Going Further: 4.3 Measuring the
Astronomical Unit

Historically, determining the astronomical unit was a multistep process. This is typical of cosmic distance determinations, as this chapter shows. The first step to cosmic distances was relating the distances of other planets to the size of Earth’s orbit. Venus was the critical planet for this technique.

Figure B.4.1 shows Earth, Venus, and the Sun in a configuration in which Venus is at its largest separation from the Sun, as seen in Earth’s sky. This configuration is called greatest elongation. It occurs in two forms, one is when Venus is east of the Sun in the sky, and one is when Venus is west of the Sun. For our purposes, either of these will suffice.

Figure B.4.1. In this figure, the Sun is at the center and the positions of Venus and Earth are labeled. dV indicates the distance from the Sun to Venus, and dE indicates the distance from the Sun to Earth. The figure is not to scale. Credit: NASA/SSU/Aurore Simonnet.

Notice how, in this planetary configuration, the Sun, Venus, and Earth form a right triangle with Venus situated at the right angle. The hypotenuse of the triangle, dE, is Earth’s orbital distance, i.e., the astronomical unit. The side of the triangle connecting Venus to the Sun, dV, is the size of the orbit of Venus, of course. So, using some trigonometry, we see that the ratio of Venus’s orbital radius to the astronomical unit is the sine of the angle (e) that Venus makes with the Sun:

\begin{equation} sin(e) = d_{V} / d_{E} \end{equation}

Since we can easily measure the elongation angle by observing Venus as it moves through the sky during the year (it’s about 42 degrees), we can determine the size of Venus’s orbit in AU. But this is only part of the task. What we want is the size of the AU in meters or some other known unit. To get that, we must view Venus when it is crossing the Sun.

Because Venus’s orbit lies between Earth and the Sun, the planet periodically crosses in front of the disk of the Sun as seen from Earth. Such transits, as they are called, do not happen as often as we might think. This is because Venus and Earth orbit in slightly different planes. The orbital plane of Venus is tilted by a bit more than 3 degrees with respect to the ecliptic, the name given to Earth’s orbital plane. The Sun, however, is only about half a degree in diameter. That means that Venus must pass between Earth and the Sun while it is within a quarter degree of the ecliptic. This alignment is actually quite rare. Transits come in pairs separated by 8 years, but successive pairs are separated by about 120 years. Usually, Venus passes above or below the Sun, and no transit is visible.

However, occasionally, Venus does pass directly in front of the Sun. When that happens, two observers on Earth, at points A and B in Figure B.4.2 for example, might view the event as shown. Notice that they observe a small angular shift (exaggerated greatly in the figure) of Venus against the Sun’s surface. This shift is the key to determining the size of the astronomical unit.

Figure B.4.2. The Sun is at the left-hand side of the figure, and Earth is at the right. Venus is at the point in the middle. The distance from the Sun to Venus is again given by dV, and the distance from Venus to Earth is therefore dE–dV. The distance between two different observers on Earth is given by the line AB. The figure is not to scale. Credit: NASA/SSU/Aurore Simonnet.

In the triangle made between Earth and Venus, the short side, AB, can be measured if the latitude and longitude of each site are known. The angle can be determined by comparing the two observations. Then, using the small-angle approximation, we can deduce the long sides of the triangle. This is the difference in size of the orbit of Venus and Earth’s orbit.

\begin{equation} \theta = AB / (d_{E} – d_{V}) {\rm\ \ \ or\ \ \ } (d_{E} – d_{V}) = AB / \theta \end{equation}

But, we have already determined the ratio of these two distances, so we have two equations with two unknowns, and we can solve for either or both unknown. Using the ratio to eliminate dV, we get:

\begin{equation} d_{E} – d_{E}sin(e) = \frac{AB}{\theta} {\rm\ \ \ or\ \ \ } d_{E} = \frac{AB}{\theta[1–sin(e)]} \end{equation}

This is how the astronomical unit was originally determined. The method was first employed, without great success, because of the small separation between observing points, in 1639. More successful measurements, the result of large international expeditions, were done in 1761 and 1769, and again in 1874 and 1882. The last two transits of Venus were on June 8, 2004 and June 6, 2012. There will not be another transit of Venus until December 2117, and then 8 years later in 2125.

Much more precise determination of the astronomical unit is now possible by bouncing radio signals off of Venus and using time of flight to measure its distance.