The Interstellar Medium [Reprint 2014 ed.] 9780674493988, 9780674493971

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The Interstellar Medium

The Interstellar Medium

S. A. Kaplan and S. B. Pikelner

Harvard University Press

Cambridge, Massachusetts, 1970

Copyright © 1970 by the President and Fellows of Harvard College All rights reserved Library of Congress Catalog Card Number 70-85076 Standard Book Number 674-46075-8 Printed in the United States of America Translated from Mezhzvezdnaya Sreda (Moscow, 1963), with corrections and additions by the authors

Preface

The study of the interstellar medium occupies an important place in astrophysics. This can be attributed on the one hand to the peculiarity of the physical conditions, which often permit a simple interpretation; on the other hand, especially recently, there is the important role of the interstellar medium in the over-all picture of the formation and evolution of the Galaxy. The interstellar medium in the widest sense of the word is not limited to the " classical" components—gas and dust. Radiation fields, cosmic rays, and magnetic fields must all be equally considered. All these components contain energy comparable to the thermal energy of the gas and all interact closely with one another. At the same time, methods for investigating each component vary, and so does the extent of our knowledge of each. This book presents in an up-to-date manner, as far as possible, all aspects of the physics of the interstellar medium. Differences in apparatus employed and in the degree of our knowledge are indicated by the heterogeneous nature of the relevant chapters. The physical conditions in interstellar gas, its emission, degree of ionization, and temperature, can be studied by sufficiently reliable methods. Observations yield sound, quantitative data, while the theory linking these processes is even classical. A study of the properties of cosmic dust offers less certain results. For example, it is still impossible to say with confidence what interstellar dust particles consist of. Here, theory uses different hypotheses and the application of refined mathematical methods is not always justified. This is even more true in the case of gaseous motion, cosmic rays, and magnetic fields, where, as a rule, only qualitative methods can be employed. In the chapter on cosmic rays and radio emission, only aspects of these problems connected with the interstellar medium are considered. There are more detailed studies of these problems in special monographs, such as V. L. Ginzburg and S. I. Syrovatskii, Origin of cosmic rays (Izd. AN SSSR, 1963; Pergamon, New York, 1965), and I. S. Shklovsky, Cosmic radio waves (Fizmatgiz, 1956; trans, by R. B. Rodman and С. M. Varsavsky, Harvard University Press, Cambridge, Massachusetts, 1960). The latter book is directed to the expert reader. As an introduction,

Vi

I

PREFACE

S. В. Pikelner's book The physics of interstellar space (Izd. AN SSSR, 1959; Foreign Language Publishing House, USSR, 1961; Philosophical Library, New York, 1963, "Soviet science of interstellar space") may be used. Considerable assistance was given to the authors by their colleagues in this work who read through particular chapters and made many useful comments: A. A. Boyarchuk, V. L. Ginzburg, N. S. Kardashov, Yu. N. Pariiskii, I. I. Pronik, S. I. Syrovatskii, and I. S. Shklovskii. Special help in every chapter of the book was given by R. E. Hershberg, V. V. Ivanov, and V. I. Pronik. The authors extend to all of them their hearty thanks.

Contents

1. Interstellar Hydrogen 1. The Ionization of Interstellar Hydrogen Ionization zones of interstellar hydrogen The ionization equation of interstellar hydrogen

1 1 2 8

2. Emission from Interstellar Hydrogen Population of levels Line emission The continuous spectrum Helium emission

11 12 18 21 25

3. The Propagation of Lc and La Radiation in the Interstellar Medium The propagation of L c radiation The propagation of La radiation

29 29 32

4. Radio Emission from Interstellar Hydrogen The transfer equation for thermal radio emission Optical thickness of radio emission The λ = 21 cm radio line Other radio lines

42 43 46 50 56

5. The Distribution of Interstellar Hydrogen Ionized hydrogen Neutral hydrogen The distribution of hydrogen in the Galaxy

61 61 64 68

2. The Physical State of the Interstellar Gas

78

6. Interstellar Radiation and Ionization of the Interstellar Gas The spectral density of radiation in interstellar space Ionization of the elements Energy acquired by electrons during ionization

78 78 82 87

viii I

CONTENTS

7. The Formation of Spectral Lines in the Interstellar Gas Observational conditions for absorption lines Analysis of absorption lines Forbidden lines Population of levels The determination of temperature and electron concentration from observations of forbidden lines

88 88 92 98 104

8. The Temperature of the Interstellar Gas The energy balance in Η II zones The determination of temperature from observations Temperature in Η I regions

112 113 118 120

9. Molecules in Interstellar Space The analysis of interstellar molecular lines The formation and dissociation of molecules CH and CH + molecules Molecular hydrogen Molecular radio lines Complex molecules and diffuse lines

128 128 129 132 135 138 140

10. The Morphologic Features of Diffuse Nebulae Individual nebulae

143 147

3. Interstellar Dust

107

155

11. Selective Interstellar Absorption and the Optics of Dust Particles The dependence of absorption on wavelength Scattering of light by dielectric particles Comparison with observations

155 155 161 170

12. Distribution of Interstellar Dust The flat dust layer Statistical analysis of a cloudlike structure Results of a detailed study of the distribution of cosmic dust The connection between dust and gas

171 171 173 178 181

13. Reflection Nebulae The statistical connection with stars Luminosity of reflection nebulae The role of higher-order scattering The connection between emission and reflection nebulae

183 184 186 192 196

14. Interstellar Polarization of Light Observational data Scattering by elongated oriented particles

198 198 201

CONTENTS I ΪΧ

Orientation mechanisms Degree of polarization The Stokes parameters Polarization of reflection nebulae 15. Physical Properties of Cosmic Dust Particles Particle temperature Adsorption of gas and evaporation from particles The chemical composition and charge of dust particles The formation and disintegration of interstellar dust particles Other ideas on the nature of particles

