Citation

## Material Information

Title:
Magnetic braking during star formation
Creator:
Fleck, Robert Charles, 1949-
Publication Date:
1977
Language:
English
Physical Description:
xi, 96 leaves : ill. ; 28 cm.

## Subjects

Subjects / Keywords:
Angular momentum ( jstor )
Braking ( jstor )
Cooling ( jstor )
Cosmic rays ( jstor )
Ionization ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnets ( jstor )
Main sequence stars ( jstor )
Stellar rotation ( jstor )
Astronomy thesis Ph. D ( lcsh )
Dissertations, Academic -- Astronomy -- UF ( lcsh )
Stars -- Evolution ( lcsh )
Stars -- Rotation ( lcsh )
Stars, New ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

## Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 87-96.
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Robert Charles Fleck, Jr.

## Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026188182 ( AlephBibNum )
03887464 ( OCLC )
AAW8374 ( NOTIS )

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Full Text

NORTOUN STAR FOPMtATION

BY

T CHLES FUICK, JR,

'SPNMT TO 'TH GRADUATE COUNCIL OF
UNIVERSITY OF FLORIR-k
TLUINTe OF 1ri1E REQU I RBtNTIS FOR ITE
SOF LCOMR 01 PHILOSOPHlY

rB S.'ITY OF FLORIDA

UNIVERSITY OF FLORIDA
I 6II5ffRl MMI II2I I
3 1262 08552 4626

a

To Sherrn

.., and my Parents

We are stardust...

Joni Mitchell, Woodstock

ACOU-NELD(Ti-iFS

Sincere thanks go to my advisor, Professor James H. Hunter, Jr.,

who first introduced me to the topic of star formation, and has since

guided ne through my MS thesis as well as the present dissertation. His

friendship and scientific counsel are most deeply appreciated.

Thanks also to the other committee members, Professors Edward E.

Carroll, Jr., Kwan-Y. Chen, and Charles F. Hooper, Jr., and

Drs. Jean-Robert Buchler and Hugh D. Campbell, whose comments throughout

the course of this work greatly improved both its content and style.

Special thanks to Dr. Jean-Robert Buchler for teaching me the "physics"

of astrophysics, and for sponsoring my tenure as a graduate research

student during the 1975-1976 academic year.

I thank Drs. Frederick W. Fallon, Robert B. Dickman, Stephen T.

Gottesman, Robert B. Loren, and Telemachos Ch. Mouschovias for privately

communicating to me their thoughts on some of the topics treated here.

I wish to express my appreciation to Professor Heinrich K. Eichhorn-

von 'urmb for "looking out" for me while I was a graduate student at the

University of South Florida, and to Professor Frank Bradshaw Wood for

doing the same during ray tenure at the University of Florida. Their

friendship and advice shall always be remembered. Special thanks to

Professor Wood, who along with Professor Kwan-Y. Chen, provided me with

a research assistantship for part of'the 1974-1975 and 1976-1977 academic

years.

I thank the State of Florida for sponsoring me as a graduate teaching

assistant at the University of South Florida from 1972-1974, and at the

University of Florida during my final two quarters of residence.

Computer time was donated by the Northeast Regional Data Center of

the State University System of Florida and is grat.efully acknowledged.

I wish to thank the Nato Advanced Study Institute Program for

their most generous support which enabled me to attend the conference

on the Origin of the Solar System held during the spring of 1976 at the

University of Newcastle upon Tyne, England. Discussions with other

participants helped to refine some of the ideas presented in this work.

I thank my wife, Sherry, for typing the various drafts of this thesis.

Thanks also to Beth Beville for her diligent and accurate typing of the

final draft.

Most importantly, I express my deepest appreciation to my wife,

Sherry, and to my parents for their much needed support and encouragement

over the years. Thank God for the many super weekends spent camping on

the beach and surfing in Cocoa: without the welcome diversions from my

work provided by family, freinds, and The Sea, I might not have lasted.

Finally, thank God for collapsing interstellar clouds, from which

we originated.

Thank God it's over.

TABLE OF CONTENT'S

PAGE

ACINOWLEDGMENTIS ....................................... ........... iv

LIST OF TABLES .................................................... viii

LIST OF FIGURES .................................. ... . ............ ix

ABSTRACT ........................................................ x

IAPFTER

I INTrRODUCTION ............................................. 1

Angular I.M mentum Problem ............................... 1
Present Work ........................................... 8

II M4AGNETIC BRAKING ......................................... 10

Magnetic and Velocity Fields ........................... 10
Torque Equation ........................................ 12
Toroidal Magnetic Field ................................ 14
Rotational Deceleration ................................ 19

III STAR FORMATION ........................................... 23

Shock-induced Star Formation ........................... 23
Thermal Instabilities .................................. 25
Physical Conditions in Dark Clouds ..................... 27
Initial Conditions for Collapse ........................ 33

IV MAGNETIC BRAKING OF COLLAPSING PROTOSFARS ................ 38

V DISCUSSION OF RESULTS: COMPARISON WITH OBSERVATIONS ...... 49

Specific Angular Momenta of Single and Binary Stars .... 49
Rotation of Main-sequence Stars ........................ 53
Angular Momentum of the Protosun ....................... 58

VI CONCLUDING REMARKS ....................................... 60

APPENDIX A: HEATING AND COOLING RATES IN DENSE CLOUDS ............. 62

Heating ................................ ............... 62
Molecular Cooling ............ .......................... 63
Molecular Hydrogen ................................... 67

PAGE

Hydrogen Deuteride ................................... 68
Carbon Monoxide ...................................... 69
Grain Cooling .......................................... 70
Cloud Temperature ...................................... 72

APPENDIX B: FRACTIONAL IONIZATION IN DENSE MAGNETIC CLOUDS ........ 76

APPENDIX C: m.IENT OF INERTIA FOR DIFFERENTIALLY ROTATING
MAIN-SEQUENCE STARS ................................... 84

LIST OF REFERENCES ................................................ 87

.BIOGRAPHICAL SKETCH ................................................ 97

LIST OF TABLES

;.ABLE PAGE
-3
1 Initial values of particle density n (cm ), cloud radius
Ro (cm), surface magnetic field streRgth Bo/BG, and
angular velocity wo (s-l) for various cloud masses mr/m
marginally unstable to gravitational collapse. Numbers in
parentheses are decimal exponents ......................... 36
-1
2 Cloud radius Ru (cm), angular momentum Ju (g an- s ),
and poloidal surface magnetic field strength BF (pG or mG)
at the uncoupling epoch for ionization by 40K only (c )
and ionization by cosmic rays (RC\). Nmunbers in
parentheses are decimal exponents ......................... 43

3 Predicted equatorial rotational velocities v, (kn s ) for
uniformly rotating (K=O.OS) and differentially rotating
(:=0.28) main-sequence stars for the two limiting ioni-
zation rates, Qg and tCR. The last column, taken from
Allen (1973), gives mean values for observed stars ........

LIST OF FIGURES

FIGURE PAGE

1 Observed magnetic field strength B (gauss) in inter-
stellar clouds and OH maser sources as a function of
their particle density n (ci-3). BG=3 microgauss is the
strength of the large-scale magnetic field of the Galaxy... 31
7 -1
2 Specific angular moment j (cm" s ) for binary systems
(visual, spectroscopic, and eclipsing), single main-
sequence stars, and the solar system (SS). Also shown is
the specific angular momentum predicted for the two
limiting cases of 40K (jK) and commic-ray (jCR) ionization,
sis well as j for the case of angular momentum
conservation .............................................. 45

3 Specific angular momentum (m s ) for the 40K
ionization rate jj, and threshold angular momentum
necessary for fission, designated by R and BO. Also shown
are the specific angular moment for a number of eclipsing
binary systems. The location of ESS represents the
specific angular momentum of the early solar system, and
SS designates its present value ........................... 52

4 Predicted equatorial rotational velocities v, (km s1) for
uniformly (K-=0.08) and differentially (<=0.28) rotating
main-sequence stars for the cosmic-ray ionization rate,
ECR. Also shown for comparison are mean values of v, for
observed stars ............................................ 57

Al Cosmic-ray (FCR) iad compressional (rc) heating rates (erg
s-1 c-3) ana molecular (A,) and grain (Ag) cooling rates
(erg s-1 cmi,-) as a function of kinetic gas temperature T
(K) for gas densities (a) n=103, (b) n=]06, and (c) n=109
cm-a. For lA.j, the solid line represents the molecular
cooling rate for a one-solar-mass cloud, and the broken
line is the cooling rate for a 40 solar-mass cloud ........ 74

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

MG-NETIC BRAKING DURING STAR FORMATION

By

Robert Charles Fleck, Jr.

August 1977

Chairman: James H. Hunter, Jr.
Major Department: Astronomy

Angular momentum is prima facie a formidable obstacle in the theory

of star formation: without rotational braking during star formation,

stars would rotate with speeds close to that of light. The present

investigation suggests that magnetic torques acting on a rotating,

contracting, cool interstellar cloud which is permeated by a frozen-in

magnetic field coupling the cloud to its surroundings, rotationally

decelerate a cloud, constraining it to co-rotate with the background

mediLn. Angular momentum is thus efficiently transferred from a collapsing

cloud to its surroundings.

We examine angular momentum transfer from cool, rotating, stellar-mass

condensations, collapsing isothermally or. a magnetically-diluted dynamic

time scale. Some mechanisms are discussed for forming gravitationally-

bound protostcllar condensations within cool, dense, molecular clouds.

Rotation induces a toroidal magnetic field and the accompanying magnetic

stresses generate a set of Alfvdn waves which propagate into the background

medium, thereby transporting angular momentum from a clouJ to its sur-

roundings. A modified virial approach is employed to calculate time-

dependent quantities of interest at the cloud's surface in order to

estimate the braking efficiency of the magnetic torques.

It is found that so long as a cloud remains magnetically coupled

to its surroundings, the magnetic torques constrain a cloud to co-rotate

with the background medium. Centrifugal forces are always kept Awell

below gravity. The one single factor most important in determining

the angular momentum of a protostar is the ionization rate in dense

magnetic clouds: the degree of ionization controls the coupling of a

cloud to the galactic magnetic field. The fractional ionization in

dense magnetic clouds is therefore discussed in some detail.

The angular momentum of magnetically-braked protostars is shot.n to

be consistent with the observed angular moment of close binary systems

and single early-type main-sequence stars. The hypothesis of magnetic

braking offers support to the fission theory for the formation of close

binary systems, and is able to account for the relative paucity of single

stars. The calculations also suggest a common mode of formation for

(close) binary and planetary systems.

This investigation shows that magnetic fields do indeed play an

important, if not dominant, role during the early stages of star

formation. Detailed numerical hydrodynamic collapse models have, as yet,

ignored the possible effects that magnetic fields may have on the

structure and evolution of a protostar. Such models are therefore highly

suspect and probably not physically realistic.

SECTION I

INTRODUCTION

Angular MomentuLT, Problem

Traditionally, magnetic fields and angular momentum have presented

formidable problems to the theory of star formation (cf. Mestel 1965).

Dae to the high conductivity of the interstellar medium, the frictional

coupling between plasma and neutral gas is sufficient to cause the large-

scale galactic magnetic field to become 'frozen' into the fluid and

dragged along with it. Accordingly, the magnetic energy density of a

collapsing interstellar cloud (or fragment) increases as the cloud contracts,

and the collapse is retarded and subsequent fragmentation may be prevented.

Condensations in the interstellar medium will also possess angular

momentum by virtue of local turbulence or galactic rotation. A simple

calculation shows that a main-sequence star would rotate with an equa-

torial speed close to that of light if it were formed by isotropic

compression from the interstellar gas, conserving angular momentum during

contraction. Of course it is doubtful that stars could ever form under

such conditions since centrifugal forces at the equator will increase

faster than the gravitational forces, ultimately resulting in a

rotational instability.

Mestel and Spitzer (1956) and Nakano and Tademaru (1972) have

shown that the 'magnetic field problem' is only temporary. Ambipolar

diffusion allows the field to uncouple from the gas when the fractional

ionization is reduced. Furthermore, Mouschovias (1976a,1976b) has shown

that, at least for relatively low gas densities, some material may stream

preferentially down the magnetic field lines, thereby increasing the

ratio of gravitational to magnetic energy within a condensation.

Radio-frequency observations of molecular clouds do not show any

clouds rotating much faster than the Galaxy (e.g. Heiles 1970; Heiles and

Katz 1976; Bridle and Kesteven 1976; Kutner, ct al. 1976; Loren 1977, and

private communication; Lada, et al. 1974). Main-sequence stars are

observed to rotate with equatorial velocities ranging from a few hundred

ki-lometers per second for the early-type stars to just a few kilometers

per second for stars later than spectral type F5 (Stnive 1930; Abt and

Hunter 1962). Evidently, nature has found a solution to the angular

momentum problem.

