Measuring the Astronomical Unit - part i

Updated 29 May 26

2004 transit of the Sun by Venus, projected on a
wall through one side of 10x50 handeld binoculars.

Having seen that trying to estimate the size of the AU in terrestrial units by measuring the Moon's penumbra during a solar eclipse is a non-starter, let's look instead at the tried-and-tested method of parallax.

Consider Mars, for example, when it is at opposition and about half an AU from the Earth. Let's create a baseline of 5 thousand miles, say, measured through the Earth, with at each end, an experienced observer and telescope pointing at the planet. Now arrange for readings to be taken at an agreed instant when the baseline is perpendicular to a line between Earth and Mars and thereby maximizing parallax.

The method needs:

• accurate and precise local time-keeping - only possible with the invention of the long pendulum clock in the late 1600s
• an accurate and precise measurement of the difference in longitude to determine the baseline and time-difference - actually possible on land with the use of astronomical tables before the invention of the marine chronometer.

If Mars were say 5 million miles away, then the angle of parallax would be one thousandth of a radian. One radian is roughly 60 degrees or 60 x 60 minutes of arc. One thousandth of that is roughly three and one half minutes of arc. That's about ten times the angular size of Mars itself. Now if the observers can draw Mars against the background of stars with reasonable accuracy, the difference in position should be measurably accurate.

If Mars were instead say 50 million miles away, then the angle of parallax would now be one tenth of that and about the same as the angular size of Mars itself. Perhaps this would be too small to discern. However, if the planet were seen to obscure a star (an example of an occultation), from one location and to touch it at the other, then that would provide a measurement of parallax.

Even if the angle were too small to measure, it should at least be possible to establish a minimum size for the AU.

Venus is closer to us at conjunction, just 0.28 AU away, a little over half the distance to Mars at its closest. Its maximum angle of parallax is therefore about twice that of Mars and hence easier to measure. How to observe it in daylight? Just wait for when it transits the Sun! Then during the transit, our observers can record the progress of Venus across the disk of the Sun, noting the times as they do so. Afterwards, they can compare their findings, noting the different positions of Venus with the same 'timestamps', allowing for difference in longitude.  Armed with this information and the length of their baseline (and its orientation at each instant to the line between the Earth and Venus), they will now have an angle of parallax for Venus measured against the Sun's disk. However, this will be smaller than the true figure, had they been able to measure it against the fixed stars. The reason being that there will also be some parallax for the Sun - 28% of that measured for Venus. So, the true figure for the parallax of Venus will be the measured figure times 100/72. While it sounds easy in principle, I won't underestimate the difficulty of doing this in practice, especially before the invention of photography.