Download Measuring the Distance to the Sun: Final Report

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
4.1 La Silla, Chile, November 19/20, 1996 . .
4.2 COAA, Portugal, November 20/21, 1996
4.3 OHP, France, November 20/21, 1996 . .
4.4 La Silla, Chile, November 20/21, 1996 . .
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5 Evaluation
5.1 Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Klio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Semiramis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Yerkes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Position and Parallax Determination . . . . . . . . . . . . . . . . . .
5.2.1 Klio, November 19/20, La Silla . . . . . . . . . . . . . . . . .
5.2.2 Semiramis, November 20/21, Portugal . . . . . . . . . . . . .
5.2.3 Semiramis, November 20/21, Haute-Provence . . . . . . . . .
5.2.4 Semiramis, November 20/21, La Silla . . . . . . . . . . . . . .
5.2.5 Combination of Semiramis’ positions due to Chile and France
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6 Conclusion
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12
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.
1
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 amateur astronomers,
• the acquisition of pictures and astrometric data allowing the derivation of the Astronomical 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 description1 and published the mathematical details2 and the pascal source code of the corresponding computer algorithm3 in 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 Osnabrück4 (the details of theses tests are described in the document FirstTests5 . 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:
The four brightest stars in this picture are:
1
http://www.eso.org/astronomyonline/market/collaboration/solpar
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
2
2
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:
πsun = 9.6
• By combinating one of the topocentric positions with the corresponding geocentric
position we got:
πsun = 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 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!).
3
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 OnLine. And the Astronomische Arbeitsgemeinschaft Osnabrück 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)6 in 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-Provence7 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.
4
Results of Observation
For further evaluations we collect here the main images (or a combination of them, respectively) 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
7
http://www.algarvenet.pt/coaa
http://www.obs-hp.fr
4
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: Klio8 , Semiramis9 , Yerkes10 .
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 url11 .
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 here12 : 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
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/
9
5
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 = 2.361AE, 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 = 2.307AE, 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 = 2.672AE, 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.
6
5.2
5.2.1
Position and Parallax Determination
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, respectively.
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)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:
πsun = 9.2
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.
13
http://archive.eso.org/dss/dss
7
5.2.2
Semiramis, November 20/21, Portugal
We didn’t spend many time for evaluation the COAA images because there are no suitable
reference images. But of course, we tried to combine the Semiramis picture with those
made in La Silla by extrapolating the topocentric positions due to La Silla to the exposure
time of Portugal (the procedure is described below). The result is, not surprisingly because
of the five hour time difference, quite bad. Nevertheless, we give here our for illustration:
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:
πsun = 14.4
The error is mainly due to the 2” extrapolation error in declination.
5.2.3
Semiramis, November 20/21, Haute-Provence
With Semiramis the astrometry is much easier: Five GSC stars can easily be identified
on the images. The position of a sixth one we got (for higher accuracy) with the help of
a DSS image. The five positions of Semiramis due to the images of OHP are listed below.
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"
8
5.2.4
Semiramis, November 20/21, La Silla
Because of the fact that reference images exist for Semiramis only, we will now concentrate
on that asteroid.
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
Semi4
01:52:56
Semi5
01:56:58
Semi6
02:00:38
Semi7
02:05:29
Semi8
02:09:43
Semi9
02:12:42
Semi10
02:15:36
5.2.5
UT RA 02h33m21.288s DEC 32d23’10.43"
UT RA 02h33m21.161s DEC 32d23’08.05"
UT RA 02h33m21.034s DEC 32d23’06.04"
UT RA 02h33m20.918s DEC 32d23’03.98"
UT RA 02h33m20.770s DEC 32d23’01.50"
UT RA 02h33m20.642s DEC 32d22’59.34"
UT RA 02h33m20.551s DEC 32d22’57.86"
UT RA 02h33m20.453s DEC 32d22’56.21"
Combination of Semiramis’ positions due to Chile and France
If you compare the images taken at OHP and La Silla, respectively, you can see in the
OHP pictures a gap between the positions of Semiramis. And the positions due to the
images of La Silla fill this gap exactly. Therefore, there are no two pictures belonging to
the same time with sufficient accuracy and no direct comparison of two images is possible.
For this reason, we have to reduce the two series of topocentric positions to a common
time. We take 2.00 UT for this purpose.
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"
9
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:
πsun = 5.6 !
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,
∆α = 0.208s, ∆δ = 3.81 .
Because of the proper motion of Semiramis these differences correspond to a time
difference of
∆t = −7min!
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:
πsun = 8.5 .
10
With a correction by ∆t = −8min (which yields a similar difference between measurement and calculation for the ESO and the OHP images) we even get
πsun = 8.9 .
An e-mail gave us the confirmation:
• Question (U. Backhaus): “I think we found the reason (for the difference described
above): You should take care for the clock in your observatory (or, at least, of the
computer): Most likely, it gaines about 7-8 minutes. . . . Are we right?”
• Answer (M. Dennefeld): “I had indeed noticed a 7 minutes difference between the
computer clock and the UT! . . . My correction of 7 min is approximate only, it could
be a little more. . . . ”
This confirmation was very satisfying and exciting for us: Our procedures have been
proved to be precise enough for determing the distance to the sun. Moreover, we found
a possibility to control the clocks of all observatories of the world: We only must have
a look to their sky via internet. The necessary corrections to the sun’s parallax derived
from that look will show the difference between the observatory’s clock and the time!
We, therefore, summarize our results due to Semiramis with the following pictures in
which we 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.
declination δ
32◦ 23 30
p
p
p
p
p p
qp
p
p p
p
32◦ 23 00
q
p
2h33m22s
right ascension α
p
2h33m20s
Proper motion of Semiramis as seen from Haute-Provence and La Silla, respectively
11
right ascension α
p
2h33m22s
p
p
p
p
p qp
p
q
p p
p
2h33m20s
p
1h30m
p
2h00m
time
Parallax effect of Semiramis in right ascension
declination δ
32◦ 23 30
p
◦
p
p
p
p
p qp
32 23 00
q
p
p p
p
p
1h30m
p
2h00m
time
Parallax effect of Semiramis in declination
By the above reduction we got the following measure for the solar parallax:
πsun = 8.7 ± 0.2
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).
6
Conclusion
It had been proved to be possible to determine the distance to the sun by means of
parallax measurements on minor planets with geocentric distance of about 1 AU. The
12
focal length of the telescopes used were 7-9 meters, approximately. Unfortunately, there
was no chance for testing the applicability of amateur instruments of shorter focal lengths.
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:
• Take some pictures of a minor planet taken by different observatories during a
common time interval of about one hour.
• Determine the corresponding topocentric positions of the asteroid, fit them to a
straight line and reduce them to a common point of time.
• The two corresponding simultaneous topocentric positions allow an easy determination of the asteroid’s parallax.
• The geocentric distance of the asteroid needed for deriving the solar parallax can be
calculated by means of a simple ephemeris data calculation. This calculation must
not be very precise: The solar parallax depends on its result only linearily.
• Perhaps, the orbital elements needed for this calculation can be determined by own
observations.
In this way, we combined the positions of 584 Semiramis as observed from La Silla and
Haute-Provence, respectively. These observatories are seperated by an central angle of
101 degrees. Thus, the length of our basis was about 1.5 times the earths radius (see the
above picture).
And this is our result:
πsun = 8.7 ± 0.2
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
13
dsun = 23700rE ± 2.3%
or, taken the equatorial radius of the earth as known,
dsun = 151000000km ± 3000000km.
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.
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