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The New Cosmos

An Introduction to Astronomy and Astrophysics

Bearbeitet von Albrecht Unsöld, Bodo Baschek

Neuausgabe 2005. Buch. xiv, 562 S. Hardcover ISBN 978 3 540 67877 9 Format (B x L): 21 x 28 cm Gewicht: 1820 g

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Astronomy, the study of the and other celestial telescope, and of space travel, allowing observations to objects, is one of the exact sciences. It deals with the be made over the entire range of the electromagnetic quantitative investigation of the cosmos and the physical spectrum. laws which govern it: with the motions, the structures, In the 19th and particularly in the 20th centuries, the formation, and the evolution of the various celestial physics assumed the decisive role in the elucidation bodies. of astronomical phenomena; astrophysics has steadily Astronomy is among the oldest of the sciences. The increased in importance over “classical astronomy”. earliest human cultures made use of their knowledge of There is an extremely fruitful interaction between as- celestial phenomena and collected astronomical data in trophysics/astronomy and physics: on the one hand, order to establish a calendar, measure time, and as an aid astronomy can be considered to be the physics of the to navigation. This early astronomy was often closely cosmos, and there is hardly a discipline in physics which interwoven with magical, mythological, religious, and does not find application in modern astronomy; on the philosophical ideas. other hand, the cosmos with its often extreme states of The study of the cosmos in the modern sense, matter offers the opportunity to study physical processes however, dates back only to the ancient Greeks: the under conditions which are unattainable in the labora- determination of distances on the Earth and of positions tory. Along with physics, and of course mathematics, of the celestial bodies in the sky, together with know- applications of chemistry and the Earth and biological ledge of geometry, led to the first realistic estimates of sciences are also of importance in astronomy. the sizes and distances of the objects in outer space. The Among the sciences, astronomy is unique in that no complex orbits of the , the , and the experiments can be carried out on the distant celes- were described in a mathematical, kinematical picture, tial objects; astronomers must content themselves with which allowed the calculation of the positions of the observations. “Diagnosis from a distance”, and in par- planets in advance. Greek astronomy attained its zenith, ticular the quantitative analysis of radiation from the and experienced its swan song, in the impressive work cosmos over the widest possible spectral range, thus of Ptolemy, about 150 a.D. The name of the science, as- play a central role in astronomical research. tronomy, is quite appropriately derived from the Greek The rapid development of many branches of astron- word “αστηρ” = staror“αστρoν” = or omy has continued up to the present time. With this heavenly body. revised edition of The New Cosmos,wehavetriedto At the beginning of the modern period, in the 16th keep pace with the rapid expansion of astronomical and 17th centuries, the Copernican view of the universe knowledge while maintaining our goal of providing became generally accepted. Celestial mechanics re- a comprehensive – and comprehensible – introduc- ceived its foundation in Newton’s Theory of Gravitation tory survey of the whole field of astronomy. We have in the 17th century and was completed mathematically placed emphasis on observations of the manifold ob- in the period immediately following. Major progress in jects and phenomena in the cosmos, as well as on the astronomical research was made in this period, on the basic ideas which provide the foundation for the var- one hand through the introduction of new concepts and ious fields within the discipline. We have combined theoretical approaches, and on the other through ob- description of the observations as directly as possible servations of new celestial phenomena. The latter were with the theoretical approaches to their elucidation. Par- made possible by the development of new instruments. ticular results, as well as information from physics and The invention of the telescope at the beginning of the the other natural sciences which are required for the 17th century led to a nearly unimaginable increase in understanding of astronomical phenomena, are, how- the scope of astronomical knowledge. Later, new eras ever, often simply stated without detailed explanations. in astronomical research were opened up by the devel- The complete bibliography, together with a list of im- opment of photography, of the spectrograph, the radio portant reference works, journals, etc., is intended to 1. Introduction

