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Measuring the Distance to the Sun” The Internet Project “Measuring the Distance to the Sun” Udo Backhaus, Universit¨at Koblenz November 26, 2007 We will try to determine the magnitude of the solar system by measuring the distance of a nearby minor planet. Therefore we are looking for groups inter- ested in parallax measurements by means of ccd imaging. If we will find groups (spread all over the earth, if possible) we hope to become able to determine the asteroid’s parallax and by that to measure the distance of the sun. Contents 1 Introduction 2 2 Why is it important to know the sun’s parallax? 2 3 A short history of the sun’s parallax 4 4 Methods of measurement 5 5 The project 6 5.1 The goal of the project . 6 5.2 Determination of the distance of a minor planet . 6 6 How to do measurements 6 6.1 Determination of the asteroid’s topocentric position . 8 6.2Combiningtwosimultanouslymeasuredresults................ 8 6.3 Combining two positions measured at the same observatory . 9 6.4 Comparison of one topocentric result with the corresponding geocentric position..................................... 10 6.5Shortdiscussionofthethreemethods..................... 10 7FirstTests 11 References 11 1 1 Introduction The radius of the earth’s orbit around the sun is one of the most important constants in astronomy: the Astronomical Unit (AU). Its value is the basis not only for determing the dimensions and structures of space but for measuring the astrophysical properties of planets and stars. Because of the large distance to the sun all related effects are very small. For this reason, the measurement of the Astronomical Unit is very difficult and was one of the main problems of astronomy over hundreds of years. Up to now, there exists no possibility to determine it at school by own measurements. In the second half of 1996 the European Association for Astronomy Education (EAAE) in cooperation with the European Southern Observatory (ESO) organized a world wide internet project, “the World’s biggest Astronomy Event on the World Wide Web”: As- tronomy On-Line1. It offered the possibility to schools and amateur astronomers to communicate with professional astronomers and observatories and to experience the chal- lenge of international “real time” cooperation. The project offered a unique framework for getting simultaneously taken pictures of minor planets from all over the world thus allowing parallax determinations of the minor planets and, finally, of the sun. 2 Why is it important to know the sun’s parallax? During the 18th and 19th century many expeditions were undertaken to all regions of the world from which astronomers hoped to be able to observe one of the transits of the planet Venus through the sun’s disc an event which happen very seldom. By these observations they hoped to get a better measure of the sun’s distance. Why did different governments spend a lot of money? Why did astronomers undergo the hardships due to such an expedition? And why is it important even today to know not only the value of the Astronomical Unit but also something about the methods by which astronomers found it? We think there are mainly five reasons: 1. If one knows the sun’s distance one can determine the magnitude of the solar system: • If you observe the motion of Venus you will find that it remains relatively close to the sun. To be more precise the largest angular distance between venus and the sun seen from the earth is about 45 degrees. From this you can calculate the ratio of the radii of the orbits of Venus and Earth: rVenus =sin45◦ rEarth (Take a sheet of paper, draw the sun and the cicular orbits of Venus and Earth around it: When the angular distance between sun and Venus is maximum the 1All informations and project descriptions are available up to today ([1]). 2 line of view from the earth to Venus is tangential to Venus’ orbit thus forming a right angle with the radius Sun-Venus!) The distance between Sun and Venus is thus about 70 percent of the radius of the earth’s orbit. • In a similar but not as easy way it is possible to determine the radii of the outer planets as multiples of rEarth for instance by observing them during their retrograde motion (see for instance W. Schlosser [4]). • Since Kepler this is easier by means of his third law which connects the radii of the planet’s orbits with their orbiting period. The latter is easy to measure. (1AU)3 r3 = T 2 planet (1a)2 planet But by all these methods astronomers got the distances as multiples of the earth’s radius of orbit - and this unit was not well known! A magnification of the first value by the factor two would have yielded the same magnification of the whole solar system. And the ancient value for the sun’s distance was wrong by a factor of about 20! 2. When the distances in the solar system are known it is possible to determine the astrophysical properties of the sun and the planets. For instance: • You can calculate the absolute size of the sun and the planets when you know their angular diameter and the corresponding distance. diameter tan(angular width)= distance • When you know the universal constant of gravity you can determine the mass of a central body by measuring the orbit’s radius and the period of the satellit orbiting around it. 3 4π rsat m = 2 γ Tsat • By measuring the so called solar constant f (that means the solar energy gathered per second by one square meter of the earth’s surface perpendicular to the direction of the radiation) you can calculate the radiation power of the sun – provided the sun’s distance is known. 2 Psun =4π(1AU) f 3. When the absolute distances are known it is possible to take into account the per- turbations due to gravitational interactions between the different planets. By this better predictions of the planet’s positions and especially of those of the moon are possible. This is a very important condition for astronomical navigating (see for instance D. Sobel [5]). 3 4. According to the earth’s motion around the sun stars are changing their positions relativ to each other during the year. Measurements of this parallactic motion enables to determine the star’s distance as a multiple of the radius of the earth’s orbit. For instance, the result of the first measurement of a star’s parallax by Bessel was 0.35 arcseconds and in this way the first measured distance of a star outside the solar system was about 60.000 times as large as the sun’s distance! dsun 1AU dstar = = ≈ 60.000AU sin πstar sin(0.35arcsec) Therefore the distance between the earth and the sun is not only a measure of the size of the solar system (and for the mass of its members, for instance) but for the dimensions of space. For this reason, it is called the Astronomical Unit. By hearing something about the Astronomical Unit and the development of the interest in its value and the increasing refinement of the methods of determination you can learn something about “what it means to do physics and astronomy” and “how it was (and still is) possible to know those things” (Wagenschein). 3 A short history of the sun’s parallax Until the middle of the 16th century most of the astronomers were not very interested in knowing the exact value of the sun’s distance because it was one astronomical distance among many others. This fact was due to the geocentric system of the world that did not allow to deduce the radii of the other planet’s spheres from that of the sun’s sphere. Therefore the result of the greek astronomer Aristarchus, wrong by about the factor 20, was taken unproofed over nearly two thousand years. Things changed completely with the development of the heliocentric system by Coper- nicus: Now the size of the sun calculated by its distance became an important argument for its central position in the planetary system. As described above in this system it is possible to determine all distances between the planets as multiples of the distance be- tween the earth and the sun. Furthermore, it became clear that the stars must reflect the earth’s motion around the sun. But no parallactic motion of the stars could be observed and thus the farther the sun was found the larger the distance of the stars had to be. Therefore, from the beginning of the 17th century increasing exertions were undertaken to measure the sun’s distance. Already the first attempt by Kepler who tried to measure the parallax of Mars at its opposition time without using a telescope prooved the ancient value to be too small at least by a factor of three. But it took further 70 years until it became possible to measure the sun’s distance by observing an parallactic effect in the position of a planet (Cassini and Richer observed the position of Mars from Paris and Cayenne simultanously.). And due to that result the solar system was about twenty times as large as men had believed until Kepler! But the measurements were very difficult and it took further 200 years until the Astronomical Unit was known with an accuracy better than one percent. Therefore it 4 is not surprising that up to today there exists no school experiment to measure it and almost no pupil and no student knows the way astronomers found the correct measure. 4 Methods of measurement Even today the best method of measuring the distance of a far away object in astronomy is to determine its trigonometric parallax.
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