Astrophysics E-Book

Judith Ann Irwin

• Herausgeber: John Wiley & Sons
• Kategorie: Lebensstil
• Sprache: Englisch

Seitenzahl: 1033

Veröffentlichungsjahr: 2021

Beschreibung

ASTROPHYSICS The new edition of the popular textbook for undergraduate astronomers, covers the "how" of astrophysics Astrophysics: Decoding the Cosmos, Second Edition, describes how information about the physical nature of stars and other celestial bodies is obtained and analyzed to gain a better understanding of the universe. This acclaimed introductory textbook makes the complex principles and theories underlying astrophysics accessible to students with basic knowledge of first-year calculus-based physics and introductory astronomy. Reader-friendly chapters explore physical processes using relevant examples and clear explanations of how radiation and particles are analyzed. Such analysis leads to the density, temperature, mass, and energy of astronomical objects. In the time since the first publication of Astrophysics, the power of telescopes has increased considerably. Reflecting advancements in the field, this new edition includes carefully reviewed and updated material throughout, including recent GAIA satellite results, new information from subatomic particles, neutrinos, and cosmic rays, and brand-new case studies on Gamma-ray bursters, soft repeaters, fast radio bursts, exoplanets, and signals from exoplanetary atmospheres. Retaining its focus on electromagnetic radiation, the second edition now covers more of the ways that information about the universe is acquired, such as particles, gravitational radiation, and meteoritics. This textbook: * Describes complex processes in a clear and accessible manner * Provides relevant background information on the physics and examples of the theory in practice to place the subject into context * Includes new figures, case studies, examples, further readings, end-of-chapter problems of varying difficulty levels, and open-ended "Just for Fun" problems * Features a companion website containing information required to solve the designated web-based problems in the text and a range supplementary learning material Astrophysics: Decoding the Cosmos, Second Edition, is the ideal intermediate textbook for second- and third-year undergraduate students in Astrophysics courses, as well as a useful resource for advanced undergraduate and graduate students looking to refresh their knowledge in basic concepts.

Leseprobe

ASTROPHYSICS

Decoding the Cosmos

Second Edition

JUDITH A. IRWIN

Queen's University, Kingston, Canada

This second edition first published 2021

© 2021 John Wiley & Sons Ltd

Edition History

John Wiley & Sons Ltd (1e, 2007)

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Cover Image: © Courtesy of Caltech/MIT/LIGO Laboratory

To my dear Richard – this one is for you

Preface to the 1st edition

Like many textbooks, this one originated from lectures delivered over a number of years to undergraduate students at my home institution – Queen's University in Kingston, Canada. These students had already taken a first year (or two) of physics and one introductory astronomy course. Thus, this book is aimed at an intermediate level and is meant to be a stepping stone to more sophisticated and focussed courses, such as stellar structure, physics of the interstellar medium, cosmology, or others. The text may also be of some help to beginning graduate students with little background in astronomy or those who would like to see how physics is applied, in a practical way, to astronomical objects.

The astronomy prerequisite is helpful, but perhaps not required for students at a more senior level, since I make few assumptions as to prior knowledge of astronomy. I do assume that students have some familiarity with celestial coordinate systems (e.g. Right Ascension and Declination or others), although it is not necessary to know the details of such systems to understand the material in this text. I also do not provide any explanation as to how astronomical distances are obtained. Distances are simply assumed to be known or not known, as the case may be. I provide some figures that are meant to help with ‘astronomical geography’, but a basic knowledge of astronomical scales would also be an asset, such as understanding that the Solar System is tiny in comparison to the Galaxy and rotates about the Galactic centre.

As for approach, I had several goals in mind while organizing this material. First of all, I did not want to make the book too ‘object-oriented’. That is, I did not want to write a great deal of descriptive material about specific astronomical objects. For one thing, astronomy is such a fast-paced field that these descriptions could easily and quickly become out of date. And for another, in the age of the Internet, it is very easy for students to quickly download any number of descriptions of various astronomical objects at their leisure. What is more difficult is finding the thread of physics that links these objects, and it is this that I wanted to address.