204 206 208 210 210 211 213 216 217 221

4. Interstellar Magnetic Fields and Nonthermal Radio Emission 225 16. Magnetic Fields in Interstellar Space Electric conductivity of the interstellar medium Freezing of the magnetic field Bessel fields The Zeeman effect in the line A = 21 cm 17. The Motion of Charged Particles in Interstellar Space (Cosmic Rays) Primary cosmic rays The motion of charged particles in interstellar magnetic fields The acceleration and deceleration of charged particles in interstellar space X-rays and y-rays in the interstellar medium Motion of cosmic rays in interstellar space Relativistic electrons

225 225 228 231 232

234 235 237 241 247 249 251

18. Nonthermal Radio Emission from the Interstellar Medium Magnetobremsstrahlung The intensity of nonthermal radio emission The distribution of nonthermal radio emission The polarization of nonthermal radio emission

255 256 264 265 268

19. The Spiral Arms, the Corona, and the Magnetic Field of the Galaxy The magnetic field and the equilibrium of the spiral arms Structure of the galactic magnetic field in the solar neighborhood The galactic corona The boundaries of the corona and the intergalactic medium Coherent synchrotron radiation

270 270 274 277 278 280

I

CONTENTS

20. Discrete Galactic Sources of Nonthermal Radio Emission The Crab Nebula Cassiopeia A The Loop in Cygnus The galactic Spur

281 282 291 293 296

Interstellar Gas Dynamics and the Evolution of the Interstellar Medium

298

21. Shock Waves in Interstellar Space The structure of a shock wave Luminescence and ionization in shock waves Heating of gas by shock waves in Η I regions High-compression shock waves Radiation from interstellar shock waves Shock waves in interstellar magnetic

298 299 303 306 309 310 313

fields

22. Ionization Fronts The parameters of the ionization front The motion of ionization fronts

317 318 323

23. Motion of the Interstellar Gas The equations of motion of the gas The method of self-similar solutions Interstellar turbulence Observations of interstellar turbulence

327 328 329 336 340

24. The Evolution of the Interstellar Medium Galactic structure Stellar evolution The connection between gas and stars Gravitational condensation The contraction of a cloud The role of rotation and the magnetic field The gravitational condensation of the interstellar medium

347 347 351 354 358 361 363 365

25. Formation and Evolution of Galaxies 370 Formation of galactic subsystems 370 Classification of galaxies 374 Radio galaxies and typical galaxies of various classes 379 The general properties of radio galaxies 389 Clusters and groups of galaxies 391 Gravitational condensation of clusters and galaxies 393 The magnetic fields of galaxies and the formation of spiral arms 396 Barred spirals and the annular structure of galaxies 398 The extragalactic magnetic field 401 Addendum 402

CONTENTS

Appendix I. The Probability of Elementary Processes for Hydrogenlike Atoms and Ions Probability of discrete transitions Ionization and recombination Free-free transitions Excitation and ionization of hydrogen atoms by electron collision

I

XI

406 406 411 415 417

Appendix II. Coefficients of Excitation and De-excitation of Metastable Levels by Electron Collisions

421

Appendix III. Electronic Terms of Simple Diatomic Molecules

426

References

430

Index

457

The Interstellar Medium

1. Interstellar Hydrogen

1. The Ionization of Interstellar Hydrogen The physical properties of the interstellar medium are marked above all by its extreme diluteness—1 cm 3 contains only a few atoms. A further characteristic feature is that interstellar gas is found in a very attenuated interstellar radiation field, the density of which is of the order of a few photons per cubic centimeter. The concept of thermodynamic equilibrium is generally inapplicable to interstellar gas, although it may be employed in isolated cases, so that kinetic methods form the basis for the physical study of the interstellar medium. Fortunately, because of the rarefaction of the interstellar gas and the radiation mentioned above, the kinetics of elementary processes are here relatively simple. For example, atoms absorb quanta only from ground states and cascades of permissible downward transitions are virtually never interrupted by collisions or absorption of quanta, except at very high levels. The simplicity of the kinetic processes has made it possible to form a more or less complete theory of emission from interstellar gas, in both the optical and the radio bands, and to obtain sufficiently reliable determinations of its temperature and degree of ionization. The theory of kinetic processes in interstellar gas depends essentially on the accuracy of determining the probability of elementary transitions in atoms. These values are known best for hydrogen and for hydrogenlike atoms. Moreover, hydrogen is a basic component of the interstellar gas—the hydrogen content exceeds 80 percent of the total number of atoms. Therefore, the state of the hydrogen—primarily, its degree of ionization—frequently determines the physical condition of the interstellar medium in general. For these reasons we think it best first to investigate the kinetic processes of and the emission from hydrogen atoms in interstellar space, and then to take into account the presence of other atoms and molecules. In the first chapter, we shall study only interstellar hydrogen. The essential data on the potentials of elementary processes in hydrogen atoms are given in Appendix I. (References to the equations in the appendix will be found in the text; in numbering these equations, a Roman

2

I

THE INTERSTELLAR MEDIUM

numeral is used, indicating the appendix number; for example: (1.20) is Eq. (20), Appendix I.) Ionization Zones of Interstellar Hydrogen The determination of the degree of ionization of interstellar hydrogen is one of the main problems in the physics of the interstellar medium. It was first made by Strömgren in 1939 [22]. His main conclusion was that interstellar space is divided into rather sharply defined regions of ionized (Η II) and nonionized (Η I) hydrogen; he was in many respects responsible for the further development of the physics of the interstellar medium. Interstellar hydrogen is ionized by radiation in the far-ultraviolet region (Lyman continuum, λ < 912 Ä) of the spectrum of hot stars with high luminosities about which little is known at present. Therefore, it seems expedient to consider first a simple estimate of the degree of ionization of interstellar hydrogen, and then to develop more accurate equations. Suppose a hot star is situated in a region of interstellar hydrogen of constant density. It is clear that radiation in the Lyman continuum travels a distance s0 that corresponds to an optical thickness τ = a^N^ equal approximately to unity. Here a,(v) is the absorption coefficient and Ni is the average number of hydrogen atoms per unit volume in the first level. Using Eq. (1.20) with g f ] = 0.8 and neglecting temporarily the dependence of τ on frequency, we obtain τ = 6.3 χ 10" 18 A f 1 i ü , or, expressing s0 in parsecs,

