 # HM Nautical Almanac Office: Transits of Venus

## Cyclical Nature of the Transits of Venus

At inferior conjunction, Venus lies in the same direction as the Sun. If the orbit of Venus was in the same plane as the orbit of the Earth, a transit would occur at every inferior conjunction. However, the orbit of Venus is inclined at approximately 3.4° to the ecliptic, and an alignment of the Sun, Venus and the Earth can only take place along the line of nodes, where the plane of the orbit of Venus crosses that of the Earth.

The sidereal period of Venus is 224.701 days and that of the Earth is 365.256 days. The synodic period for Venus, or the period between successive appearances of Venus at the same point relative to the Sun as seen from the Earth, is 583.924 days. It can be shown that 5 synodic periods of Venus (2919.62 days) corresponds very closely to 8 Earth orbital periods (2922.05 days) or indeed 13 orbital periods of Venus (2921.11 days). This underpins the 8 year periodicity in the transits of Venus.

In general, for alignments to take place at one of the nodes, we need to solve the equation
224.701.n ≈ 365.256.m,

where n and m are integer number of orbital periods for Venus and the Earth respectively. Here we also assume that the orbits of the Earth and Venus are circular. There are no exact solutions to this equation, we can only find close approximations. The first good approximation is n=13 and m=8, the difference between n and m being the number of synodic periods. However, a better fit can be obtained with n=382 and m=235 and better still, n=395 and m=243. Alignments at each node occur at intervals of 243 years.

When considering the change from one node to the opposite node, we need to solve the equation
224.701.(n+½) ≈ 365.256.(m+½).

A good fit can be obtained with n=197 and m=121, or 197.5 Venus orbits is a close match to 121.5 Earth orbits. An ascending node pair of transits will be followed 121.5 years later by a descending node pair of transits. The diagram below shows that corresponding members of each pair of transits are separated by 243 years and can be seen as members of a series of transits in much the same way as solar eclipses are members of Saros series. The transits of 1518, 1761 and 2004 are members of one series, the transits of 1396, 1639 and 1882 are members of another. We can see that the transit of 1631 is the beginning of a new series as the separation of the "near" transit of 1388 was just too large for a transit to take place. In the diagram above, Venus moves from left to right (east to west) in all cases and north is at the top. Both the 2004 and 2012 transits occur at the descending node whereas both of the 19th-century transits occur at the ascending node. Six transit "seasons" are shown in the plot e.g. 1388-1396, 1518-1526, 1631-1639, 1761-1769, 1874-1882 and 2004-2012. However, only eleven transits actually take place. The "near" transit of 1388 is shown to illustrate the fact that not all of the transit seasons have two transits. The plot clearly shows that four transits can occur in a 243 year period. Two transits at the ascending node (denoted by Asc.) separated by eight years, then a 121.5 year gap before two more transits at the descending node (denoted by Desc.) separated again by eight years. After another 105.5 years, the sequence starts again. This pattern holds true at current epochs but has not and will not always be the case.

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