Fundamental astronomy 6th ed h karttunen, p kröger, h oja (springer, 2017)

Hannu Karttunen
Pekka Kröger
Heikki Oja
Markku Poutanen
Karl Johan Donner
Editors

Sixth Edition


Fundamental Astronomy


Hannu Karttunen r Pekka Kröger
Heikki Oja r Markku Poutanen r
Karl Johan Donner

r

Editors

Fundamental Astronomy
Sixth Edition
With 419 Illustrations
Including 34 Colour Plates
and 83 Exercises with Solutions


Editors
Hannu Karttunen
Tuorla Observatory

University of Turku
Piikkiö, Finland
Pekka Kröger
Helsinki, Finland
Heikki Oja
Observatory and Astrophysics
Laboratory
University of Helsinki
Helsinki, Finland

Markku Poutanen
Dept. Geodesy & Geodynamics
Finnish Geodetic Institute
Masala, Finland
Karl Johan Donner
Finnish Geodetic Institute
Helsinki, Finland

ISBN 978-3-662-53044-3
DOI 10.1007/978-3-662-53045-0

ISBN 978-3-662-53045-0 (eBook)

Library of Congress Control Number: 2016957787
Springer Heidelberg New York Dordrecht London
© Springer-Verlag Berlin Heidelberg 1987, 1994, 1996, 2003, 2007, 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole
or part of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,
and transmission or information storage and retrieval, electronic adaptation, computer software,

or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in
this book are believed to be true and accurate at the date of publication. Neither the publisher
nor the authors or the editors give a warranty, express or implied, with respect to the material
contained herein or for any errors or omissions that may have been made.
Cover illustration: Atacama Large Millimeter/submillimeter Array (ALMA) is an interferometer telescope composed of 66 antennas. ALMA observes molecular gas and dust of the cool
Universe—building blocks of stars, planetary systems, galaxies and life itself. Credit: ESO/
Y. Beletsky
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)


Preface to the Sixth Edition

As the title suggests, this book is about fundamental things that one might
expect to remain fairly the same. Yet astronomy has evolved enormously over
the last few years, and only a few chapters of this book have been left unmodified.
Since the book is used also by many amateurs, the introductory chapter
has been extended to give a brief summary of different celestial objects to
“soften” the jump to rather technical topics.
The chapter on the solar system was very long. It has now been split into
two separate chapters. Chapter 7 deals with general properties of the solar
system. Individual objects are discussed in Chap. 8, which is more prone
to change when new data will accumulate. Also, new data on exoplanets is
obtained at an increasing rate. Therefore exoplanets are given a chapter of
their own; it is at the end of the book, since it is closely related to astrobiology,
already included in the previous edition. These last chapters may change more

than the rest of the book in the future.
These changes mean that the numbering of formulas and figures has
changed quite extensively after the previous version of the book.
Cosmology and galactic astronomy have still been evolving rapidly. Therefore there are many revisions to the chapters on the Milky Way, galaxies, and
cosmology.
In addition, several other chapters contain smaller revisions and many of
the previous images have been replaced with newer ones.
Helsinki, Finland
April 2016

Hannu Karttunen
Pekka Kröger
Heikki Oja
Markku Poutanen
Karl Johan Donner

v


Preface to the First Edition

The main purpose of this book is to serve as a university textbook for a first
course in astronomy. However, we believe that the audience will also include
many serious amateurs, who often find the popular texts too trivial. The lack
of a good handbook for amateurs has become a problem lately, as more and
more people are buying personal computers and need exact, but comprehensible, mathematical formalism for their programs. The reader of this book is
assumed to have only a standard high-school knowledge of mathematics and
physics (as they are taught in Finland); everything more advanced is usually
derived step by step from simple basic principles. The mathematical background needed includes plane trigonometry, basic differential and integral
calculus, and (only in the chapter dealing with celestial mechanics) some vector calculus. Some mathematical concepts the reader may not be familiar with

are briefly explained in the appendices or can be understood by studying the
numerous exercises and examples. However, most of the book can be read
with very little knowledge of mathematics, and even if the reader skips the
mathematically more involved sections, (s)he should get a good overview of
the field of astronomy.
This book has evolved in the course of many years and through the work
of several authors and editors. The first version consisted of lecture notes by
one of the editors (Oja). These were later modified and augmented by the
other editors and authors. Hannu Karttunen wrote the chapters on spherical
astronomy and celestial mechanics; Vilppu Piirola added parts to the chapter
on observational instruments, and Göran Sandell wrote the part about radio
astronomy; chapters on magnitudes, radiation mechanisms and temperature
were rewritten by the editors; Markku Poutanen wrote the chapter on the solar system; Juhani Kyröläinen expanded the chapter on stellar spectra; Timo
Rahunen rewrote most of the chapters on stellar structure and evolution; Ilkka
Tuominen revised the chapter on the Sun; Kalevi Mattila wrote the chapter
on interstellar matter; Tapio Markkanen wrote the chapters on star clusters
and the Milky Way; Karl Johan Donner wrote the major part of the chapter on galaxies; Mauri Valtonen wrote parts of the galaxy chapter, and, in
collaboration with Pekka Teerikorpi, the chapter on cosmology. Finally, the
resulting, somewhat inhomogeneous, material was made consistent by the
editors.
The English text was written by the editors, who translated parts of the
original Finnish text, and rewrote other parts, updating the text and correcting

