by Colin Pask
6.5.1 Transits of Mercury and Venus
When Mercury or Venus passes between the earth and the sun, the result is not an eclipse of the sun but the passage of a small dark spot across its face. This is termed a transit. Transits of Mercury were observed (for example, by Edmond Halley in 1677, acting on the suggestion of James Gregory), but because Mercury is small and close to the sun, it turns out not to be so useful for precise astronomical measurements. That leaves Venus for consideration.
The orbit of Venus does not lie in the ecliptic plane (where we find the sun and the earth), but rather cuts the plane at the ascending and descending nodes. Examination of the orbits in figure 6.6 shows why the transit of Venus is quite a rare phenomenon. For a transit to occur, the earth, Venus, and the sun must line up: Venus must be at the ascending node Na and the earth at E1 to give the line E1NaS; or Venus must be at the descending node Nd with the earth at E2 to give the line E2NdS. The last eight transits of Venus occurred in 2012, 2004, 1882, 1874, 1769, 1761, 1639, and 1631. Transits occur in pairs separated by about eight years and pairs occur about every 120 years. Kepler famously predicted the 1631 transit, but it was not observed; using Kepler's work, Jeremiah Horrocks predicted and observed the next transit in 1639.
Figure 6.6. Orbits of Earth E and Venus V around the sun S. The orbit of Venus cuts the ecliptic plane at the ascending and descending nodes, Na and Nd. (Orbit shapes are greatly exaggerated.) Figure created by Annabelle Boag.
6.5.2 Halley Shows How to Proceed
Halley was an assistant to Astronomer Royal John Flamsteed for parallax measurements on Mars, and he concluded that the parallax method for finding the astronomical unit was basically useless. Subsequently, in 1678, he wrote:
There remains but one observation by which one can resolve the problem of the distance of the Sun from the Earth, and that advantage is reserved for the astronomers of the following century, to wit, when Venus will pass across the disc of the Sun, which will occur only in the year 1761 on May 26.10
Halley gave a full discussion of the proposed observations and calculations in his 1716 Philosophical Transactions paper (see bibliography). The essentials of Halley's method are given with reference to figure 6.7. The transit is to be observed by astronomers at points A and B on the earth where spots at M and N on the sun will be observed. Very large distances are involved, and the angles AVB and MVN in figure 6.7 (a) are equal, leading to
We can use the Venus orbit radius Rv for VN and the difference in Earth and Venus radii (Re – Rv) for VA. As mentioned above, Kepler's third law tells us that Rv = 0.72 Re. Using those ideas in equation (6.3) tells us that
We can find AB (by surveys of the earth), so MN will be known.
Figure 6.7. (a) Observers A and B on Earth see a shadow spot cast by Venus V on the sun at points M and N. (b) The spots trace out the lines M1M2 and N1N2 as Venus moves in front of the sun. Figure created by Annabelle Boag.
Next we find the diameter of the sun dsun from MN. Now we are almost there: the angular size of the sun θsun is known by observation, and dsun is simply θsun multiplied by the AU, the Earth-sun distance. Thus knowing θsun, and now having found dsun, at last we can find the astronomical unit by calculating dsun/θsun.
However, there is one problem in all of that: Exactly how do we “find the diameter of the sun dsun from MN”? The angle AVB in figure 6.7 (a) is very small, and it is extremely hard to measure things accurately and carry out the necessary calculations. The whole method appears to be doomed.
Halley made the brilliant suggestion that the two observers should measure not the angles, but rather the time it took for Venus's shadow spot to travel across the sun. Observer A would time the spot moving from N1 to N2, and observer B would time the spot moving from M1 to M2. See figure 6.7 (b). The angular drift rate for Venus's shadow spot to move across the sun is then known. From these times, the arc lengths can be found and then MN related to the full size of the sun, as required. (See the article by Phillips for the details.)