Another goal was to keep the book practical, focussing on how we obtain information about our Universe from the signal that we actually detect. In the process, many equations are presented. While this might be a little intimidating to some students, the point should be made that the equations are our ‘tools of the trade’. Without these tools, we would be quite helpless, but with them, we have access to the secrets that astronomical signals bring to us. With the increasing availability of computer algebra or other software, there is no longer any need to be encumbered by mathematics. Nevertheless, I have kept problems that require computer-based solutions to a minimum in this text and have tried to include problems over a range of difficulty.

A solutions manual to the problems is also available. I invite readers to visit the Wiley website. It is my sincere desire that this book will be a useful stepping-stone for students of astrophysics and, more importantly, that it may play a small part in illuminating this most remarkable and marvellous Universe that we live in.

Judith A. Irwin

Kingston, Ontario

October 2006

Preface to the 2nd edition

Much has happened in more than a decade, since the first edition of this textbook was printed. The sentiments from the 1st edition have not changed, but astronomy is fast-moving and it is sometimes difficult to keep pace. Fortunately, the physics is a constant, but the technology has moved ahead in leaps and bounds. The power of our telescopes and the detail, sensitivity and clarity with which we can probe astronomical sources has shown huge improvements. I hope that this text has adequately highlighted a few of them.

Along with technological improvements come discoveries and puzzles. Our knowledge of exoplanets has become a strong and fascinating subfield, to the point that atmospheres and magnetic fields of exoplanets are being probed. And mysterious γ-ray bursters and radio transients are opening up the world of ‘temporal astronomy’ as it relates to the high energy universe.

A change from the 1st edition is the introduction of Part I: The Non-electromagnetic Signal. This includes a discussion of ‘particles’, from rocks to neutrinos, in Chapter 1. Detections of these particles are now leading to important and sophisticated science. Also, Chapter 2 in this part discusses gravitational radiation. Predicted for decades, its long-awaited discovery in 2016 has opened up a new window that we can finally peak through.

The remainder of the text follows the format of the 1st edition for electromagnetic (EM) radiation, but with many updates and improved figures. We step through the parts, working ‘backwards’ along the signal path. Part II: The EM Signal Observed, Part III: Matter and Radiation Essentials, Part IV: The EM Signal Perturbed, Part V: The EM Signal Emitted, and Part VI: The Signal Decoded.

Although obtaining distances is still not a focus of the text, I do include some examples as to how distances can be found (e.g. Sects. 2.6.2, 9.1.1.1, and 9.1.2). Another change is that most of the appendices from the 1st edition have now been put online as supplementary material. Some new supplementary material is also included online (www.wiley.com/go/irwin/astrophysics2e). Whenever problems at the end of each chapter require information from the online material, the problem will start with ‘[online]’. Tables that are used often, however, are collected into a single appendix (Appendix T) at the end of the main text in this edition.

Finally, new to this edition are small sections at the end of the problems in each chapter that I've called, Just for Fun. These contain a few fanciful problems that are more open-ended. Hopefully, these will stir the imaginations of the students that engage with them.

Judith A. Irwin

Township of South Frontenac, Ontario

July 2020

Acknowledgments – 1st Edition

There are some important people that I feel honoured to thank for their help, patience, and critical assistance with this book. First of all, to the many people who generously allowed me to use their images and diagrams, I am very grateful. Astronomy is a visual science and the impact of these images cannot be overstated.

Thanks to the students, past and present, of Physics 315 for their questions and suggestions. I have more than once had to make corrections as a result of these queries and appreciate the keen and lively intelligence that these students have shown.

Thanks to Jeff Ross for his assistance with some of the ‘nuts and bolts’ of the references. And special thanks to Kris Marble and Aimee Burrows for their many contributions, working steadfastly through problems and offering scientific expertise.