#ιί0*Α·

О· 1 )

We note that the value of may change appreciably over the distance s0 from the star to the zone boundary; however, the use of the mean value Λ^ in Eq. (1.1) is entirely justified. It is easy to see that L c radiation is rarely present beyond the limits of the zone of radius s 0 , and the hydrogen remains nonionized. Actually, most of the L c quanta are changed after absorption into quanta of the Balmer and other series and therefore disappear. We shall show later that transition from almost complete ionization to the neutral state occurs within a thin layer. Therefore, the idea of an Η II zone of radius s0 and an Η I zone outside this sphere suggests itself. The numerical value of s0 is obtained as follows. In equilibrium the number of L c quanta emitted per second by a star and absorbed in a sphere of volume f ns§must be equal to the number of recombinations in the same volume at the second and higher levels. Recombinations at the first level must be excluded, since an Lc quantum is again formed and again ionizes a hydrogen atom. Thus, calling N L the number of Lc quanta emitted per square centimeter of stellar surface per second, we obtain

INTERSTELLAR HYDROGEN

4 nR\Nb

F

= AnR\

I

3

4ns30 »

dv

Xi/h 3

=

(a, -

at).

(1.2)

Here kFv is the energy flux at frequency ν per square centimeter of stellar surface, Rχ is the stellar radius, a„ is the recombination probability at the nth level, a, is the recombination probability at all levels (see Appendix I), and 7Ve and Np are the numbers of electrons and of protons per cubic centimeter. The temperature in ionized hydrogen regions is about 10,000°. Then, in agreement with Eq. (1.25), we have a, - o^ = 2.45 χ 10~ 3 cm 3 /sec, and hence 1/3

s0N2J3 =

a, - a.

= U (sp. cl.) = 1.23 χ 10"

* Л 2 / 3 ΝI 13 pc/cm 2 , Яе/ (1-3)

where, when hydrogen is almost completely ionized, Ne » Np. The expression 5 0 = U(sp. cl.)/V e ~ 2/3 means that the radius of the ionized region depends upon the star's spectral class (R* and N L ). The magnitude of ,y0 is called the radius of the Strömgren sphere. So far, there are no direct observations of the far-ultraviolet spectrum of stars; the value of N L can be determined only theoretically. Furthermore, because of absorption by interstellar hydrogen, we shall probably not be able to observe stellar radiation beyond the Lyman limit. The study of ionization zones can provide a check on the theoretical calculations of spectra of hot stars. A first approximate estimate may be obtained by assuming a Planck distribution at Я < 912 A with a stellar surface temperature Ί \ . Actually, the absorption coefficient in the Lyman continuum is so great that the radiation comes from a relatively thin surface layer. Using Planck's formula, we obtain

N, =

dv 2π I f , J- = ~2 hv &

2

dv

Jiv/tT,

- 1

2nkT^ he2

-hvc/kT* .

(1.4)

Here vc is the frequency of the limit of the Lyman continuum (vc = 3.29 χ 10 15 s e c - 1 ) . It is assumed in the integration that hvc » kTThe value of N L was originally estimated from this equation. However, calculations of model atmospheres of hot stars [23-31] show that the distribution of energy in the ultraviolet part of the spectrum of these stars differs markedly from the Planck distribution. Examples are shown in Figs. 1(a) and 1 (b), where the calculated and the Planck distributions are plotted for two model atmospheres. It is clear that radiation beyond the Lyman limit, directly responsible for the ionization zone Η II, is considerably less than that given by the Planck distribution. Observational verification of these calculations is not possible, of course, since all Lyman radiation is observed in the interstellar medium.

4

I

THE INTERSTELLAR MEDIUM

— eCa Majoris Β, Π ~ T e = 28470, log g=3.80

•35

.40

\(M)

FIG. 1. Energy distribution in the spectra of hot stars and black-body radiation at the effective star temperature : (я) calculated [26]; (6) observed [24]; (c) solid curve, ε Canis Majoris B1 II; broken curve, Te = 28,470°; log g = 3.80 [64]; (d) solid curve, α Leonis B8 V; broken curve, Te = 15,500°; log g = 3.89 [64].

INTERSTELLAR HYDROGEN

I

5

Spectra of hot stars up to the Lyman limit can be measured from rockets. The first results of this kind [64] showed that for most stars, in particular for В stars, a depression occurs at wavelengths λ < 2400 Ä. The observed intensity is less, sometimes by several orders of magnitude, than the calculated value. Examples of observed and calculated stellar energy distributions are shown in Figs. 1(c) and 1 (d). This discrepancy is explained, obviously, by neglect in the calculation of absorbers, particularly quasi molecules. When two atoms or ions approach each other, a system is established that is similar to a molecule, even though it is unstable. From the moment of approach, such a system can absorb a quantum, which corresponds to a molecular electronic transition. Quasi

Spectral class FIG. 2. The function [/(sp. cl.) [32].

molecules may arise from atoms and ions of H, He, and other abundant elements. Other analogous absorption mechanisms have been suggested for continuous spectra, such as collisions between hydrogen atoms in the ground and excited state [64a]. On the other hand, Pecker [64b] considered absorption by circumstellar dust. A decrease in flux in the near ultraviolet region must lead to an increase in flux beyond the Lyman limit. Although О stars were not observed, additional absorption is in principle possible for them as well. Thus Fv and hence {/(sp. cl.) may be increased. It is true that since s0 ~ N L 1 / 3 a change in N l has little effect on the determination of s0. In reference [32], the total number of Lc quanta radiated per second was calculated for given models [23-31], which yielded by means of Eq. (1.3) a value of the function U( sp. cl.) for stars of various spectral classes. Results of the calculation are shown in Fig. 2 and tabulated in Table 1.1. The points correspond to the different models and the average values are given in the table (the solid curve in Fig. 2). The last two lines of Table 1.1 give the observed values of the quantity

