One evening at the observatory there was a discussion about transits of Earth from Mars. Later, I decided to pen a few thoughts on this topic.
On Earth, one of the rarer astronomical phenomena is a transit of Venus or Mercury. Such an event occurs when the planet concerned crosses the disk of the Sun. Clearly, the angular size of the planet is much less than that of the Sun and so an observer sees a small black dot take several hours to cross from one side of the Sun to the other. In the case of Mercury, the small black dot is a sharp one but in the case of Venus, the dot is somewhat fuzzy due to the Venusian atmosphere.
Predicting transits of Venus or Mercury is not as fearsome as may be thought at first. The phases of Venus repeat every 1.5987 years. Put another way, once every 1.5987 years, Venus passes between the Earth and the Sun. Normally Venus will pass north or south of the Sun as seen from the Earth but occasionally, Venus will indeed pass directly in front of the Sun and a transit will be observed. The condition for this is that both Earth and Venus must be close to the nodes of their orbits i.e. the places where the two orbital planes cross.
The orbits of Earth and Venus are tilted with respect to Each other by 3.395 degrees. Because of the differences in the orbital planes, the perfect line-up required for a transit will only occur when the planets are close to the points where they pass through the orbital plane of the other. There will be two occasions per year when this happens; for Venus, these are periods of maximum 9 days in early June and early December.
As the synodic period of Venus, is close to 1.6 years, the whole situation tends to repeat after multiples of 1.6 years. Thus, 8 years after a transit of Venus, a further transit takes place. After a further 8 years, however, the alignment has been lost (due to the 1.5987 being not exactly 1.6) and it is necessary to wait a long time for the next transit. The pattern of transits of Venus is thus:
A December transit will be followed 8 years later by another December transit before a gap of 121.5 years before a June transit. Eight years later there will be another June transit and then a gap of 105.5 years before the next December transit. Thus in 243 years, there are 4 transits and the whole cycle then repeats itself. This period of 243 years represents 152 synodic periods of Venus.
The mean interval between transits of Venus can be worked out from the formula
T = 360 S sin (i / d)
where T is the average time between transits, S is the synodic period ( = 1.5987 years), i is the inclination of the orbit and d is the angular diameter of the Sun in degrees. This gives the mean time between transits as 64 years. The slight discrepancy between this figure and the four transits in 243 years is because sometimes the repetition on an eight-year basis will not occur.
Similar calculations can be made for Mercury giving an average time between transits of 26 years and transits repeating on periods of 7, 13 or 33 years. However, the eccentricity of Mercury's orbit introduces complications to the calculations.
From Mars, it would also be possible to see transits of Mercury and Venus; however, it would also be possible to see transits of Earth. These would occur, on average, every 71 years. [ = 38 Martian years ]. They are perhaps more common than may be supposed as the angle between the orbits of Earth and Mars is only 1.85 degrees (as opposed to 7 degrees for Mercury and 3.395 for Venus).
From Mars, transits of Earth do repeat on cycles. As 7 synodic periods (14.947 years) is close to 15 years, that is one candidate for a cycle. However, the alignment is not good enough. However, longer cycles could include 79 years (4 cycles), 204 years (3 cycles) and 284 years (around 16 cycles).
Transits of Earth from Mars would occur around May 12 or November 13 (Earth calendar). The May transits would occur with Mars not long past perihelion i.e. around 1.39 AU from the Sun (with Earth 1.01 AU from the Sun). In November Mars is 1.65 AU from the Sun with the Earth 0.99 AU from the Sun.
So, what would a transit of Earth look like as viewed from Mars? The great asset is that from Mars, the Moon lies close to the Earth in the sky and so there would be a good chance of seeing the two bodies cross the Sun's disk together. From Mars, in May, the Sun would appear to be 23 arc minutes ( or 0.38 degrees) across. By a strange coincidence, the length of the Earth's orbit would also be 23 arc minutes with a maximum width of 2 arc minutes (this figure would change from year to year as the Moon's orbit slewed round). Thus at maximum separation, the Moon could be a whole Sun's disk ahead or behind the Earth. Given that the transit may not be central, it is likely that the Moon could leave the Sun's disk before the Earth moved onto it (or vice versa).
In November, the Sun would appear to be 19 arc minutes across. The Earth's orbit would appear 13 arc minutes long by a maximum of around 1 minute across. Only when the Moon were at the extreme points in its orbit and when the transit was a non-central one would the Moon leave the Sun's disk before the Earth entered etc.
The Sun is the large circle with the Earth being the small circle. The Moon can appear anywhere on the ellipse. When the transit is central (lower case) the Moon cannot leave the Sun's disk before the Earth enters. However, should the transit be non-central, it is easy to find circumstances where the Moon can leave the Sun's disk before the Earth enters. In November, however, (right panel) it is harder to find circumstances where the Moon and leave the Sun's disk before the Earth enters. It would be necessary for a transit to occur near the Sun's pole with the Moon furthest away from the Earth.
Also worth mentioning is that May transits would outnumber November transits by a ratio of about 6:5 i.e. May transits would occur on average every 65 years while November transits would average 77 years). From Mars it would also be possible to see transits of Venus and Mercury. Those of Mercury would occur on average every 25 (Earth) years with those of Venus occurring on average every 32 years. One cycle occurring for transits of Mercury would be of 34 synodic periods which would equal 4.993 Mars years, 38.993 Mercury years i.e. 9.4 Earth Years. A cycle for Venus would be 72 Synodic Periods which equal 34.997 Mars years, 106.997 Venus years i.e. 65.8 Earth Years.
An interesting fact is that the nodes of the orbits of Mercury and Mars are very close together. A consequence is that it is theoretically possible for Mercury and Earth to transit the Sun at the same time as seen from Mars. The diagram shows the orbital planes of Mercury and Earth as seen from Mars just after Mars has crossed Earth's orbit and is about to cross Mercury's orbit. Please note that the scales are reasonable although not quite true in order to show an angle of less than 2 degrees.
The Sun will be moving 'horizontally' at a minute of arc every 46 minutes i.e. it moves its own diameter in 16 hours. Thus a rough rule for when this occurrence occurs would be that Earth and Mercury would reach inferior conjunction within about 16 hours of each other around May 12 or November 13. Another rule, considering events related to Earth would be that an opposition of Mars would occur within about 16 hours of a transit of Mercury.
It is interesting to note how this phenomenon would appear from the various planets involved. From Mars, it would be possible to see Mercury, Earth and (probably) the Moon all crossing the Sun's disk at the same time. From Earth, a transit of Mercury would occur with Mars at opposition on the ecliptic, while from Mercury, the planets Earth and Mars would appear very close together in the sky at opposition. If it were possible to observe from the Sun, the three planets would all appear close together in the sky (less than a degree apart).
There is one more vantage point of interest i.e. beyond Mars. From a suitable distance from Mars away from the Sun it would be possible to see Mercury, Earth, Moon and Mars all against the Sun's disk. Unfortunately none of the outer planets would be in the correct position so a space-borne observation would be necessary. Note that all the calculations above are using the current orbital elements for the planets. Over time-scales of tens of thousands of years, the elements will change slightly making the conclusions invalid in the long term.
by Colin Steele