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- Dr Rodney Hillier Bristol University on Friday 7 May 2004

[Preamble: A very rare event will take place on 8 June 2004. The planet Venus will pass between us and the Sun and will be seen in silhouette between about 6.19 a.m. and 12.24 p.m. (both times BST).

The event will not be spectacular but a once-in a lifetime experience as it has not occurred since 1882 It has never been possible to see the complete phenomenon from England for the last 700 years and even then no one saw it because astronomical calculations were not accurate enough to predict it nor had the telescope been invented. - RHP]

The Earth orbits the Sun in 365.26 days. As Venus does the same in only 224.70 days, it overtakes the Earth from time to time. When this happens and the Sun is in line, as on 8 June, Venus is said to be in ‘inferior conjunction’. How often does this occur? Well, as the Earth passes through 360/365.26 = 0.98560 degrees per day while Venus covers 1.60214 degrees in the same time, it outstrips the Earth by 0.61654 degrees each day. Therefore, to lap the Earth will take 360/0.61654 = 583.91 days. This is called the ‘synodic period’ of Venus and is the time between each inferior conjuction. In this time, the Earth has passed through 575º and Venus through 935º so that they are both at 575º – 360º = 215º and 935º – 2 x 360º = 215º from the last position of inferior conjunction. The five inferior conjunctions following the first are described below:

b: 2nd conjunction

t =584 days

(1 synodic period)

Venus: 935º

i.e. 2 revs. + 215º

Earth: 575º

i.e. 1 rev. + 215º

c: 3rd conjunction

t = 1168 days

(2 synodic periods)

Venus: 1871º

i.e. 5 revs. + 71º

Earth: 1151º

i.e. 3 revs. + 71º

d: 4th conjunction

t = 1752 days

(3 synodic periods)

Venus: 2807º

i.e. 7 revs. + 287º

Earth: 1727º

i.e. 4 revs. + 287º

e: 5th conjunction

t = 2336 days

(4 synodic periods)

Venus: 3743º

i.e. 10 revs. +143º

Earth: 2303º

i.e. 6 revs. + 143º

f: 6th conjunction

t = 2920 days

(5 synodic periods)

Venus: 4678º

i.e. 12 revs. + 358º

Earth: 2878º

i.e. 7 revs. + 358º

So this last conjunction, after 2920/365.26 = 7.99 (= 8 approx.) years is only about 2º (more exactly 2.4º) away from that of the first!

Why does the transit not occur at every conjunction (every 584 days)?

We need to take a 3-dimensional view of the orbits. The Venus’ orbit is in a plane tilted at 3.4º to that of the Earth so the conjunctions occur mostly when the planet is too high or too low. Only when the conjunction occurs close to the point where the orbit crosses the plane of the Earth’s orbit i.e. near a ‘node’ will a transit take place. As we have seen see, a transit could take place 8 years after the last one. Once this has happened it will not do so again, because 16 years after the first transit, the conjunction will be 4.8º away from the node and the planet will be too high (or low) for transit.

The five-point pattern of inferior conjuctions will, each 8 years seem to move clockwise by about 2.4 º i.e. 0.3º per year.

As (f) passes the ascending node there will be two transits 8 years apart.

There will be no further transits at that node until (c) arrives 152 conjunctions, or 243 years later.

Before this happens (b) will pass the descending node after only 81 conjunctions or 129½ years.

The cycle will then repeat itself, starting with (c) at the ascending node.

These events are summarised as in the following diagram:

Ascending Descending

node (Dec) node (Jun)

The speaker then went on to explain the importance of the transit of Venus to 18th century astronomers.

To measure the distance of any object, it is sufficient to measure its parallax. This is the difference in the direction of the object as seen from each end of a baseline. The distance then can be determined by drawing a scale diagram or using trigonometry – the mathematics of triangles.

The longest baseline available for solar system measurements is the diameter of the Earth but even for the nearest planets the parallax is very small: about 1/100 degree.

Edmund Halley realised, in 1677, that the parallax of Venus could be derived from measurements of the durations of transits observed from sites at very different latitudes on the Earth.

Seen from different latitudes, Venus moves along different chords of the Sun.

The distance between the chords is the parallax of Venus. This small angle is related, by simple geometry, to the difference in the lengths of the chords which can be derived from very accurate (to the nearest second) timings of the duration of the transit.

Once the distance of Venus from the Earth was known, then the distances of all the planets (including the Earth) from the Sun would follow from the work of Kepler whose third law derived their ratios. The distance of Venus would provide a scale to the model.

The expeditions in the 18th and 19th centuries were designed to make these measurements. However, the determination of the parallax of Venus was only partially successful because of the difficulty of determining the end points because of the ‘black drop’ phenomenon.

From radar measurements we now know the semi-diameter of the Earth’s orbit, 1 Astronomical Unit (AU) = 149,597,870 km.

Richard H Phillips