vii


viii

Preface to the First Edition

errors found in the original edition. The parts of text set in smaller print are
less important material that may still be of interest to the reader.
For the illustrations, we received help from Veikko Sinkkonen, Mirva
Vuori and several observatories and individuals mentioned in the figure captions. In the practical work, we were assisted by Arja Kyröläinen and Merja
Karsma. A part of the translation was read and corrected by Brian Skiff. We
want to express our warmest thanks to all of them.
Financial support was given by the Finnish Ministry of Education and Suomalaisen kirjallisuuden edistämisvarojen valtuuskunta (a foundation promoting Finnish literature), to whom we express our gratitude.
Helsinki, Finland
June 1987

Hannu Karttunen
Pekka Kröger
Heikki Oja
Markku Poutanen
Karl Johan Donner


Contents

1

Introduction . . . . . . . . . . . . . . .
1.1
Celestial Objects . . . . . . . . .
1.2
The Role of Astronomy . . . . .
1.3
Astronomical Objects of Research
1.4
The Scale of the Universe . . . .

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1
1
3
4
10

2

Spherical Astronomy . . . . . . .
2.1
Spherical Trigonometry . .

2.2
The Earth . . . . . . . . . .
2.3
The Celestial Sphere . . . .
2.4
The Horizontal System . . .
2.5
The Equatorial System . . .
2.6
Rising and Setting Times . .
2.7
The Ecliptic System . . . .
2.8
The Galactic Coordinates . .
2.9
Perturbations of Coordinates
2.10 Positional Astronomy . . .
2.11 Constellations . . . . . . . .
2.12 Star Catalogues and Maps .
2.13 Sidereal and Solar Time . .
2.14 Astronomical Time Systems
2.15 Calendars . . . . . . . . . .
2.16 Examples . . . . . . . . . .
2.17 Exercises . . . . . . . . . .

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11
11
14
16
16
17
20
21
22
22
27
31
31
34
38
40
45
49

3

Observations and Instruments . . . . . .
3.1
Observing Through the Atmosphere
3.2
Optical Telescopes . . . . . . . . .
3.3
Detectors and Instruments . . . . .

3.4
Radio Telescopes . . . . . . . . . .
3.5
Other Wavelength Regions . . . . .
3.6
Other Forms of Energy . . . . . . .
3.7
Examples . . . . . . . . . . . . . .
3.8
Exercises . . . . . . . . . . . . . .

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51
51
54
64
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85
89
90

4

Photometric Concepts and Magnitudes . . . . . . . . . . . . .
4.1
Intensity, Flux Density and Luminosity . . . . . . . . . .

91
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ix

x

Contents

4.2
4.3
4.4
4.5
4.6
4.7

Apparent Magnitudes . . . . . . .
Magnitude Systems . . . . . . . .
Absolute Magnitudes . . . . . . .
Extinction and Optical Thickness
Examples . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . .

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. 93
. 94
. 96
. 97
. 99
. 101

5

Radiation Mechanisms . . . . . . . . . . . . . . . . . . . . . .
5.1
Radiation of Atoms and Molecules . . . . . . . . . . . . .
5.2
The Hydrogen Atom . . . . . . . . . . . . . . . . . . . .
5.3
Line Profiles . . . . . . . . . . . . . . . . . . . . . . . .
5.4
Quantum Numbers, Selection Rules, Population Numbers

5.5
Molecular Spectra . . . . . . . . . . . . . . . . . . . . .
5.6
Continuous Spectra . . . . . . . . . . . . . . . . . . . . .
5.7
Blackbody Radiation . . . . . . . . . . . . . . . . . . . .
5.8
Temperatures . . . . . . . . . . . . . . . . . . . . . . . .
5.9
Other Radiation Mechanisms . . . . . . . . . . . . . . . .
5.10 Radiative Transfer . . . . . . . . . . . . . . . . . . . . .
5.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

103
103
105
107
108
111
111
111
114
116
117
118
120

6

Celestial Mechanics . . . . . . . . . . . . . . . . .
6.1
Equations of Motion . . . . . . . . . . . . .
6.2
Solution of the Equation of Motion . . . . .
6.3
Equation of the Orbit and Kepler’s First Law
6.4
Orbital Elements . . . . . . . . . . . . . . .
6.5
Kepler’s Second and Third Law . . . . . . .
6.6
Systems of Several Bodies . . . . . . . . . .
6.7
Orbit Determination . . . . . . . . . . . . .
6.8
Position in the Orbit . . . . . . . . . . . . .
6.9
Escape Velocity . . . . . . . . . . . . . . . .
6.10 Virial Theorem . . . . . . . . . . . . . . . .
6.11 The Jeans Limit . . . . . . . . . . . . . . . .
6.12 Examples . . . . . . . . . . . . . . . . . . .
6.13 Exercises . . . . . . . . . . . . . . . . . . .

