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Part 5 - Predictions of Transits of Venus

The key to Horrocks' success was his accurate prediction of the 1639 transit of Venus. Such transits occur with what may seem at first sight a strange periodicity. The dates (new style) of all the transits from the 17th Century through to the 21st Century are as follows:-

Date	Node	Interval (Years)
1631 December 7	A
		8
1639 December 4	A
		121½
1761 June 6	D
		8
1769 June 3-4	D
		105½
1874 December 9	A
		8
1882 December 6	A
		121½
2004 June 8	D
		8
2012 June 6-7	D

Table 1: Dates of the Transit of Venus

The explanation of this pattern is as follows:-

(a) If at time zero the Earth, Venus and the Sun are in line at the ascending node (i.e. a perfectly central transit occurs), then this will occur again only when Venus has completed an integral number of orbits and the Earth too has completed an integral number of orbits. If these integral numbers are n and N respectively, then:

224.701 n = 365.25636 N

since the sidereal year of Venus is 224.701 days and that of the Earth is 365.25636 days.

(b) These expressions are not commensurable and consequently the equation is never exactly satisfied. Equality almost occurs however for the following values of n and N:

Ascending Node Near-alignments

Earth Orbits	Venus Orbits	Difference (hours)
0	0	0
8	13	22.5
235	382	12.9
243	395	9.6
478	777	3.3

(c) The gaps between these alignments are too large to explain the dates in Table 1, but we must remember that alignments can also take place at the descending node. These must occur at an integral number of orbits (of Earth and Venus), plus a half orbit, after passing the ascending node at time zero. The equivalent equation is then:

224.701 ( n + ½ ) = 365.25636 ( N + ½ )

and the corresponding list of near-equalities is:-

Descending Node Near-alignments

Earth Orbits	Venus Orbits	Difference (hours)
113½	184½	17.7
121½	197½	4.8
356½	579½	8.1
364½	592½	14.4

(d) The list for the descending node near-alignments assumes that the orbit of Venus has negligible eccentricity and hence the time between ascending and descending node is exactly half a sidereal period of Venus. Note also, that, in addition to assuming that the orbits of Venus and the Earth are concentric circles, we have also assumed that the positions of the nodes do not change. None of these assumptions is strictly true, but the above figures are illustrative only.

(e) The above lists, between them, fit the sequence of dates in Table 2.

6. To the Four Corners of the Earth