Measuring the Distance to the Sun: Final Report
Udo Backhaus November 26, 2007
Contents
1 Goal of the Project 1
2 Preparation 2
3 The Hot Phase 4
4 Results of Observation 4
4.1 La Silla, Chile, November 19/20, 1996 . . . . 5
4.2 COAA, Portugal, November 20/21, 1996 . . . . 5
4.3 OHP, France, November 20/21, 1996 . . . . 5
4.4 La Silla, Chile, November 20/21, 1996 . . . . 5
5 Evaluation 5 5.1 Orbit Determination . . . . 5
5.1.1 Klio . . . . 6
5.1.2 Semiramis . . . . 6
5.1.3 Yerkes . . . . 6
5.2 Position and Parallax Determination . . . . 7
5.2.1 Klio, November 19/20, La Silla . . . . 7
5.2.2 Semiramis, November 20/21, Portugal . . . . 8
5.2.3 Semiramis, November 20/21, Haute-Provence . . . . 8
5.2.4 Semiramis, November 20/21, La Silla . . . . 9
5.2.5 Combination of Semiramis’ positions due to Chile and France . . . 9
6 Conclusion 12
1 Goal of the Project
The distance to the sun is one of the most important constants of astronomy. Up to
the thirties of our century parallax determinations for planets and asteroids led to the
best results. For these measurements expensive and sometimes dangerous expeditions all
over the earth had to be undertaken - and the abilities of the best astronomers had been
necessary to derive the sun’s parallax from the experimental data.
Due to modern CCD imaging and electronic computing and due to the possibilities of international cooperation offered by the internet it should be possible to get appropriate data by amateur equipment and without the necessity of worldwide expeditions.
Main goals of this project were
• the incorporation of collaboration between groups of pupils, students and/or ama- teur astronomers,
• the acquisition of pictures and astrometric data allowing the derivation of the As- tronomical Unit as easy as possible,
• the understanding of the basic evaluation procedures,
• the development and testing of appropriate computer algorithms of our own and, of course,
• the derivation of our own measure of the distance to the sun by means of the results of the cooperating groups.
By this way we hoped to get a better insight into the underlaying mathematics, to astrometry and to the corresponding technical problems. Last but not least, we hoped to find other groups with similar interests.
2 Preparation
We described the basics of the parallax method in our project description
1and pub- lished the mathematical details
2and the pascal source code of the corresponding com- puter algorithm
3in seperate papers.
Because of the difficulties of finding cooperative groups we tested the algorithms and the possibilities of our own equipment in close cooperation with the Astronomische Arbeitsgemeinschaft Osnabr¨ uck
4(the details of theses tests are described in the doc- ument FirstTests
5. Erwin Heiser took several CCD pictures showing the three minor planets 84 Klio, 584 Semiramis and 990 Yerkes, respectively.
In a first step, these pictures allowed us to test the abilities of our astrometry pro- grams (Mira resp. MiPS) and the problems due to them. The results derived with the respective programs differed slightly but both proved to be appropriate for our purpose.
The positions of the reference stars we got from the Guide Star Catalogue which is due to the astronomy program Guide.
For instance, by combinating two CCD images we got the following picture showing the retrograde motion of Klio in a time interval of only one hour:
The four brightest stars in this picture are:
1
http://www.eso.org/astronomyonline/market/collaboration/solpar
2
http://www.eso.org/astronomyonline/market/collaboration/solpar/solpar-det.html
3
http://www.eso.org/astronomyonline/market/collaboration/solpar/solpar-par.html
4
http://www.physik.uni-osnabrueck.de/students/ahaenel/aol/first.html
5
tests.html
1. GSC 2329 624: RA 2h43m 1.624s, DEC 33d11’53.77”
2. GSC 2329 573: RA 2h43m 0.434s, DEC 33d11’03.91”
3. GSC 2329 084: RA 2h42m46.190s, DEC 33d10’02.24”
4. GSC 2329 812: RA 2h42m51.110s, DEC 33d08’45.82”
With these reference stars we found the following topocentric positions of Klio:
1. 19:16:56 UT: RA 2h42m47.740s, DEC 33d09’00.87”
2. 20:10:01 UT: RA 2h42m45.567s, DEC 33d09’06.75”
The calculated geocentric positions are due to Ceres
1. 19:16:57 UT: RA 2h42m47.518s, DEC 33d09’06.54”, r = 1.028148 AU 2. 20:10:01 UT: RA 2h42m45.377s, DEC 33d09’11.67”, r = 1.028072 AU
In a second step, we tested our algorithms by evaluating these data:
• Combinating the two positions of Klio, thus regarding them as having been taken from different positions in space (due to the earth’s daily revolution) we got our first estimate for the sun’s parallax:
π
sun= 9 . 6
• By combinating one of the topocentric positions with the corresponding geocentric position we got:
π
sun= 8
In this way we became shure that the pictures taken at the Oldendorfer Berg, the astrometry and our algorithms are precise enough (and perhaps more than this) for our goal.
