The Transit of Venus

On June 5, 2012, a rare astronomical event will occur: a transit of Venus.  This just means that Venus will be between the Earth and Sun, so that Venus will appear as a small dot on the Sun’s surface.  Figure 1 is a photograph made during the last transit, which was June 8, 2004. No one who sees the 2012 transit is likely to see another; the next will occur December 11, 2117.

Figure 1

Figure 1

During the transit, the relative motion of Venus and the Earth will cause the silhouette of Venus to drift across the Sun’s face over a period of several hours. Venus transits happen in pairs, eight years apart, so that the 2012 event is the second of a pair.  Once we have a pair of these transits, it is either 105 or 121 years until the next one.  This odd timing is due to the vagaries of the orbits of the Earth and Venus.  If both orbits were exactly in the same plane, there would be a transit whenever Venus and Earth were in line with the Sun; roughly once every two years. As it happens, in almost every Earth-Venus-Sun alignment, Venus is in the sky either above or below the Sun’s disk.

Astronomers will observe the June event mostly as a curiosity, but the Venus transits that occurred in the eighteenth century were of intense interest, and the efforts to observe them constituted the biggest scientific enterprise of the century.   Britain, Austria, and France each launched expeditions to observe and time the transits from several locations throughout the world. In all, 250 observers were dispatched to over 100 locations, ranging from the Arctic Circle to the South Pacific.

The ultimate purpose of these transit studies was to collect data that would allow the size of the Solar System, including the Earth-Sun distance, to be calculated. This was one of the most important scientific questions of the age, because only the relative distance of each planet from the Sun was known. If just one distance could be determined, then the actual size of the whole Solar System would be known.

These efforts succeeded, and in this post, I will outline how these calculations were made.

Halley’s Great Idea

In 1716, the English astronomer Edmund Halley (1656-1742) described a method for using a Venus transit to measure the solar system. The next such transit was due in 1761. As with the comet whose return he had forecast for 1758, Halley knew that he would not be alive to see his idea tested.

To understand what Halley proposed, consider Figure 2.

Figure 2

Figure 2

An observer in the northern hemisphere at \(A\) would see the disk of Venus on the Sun at Point \(A’\).  Observer \(B\) in the southern hemisphere would see Venus at location B’ on the Sun. By later comparing these observations, the angular separation between \(A’\) and \(B’\) (angle \(\theta\)) can be determined. Because we know the size of the Earth and the observers’ locations, we can find distance AB, and AB is the base of isosceles triangle ABV. From the base length and the angle \(\theta\), we can compute the size of the triangle – that it, we can find the distance to Venus.

Halley already knew that the Sun-Venus distance was 0.72 times the Sun-Earth distance (in Note 1 below, I describe how these proportional distances were known).  With distance to Venus, call it \(v\), in hand, it is easy to compute the Earth-Sun distance using \(E = 0.72E + v\).

Unfortunately, it’s not that easy in practice. The two paths across the Sun that the observers see turn out to be very close together; so close that it is impossible to measure an accurate separation between them.

Halley suspected that this measurement problem would occur, and he thought of an ingenious way around it.  He proposed that, instead of measuring positions directly, that each observer would time the duration of the transit. To see how this helps, see Figure 3.

Figure 3

Figure 3

For both observer \(A\) and observer \(B\), Venus will appear to drift across the Sun’s disk at the same rate. However, for observer B, Venus will be on the disk longer because chord \(B’\) is longer than chord \(A’\). These times might differ by only a few minutes out of the several hour duration of the transit, but each transit time can be measured with an accuracy of a few seconds. This means that even if the chords are extremely close together, we can still measure the small difference in chord length.

To find the chord lengths, suppose that \(d_v\) is the rate of Venus’ drift across the Sun’s disk, measured in arc minutes per minute of time. The length of chord \(A’\) is then \(T_A \times d_v\), where \(T_A\) is the transit duration in minutes for Observer \(A\). Likewise, chord \(B’\) is \(T_B \times d_v\). Note that we are measuring length in terms of the angular separation, not actual distance.

Referring to Figure 4, we have drawn two right triangles, and the base of each one is half of one of the computed distances.  What we really want is \(\theta\), the angular separation between the paths, but that is simply the difference between the altitudes of the triangles. Using the Pythagorean Theorem for each triangle, we can write that distance as:

\[\theta=\sqrt{R^2-\left(\frac{T_A d_v}{2}\right)^2}-\sqrt{R^2-\left(\frac{T_B d_v}{2}\right)^2}\]

Figure 4. Calculating the Parallax at Venus

Figure 4. Calculating the Parallax at Venus

We can measure \(R\) quite accurately by  just measuring the apparent angular size of the Sun. We must also know Venus’ rate of drift across the Sun, \(d_v\),  to high precision. In Note 2 below, I show how that can be derived.

