Lecture 12 - Tycho and Kepler (2/12/96)

Seeds: Chapter 4

1. Tycho Brahe
   • born of Danish nobility, (1546-1601)
   • patron was Frederick II, King of Denmark
   • established observatory on Baltic island of Hveen
   • carried out the most complete and accurate astronomical observations for 20 years
   • In 1572 Tycho observed a bright "new star" that appeared suddenly in the sky, then faded away. This is known as Tycho's supernova. Supernovae are now known to be the result of titanic explosions of dying stars.
   • In 1577 Tycho had seen bright comet for which no parallax was measureable. He concluded that the comet must be at more than 3 times the distance of the Moon. This was surprising, since it was thought that comets and all changeable phenomena were part of the corruptible Earth, not the perfect heavens.
   • The supernova and comet showed that the heavens were changing.
   • left Hveen after death of Frederick, whose successor Christian IV discontinued the royal support
   • Tycho became court astronomer for Rudolph II of Bohemia and spent remaining years of his life analysis the 20 years of measurements
   • excellent observations of the stars and planets with naked eye and measuring instruments
   • positions of hundreds of stars to 4 arcminutes
   • positions of nine fundamental stars to nearly 1 arcminute accuracy
   • continuous record of motions of Sun, Moon and planets
   • determined length of year to 1 second
   • precise measurements showed variations from predicted positions given in the Ptolemaic Alphonsine Tables
   • was able to find regularities in these variations, and set out to make his own Rudolphine Tables
   • devised his own "Tychonic" cosmology, with the Earth unmoving at the center and the Sun circling the Earth, but all the other planets orbited the Sun
   • In 1600, a year before his death, he took on a young mathematician named Kepler to help analyse his data.
2. Johannes Kepler
   • born in southwestern Germany, (1571-1630)
   • taught mathematics and astronomy in Austria
   • was an early convert to the Copernican heliocentric hypothesis
   • tended toward philosophy and mysticism, published the Mysterium Cosmolographium in 1596
   • In 1600, he became assistant to Tycho Brahe on the strength of his mathematical skills.
   • Tycho set him to work to find a satisfactory theory of planetary motion using parts of his data from Hveen.
   • Tycho oddly enough did not supply Kepler with enough of the measurements to make substantial progress. Perhaps he was worried about being "scooped" by the Copernican Kepler and wanted to prove his own Tychonic model for the solar system.
   • After Tycho's death in 1601, Kepler succeeded him as Rudolph's court astronomer and mathematician, and most importantly, was able to obtain most of Tycho's records for himself.
   • Kepler first studied the extensive data on the orbit of Mars. He first tried fitting every combination of uniform circular motions he could think of with epicycles, equants, eccentrics, and epicyclets. His best results only agreed with the measurements to 8 arcminutes, and he knew Tycho's positions were better than that.
   • He boasted to a colleague that he would solve the problem of the orbit of Mars within 2 weeks - it took him 5 years.
   • He took time out from Mars to write a treatise on the theory of Optics that set down the fundmentals for what later became modern geometric optics.
   • Mars was know to return to the same place in its orbit every 687 days (its sidereal period). The Earth was at different positions so the orbit of Mars could be triangulated. It was seen to be clearly non-circular.
   • He tried various things like ovals. He eventually discovered the the orbit was excellently fit by an ellipse with the Sun located at one of the foci
   • The ellipse is basically a circle that has been stretched in one direction. If you look at the picture of an ellipse in your book and tilt the page, you will see that it looks like a circle at some particular angle.
   • The ellipse is part of the family of curves known as conic sections, the geometric figures formed by intersecting a plane with a cone at various orientations. The conic sections are circles, ellipses, parabolas, and hyperbolas.
   • The ellipse has two points on either side of the center called foci. The total distance from one focus to any point on the ellipse then to the other focus is the same.
   • The long diameter of the ellipse is the major axis. The short diameter is the minor axis. Half the major axis (from the center to the farthest point) is the semimajor axis.
   • The family of ellipses is parameterised by the eccentricity, usually symbolised by e, which is the ratio of the distance between the foci to the major axis. A circle has e = 0 while a parabola has e = 1.
   • The ratio of semiminor axis b to semimajor axis a is given by b^2 / a^2 = 1 - e^2 .