A variety of mechanisms have been proposed to reduce the angular

momentum of collapsing clouds and protostars. Hole (1945) and McCrea

(1960, 1961) have suggested that condensation may take place in regions

where the local turbulence is abnormally small. However, each object is

still likely to have somewhat more angular momentum than is found in

single main-sequence stars. Preferential mass flow along the rotational

axis would increase the gas density at constant angular velocity. However,

this process is not without its own difficulties (Mestel 1965; Spitzer 1968a).

More 'attention has been given to the possibility of transforming the

A
Ambipolar diffusion ordinarily refers to the process of charged-
particle diffusion due to a balance between a space-charge electric field
and density gradients (cf. Krall and Trivelpiece 1973). In the astro-
physical literature, ambipolar diffusion refers to the drift of a weakly
ionized plasma across a magnetic field.

_ .. .................. ..... .........

3

intrinsic (spin) angular momentum of a single massive protostar into the

orbital angular momentum of a multiple star system (Larson 1972a; Black

and Bodenheimer 1976). However, as Mouschovias (1977) points out, the

angular momentum of such a hypothetical system is still some ti.'o orders

of magnitude greater than that observed for the long-period (visual)

binaries. Dicke's (1964) claim that the interior of the Sun is in

rapid (differential) rotation suggests that single stars may store a

large amount of angular momentum beneath their surface. Although not

accounting for the possible stabilizing effect of toroidal magnetic fields,

Goldreich and Schubert (1967) have shown that a necessary condition for

stability in differentially rotating stars of homogeneous chemical

composition is that the specific angular momentum (i.e. angular momentum

per unit mass) should increase with increasing distance from the rotational

axis. Thus it appears unlikely that a differentially rotating main-

sequence star can have an angular momentum much in excess of a uniformly

rotating star. Furthermore, convective mixing and poloidal magnetic

fields redistribute angular momentum in the direction of rigid-body

rotat icn.

It has often been suggested that the angular momentum of a con-

tracting cloud or protostar may not be conserved. That is, angular

momentum may be transferred in some manner to the surrounding interstellar

material. Weizsacker (1947) has argued that a rapidly rotating star will

be rotationally decelerated as angular momentum is transferred from the

star to its surroundings by turbulent viscosity. Ter Haar (1949)

subsequently showed that Weizsacker's purely hydrodynamic mechanism for

angular momentum transport is probably not very efficient. Recently,

Sakuraj (1976) has calculated the braking torque on a Jacobian ellipsoid

by a tidal acoustic wave which is generated in the surrounding medium

by the rotating configuration. However, as Sakurai points out, the

effectiveness of the braking for pro-main-sequence stars is uncertain

because the braking time is of the same order of magnitude as the time

scale of evolution.

Hydromagnetic braking appears to be more efficient in disposing of

angular momentum. In an attempt to account for the sun's observed slow

rotation, Alfvdn (1942) suggested that the interaction of the sun's dipole

magnetic field with the surrounding 'ion cloud' would produce a torque

on the sun tending to brake its rotation. Ter Haar (19,19) generalized

this concept to include all stars magnetically coupled to HII regions.

Lust and Schliiter (1955) examined in some detail, particularly for the

special case of torque-free magnetic fields, the transport of angular

momentum by magnetic stresses acting on a rotating star.

Hoyle (1960) has proposed three stages of development for star

formation: (1) the initial stage when a condensation is magnetically

coupled to its surroundings by the frozen-in galactic magnetic field.

Angular momentum is efficiently transferred from the contracting conden-

sation to the surrounding medium with the condensation being constrained to

co-rotate with the surroundings; (2) a subsequent phase when the fractional

ionization becomes low enough so that the condensation uncouples from the

galactic magnetic field via ambipolar diffusion, afterwhich angular momentum

is effectively conserved; and (3) a recoupling with the galactic field

during the final stage of slow contraction to the main sequence. Hoyle

was able to explain the anomolous distribution of angular momentum within

the solar system (98% of the total angular momentum of the solar system

is concentrated in the planets which comprise less than 1% of the total

mass of the system) as being the result of a hydromagnetic transfer of

angular momentum fran the primitive solar nebula to the planetary

material. Hoyle's calculations were confirmed in a more quantitative

fashion by Dallaporta and Secco (1975).

Following Hoyle's (1960) paper on the origin of the solar system

(for a review of this and other theories of solar system formation, see

Williams and Cremin 1968), it was generally believed (McNally 1965;

Huang 1973, and references cited therein) that single main-sequence stars

of spectral type F5 and later were likely to have planetary systems, and

that their observed slow rotation w.as thus explained ipso facto.

Schatzman (1962) pointed out that the transition between stars with deep

envelopes in radiative equilibrium and those with well-developed sub-

surface hydrogen convection zones occurred among the F types. He

introduced a theory in which the gas emitted by jets and flares associated

with the active chromospheres of the later-type stars (those stars having

subphotospheric convective zones) is magnetically constrained to co-rotate

with the star out to very large distances where it carries away a large

amount of angular momentum per unit mass. This theory is consistent with

observational evidence. T Tauri stars undergoing pre-main-sequence

contraction are ejecting matter (Herbig 1962; Kuhi 1964, 1966; see,

however, Ulrich 1976). The observations of Wilson (1966) and Kraft (1967)

show. a connection between the rotation of stars and their age as

detennined by chromospheric activity (measured by the presence of II and

K emission lines of Call) which is usually associated with the hydrogen

convection zone.

Dicke (1964), Brandt (1966), Modisette (1967), and Weber and Davis

(1967) have calculated the solar-wind induced torque on the Sun. They

conclude that the torque is sufficient to halve the sun's rotation on

a cosmological time scale. Elaborating on the ideas of Schatzman,

Iestel (1968) has formulated a theory of magnetic braking by a stellar

wind. Using Mestel's results, Schwartz and Schubert (1969) have shown

that the Sun may have lost a considerable amount of angular momentum

if it passed through an active T Tauri stage. Assuming that stars in

the pre-main-sequence stage are wholly convective (Hayashi 1961),

Okaloto (1969, 1970) has shown that solar-type stars may lose almost all

of their angular momentum via a Schatzman-type braking mechanism during

prc-main-sequence contraction.

The Schatznan-type magnetic braking mechanism would apply only to

those stars having appreciable subsurface convection zones and therefore

enhanced surface activity (e.g. mass loss). This may account for the

break in stellar rotation on the main sequence at spectral type F5,

although as mentioned earlier, it may be that in some cases, angular

momentum has been transferred to a surrounding planetary system. It is

not clear that early-type stars ever develop a fully convective structure

during pro-main-sequence contraction (Larson 1969, 1972b). Accordingly,

for these stars in particular, we must examine the possibility of

rotational braking during the early pre-opaque stages of star formation.

Ebert, et al. (1960; see also Spitzer 1968b and Rose 1973), in a

pioneering study' have investigated the transfer of angular momentum

from a contracting interstellar cloud which is magnetically linked to the

surrounding interstellar medium by the frozen-in galactic magnetic field.

Kinks in the field lines introduced by the rotation of the cloud propagate

into the surrounding medium in the form of magnetohydrodynamic (M4D) waves

(in this case, the transverse Alfvyn mode is excited), thereby rotationally

decelerating the cloud. This mechanism is expected to be operative only

so long as the cloud remains magnetically coupled to the background.

As the collapse proceeds to higher densities, ambipolar diffusion

(Mestel and Spitzer 1956; Nakano and Tademaru 1972), 1W-D instabilities

(Mestal 1965), or perhaps intense Ohmic dissipation (Mestel and

Strittmatter 1967) may act to uncouple the cloud's field from the

surrounding medium. Although their results remain somewhat tentative due

to the uncertainties in the formulation of the problem (e.g. assumed

cylindrical symmetryy, it appears that the magnetic torques may be

sufficient to brake the cloud's rotation so that Hoyle's (1960) argument

for efficient angular momentum transfer during the initial stage (Hoyle's

stage 1) of star fonration is supported. In a more detailed general

analysis, Gillis et al. (1974) find, in one particular application of

their somewhat artificial time-independent pseudo-problem, that the

magnetic braking is "embarrassingly efficient" although they admit that

their mathematical approximations introduce some degree of uncertainty.

Kulsrud (1971) has calculated the rate of emission of energy in the

form of MHD waves (specifically, the fast magnetosonic mode) for a

rotating, time-dependent, point magnetic dipole. Kulsrud has shown that

stars with very large magnetic fields (e.g. magnetic A stars) and

initially small rotation may be decelerated to very long periods. Indeed,

this mechanism may explain the anticorrelation of rotational velocity and

surface magnetic field strength observed for the magnetic stars

(Landstreet at al. 1975; Hartoog 1977). The magnetic accretion theory

of Havnes and Conti (1971) and the centrifugal wind theory of Strittmatter

and Norris (1971) have also been proposed to account for the long-period

Ap stars. Nakano and Tadenanr (1972), Fleck (1974), and Fleck: and

I _
iii iii---*ii

Hunter (1976) have adapted Kulsrud's result (even though Kulsrud's

formulae are strictly applicable only to a periodically time-varying

dipolar field) in order to estimate the efficiency of braking for

collapsing interstellar clouds. The results of Fleck and Hunter are in

good agreement with observations of molecular clouds and stellar rotation

on the main sequence.

Prentice and ter Haar (1971; see also Krautschneider 1977) have

suggested that a collapsing grain-cloud may lose angular momentum to

the neutral gas component. However, this mechanism assumes that the

grains are electrostatically neutral, and it ultimately relies on a hydro-

magnetic transfer of angular momentum to the outside.

Present Work

The purpose of the present investigation is to show that magnetic

fields do indeed play an iinportant, if not dominant, role during the

early stages of star formation. We examine angular momentum transfer

from a cool, rotating, magnetic cloud, magnetically coupled to its

surroundings prior to the epoch of ambipolar diffusion, and undergoing

essentially pressure-free collapse on a magnetically-diluted dynamic

time scale. Rotation induces a toroidal magnetic field in the neighborhood

of the cloud and the accompanying magnetic stresses produce a net torque

acting on the cloud tending to keep the cloud in a state of co-rotation

with its surroundings. We do not attempt a detailed solution of the

coupled hydrodynamic and electrodynamic equations as to do so would

require a sophisticated computer code to handle the problem numerically.

Such a formidable (if not impossible) task is hardly justifiable in

view of our lack of understanding of many of the details of the star

formation process. Instead, we employ a modified virial approach to

calculate time-dependent quantities of interest at the surface of a

cloud in order to estimate the efficiency of the magnetic torques in

de-spinning the cloud. We compare our results with observed properties

of molecular clouds, the specific angular monmenta of (c]ose) binary

systems, the angular momentum of the protosun, and with stellar rotation

on the main sequence.

Uncertainties in some of the physical processes of star formation and

complexities in the mathematical formulation o-'thle problem do, of course,

necessitate some degree of approximation and simplification in order that

tle problem remain tractable. We cannot hope to improve on the approximate

nature of any theoretical study of star formation until we better

understand the observations that are just now becoming available.

SECTION IT

MArGNETC BRAKING

Magnetic and Velocity Fields

It has been established (Heiles 1976, and references cited therein)

that a large-scale magnetic field pervades the "Galaxy. Due to the high

conductivity of the interstellar medium, this field is 'frozen' into

the fluid (Mestel and Spitzer 1956). Consider a uniform, spherical,

interstellar cloud with radius R which is contracting isotropically.*

Strict flux conservation implies that the magnetic field strength B

within a radially contracting cloud increases according to

B = Bo(R /R)2 (, 1)

where the subscripts denote initial values. As a consequence of flux:-

freezing during an isotropic collapse, the initially uniform (galactic)

field lines are drawn out from the cloud into a nearly radial structure

0M-estel 1966). Accordingly, we approximate the magnetic field outside

the cloud by the spherical polar coordinates B = (B ,B ,B ) where

R2
Br B o(- +l)cosO (2)
r

Observations of condensations in the interstellar median spanning a
range in mass from the massive molecular cloud complexes down to the stellar-
mass Bok globules (Zuckerman and Palmer 1974, and references cited therein)
indicate an approximate spherical geometry. Isotropic contraction will
be partly justified and partly relaxed in a later section of this paper.
Of course, the simplifying assumption that a cloud is unifonn and has a
well-defined boLmdary at R is somewhat artificial although it does simplify
the calculations and it is not expected to affect the validity of the results.

11

B- = -B sinO (3)

B = 0 (4)

For r>>R the field becomes uniform and is described by the equation

o = B (cos6,-sinO,0) (5)

The velocity fields outside a radially contracting, rotating cloud

are given by

v V ) (6)
V = (vr,0,) (6)

where

v = R(r/R)n (7)

and

v, = rsin8 (8)

SdR
In Eq. (7), R -= the collapse velocity at the cloud surface (r=R),

and we set the exponent n=l in accordance with the findings of Gerola

and Sofia (1975) and Fallen et al. (1977, and private communication)

for the Orion A molecular cloud. However, the exact value of the

exponent is somewhat uncertain (cf. Loren at al. 1973; Loren 1975, 1977;

Snell and Loren 1977). In Eq. (8), w=ao(r,t)^ is the angular velocity of

the material, and we have taken the axis of rotation to be parallel to

o i.e., =B0 =1, the unit vector along the positive z-axis. The z-axis
thus becomes the axis of symmetry and derivatives with respect to the

azimuthal coordinate vanish, i.e., = 0 Mouschovias (private

communication) is investigating magnetic braking for the case of wrB ,

and believes that the braking may be more efficient in this case.

