Astronomy

The Middle Ages

After Ptolemy and Before Copernicus

The extraordinary advances of Greek scientists like Aristarchus and Hipparchus were largely abandoned and nearly lost during the Middle Ages. The Great Library at Alexandria was burned in 391 CE.
For ~1000 years all intellectual activity in Europe was dominated by the Catholic Church, which incorporated the ideas of Plato and Aristotle into a static, geocentric worldview that was not open to question.
In the same time period, Indian astronomers and mathematicians and Muslim scientists in the Middle East made significant advances. The Arabs preserved old Greek writings which they had found in the lands they conquered. These eventually found their way back to Europe via Spain.
Even educated Europeans like Dante Alighieri (around 1300 CE) adhered to a worldview that had changed very little since Aristotle. In astronomy, the incredibly complex system devised by Ptolemy was considered the ultimate authority (Geocentrism + uniform, circular motion).
By 1400 CE, change was afoot: Universities, re-urbanization, prosperity

Dante's Universe

Dante Alighieri (1265-1321), Italian poet, considered the 'father of the Italian Language'.
His masterpiece The Divine Comedy showcases the world view of an educated European in the 13th century CE.
Following the teachings of St. Thomas Aquinas, Dante assigned Heaven and Hell physical places in the Universe.
The rest is almost straight from Aristotle (1500 years later!): concentric spheres, with more distant planets being more pure, and closer ones more corruptible.
The last (9th) Sphere is the Prime Mover, and beyond is the Divine (Paradise).

Nicolaus Copernicus

Nicolaus Copernicus (1473–1543). Born in Poland of German ethnicity, contemporary of Christopher Columbus.
True 'Renaissance man': mathematician, astronomer, physician, polyglot, classics scholar, translator, diplomat, and economist.
Attended the University of Kraków (now Jagiellonian University), and spent three years at the University of Bologna in Italy, where he studied the classics in great detail (including Aristarchus!) and worked with astronomer Domenico Maria Novara.
Spent most of his life as a canon of the bishopric of Warmia (i.e., was an employee of the Catholic Church).

What Bothered Copernicus

It is important to remember that Ptolemy's model matched the observations available at the time.
However, the model was very complex. More importantly (to Copernicus), it was aesthetically unpleasant. Copernicus felt that the introduction of equants violated the principle of uniform, circular motion (Plato).
Why are Mercury and Venus never seen far from the Sun, unlike the other planets? How come the Sun does not need epicycles? Why is it all so complicated?
Copernicus knew about the work of Aristarchus and the idea of heliocentrism.

The Commentariolus

Around 1512, Copernicus wrote a 40-page outline of his ideas and circulated it privately to his friends. The Commentariolus had seven propositions:

Celestial bodies do not all revolve around a single point.
The Earth is only the center of the lunar sphere (Moon’s orbit).
All other spheres rotate around the Sun, which is, therefore, the center of the Universe.
The distance between the Earth and the Sun is an insignificant fraction of the distance to the stars, so stellar parallax is not observed.
The stars are immovable; their apparent daily motion is caused by the daily rotation of the Earth.
Earth is moved in a sphere around the Sun, causing the apparent annual migration of the Sun (the Earth has more than one motion).
Earth's orbital motion around the Sun causes the retrograde motions of the planets.

Parallax in a Heliocentric Universe

In a heliocentric Universe, the Earth moves around the Sun every year. This should cause an effect called parallax: nearby stars should move with respect to more distant ones.
This was a common critique of Aristarchus' model in antiquity, and Copernicus anticipated it in the Commentariolus. Parallax is very hard to see only if all stars are very far away.

Elongation (angular distance from the Sun) of the Inner Planets

What about the elongation of the inner planets, Mercury and Venus? They are never seen far from the Sun.
Ptolemy solved this by 'tying' the motion of their spheres to that of the Sun – the only planets with this feature!
This is explained naturally in a heliocentric model. It is also obvious why these are the only planets with transits

Retrograde Motion

Retrograde motions are also explained without the need for epicycles.
The orbital period of the planets is longer the farther they are from the Sun: 1 year for Earth, 1.9 years for Mars, etc.
As the faster moving Earth overtakes the slower moving outer planet, the position of the planet against the stars seems to 'loop'.
Note that the planet appears brightest during the retrograde loop!

The Phases of Venus

The Ptolemaic and the Copernican systems make different predictions about the phases of the inner planets.
In the Ptolemaic system, Venus will never show a half, gibbous, or full phase. Venus will only go through a new or crescent phase.
In the Copernican system, Venus will go through all phases.
Since these phases cannot be seen with the naked eye, the resolution of this problem had to wait for Galileo’s use of his telescope to view Venus.

De Revolutionibus Orbium Coelestium

On the Revolutions of the Heavenly Spheres

Most of the scholars who were shown the Commentariolus were impressed by the elegance of Copernicus' model.
However, Copernicus was very reluctant to publish his findings, partly because he was a bit of a perfectionist, and partly because he was afraid of being criticized or (much worse!) accused of heresy.
De Revolutionibus was finally published in 1543, just as Copernicus lay on his deathbed. It included a note by theologian Andreas Osiander warning the reader that 'these hypotheses need not be true, nor even probable’.