Halley had shown that the Earth-sun distance could be found by the relatively simple process of measuring two times. Furthermore, he knew from his earlier experience with Mercury that such times can be accurately recorded. As he wrote in his 1716 Philosophical Transactions paper:
I discovered the precise quantity of time the whole body of Mercury had then appeared within the Sun's disc, and that without an error of one single second of time.11
However, some care must be taken with the calculations because the movement of Earth in its orbit and its rotation must also be considered over the period of hours involved in the transit. For Venus's spot to cover a solar diameter, it takes almost eight hours. (See Phillips for an introduction to the calculations used when discussing a transit.)
6.5.3 Results
The French astronomer Joseph-Nicolas Delisle (1688–1768) met Halley in 1724 and became a champion for his methods as the latter aged (and died in 1742). In his 1716 paper, Halley discussed suitable sites for observers, and, in May 1760 (so just in time), Delisle published a world map showing regions where the transit could be observed with instructions for observers. As the transit date of June 1761 approached, various scientific organizations began sending expeditions to sites all over the world. (This is a fascinating piece of scientific history, and the entertaining books by Woolf and Lomb relate the full and multifaceted story.) In the end, 120 observations of the 1761 transit of Venus were made at locations spread worldwide by astronomers mostly from France and Britain, but Sweden, Germany, Denmark, Italy, Russia, and Portugal were also represented. (Woolf gives a table with all the details.)
The accuracy of the results varied with the expertise and location of the observers. It became apparent that there are also technical problems. For example, it is difficult to decide exactly when the shadow spot encounters the sun's disc, the so-called black-drop effect. Results at the time were expressed in terms of the solar parallax, and values ranged from 8.28" to 10.60". Since Halley had predicted an error “within the 40th part of one second” this was a disappointing outcome. (An angle is measured in degrees; a degree is divided into 60 minutes and each minute of angle is divided into 60 seconds, or 60 arcseconds.)
For the 1679 transit, Woolf tabulates 150 observers, again spread far and wide across the globe. The results were generally better this time, but examination of the observations and analysis by different astronomers produced inconsistent final values. For example, the eminent French astronomer Lalande claimed the answer was between 8.55" and 8.63". English observations suggested 8.8" and the Swedish Academy settled on 8.5".
Whatever the final outcome details, these calculations based on Halley's ideas finally showed that the earth is around 149 million kilometers from the sun. Setting the scale of the solar system was a major achievement; it is a fine example of how calculations can take simple observations to obtain a physically important parameter. I add calculation 19, finding the astronomical unit to my list. It is one of the sad points in physics where we realize that Edmond Halley used calculations to make two great predictions (the return of Halley's Comet and the determination of the solar parallax or Earth-sun distance using a transit of Venus), but did not live to see either of them wonderfully proved correct.
6.6 ROTATING ORBITS
Kepler's first law states that the orbit of a planet is an ellipse with the sun at one focus as shown in figure 6.8 (a). In his Principia, Newton showed that his law of gravity, equation (6.1), gives rise to orbits that are conic sections (thus including ellipses). He also looked at the inverse problem: Which forces give an elliptical orbit? He showed that only the inverse square law force, as in equation (6.1), or a force depending on distance linearly, would do the job. (Newton also refined the dynamics of systems comprising two or more bodies by building in center of gravity concepts. See Pask for details.)
Figure 6.8. (a) The planet P moves in an ellipse with the sun S at one focus. A1 and A2 are the orbit apsides. (b) When the force is not an inverse squa
re law force, the orbit does not close but slowly rotates, and there are no fixed apsides. The apsides are the points on the orbit that are nearest and most distant from the sun. Figure created by Annabelle Boag.
Newton also wished to find properties of the inverse square law force that would enable him to further support his claim that it represented the force of gravity. This led him to study the behavior of orbits when the force deviated from inverse square form, so instead of r2 in equation (6.1), we would have rp. The results are in his Principia (Section 9: The Motion of Bodies in Movable Orbits; and the Motion of the Apsides).