I I ~1 1111111 1 ~

12

Paris (1971) has shown a tendency for the torques exerted by an undetached

magnetic field to rotate the angular momentum vector into parallelism

with the overall direction of the field. Thus, our assumption that

w.B =l probably more closely approximates reality. Velocities at the

cloud's surface can be found by setting r=R. The cloud is assumed to

rotate rigidly at a uniform rate w(rIR)=`(R). The magnetic viscosity

due to the cloud's frozen-in magnetic field constrains the cloud to rotate

uniformly as long as the travel time of an Alfv6n wave through the cloud

is less than the collapse time.

Torque Equation

The shear at R due to the cloud's rotation generates a toroidal

field B and the resulting magnetic torques react on the rotation field.

The magnetic stresses acting to minimize B generate a set of Alfv6n waves

which propagate into the surrounding medium, thereby transporting angular

momentum from the cloud to its surroundings. As pointed out by Lust

and Schliiter (1955; see also Mestel 1959), the magnetic torque exerted

on currents within a given volume can be described by means of a tensor

D e jx.T (9)
kU kij i j2.

analogous to the Maxwell stress tensor*

Tk, = 6 B ) (10)

where E.ij and 6ki are, respectively, the Levi-Civita tensor and

Kronecker delta, and x a=i,j,k denotes the Cartesian coordinates.

We employ the Gaussian system of electromagnetic units.

If (k,i,j) is a cyclic permutation of Eqs. (9) and (10), then
2 B
D = xiT -xj.T. = (xi6 -xj.)- -(xiBj-x.B.)-- (11)
U =1 jC 3 1i2 i JtZ Ji91T 1J j 17 471

The k-component of the magnetic torque density about the origin is
given by

D-k (12)
S ax k

so that the magnetic torque acting on a volume V may be transformed into

a surface integral:

SdkdV= Dkn dS (13)

\:here n9 is the unit normal outward from the surface element dS. If the

surface S is a sphere centered on the origin, then the total outflow of

angular momentum is

lDLnZdS = -f (-Br)(xiBj-xjB.)dS (14)

where Br is the radial component: although the magnetic pressure (B2 /n)

can interchange angular momentum between field streamlines, only the

tension along the field.lines B.B./4u contributes to the flux across

the sphere S because the pressure acting normally to each surface element

has zero moment about the center. The total torque can have only a

z-component since our system is symmetric about the z-axis. Letting B

denote the poloidal component of the magnetic field (i.e., P=B r +

the total torque T is (Iust and Schliiter 1955)

S= -L BB rsinOdS (15)
4- f 41

111 111

v..ere the surface element for a sphere of radius r is just

dS = r sinOdOd. (16)

For a rotation field described by Eq. (8), it is intuitively clear-

that the toroidal magnetic field vanishes along the z-axis where v is

zero, and in the xy-plane where B changes sign. Thus one can write

the toroidal field as

B (r,9) = B (r)sin6cosO (17)

Since the poloidal magnetic field outside a collapsing cloud has an

almost purely radial structure, we set B =B so that using Eqs. (16) and
p r
(17) in Eq. (15) and carrying out the appropriate integration, the torque

at the surface of the cloud becomes

S= 2 R3B (F B (R) (18)

where B (R) is the surface poloidal field and is given by Eq. (1).

Toroidal Magnetic Field

The time dependence of a frozen-in magnetic field is given by

(cf. Jackson 1975)

at
t= V. (') (19)

Provided that the ratio IB /Bp does not become large, the temporal

behavior of the surface poloidal field should be adequately described

by Eq. (1) if R=R(t) is Ikown. For the v and B fields described by

Eqs. (2), (3), and (6)-(8), the time dependence of the toroidal field,

given by Eq. (1) is

...... ....... ...

a [1 (iv B -rv B) + (B)] (20)
? t r ar r r aeo

which becomes

dB R
S 2R tB B sinecose( r) a (21)
tt R o r -r

where we have made use of the convective derivative,

d a -_ a a
S= =- + v (22)
dt at at r ar

In Eq. (21), the first term represents the convection of B due to v

while the second term shows clearly the expected dependence of dB /dt

on the shear in the azimuthal velocity field a/Dr .For Ir,'VA ,

where

v B (23)
vA "

is the Alfv6n speed in a plasma having a mass density p, the convection

term is unimportant and the rate of growth of B is determined sole)' by

the rotational shear ai/-r .

We now derive an approximate expression for w(r), and finally,

aw//r The equation of motion in a fixed non-rotating inertial frame is

P = -V(P++p) + (24)
fixed

Here, P is the thermal gas pressure, B denotes the gravitational

potential and

fM = .B (25)
'N c

is tho magnetic force density, j being the electric current density,

and c is the speed of light. The transfonnation of v between a fixed

---- II

frame and a frame rotating with angular velocity w is given by

(cf. Marion 1970)

dt + t y ., (26)
t dt
fixed rotating

Thus, in the reference frame of our rotating cloud (w being the cloud's

angular velocity) the equation of motion for the velocity field given

by Eqs. (6)-(8) reads

p = -v(Pg +p)+ m- r -px(-xr) (27)
rotating

The 0-component of this equation is

dv
p dt -V ( 'g+P) c+f -fcO (28)
dt 0 g M, c

where V0-E/S9 fI\0 is the 0-component of the magnetic force density and

fce = p[,x(ur)]0
2
= pw rsin0cose (29)

is the 6-component of the centrifugal force density. The first term on

the right-hand-side of Eq. (28) vanishes for a spherically syNimetric cloud.

The left-hand-side of the equation vanishes as well since the velocity

field given by Eq. (6) assumres vo=0 Thus, Eq. (2S) reduces to

f = f (30)
me co

Using Ajnphre's law

x 4 t (31)
c

in Eq. (27), the magnetic force density becomes

t 4- I xx) (32)

whence the 6-component

B aB B
rM =- I r )_ s ina)] (33)
rbr aU

Combining Eqs. (29) and (33) in accordance with Eq. (30), using Eqs. (2)

and (3) for Br and B0, respectively, yields the following expression for
2

R2
L2 -- [o2 (- +1)+2B (r)(sin2 -cos20)] (34)
pr r

where we have used Eq. (17) to write out the explicit r and 0 dependence

of B The effect of the second term in brackets is to increase w in

the equatorial zones (i.e. the xy-plane) and decrease w near the poles

(i.e. along the z-axis). Averaged over a sphere of radius r which is

concentric with the cloud, this term vanishes, i.e.,

7 (sin20-cos6 )dO
= 0 = 0 (35)
f de
0

so that an approximate (average) angular velocity for the material

surrounding the cloud is

BR R'
S( -- (7- (36)
rp I

This procedure, which is equivalent to neglecting currents in the radial

direction, is similar to that employed by Alfv6n 1967; also Alfv6n and

Arrhenius (1976) in deriving his 'law of partial co-rotation' for a

magnetized plasma.

The gas density p outside the cloud will be, in general, some

function of r. The theoretical collapse models of Hunter (1969) and

the observational findings of Loren (1977) for the Mon R2 molecular

cloud suggest

P = P '-)- (37)

where p = 3m/4rTR is the density at the surface of a uniform spherical

cloud having a mass m. Using this result in Eq. (36) and differentiating

with respect to r yields

SBR [R ,R R2 R2
-u -4 ( 2 ) +1) + o ( +1) (38)
-r m 2 -Z 2 2
r r r r

As expected, -<0 Using this expression for uw/8r in Eq. (21),
dr
R2
taking -->> 1 (which should be true as the collapse proceeds and ias
r
the virtue of somewhat underestimating the rate of growth of B, initially),

and evaluating the result at the surface of the cloud, -=R, yields

dB (R,0) B2R4
t - ) 3.5 R sincose (39)
R B (R,Gc )

This is a linear first-order differential equation which can be cast

into the form

2R constant
B(R,0) = B(P., ) 35 (40)
_3

An integrating factor is R-2. Making use of the free-fall collapse

velocity (in a later section, we modify this equation to take into

account pressure gradients within the cloud)

dR 1 41)
-= -[2Gm( 1--)] 2 (41)
0

:: ..............iilii

G being the Newtonian gravitational constant, and defining

S- R/R (42)

the solution to Eq. (40) assumes the form

S, 2B RR2sinOcose (n
B(R,6) -n2 B (R ,6)-( 8 -' o o n5(- (43)

Evaluating the integral and writing B (R,6) = B (R)sinecose in accordance

with Eq. (19) gives
-33 I2 2 (2)
_q2 R 1.o5" Ro (1-n)"% 7 35 2+ 35
B(R) -n2 R)- 0 [10+ 1- + 3- nr +2 n3]
G (m/m ) 4n"

2 35 -(1- (44)
1 0 Ll+(l-n]OJ

where m = 2.0x'103g is the mass of the Sun. Asymptotically, as rr0,
-2
B (R)- n .

Rotational Deceleration

The net torque T acting on a rotating cloud is related to the time-

rate-of-change of angular momentum by

dT
(I (45)
dt (

wh4iere

J = K R2w (46)

is the cloud's total angular momentum, K being the gyration constant

(K=0.4 for a homogeneous unifonply rotating spherical cloud). Combining

these two equations yields an expression for the rotational deceleration

of the cloud:

do 2R rT
dt R= + --- (47)
bMR

The collapse velocity at the cloud surface R is given by Eq. (41), and

the magnetic torque acting on the cloud is determined from Eq. (18)

using Eq. (44) to evaluate B (R). Notice that for T--O, the above

expression reduces to angular momentum conservation. Angular momentum

is transferred from the cloud to its surrounding's so that r is intrin-

sically negative and the cloud is rotationally decelerated.

A discussion of some of the relevant time scales is in order. From

Eq. (47) it is apparent that braking will be efficient only if the second

tenn on the right-hand-side dominates the first. Using Eq. (23) for the

Alfvn speed, one can easily show that this is equivalent to the following

condition:

vA > VR| (48)

where vR=R and v =wR is the cloud's surface rotational velocity in the

equatorial zones. The combined radial and azimuthal mass motion must not

exceed the wave speed at the surface if the magnetic stresses are to

transport angular momentum to the surrounding medium. A crude estimate
,I
of the power radiated away via FID waves is given by

E
p rot
tA

1 2
= 1 IKMRvA2 (49)

where

E 1 MR2 2 (50)
rot =2

is the cloud's rotational kinetic energy and

tA R/vA (51)

is a measure of the characteristic hydromagnetic time scale, i.e., the

travel time for an Alfv6n wave traversing the clotid. (Interestingly

enough, this order-of-magnitude estimate for the power-loss is, excepting

for a constant of order one, just that predicted by the magnetic braking

model of Ebert, et al. (1960)). Since P = -Tw, Eq. (47) becomes

V
d -w + 2R) (52)

whence the condition

vA IVRl (53)

in order that braking be efficient. A measure of the characteristic time

t (54)
f v
SVR

so that the condition for efficient braking becomes

tA tf (55)

For a marginally unstable cloud collapsing from rest, this condition is

satisfied during the initial contraction stage since VA is typically a

few kilometers per second in the interstellar medium. In fact, Mouschovias

(private communication) believes that the magnetic stresses acting on the

surface of a contracting cloud will prevent vR from ever exceeding v..

It is sometimes argued that because the magnetic energy of a gravitationally-

bound condensation can never exceed the gravitational energy, the travel

time of Alfv6n waves through the condensation is at least equal to, and

may well be considerably longer than, the free-fall time which is given by

.. .. .. .. .. .

22

t = (56)

However, this is the time required for complete collapse to a zero-

radius singularity (cf. Hunter, 1962). It is more appropriate to compare

time scales of interest with the 'instantaneous' dynamic time scale as

given by Eq. (54). As pointed out by Mcstel (1965) and Mouschovias

(1976a, 197Gb, 1977), the free-fall time as defined by Eq. (56) may have

but an academic significance for clouds with frozen-in magnetic fields.

SECTION III

STAR FORMATION

Shock-induced Star Formation

The fact that young stars are frequently found in clusters suggests

that stars are formed by a fragmentation process which occurs during

the gravitational collapse of large interstellar clouds. According to

the Jeans (1928) instability criterion, the minimum unstable mass is

related to the gas temperature T (K) and particle density n (cm-)

through the relation

mIj 10 (i 2 (57)
m n

Due to the isothermal behavior of the interstellar medium at relatively

low gas densities (Gaustad 1963; Gould 1964; Hayashi and Nakano 1965;

Hattori et al. 1969; see also Appendix A of this paper), the minimum

unstable mass decreases as the collapse proceeds to higher gas densities

so that a large collapsing cloud is expected to fragment into a number of

smaller stellar-mass condensations.

