## The universal theory of gravitation

When it comes to our Solar System, pretty much the whole of modern-day predictive astronomy is based on the law of universal gravitation, introduced by Isaac Newton. Newton's work was the crowning achievement of a century of investigations into the subject of mechanics, which began with the work of Galileo. A detailed history of the development of dynamical ideas in the seventeenth century is beyond the scope of this book, but we will discuss briefly the main ideas and attitudes that shaped...

## Ds2 gV dx dxV125

Where g V (which depends on the coordinates xis a symmetric tensor of rank 2.62 Here we have replacedx, y, z, and ct,by x1, x2, x3, andx4, respectively, 60 For a thorough discussion of Einstein's interpretation of the principle of equivalence, see Norton (1985). See Whitrow and Morduch (1965), an article that tabulates the predictions of numerous relativistic theories of gravitation for redshift, light deflection, and perihelion advance. A similar comparison is given in Harvey (1965)....

## Info

A detailed analysis of the numerical accuracy of the polyhedral hypothesis, including corrections to Mastlin's calculations, is given in Brackenridge (1982). A second edition of the Secret of the Universe (with extensive annotations) was published in 1621, after Kepler had made the discoveries of his three laws of planetary motion. This suggests that Kepler still saw some value in his early work (see Field (1988)). 46 It is often written that at this time Kepler suggested that the relationship...

## Universal gravitation

A number of phenomena had caused Newton to consider that gravity was a property of all celestial bodies. In order to make the leap to universal gravitation, an understanding was required of how a large body like the Earth would attract an external object if its pull was the result of attractions from all its constituent parts. Newton answered these questions by proving some results concerning the attraction of thin spherical shells. His approach was hard-going the description below is a modern...

## Technical modifications to Ptolemaic astronomy

In Ptolemy's theory, the motion of the Sun plays a fundamental role in determining the motion of all the other heavenly bodies, and so all of the calculations in the Almagest are based on the parameter values Ptolemy used for his solar theory. Ptolemy failed to improve on the values of Hipparchus and so used 23 51' 20 for the obliquity of the ecliptic, 365 14, 48 days for the length of the tropical year, 65 30' for the longitude of the solar apogee, and 1 per century for the value of...

## R 4 r VR116

Where R is the disturbing function, considered small, and expanded in terms of small parameters. If R 0, we have the standard two-body problem with (setting G 1 for simplicity) 4, the sum of the masses of the bodies, and r, the position of one of the bodies with respect to the other. In the simplest case, the perturbation comes from one other body, the position of which is r' and the mass of which is m, for example, and then, from Eqn (9.3), In the problems of planetary motion, R is small...

## Ptolemy and the Almagest

Ptolemy (Latinized as Claudius Ptolemaeus) was one of the great scholars of antiquity, and mathematical astronomy was dominated by his ideas for nearly 1500 years following his death. Little is known of his life, but he taught in Alexandria and quoted the results of his observations made between AD 127 and 141. He was responsible for a number of great works, each of which place him among the most important ancient authors. The earliest of these is his masterpiece of mathematical astronomy, the...

## Babylonian astronomy

The heavenly phenomena were of great importance to the Babylonians, as they were perceived as omens and just about every possible astronomical event had some significance. For example, when it came to the retrograde motion of the planets it was not simply the retrograde motion itself, but also where it took place with respect to the stars, that was important When Mars comes out of the constellation Scorpius, turns and reenters Scorpius, its interpretation is this do not neglect your guard the...

## M22 y2 Vx 1 m22 y2

And the Figure 11.1 shows a selection of such curves when m2 0.1. The points A and B are the two primaries and the curves are obviously symmetric with respect to the line AB. The value of C is a minimum at the point labelled L4 and increases without bound as one approaches either A, B, or infinity. It follows, therefore, that if, for a particular system, C is sufficiently large and the mass m3 is close to either A or B, then it will remain in the vicinity of that primary. The points L1-L 4 are...

## Early Greek astronomy

Early cosmologies were based heavily on man's earthly experiences and owed little to accurate astronomical observation. They fulfilled a well-documented psychological need, providing a stage for the drama of daily life, for the actions of the gods and also supplying meaning to man's existence. Before the ancient Greek civilization, cosmologies were constructed that made no attempt to explain any but the most rudimentary of astronomical phenomena. For example, the Egyptians explained the motion...

## The revival of learning in Western Europe

While Islamic scholarship flourished, learning in Western Europe stagnated. The disintegration of the Roman Empire resulted in the almost total disappearance of the Greek language and, thus, most of the great Greek scientific works were inaccessible to European scholars. A few secondary works had been translated into Latin, including the first part of Plato's Timaeus - which Described in detail in Kennedy (1990). represented the main source of knowledge concerning Greek cosmology - and a few...