6

I

THE INTERSTELLAR MEDIUM

U(sp. cl.) from the flux of thermal radio emission [33], and from optical measurements from diffuse nebulae (a comparison is made in references [34] and [33a]). We see that the agreement between the theoretical calculations and the observed data obtained by independent methods is quite satisfactory. This means that the estimates of jVl are more or less correct. Table 1.1 Spectral class N l χ ΙΟ" 2 3 U (sp. cl.) (pc/cm 2 ) Calculated Observed, radio Observed, optical

05 8.7

Об

88 90 80

80 73 67

—

07 5.8

08

62 54 44

46 39 48

—

09 2.3

09.5 0.92

31 27

25 22

—

—

BO 0.36 20 17.5 15

B1 —

11 13 16

B2 0.015 4.6 — —

If the gas density in Η II zones is of the order of 10-100 particles per cubic centimeter, they are observed as emission nebulae, and if the density is somewhat lower, as extended weak hydrogen fields. In the following section we shall show that the intensity of the internal emission from a nebula is proportional to the so-called emission measure M E [35]. For an Η II zone '

ME =

2 s

°

_

_

ΝI dr » Nl 2s0 χ 2t/(sp. cl.)Wt /3 ,

(1.5)

where the integral is over the line of sight through the whole Η II region and r is expressed in parsecs. Regions with ME < 50-100 are difficult to observe against the background of the night sky. Therefore, it follows from Eq. (1.5) that only the denser Η II regions can be observed. We note that, since Eq. (1.3) or (1.5) connects one stellar parameter (the spectrum) with two parameters of the interstellar gas and 7VJ, if we can measure two of them a third can be found. For example, by measuring s0 and Nc, the spectral class of the ionizing star can be determined if we can be assured that the measured s0 does in fact represent the boundary of the Η II zone. The average degree of hydrogen ionization is found by comparing Eqs. (1.1) and (1.3); N

A ~ NJ

ΤΓ I x \N

20A

U > ~ 20L'(sp. cl.)A,c1,3.

(1.6)

In typical Η II zones, where Ne χ 1-10 c m " 3 , the ratio of the number of protons to neutral atoms is of the order of 103. In the transition layer between the ionized (Η II) and nonionized (Η I) hydrogen regions, the degree of ionization falls very rapidly. Since τ increases with distance, increases correspondingly, which in turn leads to a still further rapid increase in τ. In the transition zone of thickness Δί 0 , the number of

INTERSTELLAR HYDROGEN

I

7

neutral atoms increases from к 10"37VH to Nl » 7VH; here NH is the total number of neutral and ionized atoms per unit volume. Let us determine the thickness of the transient layer in which τ changes, for example, by two units, and where the number of neutral atoms is on the average of the order of Obviously, we have 2 0.2 б . З х К Г " ^ * ^

(L7)

Hence, the transition layer is indeed relatively thin (Aä0 « s 0 , when Лгн > 1) for stars of spectral classes О and B. The thickness of the transition layer, in contrast to s 0 , depends very little on the radius of the ionization zone or on the general characteristics of the ionizing star. The structure of the transition layer is discussed again in Chapter 5. The mass of hydrogen Μ that a single star can ionize is determined in the following way. Assuming constant density, we have obviously Μ = у slNpmH = ^

[£/(sp. cl.)]3 =

[U(sp. cl.)]3,

(1.8)

where mH is the mass of a hydrogen atom. For stars of spectral class 07, for example, U(sp. cl.) = 62 pc/cm2 and Eq. (1.8) yields Μ _ 24,000 К Masses of Η II regions are comparatively large. The inverse relation of Μ to density is explained by the fact that a reduction in Ne in a given mass reduces the number of recombinations. Therefore, a flux of Lc quanta from a star can maintain a larger number of hydrogen atoms in the ionized state. Direct observations show that the distribution of gas in interstellar space, and in Η II regions.in particular, is extremely heterogeneous. The foregoing relations were obtained under the assumption of a homogeneous hydrogen distribution. The effect of heterogeneity (or " porosity " as it is sometimes called) in the medium can be allowed for by the following method [34]. Assume that a certain volume δ is occupied by ionized hydrogen of density Np (in this case, for purposes of neutrality, Ne — Np), while the remaining volume 1 — L An,k k= 1 *

(Λ„ι +

1)) *

9

X

n4

10

·

(2.5)

In this way, when η > nk = 7STi/i6Ne~l's x 1 4 0 N e ' 1 , s , the populations of levels must be determined by collisions with free electrons possessing a thermodynamically stable (Maxwellian) velocity distribution. Therefore, the population of these levels also appears as a characteristic condition of thermodynamic equilibrium. In other words, bn approaches unity with increase in η faster than predicted by the solution of the system (2.1). Without serious loss in accuracy, we can set b„ = 1 for η > 2nk. We note that in dense nebulae, where, for example, Ne = 1 0 4 c m " 3 , we have tik « 46. A more detailed study of the system (2.1) with additional terms allowing for transitions under the influence of electron collisions was given by Seaton [49d]. It was shown that for η < nk the role of collisions is not great, whereas for η > nk the parameter is 0.9 b„ < 1. However, this increased precision has practically no influence on the intensity

INTERSTELLAR HYDROGEN

I

15

of the lines in the optical region of the spectrum. Transitions between neighboring levels when η > 60 fall into the radio band, and here we know that bn = 1 approximately without even solving the equilibrium equations. Before proceeding to the numerical results of the solution of (2.1), let us briefly consider azimuthal degeneracy. We can readily see (remembering that l' = l + 1) that in this case the system of equations analogous to (2.1) has the form oo n" — η

(2.6) =

^nl

Σ

^nln'U+l) + Σ Λ„1„·(1-

Vn' =ί+2

η = 3, 4, 5, 6 , . . . , oo,

η' — I

/ = 0, 1, 2 , . . . , η — 1.

