J.E. Arlot, P. Rocher,
We now have received all the data of timings of the contacts and we are able to make the final calculation of the Astronomical Unit (AU) through several algorithms after selection of data. What is the final value of the AU? Can we compare the results with those of the former centuries? What is the precision of today measurements? You will find below some answers to these questions.
The database
We received
4500 timings of contacts from 2500 registered observers. Most of the
observations were made in

T1 
T2 
T3 
T4 
all 

676 
1105 
1297 
1137 
4215 

8 
14 
21 
20 
63 

3 
3 
30 
27 
63 

35 
59 
60 
32 
186 

9 
14 
0 
0 
23 
all 
731 
1195 
1408 
1216 
4550 
Number of timings received
Note that the observations of T1, T2 from the
The calculations
The calculation of the Astronomical Unit will
be made of several manners, either using new possibilities available in the XXIth century or using old methods as it was in the past
centuries.
The first method is the calculation in real time of the AU as
the data enter the database. The calculation starts with the first entry which provides a value of the AU (the solution
to the question: what should be the AU in order to minimize the difference
between the observed timing and the theoretical one?). A sample average is made
with the next entries. Since we did not know in advance the accuracy of the
entries, we had to suppose known several parameters introducing a constraint in
the algorithm (the relative distances VenusEarthSun were supposed known, i.e.
the Sun was supposed to not be at the infinite); if not the averages of the
calculated AU were not possible.
The second method is possible when we have the whole database.
From a starting AU we make predictions and keep only data near the predictions.
These data allows to determine a new AU allowing to determine new predictions
and so on. The process will converge to a final AU. If starting from the
wellknown AU determined by radar, the process does not need to be iterate but
leads to the same value of the AU.
The third method is the use of the Delisle’s
or Halley’s method: the use of couples of observers to determine the distance
EarthSun through the parallactic effect. This method
needs also the knowledge of all the data in order to choose the optimum ones.
However, since the observers were not situated on optimum sites (where the
parallax is maximum) as proposed by Delisle or
Halley, the calculations will be difficult.
The online calculation in real time
on
During the
transit on June 8, we received the observations of timings of contacts and we were
able to make a calculation of the AU in real time (see the information note
n°31a for more details). We were not able to use an algorithm
based upon Delisle’s method since we
had not all the data and since the
observations were not made from
selected sites. So, we made comparison to the theoretical predictions in order to
determine an AU for each timing (the AU for which the difference between the
observed timing and the calculated one is the smallest) and we facilitate the
convergence of the calculations in order to accept all the data. The result,
using the 4367 timings entered in the database, is as follows:
AU =
149 529 684 km +/ 55 059 km ; difference to the « true »
AU : 68 186 km
We facilitate the convergence of the calculation through a constraint avoiding the Sun to be at the infinite (that will falsify the average).
Note that
nowadays we know the « true » value of the AU thanks to radar
observations and we may calculate the difference to the true value anytime.
The calculations made using all the data
When having
all the data, we may decide which observations should be kept and which ones
must be rejected. Then, it will no more be necessary to put a constraint on the
calculation of the AU. But how to decide that some observations are good and
some are not? Starting from a value of the AU, we calculate the theoretical
timings. We keep only the data near the predicted timings (within a few seconds
of time providing the distribution of the kept data be Gaussian) and determine
a new AU allowing tho determine new theoretical
timings, and so on. In fact, starting with the known “true” value of the AU
does not change the final result but avoid any iteration of the process (we
provide more explanations in the information note n°31b).
First criterion : we eliminate the timings being outside
an interval of 16 s centered on the prediction. Results
are as follows :
contacts
T2, 262 timings, AU = 149 590 268 km +/ 108 359 km, diff. to AU
7602 km
contacts
T3, 421 timings, AU= 149 226 725 km +/ 324 822 km, diff. to AU
371 145 km
all
contacts, 1066 timings, AU= 149 421 803 km +/ 252 081 km, diff.
to AU 176 067 km
all contacts
weighted, 1066 timings, AU= 149 507 347 km +/ 173 437 km, diff.
to AU 90 523 km
In the case
of weighted contacts, the more weighted contacts correspond to wellsituated
observations (with the largest parallax).
Second criterion : we eliminate the timings being outside
an interval of 8 seconds centered on the prediction. Results
are as follows :
all
contacts, 583 timings, AU= 149 608 708 km +/ 11 835 km, diff. to AU 10 838 km
What conclusions from these results ?

the best results are those with the smallest error, not the one the
closest to the “true” value of the AU; it is the one corresponding to the
“best” observations (within 4 seconds to the prediction) that provides an error
of only 10 000 km.

note that the difference to the « true » AU is smaller for the
contacts T2 within 8 seconds ; this comes from the large parallax
observable during T2 by most of the european
observers (sunrise) but the bad seeing and the difficulties at sunrise led to a
large error on the result (+/ 108 359 km)

the timings of contacts T3, more numerous,do
not provide good results, most of the european
observers observe dit near the meridian and the
zenith when the parallax to be measured is the smallest. Even a good
measurement of the timings provide a poor result.