## Developments in trigonometry

The last great representative of the House of Wisdom in Baghdad was the tenth___13 century astronomer and mathematician Abu al-Wafa. He made significant 11 See van Helden (1985), p. 32. Copernicus later based his own determination on parameters adapted from al-Battani (see Swerdlow (1973)). In his L' histoire de l'astronomie du moyen age (1819), Delambre devotes fifty-three pages to a thorough analysis of al-Battani's zij, but the best modern work is held widely to be the Latin translation and...

## E

Copernicus' theory for the orbit of the Earth. The Earth E orbits uniformly about O, completing 1 revolution in 1 sidereal year, which Copernicus computes to be 365 days 6 h 9 min 40 s. The centre of the orbit of the Earth is no longer fixed but rotates around C with CO r2 say, completing 1 revolution in 3434 years, and C rotates around the Sun S with SC ri, completing 1 revolution in just over 53 000 years. The motion around the Sun is linked to the precession obliquity mechanism...

## D2 xk x dxv dxv

Dixr + ri ST 5T 0 (l27) is the Christoffel symbol of the second kind, also called the 'affine connection', or simply the 'connection coefficient'. Applying the same transformation to the line element ds - which must be invariant under a coordinate transformation, since it is a property of the underlying spacetime - we find that ds2 gvv dxv dxv, where the metric tensor takes the form _ 3 a gvv 3 3 nap' Within the framework of relativity, there are no forces acting on an object in free fall, and...

## Aristotle

Aristotle was a pupil of Plato, but the two men differed significantly in their approach to the understanding of the natural world. Plato was an idealist who focused on mathematics as the underlying reality, whereas Aristotle took a more pragmatic approach and emphasized physical aspects (e.g. cause and effect) he searched for reasons why things were as they were. Aristotle's modification of Eudoxus' system (he described his cosmology in the work On the Heavens, or De caelo in Latin) was not...

## The Maragha School

Theoretical developments in astronomy were not restricted to the astronomers of southern Spain. In the thirteenth century, another centre of theoretical activity grew up in the eastern part of the Islamic world, the first major figure being Nasir al-Din al-TusI(better known in the West as Nasir Eddin). Nasir al-Din was an advisor to the Mogul conqueror Hiilegii, a grandson of Genghis Khan and (despite his other excesses) a patron of the sciences, and was a man of enormous influence, both during...

## Hipparchus

Hipparchus, who lived in the second century BC,1 built an observatory and performed most of his work on the island of Rhodes and was perhaps the greatest astronomer of antiquity. He used observations to produce geometrical models with real quantitative predictive power. His theory of the motion of the Sun was extremely accurate and he produced a model for the Moon that worked well at new and full moons, thus enabling him to produce a theory of eclipses which, in the case of lunar eclipses, was...

## F 0o1 e cos 0o

This process can then be repeated until the desired accuracy is reached.6 For the case of planetary motion, where e is very small, Newton suggests a more straightforward alternative - one clearly having its origins in Boulliau's empty focus equant hypothesis. From Kepler's first and second laws, Newton deduced that the angle subtended by the planet at the empty focus, u, is related to the mean anomaly t via u t + 4e2 sin2t + f e3 sin31 + O(e4), and that, even for Mars, 'the error will hardly...

## Peurbach and Regiomontanus

The history of astronomy in the fifteenth century is dominated by two men, Georg Peurbach and Regiomontanus. The significance of their contributions comes, not so much from the technical content their astronomy continued the medieval tradition , but from the fact that with the introduction of printing into Europe, their books became the first astronomy textbooks to achieve what might be described as a mass circulation.51 After receiving his master's degree from the University of Vienna in 1453,...

## Eudoxus system of concentric spheres

The Babylonians were concerned with predicting the time at which a particular phenomenon e.g. a planetary opposition would occur, since it was the date that was ominous. Their astronomy was thus concerned with the analysis of discrete processes. On the other hand, as we shall see, the Greeks in their astronomy focused on predicting where a celestial body would be at a given time, and they were thus concerned with modelling a continuous process, which naturally leads to the use of geometrical...

## Eccentric circles and epicycles

Some 150 years after Eudoxus, Apollonius of Perga who spent most of his life in Alexandria and is now most famous for his work on conic sections devised a new solution to Plato's challenge of saving the phenomena using uniform circular motions. No astronomical works of Apollonius survive we know of his writing only through the later work of Ptolemy. It was well known that Eudoxus' scheme of homocentric spheres failed to account for certain readily observable phenomena e.g. the varying...