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123
123
124
126
127
128
130
131
132
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136
140

7

The Solar System . . . . . . . . . . . . . . . . .
7.1
Classification of Objects . . . . . . . . . .
7.2

Planetary Configurations . . . . . . . . . .
7.3
Orbit of the Earth and Visibility of the Sun
7.4
The Orbit of the Moon . . . . . . . . . . .
7.5
Eclipses and Occultations . . . . . . . . .
7.6
The Structure and Surfaces of Planets . . .
7.7
Atmospheres and Magnetospheres . . . . .
7.8
Albedos . . . . . . . . . . . . . . . . . . .
7.9
Photometry, Polarimetry and Spectroscopy
7.10 Thermal Radiation of the Planets . . . . . .
7.11 Origin of the Solar System . . . . . . . . .
7.12 Nice Models . . . . . . . . . . . . . . . .

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141
141
143

145
147
149
151
154
160
162
166
167
174

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Contents

xi

7.13
7.14

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8

Objects of the Solar System . . . . . . . . .
8.1
Mercury . . . . . . . . . . . . . . . . .
8.2
Venus . . . . . . . . . . . . . . . . . .
8.3
The Earth and the Moon . . . . . . . .
8.4
Mars . . . . . . . . . . . . . . . . . .
8.5
Jupiter . . . . . . . . . . . . . . . . . .
8.6
Saturn . . . . . . . . . . . . . . . . . .
8.7
Uranus . . . . . . . . . . . . . . . . .
8.8
Neptune . . . . . . . . . . . . . . . . .
8.9
Dwarf Planets . . . . . . . . . . . . . .
8.10 Minor Bodies . . . . . . . . . . . . . .
8.11 Asteroids . . . . . . . . . . . . . . . .
8.12 Comets . . . . . . . . . . . . . . . . .

8.13 Meteoroids . . . . . . . . . . . . . . .
8.14 Interplanetary Dust and Other Particles
8.15 Examples . . . . . . . . . . . . . . . .
8.16 Exercises . . . . . . . . . . . . . . . .

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181
181
185
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196
199
205
208
211
212
214
214
219
222
224
224
225

9

Stellar Spectra . . . . . . . . . . . . . .
9.1
Measuring Spectra . . . . . . . . .
9.2
The Harvard Spectral Classification

9.3
The Yerkes Spectral Classification .
9.4
Peculiar Spectra . . . . . . . . . .
9.5
The Hertzsprung–Russell Diagram .
9.6
Model Atmospheres . . . . . . . .
9.7
What Do the Observations Tell Us?
9.8
Exercise . . . . . . . . . . . . . . .

10 Binary Stars and Stellar Masses
10.1 Visual Binaries . . . . . .
10.2 Astrometric Binary Stars .
10.3 Spectroscopic Binaries . .
10.4 Photometric Binary Stars .
10.5 Examples . . . . . . . . .
10.6 Exercises . . . . . . . . .

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227
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239

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241
241
242
243
244
246
247

11 Stellar Structure . . . . . . . . . . .
11.1 Internal Equilibrium Conditions
11.2 Physical State of the Gas . . . .
11.3 Stellar Energy Sources . . . . .
11.4 Stellar Models . . . . . . . . .

11.5 Examples . . . . . . . . . . . .
11.6 Exercises . . . . . . . . . . . .

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249
249
252
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257

260
262

12 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 263
12.1 Evolutionary Time Scales . . . . . . . . . . . . . . . . . 263
12.2 The Contraction of Stars Towards the Main Sequence . . . 264


xii

Contents

12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10

The Main Sequence Phase . . . . .
The Giant Phase . . . . . . . . . .
The Final Stages of Evolution . . .
The Evolution of Close Binary Stars
Comparison with Observations . . .
The Origin of the Elements . . . . .
Example . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . .

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267
269
271
272
275
277
280
281

13 The Sun . . . . . . . . . . . . . . .
13.1 Internal Structure . . . . . . .
13.2 The Atmosphere . . . . . . .

13.3 Solar Activity . . . . . . . . .
13.4 Solar Wind and Space Weather
13.5 Example . . . . . . . . . . .
13.6 Exercises . . . . . . . . . . .

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283
283
286
290
296
297
297

14 Variable Stars . . . . . .
14.1 Classification . . .
14.2 Pulsating Variables
14.3 Eruptive Variables
14.4 Supernovae . . . .
14.5 Examples . . . . .
14.6 Exercises . . . . .

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299
299
301
303
308
312
312

15 Compact Stars . . .
15.1 White Dwarfs .
15.2 Neutron Stars .
15.3 Black Holes . .
15.4 X-ray Binaries
15.5 Examples . . .
15.6 Exercises . . .

