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On 06 Oct 2018 at 11:36:56 user 'jcobban' wrote: Greek Theory of Planetary Motion

The objective of a theory of planetary motion is to match historical observations and then to use the theory to predict future motion of the planets. Most commonly throughout history this has been for the purpose of astrological forecasting.

Popular culture interprets the word “theory” most commonly using definition 3a from the Merriam-Webster Dictionary: “an unproven assumption”. But in this discussion, as in any scientific analysis, the first definition is used: “a plausible or scientifically acceptable general principle or body of principles offered to explain phenomena.” In particular a theory of planetary motion endeavors to specify the position in the heavens of each of the planets relative to the non-rotating spherical coordinate system which is fixed to the background of stars. Most commonly what is desired is to determine the latitude relative to the path followed by the Sun around the sky, called the ecliptic, and the longitude along the ecliptic from the point where the path of the Sun crosses from below the extension of the Earth’s equator to the celestial sphere to above at the Spring equinox1. In Astrology the point where the ecliptic crosses the equator is called the first point of Aries although the location is now in the constellation of Aquarius2. In this discussion the word planet is used in its original meaning, from the Greek πλάνητες (planetes) meaning wanderers, of any celestial object that was not fixed to the celestial sphere.  The seven planets were the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn.  In the Western Roman Empire each of these planets was believed to be associated with one of the days of the week.

A basic problem is that a theory based simply upon Platonically “saving the phenomena” cannot be better than the quality of the observations and if the underlying understanding of the theory is flawed then the resulting predictions will be unreliable. Specifically the Hellenistic clockwork style theory, most popularly implemented using a deferent and epicycles, is mathematically equivalent to the determination of the coefficients of a sum of trigonometric sine functions as developed by Joseph Fourier (1768-1830). A Fourier series is an efficient way to express any function which has repetitive or periodic characteristics. For example a Fourier series is used to describe modulated radio transmissions as modifications of the electromagnetic carrier wave which is expressed by a sine function, and Fourier series are also used in analyzing and compressing images. However a trigonometric series can be established to describe the past behavior of any phenomenon and yet not reveal anything about the actual nature of the phenomenon. For example you can generate a Fourier series which matches the hour by hour temperatures at a location or the value of a stock with any desired level of accuracy. But when that series is evaluated for future times you will find that the predicted values will increasingly disagree until after just a short time there will be almost no correlation. This is what is known as chaos theory. Even the motions of the planets are subject to chaos, it is just that the time frame for chaotic divergence is much longer than human time.

A major problem with all theories of planetary motion prior to the European Renaissance is that the observations upon which the theories were based were poor. Comparison of the observations recorded in the Venus Table of Ammisaduqa, from the 17th century BCE, and the planetary and stellar coordinates recorded by Hipparchos of Nicaea, 2nd century BCE, as preserved by Claudius Ptolemaios (100-170), demonstrate that the observational methods were only accurate to about 1/12th of a degree. Hipparchos was probably using an armillary sphere or possibly an astrolabe, devices for the purpose of measuring angles, which he may have invented and was certainly the first astronomer to exploit, and the observational errors are consistent with the limitations of those instruments. One issue was that using these tools the observer could not directly measure the position of an object relative to the coordinate system. You could only measure the position of each object relative to the objects near it. Another problem faced by Hellenistic astronomers was the lack of an accurate clock, since they had nothing more consistent than a clepsydra. A clepsydra measures time by measuring the rate at which water drips out of a small hole in a tank of water where the depth of the water was kept constant. A clepsydra was used to control the display of the time of day and the phase of the moon in the first public market clock, the Tower of the Winds in the Agora of Athens.  The inaccuracy of the observations made not just by Hipparchos but by every subsequent astronomer until the 16th century is difficult to understand as Tycho Brahe (1546-1601) did not have access to any technology, aside from the clock escapement, or mathematical theory beyond what he inherited from Islamic astronomy which itself was largely derived from the Μαθηματικὴ Σύνταξις of Claudius Ptolemy. In particular Brahe did not have the use of a telescope, invented by Hans Lippershey in 1608, or the pendulum-governed clock, invented by Christian Huyghens in 1656, and yet he published observations of the motion of the planets which were accurate to the limit of his excellent vision, about 1 arc-minute or five times better than the Hellenistic and Islamic observations.

