HOW TO PREDICT TRANSITS OF VENUS

by Knut Jørgen Røed Ødegaard

Accurate predictions of transits of Venus require advanced computer programs and lots mathematics. The transits are however, regular enough to be predicted quite good by studying statistics.

TRANSITS OF VENUS FROM 1631 - 2012

Date	Node	Interval to next (years)
1631, December 7	Ascending	8
1639, December 4	Ascending	121.5
1761, June 6	Descending	8
1769, June 3	Descending	105.5
1874, December 9	Ascending	8
1882, December 6	Ascending	121.5
2004, June 8	Descending	8
2012, June 5-6	Descending

Let us for simplicity assume that the Earth and Venus move in circular orbits outside each outer. In reality the orbits are slightly eccentric, but not enough to ruin our estimates.

When are the Earth, Venus and the Sun aligned at the ascending node of Venus? Observed from the Earth Venus will then cross the solar disk.

Such an exact alignment will occur again when the Sun has completed a number of orbits. Both planets will then be back in the same position. If will call these integers n and N, the following expression gives us the number of orbits:

224.701 n = 365.25636 N

The siderian year of Venus is 224.701 days, while the siderian year og the Earth is 365.25636 days.

This equation is never fullfilled completely fo rinteger values of n and N. Equallity almost occur for the values of n and N given below:

Number of Earth orbits	Number of Venus orbits	Difference (hours)	Interval (years)
0	0	0	8
8	13	22.5	227
235	382	12.9	8
243	395	9.6	235
478	777	3.3

The time intervals between the alignments are too long to explain all transits. However, transits also occur at descending node, that means on the opposite side of the Sun. There Venus is crossing the orbital plane of the Earth from above.

These alignments must in the same way as at descending node occur with an interval of an integer number of orbits, but half an orbit after the starting point.

The equation becomes

224.701 (n + 1/2) = 365.25636 (N + 1/2)

For 121.5 Earthly orbits and 197.5 Venus orbitals the equation is almost satisfied. The difference between the left and right side of the equation is only 4.8 hours.

Transits occuring at descending node come with the same time intervals of 8, 235 and 243 years as the transits at ascending node do. A couple of ascending and a couple of descending nodes are separated by 121.5 years.

This give us the pattern in the first table.

Venus creates a giant shadow when moving through space. The shadow is racing past the Earth with more than 60 000 km/h, but once in a very seldom while it hits the Earth.

The shadow is nearly half a million kilometers in diameter when crossing the Earth. It is therefore not necessary for Venus to be exactly in one of the nodes for a transit to occur. Venus may be two days away for the nearest node and still cause a transit. If the distance is much more than two days, the shadow will miss the Earth - it will move either below or above our planet.

During a transit of Venus the Earth is moving through the penumbra behind Venus. The planets are not to scale. Illustration: Institute of Theoretical Astrophysics, University of Oslo

The showdow cast by Venus consists of an inner umbra and an outer penumbra. (See also illustrations in these articles about eclipses of the Moon (in Norwegian) November 9, 2003 and May 4, 2004).

The umbra is conical with it top point turned away from Venus. The reason is the angular size of the Sun. The umbra is far too short to reach the Earth. If it had reached us, we would have had a total eclipse of the Sun! The penumbra is also formed like a cone, but the top point is in Venus and it is widening away from Venus.

It is there the penumbra that can hit the Earth. Seen from Earth Venus is a small , black dot against the large solar disk.

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