Transits of Venus are not the only way of finding the AU. Although, by the end of the nineteenth century, useful accuracies of about 2% were eventually achieved by transits, it came at the expense of mounting many lengthy international expeditions, and pooling their results. Better but less glamorous methods somehow struggled to be taken so seriously. Yes, such bias still occurs in funding research nowadays!

There is an excellent historical review of methods in Prof D. W. Hughes article "Six stages in the history of the astronomical unit" in the Journal of Astronomical History and Heritage Vol 4 pp15-28 (2001) which shows in particular that early measurements all underestimated the AU. Nobody wanted to believe that the Sun was so far away!

The following draws heavily on that article, but with emphasis on the types of method, in relation to the rest of this website.

1 GEOMETRICAL METHODS

1.1 Transits

Edmond Halley was the first (1677) to propose measuring the difference between transits of Venus (or Mercury) as seen from the north and south hemispheres to find the AU.

There are several methods of using the data which can be obtained in a transit. Halley's analysis seems eminently sensible, to measure the total time of the transit as seen from the two sites in order to determine the lengths of the chords traced out on the Sun. The ratio of the lengths can then be used to calculate their angular separation, knowing the angular diameter of the Sun. Alternatively, drawings of the position of the chord from the two sites, carefully matched in size, could be superimposed and the displacement measured, and again converted into an angular separation.

Which is better, drawing or timing? The drawings would work well if the separation were large but it suffers from the fact that all, or at least most, of the Sun's disk must be included to allow the superposition, and this means a magnification which is too low to get an accurate result, given that the actual separation is less than the angular diameter of Venus. So timings are better and the analysis is identical to the analysis of the single contact N-S baseline method given elsewhere on this website, except that here the time differences are twice as big, since both ends of the transit are used rather than just the one end.

Delisle suggested using an E-W baseline, that is, two observers at approximately the same latitude but spaced as far as possible in longitude. This limits the baseline to less than a radius. For each contact, there will be (weather permitting!) a pair of timings. The calculation is simpler than for N-S baselines, since one only has to consider that the silhouette of Venus moves at a speed given by the difference of the speeds of Venus and Earth. This is what astronomers call the synodic rate, 0.0257 degrees per hour, so for instance a 3 minute difference between the two ends (= 0.05 hr) corresponds to 0.00128 degrees for that baseline.

For the purposes of this website we have chosen a N-S baseline, concentrating on the third contact. This is a hybrid of the two methods, but seemed preferable in simplicity and practicality.

It would be much easier if one could use measurements by a single observer, and so not have to combine results from different instruments. Suppose for a moment that the Earth were not rotating. One would indeed see a transit, but it would be due purely to the different speeds of the Earth and Venus and its duration would contain no information about the solar parallax. However the Earth is rotating, and so an observer gets a free ride during the transit, covering an arc shaped baseline. So the observed duration is changed by an amount which depends on the solar parallax. The problem is how to separate out the few minutes due to the parallax from the several hours due to the differing speeds of Venus and the Earth. Since the observer's path is curved, the component of the observer's velocity projected on to the Earth's orbit changes with time. For instance if the transit started at dawn, that component would initially be zero. The speed of the silhouette of Venus passing across the Sun increases very slightly during the transit. Measuring an acceleration is much more difficult than measuring a time, and the baseline length projected on to the Earth's orbit is necessarily small, so as a method it is not competitive. That is why so much money was spent on international expeditions in the eighteenth and nineteenth centuries for the Halley and Delisle methods.

Whatever the method chosen, the limiting factors are the poor images which do not allow accurate measurement, and the practicalities of finding or setting up a suitable pair of observatories.

1.2 Mars

The poor image problem can be avoided by a different method, using Mars. Mars comes almost as close to the Earth as Venus. It was particularly close in 2003, and became a media event by looking, for several weeks, brighter in the sky than any person alive had seen before. It was an excellent time to make a measurement of the AU, but nobody bothered! The Mars method had however been used in the past, from 1672 onwards. The method is similar to a transit, but simpler. The orbit of Mars is outside that of the Earth so it never transits the Sun, our own star. It does however pass by the line of sight to many other distant stars, which provide sharp clear reference points. To make the measurement practicable, it needs a fairly bright star (so that its light is not drowned out by Mars) in the same field of the telescope as Mars itself. All that is needed is to view both objects at the same time from different parts of the Earth, making an accurate measurement of the angle between them. The distant star appears as a point of light, so that is easy. As for Venus both E-W and N-S baselines can be used, the choice is more a matter of practicality rather than astronomy. There will usually be other stars in the same field, so relating the measurements of the two observers after the event is not too difficult. On this point however an astronomical judgement is needed - to get high accuracy of the measurement between Mars and the chosen star needs high magnification which implies a field of only a few minutes of arc, but to get enough reference stars needs a wider field and lower magnification.

