The passage of Venus across the disk of the Sun and determining the distance between them

Fig. 1: Earth (blue), Venus (gray) and the Sun (orange), not to scale.

Many articles have already been written about the passage of Venus across the disk of the Sun of 2012. How rarely this event happens, and why exactly: according to the idea, Venus moving around the Sun more often than the Earth should pass between the Earth and the Sun during each of its revolution (Fig. 1), but because the orbits of the two planets are not aligned (not in the same plane, see Fig. 2), Venus often passes above or below the Sun from the point of view of the Earth.

But instead of repeating the words of others, I want to add a few details that are not so easy to find on the Internet.

You may have read that with the help of a technique based on the reasoning of the astronomer Edmund Halley (known by Halley’s comet) made by him from 1678 to 1716, as well as James Gregory before him, the passage of Venus of 1716 was used to determine the distance from Earth to The Sun (and to Venus, and all other planets) with an error of 2% is the highest of those achieved at that time. They hoped that the accuracy would be 10 times higher, but an unexpected optical effect called the “ black drop effect ” interfered in the process - there are still controversies about the exact reasons for its occurrence. But you could not read that this measurement - and many other measurements of distances in astronomy, up to fairly closely spaced stars - is based on the principle of parallax , on the same geometric fact that is used by our eyes and brain to perceive depth, or our ability to feel how far objects are from us just by looking at them.


Fig. 2: Earth (blue), Venus (gray) and the Sun (orange), not to scale. The orbit of Venus (black circle in the gray rectangle) is inclined relative to the Earth's orbit (blue circle in the blue rectangle). The degree of slope is greatly exaggerated. Since the Earth and Venus revolve around the Sun at different speeds, they can pass by each other at any point in the orbits.

Top: most of this passage Venus is above or below (green line) the line connecting the Earth and the Sun (red line), so the passage of Venus on the solar disk does not occur.

Below: In rare cases, the line connecting the Earth and the Sun coincides with the line of intersection of the planes of the orbits, and Venus is located near the same line, which leads to the passage.

Without parallax, it is also easy to determine the relative distance from Venus to the Sun - that is, the ratio of the radius of the orbit of Venus L _V to the radius of the Earth's orbit L _E. Therefore, in astronomy of the Renaissance, the relative distances from the planets to the Earth and the Sun were calculated quite early. But in order to determine L _V and L _E separately, it is necessary to measure the parallax, and the passage of Venus can provide it. The passage of Venus in the 1760s gave a fairly accurate measurement of the value of L _E - L _V , the "absolute" distance from Earth to Venus; this allowed us to find out L _E , L _V , and the distances to all other planets with an error of a couple of percent. Before that, at the end of the 17th century, a measurement was made of the distance from the Earth to Mars, which had an error of about 10%; it was also based on parallax, but this is a completely different story.

Preliminary note: the Earth and Venus, and even the Sun are very small compared to the distances between them, so it is almost impossible to draw accurate images. The illustrations all the time have to draw planets larger than they really are, in relation to the distances between them, just so you can understand the concept. Keep this in mind! All my illustrations are not to scale.

The relative size of the orbits of Venus and Earth

Fig. 3

To understand the main reason for the definition of L _V / L _E , suppose that the orbits of the Earth and Venus are circular and aligned - they lie in the same plane (as shown in Fig. 1, isometrically, and in Fig. 3 - the “top” view). In fact, the orbits of the Earth and Venus are slightly elongated and not aligned (Fig. 2). But the ellipticity and the mismatch of the planes are not very important for our reasoning, so first we can ignore them, and then recall them again in order to get more accurate answers.

Here we will apply the technology that is classical for physics: we will make an approximation sufficient for the current problem, and we will not go further than necessary. This is a very powerful way of thinking about science and about knowledge in general - it is enough to answer any question with a certain level of accuracy, so you can use the simplest technique among those that will give you the desired level of accuracy. This method has been used beautifully for centuries and is applicable not only to physics.

Therefore, we will take an approximation in which the orbits are circular and aligned, and we get approximately the correct answers, with an error of several percent. This will be enough to demonstrate the basic concepts, which I am trying to achieve. Believe me, you can make much more accurate calculations - or you can become an expert in this field yourself. But our approximation will not only give a very good answer, but also be able to show why it is so easy to calculate the ratio of L _V to L _E , but not the values ​​of L _V and L _E themselves.