2 help the reader to gain access to the more detailed and The treatment of the physics of individual stars occupies specialized literature. an important place in Part III. Along with the theory of We begin our study of the cosmos, its structure and radiation, atomic spectroscopy in particular forms the its laws, “at home” by considering our in basis for quantitative investigation of the radiation and Part I, along with classical astronomy. This part, like the the spectra of the Sun and other stars, and for the un- three following parts, starts with a historical summary derstanding of the physical-chemical structure of their which is intended to give the reader an overview of the outer layers, the stellar atmospheres. Understanding of subject. We first become acquainted with observations the mechanism of energy release by thermonuclear reac- of the heavens and with the motions of the Earth, the tions and by gravitation is of decisive importance for the Sun, and the Moon, and introduce celestial coordinates study of stellar interiors, their structures and evolution. and sidereal time. The apparent motions of the planets We then discuss the development of the stars of the and other objects are then explained in the framework of , which includes the phase of intensive the Newtonian Theory of Gravitation. Before consider- stellar hydrogen burning, continuing to their final stages ing the planets and other objects in the Solar System in (, neutron or black hole). The forma- detail, we give a summary of the development of space tion of stars and their earliest development are treated research, which has contributed enormously to know- in the following sections in connection with the inter- ledge of our . Part I ends with a discus- stellar material in our . At the end of Part III, we sion of the individual planets, their , and other deal with strong gravitational fields, which we describe smaller bodies such as asteroids, comets and meteors. in the framework of ’s General Relativity the- Prior to taking up the topic of the Sun and other ory; here, we concentrate in particular on black holes, stars, it is appropriate to describe the basic principles gravitational lenses, and gravitational waves. of astronomical observation methods, and we do this in In Part IV, we take up stellar systems and the macro- Part II. An impressive arsenal of telescopes and detec- scopic structure of the universe. Making use of our tors is available to today’s astronomer; with them, from knowledge of individual stars and their distances from the Earth or from space vehicles, he or she can inves- the Earth, we first develop a picture of stellar clusters tigate the radiation emitted by celestial bodies over the and stellar associations. We then discuss the interstellar entire range of the electromagnetic spectrum, from the matter which consists of tenuous gas and dust clouds, radio and microwave regions through the infrared, the and treat . Finally, we develop a picture visible, and the ultraviolet to the realm of highly ener- of our own Milky Way galaxy, to which the Sun be- getic radiations, the X-rays and gamma rays. The use longs together with about 100 million other stars. We of computers provides an essential tool for the modern treat the distribution and the motions of the stars and astronomer in these observations. star clusters and of the interstellar matter. After mak- Part III is devoted to stars, which we first treat as in- ing the acquaintance of methods for the determination dividual objects. We give an overview of the different of the enormous distances in intergalactic space, we types of stars such as those of the main sequence, gi- turn to other , among which we find a variety ants and supergiants, brown dwarfs, white dwarfs and of types: spiral and elliptical galaxies, infrared and star- neutron stars, as well as the great variety of variable burst galaxies, radio galaxies, and the distant quasars. In stars (Cepheids, magnetic stars, , , pul- the centers of many galaxies, we observe an “activity” sars, gamma sources ...) and of stellar activity, and involving the appearance of extremely large amounts of become acquainted with their distances, magnitudes, energy, whose origins are still a mystery. colors, temperatures, , and masses. In this Galaxies, as a rule, belong to larger systems, called part, the Sun plays a particularly important role: on the galactic clusters. These are in turn ordered in clusters of one hand, as the nearest star, it offers us the possibility of galactic clusters, the superclusters, which finally form making incomparably more detailed observations than a “lattice” enclosing large areas of empty intergalac- of any other star; on the other, its properties are those of tic space and defining the macroscopic structure of an “average” star, and their study thus yields important the Universe. Like individual stars, the galaxies and information about the physical state of stars in general. galactic clusters evolve with the passage of time. The 1. Introduction

3 mutual gravitational influence of the galaxies plays an Finally, after pressing out to the far reaches of the important role in their development. cosmos, we return at the end of Part IV to our Solar At the conclusion of Part IV, we consider the Uni- System and take up the problems of the formation and verse as a whole, its content of matter, radiation, and evolution of the Sun and the planets as well as the ex- energy, and its structure and evolution throughout the istence of planetary systems around other stars. In this expansion which has taken place over the roughly 2 · 109 section, we give particular attention to the development years from the “big bang” to the present time. of the Earth and of life on Earth.