However, rather compelling theoretical arguments and observational

evidence have been presented suggesting that gravitationally-bound

stellar-mass condensations (i.e. protostars) may form directly out of

the interstellar medium without recourse to fragmentation. Ebert (1955)

and McCrea (1957; see also the discussion in Mestel 1965) have shown that

external pressures of the order 104 to 105 cm -3K can reduce the minimum

unstable mass to stellar order. Such extreme pressure variations are

kno;,n to exist in the interstellar medium (Jura 1975).

Shock waves propagating in the interstellar medium can increase

the gas density up to two orders of magnitude, and thus reduce the Jeans

mass by a factor of ten. Because of the cooling efficiency of the

interstellar medium at low densities, the cooling tine behind a shock

in an HII region is typically two orders of magnitude less than the dynamic

time scale (Field et al. 1968; Aanestad 1973), so that the shock propagates

*isothermally. The jim.p in density across an isothermal shock front is

approximately (Kaplan 1966)

n v
n 2 s ), 5S)
1 2 km-) (5

where v is the shock velocity, (measured in kn s 1) and may be as large

as 20 Ik s- for a strong shock.

Various mechanisms have been proposed for producing and maintaining

interstellar shocks, and the possibility of shock-triggered star formation

has been examined under a variety of physical conditions. The hydro-

dynamical models of Stone (1970) indicate that star formation may be

enhanced by shocks generated during collisions between interstellar clouds.

Indeed, Loren (1976) believes that ongoing star formation in the NGC 1333

molecular cloud is the result of such a cloud-cloud collision. Roberts

(1969), Shu et al. (1972), and Biermarn et al. (1972) have suggested that

shock waves associated with density waves in spiral galaxies may induce

the gravitational collapse of gas clouds thus leading to star formation.

Giant 1111 regions associated with young early-type stars often line up

'like beads on a string' along the spiral arms of our Galaxy, and there

is recent evidence for star formation by density wave shocks in M33 as

well (Dubour-Crillon 1977).

The shock front associated with the advancing ionization front of

an HII region may trigger the collapse of stellar-mass condensations

(Dyson 1968). Large OB associations may be caused by a sequential

burst of HII regions in a dense cloud (Elmegreen and Lada 1977), or

perhaps by a supernova cascade process (Ogelman and Maran 1976).

Observations of the Origem Loop supernova remnant (Berkhvijsen 1974)

and the expansion of the Gum Nebula (Schwartz 1977) suggest that the

strong shock from a supernova explosion may trigger star formation.

Cameron and Truran (1977) explain various isotopic anomalies and traces

of extinct ratioactivities in solar system material as being the result of

a nearby Type II supernova that triggered the collapse of a cloud which

led eventually to the formation of the solar system. The detailed two-

dimensional numerical hydrodynamic calculations of Woodward (1976)

demonstrate the validity of the shock-induced mechanism of star formation,

particularly when the effects of self-gravitation, thenual instabilities,

and dynamical instabilities of the Kelvin-Helmholtz and Rayleigh-Taylor

type (cf. Chandrasekar 1961) which are triggered by the shock, are taken

into account.

Thermal Instabilities

Thermal instabilities in the interstellar medium can result in the

fcoiation of non-gravitational condensations of higher density and lower

temperature than are found in the surrounding medium (Field 1965).
2
Basically, this is because cooling rates at low densities var) as n

while heating rates vary only as n. Following a thermal instability, as

the density increases and the temperature drops (pressure equilibrium

obtaining for short-wavelength perturbations), the critical Jeans mass

given by Eq. (57) decreases rapidly. Theoretical studies by Hunter

(1966, 1969) and Stein and McCray (1972) have shown that self-gravitating,

primary, stellar-mass condensations can form out of the medium directly,

without the occurrence of fragmentation, by the two-step process of

thermal instability at pressure equilibrium followed by gravitational

collapse. Observationally, isolated primary condensations having stellar

masses are known to exist (Aveni and Hunter 1967, 1969, 1972; Herbig 1970,

1976). Replacing the usual assumption of isothennal compression with the

condition of energy balance, Kegel and Traving (1976) have generalized the

Jeans criterion for gravitational instability, and they find that the
dinP
dnPo 3/2
minimum unstable mass is reduced by a factor (- i ) <1, with Po and

being the pressure and density at energy equilibrium.

Thermal-chemical instabilities may also lower the Jeans mass. The

formation of hydrogen molecules on grain surfaces in interstellar clouds

may result in pressure instabilities leading to the formation of protostars

(Schatzman 1958; Reddish 1975). Because the cooling efficiency is greater

for CO than for CII, the conversion of CII to CO during the evolution of

dense interstellar clouds (cf. Herbst and Klempercr 1973; Allen and

Robinson 1977) may lead to instabilities (Oppenheimer and Dalgarno 1975;

Glassgold and Langer 1976). Generalizing Field's (1965) work to include

chemical effects, Glassgold and Langer find unstable masses of stellar order.

Oppenheiner (1977) has demonstrated that the interstellar gas may be

unstable to the isentropic growth of linear perturbations in dense,

optically-thick regions where the molecular transitions governing the

cooling of the gas are thennalized, and where strong heat sources are

present. Such instabilities may also lead to the formation of protostars.

The criterion for thermal instability becomes modified in the

presence of magnetic fields (Field 1965). Just as in the case of

a4~ _... ~ I II I

shock-induced density growth in a magnetized plasma (cc. Kaplan 1966),

magnetic pressures inhibit compression of the gas in directions

perpendicular to the field lines. Even so, Mufson (1975) has shown for

a wide variety of physical conditions, that the post-shocked gas is

likely to become thermally unstable and that condensation modes can

grow across magnetic field lines. High resolution radio observations of

the supernova remnant IC 443 by Duin and van der Laan (1975) give evidence

for condensation perpendicular to field lines.

Physical Conditions in Dark Clouds

Young stars (e.g. T Tauri stars, Herbig Ae and Be stars, and Herbig-

Haro objects) and pre-stellar objects (e.g. IR and maser sources) are

frequently associated with dense molecular clouds (cf. Strom at al. 1975).

The apparent location of newly formed stars and HII regions on the outsides

of dense, massive clouds and not at their centers (Zuckenran and Palmer

1974; Kutner at al. 1976; Elmegreen and Lada 1977; Vrba 1977) suggests a

star-formation scenario wherein a shock-driven implosion at the boundary of

a cloud initiates a thennal-gravitational instability, ultimately resulting

in gravitationally-bound condensations.

Theoretical studies by Solomon and Wickramasinghe (1969) and Hollenback

et al. (1971), and the dense cloud chemical models of Herbst and Klemperer

(1973) and Allen and Robinson (1977), indicate that hydrogen is pre-

dominantly molecular in dense (n 10 cm ) clouds. Rocket observations

by Carruthers (1970) and Copernicus satellite observations by Spitzer et al.

(1973) support this conclusion. Other major chemical constituents include

lie, CO, mI3, Ii, H, HD, Ol, H2CO, and H20. A representative mean molecular

weight for dense cloud material would be p=2.5. Dark clouds typically

II ~

3 -3
have particle densities rrn2l =10 cm and kinetic gas temperatures
2
T=10K (lceiles 1969; Penzias et al. 1972; Zuckerman and Palmer 1974,

and references cited therein, see also Appendix A of this paper for a

detailed calculation of cloud thermodynamics).

Observations do not show any interstellar clouds rotating much

faster than the Galaxy (cf. Heilcs 1970; Heiles and Katz 1976; Lada et al.

1974; Kutner ct al. 1976; Bridle and Kesteven 1976). In the solar
-15 -1
neighborhood, the Galaxy rotates with an angular velocity wJG=10 s

Observed line widths of molecular transitions originating in dense

molecular clouds are almost invariably too wide to be explained by thermal

motions, and they frequently imply supersonic velocities. The line widths

have commonly been attributed to microturbulence (Leung and Liszt 1976)

or macroturbulence (Zuckerman and Evans 1974), but difficulties with line

profile interpretation (Snell and Loren 1977) and energetic difficulties

associated with supersonic turbulence (Dicknan 1976, and private conmmui-

cation) have led to the supposition that the line widths reflect

systematic motions within the clouds, probably large-scale collapse

(Goldreich and Kwan 1974; Scoville and Solomon 1974; Liszt et al. 1974;

Gerola and Sofia 1975; de Jong et al. 1975; Snell and Loren 1977; Plamnbeck

et al. 1977; Fallen ot al. 1977). However, in favor of a turbulent origin,

Arons and Max (1975) have suggested that the observed large line widths may

be due to the presence of moderate-amplitude hydromagnetic waves in

molecular clouds. Such waves may be generated by the magnetic braking

process.

Magnetic field strengths in interstellar clouds are very uncertain.

Measuring Zeeman splitting in the 21 an line of neutral hydrogen and the

18 cm OH line, Vcrschuur (1970) has obtained field strengths in a number

3 -3
of diffuse (n<10 nm ) clouds. Clark and Johnson (1974) have suggested

that the apparently anomalous broadening of millime.eter-wavelength SO

lines observed in Orion is caused by the Zeeman effect in 6-gauss

~agnctic fields. However, Zuckerman and Palmer (1975) believe that the

large line widths are probably due to cinematic rather than magnetic

broadening. Beichman and Chaisson (1974) find evidence frcim infrared

polarization measurements and OH Zeeman patterns for milligauss fields in

the Orion infrared nebula. Rickard et al. (1975) and Lo ct al. (1975)

have derived milligauss field strengths for a number of OH maser sources.

However, well-known observational and theoretical problems in interpreting

CH spectra in terns of Zeeman patterns (Zuckerman and Palmcr 1975; Heiles

1976) make these results tentative. Magnetic field strengths obtained by
3 -3 6 -3
Verschuur (n<103 cm ), Beichman and Chaisson (n=10 cm3 ), and Lo et al.

(n=10 cm ), are plotted in Figure 1 as a function of inferred particle

density in the magnetic region. We employ Zuckerman and Palmer's estimate

of the gas density in the Beich-ian-Chaisson source, and for the density

in the neighborhood of the source discussed by Lo et al., we take

n=108 cm as suggested by Mouschorias (1976b).

At low densities (n<10 cm 3), the magnetic field strength reflects

the large-scale galactic field BG=3pG. The constancy of the field

strength for these low-density clouds suggests that material may stream

preferentially along the field lines until higher gas densities are reached.

.Anisotropic gas flow along magnetic field lines increases the thermal gas

pressure P =nkT, k=l.38x106 erg deg being Boltzmann's constant, while

holding the magnetic pressure PM=B /8x constant. Neglecting inertial

forces, these two pressures will come into balance when the gas density

reaches a critical value given by

c rU -
.

( 0

-4 rJ-
-i 4-J

B *- o
*r I 4-4 4-1
o m

rto ti

icB3 0 u1o
n U)

uth E b

"0 3 0

Su E 7c

ir 0 r- i
SrI L)
r: 4-4

0 u 7

Q,
*H > ( C

*) a )

.Drl 0 c(
OU 3 rl

: p

w

31

i-
--- ^ --- ---.--- ---C---o---

Vo

0
-\J
-I-
7, 4

CD)
_-0
CD

(SSV) a 001

ncr B2/8T (59)

so that for B=BG=31pG and T=10K,

-3
N = 260 an (60)
cr X

which is in good agreement with Figure 1. Indeed, the Parker (1966)

instability (a magnetic Rayleigh-Taylor instability) may provide a

mechanism for preferential gas flow along magnetic field lines at low

densities, and the observational findings of Appenzeller (1971) and Vrba

(1977) support Parker's predictions.

Assuming pressure equilibrium maintains for n>ncr, the magnetic field

strength should scale with the gas density according to Eq. (59):

B = (8TkT) n (61)

so that

B n (62)

for an isothermal compression, in agreement with the detailed equilibrium

models of Mouschovias (1976a, 1976b) for self-gravitating magnetic clouds.

This result is to be compared with Eq. (1) which obtained for the case of

isotropic contraction and strict flux-freezing:
2
3
B n (63)

where we have used

p = /n = 3m/4rR (64)

to relate the radius of a spherical cloud to its particle density,

m1=l.67x10 24g being the mass of the hydrogen atom. Because of the

itcertai,-ites in determining B and n for Figure 1, it is not possible

to deterrd,-.e precisely whether a slope of 1/2 or 2/3 best fits the

observn ;t.~-s: neither is inconsistent. However, as Mouschovias (1976b)

points 5r., a slope of 2/3 may be incompatible with certain properties

(e.g. s;h, density, inferred magnetic fields) of maser sources.

Furtlier:.-;::;, Scalo (1977) has shown that the heating of dense inter-

stellr colors by ambipolar diffusion imposes a constraint on cloud field

strcrgtC,: 3 must not increase faster than n7055 so that predicted gas

tcpl)er;:i}r,3 do not exceed those observed in dense clouds.