   • This was Kepler's First Law of Planetary Motion: The orbits of planets are ellipses with the Sun at one focus.
   • He found that he had to abandon not only circular motion, but uniform motion. The planets move faster in their orbits when closer to the Sun than when farther away.
   • Kepler found this could be expressed by imagining the line from the Sun to the planet sweeping out an area as the planet orbited. Kepler's Second Law of Planetary Motion: A line from the planet to the Sun sweeps out equal areas in equal intervals of time.
   • He published these results in the 1609 book Astronomia Nova or the "New Astronomy". He came close to expressing the prinicple of mutual gravitiation, and he discussed the force holding the planets in their orbits as a sort of magnetism.
   • He later published reports on a new star (Kepler's Supernova), and in 1609 the book Harmonia Mundi (Harmony of the World).
   • Harmonia Mundi was mostly mysticism, concerning itself with the musical notes played by the planets as they orbited. He supposed the Earth to play the notes mi fa mi, for misery famine, misery.
   • Also in this book, he set down the mathematical relation between the period and size of the orbit. Kepler's Third Law of Planetary Motion: A planets orbital period squared is proportional to its average distance from the Sun cubed.
   • The proportionality can be scaled for all the planets from the Earth's orbit. If we measure the period P in years, and the semimajor axis a in astronomical units (AU), then we can write simply P^2 = a^3. Note 1 AU is just the semimajor axis of the Earth's orbit (1.49 x 10^8 km).
   • For example, Saturn has a period of about 29.5 years, so its semimajor axis is about 9.5 AU.
   • Kepler's Rudolphine Tables were exquisitely accurate, and a tribute to his theory.
   • Because he lived in Protestant northern Europe, Kepler was not persecuted for promulgating the Copernican hypothesis. It would be Galileo, his Italian contemporary, who would take the brunt of the Catholic Church's anger.
3. The Church and the Copernican Hypothesis
   • At first, Protestant and Catholic response to Copernicus' De Revolutionibus was muted.
   • Luther himself made the comment that "this fool would turn the whole art of astronomy upside down" - nowadays we would say the same thing as "to seem clever one would have to come up with something new". But in general, the Lutheran judgement was to hold Copernicus' work in high regard. It was extensively taught and discussed in Lutheran universities, and this is how Kepler was exposed to the heliocentric cosmology.
   • The potential ecclesiastical reaction to Copernicus' "radical" hypothesis was tempered by the preface added by the publisher, who took an instrumentalist stance by stating that the hypotheses of the work "need not be true or even probable" and that the essential point was to furnish a model whereby planetary positions could be calculated for any conceivable time. It was also helped by the fact that Copernicus didnt really state in the text whether his model was to be believed as the true nature of reality.
   • However, the young Kepler was attracted to heliocentric cosmology not by geometric abstraction, but expressly as a description of physical reality itself.
   • Kepler noted that there was a like-minded Italian astronomer with "the same first name as the last" - Galileo Galilei. Kepler began correspondence with Galileo.
   • In the opening decades of the 1600's both Kepler and Galileo challenged this instrumentalist view of astronomy, adopting the Copernican system not as geometrical convenience but as physical reality. At the same time they argued that the new interpretation of nature did not violate the inerrancy of Holy Writ.
   • Primarily because of Kepler and mostly Galileo, the Roman Catholic Church sought to make sure Copernicus' work was seen to be hypothetical and not as reality. The Church's battle was on the interpretation, not the content.
   • In the Decree XIV of the Holy Congregation in Rome (the Roman Inquisition) issued March 5, 1616, the De Revolutionibus was placed on the prohibited list "until corrected". The term until corrected was a standard vocabulary of the Inquisition, but only in this one case were the specific corrections actually announced.
   • The corrections to De Revolutionibus were spelled out in Decree XXI issued in 1620. They all concerned striking text which purported to interpret the heliocentric system as divinely inspired or constructed. None of the pertinent description of the model was changed. The censored versions of the book, printed after the Decree, were issued in Italy at this time, though the original version were predominant in Protestant northern Europe and in England. About 1 in 12 existing copies of the book are of the censored Italian version.
4. Midterm #1 Next Wednesday 2/14/96
   • Here are some Sample Problems for the midterm.

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