5 Classical Astronomy I. and the Solar System I 6 Humanity and the Stars: Observing and Thinking Historical Introduction to Classical Astronomy

Unaffected by the evolution and the activities of with the aid of systematic measurements. The numerical mankind, the objects in the heavens have moved along results are less important for us today than the happy re- their paths for millenia. The starry skies have thus al- alization that the great Greek astronomers made the bold ways been a symbol of the “Other” – of Nature, of leap of applying the laws of geometry to the cosmos! deities – the antithesis of the “Self” with its world of Aristarchus of Samos, who lived in the first half of the inner experience, striving and activity. The history of 3rd century B.C., attempted to compare the distances of astronomy is at the same time one of the most excit- the Earth to the Sun and the Earth to the Moon with the ing chapters in the history of human thought. Again diameters of the three bodies by making the assumption and again, there has been an interplay between the ap- that when the Moon is in its first and third quarter, the tri- pearance of new concepts and ways of thinking on the angle Sun-Moon-Earth makes a right angle at the Moon. one hand and the discovery of new phenomena on the In addition to carrying out these first quantitative esti- other, the latter often with the aid of newly-developed mates of dimensions in space, Aristarchus was the first observational instruments. to teach the heliocentric system and to recognize its im- We cannot treat here the great achievements of the portant consequence that the distances to the fixed stars ancient Middle Eastern peoples, the Sumerians, Babylo- must be incomparably greater than that from the Earth to nians, Assyrians, and the Egyptians; nor do we have the the Sun. How far he was ahead of his time with these dis- space to describe the astronomy of the the Far Eastern coveries can be seen from the fact that by the following cultures in China, Japan, and India, which was highly generation, they had already been forgotten. Soon af- developed by the standards of the time. ter Aristarchus’ important achievements, The concept of the Universe and its investigation in carried out the first measurement of a degree of arc on the modern sense dates back to the ancient Greeks, who the Earth’s surface, between Alexandria and Syene: he were the first to dare to shake off the fetters of black compared the difference in between the two magic and mythology and, aided by their enormously places with their distance along a much-traveled car- flexible language, to adopt forms of thinking which al- avan route, and thereby determined the circumference lowed them, bit by bit, to “comprehend” the phenomena and diameter of the Earth fairly precisely. However, of the cosmos. the greatest observer of ancient times was Hipparchus How bold were the ideas of the pre-Socratic Greeks! (about 150 B.C.), whose stellar catalog was still nearly Thales of Milet, about 600 B.C., had already clearly unsurpassed in accuracy in the 16th century A.D. Even understood that the Earth is round, and that the Moon though the means at his disposal naturally did not al- is illuminated by the Sun, and he predicted the Solar low him to make significantly better determinations of eclipse of the year 585 B.C. But is it not just as important the basic dimensions of the Solar System, he was able that he attempted to reduce understanding of the entire to make the important discovery of precession, i.e. the universe to a single principle, that of “water”? yearly shift of the equinoxes and thus the difference The little that we know of Pythagoras (in the middle between the tropical and the sidereal years. of the 6th century B.C.) and of his school seems sur- The theory of planetary motion, which we shall treat prisingly modern. The spherical shapes of the Earth, the next, was necessarily limited in Greek astronomy to Sun, and the Moon, the Earth’s rotation, and the revo- a problem in geometry and kinematics. Gradual im- lution of at least the inner planets, Venus and Mercury, provements and extensions of observations on the one were already known to the Pythagorans. hand, and new mathematical approaches on the other, After the collapse of the Greek states, Alexandria formed the basis for the attempts of Philolaus, , became the center of ancient science; there, the quanti- Heracleides, Appollonius, and others to describe the ob- tative investigation of the heavens made rapid progress served motions of the planets; their attempts employed Humanity and the Stars: Observing and Thinking I 7 the superposition of ever more complicated circular mo- nicus. The result was a turning-away from the rigid tions. Ancient astronomy and planetary theory attained doctrine of the Aristotelians in favor of the much more its final development much later, in the work of Claudius lively and flexible thinking of the schools of Pythago- Ptolemy, who wrote his 13-volume Handbook of As- ras and Plato. The “Platonic” idea that the process of tronomy (Mathematics), Mαθηµατιχης Συνταξεως, understanding the Universe consists of a progressive in Alexandria about 150 B.C. His “Syntax” later ac- adaptation of our inner world of concepts and ways quired the adjective µεγιστη, “greatest”, from which of thinking to the more and more precisely-studied the arabic title Almagest is derived. The Almagest is outer world of phenomena has become the hallmark based to a large extent on the observations and research of modern research from through Kepler to of Hipparchus, but Ptolemy also added much new ma- Niels Bohr. Finally, with the blossoming of a practical terial, particularly in the theory of planetary motion. At approach to life exemplified by the rise of crafts and this point, we need only sketch the outlines of Ptole- trades, the question was no longer “What did Aristotle my’s geocentric system: the Earth rests at the midpoint say?”, but rather “How can you do this ... ?”. of the Universe. The motions of the Sun and the Moon In the 15th century, a completely new spirit in science in the sky may be represented fairly simply by circular and in life arose, at first in Italy and soon thereafter in orbits. The planetary motions are described by Ptolemy the North as well. The sententious meditations of Car- using the theory of epicycles: each moves on a dinal Nicholas Cusanus (1401–1464) have only today circle, the so-called epicycle, whose nonmaterial center begun to be properly appreciated. It is fascinating to moves around the Earth on a second circle, the defer- see how his ideas about the infinity of the Universe and ent. We shall not delve further into the refinements of about quantitative scientific research arose from reli- this system involving additional, in some cases eccen- gious or theological considerations. Near the end of the tric circular orbits, etc. The intellectual posture of the century (1492), the discovery of America by Christo- Almagest clearly shows the influence of Aristotelian pher Columbus added the classic expression “il mondo philosophy, or rather of Aristotelianism. Its modes of e poco” to the new spirit. A few years later, Nico- thought, originally the tools of vital research, had long las Copernicus (1473–1543) founded the heliocentric since hardened into the dogmas of a rigid school; this system. was the principal reason for the remarkable historical About 1510, Copernicus sent a letter to several noted durability of the Ptolemaic world-system. astronomers of his time; it was rediscovered only in We cannot go into detail here about how, following 1877, and was entitled “De Hypothesibus Motuum the decline of the academy in Alexandria, first the Nesto- Caelestium A Se Constitutis Commentariolus”. It fore- rian Christians in Syria and later the Arabs in Bhagdad shadowed the major part of the results which were later took over and continued the work of Ptolemy. published in his major work, “De Revolutionibus Or- Translations and commentaries on the Almagest bium Coelestium”, which appeared in Nuremberg in were the basic sources of the first Western textbook 1543, the year of his death. on astronomy, the Tractatus de Sphaera of Ioannes de Copernicus held fast to the idea of the “perfection of Sacrobosco, a native of England who taught at the Uni- circular motion” which had formed the basis for astro- versity of Paris until his death in the year 1256. The nomical thought throughout antiquity and the Middle Sphaera was issued again and again and often com- Ages; he never considered the possibility of another mentated; it was still “the” text for teaching astronomy form of motion. in Galileo’s time, three centuries later. It was Johannes Kepler (1571–1630) who, starting The intellectual basis of the new thinking was pro- from the phythagorian-platonic traditions, was able to vided in part by the conquest of Constantinople by the break through to a more general point of view. Making Turks in 1453: thereafter, numerous scientific works use of the observations of Tycho Brahe (1546–1601), from antiquity were made accessible to the West by which were vastly more precise than any that had pre- Byzantine scholars. For example, some very fragmen- ceded them, he discovered his three Laws of Planetary tary texts concerning the heliocentric system of the Motion. Kepler derived his first two laws from an enor- ancients clearly made a strong impression on Coper- mously tedious trigonometric calculation of the motions I Humanity and the Stars: Observing and Thinking 8 of Mars reported by Tycho in his “Astronomia ” An entirely new era of natural science be- (Prague, 1609). The third law is reported in his “Har- gan with Isaac Newton (1642–1727). His major monices Mundi” (1619). We can only briefly mention work, “Philosophiae Naturalis Principia Mathemat- Kepler’s ground-breaking works on optics, his Kep- ica” (1687), begins by placing theoretical mechanics lerian telescope, his Rudolphinian Tables (1627), and on a firm basis using the calculus of infinitesimals numerous other