With much gratitude, I wish to acknowledge those individuals who have read sections or chapters of this book and offered constructive criticism – Terry Bridges, Diego Casadei, Mark Chen, Roland Diehl, Martin Duncan, David Gray, Jim Peebles, Ernie Seaquist, David Turner, and Larry Widrow.

To my dear children, Alex and Irene, thank you for your understanding and good cheer when mom was working behind a closed door yet again, and also thanks to the encouragement of friends – Joanne, Wendy, and Carolyn.

My tenderest thanks to my husband, Richard Henriksen, who not only suggested the title to this book, but also read through and critiqued the entire manuscript. For patience, endurance, and gentle encouragement, I thank you. It would not have been accomplished without you.

Acknowledgments – 2nd Edition

I am deeply indebted to my generous friend and colleague, Theresa Wiegert, for her extraordinary support with proofreading and other scientific feedback. Many thanks to Dominic Rochfort for his careful reading of Chapter 2 and helpful comments about gravitational radiation. Peter Brown, a world expert on meteoritics, kindly reviewed that section for me. Thanks to Aaron Vincent for pointing me towards crucial information about neutrinos. Alex Wright sacrificed valuable time out of his busy schedule to read the cosmic ray and neutrino sections. I am lucky to be in a department where there is so much expertise on astroparticle physics.

Thank you to all the people who so kindly allowed me to use their amazing images. This text would be dull indeed, without such visual enhancement. And special thanks to Jayanne English, for providing her gorgeous picture of the cygnus star forming region in Chapter 12.

Last but not least to Richard for, again, reading the whole thing.

J.A.I

List of Symbols

Symbol

Meaning

a

radius, acceleration, radiation constant

a(t)

scale factor of the Universe

A

atomic weight, total extinction

A

area, albedo

A

j,i

Einstein

A

coefficient between levels j and i

b

impact parameter, velocity parameter

B

magnetic flux density, magnetic field strength

B

(T)

intensity of a black body (or specific intensity if subscripted with

ν

or λ)

c, c

s

speed of light, speed of sound, respectively

D

p

degree of polarization

e

charge of the electron, eccentricity

E

,

energy, selective extinction, respectively

E

electric field strength

ℰ

ℳ

emission measure

f

i,j

oscillator strength between levels, i and j

f,

f

correction factor, fraction, respectively

F, S, f

flux (or flux density, if subscripted with ν or λ)

F

force

f

GW

,

frequency and rate of change of frequency of a gravitational wave, respectively

g

n

statistical weight of level, n

g

ff

,

g

bf

Gaunt factor (free–free, bound–free, respectively)

G

universal gravitational constant

h

Planck's constant, gravitational strain

H

scale height

i

inclination

I

intensity (or specific intensity, if subscripted with ν or λ)

I

moment of intertia

j

v

emission coefficient

J

mean intensity (or mean specific intensity, if subscripted with ν or λ)

J

(

E

)

‘specific intensity’ of cosmic ray particles

k

,

k

e

Boltzmann constant, Coulomb's constant, respectively

mean free path

L

luminosity (or spectral luminosity, if subscripted with ν or λ)

L

n

angular momentum of orbital, n

m

,

M

apparent magnitude, absolute magnitude, respectively

M, m

Mass

m

complex index of refraction

ℳ

,

Mach number, chirp mass, respectively

n

,

N

number density and number of (object), respectively

n

index of refraction, principal quantum number

N

map noise

number of moles, column density

p

momentum, electric dipole moment

P

power (or spectral power, if subscripted with ν or λ), probability

P

i

production rate of species,

i

P

pressure

q

charge

Q

efficiency factor

r, d

,

D

,

s, x, y, z, l,

h,

R

Position, separation, or distance

R

Rydberg constant

ℛ

collision rate

S

ν

source function

t

,

t

1/2

time, half-life, respectively

T

kinetic energy

T

temperature

period

u

energy density (or spectral energy density, if subscripted with ν or λ)