Copernicus: Summary

The Renaissance triggered a widespread intellectual revolution. Humanism became the leading philosophical system, shifting the focus from God to Man. This led to many profound changes.
Nicolaus Copernicus was not satisfied with Ptolemy's complex geocentric system of epicycles, deferents, and equants, with its special cases and exceptions. Although it reproduced the observations available at the time, Copernicus felt that it strayed too far from the Platonic ideal of uniform, circular motion.
He suggested a much simpler heliocentric system that could explain a wide variety of phenomena (retrograde motions, limited elongation of the inner planets) with a unified framework. In this system, the Earth is just another planet that goes around the Sun once a year and rotates every day.
At first, Copernicus' system was only accepted as a device to facilitate calculations.

After Copernicus

For decades after the publication of De Revolutionibus in 1543, scholars did not know what to make of it. Copernicus' model was often discussed in academic circles, but its heretical implications were swept under the rug.
Copernicus made calculations simpler, but Ptolemy could also explain the observations available at the time, and was not in conflict with Aristotelian doctrine.
However, more accurate observations were about to become available...
The key test of any scientific theory is to falsify it using specific and quantitative predictions.
There is one fundamental aspect in which the Ptolemaic and the Copernican systems make different predictions: the phases of the inner planets.
Unfortunately, these phases are not visible to the naked eye. It took Galileo’s telescope to confirm Copernicus’ theory

Is the Copernican Universe Infinite?

One past objection to the concept of an infinite Universe had to do with the impossibility of spheres at infinity moving (i.e., a sphere at infinity would have to move at infinite velocity, which was not possible). This problem was eliminated in the Copernican Universe since it was the Earth that rotated.
Thomas Digges (1546-1595) introduced the concept of an infinite Copernican Universe. This idea slowly created a major religious controversy - if the Universe is infinite, where is Heaven? Were the stars in Heaven?

Giordano Bruno (1548-1600)

A member of the Catholic clergy, Bruno introduced many modern ideas.
Space extends without limit in all directions, and it has no central point.
Bruno distinguished between the Earth and the planets known at the time (i.e., our Solar System) and the entire Universe, which he said consisted of countless worlds. The stars were like the Sun.
Bruno not only suggested that there were countless worlds, but that life exists on other planets.
However, Bruno was primarily a mystic, and his approach was not very scientific. Bruno's approach was basically speculation.
The Church condemned Bruno, but Bruno remained religious and claimed that there was no limit to the power of God and that God could have created such a Universe.
Nevertheless, Bruno was burned at the stake in 1600 for espousing these ideas.

Tycho Brahe

Tycho Brahe (1546–1601), Danish astronomer, made the most accurate astronomical observations before the invention of the telescope.
Of noble birth, at age 2 he was kidnapped and adopted by his uncle Jørgen, who wanted to make a scholar of him. Attended the Universities of Copenhagen (at age 12!) and Rostock (in Germany).
Became interested in astronomy after witnessing a solar eclipse in 1560 – he was amazed by the fact that the eclipse had been predicted.

Tycho Brahe's Stella Nova of 1572

When Tycho was 26, he discovered a new star in the constellation of Cassiopeia. He observed it for several months as it brightened and faded away, carefully measuring its parallax. He of course, could not find any, concluding that the new star was in the sphere of fixed stars. The heavens were not immutable after all!
He published his findings in a book, De Stella Nova, which appeared in 1573. The book earned him considerable fame and King Frederik II of Denmark granted him a large estate on the Island of Hven and funds to build an observatory.
Tycho's stella nova was a supernova that exploded in the Milky Way
Still observable today

Tycho Brahe's Colorful Life

At age 20, he lost his nose in a duel with his distant cousin, a fellow student at the University of Rostock, arguing about who was the better mathematician. Afterwards, he wore a prosthetic nose made of gold.
Did not marry the mother of his eight children.
Kept a pet elk, which died after breaking a leg while going down stairs drunk.

Tycho Brahe : Observations

In 1577, Tycho observed a comet and also failed to find any parallax, implying that it must be beyond the Moon. He noticed for the first time that the comet's tail always pointed away from the Sun.
Tycho excelled at building ingenious instruments that greatly improved the accuracy of naked eye astronomical observations. He built many of them at Uraniborg - some were several stories high. He described them in his book Astronomiae Instauratae Mechanica (1598).
Tycho achieved an unprecedented precision of about one arcminute (1' = 1/60th of a degree!) in his stellar and planetary positions

The Tychonic System

Tycho wanted to produce a definitive catalog of planetary positions that could be used to decide between the systems of Ptolemy and Copernicus.
He recognized the great simplicity of Copernicus, but could not bring himself to accept heliocentricity – he devised his own Tychonic system, where the planets orbit the Sun, but the Sun orbits the Earth. This required abandoning the concept of 'crystalline spheres'!
His observations were so good that he could make none of these three systems work. In desperation, he invited Johannes Kepler to join him at Prague in 1600.

