calculation of the astronomical unit

# Final results for the calculation of the AU :

J.-E. Arlot, P. Rocher, October 3, 2004.

### Introduction.

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 Europe but data from the U.S.A., from Africa, from Asia and Australia also arrived (see the maps available on the web site). Unfortunately, the observers were not situated on the best sites –i.e. the sites with the largest parallaxes- and the Delisle’s and Halley’s method was difficult to use.

 T1 T2 T3 T4 all Europe 676 1105 1297 1137 4215 Africa 8 14 21 20 63 Americas 3 3 30 27 63 Asia 35 59 60 32 186 Australia 9 14 0 0 23 all 731 1195 1408 1216 4550

Note that the observations of T1, T2 from the Americas correspond to wrong entries of coordinates since these contacts were not observable from there. Note also that some observers entered several times the same observations because of problems in the internet connection! The number of data above corresponds to the full database including wrong entries. When excluding the obviously meaningless data, the total number becomes 4509. The final calculations will be made using a “clean database” selecting only the “good” observations through several criteria as we will see below.

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 Venus-Earth-Sun 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 well-known 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 Earth-Sun 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 on-line calculation in real time on June 8, 2004

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 well-situated 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 well-situated sites is too small (Australia, Siberia, …) to get a large improvement.

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 Australia with a high accuracy leading to a very good value of the AU; because of the strong correlation between the combinations, the error on this result is very large

-         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 well-situated 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 time-scale 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 June 8, 2004, about 2000 registered observers were able to see the beginning and the end of the transit. Unfortunately, they were mostly in Europe: only 10 were well situated for Halley’s method and none of them having a sufficient accuracy.

In fact, Halley’s method may be applied only if the maximum of visibility of the transit takes place above the Americas. In that case, it is possible to find observing sites, the American continent laying from the north to the south. In 2004, the maximum was from Siberia to the Indian ocean, not favourable for Halley’s method.

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 well-known 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, VT-2004 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.

last update: November 20, 2007
contact: vt2012 (at) imcce.fr