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Historical lunar motion theories by Jacek Szubiakowski
- 1. Historical lunar motion theories and lunar samples in the Olsztyn Planetarium and Astronomical Observatory Jacek P. Szubiakowski Galaxy Forum Europe 2020 Vienna Astronomy from the Moon International Lunar Observatory Association and University of Vienna - Institute for Astronomy September 18th 2020 Olsztyn Planetarium and Astronomical Observatory 1
- 2. 2 Nicolaus Copernicus for four years, from 1517, was the administrator of the Warmia Chapter estates and he resided in the Olsztyn castle. Photo courtesy of the OPAO
- 3. The location of the astronomical table F. von Quast, „Denkmale der Baukunst im Ermeland“ in: Denkmale der Baukunst in Preussen, Ernst & Korn, (Berlin, 1852), bl. XXI 3
- 4. Copernicus’ solar dial in the Olsztyn Castle 4 Source: Photo courtesy of the Museum of Warmia and Mazury.
- 5. Historical lunar theories • Hipparchus • Ptolemy • Ibn al-Shatir – Copernicus 5 This work has been released into the public domain by its author, Tomruen.
- 6. Hipparchus’ lunar theory • The epicycle would move uniformly over the deferent. Earth Moon 6
- 7. Ptolemy’s lunar theory • The center of the mobile eccentric turns around the Earth. • The segment of a straight line connected the current position of the eccentric crank circle and the center of the epicycle. 7
- 8. 33 Ptolemy’s lunar theory • The large variation of the distance. • The apparent size of the Moon changes in the ratio as 33/17. Quadrature First or Last Quarter Conjunction or opposition Full Moon or New Moon MeanSun 33:17 740 51 8
- 9. Domenico Novara and Nicolaus Copernicus observed the occultation of the star Aldebaran ( Tauri). In Bologna on March 9th, 1497. The Moon near Aldebaran as presented by the Stellarium 0.20.2 planetarium software. 9
- 10. Copernicus’ lunar theory • The system of two epicycles with diameters in the ratio 1097:337. Earth Moon 1 2=21 10
- 12. Apollo 11 Neil Armstrong, Michael Collins, Buzz Aldrin 22 kg of lunar rocks 12 Photos courtesy of NASA
- 13. The only lunar crumbs from the Apollo mission in Poland are in Olsztyn 13
- 14. Pilbara Craton Soil as A Possible Lunar Soil Simulant for Civil Engineering Applications, Janusz Kobaka, Jacek Katzer and Paweł K. Zarzycki, Materials 2019, 12(23), 3871; https://doi.org/10.3390/ma12233871 14
Editor's Notes
- Astronomical traditions of Olsztyn go back to the sixteenth century and they are related to Nicolaus Copernicus. For four years from 1517 was the administrator of the Warmian Chapter estates. He resided in the Olsztynian castle. He carried out the astronomical observations and work on his main opus De Revolutionibus that embraces the heliocentric theory of the Universe.
- You can find there a rarity an astronomical instrument made by Copernicus himself. The dial, preserved partially on the wall of the cloister and presumably designed to determine the time of equinoxes, served as an astronomical instrument mapping the daily paths of the sun in the sky. The construction of the instrument is characterized by substantial originality. Due to its location, on the north-eastern wall of the cloister of the castle, the author was forced to use a novel method employing reflection of the rays of the sun.
- It has a form of several lines drawn on the wall in the castle’s gallery. The chart recorded observations made by Copernicus from January 25th to April 20th, 1517. In this way, he determined the moment of the spring equinox and as well as the duration of a tropical year.
- The Moon has been observed using different techniques for millennia from times of the Babylonian and Greek astronomers, down to modern lunar laser ranging. For a long time, it had been known that the motion of the Moon is not uniform. Its speed varies and the orientation and the shape of Moon’s orbit changes. In the past the motion of the Moon was also a great challenge, the solution of which was sought in various ways in the Hipparchian, Ptolemaic, and Copernican systems of astronomy. The problem of motions of the Earth-Moon system is sophisticated and even today it is not fully solved theoretically. There are in use two methods, which precisions is verified by Lunar Laser Ranging measurements. The first is the "semi-analytical" lunar theory (Éphéméride Lunaire Parisienne) based on a series expansion of the orbital elements of the Moon. The second based on numerical algorithms that takes account not only of gravitational forces and their relativistic corrections but also of many tidal and geophysical effects.
