by Colin Pask
Figure 5.7. The opening page from Kepler's workbook on Mars. Courtesy of Owen Gingerich.
Kepler's results were published in his 1609 Astronomia Nova, known today with its subtitle as New Astronomy Based upon Causes, or Celestial Physics, Treated by Means of Commentaries on the Motion of the Star Mars, from Observations of Tycho Brahe. After trying many mathematical forms and thinking about how a magnetic force exerted by the sun might move the planets, Kepler came to chapter 58 where he wrote:
I was almost driven to madness in considering and calculating this matter…. With reasoning derived from physical principles agreeing with experience, there is no figure left for the orbit of the planet except for a perfect ellipse.9
At last, the tyranny of the circle was vanquished. The uniform motion constraint was also replaced with something completely new: the equal areas in equal times law. Since Kepler believed there was a universal underlying mechanism, he could state that what he discovered for Mars would be true for all planets. So it is that we have Kepler's laws (see figure 5.8), in modern form:
Law 1: The orbit of each planet is in the shape of an ellipse with the Sun at one focus.
Law 2: In any equal time intervals, a line from the planet to the Sun will sweep out equal areas.
Figure 5.8. The orbit of a planet around the sun according to Kepler's laws. The orbit is an ellipse with the sun at one focus. If the planet moves from A to B in the same time that it takes to move from C to D, then the shaded swept-out areas are also equal. (The elliptical shape is greatly exaggerated, and the actual orbits are quite close to circular.) Figure created by Annabelle Boag.
Kepler continued to look for the overall harmony of the solar system, sometimes using musical analogies. In 1618, he published Harmonice Mundi, in which he reported that he had finally found the rule linking all planetary orbits in what we now call Kepler's third law:
Law 3: The ratio of the squares of times of revolution of any two planets around the Sun is proportional to the ratio of the cubes of their mean distances from the Sun. (Or as sometimes stated, the square of the period is proportional to the cube of the orbit size.)
Using his new theory, Kepler produced the Rudolphine Tables in 1627, and astronomers now had a new resource of unprecedented accuracy. (Incidentally, in producing those tables, Kepler was able to use the idea of logarithms introduced in section 3.2.) The equation giving orbit position for a specified time is now known as Kepler's equation, and it is notoriously difficult to solve (see Pask, chapter 12); Kepler tackled this problem and included in his tables results that could be used in an interpolation scheme.
5.2.4 The Achievement
It would be hard to overestimate the significance of Kepler's achievements; he changed the course of science with his ideas about physical reasoning, and his insistence upon fitting data with great accuracy set the standard for science ever after. Most people hear of Kepler's laws as a product of Newton's dynamics and theory of gravity; it is too easily forgotten that they were discovered on the basis of vast and truly heroic calculations. I add calculation 14, Kepler's astronomical calculations to my list of great calculations.
Kepler set out what has become our modern view of the solar system. The next steps required the discovery of the underlying mechanism and the proof that the theory matched centuries of observations. This takes us into the next chapter.
in which we see the roles played by six calculations in establishing our modern theory of the solar system; and meet some of the great scientists involved.
The previous chapter ended with Kepler completing the revolution begun by Copernicus and suggesting the next steps that needed to be taken. We now move into the era of Newton and Einstein.
6.1 THE REVOLUTION CONTINUES
Kepler had stressed that astronomy should become a branch of physics by investigating the underlying causes of orbital motion. He had no proper theory of dynamics and struggled to use William Gilbert's ideas about magnetic forces on a large scale. The next step was taken by Isaac Newton. In his 1687 Philosophiae Naturalis Principia Mathematica (generally referred to as the Principia), Newton developed his theory of dynamics and then showed that if it was applied to the solar system with the force of gravity, it would supply the physical basis that Kepler demanded. Thus Kepler's laws became a consequence of Newton's dynamical theory of the solar system. (See Pask for a detailed introduction.)
Newton discovered that his theory of the solar system was successful if the attractive force of gravity between two bodies with masses m and M is given by
where r is the distance between the bodies and G is the gravitational constant. However, the question remained: Was the force acting in the heavens (the force acting between the sun and the planets), the same as the force we know as gravity on Earth (the force that accelerates a falling body)? In the Principia, Newton set out his “Rules of Reasoning in Philosophy,” which include:
Rule I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
Rule II. Therefore to the same natural effects we must, as far as possible, assign the same causes.
In his discussion of Rule II, Newton writes: “As to respiration in man as in beast; the descent of stones in Europe and America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets.”1 In other words, while he is wrong about the sun being a fire, he is clearly saying that we should assume the same causes for phenomena whether they occur on Earth or in the heavens. This is a major assumption; in fact, it is the central assumption for the whole of astrophysics. Can Newton give evidence to back up his assumption?
6.1.1 The Moon Test
Newton devised the ingenious Moon Test as a way of comparing how bodies fall in different places. This would then allow him to suggest properties of the gravitational force in those places.
