Parallax effects: if the earth moves around the sun, the angular separation of the stars should vary (C&N Fig. 3-11). The observed absence of this effect is consistent with the heliocentric model only if the stars are very distant.
Early Greeks assumed the size of the whole universe was just a few miles, but estimates kept getting larger as people travelled longer distances.
A later Greek estimate is that of Archimedes (287-212 BC).
In his book The Sand Reckoner, he estimates the number of grains of sand which would fill the universe, in order to demonstrate use of a mathematical notation which he had developed for representing large numbers. (Using modern notation his result was 10^63, where ^ denotes "to the power of").
To make his result as impressive as possible, Archimedes took the biggest universe which had been imagined: the heliocentric model of Aristarchus. To explain why parallax of the stars is not observed, Aristarchus had realized that the stellar sphere would have to be very distant.
Archimedes' subsequent calculation placed the distance to the stellar sphere about 6x10^12 miles (1 light year). This is a factor of 10^5 larger than Ptolemaic model, but a factor 10^12 less than modern values.
Measuring distances to either the stars or planets is not easy. On the basis of his model, Copernicus deduced the relative radii of planetary orbits to within 5%. So if the distance between any two planets could be measured (at known points in their orbits), all the distances would be known.
In principle, the easiest method of measuring distance is by triangulation (use of the parallax effect). We need to measure the position of a planet from two points on the earth a known distance apart. This is most conveniently done by observing the planets position relative to the stars (whose location does not change for observers on different parts of the earth, since they are so far away) simultaneously, otherwise the planet will have moved along its path (relative to the stars)
From geometry, D= d / theta (see Figure 1 below)
Problems:
By considerable effort, these problems were solved in 1700's, following many shipwrecks and subsequent offers of rewards for inventions which would improve the accuracy of navigation at sea (which depended on measurement of the elevation of stars at a known time).
Triangulation can also be used to measure the distance to nearby stars, in terms of the mean distance of earth to sun: the astronomical unit (a.u.) D= 2 a.u. / theta (see Figure 2 below)
But the effect is very small; the largest value (alpha-centauri) is 0.3 seconds of arc. (60 arc sec = 1 arc min, 60 arc min = 1 degree). The required accuracy was not achieved until 1838.
As a result, most intellectuals of the 16th century either rejected the heliocentric scheme or accepted it only as a calculational tool (planetary tables were based on Copernicus' model).
One exception was the Italian philosopher Giordano Bruno (1548-1600). He extended Copernicus' conception, arguing that the universe might be infinite, that life might exist elsewhere in the universe etc. Arrested by the Inquisition, he was charged with heresy and (after a 7-year trial) burned at the stake in Rome. The tide began to turn with
Son of a Danish nobleman. Intended for a career as a statesman; entered University of Copenhagen at age 13. But at age 14, he witnessed a solar eclipse and decided to study mathematics and astronomy. At age 17, made his first recorded observation: he noticed that Saturn and Jupiter so near together that they appeared indistinguishable. Upon checking the Alfonsine tables (1252), he found that this prediction was several days in error.
1572: a supernova became visible, even during the day ("new star", but really an exploding one). He observed it over a period of 16 months (> orbital period of earth) and established that there was no parallax , so the object had to be in the stellar sphere. This argued against the Greek idea of the heavens as perfect and unchanging, the original basis of geocentric models.
Tycho wrote a 52-page book on the supernova (now called Tycho's star) including its astrological significance. Retained a lifelong interest in astrology, as many astonomers did, partly because is was lucrative.
To dissuade Tycho from emigrating to Germany, Danish King Frederick built him a magnificent observatory on island of Ven (between Denmark and Sweden) with lavish buildings, outfitted with the best instruments of the day (not telescopes: C&N Fig.4-9). King Frederick gave an annual subsidy; first true observatory; cost several $million in today's money. As a result, Tycho's reputation continued to grow; he attracted scholars from all over Europe.
1577: spectacular comet appeared. Traditionally regarded as an atmospheric phenomenon, like meteorites. Also as a bringer of disaster, causing panic. Tycho studied it carefully and by parallax concluded that it was at least 3 times more distant than the moon. This again weakened the Greek idea of permanence of the heavens.
From the comet's motion, Tycho concluded that it could not be in a circular orbit: must be passing through the planetary spheres. As a conservative astonomer, he found this difficult to accept (can the spheres exist?). Up to his death, he rejected heliocentric theory; his main argument was the absence of stellar parallax; would require "too large" a stellar sphere (in this he disagreed with Galileo, with whom he corresponded).