Here also parameters bnl, which characterize the degree of deviation from thermodynamic equilibrium, may be introduced: N, — b ,Ν Ν

^21

+

eXnlkT •

(2 7)

Equation (2.7) follows directly from Eqs. (1.3) and (2.4). At thermodynamic equilibrium, bnl = 1. System (2.6), and similarly system (2.1), were approximately solved initially [47, 48, 49] as a system of algebraic equations, where, because of their clumsiness, only a small number of equations were usually taken. A solution was obtained by Pengelly [49b] with the aid of cascade matrices. The role of electron collisions described above remains valid here. What is more, since these collisions are more effective in the case of transition with no change in principal quantum number, the populations of sublevels conform to thermodynamic equilibrium at lower values of η [49с]. Using Eq. (1.47) and proceeding as in the derivation of Eq. (2.5), we find that for η > n'k к 45Ne ~1/8 the distribution over sublevels must correspond to that for thermodynamic equilibrium. That is, for n'k < η < nk we have bnl = bn φ 1. The solution of system (2.1) in this region of principal quantum numbers gives a more accurate result than the solution of system (2.6). Values for the parameters b„eXn/kT at different temperatures, obtained by the numerical solution of Eqs. (2.1) [42], and parameters bnl when Τ = 10,000°, but for different values of I [49b], are given in Tables 2.1 and 2.2. From these tables we can see that the population of levels (more accurately, the degree of deviation from thermodynamic equilibrium) depends relatively little on temperature. The dependence on azimuthal quantum number is much stronger; s states are overpopulated, while ρ and d states, on the contrary, are underpopulated. This is explained by the high probability of ρ —> s and p—>d

16

I

THE INTERSTELLAR MEDIUM

transitions and the low probability of s—> ρ transitions. However, since the mean populations 21 + 1 bn=

Σ

1= 0

n

calculated from the data of Table 2.2, differ by not more than 10-20 percent from b„ obtained by solving (2.1), the error caused by neglecting azimuthal degeneracy is not great and has no effect on the interpretation of the observational results. Table 2.1

Γ(°Κ) η

3 4 5 6 7 8 9 10 15 20 25 30

2500

5000

10,000

20,000

0.257 .218 .220 .231 .245 .259 .274 .287 .344 .386 .417 .442

0.422 .350 .346 .355 .369 .384 .398 .412 .466 .503 .531 .552

0.668 .540 .519 .520 .529 .540 .552 .563 .605 .635 .656 .673

1.013 0.792 .739 .725 .722 .725 .730 .735 .756 .772 .785 .795

To determine the populations of levels excited by electron collisions we can rewrite Eq. (2.3) in the form 1

1:0

3 2 n" = 3

2.28 χ 10" 8 (tf 1 3 + O.5160 14 + 0.484tf 15 + •• •),

NeNt 5.78 χ 1 0 " 8 ( ? 1 4 + 0.363tf 15 + 0.306^ 16 + • · · ) . and so on. The introduction of the parameters bn does not lead to any simplification here, since values of qin.. are given in tabular form. In the case of high temperatures, by taking into account the condition (1.44), that is, the proportionality of q to the corresponding oscillator strengths, these expressions may be written in the form

INTERSTELLAR HYDROGEN

i N
50,000° the role of electron collisions increases substantially. H o w e v e r , in the steady state of interstellar gas, such t e m p e r a t u r e s are n o t reached. E q u a t i o n s f o r determining the p o p u l a t i o n s of levels are linear in N„. Therefore, if b o t h excitation mechanisms are present a n d N„ « Nu we can find

18

I

THE INTERSTELLAR MEDIUM

separately for each of them the corresponding population and the resulting values for N„· (recombinations) and N„~ (electron collisions), which are then combined. First of all, of course, we must determine the degree of ionization. So far we have been discussing the third and higher levels. Populations of the 2s and 2ρ sublevels are estimated in another way, since the optical thickness in the L a line is large. The number of atoms in the 2ρ sublevel is found by studying the diffusion of L a quanta. This will be treated in the next section. We must remember that the results obtained above are correct only in the case of small optical thickness in lines of the Balmer series. The population of the 2s sublevel is determined by the condition of equilibrium between, on one hand, the number of atoms arriving at this level by means of " u p w a r d " spontaneous transitions, by direct recombination into the 2s state, and by excitation by electron collisions, and, on the other hand, the number leaving the level by means of " downward " transitions to the b state during two-quantum emission and by 2s —> 15 excitation due to electron collisions. In this way we obtain { N

t

q

2 s

u

+

N ^ i s i p

+

A

) N

2 s

NtN

p

2 s U

=

oc

2 s

(T) +

N

e

N

+

i q i s l s

£

N

n p

A

n p 2 s

.

(2.14)

π=3 Here we also take into account the probable 2s —> 2 ρ transition due to electron collision. It is convenient for solution to introduce a parameter X, which describes the fraction of all the recombinations that bring atoms to the 2s state [50]. Neglecting excitation and de-excitation due to electron collisions, we can write N

2 s

=

X N

e

N

p

Σ n=2

oi„(T)/A

2 s l s

=

X N

c

N

(2.15)

p

2s Is

The parameter may be calculated from the data of Table 2.2. It turns out that Χ к 0.34, while its value is almost independent of temperature [49]. Hence we find at once the population of the 2s sublevel (T = 10,000°): 2 45 χ 10" 1 3 N2s * — — 0.34N e N p » 1.0 χ 10" 14/VejVp cm~ 3 .

(2.16)

Thus, the metastable 2s sublevel is overpopulated compared to all the remaining sublevels of the hydrogen atom (except 2p) by five or six orders of magnitude. In dense nebulae the population of the 2ρ sublevel may be large. In this case 2p-*2s transitions must also be taken into account [51].

Line Emission Calculation of radiation intensity for known populations offers no difficulty. For example, the energy emitted in the Н„ line (transition 4 • 2) per unit

INTERSTELLAR HYDROGEN

I

19

volume per second per steradian (the so-called volume emission coefficient), is determined by the expression 2p) are forbidden by selection rules. Calculations with Eq. (2.17), both with and without taking into account azimuthal degeneracy (Tables 2.1 and 2.2, respectively), give the same result; for Te = 10,000°, ε(Η„) = 0.97 χ 10"2бЛГеЛГр erg/cm 3 sec ster.