when the timings are weighted by the site (i.e. by the parallax), the
result is slightly improved but the
number of wellsituated sites is too small (
Our
final result may be
the one obtained thanks to the timings within 4 seconds of time from the
prediction :
AU =
149 608 708 km +/ 11 835
km (diff. to AU 10 838 km)
This result
is better than the one obtained in real time since we eliminated the bad
observations. The elimination of more observations does not provide better
results (too few data, without a Gaussian distribution).
The calculation using
Delisle’s method
Since we
have now all the data, we may use the Delisle’s
method to calculate the AU, i.e. we will associate observations by two to
calculate the parallax. Please see the information note n°31b for more details
on these calculations. Below, the main results using the observations within 8
seconds of time from the prediction (interval of 16 s centered
on the prediction). Same as above, without the prediction made with the “true”
value of the AU, we should iterate the calculations.
contacts T1, 103 couples, AU= 149 593 369 km +/ 1 308 668 km, diff. to AU 4501 km
contacts T2, 1531 couples, AU= 149 604 208 km +/ 535 661 km, diff. to AU 6338 km
contacts T3, 1979 couples, AU= 150 623 168 km +/ 423 861 km, diff. to AU 1 025 298 km
contacts
T4, 773 couples, AU= 148 904 105 km +/ 534 664 km, diff. to AU
693 765 km
all
contacts, 4386 couples, AU= 149 840 958 km +/ 310 577 km, diff.
to AU 243 088 km
Which conclusions to draw from these results ?

the contact T1 was observed only once in

the result for the contacts T2 is good, the parallax is well observable
from most of the observing sites (see maps on information note n°31b)

the result for the contacts T3 is not good, the parallax is too faint
for this method since observers observed when the Sun was near the
meridian ; the « true » AU is outside the error interval that is
to say that the distribution of the errors is not Gaussian.

same for the contacts T4

when using all the contacts, the error decrease : we will choose
the value of the AU deduced from all contacts for our final result of the AU.
Calculations with Halley’s method
Halley’s
method is based on measuring the duration of the transits from several
wellsituated sites, i.e. sites from where the durations of the transits are
very different. A great advantage of this method is to avoid the knowledge of
any absolute timescale such as the Universal Time. The inconvenient is the
necessity to observe the beginning AND the end of the transit. This reduces the
number of possible sites and makes the observations too sensitive to
meteorological conditions.
On
In fact,
Halley’s method may be applied only if the maximum of visibility of the transit
takes place above the
Other possible calculations
Since the
Astronomical Unit was well known nowadays, we thought that it would be possible
to consider the AU as a known parameter and to introduce the radii of the Sun
and Venus as unknowns. The calculations were made with the best 1066 selected
observations of the database and the results are as follows:
Radius of
the Sun: 695980.7 km +/ 806km, diff. to present value: 8.6 km
Radius of
Venus: 6052.3 km +/ 7.0km, diff. to present value: 0.5 km
In
conclusion, the radii of Venus and the Sun are wellknown nowadays and our
determinations are affected by large errors regarding the calculated
differences to the present values.
The comparison with the calculations made with the transits of the past
centuries
When
compiling the results obtained from the observations made during the past
transits of Venus, we may deduce the following results :
for the XVIIth century :
 Horrocks, AU= 94 000 000 km, diff. to AU :
55 597 870 km
for the XVIIIth century :
 1761, Pingré et Short, AU= 138 540 000 km +/
14 400 000 km, diff. to AU 11 057 871 km
 1761
& 1769, Lalande et Pingré,
AU= 151 000 000 km +/ 1 500 000 km, diff. to AU : +1 402 130
km
 1761
& 1769, Newcomb (1890), AU= 149 670 000 km +/ 850 000 km,
diff. to AU : +72 130 km
for the XIXth century :
 1874
& 1882, Newcomb (1890), AU= 149 670 000 km +/ 330 000
km , diff. to AU +72 130 km
for the XXIth century, our final “official result”:
 2004, VT2004
programme, AU = 149 608 708 km +/
11 835 km, diff. to AU +10 838 km
The
comparison with the results of the XXIst century
shows :

before the XVIIIth century, the AU was
strongly underestimated

after the transit of 1761, the AU was not well determinated
but after the transit of 1769, the results were improved. More, a new
calculation by Newcomb in 1890, correcting the wrong longitudes used during the
XVIIIth century led to an accurate value of the AU
which will not be improved after the transits of the XIXth
century (only the errors will decrease).

the results of the XXIth century are the
best ; in spite of the inexperience of the observers and in spite of the
random distribution of the observing sites, the use of GPS, of Universal Time,
of CCD receptors and of good quality optics (very few black drop), allows to
get good timings

The error of measurement decreases century after century thanks to
technological progresses
Note that
even many observations were made in the past centuries (100 in 1761, 120 in
1769, 71 in 1874 and 85 in 1882), only a few – a selection of the best ones –
were used for the calculations (from 10 to 30 depending on the authors). The
errors provided in the table above are estimations: no error calculation are
available since the theory of errors was developed at the beginning of the XXth century. The values of the AU for the past centuries
are based upon a value of 6378.14km for the radius of the Earth.
Conclusion
We reached
our goal : to show to young pupils, students, amateurs and general public
that an international collaboration allows to make a scientific measure of good
quality. An interesting and surprising
conclusion is the quality of the observations carried out, quite higher than
that of the previous centuries. The good
knowledge of the longitudes, the availability of the Universal Time everywhere,
the recording of the event thanks to CCD cameras and the quality of the
optics avoiding a too large “black drop”
effect, explain the good result on the AU.
We will be able
to start again in 2012, and, as for the previous centuries, we will take
benefit from the 2004 experiment which will enable us to improve the result:
 by optimizing
the position of the observers on the Earth, requesting the sites well located
 by using a
criterion of selection of data for the calculation in real time (founded on the
observing precision noted in 2004) allowing to use a no constrained algorithm.
If you wish
to make yourself the calculation, databases are on line and softwares
are available to calculate the AU. See our web pages at http://vt2004.imcce.fr/vt2004i.
The list of
information sheets is available at http://www.imcce.fr/vt2004/en/fiches_eng.html.