1.3 Using Asteroids

Although the asteroids are clustered between Mars and Jupiter, some of them have quite elongated orbits and so can come closer to the Earth than Mars. They can be used in exactly the same way as Mars itself. The asteroids Flora and Eros have provided suitable opportunities when they approached within 0.2 AU. They appear point like and not too bright in a modest telescope, and so are somewhat easier to measure than Mars. Closer would be even better, because it makes the angular displacements bigger. If and when an asteroid is found to be on a collision course with the Earth - another media event - the day before the disaster will be an excellent time to make the (final) measurement of the AU!

2. GRAVITATIONAL PERTURBATIONS

The planets and asteroids in the Solar System are retained in orderly motion by the gravitational pull of the Sun, but it is not in sole and total control. When a body comes close to the Earth it reacts to the gravitational pull of both the Earth and the Sun, to an extent depending on their relative distances. An obvious case is the Moon. The neap tides register the effect, as Newton was quick to realise. However the orbital effects are small and anyone looking at the trigonometry involved nowadays might well conclude that the analysis of the effects would be impossible without modern computers. That is not true, however, various interplanetary gravitational effects were used to make sensible estimates of the AU, although it has never been a prime method.

3. MEASURING SPEEDS RATHER THAN DISTANCE

This method not only measured the AU, but uncovered new fundamental physics. Since the orbital periods of the planets (their "years") are well known, a measurement of the speed of a planet immediately gives its distance from the Sun - it is just the product of the speed and the period, divided by 2 pi. So we can find the value of the AU from the speed of the Earth. But how can we measure the speed of the Earth? - nothing in our normal experience betrays the fact that we are travelling at over 150,000mph! Bradley discovered a systematic displacement of the apparent positions of stars caused directly by the speed of the Earth in its orbit. The displacement angle, in radians, is approximately the ratio of the speed of the Earth to the speed of the incoming light. The angle was measured, with commendable accuracy, as being close to 20 arc seconds, which is 20/(60 x 60 x 57.1) radians, =0.000097 radians (i.e. just under 1 in 10,000). Since, as we now know, the speed of the Earth is 30 km per second and the speed of light is nearly 300,000 km per second, it is clear that this was a very good result. Incidentally the Michelson-Morley experiment was designed nearly 200 years later, (in 1900) to detect the motion of the Earth "through the ether" that is, in absolute terms, using a very sensitive interferometer. Instead it became famous for finding no effect at all, a result which Einstein was, at the same time, in the process of predicting as a consequence of special relativity. Einstein's name is indissolubly linked with orbital mechanics, since he showed that the planets move round the Sun in accordance with general relativity, not with Newton's Laws. The difference is small in percentage terms, and at first only detectable for Mercury. The change it required in the basic concepts of physics was, however, immense. One could hardly have a better example of the fact that fastidious measurements, to seemingly unnecessary accuracy, can reveal matters of fundamental importance.

4. RADAR

Moving closer to modern times, Eros comes into the story for the second time. Radio telescopes are normally used to collect incoming radiation and focus it on to a detector. If the detector is removed and replaced with a radar transmitter, a narrow beam of powerful radar pulses can be sent into space, in any chosen direction. Reflections from Eros were used in 1960 to deduce its distance and hence its orbit to high accuracy. With elliptical orbits it is the length of the major axis which is important. Kepler's ratios of orbits then applies and the AU is deduced. In fact, Kepler's Laws were written in terms of the major axis, but for a circle the major axis is the same thing as the radius. The circle, after all, is just an ellipse of zero eccentricity. As we know (all too well!) radar can measure speed, which gives a second and independent way of deriving the AU from Eros, but working from the distance is more accurate. Why not measure the distance of the Sun directly by radar? Whilst that is now possible, the echo comes back as a mixture of pulses from a spread of depths in the solar atmosphere so the sharply defined solid surface of Eros is a better target. In the same vein, a telescope in Greenwich has been modified to send pulsed laser signals to the Moon. For each pulse only a few photons return, about 1.5 seconds later, to the telescope, but that is sufficient to routinely make measurements of the Moon's distance to within a few metres. With this accuracy it is easily seen that the distance is continuously varying because of the elliptical shape of the orbit, but on top of that there are also several subtle processes hidden within the fine detail of the orbit, which is why it is worth measuring it so exactly. As an example, the orbit provides the most accurate way of checking whether the gravitational constant is indeed constant. Maskelyne would have been delighted by this measurement .