The Results

The transit expeditions were a success, although several observers traveled half way around the Earth, only to have clouds spoil their view of the transit. When the numbers were put through the above formulas, the computed Earth-Sun distance ranged from 146,000,000 km to 151,000,000, within 2.6% of the correct value of 149,597,900 km.

This was a major step in our efforts to measure the universe. Within a few decades, astronomers used parallax to find the distance to nearby stars. They did this by sighting on a star with a telescope, then repeating the sighting six months later, when the Earth was on the opposite side of the Sun, and the angle to the star was slightly different. Because the base of the huge triangle formed was just twice the Earth-Sun distance, knowing this distance was necessary to find the stellar distances.

Some Omitted Details

  • I have described the transit as if the disk of Venus on the Sun was a point, when in fact it is an extended disk about 1/30 the size of the Sun. This makes some difference in the geometry calculations, but it changes nothing fundamental.
  • The location of each observer on the Earth interacts with the Earth’s rotation, and changes the transit timing. To see why, suppose observer A was at the North Pole, and observer B was on the equator. During the several hours of the transit, A’s position relative to the Earth’s center will not change.  However, rotation  will move observer B several thousand kilometers to the east, and affect the view (and perceived end time) of the transit. In a similar way, differences in the longitude of the observers means that there is an east-west offset that will affect when they see Venus enter and leave the Sun’s disk. These effects can be calculated out with some tedious trigonometry.
  • To obtain the baseline AB, it is not sufficient to compute the straight-line distance between the two. The plane of the Earth’s equator is tilted 23 degrees relative to the plane of the orbit (Figure 4).  The distance that the calculations require is the distance perpendicular to the plane of the orbit, so we must multiply the straight line distance by cos(23 degrees).
  • Halley’s method, as described here, requires that each observer time the entire transit. The French astronomer Delisle later suggested an alternative: each observer would record the exact Greenwich Mean Time when the transit started (or ended). Differences in location would cause these times to differ by a few minutes. These time differences, and the observers’ position differences, allow trigonometry to be applied to compute the Venus distance. The Delisle method was used to process some of the observers’ results, especially where they saw only the beginning or end of the transit.
Figure 5

Figure 4

Note 1 – Relative Planet Distances

Copernicus used observations and some simple geometry to compute, at least roughly, the relative distance of the planets from the Sun. Really precise relative distances became possible when Kepler discovered his third law, which relates a planet’s orbital period to its distance from the Sun.  The law says that the period and distance for each planet in the Solar System is related by:

\[\frac{T^2}{R^3} = constant\]

Suppose we use as our unit of time one Earth year, and the unit of distance is the Sun-Earth distance. That is,

\[\frac{T^2}{R^3}=\frac{1^2}{1^3}=1\]

So, for any planet,

\[R=T^{2/3}\]

A Venus year can be determined precisely as 224 days or 224/365 = 0.614 Earth year. So

\[R_{venus} = T^{2/3} = .614^{2/3} = 0.72 AU\]

In like manner, Kepler determined the relative orbit sizes of all the known planets. Once the transit measurements gave us the size of one AU, the actual distances of all the planets was known. Then, knowing their distance and their apparent size in the sky, their actual diameter could be computed. They found, for example, that Jupiter was so large that it has a volume of one thousand times that of Earth.

Note 2 – Venus’ Rate of Drift Across the Sun

During the transit, both Venus and Earth are moving through their orbit, each at different speed, and at different distances from the Sun.  The result is that we see Venus move relative to the Sun’s disk at some rate, which we called \(d_v\) (measured in seconds of arc per second of time). Calculating \(d_v\) makes for an interesting problem.

As a first step, note that Venus moves around its orbit at a rate of \(360 / 224\) degrees per day, and Earth moves at \(360 / 365\) degrees per day. Figure 5a shows this, with the rates converted to a more convenient arc seconds per second of time.

Figure 5

Figure 5

In Figure 5b, we use the Sun-Earth line as a rotating reference. Relative to this line, Venus rotates around the Sun at 0.0670 – 0.0411 = 0.026″/sec. This rate is based on the Sun being the center of rotation, and the rate is not the same as viewed from the Earth. We need Venus’ rate relative to the Earth-Sun line as seen from the Earth.