During the year when the Earth and Venus rotate around the Sun at different speeds, the relative position of the Earth and Venus relative to the Sun changes. If on a certain day (day, month, year) I decide to draw a picture with the Sun in the center and the Earth on the left, as in Fig. 2, then Venus can be anywhere in its orbit. This means that from the point of view of the Earth, the angle between Venus and the Sun in the sky will vary depending on the date. This is shown in fig. 3, where the angle is called γ. The angle is easy to measure; find Venus in the sky after sunset or before sunrise and measure the angle between Venus and the Sun; see fig. four.


Fig. four

From fig. 3, it can be seen that γ has a maximum size — the angle between the orange and violet lines. Moving in orbit, Venus with each sunset will appear in a different place; for some time it will rise several times over the horizon, and then gradually begin to appear below. Observing Venus for several nights in a row and measuring γ, we can determine the maximum value of γ, which I will call γ _max .

From fig. 3, it is obvious that (as shown in Fig. 4), γ _{max is} less than 90 °, since the purple line must lie between the orange and red perpendicular. Geometrically, this is due to the fact that Venus is always closer to the Sun than the Earth. These angles explain why Venus is always visible either immediately after sunset or just before dawn (with the exception of those days when it is located behind the Sun). Venus cannot be at its zenith after dark, because for that it would have to be to the left of the red line.


Fig. five

Now we can determine the ratio of the radii of two orbits - L _V to L _E - using γ _max . This is the simplest geometry, see fig. 5. The point is that when Venus is at the maximum angle from the Sun, the line between the Sun and Venus is perpendicular to the line between the Earth and Venus, therefore the lines connecting these three objects form a right triangle. From here we get using standard trigonometry:

$$

$$L_V / L_E = \ gamma_ {max}$$

And from here, with the help of other simple geometric formulas, we get the relationship between the distances to other planets.

This is not entirely accurate, for the reasons given at the beginning; the orbits of the planets are ellipses, and do not lie on the water plane. In other words, L _V and L _{E are} not saved for a year, and γ _max is applied a little more difficult, in three dimensions, as in fig. 2, but not in two, as in fig. 1, 3 and 5. But with the help of accurate measurements of the position of Venus and the Sun in the sky, it is possible to determine the exact orbits of Venus and the Earth around the Sun and improve the calculations. The meaning is the same; All measurements of the position of Venus and the Sun in the sky only allow to measure the relative sizes of the orbits of Venus and the Earth. But the exact values ​​of L _V and L _E cannot be determined this way. A different approach is needed here.

The passage of Venus, parallax and distance to the Sun

The reason why the passage of Venus allows you to measure the absolute values ​​of the orbits of the Earth and Venus - this process can be observed with high accuracy from different places of the globe, with the result that you have two perspectives of the apparent location of Venus relative to the Sun, measured from different places with known distance between them. Parallax measurement allows you to determine the absolute value of the distance from Earth to Venus from the angle of parallax and the distance between two measurement points on Earth - just as a different type of object for the left and right eyes allows our brain to give us a sense of depth - a sense of distance to objects.


Fig. 6

To demonstrate, let me draw what it would look like from a large planet. In fig. 6 shows the planet from which we will observe the passage (this will be the Earth) and the planet passing in front of the star (this will be Venus). I will present a simplified situation (just to make geometry simpler and easier to see the basic concept) in which the planets and the star are aligned, therefore, from the observer’s point of view at the equator, the passing planet will pass along the star's equator. On top of fig. 6 shows a side view; notice the red line from the equator of the observing planet to the star through the equator of the planet passing through the star's disk.

In the case of a perfect alignment, the observer at the equator of the outer planet will see the inner planet pass along the equator of the star. This is shown as a red line at the bottom of fig. 6. But an observer from the south pole of the outer planet will see the inner planet pass the star along the path (purple line) north of the star's equator (in the case of the north pole everything will be the other way around). If we measure the angle α in the sky between the paths along which the passing planet moves and know the radius R of the observing planet, we can draw a right-angled triangle connecting the passing planet, the center of the observing planet and the pole of the observing planet with a small angle α. Simple trigonometry will give us the distance D between the planets during the passage, where

$$

$$D = R / tan \; \ alpha$$

Fig. 7

The same is true for the Earth, Venus and the Sun, except that the Earth and Venus are so small compared with the distance between them and the Sun that the angle α will be about 1/20 °! (This is a rather small size, but quite measurable, although to accurately measure the distance to the Sun, which astronomers of the 18th century wanted to get, would require a rather complicated and technically accurate measurement of the small angle). I will not draw such a small angle, so you will have to take my word for it that what is happening is the very version of what I have depicted in fig. 6, with planets and a star (the Sun) much smaller than they were drawn there, in relation to the distances. Even the image in fig. 7 makes the planet much larger than they are. But the idea remains the same: the distance D _EV between the Earth and Venus during the passage can be determined by measuring the parallax angle α (at the bottom of Fig. 7; note that the angular diameter of the Sun is about 1/2 °).