U, V, B, etc.

apparent magnitudes

U

gravitational binding energy, partition function

excitation parameter

velocity, speed

V

volume

W

work

X, Y, Z

mass fraction of hydrogen, helium, and heavier elements, respectively

z

redshift, zenith angle

Z

atomic number

α

synchrotron spectral index, fine structure constant

α

ν

absorption coefficient

α

r

recombination coefficient

β

speed relative to

c

γ

Lorentz factor

γ

coll

,

γ

a

collision rate coefficient, adiabatic index, respectively

Γ

spectral index of cosmic ray power law distribution, damping constant

ε

permittivity, cosmological energy density

θ

,

φ

one-dimensional angle

κ

ν

mass absorption coefficient

λ

,

λ

i

wavelength, decay constant of species,

i

, respectively

μ

permeability, mean molecular weight, reduced mass, proper motion

ν

frequency, collision rate per unit volume

ρ

mass density

σ

cross-sectional area, Stefan–Boltzmann constant, Gaussian dispersion

τ

ν

,

τ

optical depth, timescale, respectively

Φ

line shape function

χ

ionization potential

ω

angular frequency

Ω

solid angle, cosmological mass, energy density

About the Companion Website

This book is accompanied by a companion website:

https://wiley.com/go/irwin/astrophysics2e

The website includes:

Supplementary Material

Solutions Manual

Introduction

Decodeb: to convert (a coded message) into ordinary language.

Knowledge of our Universe continues to grow exponentially in modern times. We see newly found planets around other stars, detections of powerful gamma ray bursts, galaxies in the process of formation in the infant Universe, evidence of a mysterious force that appears to be accelerating the expansion of the Universe, and now the discovery of gravitational radiation. From exotic black holes to the microwave background, the modern understanding of our larger cosmological home could barely be imagined just a generation ago. Headlines exclaim astonishing properties for astronomical objects – stars with densities equivalent to the mass of the sun compressed to the size of a city, energy sources of incredible power, luminosities as great as an entire galaxy from a single dying star, and distortions in the very fabric of space–time.

How could we possibly have reached these conclusions? How can we dare to describe objects so inconceivably distant that the only apparent influence they have on our lives is through our very astonishment at their existence? With the exception of the Moon, no human being has ever set foot on another astronomical object. Because of the vast distances involved, no space probe has ever reached a star other than the Sun, let alone returned with material evidence. Yet we continue to amass information about our Universe. How?

In contrast to our attempts to reach outwards into the expanse of space, the natural Universe itself has been continuously and quite effectively reaching down to us, communicating in its own language. Our challenge, in the absence of an ability to travel amongst the stars, is to find the best ways to detect and decipher such signals. These signals reach us via matter, gravitational radiation, and electromagnetic radiation.

Matter includes the high energy subatomic particles and nuclei which make up cosmic rays that continuously bombard the earth (Section 1.2) as well as an influx of meteoritic dust, occasional meteorites and those rare objects that are large enough to create impact craters. In this category, we also include the charge-less subatomic particle, the neutrino. Such incoming matter provides us with information on a variety of astronomical sources, including our Sun, our Solar System, supernovae in the Galaxy1, and other more mysterious sources of the highest energy cosmic rays. Particles will be discussed more fully in Chapter 1.

Gravitational waves, weak perturbations in space predicted by Einstein's General Theory of Relativity, is a new and exciting field of astrophysics. The monumental discovery of gravitational waves, to be described more fully in Chapter 2, has opened up a window on our universe unlike any previously explored. As this field rapidly matures, we can expect a new understanding of a variety of phenomena, from regions around black holes to fundamental physics.