achievements. (“fluxions”), which he developed for the purpose. Its About the same time, the Italian Galileo Galilei connection with the Law of Gravitation explains Ke- (1564–1642) directed the telescope which he had built pler’s Laws and in one stroke provides the justification in 1609 to the heavens and discovered, in rapid suc- for the whole of terrestrial and celestial mechanics.In cession: the “maria”, the craters, and other mountain the area of optics, he invented the reflecting telescope formations on the Moon; the numerous stars of the Plei- and investigated the interference phenomena known des and the Hyads; the four largest moons of Jupiter and as “Newton’s Rings”. Almost casually, he developed their free orbits around the planet; the first indication of the basic approaches leading to numerous branches of the rings of Saturn; and sunspots. His “Galileis Sidereus theoretical physics. Nuncius” (1610), in which he describes the discoveries Only the “Princeps Mathematicorum”, Carl with his telescope, the “Dialogo Delli Due Massimi Sis- Friedrich Gauss (1777–1855), is of comparable im- temi Del Mondo, Tolemaico, e Copernico” (1632), and portance; to him, astronomy owes the theory of orbit the “Discorsi e Dimonstrazioni Matematiche Intorno a calculation, important contributions to celestial mech- Due Nuove Scienze” (1638), which was written after anics and advanced geodesics as well as the method of his condemnation by the Inquisition and contained the Least Squares. Never again has a mathematician shown beginnings of theoretical mechanics, are masterworks such a combination of intuition in the choice of new not only in the scientific sense but also as works of art. areas of research and of facility in solving particular The observations with the telescope, Tycho Brahe’s ob- problems. servation of the of 1572 and that of 1604 Again, this is not the place to pay tribute to the great by Kepler and Galileo, and finally the appearance of theoreticians of celestial mechanics, from L. to several comets required what was perhaps the most es- J.L. Lagrange and P.-S. Laplace to H. Poincaré; how- sential scientific insight of the time: that, in contrast ever, to finish this historical overview, we describe to the opinion of the Aristotelians, there is no funda- briefly the discovery of those planets which were not mental difference between cosmic and earthly matter known in ancient times. and that the same natural laws hold in the realms of The planet Uranus was discovered quite unexpect- astronomy and of terrestial physics (this had already edly in 1781 by W. Herschel. Kepler had already been recognized by the ancient Greeks in the case of supposed that there should be a celestial body in the the laws of geometry). This leap of thought, whose dif- gap between Mars and Jupiter (Fig. 2.15); the first plan- ficulty only becomes clear to us when we look back at etoid or asteroid, Ceres, was discovered in this region on Copernicus, gave impetus to the enormous upswing of 1.1.1801 by G. Piazzi, but in mid-February, it was “lost” scientific research at the beginning of the 17th century. when it passed near the Sun. By October of the same W. Gilbert’s investigations into electricity and mag- year, the 24-year-old C.F. Gauss had already calculated netism, Otto v. Guericke’s experiments with vacuum its orbit and ephemerides, so that F. Zach could find it pumps and electrification machines, and much more, again. Following this mathematical achievement, Gauss were stimulated by the revolution in the astronomical solved the general problem of determining the orbit of a worldview. planet or asteroid based on three complete observations. We have no space here to pay tribute to the many Today, several thousand asteroids are known, most of observers and theoreticians who developed the new them between Mars and Jupiter (Sect. 3.3). astronomy, among whom such important thinkers as From perturbations of the orbit of Uranus, J. Hevelius, C. Huygens, and E. Halley are particularly J.C. Adams and J.J. Leverrier concluded that there must prominent. be a planet with a still longer orbital period, and calcu- Humanity and the Stars: Observing and Thinking I 9 lated its orbit and ephemerides. J.G. Galle then found Lengthy search programs for a “planet X” beyond Neptune near the predicted position in 1846. the orbit of Pluto have remained unsuccessful; there are Perturbations of the orbits of Uranus and Neptune led no indications for the existence of a further large planet. to the postulate that there was a transneptunian planet. However, in 1992, D. Jewitt and J. Luu succeeded in The long search for it, in which P. Lowell (d. 1916) discovering a small object outside Pluto’s orbit, whose played a decisive role, was finally crowned with success: size is comparable with that of many of the asteroids. C. Tombaugh discovered Pluto in 1930 at the Lowell Soon thereafter, a number of “planets” were observed Observatory as a “faint star” of 15th . outside the orbits of Neptune and Pluto. I 10 2. Classical Astronomy