However, there are still a lot of questions:

• I told you how to measure D _EV , the distance from Earth to Venus during the passage. But wasn’t our goal not to measure L _E and L _V , the distance from Earth to the Sun and from Venus to the Sun?
• No one went to the South Pole of the Earth to observe the passage of Venus in 1761 or 1769.
• I assumed ideally aligned orbits of the Earth, Venus and the position of the Sun, such that from a point on the Earth's equator one could see Venus moving along the equator of the Sun. But this is actually not the case, and not even closely resembles a typical passage (and this was not the case in 2012 either).
• The angle α is small enough to be accurately measured - especially in times before photography and instant messaging, in the absence of clear indications of the location of the north pole of the Sun, making it difficult to accurately compare measurements of the path of Venus taken from two different points of the Earth. However, the primary goal was to measure the angle no worse than 1 part out of 500 (0.2%) (although due to the black drop effect, the result was closer to 1 part out of 50 (2%)).

How to cope with these problems?

First, how to go from measuring D _EV to measuring the desired values, L _E and L _V ? It's simple - we already know all the relationships, in particular, we already know L _E / L _V (approximately, from Fig. 4, or, to be more careful, we can calculate it more precisely) from the maximum angle γ _max between Venus and The sun from the point of view of the earth. We also know D _EV = L _E - L _V = L _E (1 - L _V / L _E ) from fig. 7. Therefore, we can obtain an approximate value of L _E with:

$$

$$L_E = D_ {EV} / (1 - L_V / L_E) = R / tan \; α / (1 - sin \; \ gamma_ {max})$$

where α is the parallax angle measured during the transit, and γ _max is the maximum angle between Venus and the Sun (Fig. 5). More accurate measurements require more complex geometry, but with the same basic idea.

Second, even if the orbits of the planets were perfectly aligned, the two dimensions of the path of Venus do not need to be measured from the equator and the poles of the Earth. They can be measured from any two latitudes. Geometry becomes a bit more complicated, but not much, but the principle remains (see Fig. 8).


Fig. eight

Third, even without perfect alignment, a small parallax angle will appear when measuring values ​​from two different points on Earth, and if this angle is well measured, this measurement can be turned (through slightly more complex equations) into dimension D. This is shown in Fig. 8, below.

The fourth question is the historically difficult problem of measuring the angular shift of the path of Venus during the passage through the angle α leads us to an alternative attempt to measure the time — either the transit time or just the beginning and end of the passage, rather than the angles. The first version was proposed by Halley on the basis of Gregory's ideas, and the second, as a further improvement, was proposed by Joseph Nicola Delille . The method of Halley did not require synchronization of clocks in different places of the Earth; The Delilia method was required, therefore, it was based on more advanced watch technology.

Even in the 17th or 18th century, it is much easier to make an accurate measurement of the interval, or the beginning and end of an eclipse, than to accurately measure the location of Venus relative to the solar disk, especially in the absence of a photograph. In fig. 9 you can see that the purple and red paths of Venus crossing the Sun are slightly different lengths because they do not cross it in one place, which means that the duration of the passage will differ by the time associated with the parallax angle. Unfortunately, everything turns out to be more complicated than it looks at first glance - the Earth rotates and moves around the Sun, so the observer travels quite a considerable distance during the passage of Venus across the solar disk. Therefore, a lot of effort is required (calculations are quite complicated, although with modern computers they are much simpler) to determine the difference in time intervals between the beginning and the end of the passage, observed by two different observers on the Earth, depending on the distance to the Sun.

At the beginning of the 18th century, Halley understood all the necessary geometrical principles (if you subtract outdated English phraseology and style from his texts, you will be surprised how modern his complex statements sound, and you will see that scientists were very similar to today's scientists three hundred years ago, possessed the same intelligence and they lacked only today's scientific technology).


Fig. 9

All this suggests that parallax — the difference in the visible position attributed to Venus relative to the Sun from the point of view of observers measuring it at the same time but from different places on Earth — was historically a very important method by which determined by the size of the solar system. More powerful methods are available to us today, but it may be interesting to you that what you see in the sky today is of the greatest historical importance, or you can simply enjoy the view of Venus moving majestically around our star.