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313
313
315
320
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325
326

16 The Interstellar Medium . . . . . . . . . . . . . . . .
16.1 Interstellar Dust . . . . . . . . . . . . . . . . . .
16.2 Interstellar Gas . . . . . . . . . . . . . . . . . .
16.3 Interstellar Molecules . . . . . . . . . . . . . . .
16.4 The Formation of Protostars . . . . . . . . . . .
16.5 Planetary Nebulae . . . . . . . . . . . . . . . .
16.6 Supernova Remnants . . . . . . . . . . . . . . .
16.7 The Hot Corona of the Milky Way . . . . . . . .
16.8 Cosmic Rays and the Interstellar Magnetic Field

16.9 Examples . . . . . . . . . . . . . . . . . . . . .
16.10 Exercises . . . . . . . . . . . . . . . . . . . . .

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327
327
337
345
348
349
350
353
354
356
357

17 Star Clusters and Associations
17.1 Associations . . . . . . .
17.2 Open Star Clusters . . . .
17.3 Globular Star Clusters . .
17.4 Example . . . . . . . . .

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359
359
361
363
364

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Contents

xiii

17.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18 The Milky Way . . . . . . . . . . . . . . . . . . . .
18.1 Methods of Distance Measurement . . . . . .
18.2 Stellar Statistics . . . . . . . . . . . . . . . . .
18.3 The Rotation of the Milky Way . . . . . . . .
18.4 Structural Components of the Milky Way . . .
18.5 The Formation and Evolution of the Milky Way
18.6 Examples . . . . . . . . . . . . . . . . . . . .
18.7 Exercises . . . . . . . . . . . . . . . . . . . .

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367
367

371
375
380
383
385
386

19 Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1 The Classification of Galaxies . . . . . . . . . . .
19.2 Luminosities and Masses . . . . . . . . . . . . . .
19.3 Galactic Structures . . . . . . . . . . . . . . . . .
19.4 Dynamics of Galaxies . . . . . . . . . . . . . . .
19.5 Stellar Ages and Element Abundances in Galaxies
19.6 Systems of Galaxies . . . . . . . . . . . . . . . .
19.7 Active Galaxies and Quasars . . . . . . . . . . . .
19.8 The Origin and Evolution of Galaxies . . . . . . .
19.9 Exercises . . . . . . . . . . . . . . . . . . . . . .

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387
387
392
397
401
404
405
409
416
420

20 Cosmology . . . . . . . . . . . . . . . . . . .
20.1 Cosmological Observations . . . . . .
20.2 The Cosmological Principle . . . . . .
20.3 Homogeneous and Isotropic Universes .
20.4 The Friedmann Models . . . . . . . . .
20.5 Cosmological Tests . . . . . . . . . . .
20.6 History of the Universe . . . . . . . . .
20.7 The Formation of Structure . . . . . . .
20.8 The Future of the Universe . . . . . . .
20.9 Examples . . . . . . . . . . . . . . . .
20.10 Exercises . . . . . . . . . . . . . . . .

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421
421
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429

431
433
435
439
442
443

21 Astrobiology . . . . . . . . . . . . . . .
21.1 What Is Life? . . . . . . . . . . .
21.2 Chemistry of Life . . . . . . . . .
21.3 Prerequisites of Life . . . . . . .
21.4 Hazards . . . . . . . . . . . . . .
21.5 Origin of Life . . . . . . . . . . .
21.6 Are We Martians? . . . . . . . .
21.7 Life in the Solar System . . . . .
21.8 Detecting Life . . . . . . . . . .
21.9 SETI—Detecting Intelligent Life
21.10 Number of Civilisations . . . . .
21.11 Exercises . . . . . . . . . . . . .

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445
445
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455
456
457

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22 Exoplanets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
22.1 Other Planetary Systems . . . . . . . . . . . . . . . . . . 459
22.2 Observational Methods . . . . . . . . . . . . . . . . . . . 459


xiv

Contents

22.3 Properties of Exoplanets . . . . . . . . . . . . . . . . . . 461
22.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Appendix A Mathematics . . . . . . . . . .
A.1 Geometry . . . . . . . . . . . . . .
A.2 Conic Sections . . . . . . . . . . .
A.3 Taylor Series . . . . . . . . . . . .
A.4 Vector Calculus . . . . . . . . . . .
A.5 Matrices . . . . . . . . . . . . . . .
A.6 Multiple Integrals . . . . . . . . . .
A.7 Numerical Solution of an Equation .

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463
463
464
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467
469
470

Appendix B Theory of Relativity . . . . . . . . . .
B.1 Basic Concepts . . . . . . . . . . . . . . .
B.2 Lorentz Transformation. Minkowski Space
B.3 General Relativity . . . . . . . . . . . . .
B.4 Tests of General Relativity . . . . . . . . .

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473
473
474
475
476

Appendix C Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 477
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Photograph Credits . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Colour Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . 513
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533


1

Introduction

On a dark, cloudless night, at a distant location
far away from the city lights, the starry sky can
be seen in all its splendour (Fig. 1.1). It is easy to
understand how these thousands of lights in the
sky have affected people throughout the ages.
As long as human beings have existed, they
have certainly wondered the sky. In the sky,
ancient people saw figures related to religious
myths and omens sent by the gods. However,
already a couple of millennia ago the real astronomy started to evolve, separating itself from
religions and astrological superstitions. People
started to study the sky for its own sake.