Figure 5c shows how we find the Earth-based rate. In one second of time, Venus will travel, relative to the Sun-Earth line, a distance proportional to \(0.026 \times r_v\), where \(r_v\) is the Sun-Venus distance. As seen from Earth, this is an angular movement of

\[d_v= \frac{\text{arc length}}{\text{radius}}=\frac{0.026 r_v}{r_e - r_v} = \frac{0.026}{r_e/r_v - 1}= \frac{0.026}{1/.72 - 1} = 0.0669"/sec\]

Note that only relative distances are needed to compute the number, and relative distances were known in Halley’s time.

From this number, we can determine the maximum time that a transit could last. The Sun’s disk is 1920 arc seconds across, so a Venus transit through the center of the Sun would take \(1920/0.0669\) seconds or 7.9 hours.

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10 Responses to “The Transit of Venus”

  1. Nodas Says:

    Great article, thanks for compiling and posting. Can’t wait for 6/6/2012 !!

  2. Dota 2 Builds Says:

    You really make it appear so easy together with your presentation however I to find this topic to be actually something which I believe I would never understand. It sort of feels too complicated and very large for me. I’m taking a look forward in your next put up, Iâ??ll try to get the grasp of it!

  3. Dennis Ghyst Says:

    Outstanding article, Larry!

  4. John Schmidt Says:

    Larry,
    Thanks for this great explanation of the trigonometry related to the event and its results. . .I have been looking for this description on the web for a long time.

  5. Mike Gascoigne Says:

    Fantastic article to read after watching the transit earlier today.

    Looking forward to reading more of your posts.

    Many thanks.

  6. Axel Boldt Says:

    Hi, from reading Haley’s paper at http://eclipse.gsfc.nasa.gov/transit/HalleyParallax.html , it seems to me that his idea is slightly different. He isn’t interested in the transit durations viewed from different latitudes, but rather from different longitudes. He picks one point on Earth (Norway) from which the whole 8-hour transit can be viewed from start to finish, and another point (Hudson Bay) where Venus can be seen entering the sun at sunset and leaving the sun at the next sunrise. These two observed transit durations are then compared. The one measured in Norway is a lot shorter because of parallax (at the moment when Norway sees Venus entering the Sun, Hudson Bay already sees it inside the sun; when Norway sees Venus leaving the sun, because of the intermediate rotation of the earth, Hudson Bay still sees it inside the sun). The difference in durations can then be used to estimate the amount of parallax.

    Of course, if the two observation points differ in latitude, his calculations have to be adjusted like you do above, and he mentions that, but it’s not really the point; ideally he would use two observation points of identical latitude to eliminate this problem.

    I suspect that his method gives higher precision, since he exploits parallax with respect to almost twice the whole diameter of Earth, whereas your method only exploits parallax with respect to the difference in latitude of the two observation points.

  7. Abraham Bookstein Says:

    Attending the Venus transit at Chicago’s Adler, I first became aware of its historical significance. I really appreciate your lucid explanation. But I think I am overlooking something — can you help? Theta is the angle as seen from Venus. But aren’t R and the chords measured by angles as seen on Earth? Is there an implicit translation somewhere?

  8. Thierry Délèce Says:

    I like this kind of sensational sentences! : \”a rare astronomical event will occur\” :) :)

    Firstly: Assuming that there are about 100 billions of galaxies containing each 100 billions of stars with each about 100 planets (a million of billions of billions of planets!!!);

    Secondly: Assuming that the first proposal take into account only the mass we know, that to say 4% ;

    It\’s just funny to say \”a rare event\”!

    To be a little bit serious, It\’s surprising and, in fact, just unbelievable that a people like Halley determine the earth-sun distance with this kind of idea (which is simple when you know it after)!

    That\’s like Einstein who determine and prove that material are atoms and calculate size of them just observing pollen particles bouncing on the water…

    Or this man I don\’t remember the name that determinate that sun in not a simple combustion because even the best combustible is not sufficient, regarding the quantity of energy and the size of the sun…

    We look very simple people next people like that, very simple…

  9. Kevin Clark Says:

    As of i know it was French astronomer Jerome Lalande and he used data observed from the transit of Venus to determine the distance to the sun using information provided by Halley. Actually Halley came up with the formula, but knew he would need to witness it to properly come up with a figure. Knowing when the next transits would be, and knowing he wouldnt be alive for them, he left suggestions for future astronomers. Lalande used Halleys calculations along with the observance to determine the distance.

  10. Lauren bach Says:

    The earth and beautiful before it projected on Venus it should be protected.

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