Kepler course, Lesson 06). The Rudolphine Tables (1627) incorporated his star catalogue and remained the standard for about a century.
• Cosmology: the new star and the comet ended belief in an unchangeable heaven and in solid spheres. The Tychonic system, adopted by many Jesuit astronomers after 1616, remained the main rival to Copernicus until dynamics and the discovery of stellar aberration in the eighteenth century settled the question.
• Method: large calibrated instruments, systematic correction for refraction, repeated observation and team-based data reduction set the standard for observatories.
• Honours: the lunar crater Tycho, the Martian crater Tycho Brahe, the asteroid 1677 Tycho Brahe, and the supernova remnant of SN 1572 bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Tycho Brahe, MacTutor History of Mathematics.
2. Tycho Brahe, Wikipedia (with extensive references).
3. Uraniborg, Wikipedia.
4. V. E. Thoren, The Lord of Uraniborg: A Biography of Tycho Brahe, Cambridge University Press, 1990.
5. J. R. Christianson, On Tycho’s Island: Tycho Brahe and His Assistants, 1570–1601, Cambridge University Press, 2000.

Astronomy and mathematics

1. Tychonic system, Wikipedia.
2. SN 1572 and Great Comet of 1577, Wikipedia.
3. Prosthaphaeresis, Wikipedia; Tycho Brahe and mathematical techniques, Whipple Museum, Cambridge.
4. J. J. O’Connor and E. F. Robertson, Paul Wittich, MacTutor.
5. Lunar theory, Wikipedia.
6. N. M. Swerdlow, The lunar theories of Tycho Brahe and Christian Longomontanus, Journal for the History of Astronomy 40 (2009). Caltech record.
7. N. Jardine, The Birth of History and Philosophy of Science: Kepler’s A Defence of Tycho against Ursus, Cambridge University Press, 1984.
8. O. Gingerich and J. R. Voelkel, Tycho Brahe’s Copernican campaign, Journal for the History of Astronomy 29 (1998).