1.1

Celestial Objects

In the 17th century people started to realise that
the Earth is not the centre of the Universe. About
the same time emerged the current view that stars
are celestial bodies similar to our Sun. The seem
to be faint dots only due to their huge distances.
We now know that the Sun and stars are hot glowing balls of gas, producing energy when fusion
reactions convert hydrogen to helium and also to
other heavier elements (Chap. 11).
Although stars actually move at enormous
speeds, the sky does not seem to change even
in thousands of years, due to the vast distances
of the stars. In addition to the Sun and the Moon
there are some other objects that move with respect to the stars. Since the antiquity, these moving objects have been called planets, from the

Greek word meaning a wanderer.

The rapid motions of the planets reveal that
they are much closer than the stars. Indeed they
are objects orbiting the Sun. According the current definition (Chap. 7) there are eight planets
orbiting the Sun: Mercury, Venus, Earth, Mars,
Jupiter, Saturn, Uranus and Neptune. In addition
to these relatively big bodies a lot of different
smaller objects move around the Sun: dwarf planets, asteroids, comets and meteoroids (Chap. 8).
Mosts planets also have their own satellites or
moons. Planets, moons and minor bodies do not
produce light by nuclear fusion; instead they
shine just by reflecting the sunlight.
At the centre of the solar system shines the
Sun, producing energy by fusion reactions
(Chap. 13). It is the nearest star, and studying its
properties reveals also a lot about other stars.
A few thousand stars can be seen by the naked
eye, but even a small telescope reveals millions
of them. Based on their properties, stars can be
divided into different categories. A great majority of them is main-sequence stars, like our Sun.
Some of them, though, are much bigger, giants
or supergiants, and some are much smaller, white
dwarfs. Different stars are usually related to different evolutionary stages in the lives of stars.
Many stars are variable stars, whose brightness
varies with time.
Rather recently found objects are compact
stars: neutron stars and black holes (Chap. 15).
Their material is squeezed into such a compressed form and their gravitational field is so
strong that Einstein’s general theory of relativity

must be used to describe their matter and space
around them.

© Springer-Verlag Berlin Heidelberg 2017
H. Karttunen et al. (eds.), Fundamental Astronomy, DOI 10.1007/978-3-662-53045-0_1

1


2

1

Introduction

Fig. 1.1 The starry sky in
all its splendour can be
seen only far away from
the light pollution of cities.
(Pekka Parviainen)

Fig. 1.2 The Pleiades is
one of the best-known open
star clusters. The six
brightest stars can easily be
seen with the naked eye.
Photographs reveal also
interstellar gas reflecting
the light of the stars.
(NASA, ESA,

AURA/Caltech, Palomar
Observatory)

The Sun is a solitary star. Many stars appear in
pairs, they are binary stars, orbiting around their
common centre of mass (Chap. 10). Also systems
of several stars are relatively common.
Bigger groups of stars are star clusters
(Chap. 17). Open clusters (Fig. 1.2) usually contain a few tens or hundreds of stars, that were born
in the same area, usually quite recently. Eventually the stars will diverge to their own paths.
Globular cluster (Fig. 1.3), on the other hand,
may contain hundreds of thousands or millions of
stars, which are usually very old.

The interstellar space corresponds pretty well
to our idea about a perfect vacuum. However, it
is not totally empty but contains interstellar matter, mainly hydrogen and helium, but also minute
amounts of heavier elements, molecules and dust
(Chap. 16). The interstellar medium does not
fill the space as a uniform mist, but forms huge
clouds (Fig. 1.4).
New stars are born by condensing from the interstellar matter. When the density, pressure and
temperature of the condensing cloud have risen
high enough, fusion reactions start and a new star


1.2

The Role of Astronomy

3

Fig. 1.3 The globular
cluster M13 in the
Hercules constellation
contains over a million
stars. The cluster can even
be seen with the naked eye
as a small nebulous spot.
(Palomar Observatory)

begins to radiate the energy released in the reactions (Chap. 12). After millions or billions of
years the energy resources will be exhausted. The
evolution then depends on the mass of the star.
The smallest stars just cool down and fade away,
but more massive ones either eject part of their
mass back to space as planetary nebula or explode as supernovas. Thus matter converted by
the nuclear reactions of the stars is mixed with
the interstellar matter.
All stars visible as separate objects to the
naked eye or with binoculars belong to the Milky
Way (Fig. 1.5, Chap. 18). The Milky Way is a
system containing a couple of billion stars, a
galaxy (Figs. 1.6 and 1.7, Chap. 19). It takes
about 100,000 years to travel across the Milky
Way with the speed of light.
Milky Way is not the only galaxy, but just
one of very many similar systems. Galaxies are
the basic building blocks of the Universe. They
do not spread out evenly but form small galaxy

groups, bigger galaxy clusters and even bigger
superclusters.
Galaxies are observed close to the edge of the
visible universe. Nuclei of some galaxies are seen
as quasars; the most distant of them have radiated
the light we detect now when the age of the Universe was only one tenth of the current value.