(2.18)

When the temperature is doubled (T = 20,000°), the emission coefficient ε(Η β ) is reduced approximately to half, while at a temperature of 5,000°, я(НД on the contrary, is times as large [ε(Η,,) ~ T~3/2b4ex"/kT. In the majority of cases, absorption in the H ß line is negligible since the optical thickness in this line is much less than unity. The intensity of the H^ line is w

=

ε(Η ß )dr = 0.97 χ 10"

NeNpdr = 3 χ 1 0 " 8 M E erg/cm 3 sec ster.

(2.19)

Here the integral is taken along the line of sight. The radiation intensity at a given temperature is determined by a single parameter—the emission measure [Eq. (1.5)]. Therefore, the concept of the emission measure, introduced for the first time by Strömgren in 1948 [35], proved to be very useful in the physics of interstellar gases. At the present time, with the aid of wide-band filters, nebulae can be observed that have an emission measure of about 400, and with narrow-band filters nebulae with emission measures of 50-100 are seen. The possibility of observing very weak emitting regions is limited, not by apparatus, but by the radiation from circumterrestrial hydrogen (geocorona). Its emission measure, as P. V. Shcheglov showed, varies over the range from 5 to 50. The intensity of other hydrogen lines and also the intensity of the continuous spectrum are usually given relative to H^. It is obvious that when the ratio /„„.//(Hp) is calculated common factors cancel, the emission measure among them, and this ratio turns out to be dependent only on temperature. In particular, the ratio of intensities of lines of the Balmer series, the so-called Balmer decrement, is determined by the equation Jn2_ /(Hp)

=

^AniNn ^Mg^h V 42 Λ42 N, пг 0 4 2 bA

{Xn.ti)lkT

20

I

THE INTERSTELLAR MEDIUM

In this case, Eqs. (1.2) and (1.5) are used. Calculation of the ratio (2.20) from the data of Table 2.1 is not difficult. Results of the calculation of relative line intensities for two temperatures are given in Table 2.3 ([42, 49a]; the mean Table 2.3 Determination

Neglecting azimuthal quantum number Considering azimuthal quantum number Observed

Temperature (°K)

H„

Щ

H,

H,

Ηε

10,000 20,000 11,000 20,000

2.71 2.79 2.87 2.76 2.55

1.00 1.00 1.00 1.00 1.00

0.506 .421 .466 .474 .50

0.298 .262 .256 .262 .29

0.192 .178 .158 .162 —

observational results are also taken from these papers). The data show clearly that the Balmer decrement is sensitive neither to temperature change nor to azimuthal quantum number. Therefore it is impossible to use the observed Balmer decrement to determine the temperature of an electron gas. The mean observed Balmer decrement agrees well with theoretical calculations. However, in a number of cases, especially in emission nebulae, considerable deviations are detected—the observed ratio /(H a )//(H^) is greater than the theoretical value. In principle, such deviations can be explained by the fact that the optical thickness of these nebulae for the H a and lines is greater than unity. This situation was analyzed by Pottasch [53, 54], who showed that the ratio /(H C [ )//(H i ) increased with increase in optical thickness, since in this case some of the H^ quanta are divided into Η„ and P, quanta. Generally speaking, the above-calculated population of the 2s sublevel is inadequate to make a nebula optically thick in the H a line under usual conditions. For example, in a nebula of about 3 pc in extent, the optical thickness for the H , line is τ(Η α ) « 10" 1 4 i\yV p χ ΙΟ" 1 3 χ 10 19 « lO~sNeNp

« 1.

Here the absorption coefficient for H a is assumed equal to 10" 1 3 cm 2 . In denser nebulae, N e > 104, but as a rule such nebulae are not large. Moreover, in dense nebulae, the population of the 2ρ sublevel increases rapidly because of the repeated scattering of L a quanta, which can also increase the optical thickness for H , and Η β lines. We return to this question in Sec. 3. We must stress that observations are still not reliable enough to allow us to propose a hypothesis concerning the large optical thickness of nebulae for lines of the Balmer series. In particular, it is difficult to take into account correctly selective interstellar absorption, which also makes the observed Balmer decrement steeper. If hydrogen emission is excited by electron collisions, then the Balmer decrement is also extremely steep. Using Eqs. (2.20) and (2.11), we obtain for the

INTERSTELLAR HYDROGEN

I

21

ratio /(H a )//(H„) 10.8 at 10,000°, 4.7 at 50,000°, and 4.2 in the limit temperatures ( 10 4 ), collisions of the second kind begin to quench the excited 2s state. In this case, the number of two-quantum transitions is diminished, and consequently the magnitudes of the Balmer and Paschen discontinuities are increased. Therefore, the density may be determined f r o m the value of these discontinuities, although with no great degree of reliability. Moreover, account must be taken of the recombination emission from helium. Figure 3 shows the theoretical distribution with respect to Н л of the radiation intensity of nebulae in the continuous spectrum at various temperatures and densities of the electron gas [57]. Attention is drawn to the fact that at low temperature the intensity of the continuous spectrum depends essentially on the density of the electron gas. The dependence of the Balmer (D B ) and Paschen (DP) discontinuities on the density and temperature is shown in Fig. 4. Let us stress again that the emission from interstellar hydrogen, in both the

INTERSTELLAR HYDROGEN

I

25

log Α .

