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Wednesday, 28 June 2017 20:52

Josée SERT

"EAAE Summerschools" Working Group

CLEA, France

I. The Ecliptic and the Zodiacal Constelations

Along one year, the Earth performs one revolution around the Sun. From a geocentric point of view, the Sun is seen achieving a complete turn from West to East on a large circle of the celestial sphere: the ecliptic, inclined to the celestial equator. Meanwhile, it crosses several constellations (that the Ancient delimited very early), which is particularly perceptible with the heliac risings.

Activity:

From an Ephemerides book, one can try to find out the dates at which the Sun enters each of these constellations. First, take a map of the sky with the constellations well delimited and the ecliptic drawn. You can see that the ecliptic crosses 13 constellations, and that the path of the Sun crosses their limits most of the time on a line of constant right ascension, sometimes of constant declination. The first work is to determine the good co-ordinate to get the dates of the passage of the Sun in them: that can be done using a table giving the situation of the constellations (right ascension and declination) - sometimes you have to make extrapolations- and a table giving day by day the co-ordinates of the Sun. So, you get a table which allows you to calculate how many days the Sun spends in each constellation.

That can be done by using software instead of an Ephemerides book.

Model:

This table allows the pupils to build a model: let's have a small Sun and a small Earth the axis of which will be inclined by an angle of 23°: these balls (polystyrene for example) will be set down on the floor. Let's say the radius of the Earth's orbit is 10 cm (Attention: at that scale, the Earth's diameter would be 0,001 cm and the Sun's 1 mm) and the radius of the celestial sphere 1 metre (with that scale, Proxima Centauri would be 27 km far away…). Then, you have to cut out of cardboard bands for the constellations with a length proportional to the time the Sun passes in them, and put them on the floor:

  • in which sense do they follow one behind the other, according to the orbital sense of the Earth?
  • in which relative position have they to be with the Earth and its axis?
  • where is g (place it on the cardboard with a coloured paper clip for example)?
  • where should the North celestial pole be (you can pin it on the ceiling)?
  • where should the celestial equator be?

Questions:

γ was the symbol of Aries, but nowadays, γ is in the constellation of Pisces…

Pick out of a newspaper the dates of the astrological Zodiac signs; write them down on a paper beside the dates of the entrance of the Sun in the Zodiacal constellations you computed before. Compare.

Kheops' pyramid (built about 2500 BC) was orientated North-South according to the direction of the polar star: α Draconis…

II. Precession of equinoxes

The inclination of the axis of the Earth on the ecliptic plane can be considered as constant: 23,5°. If on the model we suppose γ has moved of at least 30 ° since the astrological signs were defined, and if we assume that that motion will continue, we can notice that on the ceiling the pole will move in the same sense on a large circle the centre of which is the celestial North pole of the ecliptic. If we look at a sky map of that area, we can draw a circle which corresponds to an ecliptic latitude of 90°-23,5° : that circle passes very close to α Ursae Minoris, to α Draconis, and not very far from α Cygni (Deneb) or α Lyrae (Vega).

But what is the exact period of that phenomenon? Two types of years can be defined: the tropical year: mean duration between two consecutive passages of the Sun at g (or between two Spring equinoxes): 365 days 5 hours 48 minutes 46 seconds the sidereal year: mean duration between two consecutive passages of the Sun at the same point in the sky (located according to the fixed stars): 365 days 6 hours 9 minutes 10 seconds You have first to visualise that difference on the model: what does it mean for γ from year to year? Calculate the period of the motion. We can thus graduatethe figure for the displacement of the pole on the celestial vault, and so, answer to the questions : where were γ and the North pole for Hipparchus? Where was the pole for Kheops ? Can we understand the fact that for the Chaldeans, the bull (Taurus) symbolised the Sun at Spring about 3000 BC?

III. Ancient as astronomy

b) Egyptians and Mesopotamians

1. The Egyptians had a device to determine the time by observing the passage of some bright stars in the meridian during the night according to the date; during the day, they used vertical rudimentary sun dials and clepsydra, which gave not very precise results. When the pyramids were built, some corridors were oriented according to the North-South direction towards a Draconis. They did not collect observations (eclipses for example).