1.2

The Role of Astronomy

Already a long time ago man was interested in
celestial phenomena. Several bone carvings made
by the Cro-Magnon men as early as 30,000 years
ago have been found, possibly recording Lunar
phases. In that case these calendars would be
the oldest astronomical documents, predating the
skill of writing by some 25,000 years.
Agriculture required a good knowledge of
the seasons. Religious rituals and prognostication were based on the locations of the celestial
bodies. Thus time reckoning became more and
more accurate, and people learned to calculate
the movements of celestial bodies in advance.
During the rapid development of seafaring,
when voyages extended farther and farther from
home ports, position determination presented
a problem for which astronomy offered a practical solution. Solving these problems of navigation were the most important tasks of astronomy in the 17th and 18th centuries, when the first
precise tables on the movements of the planets
and on other celestial phenomena were published.
The basis for these developments was the discovery of the laws governing the motions of the planets by Copernicus, Tycho Brahe, Kepler, Galilei

and Newton.


4

1

Introduction

Fig. 1.4 The North America nebula in the constellation
of Cygnus is a large cloud of interstellar gas. The nebula
appears brighter than the background because the radiation from the nearby stars makes it shine. The nebula is,

however, very faint and difficult to observe visually. The
brightest star on the right is α Cygni or Deneb. (Photo
M. Poutanen and H. Virtanen)

Astronomical research has changed man’s
view of the world from geocentric, anthropocentric conceptions to the modern view of a vast universe where man and the Earth play an insignificant role. Astronomy has taught us the real scale
of the nature surrounding us.
Modern astronomy is fundamental science,
motivated mainly by man’s curiosity, his wish to
know more about Nature and the Universe. Astronomy has a central role in forming a scientific view of the world. “A scientific view of the
world” means a model of the universe based on
observations, thoroughly tested theories and logical reasoning. Observations are always the ultimate test of a model: if the model does not fit
the observations, it has to be changed, and this

process must not be limited by any philosophical,
political or religious conceptions or beliefs.

1.3

Astronomical Objects of
Research

Modern astronomy explores the whole Universe
and its different forms of matter and energy. Astronomy can be divided into different branches in
several ways, based e.g. on the object of research
or the method used.
The Earth (Fig. 1.10) is of interest to astronomy for many reasons. Nearly all observations
must be made through the atmosphere (Fig. 1.9),
and the phenomena of the upper atmosphere and
magnetosphere reflect the state of interplanetary


1.3

Astronomical Objects of Research

5

Fig. 1.5 The Milky Way
appears as a nebulous band
stretching across the sky.
A telescope reveals that it
consists of myriads of stars
as observed already by
Galileo Galilei 400 years
ago. The Milky Way is a
flat disclike stellar system.

Our solar system is close to
the plane of the disc, and
looking in the direction of
the plane we see a lot of
stars. But if we look away
from the disc the stellar
density is much lower. The
disc contains also unevenly
distributed interstellar gas
and dust, which obscures
the view in some
directions. Near the lower
edge of the picture the
Milky Way seems to split
into two branches because
the distant stars are behind
the intervening obscuring
matter. (Pekka Parviainen)

space. The Earth is also the most important object of comparison for planetologists.
The Moon is still studied by astronomical
methods, although spacecraft and astronauts have
visited its surface and brought samples back to
the Earth. To amateur astronomers, the Moon
is an easy and interesting object for observations.
Space probes have already studied all planets, many of their satellites, some asteroids and

comets. The most distant planets, Uranus and
Neptune, have been observed only by fly-bys, but
all the other ones also by orbiters. Spacecraft have

softlanded on Mars, Venus, Saturn’s moon Titan and some minor bodies. Exploration by such
probes has tremendously added to our knowledge
about the conditions of these objects. Continuous
monitoring of the planets, however, can still only
be made from the Earth, and many small bodies
of the solar system still await their spacecraft.


6
Fig. 1.6 The galaxy M31
in the Andromeda
constellation is a star
system resembling our own
Milky Way. Stars and
interstellar matter
concentrate in spiral arms.
The shape of M31 is a
round, flat disc, but due to
the oblique view it looks
oval. In good conditions
the centre of the galaxy can
be seen even with the
naked eye as a faint
nebulous spot. M31 has
two small elliptic
neighbour galaxies seen
here as bright ellipses. M32
is below the centre of M31
and M110 towards
northeast from the centre.

North is up. The dots are
stars of the Milky Way.
(Bill Schoening, Vanessa
Harvey/REU program/NOAO/AURA/NSF)

Fig. 1.7 In addition to the
big galaxies like the Milky
Way there are numerous
much smaller dwarf
galaxies, which are often
irregular in shape. One of
them is the Large
Magellanic Cloud, our
nearest neighbour galaxy.
It is easily seen with the
naked eye, but it is close to
the southern pole of the
sky. (NOAO/Cerro Tololo
Inter-American
Observatory)

1

Introduction


1.3

Astronomical Objects of Research

7

Fig. 1.8 The deep-field
picture of the Hubble space
telescope is a combination
of several images exposed
altogether over 11 days.
The picture shows several
galaxies that are the most
distant ones known. When
we are looking far to space
we are also looking far to
the past since the light
proceeds at a finite speed.
Thus many of the galaxies
in the picture are also
among the oldest known
objects. When we compare
them with the objects in
our neighbourhood we can
deduce how the galaxies
evolved during billions of
years. (NASA)

Fig. 1.9 Although space
probes and satellites have
gathered remarkable new
information, a great
majority of astronomical
observations is still

Earth-based. The most
important observatories are
usually located at high
altitudes far from densely
populated areas. One such
observatory is on Mt
Paranal in Chile, which
houses the European VLT
telescopes. (Photo ESO)

An astronomer can also specialise in studying several different fields like the Sun, different kinds of stars, star clusters, the Milky Way
or galaxies (Fig. 1.11).