Initial Conditions for Collapse

I'r, equation of motion of the form given by Eq. (27), one can

derive (ct. Cox and Giuli 1968) a fairly general form of the virial

equa JonI:

=I = 2K+3U+M+P-3P V (65)

."'fr, I where I is the moment of inertia of the fluid about the

',*'r)il of coordinates, K, U, M, and n are, respectively, the total kinetic

r:,luy iof ;tiss motion, the thermal energy, magnetic energy, and gravi-

t.fimnJ ':.:ergy within the volume V, Ps is the hydrostatic pressure on

thi" :'Il.fc:; defined by V, and y is the ratio of specific heats (y=7/5 for

SlIr W- l7:j.lmrature gas of diatomic molecules). From what has been said

'*.',*rl rf::rding rotation and turbulence, we may safely ignore the mass-

"i rlri J');etic energy term. Also, although strong surface pressures may

. ,'",' flju .al.ssing shock waves, it is primarily the thermal instability

"'I, t! :,s; that drives the condensation of material to the higher densities

required for eventual gravitational collapse. Accordingly, \e neglect

the P term as well.
s
The condition for collapse is I<0 Eq. (65) then becomes

10|>3U+M (66)

For a uniform spherical mass distribution, 0 = -. Dividing

Eq. (66) by the volume of the (spherical) cloud V, noting that for a

non-relativistic gas the pressure is two-thirds the energy density, and

assuming mechanical equilibrium between the thermal pressure (Pg=nkT)

and magnetic pressure (PI-B/S7r) the condition for gravitational

collapse, Eq. (66), becomes

n > 3.7510 (m/mO)2 cm-3 (67)

or, equivalently,

R < 6.73x1016 (m/m) c (68)

where we have used Eq. (64) to eliminate n in favor of R, and the

subscripts here denote initial (i.e. critical) values for collapse. The

initial magnetic field strength at thp cloud surface is found from

Eqs. (59)-(61) to be

B = B(n /2602 (69)

*Galactic rotation (IG=105 s- ) sets a lower limit to the angular

velocity of a contracting cloud, and an upper limit is imposed by con-

servation of angular momentum, provided there are no external torques

acting on the cloud. Since the evolution of a condensation up to the

time it becomes gravitationally-bound is highly uncertain, we do not

attempt to calculate the efficiency of magnetic braking during this

stage. Therefore, the angular velocity of a marginally-unstable cloud

cannot be determined a priori. It is possible that a condensation may

derive its rotation from (subsonic) turbulence which may be generated by

the dynamical instabilities, particularly the Kelvin-Helmholtz modes

(cf. Woodward 1976), following the passage of a shock. Because of the

strong dissipation of supersonic turbulence (Heisenburg 1948), turbulent

velocities must not exceed the sound speed

co = CYT/ ) (70)

-1 7
which is about 0.3 km s for T=10K, 1=2.5, and Y= If the correlation

length of the turbulence is of the order of the cloud's diameter, then

W -- c /2R (71)
o 0

A turbulent origin for the angular momentum of protostars has the

attractive feature of explaining (1) the random orientation of rotational

axes of early-type stars (Iluang and Struve 1954) and field Ap stars

(Abt et al. 1972), (2) the lack of a dependence of inclinations in visual

binary systems on galactic latitude (Finsen 1933), and (3) the lack of

evidence (Huang and Wade 1966) for preferred galactic distribution of

orientations of orbital planes of eclipsing binaries. If stars and stellar

systems acquired their angular moment directly from galactic rotation,

angular momentum vectors would generally be aligned perpendicular to the

galactic plane.

Initial values of no, Ro, Bo, and wo appear in Table 1 for a range

of protostellar masses from 1 m up to 40 m The non-integer masses
correspond to ain-sequence spectral s for which ain-sequce
correspond to main-sequence spectral types for which main-sequence

-3
Table 1. Initial values of particle density no (cm ), cloud radius
Ro (cm), surface magnetic field strength B /BG, and angular
-1 o
velocity 0 (s ) for various cloud masses m/mr marginally
unstable to gravitational collapse. Numbers in parentheses
are decimal exponents.

m/m Spectral n (cm -3) R (cm) B /BG 0 (s-)
type

1 G2 3.75 (5) 6.73 (16) 38.0 2.23 (-13)

1.7 FO 1.30 (5) 1.14 (17) 22.4 1.31 (-13)

2.1 AS 8.50 (4) 1.41 (17) 18.1 1.06 (-13)

3.24 AO 3.57 (4) 2.18 (17) 11.7 6.88 (-14)

6.5 B5 8.88 (3) 4.37 (17) 5.8 3.43 (-14)

10 B3 3.75 (3) 6.73 (17) 3.8 2.23 (-14)

17.8 BO 1.18 (3) 1.20 (18) 2.1 1.24 (-14)

40 05 2.34 (2) 2.69 (18) 1 5.57 (-15)

rotation data have been accumulated. Spectral types later than early G

will not be considered since these stars are expected to lose most of

their primordial angular momentum during pre-main-sequence contraction

and main-sequence nuclear burning (cf. Section I). Interestingly enough,

the values of u are roughly the same as those one would calculate assnijing

conservation of angular momentum for a condensation having initial

densities of the order of a few particles per cubic centimeter, and

initially rotating with the Galaxy. Clearly then, the adopted values for

o are probably overestimated. By adopting a possibly exaggerated w we

require the magnetic braking to be correspondingly more efficient in

decelerating a protostar. (It will turn out that the calculations are

quite insensitive to a wide range of values of o .) The values of n

and to for the 1 m and 2.1 m clouds are in agreement (fortuitously)
o e o
with the initial conditions assumed by Black and Bodenheimer (1976)

in their calculations of rotating protostars. For all mases, the initial

ratio of gravitational to centrifugal forces at the equator, F /Fc=Gm/R L3
g c o0
is about ten. Thus, centrifugal forces are not sufficient to stabilize

the initial configurations, and the assumption that 2K<
reasonable.*

Goldreich and Lynden-Bell (1965) and Toomre (1964) have shown by
a generalization of the Jeans stability criterion to include the sta-
bilizing effect of rotation, that whereas in the classical Jeans case
with pressure effects stabilizing short-wavelength perturbations, long
waves are stabilized by rotation.

SECTION IV

MAGNfTIC BRAKING OF COLLAPSING PROTOSTARS

We now proceed to calculate the rotational deceleration of a

magnetically braked, contracting protostar. The value of the rotational

deceleration is given by

dw 2R T
dw 2R + T (47)

With the initial conditions given in Table 1, w--w(R) is obtained by

numerical integration of Eq. (47) together with R, U, and B (R) given by

Eqs. (41), (18), and (44), respectively. A fourth-order Runga-Kutta

integration scheme with variable step-size was employed. The accuracy

of the numerical code was tested by setting T=0 in Eq. (47) and checking

conservation of angular momentum (J=0). The value of the cloud radius

at each step in the integration was computed by a subroutine which solved

Kepler's equation for a collision orbit (i.e., a degenerate ellipse with

eccentricity e=l) by a standard iterative procedure. This was found to be

easier than integrating Eq. (41) and solving the resulting transcendental

equation for R at each step.

In order to keep the problem tractable, a number of simplifying,

although reasonable, assumptions have been made. To simplify the geometry,

the collapse is assumed to be isotropic and homologous, which implies

further, that the surface poloidal magnetic field increases according to

B (R) = B (R /R)2 (72)
S 00O

with B taken from Table 1. Furthermore, it is assumed that the magnetic
o
stresses within a cloud will constrain the cloud to rotate uniformly

as a rigid body. It may be argued that the assumption of homologous

collapse may be somewhat artificial in view of the numerical collapse

models of Larson (1969, 1972b) and Hunter (1969). However, there is little

or no observational evidence for 'Larson-type' dynamical evolution

(Cohen and Kuhi 1976), and Disney-(1976) believes the impressed boundary

conditions in the Larson approximation are probably not realistic.

A nonhomologous collapse would have the effect of lowering the moment

of inertia of a contracting cloud, as well as increasing the gravitational

(binding) energy somewhat.

The initial value of the azimrthal magnetic field B (R ) can not be

determined a priori. We adopt B (R )=0, with the understanding that the

braking efficiency will be somewhat underestimated during the initial

collapse phase since T =0.

Because the magnetic field of a cloud remains frozen-in during the

collapse, the free-fall time given by Eq. (56) which is valid only for

a pressure-free collapse, will underestimate the true collapse time.

From the equilibrium virial theorem [I=0 in Eq. (65)], one can define an

effective New-.tonian gravitational constant

G' = G (l j-2K-M-3U (7

Since the collapse proceeds isothermally at the relatively low densities

(n<10 cn-3) considered here (Gaustad 1963; Gould 1964; Hayashi and Nakano

1965; Hattori et al. 1969; see also Appendix A of this paper), the term

3U representing thermal gas pressure is not expected to contribute

significantly to Eq. (73). Furthermore, provided a cloud loses angular

momentum during collapse, the term 2K can be neglected (see discussion

in Section III). Thus, the collapse of a magnetic protostar proceeds

on a magnetically-diluted time scale given by

t = (3 ) (74)

uhere G' = G(jl-I)/|| Since In = 21.1 (see Eq. (66) and discussion

leading to Eq. (67)), and both I|2 and AM grow like ~R-1 for an isotropic

collapse with flux conservation, G'= G/2 .

The magnetic braking mechanism is operative only so long as a cloud

remains magnetically linked to its surroundings. Mestel (1966) and

Mestel and Strittmnatter (1967) have argued that, as a cloud contracts,

the almost oppositely directed field lines at the equatorial plane give

rise to strong 'pinching' forces that dissipate flLux and reconnect field

lines, so that the magnetic field of a cloud is effectively detached from

that of the background.* However, as Mestel and Strittmatter themselves

point out, the time scale for this process is so long that this process may

not be efficient. Furthermore, the equilibrium models of Mouschovias (1976b)

do not show any tendency for equatorial pinching.

The expulsion of a cloud's magnetic field by ambipolar diffusion

provides a more efficient mechanism for detaching the cloud's field from

*
Kulsrud's (1971) mechanism for the rotational deceleration of a
rotating dipole applies in this case. Kulsrud finds T-R3 Bo. How-.ever,
because the magnetic torques are then proprotional to the magnetic field
in the su-rounding medium instead of the azimuthal field, the braking
efficiency is reduced considerably. Kulsrud's formulae, derived for a
hanponically time-varying dipole, may not be strictly applicable to a
contracting protostar, the radius of which decreases secularly with
time.

that of its surroundings (Mestel and Spitzer 1956; Nakano and Tademaru

1972). The time scale for ambipolar diffusion is (Nakano and Tademaru

1972)

2 -2
t = 8TRminn B
D

= 8.26x1021 (ne/n) (75)

where we have used Eqs. (64), (67)-(69), and (72) to write the second
-9 3 -1
equality, taking =2x10 cmn s- (Osterbrock 1961). The electron

density ne is calculated from Eq. (B7) which appears in Appendix B

(see Appendix B for a discussion on the fractional ionization in dense

magnetic clouds). Because the ionozation rate in dense clouds is somewhat

uncertain (cf. Appendix B), we consider the two limiting cases: weak

ionization by radioactive 40K nuclei only, at a rate given by rK

(Eq. (B9)), and ionization by both 40K and (magnetically screened)

cosmic rays at a rate determined essentially by (CR (Eq. (B19)), bearing

in mind that the 40K rate is probably the more realistic of the two

(cf. Appendix B). As pointed out by Mouschovias (1977), if there is

tension in the field lines, Eq. (75) overestimates the diffusion time

scale. However, the field inside the cloud is assumed uniform so that

tD is probably not much less than that given here by Eq. (75).

The magnetic field becomes essentially uncoupled when tD = tc

(Mestel and Spitzer 1956; Nakano and Tademaru 1972). Afterwards,

a cloud contracts conserving angular momentum. Thus, the angular

momentum of a protostar is established at the uncoupling epoch. The

integration were therefore terminated when this condition was met. For

a uniform, spherical, rigidly-rotating protostar, the angular momentum

i-. the uncoup!ling epoch is simply

JU = 0.4 mRi u (76)

Jzsults of the calculations appear in Table 2.

In all cases considered here, tile magnetic torques rotationally

aicclcrat.c tlh clouds, constraining them to co-rotate with their

1.rroundings at an angular velocity u = G=105 s Physically, the

L-v6cn speed just outside a cloud is always greater than the collapse

Tlocity at the cloud surface, so that the torques are able to transmit

ingi.lar inlOmcntlim from a collapsing cloud on a time scale which is less

"ian the (m:anetically-diluted) free-fall time. Thus, the supposition

.:lat imgnetlic braking is efficient all the way down to the breakdown of

f -x-freezing (e.g. Hoyle 1960; Mouschovias 1977) appears to be vindicated.

The angular momentum of an initial condensation is reduced some three

Four orders of magnitude for the case of ionization by '0: and cosmic

:-yrs, respectively. The angular momentum of the 40K clouds at the

mUlnupling epoch is an order of magnitude greater than that of the cosmic-

.-yv ionized clouds because uncoupling occurs earlier in the collapse

q::ience (n1. 10 cm-3) for the weaker 40K ionization rate. Clouds ionized
n 7 -3
b.ilmalrily by cosmic rays uncouple from their surroundings later (n 107 cm )

;nre the time scale for ambipolar diffusion is longer for these clouds.

