The Internet Project “Measuring the Distance to the Sun”

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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.

1 Introduction

The radius of the earth’s orbit around the sun is one of the most important constants in astronomy: theAstronomical 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:

r_{V enus}

r_Earth = sin 45^◦

(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]).

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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 ofr_Earthfor instance by observing them during their retrograde motion (see for instance W. Schlosser [4]).

• SinceKepler 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.

r_planet³ = (1AU)³ (1a)² T_planet²

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 theabsolute sizeof the sun and the planets when you know their angular diameter and the corresponding distance.

tan(angular width) = diameter distance

• When you know the universal constant of gravity you can determine themass of a central body by measuring the orbit’s radius and the period of the satellit orbiting around it.

m= 4π γ

r_sat³ T_sat²

• 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.

P_sun = 4π(1AU)²f

3. When the absolute distances are known it is possible to take into account the perturbations 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]).

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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!

d_star = d_sun

sinπ_star = 1AU

sin(0.35arcsec) ≈60.000AU

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 astronomerAristarchus, wrong by about the factor 20, was taken unproofed over nearly two thousand years.

Things changed completely with the development of theheliocentric systembyCoper- 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 between 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 Keplerwho 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 Richerobserved 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

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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 itstrigonometric parallax. You know the underlying eﬀect from every day life: When you go by bike, car or train all things around you seem to move in the opposite direction - the nearer the more quickly².

If one tries to measure the sun’s parallax in this way that means by locating it from diﬀerent places on the earth one will have to measure angles in a triangle with a ratio of length of about 24.000:1. Try to draw such a triangle and you will be able to imagine the corresponding diﬃculties! Indeed, up to now it is impossible to measure the sun’s parallax in this way directly.

There are three principal alternatives:

1. Aristarchus had the idea to measure the distance of the sun not as a multiple of the earth’s radius but of the radius of the moon’s orbit. This requires angular mea- surements in triangle with a ratio of length of “only” 400. If the moon’s distance is known (and Aristarchus knew a quite good value!) one can calculate the Astro- nomical Unit as a multiple of the earth’s radius.

2. The second possibility to get larger and therefore more easily to measure angles is to measure a distance smaller than that of the sun. As described above it is possible to derive the sun’s distance from the result.

(a) The smallest distance between Mars and Earth is about half as large as the sun’s distance. The ﬁrst modern measurements therefore took use of parallax measurements on Mars(Kepler ca. 1600, Cassini 1671).

(b) The closest distance of Venus is only a quarter of that of the sun. Unfortu- nately, it is then in “inferior conjunction” and therefore above the horizont simultanously with the sun. But very seldom (all 110 years approximately, next time in 2006) it passes “through” the sun’s disc thus allowing precise measurements of its position relativ to the sun. This method is very similar to that used to determine the moon’s distance during a solar eclipse in [?]. Up to the end of the 19th century the best results where due to observations of transits of Venus([6]).

(c) Some minor planets come even closer to the earth (down to 0.1 AU). Parallax measurement on the minor planet Eros during its opposition in 1930 yielded the best value of the Astronomical Unit up to that time.

3. The third alternative takes use of physical laws opening the possibility of deriving the sun’s distance from measurements of other physical quantities:

2The method of parallax measurement is excellently described in the Astronomy On-Line project

“Solar Eclipse on October 12, 1996” [2].

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(a) When the speed of light is known one can change from geometric methodes to measurements of time eﬀects. Examples are the observations of the occulta- tions of Jupiter’s moon Io byRoemer(who argumented the other way round) and the discovery of the aberration of light by Bradley. Today the best re- sults are due to run time measurements with radar signals between Earth and Venus.

(b) Newton’s law of gravity allows to derive the sun’s parallax from precise observations of the perturbations of the moon’s orbit by the sun. At the beginning of our century this was the exactest method.

(c) Finally the Doppler eﬀect due to the earth’s motion gives the possibility of deriving the Astronomical Unit from a star’s spectra taken at days the earth is moving directly to or away from it.

5 The project

5.1 The goal of the project

5.2 Determination of the distance of a minor planet

The idea of Aristarchus can be understood most easily of the above methods. But, unfortunately, it has been proved to be impossible to realize it even by using a sextant (see [7], for instance). Therefore, reversing Roemer’s method of determing the velocity of light ([3]), that means to deduce the radius of the earth’s orbit from the known value of the velocity of light, seems to be the only method which can up to now be done at school by own measurements.