The largest object of research is the whole
Universe. Earlier this field, cosmology, belonged
to theologists and philosophers, but in the 20th
century it became an object of physical theories


8

and eventually of concrete astronomical observations.
Spherical astronomy is an old field of astronomy studying the coordinate systems of the celestial sphere and apparent positions and motions
of the celestial objects. Until the 17th century astronomy was mainly spherical astronomy.
When Isaac Newton published the fundamental laws of mechanics in 1687 in his Principia
mathematica, the motions of celestial objects got
a physical explanation. That was the beginning
of celestial mechanics, studying the motions from
the planets of the Solar System and satellites orbiting the Earth to distant galaxies and galaxy
clusters.

1

Introduction

Halfway the 19th century it was found out how
spectra can reveal physical properties of celestial
objects. This was the beginning of astrophysics,
studying the physical phenomena of the stars. Results from astrophysics are utilised particularly in
the research of the Sun, stars and interstellar matter.
Astronomy can be divided into different branches also by the wavelengths used. We can talk
about radio, infrared, optical, ultraviolet, X-ray or
gamma-ray astronomy, depending on the wavelength used in the observations.
Astronomers study also particles coming from
the space, like neutrinos and cosmic rays. Grav-

Fig. 1.10 The Earth as
seen from the Moon.
Thanks to spaceflights we
have seen clearly the
planetary status of the
Earth. The picture was
taken by the Japanese
Kaguya lunar orbiter in
2007. Currently the Moon
is the only celestial object
outside the Earth visited by
human beings, on the
Apollo flights in
1969–1972. (JAXA)

Fig. 1.11 Astronomy in the change. Although the numbers of astronomical articles have increased in all subfields in the last few decades, the relative proportions have
changed. Cosmology and galaxies are the greatest winners
whereas the share of stellar research has decreased. The

graph illustrates the relative numbers of articles in different fields of astronomy in the most influential journals in
1981–2009. (Adapted from the New Worlds, New Horizons in Astronomy and Astrophysics, 2010, p. 120.) Published by the US National Science Academy


1.3

Astronomical Objects of Research

9

Fig. 1.12 The dimensions of the Universe

itational waves are the most recent object of research.
Astronomy and space research may seem
to be related, although they are quite different things. Space research includes all activi-

ties in the space, but only a minor fraction of
that is astronomical research. Space research is
mainly commercial services, like communication, weather observations, navigation, remote
sensing and environmental control, and also mil-


10

1

itary reconnaissance. Space astronomy is a field
of astronomy that utilises observations made by
satellites and space probes.

1.4

The Scale of the Universe

The masses and sizes of astronomical objects
are usually enormously large. But to understand
their properties, the smallest parts of matter,
molecules, atoms and elementary particles, must
be studied. The densities, temperatures and magnetic fields in the Universe vary within much
larger limits than can be reached in laboratories
on the Earth (Fig. 1.12).
The greatest natural density met on the Earth is
22,500 kg m−3 (osmium), while in neutron stars
densities of the order of 1018 kg m−3 are possible. The density in the best vacuum achieved on
the Earth is only 10−9 kg m−3 , but in interstellar
space the density of the gas may be 10−21 kg m−3
or even less. Modern accelerators can give particles energies of the order of 1013 electron volts
(eV). Cosmic rays coming from the sky may have
energies of over 1020 eV.
It has taken man a long time to grasp the vast
dimensions of space. Already Hipparchos in the
second century B.C. obtained a reasonably correct value for the distance of the Moon. The scale
of the solar system was established together with
the heliocentric system in the 17th century. In
the old geocentric system the distances of planets

did not affect their apparent motions and could

Introduction

be chosen arbitrarily. In the heliocentric system
this is no more possible. Thus the distances of
the Solar System were known reasonably well as
early as in the 15th century. Also serious attempts
to determine stellar distances were made, but the
first successful measurements were made only in
the 1830’s, and decent estimates for the distances
to the galaxies were obtained only in the 1920’s.
We can get some kind of picture of the distances involved (Fig. 1.4) by considering the time
required for light to travel from a source to the
retina of the human eye. It takes 8 minutes for
light to travel from the Sun, 5 12 hours from Neptune and 4 years from the nearest star. We cannot see the centre of the Milky Way, but the
many globular clusters around the Milky Way
are at approximately similar distances. It takes
about 20,000 years for the light from the globular cluster of Fig. 1.5 to reach the Earth. It
takes 150,000 years to travel the distance from
the nearest galaxy, the Magellanic Cloud seen on
the southern sky (Fig. 1.7). The photons that we
see now started their voyage when Neanderthal
Man lived on the Earth. The light coming from
the Andromeda Galaxy (Fig. 1.6) in the northern sky originated 2 million years ago. Around
the same time the first actual human using tools,
Homo habilis, appeared. The most distant objects
known, the quasars, are so far away that their radiation, seen on the Earth now, was emitted long
before the Sun or the Earth were born (Fig. 1.8).

