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
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 Osnabrück (the details of theses tests are described in the document FirstTests.). 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 programs (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:
Hourly retrograde motion of Klio (lower right)
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 elements using the Gauss algorithm described by A. Guthmann (Einführung in die Himmelsmechanik und Ephemeridenrechnung). In this way we became able to calculate geocentric positions and, especially, the geocentric distances needed in parallax determination by our own (but, up to now, not very precise!).
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 succeeded 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.
All of the following pictures are minimized versions of bigger ones which you can see by clicking on them!
84 Klio, 1:53:04 UT and 5:20:46 UT, respectively
(The complete FITS files may be downloaded as a self extracting archive here).
84 Klio, 19:49:28 UT; 584 Semiramis, 20:40:33 UT; 990 Yerkes, 21:08:16 UT
You can get the FITS files as a self extracting archive here:
584 Semiramis, 1:12:17 UT, 1:21:11 UT, 1:29:02 UT, 2:25:25 UT and 2:29:37 UT, respectively
You may get the FITS files as a self extracting archive here.
84 Klio, 1:45:43 UT and 2:13:45 UT, respectively (there are six images in between)
584 Semiramis, all of the eight positions between 1:48:27 UT and 2:15:36 UT
990 Yerkes, 1:47:03 UT and 2:14:45 UT, respectively there are six images in between)
You can get the FITS files as a self extracting archive here.
Klio at 1:45:43 UT, corresponding sky map and DSS image of the same sky region
The above pictures illustrate the corresponding difficulties: In spite of the GSC catalogue'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 Hänel of the Astronomische Arbeitsgemeinschaft Osnabrück knew the internet address of the Digital Sky Survey (DSS): 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 1.090
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:
There is a little but significant difference between the results calculated in Osnabrück and Koblenz, respectively, which could not be cleared up yet. The results of Osnabrück yield a solar parallax which is a little bit closer to its correct value.
Semiramis ========= Portugal, latitude 37d11', longitude -8d36', h 75 m =================================================== ( measured ) topocentric position RA 02h33m30.257s DEC 32d25'44.08" (calculated) topocentric position RA 02h33m30.323s DEC 32d25'44.09" (calculated) geocentric position RA 02h33m29.983s DEC 32d25'45.58" delta = 0.97107 La Silla, latitude -29d15', longitude -70d44', h 2400 m ======================================================= (extrapolated) topocentric position RA 02h33m30.764s DEC 32d25'50.88" ( calculated ) topocentric position RA 02h33m30.600s DEC 32d25'48.65"
From these values we derived the following solar parallax:
The error is mainly due to the 2" extrapolation error in declination.
Haute-Provence, latitude 43.9294d, longitude 5.7125d, h 650 m ============================================================= SemiA 01:12:17 UT RA 02h33m21.938s DEC 32d23'23.28" SemiB 01:21:11 UT RA 02h33m21.677s DEC 32d23'18.46" SemiC 01:29:02 UT RA 02h33m21.430s DEC 32d23'14.24" SemiD 02:25:25 UT RA 02h33m19.754s DEC 32d22'43.82" SemiE 02:29:37 UT RA 02h33m19.639s DEC 32d22'41.48"
The images of La Silla show nearly the same part of the sky as those of OHP. Therefore, we could take the same reference stars and the astrometry was as easy as for the french images. The topocentric positions of Semiramis as seen from La Silla are listed below:
Semi3 01:48:27 UT RA 02h33m21.288s DEC 32d23'10.43" Semi4 01:52:56 UT RA 02h33m21.161s DEC 32d23'08.05" Semi5 01:56:58 UT RA 02h33m21.034s DEC 32d23'06.04" Semi6 02:00:38 UT RA 02h33m20.918s DEC 32d23'03.98" Semi7 02:05:29 UT RA 02h33m20.770s DEC 32d23'01.50" Semi8 02:09:43 UT RA 02h33m20.642s DEC 32d22'59.34" Semi9 02:12:42 UT RA 02h33m20.551s DEC 32d22'57.86" Semi10 02:15:36 UT RA 02h33m20.453s DEC 32d22'56.21"
To do this, we fit the positions to a straight line using Gauss' method of least squares. By this method, we calculate the time rate of right ascension and declination dalpha and ddelta, respectively, interpolate the positions by these time rates and, finally, get the following results:
OHP === dalpha/dt = -0.030s/min ddelta/dt = -0.54"/min ( interpolated ) position: 02:00:00 UT RA 02h33m20.516s DEC 32d22'57.53" La Silla ======== dalpha/dt = -0.031s/min ddelta/dt = -0.52"/min ( interpolated ) position: 02:00:00 UT RA 02h33m20.939s DEC 32d23'04.39"Putting these results in our algorithm together with the calculated geocentric distance of Semiramis delta=0.972 AU (our own orbit determination yields, not very satisfying, 0.906 AU; our own orbital elements are worst for Semiramis.) leeds us to the following measure of the sun's parallax:
How is such a bad result (the sun's distance following from this result is more than 50 percent too large!) possible with such excellent images? Half an hour of an intensive brain storming let us find the reason in the following way:
Comparison of the interpolated topocentric positions with calculated results
due to Ceres,
OHP
===
(calculated) position: 02:00:00 UT RA 02h33m20.295s DEC 32d22'53.72"
La Silla
========
(calculated) position: 02:00:00 UT RA 02h33m20.931s DEC 32d23'04.85",
shows in the case of the ESO images a satisfying conformity between measurement