In our project we tried to realize the early idea of Kepler and the ﬁrst successful method of Cassini. But we didn’t observe the planet Mars (it was too far from the earth at the end of November and because of its size and brightness precise measurements of its position would have been too diﬃcult). In spite of Mars we took a minor planet which was opposite to the sun and therefore relatively close to the earth (1 AU approximately) and thus could be observed during the whole night.

We looked for such asteroids by using the computer programGuide([?]) and found the following minor planets which seemed to us to be appropriate: 84 Klio,584 Semiramis and 990 Yerkes. Figure 1 shows the trails of these three minor planets during

6 How to do measurements

We are going to measure the distance of the minor planet by means of the parallax method. The principles of this method are excellently described in the solar eclipse project³of Astronomy On-Line. Because of the distance of the asteroid the corresponding parallax eﬀect will be small (less than 8 arcsec approximately). To achieve the required accuracy we will therefore use ccd photography.

3http://www.eso.org/astronomyonline/market/collaboration/soleclipse

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Figure 1: Tracks of some minor planets

• The empirical basis of the method consists of ccd images taken every 15min (22.00, 22.15, 22.30 UT etc., for instance) in a ﬁxed time intervall. The images are to be corrected due to dark current. One edge of the images should be as close as possible to the north south direction.

• Due to the pictures are the exact exposition times and the geographical po- sition of the telescope.

• It would be nice to get some information about the telescope (especially its focal length and its aperture) and the ccd camera (especially its number of pixels) by which the pictures are taken.

• We will gather the results taken at diﬀerent observarories. Please send us (by anonymus ftp, for instance. You then must let us know the name of the transmitted ﬁle via e-mail)

– the original ccd images corrected only due to dark current, – the corresponding exposition times and geographical position.

– If you are able to determine the exact position of the asteroid by means of an astrometry program please send us your astrometric results.

• We will comprise the results in a tabular and publish them as soon as possible.

• Determination of the minor planet’s distance und calculation of the Astronomi- cal Unit can be done, for instance, by the associated program⁴. The method of evaluation is described below.

4parallax.html

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• We will publish typical images and related results for the parallax of the sun in a newspaper of Astronomy On-Line.

6.1 Determination of the asteroid’s topocentric position

”Topocentric” position means the position as measured at a real (topocentric) position on earth not the hypothetical position due to the earth’s center.

To be able to determine the exact position of the asteroid from an image it must show besides of the asteroid at least three stars of known position. For this purpose, we will use the astronomy program Guide 4.0and its Guide Star Catalogue.

By these equatorial positions and the corresponding pixel positions we can calculate the so called plate constants of the image (see, for instance, O. Montenbruck et al., As- tronomy on the Personal Computer, Springer 1994). By these constants it is possible to determine the equatorial coordinates of the asteroid very precisely. We will do this by means of the astrometry programsMiPS and MIRA respectively.

The result of this ﬁrst step of evaluation will be the exact topocentric equatorial coordinates of the planet:

(α_i, δ_i)

6.2 Combining two simultanously measured results

The two topocentric positions of the asteroid give us the directions of view due to the corresponding observers, and leed us to the following situation:

Therefore, by determing the intersection of the respective lines of view we are able to calculate the distanced_P between the asteroid and the earth’s center. The details of this calculation are given in aseperate document⁵. We get this distance as a multiplem of the earth’s equatorial radius r_E:

s

e₁

e₂

d_P

Erde

Kleinplanet

ρ₁

ρ₂

Beobachter 1

Beobachter 2

5details.html

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d_P =mr_E

If the orbital elements of the asteroid are known it is possible to determine its geocentric distance as a multiple n of the Astronomical Unit by means of ephemeris data calculation:

d_P =nAU

It should be stressed that for this calculation it is not necessary to know the absolute value of the Astronomical Unit!

Combination of the two equations above give us the Astronomical Unit as a multiple of r_E:

mr_E =nAU =⇒ 1AU = m nr_E

Thus, if we know the earth’s radius in kilometers (and we do!) we know the Astro- nomical Unit in the same measure.