Figure 6.1 shows the moon in its orbit around the earth. Suppose that in one minute, the moon revolves through the angle θ. If there was no force operating, the inertia of the moon would see it move off along a straight line tangent to its orbit. However, this movement is supplemented by the fall through the distance D under the influence of the earth's gravitational attraction. Newton finds this fall, D, in one minute, to be 151/12 Paris feet. (The details are in section 6.1.2.)
Figure 6.1. A diagram showing the moon traveling through an angular distance θ and falling through a distance D as calculated in Newton's Moon Test. The earth has radius Re, and the radius R of the moon's orbit is approximately 60 Re. Figure created by Annabelle Boag.
(Incidentally, Kepler, and others, did not have that concept of inertia, which requires a force directed toward the earth to keep the moon in its orbit; Kepler wished to find a force that would push or sweep the moon around its orbit, which would imply a force tangent to the orbit rather than perpendicular to it.)
Newton next considers a body (perhaps an apple!) falling on the surface of the earth. Assume that the same gravitational force (see equation (6.1) operates on the earth and on the moon's orbit. Then, since the moon's orbit has a radius about sixty times the radius of the earth, the inverse dependence on distance squared in equation (6.1) tells us that the force of gravity is sixty squared times bigger on Earth than it is on the moon's orbit. Thus it follows that while the time taken to fall 151/12 Paris feet is one minute on the moon's orbit, it should be only one second for a body on Earth. (See below for details.)
Newton notes that measurements by Huygens reveal that in one second, a body on Earth does indeed fall 151/12 Paris feet. The force of gravity, equation (6.1), must be the same for the moon and for objects on Earth. Thus Newton triumphantly concludes:
therefore (by Rule I and II) the force by which the moon is retained in its orbit is the very same force which we commonly call gravity.
This is one on the major steps in the Principia, and from it, Newton moves on to declare that he has found the universal theory of gravitation, one of science'
s greatest discoveries. He also established that the physics we use here on Earth is the physics to use in the heavens. I add calculation 15, Newton's Moon Test to my list.
6.1.2 A Few Details
If you wish to check Newton's work, you must begin with his data:
Moon's orbit: radius R is approximately sixty times the Earth's radius Re; period is 27 days, 7 hours, 43 minutes = 39,343 minutes; angle θ revolved through in one minute is 2π/39,343 radians or (360/39,343)º.
Earth: circumference, 2πRe is 123,249,600 Paris feet so Re is 123,249,600/2π Paris feet.2
The triangle containing the angle θ (see figure 6.1) gives
So we get D = Rθ2/2 and using Newton's data this comes out as 151/12 Paris feet.
The fall of a body on Earth in time t is given by the standard formula ½gt2. Since gravity g is 602 times greater than it is on the moon's orbit, to give the same distance of fall we must make t2 602 times smaller, so t is to be 60 times smaller. Thus a fall taking one minute on the moon's orbit should be compared with a fall on Earth over the period of one sixtieth of a minute, or one second. Hence Newton concludes that if gravity is the same and follows equation (6.1), in both of these cases the fall distance should be 151/12 Paris feet. And it is!
6.2 WEIGHING THE PLANETS
I have omitted several calculations of great originality and value in the Principia, but my next one is just too unexpected and brilliant in its simplicity to ignore. It concerns the masses of the planets, including the earth, as I mentioned in section 4.2.
The force of gravity, as detailed in equation (6.1), depends on the masses of the two interacting bodies. It seemed clear that the sun is a large body and its mass msun is much larger than the masses of the planets orbiting around it. But how much larger? Would the mass of any planet be so large that it would have a significant effect on the orbits of others? Would planetary masses be so large that the center of gravity of the solar system might deviate significantly from the center of the sun? Newton came up with a beautiful way to find the masses of the earth and the large planets Jupiter and Saturn using the fact that they each have satellites, or moons, circling them.
Newton's derivation of Kepler's third law allows us to write it as an equation rather than as a ratio property. (Actually Newton worked mostly in terms of ratios, too, but to explain his approach here, it is simpler to use an equation.) If a body orbiting a large body of mass M has period T and orbit semi-major axis a, Kepler's third law gives
(If the mass of the orbiting body is not insignificant in comparison with M, a correction is needed, but that refinement is not of great importance here.)
Suppose that a planet orbits the sun with period Tp and semi-major axis ap. Further, suppose that a moon orbits that planet with period Tmn and semi-major axis amn. Kepler's third law, equation (6.2), applies to the sun-planet system (with M taken as msun) and to the planet-moon system (with M now taken as mp). The result is the two equations
Eliminating the unknown factor (4π2/G) between these two equations produces
This remarkable result tells us that if we observe the details of a planet's orbit around the sun, and the orbit details for one of its moons, then we can find the mass of the planet as a fraction of the sun's mass. There are moons around the earth, Jupiter, and Saturn, and substituting orbit data into equation (6.3) gives:
The table indicates that Jupiter and Saturn are large planets, and in refined calculations of solar system phenomena, it is necessary to take into account their gravitational effects on other bodies. This was a result of great importance, and we shall see an example in the next section. The earth is around a hundred times smaller again, and so, generally less important as a perturbation. The surprising error for the earth, as noted in the table, is due to Newton's poor value for the solar parallax and a copying error in the working. Nevertheless, Newton did establish the vitally important fact that the earth is relatively small compared with those larger planets. (For further discussion, see Pask, chapter 23, and references therein, especially the 1998 paper by Cohen.)