ollowing the historical overview of classical astron- summary of the motions of the other planets, the comets Fomy from ancient times up through the founding of etc.and of the determination of distances within the So- the heliocentric worldview and the discovery of the ba- lar System.After a brief treatment of the basic principles sic principles of celestial mechanics, we begin in Sect. 2.1 of mechanics and gravitational theory (Sect. 2.3), we give our treatment of astronomy with a description of the mo- some applications to celestial mechanics in Sect. 2.4.Fi- tions of the Sun, the Earth, and the Moon in terms of nally, in Sect. 2.5, we treat the orbits of artificial satellites the coordinates on the and of astronom- and space probes and summarize the most important ical determinations of time.In Sect. 2.2,wethengivea space research missions within our Solar System.

2.1 Spatial Coordinates and Time; order of decreasing brightness. Besides these Greek let- the Motions of the Sun, the Earth, ters, we also use the numbering system of the “Historia and the Moon Coelestis Britannica” (1725), compiled by the first As- tronomer Royal, J. . The Latin names of the are usually abbreviated to 3 letters (see As a beginning of our study of astronomy, in Sect. 2.1.1 Appendix A.2). we describe apparent motions on the celestial sphere and the coordinate system used to specify the positions of celestial objects. In Sect. 2.1.2, we treat the motions of the Earth, its rotation and its revolution around the Sun, 0 1 23 which are reflected as apparent motions on the celes- 2 22 R A tial sphere. Section 2.1.3 is devoted to the astronomical 3 [h ] 21 measurement of time. Following these preparatory top- Cassiopeia Perseus ics, we gradually become familiar with the objects in 4 Cygnus +40° our Solar System, beginning this process in Sect. 2.1.4 20 +8400 +50° +5175 5 with our Moon, its motions and its phases. We then treat +60° lunar and solar eclipses in Sect. 2.1.5. 19 +70° Auriga Lyra

6 ENP

14850 1950 18 Ursa

2.1.1 Minor The Celestial Sphere 7

and Astronomical Coordinate Systems 17 -1275

Lynx 8 Since antiquity, human imagination has combined the -4500

easily-recognized groups of stars into constellations 16 9

(Fig. 2.1). In the northern sky, the Great Bear (or the 15

Big Dipper) is readily seen. We can find the 10

14

11

13 (Polaris) by extending the line joining the two brightest 12 stars of the until it is about five times longer. Continuing about the same distance past Polaris (which Fig. 2.1. Circumpolar stars from a location having a geo- ◦ is the brightest star in the Little Dipper or Small Bear), graphic latitude of ϕ =+50 (about that of Frankfurt or we see the “W” of Cassiopeia. Using a sky globe or Prague). The coordinate lines indicate the right ascension RA and the (+40◦ to +90◦). Precession: the celes- a star map, we can readily find the other constellations. tial pole circles about the pole of the ecliptic ENP once In his “Uranometria Nova” (1603), J. Bayer named the every 25 700 years. The location of the celestial stars in each constellation α,β,γ ... , as a rule in the is indicated for several past and future dates 2.1 Spatial Coordinates and Time; the Motions of the Sun, the Earth, and the Moon I 11

Celestial Sphere. On the celestial sphere (in mathe- Nor matical terms, the infinitely distant sphere on which the th stars seem to be projected), we in addition define the zenith following quantities (Fig. 2.2): t C2 r circle hou of s 1. the horizon with the directions North, West, South, tar

e and East, d u it lt 2. vertically above our position the zenith, directly un- a r der us the nadir, la δ t o e p E m ti 3. the curve which passes through the zenith, the nadir, l a e er the celestial pole, and the north and south points is le d irc si N l c A meridian lle R S the ,and para 4. the curve which is perpendicular to the meridian and C1 -point the horizon, passing through the zenith and the east hori zon W and west points, is the principal vertical. tor ua eq In the coordinate system defined by these features, ical elest we denote the momentary position of a star by giving c m two angles (Fig. 2.2): (a) the is measured along e South rid celestial pole the horizon in the direction SWNE, starting sometimes ian from the S- and sometimes from the N-point; (b) the nadir ◦ altitude is 90 – the angle to the zenith. celestial pole to ϕ The celestial sphere apears to rotate once each day vertical around the celestial axis (which passes through the ce- N polar altitudeizon our hor lestial North and South Poles). The celestial ϕ is perpendicular to this axis. The position of a star latitude