1583: wrote a book on the comet, in which he suggested that all planets except the earth revolve around the sun. Then, remarkably, he made the sun and attendant planets revolve around the earth ("Tychonic model"; C&N Fig.4-10). Seemed to him the ideal theory: allowed the earth to remain at rest, as indicated by "common sense" arguments and Scripture.
Tycho came to believe that precision was vitally important in astonomy:
(1) for its own
sake and
(2) for testing the Copernican model (most data was very old and could
be improved upon). Also realized the value of repeated measurements:
(1) so that sources of random error could be averaged out (by taking mean)
(2) in order to obtain an estimate of the experimental error (error bars, standard
deviation). So he helped to lay the foundations of modern experimental
method.
He started off by using a crude instrument: a pair of jointed sticks to measure angular separation. Over the next 20 years, he refined his instrumentation and made very precise measurements of planetary motions. In fact, made the most accurate observations ever achieved without a telescope, many correct to 2 minutes of arc (2/60 degree), close to the "diffraction limit" for naked-eye observation.
Tycho also catalogued over 1000 stars, and some of his measurements are within 1' of modern values.
He determined the length of the year to within 1 second, and estimated distance to Saturn as 45 million miles (real value: over 800 million).
1588: King Frederick died; after a few years, his successor stopped Tycho's subsidy, so he left for Germany. However, soon moved to Prague, where he recruited Johannes Kepler.
Son of a professional soldier. Attended religious school but then went to University of Tubingen. Gifted in mathematics, he read and understood (!) Copernicus, whose heliocentric model he accepted.
First job was teaching science at the University of Graz. At same time, tried to make astrology into a genuine science, for casting horoscopes and solving Biblical questions - he worked out the date of creation as 3992 BC.
He corresponded with Tycho Brahe, and to escape religious disputes he eventually left Austria for Prague, where Tycho assigned him the job of analysing his measurements on Mars. There ensued a strained and tempestuous relationship between the two men. When Tycho died, Kepler inherited his position and finally gained access to Tycho's data on all of the planets.
Although mathematically gifted, he also had mystic tendencies. Believed in the "music of the spheres" (harmony generated by the planets' motion, if their orbital radii are in integer ratios) and reopened the question: why are there six planets (including earth) and what determines their positions? Like the Greeks, he found the answer in geometry: there are five "perfect solids" which can be constructed from regular polygons (C&N Fig.4-11). If we nest each of them between spheres (Fig.4-12), there will be six spheres, each representing the a planetary orbit. (The thickness of each sphere represents the deviation of each planet's orbit from a perfect circle, for which Copernicus had introduced epicycles; Fig 4-8).
Kepler spent many years trying to fit this model with observations. In the case of Mars (Fig.4-13), he resorted to the use of equants (rejected by Copernicus) but could never fit all the data points (accuracy = 2 mins arc) to better than 8 mins.
After five years and 900 pages of calculation, he decided that the data could not be accounted for unless he changed his assumptions. In place of circular motions, he tried first an egg-shaped oval, and then eventually an ellipse (Fig.4-14):
x^2/a^2 + y^2/b^2 = 1
This shape is characterized by two foci and mirror symmetry about the x- and y- axes. Its overall width is 2a along the x-axis and 2b along the y-axis. Then the Martian data could be fitted within 4 min, with the sun at one focus of the ellipse (within the orbit plane).
It is important to note that the flattening (relative to circle) is small: less than 0.5% difference between the major and minor axes, in the case of Mars.
In fact, he found the motions of all the planets could be fitted to ellipses, with the sun at one focus. This statement is now known as Kepler's First Law.
The fact that the sun occurs at a focus suggested to Kepler that the sun controlled the motion of each planet. To fit observation, the speed had to be non-uniform, with the planet moving faster when closer to the sun. After some trial and error, Kepler found that the radius vector between the sun and the planet (Fig.4-15) sweeps out equal areas in equal times, Kepler's Second Law. Both laws were included in Astonomia Nova (1609).
Kepler postulated that the planets are driven around their orbit by some tangential force produced by the sun.
Extending this idea to all the planets: those further away should be driven more slowly and exhibit longer rotational periods (C&N Table 4-1). He was able to derive a relationship (T proportional to r^3/2) between planet's period T and its mean distance r from the sun, now known as Kepler's Third Law (in a book dedicated to King James I, who tried to entice him to England to escape the Thirty Years War).
Kepler corresponded with Galileo, who had started manufacturing telescopes and giving them away as presents to princes. Kepler was able to borrow one for a month, to view moons of Jupiter (he invented the term satellite, which originally meant "hanger-on of a powerful man"). He wrote a paper explaining how the telescope operates, then designed a better one, also a compound microscope. In many ways, he was a founder of modern optics.