This is one of those calculations that makes me (and you, too, I hope) smile with delight and wonder just how anyone could have thought of such an elegant and powerful method. As Huygens put it at the time, Newton's feat gave us information about the planets that “hitherto has seemed quite beyond our knowledge.”3 Today, the masses of planets without moons can be found in different ways and from spacecraft fly-by data. I name calculation 16, Newton's determination of planetary masses.
6.3 A FIRST CONFIRMING TRIUMPH
The ancient view that beyond the earth there were perfect, unchanging heavens was gradually discredited. Kepler observed and wrote about the supernova that flared so brilliantly in 1604, and Galileo used his telescope to reveal that the moon has mountains and valleys just as the earth does. Over the centuries, there had been many sightings of comets, and they were said to predict the coming of dire events. In Newton's time, it was said that comets foretold the tragedies of the plague eruption in 1665 and the great fire of London the following year:
In the first place a blazing star or comet appeared for several months before the plague, as there did the year after another, a little before the fire. The old women and the phlegmatic hypochondriac part of the other sex…remarked…that those two comets passed directly over the city and that so very near the houses that it was plain they imported something peculiar to the city alone.4
Newton devoted over fifty pages of the Principia to the subject of comets. During a preliminary discussion he makes the following points:
The comets are higher than the moon, and in the regions of the planets.
The comets shine by the sun's light, which they reflect.
I am out in my judgment, if they are not a sort of planets revolving in orbits returning into themselves with a perpetual motion.
That the comets move in some sort of conic sections, having their foci in the center of the sun, and by radii drawn to the sun describe areas proportional to the times.
Hence, if comets are revolved in orbits returning into themselves, those orbits will be ellipses; and their periodic times be to the periodic times of the planets as the th power of their principal axes.5
In short, the suspicious nature of comets has been removed; they are bodies much like planets, and Newton's theory explains their motions. Newton also recognizes the computational problems involved. The orbits of the comets are enormous (see figure 6.2) and we only have access to a very small part of that orbit. He suggests that, since the conics are very similar in that small region (see figure 6.3), a parabola is most easily fitted, and the orbit can be corrected later. After reaching these conclusions, Newton presents a long discussion on observed comets and fitting orbits to the observational data, noting that “this [is] a problem of very great difficulty.”
Figure 6.2. The orbit of Halley's Comet projected onto the plane of the solar system. Note the enormous size of the orbit and that the comet has retrograde motion—it travels in its orbit in the opposite direction of the planets. Reprinted with permission, © Cambridge University Press, from D. W. E. Green, The Mystery of Comets (Cambridge: Cambridge University Press, 1986).
Figure 6.3. An illustration of the similarity of different conic-section orbits near the sun, which is taken as the focus. Reprinted with permission, © Cambridge University Press, from D. W. E. Green, The Mystery of Comets (Cambridge: Cambridge University Press, 1986).
6.3.1 Enter Edmond Halley
Edmond Halley (1656–1742) was involved in the Principia from the very beginning (see Pask chapter 3). Following a meeting with Newton, Halley began the business of extracting from him an account of his work. After much persuading and cajoling, the first edition of the Principia was completed in 1687, and it was Halley who paid for its publication. Only one person is acknowledged in Newton's Preface:
In the publication of this work the most acute and universally learned Mr. Edmond Halley not only assisted me in correcting the errors of t
he press and preparing the geometrical figures, but it was through his solicitations that it came to be published.
Halley is best known for mathematics and astronomy, but he was also involved in biology, geology, geography, physics, engineering, and deep-sea exploration using his invention of a diving bell. (We shall meet him again in sections 6.5.2 and 8.2. See Ronan for more on this remarkable man.) Newton reported their collaboration in the Principia, and the study of the comet that was observed in 1680 shows their theory in action. The fit by Halley to an elliptical orbit is tested in a table comparing calculations and observations. Newton drew this conclusion from the results:
The observations of this comet from the beginning to the end agree as perfectly as the motion of the comet in the orbit just now described as the motions of the planets do with the theories from whence they are calculated.
Comets had been removed from the realms of superstition and speculation and shown to be as readily described using Newton's dynamics and law of gravity as are the planets.
6.3.2 Halley's Comet
Halley became particularly interested in a comet that appeared in 1682. Certain features of its motion, such as the direction of its path around the sun (see figure 6.2), reminded him of other comets seen in 1531 and 1607. After studying greater details of the orbit, Halley became convinced that only one comet was involved and wrote to Newton that he was “more and more confirmed that we have seen that comet now three times since ye Yeare 1531.”6 Later he found that a comet observed in 1456 also fit the description. Thus it was that he claimed the comet had a period of around 75 or 76 years and estimated that it would appear again in 1758. Halley also suggested that a close passing of Jupiter caused the variability in the period of the comet, and he used 76 years to make his famous prediction.