Ч -2.0

-3.0

.5

1.0 \ ' V )

4ΰ

3.0

0.5 0.6 0.8 1.0 Μμ)

0.3

2.5

2.0

0.4

0.5 0.6 0.8 1.0 λ

—ι—

(о)

1.5

10Х"У) {μ)

ib)

Τ.-20000°

ι

ne 3, λ = 4686 Ä is taken as 100) are given in Table 2.5. Table 2.5

tl

5

6

7

8

9

10

I(n, 4)

21.9

/(«, 5)

—

14.6 6.8

2.9 5.0

7.3 3.7

5.4 2.8

4.2 2.1

We note that the value of the parameter ßUe for the helium line λ = 4686 Ä when Τ = 10,000° is /?He(4.3) = 2.08 χ 10" 1 3 . From what has previously been said about the β parameters, it follows that from the ratio of line intensities for helium and hydrogen the relative number of atoms and ions of these elements can be found. Actually, from the definition of line intensities as quantities proportional to hvßN, we obtain /(4861) _ 5876 ßH(4.2)Np /(5876) ~ 48бГ/?„ е (3 3D,23P)NHc+

' (2.37)

/(4861) _ 4686 jgH(4.2)jVp /(4686) " 486Ϊ jßHe(4.3)7VHe+ + '

28

I

THE INTERSTELLAR MEDIUM

Substituting numerical values, we find

Np

/(4861)

Np

I(4686) 7(4861)

K

'

The relations allow the relative abundance of helium to be determined, together with its degree of ionization, from the observed lines of its emission spectrum. In particular, in planetary nebulae ЛгНе « 0.2N p on the average. In typical diffuse nebulae, helium lines are weak or are not observed at all. So far, helium has been discovered only in the Orion and Lagoon nebulae. In the calculation of the radiation from neutral hydrogen it is necessary to take into account the presence of the metastable state 2 3 5. The direct transition 2 3S —> 1 2 S is strongly forbidden, while the two-quantum transition 2 3S—> 2q —• 1 lS has a low probability (А к 2.2 χ 10" 5 sec"'). The primary process of transferring helium atoms from this state is the transition to the state 2 iS by electron collisions, followed by the two-quantum transition to the 1 1S state. The coefficient of excitation by electron collision is estimated under various assumptions [58a] to be q{2 3 S - > 2 1S) к (0.4 - 2.5) χ 10" 7 sec" 1 . Considering that approximately half of the recombinations of helium result in the captured electron in the 2 3S state, we find the population of this level for Τ = 10,000° to be (0.4 to 2.5) χ \^1Nnc(23S)Ne

к 10" 13 W He + Ne

(2.39)

or, approximately, NHe(2 3S) « 10" 5 iV H e + . Nebulae, for such a population, appear optically thick in absorption, in particular for the lines 110,830 ( 2 3 S - » 2 3P) and 13889 (2 3S-> 3 3P). In fact, the latter line was observed in the Orion nebula in absorption [142] with an optical thickness at the center of the line of about 40. An analysis of the helium spectrum, taking this effect into account, was made by Pottasch [586]. In particular, quanta of the line 13889 must be broken down into the quanta 1 = 43 cm (3 3P —> 3 3S), 17065 (3 3S -* 2 3P), and 110,830 (2 3P -> 2 3 S ) ; this is similar to the case of the large optical thickness of lines of the Balmer series where Η„ quanta are broken down into H a and P a . But the ionization of the helium level 2 3S by L a quanta may greatly reduce its population. For instance, if the helium concentration in the center of the Orion nebula is 2 χ 103 c m " 3 , the population of 2 3 S i s , as follows from (2.39), equal to 2 χ 1 0 " 2 c m " 3 . Observational data on 13889 in this case give approximately 2 χ 10" 4 c m " 3 . The difference of two orders of magnitude may be explained by ionization by L a quanta, which may be accumulated in this region (see Sec. 3). This effect has been seen also in planetary nebulae by O'Dell [58c], He found that there is a large accumulation of L a quanta. This may be explained by the assumption that the planetary nebula is surrounded by an Η I region, which is equivalent to supposing that τ 0 » 104 in Eq. (2.39).

INTERSTELLAR HYDROGEN

I

29

3. The Propagation of L c and L, Radiation in the Interstellar Medium In the preceding section we treated emission in the continuous spectrum up to the Lyman limit and in the subordinate hydrogen lines where it is characterized by the fact that it can emerge freely from nebulae. This means that so far processes could be investigated in any element of volume, regardless of the properties and structure of the whole nebula. Quanta of the Lyman series must be treated differently. The optical thickness of all nebulae for lines of the Lyman series is much greater than unity. Therefore, Lyman quanta emitted in one element of volume are absorbed by a different volume element inside the same nebula. During the subsequent cascade transitions, all quanta of the Lyman series, except L, quanta, are broken down. Quanta of the Balmer, Paschen, and other series formed in this way escape from the nebula, while the L, quanta are repeatedly scattered and retained. To analyze the radiation field in the Lyman continuum (L c ) and in the L a line, it is essential to study the transfer problem taking into account the multiple scattering in the whole nebula. The first research in this direction was carried out by V. A. Ambartsumyan [59] and Zanstra [60] in 1933-1934. At the present time, methods have been worked out that permit the various aspects of this problem to be studied with the required accuracy. Here we shall confine ourselves to a series of questions that have a direct bearing on the physics of the interstellar medium, but only with that degree of completeness necessary for understanding the basic properties of the interstellar gas. A more complete account will be found either in the original works (certain of these are cited below) or in a book by V. V. Sobolev [61]. In order not to complicate the mathematical side of the problem, we shall consider only scattering in a one-dimensional medium. The angular distribution of L a and L c emission, for the calculation of which a three-dimensional problem must be solved, is of no interest since it cannot be observed. The solution of the onedimensional problem yields an answer correct up to some numerical factor, of approximately unity, which is not essential for further analysis. The Propagation o / L c Radiation We begin with solution of the problem of calculating the L c radiation field in a one-dimensional medium of infinite optical thickness. The scattering process in the Lyman continuum leads to the ionization of hydrogen atoms, with resulting recombination to the ground state, followed by the emission of a "scattered" L c quantum. Obviously, the probability of " survival" of an L c quantum during a single scattering event (designated in the limit as A) is equal to the ratio of the probability of direct recombination to the ground state to the total probability of recombination to all levels, that is, Λ = «ι = 0.39 «2

(3.1)

30

I

THE INTERSTELLAR MEDIUM

when Τ = 10,000°. This figure depends only slightly on temperature; when Τ = 20,000°, Λ = 0.29. Essentially, the scattered radiation in the Lyman continuum can be practically considered monochromatic [62]. In fact, since the mean energy of the recombination electron is of the order of0.5(f kT), the quantum radiated by the recombination has a frequency ν = vc + 0.5(§Ш/г) 1.078vc when Τ = 10,000°. The factor 0.5 allows for the fact that recombination of slow electrons is more probable. The absorption coefficient at this frequency is approximately 15 percent less than the absorption coefficient at the limit of the Lyman series. Therefore all the scattered L c quanta can be considered to have a frequency ν « 1.05v = 3.45 χ 10 15 Hz and the absorption coefficient at this frequency is approximately ä^v) « 0.850^,.) = 5.35 χ 10" 1 8 cm 2 . We define the optical thickness correspondingly as dr — α{(ν)Ν{dr. Primary L c radiation at frequency ν from stars is attenuated over the optical depth τ by a factor exp [ — τ(ν/ν)3]. Therefore, the number of stellar L c quanta absorbed per unit volume at this depth is Fvay(v) exp

-τ| KV

dv hv NtWa^v) Vv

Fv(-|

exp

—τ

dv.