Newton considered orbits that are near circular, which is appropriate as the ellipticity of planetary orbits is generally small. In a mathematical tour de force, he showed that unless p is exactly equal to 2, the orbit rotates as shown in figure 6.8 (b). After one rotation of the line from the sun to the planet through 360º, the orbit only returns to its original point when p = 2; a line drawn through the apsides rotates, slowly, when p is close to 2. (See Pask chapter 14 for an introduction to this work.) This property of forces and closed orbits is now known as Bertrand's theorem.
θ in figure 6.8 (a) increases by π radians (or 180º) as the planet moves between the apsides, A1 to A2, and then moves through another angle π to get back to its starting position. The line of apsides A1A2 remains fixed in space, and this is a property of the orbits generated by an inverse square law force. If the angular change in θ varies from π, the apsides move and the line joining them rotates. See figure 6.8 (b). Newton calculated that
Obviously, for n = 1, the force is inverse square and the angle is 180º. Newton gave examples for other values of n. He then points out that the formula can be used in the inverse way; if the orbit rotates, that will determine n, and then we know that the force producing the orbit varies inversely as r3–n.
6.6.1 Solar System Data
The data for the planets in the solar system reveal that their orbits are closed to a high degree of accuracy, thus giving validity to the inverse square law of gravity. Newton pointed out that the perturbations caused by the other planets will explain any small deviations. In particular, as we saw in section 6.2, Jupiter and Saturn are large planets and will have small but significant effects on the motion of the other planets.
This is a typical example of the work in celestial mechanics over the centuries after Newton. Properties of various planetary and lunar effects were calculated using equation (6.1) as the law of gravity and then any deviations between theory and observations were accounted for. Occasionally there was a crisis, and someone (even the great Euler on one occasion) suggested the deviations could only be reconciled with the theory if the inverse square law failed to be exactly correct. Recall that in section 6.4 we saw that changes to the law of gravity were suggested as a way of accounting for the strangeness of Uranus's orbit before the perturbing planet Neptune was discovered. It was invariably found that difficulties in the complex calculations were the source of the errors, and there was no evidence for interactions between the sun and the planets or between the planets themselves to be any other than those described by equation (6.1).
6.6.2 A Stubborn Discrepancy
Newton's theory of gravity is one of the greatest scientific achievements, linking and explaining phenomena observed on the earth and in the heavens. Naturally, all observations were used to test the theory, and the triumphs were cause for celebration—as with the successfully predicted return of Halley's Comet. However, in the nineteenth century, one small discrepancy between theory and observation refused to go away.
Mercury is the smallest planet. It is closest to the sun, and its orbit deviates most from a circle. As data on the orbit of Mercury increased, it was found that an apsidal line through its orbit rotated through 565" over a century. An evaluation of the effects of the other planets showed the following contributions:
Venus – 280.6"
Earth – 83.6"
Mars – 2.6"
Jupiter – 152.6"
Saturn – 7.2"
Uranus – 0.1".12
Each of these figures is reached after a major calculation using perturbation theory (see Parks section 7.6 for an example in modern notation). The major contributions come from the planet nearest to Mercury (Venus) and the most massive planet (Jupiter). Those individual contributions total to 527", leaving the small but significant amount of 38" unaccounted for. It may seem that a rotation of thirty-eight seconds of angle taken over a century is not something to quibble about, but the excellence of the theory-observation verification suggested otherwise.
After his triumph with Neptune, Leverrier was naturally interested in this odd result. In 1849, he pointed out the above anomaly and began work checking the calculations. He tried variations in the masses of the planets, but no change consistent with other results and observations could account for the missing 38". It will come as no surprise that Leverrier then suggested that there was once again an undiscovered planet, this time perturbing the orbit of Mercury. In order to leave the good fit for Venus's orbit unchanged, he suggested that the orbit of the new planet must lie between the orbit of Mercury and the sun. Eventually this undiscovered planet was given the name Vulcan.