Electromagnetic (EM) radiation is emitted by all astronomical objects with the exception of the interior of black holes. When we say ‘light’ in this text, we mean any radiation in any waveband from the radio to the gamma-ray part of the spectrum. Electromagnetic radiation can be described as a wave and identified by its wavelength, λ or its frequency, ν. However, it can also be thought of as a massless particle called a photon which has an energy, Eph. This energy can be expressed in terms of wavelength or frequency, Eph = h c/λ = h ν, where h is Planck's constant. The wavelengths, frequencies, and photon energies of various wavebands are given in Table I.1.

The wave–particle duality of light is a deep issue in physics and related via the concept of probability. Although it should be possible to understand a physical process involving light from both points of view, there are some problems that are more easily addressed by one approach rather than another. For example, it is sometimes more straightforward to consider waves when dealing with an interaction between light and an object that is small in comparison with the wavelength and to consider photons when dealing with an interaction between light and an object that is large in comparison to the photon's wavelength. In this text, we will apply whatever form is most useful for the task at hand. Some helpful expressions relating various properties of electromagnetic radiation are provided in Table I.2 and a diagram illustrating the wave nature of light is shown in Figure I.1.

If we now ask which of these information bearers provides us with most of our current knowledge of the universe, the answer is undoubtedly electromagnetic radiation. The world's astronomical volumes would be empty indeed were it not for an understanding of radiation and its interaction with matter. The radiation may come directly from the object of interest as when sunlight travels through a clear sky with little or no interaction en route, or it may be indirect, such as when we infer the presence of a black hole by the X-rays emitted from a surrounding accretion disk. Even when we send out exploratory astronomical probes, we still rely on man-made radiation to transfer the images and data back to earth.

Table I.1 The electromagnetic spectruma.

Waveband

Wavelength range (cm)

Frequency range (Hz)

Energy range (eV)

Radio

≥ 1

≤ 3 × 10

10

≤ 1.2 × 10

−4

(Microwave)

(100 → 0.1)

(3 × 10

8

 → 3 × 10

11

)

(1.2 × 10

−6

 → 1.2 × 10

−3

)

Millimetre –Submillimetre

b

1 → 0.01

3 × 10

10

 → 3 × 10

12

1.2 × 10

−4

 → 1.2 × 10

−2

Infrared

0.01 → 10

−4

3 × 10

12

 → 3 × 10

14

1.2 × 10

−2

 → 1.2

Optical

10

−4

 → 3 × 10

−5

3 × 10

14

 → 10

15

1.2 → 4.1

Ultraviolet

3 × 10

−5

 → 10

−6

10

15

 → 3 × 10

16

4.1 → 124

X–ray

10

−6

 → 10

−9

3 × 10

16

 → 3 × 10

19

124 → 1.2 × 10

5

Gamma-ray

≤ 10

−9

≥ 3 × 10

19

≥ 1.2 × 10

5

a There is some variation as to where the ‘boundaries’ of the various wavebands lie.

b In astrophysics, the radio band is taken to include microwave frequencies and occasionally to include the millimetre – submillimetre bands as well.

Table I.2 Useful expressions relevant to light, matter, and fields.

Meaning

Equation

Wavelength and frequency relation

c

 = 

λν

Lorentz factor

Energy of a photon

Equivalent wavelength of a mass (de Broglie wavelength)

λ

 = 

h

/(m

)

Momentum of a photon

p

 = 

E

/

c

Momentum of a particle

p

 = 

γ

m

Snell's law of refraction

a

n

1

sin

θ

1

 = n

2

sin

θ

2

Index of refraction

b

Doppler shift

c

Electric field vector

Electric field vector of a wave

d

Magnetic field vector of a wave

d

Poynting flux

e

Time-averaged Poynting flux

e

Energy density of a magnetic field

f

Lorentz force

g

Electric field magnitude in a parallel-plate capacitor

h

E = (4 

πN e

)/

A

Electric dipole moment

i

Larmor's formula for power

j

Heisenberg Uncertainty Principle

k

Universal law of gravitation

l

Centripetal force

m

a If an incoming ray is travelling from medium 1 with index of refraction, n1, into medium 2 with index of refraction, n2, then θ1 is the angle between the incoming ray and the normal to the surface dividing the two media and θ2 is the angle between the outgoing ray and the normal to the surface.