Any planetary theory which is based upon a sum of circular motions, or trigonometric functions, gains in complexity with the accuracy of the observations which must be “saved”. The heliocentric theory published by Nikolaus Copernicus on his death bed in 1543 was only slightly simpler than the geocentric models which had preceded him and was no more persuasive than the earlier proposal by Aristarchos of Samos because it suffered from all of the same logical problems. It was based upon the same innacurate measurements as all of the geocentric models.  The increased precision of the observations of Tycho Brahe made the number of epicycles required to “save the phenomena” untenable.

However given the limited precision of the observations that Hellenistic astronomers had access to the geocentric model was convincing. The two most obvious planets, the Moon and the Sun, clearly circled the Earth, moving in stately procession against the background stars, the size and number of epicycles required to satisfy past observations and accurately predict future phenomena such as eclipses was modest, and the system of epicycles not only explained the position of the Sun and Moon, but also was consistent with the way that the angular dimensions of the Sun and Moon changed through time indicating a change in the distance. One of the most persuasive arguments for the geocentric model was its ability to predict eclipses. Three of the planets, Mars, Jupiter, and Saturn, also appeared to move around the heavens independently of the Sun. However Mercury and Venus were different in that they always remained close to the Sun. A geocentric model does not explain why those two planets, and only those planets, have epicycles that coincidentally keep them apparently attached to the Sun’s apron-strings.

It is unfortunate that apparently no Hellenistic astronomer or philosopher chose to start the construction of a theory of planetary motions by focusing on the single most predictable of all of the planets: Venus.

Of all of the planets the heliocentric orbit of Venus is the most circular. Ever since Hipparchos the Greeks were aware that the “orbit” of the Sun was not circular and that consequently the distance between the Earth and the Sun varies throughout the year while the rate of progression of the Sun along the ecliptic also depends upon the season. It was not until Johannes Kepler (1571-1630) that it was realized that the rate of angular motion was inversely proportional to the distance between the Earth, or any planet, and the Sun. If anybody had bothered to plot the longitude of Venus relative to the Sun, rather than to the stars, it would have been trivially obvious that Venus traced exactly the same motion year after year, century after century, since the observations of the Venus Table of Ammisaduqa almost two thousand years earlier. The Maya in Meso-America observed that and made it a fundamental part of their calendar cycle! Indeed the biggest problem in Mesopotamian Bronze Age Chronology is that it is impossible to determine the exact date of the Venus Table of Ammisaduqa because the motion of Venus is so monotonous that every single appearance of Venus looks exactly the same as every other appearance so the celestial events described by the tablet would have exactly repeated every 56 years.

A weakness of a heliocentric model for the motion of Venus is that it did not explain the brightness of Venus at various stages of its path. But that is only because of the assumption that Venus is a “star” with an intrinsic brightness which will appear to vary, like the brightness of the Sun and Moon, only based upon how far away Venus is at each stage of its path. It was not until Galileo turned his telescope on Venus and observed that, like the Moon, it displayed phases based upon the angle between the Sun and Venus, that this was reconciled. Those same observations also demonstrated that the angular size of the planet varied in a way consistent with a heliocentric orbit.

The heliocentric model suffered from a fundamental problem discussed by Archimedes in Ψαμμίτης (Sand Reckoner) when he discussed the heliocentric proposal of his contemporary Aristarchos of Samos. If the Earth went around the Sun then the position of the “fixed” stars should vary through the year just as the apparent relative positions of structures in a county fair seem to change as you spin around a merry-go-round. This is an effect calls parallax. Aristarchos had already demonstrated that the Sun was much farther away from the Earth than anyone had previously thought, because otherwise the half-phase of the Moon would not be observed when the Moon was 90 degrees away from the Sun as viewed from the Earth, within the observational errors of the time. But if the Earth was orbiting a long way away from the Sun then the stars must be an incredible distance away not to show parallax. Archimedes referenced Aristarchos theory in part because it gave him a way to demonstrate the use of the exponential notation that he had invented to represent very large numbers. However if the stars were that far away then each star, whose angular size was about 1 minute of arc as observed by the naked eye, must be enormous even compared to the size which Aristarchos had already calculated for the Sun. Even when the invention of the telescope demonstrated that the angular size of the stars was much less than 1 minute of arc, this issue was not resolved because the telescope also revealed that the parallax of the stars was still indetectible even at great magnification and therefore the stars were farther away by the same proportion that the observed angular size of the stars had been reduced. Indeed this philosophical objection was not resolved until Isaac Newton demonstrated that the observed angular size of a point source of light depended upon the aperture of the telescope, not on its intrinsic size. All stars have a smaller angular diameter than can be observed in all but the largest professional telescopes. In any event the maximum parallax, observed in the closest star to the Sun, is less than 1 arc-second, 300 times smaller than the accuracy of Hellenistic observations, and so small that it was not until 1838 that Friedrich Bessel first measured it, over 200 years after Kepler described accurate heliocentric model based on Tycho Brahe's measurements, and Galileo went into house arrest for publishing Dialogo sopra i due massimi sistemi del mondo.