Kepler

Kepler’s model of the Solar System

In 1596, after many years of work, Kepler published his “Mysterium Cosmographicum”. The distances between the six planets could be represented by the 5 Pythagorean solids, nested such that each planet is on the inner sphere of a polyhedron whose outer sphere contains the next planet. The last, sixth sphere, represented Saturn, the outermost planet. The five regular solids thus rationalized the existence of six planets.
The octahedron, icosahedron, dodecahedron, tetrahedron, and cube.
He was convinced that these were the shapes that God used to support the six spheres that in turn kept the planets in their orbits.
However, Kepler could not make his model fit the observations of the time. They differed by about 10%. He hoped that better observations might help, so he sent his book to Tycho. Meanwhile, Tycho wrote to Kepler’s math professor in search of a mathematical assistant who could match his observations to his own model of the Solar System (the Tychonic model). Kepler got the job and left for Prague to work with Tycho in 1600.

Kepler and Tycho

Their relationship was somewhat strained, but they needed each other. Kepler said that 'Tycho has the world's finest observations, but he lacks the architect that can construct an edifice out of them’.
When Tycho fell ill in 1601, he is reputed to have said 'Let me not seem to have lived in vain'. After his death, Kepler more or less stole all of Tycho's data. He was appointed imperial mathematician by Rudolf II.
Kepler published all of Tycho's observations and these would become the Rudolphine Tables, published in 1627.

Kepler’s Three Laws of Planetary Motion

Planetary orbits are ellipses (not circles as the Greeks demanded). This was a great revelation given the importance that the Greeks placed on the circle.
A planet in its orbital plane sweeps out equal areas in equal amounts of time. This means that the closer a planet is to the Sun, the faster it moves.
The equation P2 = D3 applies, where P is the period of a planet's revolution around the Sun in years and D is the planet's distance from the Sun (specifically its semi-major axis) in astronomical units (AU) with an AU being the Earth-Sun distance (≈ 150,000,000 km).

Ellipses

An ellipse is the locus of points such that the sum of the distances from two points (the foci) to any point on the ellipse is fixed, d1 + d2 = constant.

a is the semi-major axis.
b is the semi-minor axis.
The distance between the center of the ellipse and one focus is c.
The closer the two foci are together, the more nearly circular the ellipse is.
The degree of ellipticity is called the eccentricity, e, of the ellipse; e = c/a
- A circle has eccentricity, e = 0.
  - the more oval that circle becomes, the higher the eccentricity
  - when it reaches a flat line, it becomes e=1.00

Kepler’s First Law

Kepler's First Law: Planetary orbits are ellipses with the Sun at one focus.

Nothing exists at the other focus.

Keplerian Orbits

The period P of an orbit is the time required to complete an orbit.
The period of Earth's orbit is one year. Its semi-major axis is one astronomical unit (1 AU = 1.5x108 km = 150,000,000 km).

Kepler's Second Law

Kepler's Second Law: A line joining a planet and the Sun sweeps out

equal areas during equal intervals of time.

The time it takes the planet to go from t1 to t2 is the same as it takes the planet to

go from t3 to t4. And Area A = Area B

The speed of a planet changes – it moves faster the closer it is to the Sun.

Sizes and Speeds

Planets closer to the Sun orbit it more quickly. This fascinated Kepler ever since he came up with the idea for his Mysterium Cosmographicum.
For example, Mercury’s period is 88 days and Jupiter’s is almost 12 years.
The circumference of Mercury’s orbit is about 3 AU, the circumference of Jupiter’s orbit is about 33.7 AU.
If Jupiter moved at the same average speed as Mercury, its period should only be 2.7 years.
There must be a reason for this!

Kepler's Third Law

Kepler's Third Law: The square of the period of an orbit is proportional to the cube of the semi-major axis.
For the Solar System, if P is in years and A is in AU, P2=A3 or P=A3/2.
Kepler called this the Law of Harmonies and published his third law in the book Harmonices Mundi in 1619

The Orbit of Comet Halley

Kepler's laws of planetary motion apply to all bodies moving in a gravitational field: planets, comets, asteroids, spacecraft...
Comets have very eccentric orbits. Halley's comet has A = 17.8 AU ⇒ P = A3/2 (with P in yr and A in AU) ⇒ P = 75 yr. The orbit has e=0.97!

Reception of Kepler's Laws

Kepler's work was not immediately accepted. Many resisted the elimination of circular, uniform motion, which had been held true for almost 2,000 years.
Many important contemporaries ignored Kepler's work, including Descartes, Galileo, and his own former teacher, Maestlin.
Kepler died in relative anonymity in 1630. He spent the last years of his life trying to convince astronomers to test his laws, and defending his mother in trials where she was accused of witchcraft.