- Let us start an analysis from the Hipparchus’ epicyclic lunar theory that was further improved in later times. In this approach, the epicycle would move uniformly over the deferent, i.e. circular orbit around the Earth with mean Moon’s motion in longitude. While the period of the Moon revolution around the epicycle was an anomalistic month. For this theory to be used in practice, it was necessary to determine the relative sizes of circles consisting of the Moon’s orbit. Hipparchus devised a geometrical technique of the estimation these parameters from observations of lunar eclipses. The result, corrected later by Ptolemy, is a ratio of 60:5 1⁄4. Hipparchus' model reproduced actual lunar longitudinal motion imperfectly, providing merely for the simple, variation in the Moon's velocity, known as elliptical inequality or equation of the center. Its size was estimated at approximately about 5° 1’. It is much smaller than the modern value. The Hipparchus' theory functioned flawlessly when the Moon was nearby conjunction or opposition, with the mean Sun, but at quadratures, the Moon’s motion was too slow.
- Ptolemy discovered the second anomaly in the Moon's motion, called evection now. However, to account for it required that the Moon's epicycle be pulled closer to the Earth when it approached quadrature with the mean Sun in order to appear it to be traveling faster. As a remedy, Ptolemy introduced a mobile eccentric in his lunar motion theory. The center of the mobile eccentric turns around the Earth making that the line of apsides slowly rotates in space. The segment of a straight line connected the current position of the eccentric crank circle and the center of the epicycle. Two new parameters should be added in his model: the radius of the crank circle described by and the period of its motion. By a superposition of circular motions this ingenious lunar model accounted for two irregularities in the Moon’s motion. At new and full Moon, the maximum deviation from the uniform course was still 5° 1', but at first and last quarter it grows to 7° 40’. The angular position accuracy has been enormous and as a part of the whole Ptolemy's geocentric model of the Universe, it was in the use for more than 1000 years.
- The ancient astronomers focused on the angular positions of objects on the celestial sphere neglecting their true distances from the observer. Assuming the radius of the deferent as equal 1. The distance of the Moon from the Earth should change significantly, from 0.79 in perigee to 1.21 at apogee in Ptolemy’s model. This large variation influenced the apparent size of the Moon that changed in the same ratio as 33/17. A glance at the Moon let to state that it is not true. The eccentricity of the lunar deferent determines the ratio of the distance in perigee and apogee at 1- 0.21 to 1+0.21. The distance of the Moon itself may changes between 1.21+0.11 and 0.79 – 0.11, so that the apparent size of the Moon changes in the same ratio i.e. 33:17.
- It was the first known astronomical observation of Copernicus. He had made it the basis of the claim that it confirmed exactly the size of the apparent lunar diameter. https://www.britannica.com/biography/Nicolaus-Copernicus The observation showed that the distance to the Moon did not change as much as required by Ptolemy's lunar model.
- Arab astronomer, Ibn al-Shatir (1304–1375), and then Nicolaus Copernicus developed the new lunar theory. They replaced Ptolemy's mobile eccentric crank mechanism with a system of two epicycles with diameters in the ratio 1097:337. They moved on each other while revolving on deferent around the Earth. The center of the first epicycle completed one revolution of the deferent during the synodic month. The center of the second epicycle circled the first of them in the one animalistic month. The Moon completed two revolutions around the second epicycle in one anomalistic month. The Ibn al-Shatir – Copernicus' model had accounted for both: the first and second lunar inequalities. Making the Moon's orbit more elliptical it improves the precision of theoretical predictions.
- The exhibition in the laboratory of meteoritics covers two narrative streams. The first concerns the genesis of the solar system. The second narrative thread concerns types of meteorites, the story of searches for them and the most famous meteorite falls in the world and in Poland. A unique specimen in the collection are the lunar rock sample brought by the Apollo 11 crew on the first ever expedition to the Moon.
- On July 20, 1969 while walking on the lunar surface for two hours, astronauts Neil Armstrong and Edwin Aldrin collected 22 kg of lunar rocks.
- Four little crumbs from this amount are in our collection now. They were presented the Polish authorities by US President Richard Nixon during his Warsaw visit in 1972. The lunar soil was donated to the Olsztyn Planetarium during the ceremonial opening of the meeting of the International Union of the History and Philosophy of Science, which took place on September 5, 1973 as part of the national celebration of the 500th anniversary of the birth of Nicolaus Copernicus.
- The very limited volume of lunar soil samples means that they can be utilized only for the most important research programs. Then there is a lack of lunar soil simulant (LSS) fit for civil engineering applications. In the future permanent human presence on the Moon will be associated with significant construction efforts. Adequate technologies and building materials should be developed and tested prior to setting the actual building site on the Moon. Using as a tool Principal Component Analysis (PCA) researchers studied different kind LSS proved that Pilbara Craton soil is a suitable material for the creation of an affordable LSS for civil engineering applications. Cratons are the old parts of Earth continental plates. • That concludes my presentation. Thank you for your attention.