In a third step, Erwin Heiser took pictures of the three respective asteroids on three
different days. These three positions made it possible for us to derive first orbital ele-
ments using the Gauss algorithm described by A. Guthmann (Einf¨ uhrung in die Him-
melsmechanik und Ephemeridenrechnung). In this way we became able to calculate geo-
centric positions and, especially, the geocentric distances needed in parallax determination
by our own (but, up to now, not very precise!).
3 The Hot Phase
For us, it was disappointing that we didn’t get any contact to other amateur or student groups neither in overseas nor in Europe during the first four weeks. Therefore, we published two “cries for help” in the last week of the warming up phase of Astronomy On- Line. And the Astronomische Arbeitsgemeinschaft Osnabr¨ uck proposed observations by the ESO observatory La Silla. Only in the last two days we got contact with observatories in Denmark and Italy and with the Centro de Observacao Astronomica no Algarve (COAA)
6in Portugal.
But, unfortunately, during the first night of the hot phase (November 19/20) the weather was very bad in Europe not allowing any astronomical observations (but some nice small talks via e-mail about this nerve-shattering fact and the related bad moods!).
For this reason, it was very exciting for us to get images from La Silla - taken especially for us!
In the second night the weather in Europe was as bad as in the first one. We lost any hope to get appropriate images and data. But, luckily, we were wrong! Two days later we got the information about pictures made by the COAA in Portugal, taken not in the proposed time interval but about four hours earlier because of the weather.
Some days later, we heard that La Silla had taken pictures of the three asteroids during the second night as well. We managed to get them before they had been published and tried to combine them with the data from Portugal. But, unfortunately, the time difference between the pictures due to Portugal and Chile proved to be too large making it impossible to extrapolate the topocentric positions due to La Silla to the observation time of Portugal with sufficient accuracy. The resulting measure of the sun’s parallax was wrong by more than the factor 2!
But some more days later, we got additional pictures of Semiramis made from Europe in exactly the same time intervall as those made in Chile! We can hardly imagine how the observers of the Observatoire de Haute-Provence
7succeeded to look through the clouds of that night! But, because of these images, we suddenly could hope after all to get able to derive the distance to the sun from pictures simultaneously taken from different observatories.
4 Results of Observation
For further evaluations we collect here the main images (or a combination of them, re- spectively) and the urls from which the complete image files can be downloaded.
All of the following pictures are minimized versions of bigger ones which you can see by clicking on them!
6
http://www.algarvenet.pt/coaa
7
http://www.obs-hp.fr
4.1 La Silla, Chile, November 19/20, 1996
The following picture shows the combination of the first and the last of the images showing Klio. The size of the image is about 3
∗ 3
.
4.2 COAA, Portugal, November 20/21, 1996
The following pictures show the three respective minor planets as observed by the COAA early in the second night:
You can get the FITS files here: Klio
8, Semiramis
9, Yerkes
10.
4.3 OHP, France, November 20/21, 1996
The following picture is a combination of all of the five images of 584 Semmiramis taken by the OHP.
You may get the FITS files via this url
11.
4.4 La Silla, Chile, November 20/21, 1996
In the pictures below we combined the first and the last of the images of “our” respective minor planets made with the dutch telescope of La Silla (or, in the case of Semiramis, all of them). The pictures besides them show the corresponding sky maps as drawn by Guide.
You can get the FITS files here
12: The images have the following numbers:
• Klio: dut0318, dut0321, dut0324, ..., dut0339,
• Yerkes: dut0319, dut0322, dut0325, ..., dut0340,
• Semiramis: dut0320, dut0323, dut0326, ..., dut0341.
5 Evaluation
5.1 Orbit Determination
During the warming up phase Erwin Heiser took several pictures of the three respective minor planets with the telescope on the Oldendorfer Berg. Thus we got the topocentric positions of three different days of each of these asteroids. Because each of them is taken at the same time, approximately, we can take them as input data for the Gauss algorithm for orbit determination (thus taking them as geocentric positions!).
