4°) Measuring the Earth-Sun distance

Several observatories throughout the world have already begun to get ready for the transit of Venus in June 2004. In France, the Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE) intends to run training courses for teachers to observe the sky, and also to centralize data collected by amateurs for calculating in real time the Earth-Sun distance on the big day. All over the world, pupils could be encouraged to take part in data collection. The method proposed below is not that of Edmond Halley which was based on the timing of transit, and which involves some difficult calculations.

During the transit, an astronomer sees a minuscule black spot moving slowly across the Sun, the solar disk serving as a sort of screen. If one plots this spot and joins its successive positions, one can draw a line over the solar disk (Figure 3).


Figure 3: View of the successive positions of Mercury crossing the Sun during the transit of 7 May 2003 (Picture obtained by the National Solar Observatory.)

As we can see from Figure 2, observer A traces line aa' and B line bb'. When they pool their observations in a single scale drawing of the Sun, the diagram will show by how much the Sun’s diameter is greater than e. Once we know the distance e, we will then be in a position to work out the size of the Sun.

If AB and A’B’ are approximately parallel, or if we take AB to be the separation of the observers perpendicular to the direction of the Sun, then AVB and A’VB’ will be similar triangles. This being the case, e = AB.d_VS/d_EV (d_VS and d_EV respectively being the Venus-Sun and Earth-Venus distances). Kepler’s Third law converts d_VS/dEV into a ratio between the sidereal periods of revolution of the Earth (T_E) and of Venus (T_V) around the Sun (about 365 and 225 days respectively):


From this relation, e equals approximately 2.6 AB.

If we assume AB equals 4 000 km, for example, e will be more or less 10 400 km: the scale drawing mentioned above should indicate the Sun’s diameter to be some 135 times “larger” than e, which corresponds roughly to 1 400 000 km. The Sun’s distance from the Earth can be then obtained by using the apparent diameter of our star: 32 arc-minutes, that is, a little under a hundredth of a radian. By combining these two items data, we can situate the Sun 150 000 000 km far away from us.

Although simple in theory, this measurement technique presents a number of difficulties, particularly if one proceeds on the basis of projecting the Sun’s image onto a screen, a method which has the advantage of being both cheap and easy to set up for a group observation. Let us use the above example to work out the size of e using a projected image of 20 cm diameter. The width of the strip on the screen will not normally exceed 1.5 mm. And it will be even narrower – almost invisible - between two cities located at a distance of only a few hundreds of kilometres from each other. There is always, of course, the possibility of cooperating with some very distant countries, say Norway and South Africa. It is difficult however to imagine undertaking the journey to these countries just for the purpose of making this sort of measurement. We could try to get data from these countries via the Internet (a long list of foreign astronomy clubs with their electronic addresses is provided in Vénus devant le Soleil – See ‘Further reading’.)

The most interesting option would be to replace the projection of the Sun method with a modern imaging technique, perhaps using a digital camera fitted to a suitably protected telescope. It is essential, for safety that this is only done on the basis of expert advice. It would however have the advantage of making it much easier to utilise sunspot as reference points for the purpose of adjusting the size and position of images received from different countries.

Clearly, projection methods will give less reliable results than digital imaging. Nevertheless, even a poor result could still show that our nearest star is an immense distance away and discussion of the errors might be more useful than a lucky ‘correct’ answer. There will be plenty of very accurate images available on the Internet after the event. Students can use these to supplement their own (available in www.venus2004.org).