bl is the speed of light in the medium, and c is the speed of light in a vacuum. Note that the index of refraction may also be expressed as a complex number whose real part is given by this equation and whose imaginary part corresponds to an absorbed component of light. See Appendix D.3 of the online material for an example.

c λ0 is the wavelength of the light in the source's reference frame (the ‘true’ wavelength), λobs is the wavelength in the observer's reference frame (the measured wavelength), and r is the relative radial velocity between the source and the observer. r is taken to be positive if the source and observer are receding with respect to each other and negative if the source and observer are approaching each other.

d The wave is propagating in the x direction, and Δ φ is an arbitrary phase shift. The magnetic field strength, H, is given by B = H μ where μ is the permeability of the substance through which the wave is travelling (unitless in the cgs system). For EM radiation in a vacuum (assumed here and throughout), this becomes B = H since the permeability of free space takes the value, 1, in cgs units. Thus, B is often stated as the magnetic field strength, rather than the magnetic flux density and is commonly expressed in units of Gauss. In cgs units, E (dyn esu −1) = B (Gauss).

e Energy flux carried by the wave in the direction of propagation. The cgs units are erg s−1 cm−2. The time-averaged value is over one cycle (see Figure I.1).

fuB has cgs units of erg cm−3 or dyn cm−2.

g Force on a charge, q, with velocity, , by an electric field, and magnetic field, .

h Here N e is the charge on a plate, and A is its area. In SI units, this equation would be E = σ/ε0, where σ is the charge per unit area on a plate, and ε0 is the permittivity of free space (where we assume free space between the plates). In cgs units, 4πε0 = 1.

i is the separation between the two charges of the dipole, and q is the strength of one of the charges.

j Power emitted by a non-relativistic particle of charge, q, that is accelerating at a rate, .

k One cannot know the position and momentum (x, p) or the energy and time (E, t) of a particle or photon to arbitrary accuracy.

l Force between two masses, M and m, a distance, r, apart.

m Force on an object of mass, m, moving at speed, , in a circular path of radius, r.

This volume is thus largely (Parts II through VI) devoted to understanding radiative processes and how such an understanding informs us about our Universe and the astronomical objects that inhabit it. It is interesting that, in order to understand the largest and grandest objects in the Universe, we must very often appeal to microscopic physics, for it is on such scales that the radiation is actually being generated and it is on such scales that matter interacts with it.

Figure I.1 Illustration of an electromagnetic wave showing the electric field and magnetic field perpendicular to each other and perpendicular to the direction of wave propagation which is in the x direction. The wavelength is denoted by λ.

We should not proceed, however, without first making a brief excursion onto the road less travelled, that is, by considering incoming particles and gravitational waves (Part I). Our historical and necessarily heavy reliance on information from EM radiation can now be supported, confirmed, and expanded upon by appealing to these other messengers. Astronomy is not only multi-waveband, but also multi-messenger as well.

Throughout the text, there are references to online material which is meant to offer supplementary information for those who wish to take a deeper look at some of the concepts. In most cases, the supplementary material is not required to understand the concepts. However, some Problems at the ends of each chapter do require that online material be accessed. In those cases, (Online) is marked next to the problem number.

This text focuses on the ‘how’ of astronomy. How do we know the temperature of that asteroid? How do we find the speed of that star? How do we know the density of that interstellar cloud? How can we find the energy of that distant quasar? Most answers are hidden in the radiation that they emit and, earthbound, we have at least a few keys to unlock their secrets. We can truly think of the detected signal as a coded message. To understand the message requires careful decoding.