2. The Mesopotamians had gnomons (to determine passages of the Sun at the meridian, solstices), clepsydra and polos (hemispherical sundials where equator and solstices could be shown).Very ancient documents (from before 2000 BC) prove that some peoples in Mesopotamia had a strong interest in Astronomy (connected with astrology). They used the heliac star risings to rule their calendar. At least in the 12th century BC, they had defined the zodiacal constellations, and in the 7th, had divided the Zodiac in 12 segments of 30° each, which led them to get the ecliptic co-ordinates of about 30 stars. In a tablet from the 6th century, they described a regression of Mars. They studied the movements of the Moon and were strongly interested in eclipses: a collection of observations of eclipses could have been sent to Aristotle after Alexander's conquests; at any rate, Ptolemy writes about a complete list of eclipses since at least 747 BC.

3. It is not unlikely that the Egyptians may have perceived that those stars, by which they had fixed the directions of certain passages in their monumental buildings, changed their places in the course of centuries as those passages no longer pointed at their rising or culmination. It is also more than likely that the Babylonians must have noticed that the Sun at equinox or solstice didn't from one century to another stand at the same position on the Zodiac.

a) The Greek world before Hipparchus

1. They used the same instruments as in Egypt and Mesopotamia: Anaximander is credited with having introduced the gnomon among the Greeks, Eudoxus used to observe the azimuth of the rising Sun,… the polos was well known too. About 300 BC, Parmenion conceived a sundial that could be used under different latitudes.

2. The Greeks tried to find out the way the world was constituted and to explain (if not, to describe in a mathematical way) the phenomena they observed. Here are some main steps: We find a first explanation of solar eclipses by Thales, the Sun been covered by the Moon. Parmenides taught the spherical form of the Earth and that the Moon gets its light from the Sun; Anaxagoras gave the correct explanation of the Moon phases. Pythagoras said that the planets move in separate orbits inclined to the celestial equator, and his successors that they hold an opposite course to the fixed stars from West to East. Philolaos thought that Moon eclipses could come from the passage of the Moon in the Earth's shadow.

Plato helped greatly to spread the Pythagorean doctrines of the spherical figure of the Earth and the orbital motions from West to East; he stated that the ecliptic was distinct from the equator.

Meton had a good estimation of the solar and the lunar cycles, and is credited to have discovered that the Sun does not take the same time to describe the four quadrants of its orbit between the equinoxes and solstices.

Eudoxus: - he observed the motion of the Moon (he noticed it had not the same path among the stars from month to month and from year to year, he valued the retrogression of the nodes of the lunar orbit in 18,5 years and was aware of the Moon's changeable velocity in longitude),

  • he estimated the length of the year at 365,25 days,
  • he was aware of the periods of revolution (good except Mars), the stations and arcs of regression of the outer planets, and of their motions in latitude (but had no knowledge of their orbital changes of velocity)
  • he estimated at 24° the angle between the equator and the ecliptic.

Kallippus, pupil of Eudoxus, improved the luni-solar cycle of Meton (accurate knowledge of the length of the Moon's period of revolution), he gave best values of the length of seasons (less than 1 day error), he improved Eudoxus' system for Mars and paid attention to lunar eclipses and to the inequality of the Moon.

Both Eudoxus and Kallippus worked under the mutual influence of theory and observation.

The generation after Aristotle had noticed the variation of brightness of Venus and Mars and the variation of diameter of the Moon (reference to total or annular eclipses); they had also observed that Mars was always the brightest when it culminated at midnight (opposite to the Sun): Herakleides of Pontus and Apollonius set the theory of movable eccentrics. For Herakleides, the Earth turns round its axis in 24 hours and Venus moves round the Sun (to take in account its variable brightness). Aristarchus of Samos observed Summer solstices and inferred distances to the Earth from his observations of the Sun and the Moon. He combined observations and Mathematics and is the first to have expressed an heliocentric theory.

But in Alexandria, a school of systematic observers arose, who determined the positions of the stars and planets with graduated instruments, as well as it developed mathematics. There Astronomy became a science.

a) Hipparchus de Nicea

Hipparchus had at its disposal the observations made in Alexandria during the previous 150 years, in addition to observations made by himself as well as much earlier Babylonian observations of eclipses.