Ilie surface magnetic field strengths at the uncoupling epoch fall

licely within the range of those observed in dense clouds (see discussion

-. Section Ill leading up to Figure 1). The ratio |B /B1s at uncoupling

':c found to be near unit' for all clouds. This is consistent with the

findingss of l;ilijs et al. (1974) who predict a similar ratio in their time-

.:ndClpcndent m:inetic braking model. If 1B /B I become much larger than

0
rs

u
*rd *

0 r-

r- L,
14 0
ro

1 .r 0 .*-
U

S0

a rt '4
FO .J

0 *-lI A

0 (U
-4 O
o.

1 0 +.

- 0-I C)

CO

0 Q 10

I

,-.

(U 4-1 r
0 IA *
?0 E
o o

r '( t

N-

U

C3

LO
U
v-

f-

'^

r-

ra

00 D r-q tO N O01 n N-
C 13 rl 1i tn t; 4 4

r~f ,--

0- r-14
'-' '-'

0 'r-

00

C MC)
co

Ur-
c--- c(
mo rs

N1 oC
, L.

N
'-4

Ln o Le t oC Ln

< C LC)
N4 ico
0 t1- 0
q 0 r-i %T

Ur- 0
o c

k u

*U
U m

4. O
4-J r- C)
mv .3=
LO *%O .ja

02 r t

.V I ..C
tnA 4J rq

r O -+-I

CU g

co

r *- A -
-H -'0 cU

CL- 0

JEU
f to

*C r:C *r

c z C) Ca
S I O
E U

U S

23

21

(f)
NU
0

0.0

0.8 1.2

LOG (m/m,)

0.4

assuming rigid rotation, and an upper bound is estimated by adopting

Bodenheimer and Ostriker's (1970) differential rotation law for the

early-type stars (see Appendix C). For rapidly rotating stars, gravity

darkening may lead to an underestimate of the observed equatorial

velocity by as much as 40 percent (Hardorp and Strittmatter 1968), so

that the transition of j between the early-type main-sequence stars and

the eclipsing binaries is relatively smooth.

The decline in j for stars later than spectral type early-F

(m/me < 2) is thought to be the result of angular momentum transfer during

pre-Iain-sequence contraction (Schatzman 1962; Mestel 1968; Schwartz

and Schubert 1969; Okamoto 1969, 1970) and main-sequence nuclear burning

(Dicke 1964; Brandt 1966; Modisette 1967; Weber and Davis 1967), or

perhaps an indication of the presence of planetary systems (Hoyle 1960;

McNally 1965; Huang 1973, and references cited therein). Tarafdar and

Vardya (1971) account for this discrepancy in j by presuming a rapidly

rotating interior for the later-type stars and/or a slowly rotating

interior for the early-t)pe stars.

The important point illustrated by Figure 2 is that if angular

momentum is conserved during star formation, then the angular momentum

of a protostellar condensation, being almost two orders of magnitude

greater than that of the widest separated visual binaries, will be incon-

sistent with the observed angular moment of stellar systems. The

two-dimensional numerical hydrodynamic models of Larson (1972a) and Black

and Bodenheimer (1976) for rotating, collapsing protostars are thus highly

suspect and probably not physically realistic since they assume strict

angular momentum. conservation throughout the collapse. Even if the

toroidal figures predicted by their models (for which there is no

observational evidence) are unstable to nonaxisymmetric breakup, as

suggested by Wong's (1974) stability analysis of equilibrium toroids,

the angular momentum of the system remains unchanged, and one is hard

pressed to find a mechanism to dispose of angular momentum.

On the other hand, the calculations presented here for magnetic

braking during star formation are consistent with the observations

presented in Figure 2. Single stars are rare (Blaauwn 1961; Heintz 1969;

Abt and Levy 1976). It is therefore not surprising that the most likely

ionization rate in dense clouds, namely, ionization by 40K radioactive

nuclei (cf. Appendix B), predicts angular moment corresponding to that

observed for close binaries, while the much less likely situation of

ionization by cosmic rays accounts nicely for the angular moment of

(rare) single stars. The minimum angular moment of single main-sequence

stars is in excellent agreement with the minimum angular monentun of a

contracting protostar, jCR Single main-sequence stars with rapidly

rotating cores and/or large equatorial velocities apparently fonr in

regions of lower cosmic ray flux. These are the stars in Figure 2 having

jCR Apparently, clouds.having a specific angular moment j, become

unstable and fragment into a multiple (e.g. binary) star system. After

the magnetic field is expelled from a contracting cloud, the cloud

continues to contract conserving angular momentum since no external

(magnetic) torques act on the cloud. Eventually, as the rotational

kinetic energy of the cloud increases, the ratio of centrifugal forces to

gravity exceeds a critical value determined essentially by the distribution

of mass and angular momentum within the configuration (see Ostrilker 1970

for references), and the cloud breaks up into two or more condensations

-- ---- 11111

48

unity, there would be the danger that hydromagnetic instabilities of

the twisted field might interfere with the assumed poloidal magnetic

topography.

The ratio F /F = Gn/R3 indicates the important result that

centrifugal forces are kept well below gravity throughout the collapse

sequence.

SECTION V

DISCUSSION OF RESULTS: COMPARISON WITH OBSERVATIONS

The brightest flashes in the world of thought are
incomplete until they have been proven to have their
counterparts in the world of fact.

John Tyndall
(British Physicist 1820-1893)

The hypothesis of magnetic braking during star formation offers a

:-iausible explanation for the observational fact that interstellar clouds,

.'i general, do not rotate much faster than the Galaxy (Heiles 1970;

I'eiles and Katz 1976; Bridle and Kesteven 1976; Kutner et al. 1976;

jIren 1977 and private communication; Lada et al. 1974). Further obser-

'-vtional evidence for angular momentum transfer during star formation is

afforded by a consideration of the angular moment of binary systems

"nid single main-sequence stars, stellar rotation on the main sequence,

-'d the angular momentum of the protosun.

Specific Angular Momenta of Single and Binary Stars

The specific angular moment j (angular momentum per unit mass) for

-~nary systems (visual, spectroscopic, and eclipsing), single main-sequence

ztars, and the solar system (SS), is illustrated in Figure 2. In

calculatingg j for the main-sequence stars, we have assumed that the stars

:.-e a mass distribution given by Eddington's standard model (polytropic

L', .nx n = 3). Taking the observed mean equatorial rotational velocities

orjT main-sequence stars (Allen, 1973), a lower bound for j is obtained

with most of the angular momentum of the original cloud going into

orbital motion of the fission fragments. The details of the 'fission'

-process have yet to be worked out.

That a cloud having a specific angular momentum jK is expected

to fission can be seen most easily in Figure 3. Here we plot the

specific angular moment for a number of eclipsing binary systems for

which absolute orbital dimensions have been determined. Most of the

data are taken from Kopal (1959). Roxburgh (1966) and Bodenheimer and

Ostriker (1970) have calculated the threshold (i.e. minimum) angular

momentum necessary for fission to occur; Roxburgl (R) for the W Ursae

Majoris systems, and Bodenheimer and Ostriker (BO) for early-type close

binary systems. Their results, reproduced in Figure 3, are shown to be

in good agreement with the specific angular moment of close binary systems,

and more importantly here, indicate that our primary condensations,

having a specific angular momentum jK, will be unstable to bifurcation

since j. is greater than the threshold j for all masses.

According to our theory of magnetic braking, jK is an upper limit

to the specific angular momentum of a cloud of given mass. How then are

the wide (i.e. long-period spectroscopic and visual) binaries formed?

Evidently, an independent mode of formation exists for these systems.

Indeed, there has been increasing evidence for two separate modes of

binary formation. Contrary to the earlier suggestions of Kuiper (1955),

Van Albada (1968a) finds that the division of early-type binaries into

close (spectroscopic) and wide (visual) pairs is probably real and not due

to the obvious selection effects. Whatever the period distribution, smooth

or bimodal, Huang (private communication to H.A. Abt and S.G. Levy, 1976)

feels that no single formation process will produce binaries with such a

O

3 oa o

O a r-I

D 44 0 i

0 LA
N-4 "3 U "
4 u- 1 *r.-)i o

*r +) 0 C
o .i i

r b.)
*O ., 0.
N U 0- -
N4 0 CEn -t
0 -4 0 vrU

) C/

S.O > -< ok

o bo 0 u:
U U 10

UC 0O

4- U.

tO
Ea -C4
0 0 0 Cr
c n> ri u u

U r- 0-
N-. c 4-1
Cjr4-4 U-0

L- U 0
0 0 E

PC E 4-

r 0
JG-- *-I o
U) 4-j -J FQ E

*'

LL

* \

O
S* \\

.\

. C" *.

6 :* *

.. .

S0 o
*l) i

ILl

6

66 o'
(*6 ^0 9-

wide range of periods (1 days PI 108 days). From a statistical analysis

of the frequency of binary secondary masses, Abt and Levy (1976) conclude

that there are two types of binaries:, those with the shorter periods are

fission systems in which a single protostar subdivided because of

excessive angular momentum, whereas the longer periods represent pairs

of protostars that contracted separately but as a common gravitationally-

bound system. This 'neighboring-condensation' or 'early-capture' mechanism

for the origin of long-period binaries is supported by the numerical

calculations of Van Albada (196Sb) and Ary and Weissman (1973), as well

as by the findings of the recent two-body tidal capture theory of Fabian

et al. (1975; see also Press and Teukolsky 1977).

It is not likely that a wide binary system can evolve into a close

system by disposing anriular momentum, thereby making unnecessary a

separate formation mechanism for close binary systems. Binary stellar

winds (Mestel 1968; Siscoe and llHinemann 1974) can operate only in the

later-type, low-nass stars which have outer convection zones. Angular

momentum loss via gravitational radiation is efficient only for low-mass

systems already in near contact (Webbink 1976), and the disposal of

angular momentum by mass-loss is not expected to occur until the late

main-sequence evolutionary stages of already close binary systems

O'ebbink 1976). Furthermore, Webbink (1977) believes that most 1- Ursae

Majoris systems have always existed in a contact state, and that fission

is the only obvious formation mechanism satisfying this requirement.

Rotation of Main-Sequence Stars

The equatorial rotational velocity of a main-sequence star having a

specific angular momentum j is

*= j/KPI,,

(77)

where K is the effective gyration constant for the star having a main-

sequence radius R,. Assuming angular momentum is conserved after the

uncoupling epoch, j is given by j,. or jCR. Since stars later than early-F

may lose large quantities of angular momentum during pre-main-sequence

contraction, we consider only the eaily-type (m/mr 2) stars. The mass

distribution within such stars is given to a good approximation by the

Eddington standard model, characterized by a polytropic index n=3. For

rigid rotation K=0.08, and for the differentially rotating models of

Podenheimer and Ostriker (1970), =-0.28 (see Appendix C for details of

these calculations). Using this information together with the angular

momentiun data in Table 2, rotational velocities are calculated from Eq. (77)

and the'results appear in Table 3. Independent of the assumed rotation
40
law, v, for the 0K ionization rate, far exceeds the critical equatorial

breakup velocities for all spectral types (Slettcbak 1966); a fortiori

these stars are expected to fission into a close pair. Within the frame-

work of our present theory of magnetic braking, single stars are believed

to acquire an amount of angular momentum determined by the cosmic ray

ionization rate. Figure 4 illustrates the reasonable agreement parti-

cularly for the case of uniform rotation (K=0.OS), of predicted rotational

velocities with those observed for single main-sequence stars (Allen

1973). Rigid rotation may result from the actions of circulation currents,

convective mixing in the core, or magnetic viscosity (Roxburgh and

Strittmatter 1966), so that the case K=0.OS may in fact represent the

most plausible situation. The anomalously high rotational velocities

predicted for the lower-mass stars on the assumption of angular momentum

conservation during contraction to the main.sequence, is evidence for

further rotational braking during the later pre-main-sequence evolutionary

stages of these stars.

-1
Table 3. Predicted equatorial rotational velocities v~k0;m s ) for
uniformly rotating (K=0.08) and differentially rotating
(<=0.2S) main-sequence stars for the two limiting ionization
rates, (K and TCR Thne last column, taken from Allen (1973),
gives mean values for observed stars.

m/mo Spectral
type -K .CR

K=0.0S K=0.2S K=0.08 K=0.28

1 G2 2500 710 134 38 2

1.7 FO 2600 740 155 44 100

2.1 AS 2300 660 143 41 150

3.24 AO 2200 630 145 41 190

6.5 B5 2200 630 170 49 230

10 B3 2200 630 180 51 -

17.8 BO 2200 630 203 58 200

40 05 1900 540 186 53 180

EM

o IM

u co

Cr U
Sth 0
v C)i

30
U)\$-i

o d

. -1 .

> Lp
0

r-l
VI I!

U CU

4J 00 En
Cl. 4o
0 ca

0 C) I

o WA
CCrU

OOI
r- I N

c)r I C.