(3.2)

We consider here that within the framework of a one-dimensional problem the flux of diluted emission is equal to WFV and not to π WF%,, and also that a, (ν) ~ v3. The probability of re-emission of an absorbed quantum is Λ/2, while its frequency will then be equal to v, independent of the original frequency. In this way, the ratio of the emission coefficient at optical depth τ to the absorption coefficient per unit volume κ — Ν, α, (ν) is equal to

g(x)

=

X

=

W 2

exp

-τ| -

dv.

(3.3)

The dilution coefficient W also changes with optical depth, although more slowly (because of the dependence on distance from the star). The quantity g(z) is called the distribution function of the sources of escaping radiation. As is known, the integral equation for determining the source function of multiscattered radiation Β (τ) in the one-dimensional problem in the stationary case has the form [61]

ß w ^ j ^ W ' - ' v + itT).

(3.4)

The physical significance of this equation is very simple: it gives the radiation found in a volume element at depth τ due to absorption and subsequent reemission of quanta that have arrived in this element from other elements of the

INTERSTELLAR HYDROGEN

I

31

medium, and due to g(z), the single scattering of direct radiation from a star. It should be noted that the probability of re-emission is A/2. Equation (3.4) has an explicit solution for an arbitrary function g(z), namely, B( τ) =

2k g(z')e-kz'dx"> + g(z).

+ e"

(3.5)

Here к = (1 — A) 1/2 . The solution of (3.5) assumes a particularly simple form if g(z) = j A F e ~ \ which corresponds to the case of a flux of external monochromatic radiation falling on the medium in the absence of internal sources. Equation (3.5) then takes the form Β(τ) = F(1 - y i -

A)e~^·

(3.6)

This equation will be used subsequently. It is apparent that the solution of the problem of the scattering of L c emission requires substitution into Eq. (3.5) of Eq. (3.3), and we take Λ = 0.39 and к = 0.78. If we neglect the dependence of the dilution coefficient on r, that is, on the curvature of the layers, the general solution has the form

Bc(r) =

Л 2 W_ 4к

(v/v)3 3 g-

+

1 - к

TTV к + (v/v)3

+

dv+gb).

(3.7)

At large optical depths, the fundamental contribution to the integral is given by the high-frequency terms. Assuming, therefore, in the integral (3.7) that (v/v)3 « к = 0.78, we obtain an approximate equation correct when τ » 1: ΒΑ τ) =

AW Γα 2(T^A)JVc

F j

v

, e

-r(v/v) :

dv.

(3.8)

To evaluate this integral we set Fv = F^v/v)". Hence, we find Bc{ t) = 0.32 WF,v

со /y\4 + n ^

, e-Wdll

0.11Γ(η + 3)/3 WF,v. (3.9) T(n + 3)/3

The index (η + 3)/3 for class В stars is approximately 2-3; for class О stars it is approximately equal to f . The density of L c emission falls relatively slowly with optical depth, since the shorter-wavelength radiation penetrates to considerable depths. Considerations for a three-dimensional medium lead to a change of к

32

I

THE INTERSTELLAR MEDIUM

to [3(1 - A)]1/2 = 1.35. This lowers Β(τ) in Eq. (3.9) by approximately a factor of three. The more detailed problems concerning the scattering of L c radiation, among them the three-dimensional case, were resolved by R. E. Gershberg [62] by a somewhat different method. The characteristic feature of the distribution of Lc radiation should be noted in the transitional region between Η I and Η II. The initial Lc radiation in this case becomes bluer because of a decrease in the absorption coefficient with frequency according to the rule a,(ν) ~ v~ 3 . However, the scattered Lc radiation remains practically monochromatic. As noted in the preceding section, the role of scattered Lc radiation in the luminosity of interstellar hydrogen is taken into account in an elementary way (by neglecting transitions to the ground state). However, in a number of cases (for example, in the study of the transition layer between the Η II and Η I zones), an explicit expression is required for the flux of diffuse L c radiation. Remember that from the known source function Bc(τ) the flux of Нс(т) radiation is found from the equation Hc(τ) =

310

-А^У··· The radiation density (within the framework of a one-dimensional problem) is numerically equal to 2Bc(x)/c. As was already noted, the extrapolation of these expressions to a three-dimensional problem only produces numerical factors of the order of unity. The Propagation of L7 Radiation Let us now proceed to a study of the diffusion of radiation in the line Lx. This problem may be investigated by various methods. We limit ourselves here to an account of the fundamental aspects of the simplest method, a study of the integral transfer equation. Under the conditions of a highly rarefied interstellar gas, the profiles of spectral lines (La among them) are almost completely determined by the Doppler effect. If turbulent motion is discounted, the distribution of atomic velocities may be considered to be Maxwellian, depending on the gas temperature. In this case, it is well known that the atomic absorption coefficient for the La line is equal to kx{v) = — -J^L. mc

e

- и» -

(3.11)

where =

°

v. /2кТ\1/2 c \ «н/

=

(2kT/muY'2 К

INTERSTELLAR HYDROGEN

I

33

f2pU = 0.4162 is the oscillator strength of the line L a , and vx = 2.42 χ 10 1 5 s e c " 1 is the frequency of the center of the line. Substituting numerical values, we obtain =

5 9 χ 10"12 * ι/а e" [