In 1989 the European Space Agency launched the Hipparcos satellite which measured the positions and paralaxes of 120,000 stars to an accuracy of better than 0.001 arc second, 300,000 times more accurate than the positions of the 850 stars in the catalogue created by Hipparchos of Niceae. It’s successor, the Gaia spacecraft, is designed to measure the positions and paralaxes of a billion stars to an accuracy of 20 micro arc seconds, 15 million times more precise than the Hellenistic observations.

It is unfortunate that the main-line of Greek philosophy did not consider an enormous advantage of the heliocentric model. If the cosmos is indeed geocentric then it is impossible to measure the distance to any planet beyond the Moon with any precision because the greatest possible baseline is the diameter of the Earth, which is tiny compared to the distances to the planets. For example the angular difference between the position of Venus as observed from Europe and from other places in the world including by a team in Tahiti supported by Capt. James Cook permitted determining in 1769 that the solar parallax was about 8 arc-seconds, forty times smaller than the errors in Hellenistic measurements. Without good telescopes all you can measure is the position of the planet against the stars, not its distance. But if the planets, with the exception of the Moon, orbit the Sun then you can determine their relative distance at each point of their path around the Sun  using the baseline of two positions of the Earth separated by one orbital period of the other planet. When Tycho Brahe did this, even though he still believed the Earth must be the stationary centre of the Cosmos, he discovered that relative to the Sun all of the trans-lunary planets followed the same path every orbit to the limit of his excellent observations and it was only from the perspective of the Earth, not at the centre of the planetary motions, that the planets seemed to follow a complex path against the stars. Frustratingly those constantly repeating paths were not the perfect circles that philosophy taught they should be following to satisfy Plato, but Brahe’s assistant Johannes Kepler determined that the paths followed by the planets were the next best thing, conic sections.

Until the 1950s the predicted positions of the planets continued to be computed using trigonometric series. That is in essence by a deferent and epicycles. However instead of the dimensions of those epicycles being determined by trying to “save the phenomena” they were calculated by application of Newton’s Theory of Gravitation. With almost four thousand years of observations behind them, the last four centuries benefiting from the use of telescopes, mathematicians such as Carl Friedrich Gauss (1777-1855) and Simon Newcomb (1839-1909)3 laboriously calculated the mutual gravitational effects of all of the planets on each other, adjusting the estimated mass of each of the planets in the calculations until the predictions matched the observations. When this mathematical model failed to accurately predict the motion of the planet Uranus the mathematician Urbain Jean Joseph Le Verrier (1811– 1877) determined that an as yet undiscovered planet must exist and predicted its position with such accuracy that the predicted planet, named Neptune, was found by astronomers at two different observatories on the very first clear night after they received the telegraph notice.  This was such a striking result that when the motion of Neptune did not match exactly to the predictions Percival Lowell built an observatory at Flagstaff, Arizona, for its clear desert skies, and launched a search for a 9th planet whose gravity would explain the deviation. When no such planet was found, tiny Pluto is not big enough, it was discovered that the assumed masses of Uranus and Neptune were slightly wrong, as later verified by direct observations by the Voyager spacecraft.

Since the introduction of computers the positions of the planets are computed by directly applying Newton’s Theory of Gravitation as enhanced by Einstein’s Theory of General Relativity. Furthermore the many space probes that have been sent out, at least one to every single major planet plus the biggest of the dwarf planets4, means that we now can predict the position of the planets to within a few metres, and we know the masses of all of the planets and all significant moons to at least 6 decimal places based upon their mutual gravitation.

1Specifically since the orientation of the equator to the fixed stars slowly changes as a result of precession, as discovered by Hipparchos, and because the plane of the Earth’s orbit even more slowly changes as a result of the gravitational pull of the other planets, the coordinates of the Sun and the other planets are measured against the equator and ecliptic of a specific date, currently 1 Jan 2000 at 12:00 Terrestrial Time.

2Hence the song in the musical “Hair”.

3Simon Newcomb was born in Wallace, Nova Scotia.

4The Dawn space probe orbited Vesta before moving on to its current position orbiting Ceres. New Horizons observed the dwarf planet Pluto in 2015 on its way to explore the Kuiper Belt.

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