2

Spherical Astronomy

Spherical astronomy is a science studying astronomical coordinate frames, directions and apparent motions of celestial objects, determination of
position from astronomical observations, observational errors, etc. We shall concentrate mainly
on astronomical coordinates, apparent motions of
stars and time reckoning. Also, some of the most
important star catalogues will be introduced.
For simplicity we will assume that the observer is always on the northern hemisphere. Although all definitions and equations are easily
generalised for both hemispheres, this might be
unnecessarily confusing. In spherical astronomy
all angles are usually expressed in degrees; we
will also use degrees unless otherwise mentioned.

2.1

Spherical Trigonometry

For the coordinate transformations of spherical
astronomy, we need some mathematical tools,
which we present now.
If a plane passes through the centre of a sphere,
it will split the sphere into two identical hemispheres along a circle called a great circle
(Fig. 2.1). A line perpendicular to the plane and
passing through the centre of the sphere intersects the sphere at the poles P and P . If a sphere
is intersected by a plane not containing the centre, the intersection curve is a small circle. There
is exactly one great circle passing through two
given points Q and Q on a sphere (unless these

points are antipodal, in which case all circles
passing through both of them are great circles).
The arc QQ of this great circle is the shortest

Fig. 2.1 A great circle is the intersection of a sphere and
a plane passing through its centre. P and P are the poles
of the great circle. The shortest path from Q to Q follows
the great circle

path on the surface of the sphere between these
points.
A spherical triangle is not just any threecornered figure lying on a sphere; its sides must
be arcs of great circles. The spherical triangle ABC in Fig. 2.2 has the arcs AB, BC and AC
as its sides. If the radius of the sphere is r, the
length of the arc AB is
|AB| = rc,

[c] = rad,

where c is the angle subtended by the arc AB
as seen from the centre. This angle is called the
central angle of the side AB. Because lengths of
sides and central angles correspond to each other
in a unique way, it is customary to give the central
angles instead of the sides. In this way, the radius

© Springer-Verlag Berlin Heidelberg 2017
H. Karttunen et al. (eds.), Fundamental Astronomy, DOI 10.1007/978-3-662-53045-0_2

11

12

2

of the sphere does not enter into the equations
of spherical trigonometry. An angle of a spherical triangle can be defined as the angle between
the tangents of the two sides meeting at a vertex,
or as the dihedral angle between the planes intersecting the sphere along these two sides. We
denote the angles of a spherical triangle by capital letters (A, B, C) and the opposing sides, or,
more correctly, the corresponding central angles,
by lowercase letters (a, b, c).
The sum of the angles of a spherical triangle is
always greater than 180 degrees; the excess
E = A + B + C − 180

◦

(2.1)

is called the spherical excess. It is not a constant,
but depends on the triangle. Unlike in plane geometry, it is not enough to know two of the angles
to determine the third one. The area of a spherical
triangle is related to the spherical excess in a very
simple way:
Area = Er 2 ,

[E] = rad.

(2.2)

This shows that the spherical excess equals the
solid angle in steradians (see Appendix A.1), subtended by the triangle as seen from the centre.
To prove (2.2), we extend all sides of the triangle
to great circles (Fig. 2.3). These great
circles will form another triangle , congruent
with
but antipodal to it. If the angle A is expressed in radians, the area of the slice S(A)

Fig. 2.2 A spherical triangle is bounded by three arcs of
great circles, AB, BC and CA. The corresponding central
angles are c, a, and b

Spherical Astronomy

bounded by the two sides of A (the shaded
area in Fig. 2.3) is obviously 2A/2π = A/π
times the area of the sphere, 4πr 2 . Similarly, the
slices S(B) and S(C) cover fractions B/π and
C/π of the whole sphere.
Together, the three slices cover the whole surface of the sphere, the equal triangles and
belonging to every slice, and each point outside
the triangles, to exactly one slice. Thus the area of
the slices S(A), S(B) and S(C) equals the area of
the sphere plus four times the area of , A( ):
A+B +C
4πr 2 = 4πr 2 + 4A( ),
π
whence

A( ) = (A + B + C − π)r 2 = Er 2 .
As in the case of plane triangles, we can derive relationships between the sides and angles of
spherical triangles. The easiest way to do this is
by inspecting certain coordinate transformations.
Suppose we have two rectangular coordinate
frames Oxyz and Ox y z (Fig. 2.4), such that
the x y z frame is obtained from the xyz frame
by rotating it around the x axis by an angle χ .
The position of a point P on a unit sphere is
uniquely determined by giving two angles. The
angle ψ is measured counterclockwise from the
positive x axis along the xy plane; the other angle θ tells the angular distance from the xy plane.

Fig. 2.3 If the sides of a spherical triangle are extended
all the way around the sphere, they form another triangle , antipodal and equal to the original triangle . The
shaded area is the slice S(A)