and calculation (less than 0.5" in declination, for instance). In the case of
the french images, however, there are significant differences (nearly 4" in
declination, for instance). Closer inspection shows a systematic difference
for all of these pictures. These differences are, on average,
Because of the proper motion of Semiramis these differences correspond to a time difference of
Indeed, by correcting all exposure times for the french images by this
difference we got the following topocentric position:
OHP
===
(corrected) position: 02:00:00 UT RA 02h33m20.307 DEC 32d22'53.74"
and, with this position, the following measure of the sun's parallax:
With a correction by delta-t = -8 min (which yields a similar difference between measurement and calculation for the ESO and the OHP images) we even get
An e-mail gave us the confirmation:
We, therefore, summarize the results due to Semiramis with the above picture combining the photos from La Silla and Haute Provence. The thick dots mark the respective interpolated positions of Semiramis due to 2.00 UT. The picture clearly shows a slight difference between the passes observed from Chile and France. Because the distance of Semiramis is nearly that of the sun we can state:
The picture visualizes the parallax effect due to one Astronomical Unit - even without the interpolated positions!
As above, in the following pictures we have corrected the observation times due to OHP by -7 min. They show the parallax effect of Semiramis due to France and Chile: The thin dots show the measured positions of the minor planet, the thick ones the interpolated ones which are due to 2.00 UT and allow us first to determine the distance of the planet and second to deduce the sun's parallax easily.
Proper motion of Semiramis as seen from Haute-Provence and La Silla, respectively
Parallax effect of Semiramis in right ascension
Parallax effect of Semiramis in declination
By the above reduction we got the following measure for the solar parallax:
The error interval given here is an estimate only. We didn't determine exactly up to now the errors due to observation unaccuracies and astrometry errors (due to GSC position errors and our different astrometry programs, for instance).
We confirmed our algorithms for determing the asteroid's distance and deriving the solar parallax to be correct and of sufficient accuracy. Our orbit determination needs further refinement and more input data.
We have shown that it is possible to get estimates for the distance to the sun by combinating observations made from only a single observatory with calculated geocentric data. The corresponding algorithm, however, is quite demanding and, because of its sensitivity, the results are not very good.
By far the simpliest parallax determination can be done by combining observations simultaneously taken at different observatories seperated by distances as large as possible. The corresponding algorithm can easily be understood and the results are very satisfying.
However, it is not easy to install the international cooperation necessary for this method - even in our days via internet and even with the help of Astronomy On-Line. It was a little bit disappointing for us to find nearly no other amateur group to cooperate with. Nevertheless, it was very exciting for us mailing with some large observatories and getting excellent images from them - made especially for us! We thank them very much.
It seems to be difficult to get images from different observatories made at exactly the same time (which would offer the simpliest method of parallax measurement). Therefore, the following procedure seems to be the simpliest practicable method:
The Earth as seen by Semiramis, November 21, 1996, 2.00 UT
And this is our result:
Due to this result, the distance to the sun (to be more precise, the major axis of the earth's orbit around the sun) is about
or, taken the equatorial radius of the earth as known,
As the result of a first cooperative parallax measurement, we think this value is very satisfying!
A very nice secondary effect of this evaluation was the experience of being able to control the clock of a distant observatory by "looking" to its sky via internet and by correcting the clock with respect to a correct measure of the solar parallax (because its determination is very sensitiv to time errors)!
The results of this paper would not have been possible without the help of the Astronomische Arbeitsgemeinschaft Osnabrück. The intensive cooperation with some of its members, especially with Erwin Heiser, was very satisfying and enjoying. We hope we will cooperate in a similar way in the future very often.
Last but not least, we thank Richard West for quick and uncomplicated help in many cases and the ESO and EAAE for the project Astronomy On-Line offering so many astronomical experiences and giving us the possibility of learning a lot about orbit determination, parallax measurements and the possibilities (and difficulties) of world wide cooperation via internet.
We plan to repeat the parallax measurement during a stable high pressure season in the near future. Perhaps we will, nevertheless, be successful in taking (and getting) exact simultaneous pictures. Due to our results, we think it is likely that observations made from Europe only will suffer to derive a satisfying measure of the distance to the sun.
We hope ESO and Astronomy On-Line will give us a framework for managing the international cooperation easily und for publishing our results.