Most often the sun’s distance is given as the sun’s parallax. That is the angular width of the earth’s radius as seen from the sun (or, vice versa, half of the maximum parallax eﬀect of the sun observed from the earth). This angle can be calculated by

tanπ_sun= 1r_E 1AU = n

m

For as small angles the tangent is equal to the angle itself expressed in radiants. Thus, the last step we have to do is to convert the parallax in arcseconds:

π_sun = n m

180

π 3600 (in arcsec)

6.3 Combining two positions measured at the same observatory

Instead of communicating with another observatory we can use the earth’s daily revolution to go to a place with another equatorial position. Unfortunately, this takes some time and during this time the asteroid will not stay in a ﬁxed position! But we will be able to take into account this proper motionat least approximately:

Ephemeris data calculation gives us not only the geocentric distance of the asteroid but its geocentric equatorial coordinates. We use these geocentric positions due to the respective times of observation

(α_C₁, δ_C₁) and (α_C₂, δ_C₂)

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to determine the proper motion of the asteroid. By this we can reduce the topocentric position due to the second observation to the time of the ﬁrst one:

α₂ = α₂−(α_C₂ −α_C₁) δ₂ = δ₂−(δ_C₂ −δ_C₁)

By this trick we’ve got two observations which are due to the same time but due to diﬀerent observer’s positions! Therefore, we can now calculate the sun’s parallax in exactly the same manner as we did above.

6.4 Comparison of one topocentric result with the correspond- ing geocentric position

It is possible even to calculate the sun’s parallax with only one observed topocentric position of the asteroid:

For this purpose we take the geocentric position we got by means of ephemeris data calculation as the second “topocentric” position of the asteroid. Thus, by locating the second observer in the earth’s center and setting

α₂ =α_C₂, δ₂ =δ_C₂

we can use the same algorithm as for two really observed positions.

6.5 Short discussion of the three methods

By the third method we combine an observed position with a theoretical one. We, therefore, do not really measure the sun’s parallax. Furthermore, we need a highly precise geocentric position. For this reason, this method is very sensitive due to numerical errors of the ephemeris data calculation.

For the second method we do not really need the exact geocentric positions but only the proper motion of the asteroid as viewed from the earth’s center. This motion is influenced by the other planets much less than its exact position (in which the perturbations are cumulated). That means that a simplier method of calculation may be used. To be more concrete, starting with good orbital elements it is sufficient to take into account only the influence of the sun. For this reason, this method is less sensitive and appropriate algorithms of calculation can easily be found.

Finally, the ﬁrst method determines the asteroid’s distance from two observed positions only. It takes use of ephemeris data calculation only for deducing the sun’s parallax.

For this goal only the geocentric distance of the asteroid is needed which can easily be calculated with sufficient accuracy. Moreover, this calculated value influences the final result only linearily. Therefore, this method is much less sensitive than the others. For this reason, we would by far prefer to get able to determine the Astronomical Unit from two positions observed from two different observatories!

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7 First Tests

The pictures at the top of this document we took in collaboration with Erwin Heiser, member of theAstronomische Arbeitsgemeinschaft Osnabr¨uck⁶ at the observatory on the Oldendorfer Berg. They show the minor planet 84 Klio at October 14, 1996 (ephemeris data⁷ of this asteroid and some others are contained in a seperate paper).

They were taken at 19:16:57 UT and 20:10:01 UT respectively. In spite of the short time diﬀerence they show clearly the proper motion of Klio!

Evaluations of these pictures are described seperately⁸.

References

[1] Astronomy On-Line: http://www.eso.org/astronomyonline/

[2] The Solar Eclise on October 12, 1996,http://www.eso.org/astronomyonline/market/

collaboration/soleclipse/solecl-2d.html

[3] ,http://www.eso.org/astronomyonline/market/experiments/advanced/skills302.html [4] Schlosser, W.: Challenges of Astronomy, Springer: New York 1994

[5] Sobel, D.: Longitude, Walker Publishing Company 1995

[6] Woolf, H.: The Transits of Venus, Princeton University Press: Princeton 1959 [7] Vornholz, D., Backhaus, U.: Wer hat recht – Aristarch oder der Sextant? (in Ger-

man), Astronomie + Raumfahrt 31, 20 (1994)

6http://www.physik.uni-osnabrueck.de/students/ahaenel/aol/first.html

7toutatis.html

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The Internet Project “Measuring the Distance to the Sun”