.. ............. ... i -;;;;;-- - I

LO

im N

co c
00 ,

O O + o" 0 CC

0. >
< 0
O -o

S0 0 0

0 0 o 0 0
C(J C 'S
Ltl c~ r0

(1 w1

Angular Momentum of the Protosun

Hoyle (1960, 1963) has estimated the angular momentum of the early
51 2 -1
solar system JESS = 4>101 g cm s- by augmenting the planets up to

normal solar composition. This value is in good agreement with our
51 2 -1
predicted J -- 3x10 g cm s (see Table 2) for a one-solar-mass cloud

ionized primarily by 40K.

The specific angular momentum of the early solar system (ESS) is

compared with that of the present solar system (SS) in Figure 3. The

fact that jESS' lying well above the threshold j necessary for fission,

is comparable to the specific angular moment of close binary systems,

suggests that both (close) binary and planetary systems may be formed

by a similar process (cf. van den Heuvel 1966; Flecl 1977) involving

the rotational instability of a single primary condensation.* Drobyshevski

(1974) has suggested a mechanism for close-binary formation wherein the

convective outer layers of a rapidly rotating protostar are thrown off

fonning a ring in the star's equatorial plane which becomes unstable and

forms a second component. This is very similar to the generally accepted

Kant-Laplace nebular hypothesis for solar-system formation. The criteria

for determining whether the end-product of such an instability will be

planetary of stellar would be of interest. Circumstellar disks commonly

In the past (cf. Brosche 1962; McNally 1965), the angular moment
presently in the solar system (SS) had always been compared with that of
single main-sequence stars. Such a comparison is probably fortuitous
because, as can be seen from Figure 3, a condensation having a specific
angular momentum jSS is not expected to become rotationally unstable. One
must then look for a mechanism, other than the popular nebular hypothesis
wherein a planetary system forms in the equatorial plane of a rotationally
unstable protostar, to account for the formation of a planetary system.

59

associated with Be stars may be the result of a rotational instability,

particularly since these stars rotate ,with near-breakup velocities

(Slettebak 1966).

SECTION VI

CONCLUDING REMARKS

Without rotational braking during star formation, single stars

would rotate with speeds close to that of light. This statement is

actually a reduction ad absurdum; centrifugal forces will halt the col-

lapse perpendicular to the rotation axis long before such speeds are

attained. However, even if the resulting highly flattened system frag-

ments with most of the angular momentum going into orbital motion of the

fragments about their center of mass, multiple star systems (e.g. binnry

stars) would have periods two to three orders of magnitude longer than

the periods typically observed for even the widest pairs.

The present investigation suggests that magnetic torques acting on

a rotating, contracting cloud which is permeated by a frozen-in magnetic

field coupling the cloud to its surroundings, rotationally decelerate a

cloud, constraining it to co-rorate with the background median. Centrifugal

forces are always kept well below gravity. The angular momentum of

magnetically braked clouds is consistent with the observed angular moment

of close binary systems and single early-type main-sequence stars. The

hypothesis of magnetic braking offers support to the fission theory for

the formation of close binary systems, and is able to account for the

relative paucity of single stars. The calculations also suggest a common

mode of formation for (close) binary and planetary systems.

Throughout this investigation, some simplifying assumptions (most

of which are physically justifiable) have been made in order to keep the

iii

-i:rl~ ct*-a ~ blc,. 1n every instance, we have deliberately underestimated

:~Ic 1of:..c y ot magitic braking. Even so, the braking is still able

-' ^=u.' -,;:* tth clouds. to co-rotate nith their surroundings; a fortiori

2 3w r -'cA. tn t ;n~.kw1vsis will reach the- same conclusion. The one single

:.t::r .,- ..? fl,%.tt'lt in determining tAe angular momentum of a protostar

i-s =:; :.1i..... ~qte in dense magnetic clouds: the degree of ionization

zn7rrriu >,q ,x-';lg of a cloud to the galactic magnetic field. For this

--saa.;-, e sltgr:csL discussed in Appenrdix B concerning the fractional

:cnl., II.,n ,i'-'." clouds is in need of further study, particularly the

ed-r; .ar .I:: : t.-cz:luding cosmic rays fcnm magnetic clouds.

\i cA.::.::rl'ns suggest that the toroidal configurations obtained

in a-.N r?:-.x :1 lapse models of Le.son (1972a) and Black and

den. .-,. ': i:, wherein angular mml ntum is consented throughout the

olsa,"i.. ,w ~sa : r appear until the lat-:- pre-nain-sequence evolution of a

pyr.t-..:.. "- -ormation is a complicated process and the (simplifying)

ar.c:; -:. ..a E' made that magnetic fields (and therefore their role in

ri.c~fr :;. '. '--: cani be ignored may n:t he physically realistic. Indeed,

it W -. "n ignore effects of rotation cn the evolution of a

prtr,. -"-.-' .* rk of Larson and Bla-k- and Erodenheimer should be

Y -.;,:. 0 *rrorating the possibl- effects that magnetic fields may

hf,/ 'r5 '" r'"''' r and evolution of a protoc-zar.

APPENDIX A

HEATING AND COOLING RATES IN DENSE CLOUDS

Heating

Cosmic-ray heating and compressional heat generated by the collapse

are the dominant heating mechanisms in dense molecular clouds. Heating

by photo-dissociation of IHt (Stephens and Dalgarno 1973), by photo-

electrons ejected from grains (Spitzer 1948), by photoionization of the

gas (Takayanagi and Nishimura 1960), and by chemical reactions (Dalgarno

and Oppcinheimer 1974) is unimportant in dense clouds because the ultra-

violet photons are mostly screened out. Heat of formation released by

newly formed H2 molecular (Spitzer and Cochran 1973) is unimportant

because of the low neutral hydrogen abundance in dense clouds. Scalo

(1977) has suggested that the action of ambipolar diffusion may generate

an appreciable amount of heat if the magnetic field in a contracting cloud
k3
grows like D-n where k > This mechanism is probably unimportant for
1
our clouds which are characterized by k = y during the initial compression
9
stages and k -- for gravitational collapse. Heating by the dissipation

of (supersonic) turbulence is ignored.

The cosmic-ray heating rate per unit volume is

SCR= cRn (Al)

The cosmic-ray ionization rate per H2 molecule CR is computed in Appendix B

and is given by Eq. (B19). Glassgold and Langer (1973a) give the mean

energy gain per ionization =17 eV. The heating rate by freely

propagating cosmic rays (f=1 in Eq. (B19)) is then
I 2
-28 -7 _r -1 -3
rCR = 2.72x1028 n exp[-1.5l0SSO (rm/m) n ] erg s cm (A2)

A measure of the heat generated by the collapse PF is the ratio of the

thermal energy density of a cloud at temperature T, 2-nkT (k=1.3Sxl0

erg deg-1 is Boltzmann's constant), to its free-fall time t. defined

by Eq. (56). Thus

31 -1 -3
= 2x10 nla' T erg s m (A3)
c

Molecular Cooling

Inelastic gas-grain collisions, and rotational transitions among

the more abundant molecular species, H2, CO, and ID, will cool the gas.

Other perspective molecular coolants such as H2CO (Thaddeus 1972), HiCl

(Dalgarno et al. 1974), and CS and SiO (Goldreich and Kwan 1974) are

probably not important compared to CO and HD. Atomic coolants such as

CI, CII, and 01 (cf. Penston 1970) are unimportant in dense clouds: due

to the attenuation of ionizing ultraviolet radiation, carbon is mostly

neutral when n>104 cm (Werner 1970), and atomic carbon and oxygen are

depleted by chemical reactions in dense clouds (cf. Allen and Robinson

1977). Furthermore, the cross section for collisional excitation of the

CI and CII fine-structure levels by H2 is much lower than for atomic

hydrogen.

The energy radiated per unit volume per second in a transition

between states u and is

n U E (A4A)u,

(A4)

where n is the number of r..lecules cm-3 in state u, EuE is the energy

difference between u and Z, Au is the Einstein A-coefficient (i.e.

the transition probability per unit time for spontaneous emission), and
-TEu
= 1-e (AS)
u19 TLu

is the photon escape probability. The optical depth in a line having a

rest frequency v, and a thermal Doppler width (Mihalas 1970)

v = (2kT (A6)

is given by (Penzias 1975)

T e A N 1 exp(- .) (A7)
Ili srgz vvLL kljI

In these equations, A is the molecule's atomic weight, gu and gE are,

respectively, the statistical weights of levels u and Z, NI-nR/2 is

the column density of molecules in state Z through a mean path length

R/2, R being the cloud radius, c=3.0x1010 cm s is the speed of light,

and h=6.626x10-27 erg s is Planck's constant. In Eq. (A7), the term in

brackets is the correction for stimulated emission.

Under steady-state conditions, the relative population of levels is

given by

n (g/g~,) xp(-E,,/kT) (AS)
nz 1+(A u/Cud)

-where

C = n< v> if < 1
u uV kT kl

= n if > 1 (A9)

is the collisional de-excitation rate, a u being the collisional

de-excitation cross section, and B is the molecule's rotational constant.

The cross sections are averaged over Maxwellian velocity distributions

to obtain rate constants for rotational excitation. For simplicity,

we assume that ao = au (Eo), and furthermore, that the rotational levels

are excited (and de-excited) by collisions with H2, the most abundant

molecular species in dense clouds. Thus, the term in brackets in the

denominator of Eq. (AS). is a measure of the deviation from thermal

equilibrium: at higher gas densities, collisional de-excitation dominates

spontaneous emission so that the levels become thermalized, and Eq. (AS)

reduces to the Boltzmann distribution. 1Mien level populations become

thermalized, the cooling rate which is proportional to n" at lower

densities, goes like n [see Eq. (A4)] The cooling efficiency is thus

reduced at high densities, and the cooling is said to be 'collisionally

quenched.'

The rotational levels of a diatomic molecule are indexed by the

rotational angular momentum quantum number J. The level J has energy

E = hBJ(J+l) (A1O)

and the level degeneracy is

g = 2J+ (All)

The energy separating levels J and J-1 (corresponding to an electric

dipole transition governed by the selection rule AJ=-l) is

S-EJ-1 = 2hBJ (A2)

For thermalized levels, the population of the Jth level is, from Eq. (A8),

n = n (2J+1)exp[-J(J+l)hB/kT] (A13)

where n is the population of the ground state. The total number

density of a particular molecular species is

"T = "J
J=0

= n C (2J+l)exp[-J(J+l)hB/kT] (A14)
SJ=0

For kT>>hB as is the case for all molecules considered here,

nT = no J (2J+l)exp[-J(J+l)hB/kT]dJ

= T (CA15)
o -1

Using this result to eliminate n in favor of n in Eq. (A13), we have

S= n (2J+1) exp[-J(J+)hkT] (A16)

The cooling rate per unit volume for electric dipole transitions is,

after Eq. (A4),

A = jn EJ,J-_AJJ-ji (A17)
J=l

with n EJ,J- and j defined by Eqs. (A16), (A12), and (AS), respec-

tively, and

A 5124 8B u J (A18)
J ,1 3hc (2J+1)

where pJ is the molecule's electric dipole moment.

For lines which are optically thick, cooling occurs from the surface

of a cloud at a rate per unit volume given by

3nB cdv
As Rn (1-j) (A19)

where nj/n is given by Eq. (A16) and the photon spectral energy distri-

bution for T>> is given by the Planck function,

3
S 2hv3 dv (A20)
B^-' = --2- hv/T (A20)
c e -1

For thermal line broadening, dv is given by Eq. (A6).

Molecular Hydrogen

The H2 molecule is a homonuclear diatomic molecule and therefore

has no permanent electric dipole moment. For low temperatures (T<100 K) ,

hydrogen molecules are predominantly in the ground rotational state, so

that cooling occurs mainly via the electric quadrapole J=2-0 transition

for para-112. At such low temperatures, para-ortho collisional conversion

is very slow. Because of the long radiative lifetimes of the excited

levels, the level populations for gas densities n> 100 cm3, are given
-14
by the Boltzmann equation. For the J=2-)0 transition, E20 = 7.07x101 erg

so that

n2 = Sn e12/T (A21)

The cooling rate by H2, assuming no=nH=n is then

A =2
AH2 = n2E2A2020

-23nB -512/T -1 -3
= 1.04x10 nB 12 erg s cm (A22)

where we have set A20 = 2.947x101 s (Thaddeus 1972). The photon

escape probability is related to the optical depth, as defined by Eq. (A7),

through Eq. (A5). For H2,

1 2
-..'J0 5T (m/mn)3 n (A23)

use /."-"i=3m/47R3 to eliminate R in favor of n and m, the
.,'"i, S cooling follows from Eq. (A19):
1 1
-19 3 -3 -512/T -1 -3
1. = T.'"/; .19T n3 (m/mn)3 eS1/ 1-B2) erg s1 n-3. (A2.1)
'(12

,- 1 tie >' temperatureses characterizing dense clouds, 12 cooling

...F:ct. .9 very efficient.