This time there was enthusiasm in many places for searches to reveal Vulcan, and several false starts were recorded. (This is a fascinating period in astronomical history, and the articles by Fernie, Morando, and especially Hanson are recommended.) There was no consistent solution to the Mercury rotation problem, and the suggested planet Vulcan remained undiscovered (as it does today!). American astronomers such as Simon Newcomb were involved in the story and checking the calculations. By the twentieth century the famous missing rotation was believed to be 43".
6.6.3 The Mystery Solved
The unexplained rotation of Mercury's orbit is an astronomical fact that does need a change in the law of gravity to be used in its calculation. In one approach, the American astronomer Asaph Hall (1829–1907) used Newton's theory to suggest that the 2 giving the power of the distance r in equation (6.1) should be changed to 2.000000157. That would give the extra 43" of orbit rotation, and the change would only be significant for Mercury as it is the planet nearest to the sun. As it turned out, the law of gravity really does need to be modified, and the effect only shows up strongly for a planet close to the massive sun. However, it is not the modification envisioned by Hall.
Albert Einstein examined the most basic assumptions and ideas on which physics is based, and he was led to his theory of relativity. In 1916, he published The Foundation of the General Theory of Relativity, which gives a different approach to the description of gravity. But still, Newton's theory emerges as the first approximation to Einstein's theory. For calculating planetary orbits that first approximation must be modified, and in the approach using forces, equation (6.1) becomes
There is an additional force term (the second term in equation (6.4)), which depends on the angular momentum constant h, the speed of light c, and the distance to the fourth power r4. Because c is so large, the magnitude of the correction term is very small.
We no longer have a purely inverse square law of gravity, and as a result, the orbits calculated with it are not closed. Here, then, is the explanation for the anomalous rotation of the orbit of Mercury. (You can even use Newton's theory to calculate the rotation effect—see Pask chapter 14.) Using the relativistically correct equation (6.4) instead of equation (6.1) gives just that missing 43" amount of rotation. According to Einstein's biographer Abraham Pais:
This discovery was, I believe, by far the strongest emotional experience in Einstein's scientific life, perhaps in all his life. Nature had spoken to him. He had to be right. “For a few days I was beside myself with joyous excitement” (Einstein in a letter to Ehrenfest). Later he told Fokker that his discovery had given him palpitations of the heart.13
These are the calculations of Newton, Einstein, Leverrier, and a host of diligent theoretical astronomers. The physics and mathe
matics are brilliant, and the conclusions they lead to are profound. Calculation 20, rotating orbits must have a place in the list of great calculations.
6.7 OTHER CONTENDERS
This part of science is full of wonderful stories and marvelous calculations, so keeping to a limited selection has not been easy. In particular, I regret not being able to include two brilliant pioneering efforts. First there is Carl Gauss's 1809 Theoria Motus, which describes the mathematical procedures to use for determining the full details of an orbit when just a few observations are known. Gauss used very limited data to find the orbit of the asteroid Ceres and predict where to find it after it was lost behind the sun. Ceres was the “missing planet” between Mars and Jupiter referred to in section 6.4.1. (See the article by Marsden for more information.) Second is the work on the extremely difficult problem of understanding the moon's orbit and drawing up tables of lunar positions. This has continued to be an important enterprise ever since Newton struggled with it, and wonderful advances were made by George W. Hill (1838–1914), one of America's great astronomers. (See Morando for further details.) It has been hard to leave Gauss and Hill off my final list.
in which we see how calculations help to answer some of the biggest questions of all.
The final step from chapters 4, 5, and 6 takes us to the ultimate question: What is the nature of the whole universe? Our earth is but a speck in the vastness of the universe, something which became ever more apparent after Galileo saw a new brightness of stars through his telescope. Cosmology is for many people the most inspiring, challenging, and mystifying topic in all of science, and every advance seems to leave us with new questions. Much of cosmology involves extremely technical matters, but in this chapter, I describe five significant calculations that illustrate how the subject progresses without, I hope, being too overpowering for the general reader. Other relevant calculations will be found in the chapters on light (see section 9.6) and nuclear physics (see section 11.7).