1. Measurements:

The Sun: the Sun: it was not possible to measure the distance of the Sun to the neighbouring stars because of its bright light, and so to place it precisely on the celestial sphere. So:

to get the declination: one could measure the length of the shadow cast by a rod at midday, so get the height of the Sun above the horizon and so the distance from the equator

to get the right ascension: one could note the time between the passage of the Sun across some fixed position in the sky (meridian for example) and that of a star at the same place, but with the precision of waterclocks…; one could use the Moon as a connecting link: position relative to the stars by night, to the Sun by day, but rapid motion of the Moon between the two observations…; in fact, Hipparchus got tables from the use of the eccentric. It is a circle the centre C of which does not coincide with the Earth.

He determined the position of the line of apses and the eccentricity because he knew very well the length of the seasons (94,5 days for the Spring, 92,5 days for the Summer, 178,25 for Autumn and Winter): it allowed him to represent the apparent motion of the Sun with an error of less than a minute of arc, which would be insensible until 1700 years…

The Moon: Hipparchus determined the 4 months (lunation, sidereal month, draconitic month, animalistic month), the angle of 5° of the orbit, the retrograde movement of the nodes in 18,67 years. For that, he saw the great interest of using eclipse observations upon a long time (Chaldeans for instance) for, as they take place near the Moon's nodes, were simultaneous records of the position of the Moon, the nodes and the Sun. But to describe all the movements, he had to use movable and inclined eccentrics, and the result was only good at new or full moon.

Concerning the planets, as Ptolemy wrote: “as he had not received from his predecessors as many accurate observations as he left, Hipparchus only investigated the hypotheses of the Sun and the Moon… he contented himself with collecting systematically the observations and showing that they did not agree with the hypothesis of the mathematicians of his time”.

2. Sizes and distances:

Hipparchus had a very correct idea of the distance and size of the Moon, he recognised that the problem of finding the Sun distance had to be left unsolved.

3. Catalogue of stars:

It included 1080 stars, with their (ecliptic) latitude and longitude and with their magnitude, gathered in the same constellations as Eudoxus and as we know to day. The construction of that catalogue led Hipparchus to a notable discovery.

IV. Hipparchus and the precession of equinoxes

1. To get co-ordinates of stars, Hipparchus made observations during moon eclipses just as his predecessors did, and he compared his results with those of Timocharis about 150 years before concerning Spica, a bright star near the ecliptic. During moon eclipses, it was easy to measure the distance between Spica and the centre of the Moon, and then, as the Sun is at exactly 180° from the Moon, to get the difference of longitude between the Sun and Spica. Especially if the eclipse happened near the equinox, the position of the Sun was precisely known, and the longitude of Spica could so be deduced precisely.

Timocharis had found for Spica a distance of 8° before the Autumn equinoctial point g. Hipparchus got 6°. The longitude of Spica had so increased of 2°.

2. Further inquiry showed that, in spite of the roughness of the observations, there was an evidence of a general increase in the longitude of the stars (measured from West to East) unaccompanied by any change of latitude. The agreement between the motions of different stars was enough to justify him in trusting his predecessors and concluding that the changes did not come from a movement of the stars, but from a change in the position of the equinoctial points (from which longitudes were measured); one of the two circles who define these points must have moved, and the fact that the latitudes did not change showed that the ecliptic must have retained its position and that the change must have been caused by a motion of the equator. Again, Hipparchus measured the obliquity of the ecliptic, as several of his predecessors had done before, and found no appreciable change. He inferred that the equator was slowly sliding from East to West, keeping a constant inclination to the ecliptic.

Increase of longitude of a star.
Increase of longitude of a star.
Movement of the equator.
Movement of the equator.

So γ had a movement from East to West on the ecliptic that Hipparchus estimated at at least 36" each year: but so, the Sun, in its annual motion from West to East on the same circle, returns to the new position of γ a little before returning to its original position with respect to the stars, and the successive equinoxes occur slightly earlier than they would otherwise.

From that derived the name of "precession of equinoxes", and Hipparchus was the first to recognise two different kind of years: the tropical and the sidereal. He fixed the length of each with considerable accuracy: after comparing his observation of the Summer solstice in 135 BC to Aristarchus' one in 280 BC, he inferred the estimate of 365,25 days for the tropical year had to be diminished by about 5 minutes, and the amount of precession gave about 10 minutes more than 365,25 days for the sidereal year.