(9c)   The Copernican Model of the Solar System

Nicolaus Copernicus

    Astronomers using the Ptolemaic system had already noted that the range of retrograde motions varied--large for Mars, smaller for Jupiter, still smaller for Saturn By proposing that retrograde motion occurred when Earth, in circling the Sun, overtook slower moving planets with larger distances from the Sun, Copernicus found a reason for it and also made it possible to estimate the distance R of an outer planet from the sun, or rather, the ratio R/R_E of that distance to the Earth's. It turned out that Jupiter's orbit was larger than that of Mars, and Saturn's was larger still.

    What follows shows how one way of deriving those ratios; I don't know whether Copernicus used it, but hope it will be clear to you, the user. (You may skip this part if you wish to avoid mathematics.) The method differs for the inner planets Venus and Mercury, and for the outer planets, among which Copernicus was aware of Mars, Jupiter and Saturn. including Earth, that gave the Sun six planets.

(1)     All planets moved in circles centered on the Sun
(2)     All planets orbited the Sun at constant speed.

    To obtain a planet's orbital period T, it is necessary to observe the synodic period T_s, the time required by the planet to return to the same position in the sky relative to the visible sun. That is not the orbital period, because all observations are from the moving Earth, which orbits the Sun with its own period T_E = 365.24 mean days (equal to 366.24 rotations of the Earth around its axis--see here). However, knowing the synodic period allows the true period to be derived.

    Because all distances are derived in units of R_E, known for that reason as the astronomical unit (AU). To derive it in kilometers or miles is a different problem and not an easy one. Even using the more accurate elliptical motion, we still only get distances in AU; to a good approximation, the average Earth-Sun distance is 1 AU ~ 150,000,000 km,.

Inner Planets

    The elongation of Venus is the angle between its direction in the sky and the direction to the center of the Sun. As the drawing shows, the elongation reaches its greatest value E when the sight line to Venus is tangential to the Venus orbit. That can happen in two positions--when Venus sets after the Sun (making it the "evening star") or (dotted line) when it rises ahead of the Sun ("morning star").

    E is about 47°. The tangent to a circle is perpendicular to the radius to the point of tangency, so we get

The greatest elongation of Mercury is about 23°, so Mercury can be visible only for a short time after sunset or before sunrise.

    How long does it take for Venus to return to the same position, viewed from Earth? Denote by T_V and T_E the orbital periods of Venus and Earth, and suppose motion is timed starting from the greatest evening elongation. A time t later, Venus in its orbit has covered an angle

(t/ T_V) 360°
and Earth has advanced a smaller angle
(t/ T_E) 360°
The angle between the lines connecting the Sun to those planets is

(t/ T_V) 360° – (t/ T_E) 360°
and it keeps increasing, until at time t = T_SV (synodic period of Venus) Venus is a full circle or 360° ahead. It is then at exactly the same relation to Earth as at t=0, i.e again at maximum evening elongation

(T_SV/ T_V) 360° – (T_SV/ T_E) 360° = 360°
Dividing both sides by 360° and then dividing again by T_SV gives
1 /T_V – 1 /T_E = 1/T_SV
T_SV can be measured by counting the days between two consecutive passages of Venus through maximum elongation, T_E is one year, so the orbital period T_V can be derived.

Outer Planets

    It is also possible to derive that ratio for an outer planet, using the retrograde motion of such planets, when they appear to reverse direction when viewed from the moving Earth. Retrograde motion (as noted before) arises because Earth moves faster than any outer planet and overtakes it. However the distance by which Earth can be ahead of the outer planet is just a fraction of the diameter 2R_E of the Earth's orbit-- sooner or later Earth turns around and its velocity in the overtaking direction shrinks to zero (while the perpendicular velocity away from Jupiter increases), and it even reverses in the half of its orbit further away from Jupiter. This allows (so to speak) to project the Earth's orbit on that of the overtaken planet, and derive the ratio of their distances.     Let Jupiter serve as an example here. Since it can rise and set at any possible separation from the Sun, let us choose at reference the time when it is most distant from the horizon (due south, viewed from the USA) at midnight. At such a time Jupiter, Earth and the Sun are lined up on the same straight line, JES.

    (How could Copernicus know when the line-up occurred at midnight, a time when the Sun is not visible? Astrologers and astronomers, already many centuries earlier, had tracked the apparent motion of the Sun around the ecliptic, using for reference constellations visible just after sunset and just before sunrise, to infer the Sun's direction in the sky relative to distant stars. They even contrived an instrument made of disks rotating around a common pivot--the astrolabe--to track such motion.)

    Again, count time t from the JES lineup, and let T_J denote the orbital period of Jupiter. After a time t had elapsed. Earth has advanced an angle

(t/ T_E) 360° and Jupiter (t/ T_J) 360°
The angle between the lines connecting the Sun to those planets is

(t/ T_E) 360° – (t/ T_J) 360°
After a synodic period T_SJ that angle has grown to 360¦. Which means Jupiter, Earth and Sun again form a straight line (it may now point towards different stars than before, but that's not important) Then

            (T_SJ/ T_E) 360° – (T_SJ/ T_J) 360° = 360° which simplifies to 1 /T_E – 1 /T_J = 1/T_SJ
By measuring the time (or counting the days, at the time of Copernicus) until Jupiter again passed south at midnight, T_J can be found. Of course, astronomers tracked the planet night after night, so that even if the sky was cloudy during the appropriate time, the position of the planet could be noted on a sky-chart.