8
portugal/a84.fit
9
portugal/a584.fit
10
portugal/a990.fit
11
http://www.obs-hp.fr/www/aol/aol-images.html
12
http://www.eso.org/data/aol/
5.1.1 Klio
With the following topocentric positions
1. Oct. 14, 1996, 19:17:07 UT: RA 2h42m47.740s, dec 33d 9’ 0.87”
2. Nov. 13, 1996, 20:14:06 UT: RA 2h 9m40.625s, dec 31d48’ 8.62”
3. Nov. 14, 1996, 18:26:40 UT: RA 2h 8m49.550s, dec 31d41’52.24”
we got the following orbital elements:
a = 2 . 361 AE, e = 0 . 241 , i = 9 . 309
◦, Ω = 327 . 940
◦5.1.2 Semiramis
With the following topocentric positions
1. Nov. 8, 1996, 21:20:27 UT: RA 2h43m41.376s, dec 34d45’50.30”
2. Nov. 13, 1996, 20:27:58 UT: RA 2h39m 5.268s, dec 33d51’13.48”
3. Nov. 14, 1996, 18:39:53 UT: RA 2h38m16.977s, dec 33d40’22.86”
we got the following orbital elements:
a = 2 . 307 AE, e = 0 . 241 , i = 9 . 990
◦, Ω = 285 . 075
◦5.1.3 Yerkes
With the following topocentric positions
1. Oct. 14, 1996, 19:29:36 UT: RA 3h14m 1.07s, dec 29d 8’24.11”
2. Nov. 8, 1996, 21:03:06 UT: RA 2h49m35.70s, dec 29d38’27.92”
3. Nov. 14, 1996, 18:01:09 UT: RA 2h43m20.97s, dec 29d25’57.00”
we got the following orbital elements:
a = 2 . 672 AE, e = 0 . 215 , i = 8 . 771
◦, Ω = 354 . 277
◦All of these elements are in surprisingly good agreement with those we got from Guide.
Therefore, we are able to calculate the necessary ephemeris data by means of self measured
and calculated orbital elements. But we must be aware that the results will be not very
precise.
5.2 Position and Parallax Determination
5.2.1 Klio, November 19/20, La Silla
Because of the cloudy sky during the first night and the lack of reference pictures from Europe we evaluated the pictures from La Silla with respect to the sun’s distance. For instance, we measured the positions of Klio due to the first and the last picture, respec- tively.
The above pictures illustrate the corresponding difficulties: In spite of the GSC cata- logue’s size the near vicinity of Klio in the sky map seems to be empty. Indeed, no star in the La Silla image can be identified without additional information to be a GSC star!
It then would have been impossible to determine Klio’s position.
But fortunately, Andreas H¨ anel of the Astronomische Arbeitsgemeinschaft Osnabr¨ uck knew the internet address of the Digital Sky Survey (DSS)
13: By this url it is possible to get a digitized image of any region of the sky, that means of arbitrary center and arbitrary size. The parameters of the right picture above which we got in this way match nearly to those of the Guide map besides it. In this picture, you can easily identify all GSC stars (for instance GSC 2308 324, GSC 2308 440, GSC 2308 300). In this way, it is possible to identify GSC 2308 300 in the lower center of the La Silla image. Now, by means of astrometry of the DSS picture, the positions of the other stars in the La Silla image can be determined - and on the basis of these data finally the position of Klio!
For the first and the last position of Klio, for instance, we got in this way the following positions (delta means the geocentric distance):
La Silla, latitude -29d15’, longitude -70d44’, h 2400 m, November 20, 1996
==========================================================================
Klio1: 1:53:04 UT
( measured ) topocentric position: RA 2h04m27.686s DEC 31d04’13.33’’
(calculated) topocentric position: RA 2h04m27.873s DEC 31d04’14.49’’
(calculated) geocentric position: RA 2h04m27.873s DEC 31d04’14.49’’ delta 0.090 Klio2: 5:20:46 UT
( measured ) topocentric position: RA 2h04m20.902s DEC 31d03’09.40’’
(calculated) topocentric position: RA 2h04m21.111s DEC 31d03’10.18’’
(calculated) geocentric position: RA 2h04m21.446s DEC 31d04’04.00’’ delta 1.091 With the help of our program these positions yield the following measure of the sun’s parallax:
π
sun= 9 . 2
There is a little but significant difference between the results calculated in Osnabr¨ uck and Koblenz, respectively, which could not be cleared up yet. The results of Osnabr¨ uck yield a solar parallax which is a little bit closer to its correct value.
13