I.1 DIMENSIONS, UNITS AND EQUATIONS

The centimetre-gramme-second (cgs) system of units is widely used by astronomers internationally and is the system adopted in this text. A summary of the units is given in Table I.3 as well as corresponding conversions to Système International d'Unités (SI). If an equation is given without units, the cgs system is understood. The same symbols are generally used in both systems, and SI prefixes (e.g. mega, micro, see Table I.4 in the online material) are also equally applied to the cgs system (note that the base unit, cm, already has a prefix).

Table I.3 Selected cgs – SI conversionsa.

Dimension

cgs unit (abbrev.)

Factor

SI unit

b

(abbrev.)

Length

centimetre (cm)

10

−2

metre (m)

Mass

gramme (g)

10

−3

kilogramme (kg)

Time

second (s)

1

second (s)

Energy

erg (erg)

10

−7

joule (J)

Power

erg second

−1

(erg s

−1

)

10

−7

watt (W)

Temperature

kelvin (K)

1

kelvin (K)

Force

dyne (dyn)

10

−5

newton (N)

Pressure

dyne centimetre

−2

0.1

Newton metre

−2

(N m

−2

)

(dyn cm

−2

)

barye (ba)

c

0.1

pascal (Pa)

Magnetic flux density (field)

d

gauss (G)

10

−4

tesla (T)

Angle

radian (rad)

1

radian (rad)

Solid angle

steradian (sr)

1

steradian (sr)

a Value in cgs units times factor equals value in SI units.

b Système International d'Unités.

c This unit is rarely used in astronomy in favour of dyn cm−2.

d See note d of Table I.2.

Table I.4 Examples of equivalent units.

Equation

Name

Units

P = 

n k

T

Ideal Gas Law

F = m

a

Newton's Second Law

dyn = g cm s

−2

W

 = F 

s

Work Equation

erg = dyn cm = g cm

2

 s

−2

E

k

 = 

m

2

Kinetic Energy Equation

erg = g cm

2

 s

−2

E

PG

 = m 

g

 h

Gravitational Potential Energy Equation

erg = g cm

2

 s

−2

Electrostatic Potential Energy Equation

erg = esu

2

 cm

−1

Almost all equations used in this text look identical in the two systems and one need only ensure that the constants and input parameters are all consistently used in the adopted system. There are, however, a few cases in which the equation itself changes between cgs and SI. An example is the Coulomb (electrostatic) force,

(I.1)

With the two charges, q1 and q2 expressed in electrostatic units (esu, see Table I.1), and the separation, r, in cm, the answer will be in dynes. Note that there is no constant of proportionality in this equation, unlike the SI equivalent (Problem I.1). Equations in the cgs system show the most difference with their SI equivalents when electric and magnetic quantities are used. For example, in the cgs system, the permittivity and permeability of free space, ɛ0 and μ0, respectively, are both unitless and equal to 1.

A very valuable tool for checking the answer to a problem, or to help understand an equation, is that of dimensional analysis. The dimensions of an equation (e.g. time, velocity, distance) must agree and therefore their units (s, cm s−1, cm, respectively) must also agree. Two quantities can be added or subtracted only if they have the same units, and logarithms and exponentials are unitless. In this process, it is helpful to recall some equivalent units which are revealed by writing down some simple well-known equations in physics. A few examples are provided in Table I.4. The example of the Ideal Gas Law in this table also shows the process of dimensional analysis, which involves writing down the units to every term and then cancelling where possible. A more complex example of dimensional analysis is given in Example I.1.

Example I.1

For a gas in thermal equilibrium at some uniform temperature, T, and uniform density, n, the number density of particles with speeds2 between and  + d is given by the Maxwell–Boltzmann (or simply ‘Maxwellian’) velocity distribution,

(I.2)

where n() is the gas density per unit velocity interval, m is the mass of a gas particle (taken here to be the same for all particles), and is the particle speed. A check of the units gives,

(I.3)

Simplifying yields,

(I.4)