    Distance R is a bit harder. Viewed from the direction of the pole star (or of the pole of the ecliptic, 23.5° away), planets orbit the sun counter-clockwise. However, when Jupiter is on the JES line, faster Earth is overtaking it in rotation around the Sun, so Jupiter appears to undergo retrograde motion. Viewed from Earth, the line of sight to it then rotates (relative to distant stars) clockwise. As astronomer, you should note its position among those background stars!

    This does not last: as the Earth goes around its orbit, its orbital velocity becomes increasingly perpendicular to that of Jupiter, and after a while even opposed to it. The apparent motion of Jupiter then becomes "prograde" again, until it passes again those same background stars as before. Measure the time T requires for this to happen.

    Because distant stars define a direction in space, at that time the Earth-Jupiter line is parallel to what it was when JES formed a straight line. Parallel lines maintain a constant separation, so let it be here denoted by D.

    Let R_J be the distance of Jupiter from the Sun. In time T, Jupiter has advanced an angle (T/T_J/)360° and therefore its distance from the original JES line is

    There existed two main problems with this theory. One arose from religious dogma: according to the psalms in the Bible, "God has set Earth on firm foundations so it cannot be moved" (in another translation, " cannot collapse"). The psalms are poetic prayers, and open minded popes realized that their words do not reflect astronomical observations. (In the Bible, too, Joshua commands the Sun to stop in its motion, suggesting that rather than Earth rotating around its axis, the Sun orbited around it.) Copernicus was quite cautious in voicing his theory and only published it at the end of his life. Because of that caution, many church scholars indeed viewed his theory as a possible alternative to Ptolemy's.

    In better times, Copernicus'es theory may have gradually gained accepted, instead of being proclaimed a heresy. But these were years of religious strife and war. The storm broke in 1609, when Galileo Galilei applied the newly telescope to the study of the heavens, and discovered extensive support for the views of Copernicus.

    The other problem was that while the Copernican model generally agreed with observations, small differences remained which could not be resolved. Consider the Earth: if it moved at constant speed along a circle, the year would be divided evenly by the equinoxes into a winter half and a summer half. Actually, the summer half (north of the equator) was two days longer.

    The answer came from very accurate pre-telescope astronomical observations by the Dane Tycho Brahe, analyzed by Johann Kepler. Kepler concluded that planetary orbits were not exactly circular, but oval (Kepler's first law), and that planetary velocities slowed down with greater distance from the Sun (Kepler's 2nd law). These laws will be described in the following section.

    Many books and plays exist on the life of Galilei, the Italian scholar who laid the foundation to the discipline known for many years as "natural philosophy," now called physics.

    He was the first to observe the planets through a telescope, and what he saw convinced him that Copernicus was right. How his agressive defense of the Copernican theory turned the Catholic church against him and cost him his freedom is a fascinating story, but it goes beyond our scope here. (For a 1636 painting of Galileo, with links to more information, see here.)

    Galileo did not invent the telescope; that was done by lensmakers in Holland and elsewhere (eyeglasses had been in use for centuries). Unlike later astronomical telescopes, which turn the picture upside down, the first version worked the way opera glasses do, combining two lenses of different types. Opera glasses magnify about 2-3 times: Galileo pushed the technology to its limits, magnifying his view 8-fold and in a later instrument 33 times.

    That was the instrument with which, in 1609-10, Galileo made his revolutionary discoveries. He observed the Moon and saw a world with mountains and "seas," and risking blindness (since the Sun should never be looked at through a telescope) he also observed sunspots. When he turned his telescope to the planet Jupiter, he saw four moons orbiting around it, all practically in the same plane, close to the ecliptic (and therefore, they and the planet all seem to lie on the same straight line; you can get the same view through good binoculars or any telescope), very much like a miniature version of the kind of solar system proposed by Copernicus.

    And when he looked at Venus, he saw its visible shape changing like that of the moon, becoming a crescent when Venus was between us and the Sun, a time when most of its sunlit half faced away from Earth. Galileo was persecuted for advocating the world view of Copernicus, but his observations, which were soon confirmed by other astronomers, convinced all scholars that this was indeed the way the Sun, Earth, Moon and the planets were related.

Exploring Further:

Site about Ptolemy.

Very detailed and long site on the history of astronomy.

Arthur Koestler, The Sleepwalkers, Arkana reprint edition 1990

A concise review, by Owen Gingerich, of "De Revolutionibus Orbium Celesium " (On the Revolutions of Heavenly Spheres) by Nicolaus Copernicus, appeared in the journal Nature--vol. 391, p. 140, 8 January 1998. A 1992 translation of the book by Rosen (452 pages) is available from Willmann-Bell ($39.95)--see http://www.willbell.com. That publisher is also offering other writings by Copernicus, "Galileo at Work" by Drake and various books on the history of astronomy.

Laura Fermi and Gilberto Bernadini: Galileo and the Scientific Revolution, Basic Books 1962

The History and Practice of Ancient Astronomy by James Evans, Oxford Univ. Press, 1998; reviewed by J.D. North, "Nature ", vol. 398, p. 385, 1 April 1999