axis equator

zenith S ze l ni Fig. 2.3. Celestial coordinates: right ascension RA and dec- pole a th c i d t is lination δ. The hour angle t = sidereal time minus the right r t e a v n = = c ascension RA.C1 lower , C2 upper culmina- e e

m tion. Lower right: the Earth (polar flattening exaggerated). i

r p Polar altitude = geographic latitude E star

altitude horizon N horizon S (Fig. 2.3) at a given time on the celestial sphere, imag- t h ined to be infinitely distant, is also described by the azimu W declination δ, which is positive from the equator to the North Pole and negative from the equator to the South

meridian Pole, and by the hour angle t, which is measured from the meridian in the direction of the , i.e. towards W. nadir In the course of a day, a star therefore traces out a circle on the sphere; its plane is parallel to the plane Fig. 2.2. The celestial sphere. The horizon with north, east, south, and west points. The (celestial) meridian passes through of the . On the meridian, the greatest the north point, the (celestial) pole, zenith, south point, and height reached by a star is its upper culmination,and the nadir. Coordinates: altitude and azimuth the least height is its lower culmination. I 2.Classical Astronomy 12

Sidereal Time. We also mark the Aries Point on the pole above the horizon) is given from Fig. 2.3 by the celestial equator; we shall deal with it in the following geographic latitude ϕ (the angle between the vertical section. It marks the point reached by the sun on the and the Earth’s equatorial plane); it can be readily mea- vernal equinox (March 21), on which the day and the sured as the average of the altitudes of the Pole Star or night are equally long. The hour angle of the Aries point a circumpolar star at the upper and lower . defines the sidereal time τ. The geographic longitude l corresponds to the hour angle. If the hour angle of the same object is measured ◦ Astronomical Coordinates. We are now in a position simultaneously at Greenwich (zero meridian, lG = 0 ) to determine the coordinates of a celestial object on the and, for example, in New York, the difference gives sphere independently of the time of day: we call the arc the geographic longitude of New York, lG. The deter- of the equator from the Aries point to the hour-circle of mination of the latitude requires only a simple angle a star the right ascension RA of that star. It is quoted in measurement, while that of a longitude necessitates hours, minutes, and seconds. 24 h (hora) correspond to a precise time measurement at two places. In earlier 360◦,or times, the “time markers” were taken from the motions 1h= 15◦, 1min= 15, 1s= 15, of the Moon or of one of the moons of Jupiter. The introduction of the “seaworthy” chronometer by John 1◦ = 4min, 1 = 4s. Harrison (ca. 1760–65) brought a great improvement, From Fig. 2.3, one can readily read off the relation: as did the later transmission of time signals by telegraph and still later by radio. Hour angle t = sidereal time τ (2.1) A few further facts: at a location having (northern) − right ascension RA . latitude ϕ, a star of declination δ reaches an altitude of ◦ The declination δ, our second stellar coordinate, has hmax = 90 −|ϕ − δ| at its upper culmination and hmin = already been defined. −90◦ +|ϕ + δ| at its lower culmination. Stars with δ> If we now wish to train a telescope on a particu- 90◦ − ϕ always remain above the horizon (circumpolar lar star, planet, etc., we look up its right ascension RA stars); those with δ<(90◦ − ϕ) never rise above the and declination δ in a star catalog, read the time from horizon. a sidereal clock, and adjust the setting circles of the in- strument to the angle hour t calculated from (2.1) and Refraction. In measuring stellar altitudes h, we must to the declination (+ north, − south). The especially take the refraction of light in the Earth’s atmosphere into precisely determined positions of the so-called funda- account. The apparent shift of a star (the apparent minus mental stars (especially for determinations of the time, the true altitude) is termed the refraction. For average see Sect. 2.1.3) are to be found, along with those of atmospheric tmperature and pressure, the refraction ∆h the Sun, the Moon, the planets, etc. in the astronomical of a star at altitude h is summarized in the following yearbooks or ephemerides; the most important of these table: is the Astronomical Almanac. h = 0◦ 5◦ 10◦ 20◦ 40◦ 60◦ 90◦ ∆h = 3450 945 516 237 109 33 0 . Astronomical Coordinates. The Copernican system at- tributes the apparent rotation of the celestial sphere to The refraction decreases slightly for increasing tem- the fact that the Earth rotates about its axis once ev- perature and for decreasing atmospheric pressure, for ery 24 h of sidereal time. The horizon is defined by example in a low-pressure zone or in the mountains. a plane tangent to the Earth at the location of the ob- server; more precisely, by an infinite water surface at the 2.1.2 The Motions of the Earth. observer’s altitude. The zenith or vertical is the direc- Seasons and the Zodiac tion of a plumb-bob perpendicular to this plane, i.e. the direction of the local acceleration of (including We now consider the orbital motion or revolution of the the centrifugal acceleration caused by the Earth’s ro- Earth around the Sun in the Copernican sense, and then tation). The polar altitude (the altitude of the celestial the daily rotation of the Earth about its own axis, as well 2.1 Spatial Coordinates and Time; the Motions of the Sun, the Earth, and the Moon I 13 as the motions of the axis itself. We first place ourselves lestial equator, the obliquity of the ecliptic. This means in the position of an observer in space. In Sect. 2.4, we that the Earth’s axis retains its direction in space relative shall derive Newton’s theory of the motions of the Earth to the fixed stars during its annual revolution around the and the planets starting from his principles of mechanics Sun; it forms an angle of 90◦ − 23◦27 = 66◦33 with and law of gravitation. the Earth’s orbital plane. A brief summary will suffice to explain the seasons Ecliptic and Seasons. The apparent annual motion of (Figs. 2.4, 5), starting with the . the Sun in the sky was attributed by Copernicus to the In the Northern Hemisphere, the Sun reaches its revolution of the Earth around the Sun on a (nearly) maximum altitude (midday altitude) at a geographical circular orbit. The plane of the Earth’s orbit intersects latitude ϕ on the 21st of June (the first day of Summer or the celestial sphere as a great circle called the ecliptic Summer solstice), h = 90◦ −|23◦27 − ϕ|. On the 22nd (Fig. 2.4). This makes an angle of 23◦27 with the ce- of December (Winter solstice), it has its lowest midday altitude, h = 90◦ − ϕ − 23◦27. It can reach the zenith ϕ =+ ◦  June 21 at up to 23 27 , the . ◦ ◦  ◦  Summer solstice ϕ ≥ − = celestial pole North of the Circle, 90 23 27 66 33 , 23°27’ the Sun remains below the horizon around the Winter pole of the solstice; near the Summer solstice, the “” ecliptic acts as a circumpolar star. 23°27’ In the , Summer corresponds

equator to Winter in the Northern Hemisphere, the Tropic of Sep. 23 ’ °27 Capricorn to the Tropic of Cancer, etc. Autumnal 23 equinox ecliptic Vernal The zodiac is the term for a band in the sky on each equinox March 21 side of the ecliptic. Since ancient times, it has been -point, divided into 12 equal “signs of the zodiac” (Fig. 2.5). It is often expedient for calculating the motions of the Earth and the planets to use a coordinate system oriented on the ecliptic and its poles. The (ecliptical) longitude is measured along the ecliptic starting from

Dec. 22 the Aries or point, like the right ascension in the di- Winter solstice rection of the annual motion of the Sun. The (ecliptical) Fig. 2.4. Annual (apparent) motion of the Sun among the stars. The Ecliptic. The seasons 21.III. 18.II. Start Name Coordinates of the The Sun enters 20.IV. date Sun the constellation 20.I. 21.V. Right Decli- ascension nation 22.VI. Gemini Sep. 23 22.XII. RA [h] δ Sum mer ◦ Dec. 22 Sun June 21

March 21 Vernal 00 Aries W inter Sagittarius Spring equinoxa March 21 ◦  Leo June 21 Summer 6 +23 27 Cancer 

Summer solstice 22.XI. 23.VII. ˙ ◦ Sept.23 Autumnal 12 0 Libra  23.X. 23.VIII. Autumn equinoxa 23.IX. ◦ 

Dec. 22 Winter 18 −23 27 Capricorn  Fig. 2.5. The orbit of the Earth around the Sun. The seasons. Winter solstice The zodiac and the signs of the zodiac. The Earth is at perihe- a On these days, the day and night arcs of the Sun are equal and each corre- lion (closest approach to the Sun) on the 2nd of January, and spond to 12 hours. at aphelion (furthest distance from the Sun) on the 2nd of July