S,. io.f.-rs': of HD as a molecular coolant in dense clouds was

,,.ustcr ;t,/ 'iDalgarno and Wright (1972). Due to vibronic inter-
-22
i t, ;': 0' has a permanent dipole moment p=5.85<10 esu cm.

.IL 1K on,/ It- first two excited rotational levels need be considered.

S,,,oling /a;''c per unit volume is then

S ,) J J-1A J,J- (A25)
J4-

,..:h) and iri;-ht give E21 = 3.54x10-14 erg, E10 = 1.7710-14 erg,
S-. -8 -1
-x/10"7 and = 2.54Y10 s-. The levels are thermalized

vely I densities, and no -ntl for Ts65 K, as can be seen from

.;'. 'T Ill) cooling rate is then

-21 -129/T -19 -385/T -1 -3
( II)n().55x10 1Be +1.28x10 B2e ) erg s an ,
(A26)

.;'-lln /n~Jl12 is the fractional abundance of HD relative to H2.
N ll 11- 2
"'. 11:1) ; ratio is 20,000:1. Accordingly, we take x(ID) = 510.

""* of 111:112 by Spitzer at al. (1973), and DCN:HCH by Wilson

"3) su:.t.st that x(HD) may be two orders of magnitude larger.

However, we follow the ideas of Watson (1973) and assume that these

ratios reflect chemical fractionation rather than true isotopic

abundances. The photon escape probabilities are determined by the

optical depths which follow from Eq. (A7):

_1 1 2

1 J 2 -129/T
T2(HD) = 2.78x(HD)T (m/m ) n c 129/ (A27)

The surface cooling rate for x(HD) = 5::10 follows from Eq. (A19), and

is given by

s -22 3 -129/1' -256/T -1 -3
A = 3.46x102 (m/m) e 29 [(1-61)+79c (1- 2)] erg s an
(A28)

Carbon Monoxide

The most abundant heavy molecule in dense interstellar clouds is CO.

Because only 5.5 K separates the first excited rotational level from the

ground state, CO is a potentially efficient coolant in the relatively

cool (T=10 K) environment of molecular clouds. Indeed, Glassgold and

Langer (1973b) have shown, neglecting optical depth effects, that the

low temperatures [T=10 K; see Zuckeiman and Palmer (1974) for references]

typically observed in dense molecular clouds can be maintained by CO cooling

alone.

The CO cooling rate follows from Eq. (A17):

CO= J n AJ-I Aj- (A29)

where the summation is discontinued at the first level (J=J ) where the

collisional de-excitation rate is less than the spontaneous transition

rate. The cross section for rotational excitation of CO by H, is

-15' 2
o=10 cma (Green and Thaddeus 1976). Because of its low dipole
-19
moment (,=1.12x101 esu cm) and high abundance, the CO molecule

thennalizes at low densities. Using Eqs. (A16), (A12), and (A18),

Eq. (A29) becomes

r J
7. 76 nB m S5 -1 -3
CO = 7.410 x(CO) T J exp[-J(J+l)hB/kT] erg s cm
J=1
(A30)
17 -5
We adopt a fractional abundance x( 'CO) = 3x10 for the main isotopic

species, and x(13CO) = 3.4x10-7 for the less abundant species; the adopted
1'
isotopic ratio is the terrestrial value 89:1.- For "CO, 1=57,700 M-z,

and for 1CO, 6=55,100 MH1-. The optical depth in a line arising from a

transition where J-J-l is, from Eq. (A7),
3 1 2
TJ(CO) = 7.57x0 6x(CO) JT (m/m ) n exp[-J(J+1)hB/kT]

*[1-exp(-J(J+l)hS/kT)] (A31)

Surface cooling in the optically thick lines occurs at a rate

s -82 85In- J
ACO = 5.62x10 8 (2J+I)(1- )
S2J-1
T2(m/m )3

Sexp[-2J(J+1)h5/kT]l eg s-1 c-3 (
r exp[ l(Ji)hB/TI:j I erg s cm (A32)
1 -exp [ -J(J+ 1) h/kT]

Grain Cooling

.Cooling by inelastic collisions of grains with the gas particles

(mostly molecular hydrogen) at temperature T occurs at a rate (Dalgarno

and McCray 1972)

3- -1 -3
A = 2x10 nT2 (T-T )0 erg s cm (A33)
g g

I ii I ....................

where T is the temperature of the grains, and 0 is the energy accom-
g
modation coefficient, and is a measure of the elasticity of the collision;

a is unity for a completely inelastic collision, and becomes zero for

elastic collisions. We take 0=1. Because of the strong density dependence,

grain cooling is expected to dominate molecular cooling at high densities.

The grain temperatures can be determined from the energy balance of a

grain (Low and Lynden-Bell 1976):

4 8kT 4
aT Q pTg) = nk(T-T )( + 1 T bQp(Tb (A34)

-5 -4 -1
where a=5.67x10 erg cm deg s is the Stefan-Boltzmann constant,

Q (T) is the Planck mean absorption efficiency, nm2 is the mass of the

hydrogen molecule, and Tb=2.7 K is cosmic blackbody radiation temperature.

The first term represents the heat loss due to radiation of a dust grain

at temperature T The second term is the collisional energy gain from

the gas, and the last term represents heating from an isotropic blackbody

radiation of temperature Tb. Eq. (A34) assumes that the cloud is

shielded from the external stellar radiation field, and that there are

no embedded stars within the cloud which might heat the grains to a

temperature T >T, in which case the grainsheat the gas (Leung 1976).
g
Kellman and Gaustad (1969) give Q (T) = 4.1x10-7 T2.67 for 0.2 micron

ice grains.

The optical depth through a cloud of radius 1 for the thermal

radiation from the grains is

T (T ) = Q (T )on R (A35)
g g p g g g

By eliminating R in favor of n and m, the mass of a uniform, spherical

cloud, and by taking gn = 6x10-22n, and Q (T =10)=10', the optical depth

-71
T (Tg=10) 10 7((m/m)n3 ... (A36)

For stellar-mass clouds, T >1 for densities n10 10 an which is in
g
good agreement with the results of Hattori et al. (1969). Since densities

encountered during the initial collapse stages are much less than this,

the thermal radiation from grains is assumed to pass freely out a

contracting cloud.

Cloud Temperature

Figure Al illustrates the temperature and density dependence of the

various heating and cooling rates described in this Appendix. The total

molecular cooling rate I =A I A HD CO As expected grain cooling com-

pletely dominates for n210 cm ". Even if gas-grain collisions are only

weakly inelastic (0O0.01 say), grain cooling will still dominate at high

densities. Grain temperatures ranged from T =6 1K for T=10 K up to

T =35 K for T=100 K. Gas temperatures much in excess of 20 K are rarely
g
encountered in dense clouds; temperatures up to 100 K are considered

merely to illustrate the rapid rise in the molecular cooling rates at

these relatively high temperatures.

Surface cooling for all molecules never amounts to more than 10% of

the total molecular cooling rate, and cooling by molecular hydrogen becomes

noticeable (-10% of the total molecular cooling) only for the highest

temperatures. Carbon monoxide dominates the cooling at T=10 K for all

densities. However, except for n=10 cm3 where A HD AC hydrogen deuteride

is by far the dominant molecular coolant for T220 K, its cooling rate

becoming three orders of magnitude greater than CO cooling for the

highest temperature.

It appears that cloud temperatures equivalent to those typically

observed in dark clouds (T=10 K; see Zuckcnnan and Palmer (1974) for

C a)
tf r-4 .rc v-4
I II 4

) C) k

U (i vi .0

vI or;
0 0 P

cr di r
*4 4 0 0 U'0
I *i-n C
En 0 i o

4- 0 0

U O *

Li C i 0"

% 0 I O
or u r-4
,0 1
cO

Socl O C) :
(L)0 97 -Z

UE U 0 C
rU 0 I

'H 4H' t4-
4 O -
O o < r- '

U o i- 11 r30
'o 5 o C O
0 n 0 '
S. 0
o

-- -- ,---I- g

0 )

!

D \
;I \

-"" \O
I

O
O

1CQ
EI I P
D E ] E 01
oo

S1-

rC \ c0

I H

C N C

(zrJ\' 2.S 9l3) *I'V 001

references) are certainly attainable at low gas densities, and should

prevail for higher densities as we'll, provided grain cooling operates

with at least a one-percent efficiency. The rapid rise of the molecular

cooling rate at.higher temperatures indicates that a cloud will collapse

approximately isothermally near some equilibrium (F=A) temperature
9 -3
T =10-20 K at least for densities n<10 am Molecular cooling becomes

even more efficient if large 'velocity gradients (e.g. due to cloud collapse)

develop in a contracting cloud (Goldreich and Kwan 1974; de Jong et al.

1975). Furthermore, as is shown in Appendix B, the cosmic-ray ionization

rate (and therefore, the heating rate) adopted here is probably greatly
overestimated for densities n106 -3
overestimated for densities n10 an --

cAI tCl'.Tf cy IWI:AT 1 \ I :? M '' I,.(ill'

., ,I.eP v of iO~ni:.iLttn *:. 4 i *.-'t'ils tiid clI1d controls lhe

Cd ., 1. When the fractional
c; .Hlllit ,I* I!},, Cl^u t. t.. i ;.l', I; s ,," l

Lt i i. diffusion is

N l tn v i II cloud having

*l.. j 's ~-, e (.l t : ,ccor i i, t 'I'i h nllllli" d Dalgarno

s

S. (Bl)
& ?. t ". h )' *; *,-t )n *- 1 0:11 )n
c c i
," ,md hleav metal
,At, ,, ,< ". .. .* .t thy de.s tt.. of no il ""1- ll llc

Sr c he rate
A +
*i .' d -... -,- rrc .rl .a-.ion of :;iolc ilI' ions m ,

...." o, .,. ,, m ct a i ions a. ind lit mi. l i action on

S ; SL- r 1 -*.;,r i. 1 s It.', id th t11 ;ll rate ,I .n fficients are

... t"- 1i c.,iily :;hown I1;:: for lden:;ities

S' rj* r ; : ; en Er.:i .ur-faces Id'; 3atc:. ,Idli t ive recom-

*'. .. :;,r-,. o.n ithe rigrht-i: -sidtk of Eq. (Bl) can he
a '" '*: -T c '.'il iriui, i '. (1: then rI ives

I
r.

.... *t

(83)

--AN

I
I II !ll~J

The equilibrium rate equation for the molecular ion density gives

(Oppenheimer and Dalgarno 1974)

n(m+) n~ (B4)
an 4 n (M)

where n(M) is the total density of heavy neutral atoms (e.g. Mg, Ca, Na,

and Fe) that undergo charge transfer with molecular ions, and ( is the

rate coefficient for the charge-transfer process. We neglect associative

ionization (Oppenheimer and Dalgarno 1977) which is probably an insig-

nificant source of electrons in cold interstellar clouds.

Putting Eqs. (B2) and (B4) into Eq. (B3) gives a quadratic equation

for n which has the solution
e
un(D+ 1 n 0 2 4C 1/2nM)
c 2 + 2 +n (BS)
g

Following Oppenheimer and Dalgarno (1974), we define a depletion factor

6 by the expression

n(M) = 4x10-5 n (6)

Observations with the Copernicus satellite (Morton at al. 1973; Mortor

1974, 1975) suggest an average depletion factor for heavy metals, 6=0.1.
-9 3 -1 -6 3 -1

-17 3 -1
a =10 cm s Eq. (B6) then becomes
r

n = 2x109n [1+ 0 ] 1 (B7)
c n

so that for densities n>>1026,

n = 1017E (B8)
e

Ions in dense molecular clouds are supplied primarily by the

ionization of molecular hydrogen by cosmic rays and 40K radioactivity;

ultraviolet radiation (Werner 1970) and X-rays (Nakano and Tademaru
3 -3
1972) are screened in the peripheral regions of dense (n>10 cm-3

clouds. Cameron (1962) has estimated the ionization rate by the 8-decay

of 40K radioactive nuclei to be

-21 -1
Ek = 1.4x10 s (9)

Nakano and Tademaru (1972) have calculated the ionization rate of atomic

hydrogen by cosmic rays. Adjusting their result for ionization of

molecular hydrogen by taking into account the difference in ionization

cross sections, OH2=1.65oH (Bates and Griffing 1953), the cosmic-ray
2 *
ionization rate in a cloud having a mass m is given by

CR = 1017 exp[-1.54x]0 (m/mr )3n3] (B10)

The exponential term reflects the attenuation of cosmic rays due to

their interaction with matter as they propagate through a cloud. The

total ionization rate

+= CR +k (Bll)

By comparing Eqs. (B9) and (B10), one can see that for stellar-mass clouds

having densities n>10 -1012 cm cosmic rays are effectively screened and

ions are produced primarily by in situ 40K nuclei.

Brown and Marcher (1977) have shown that ionization of IH and 1, in
dense clouds may be enhanced by energetic secondary electrons produced by
knock-on collisions, neutron-decay reactions, and pion-decay reactions,
iri*c-e, following the interaction of fast (primary) cosmic rays with the
material in a cloud. This effect, which may be significant only when lo'w-
energy cosmic rays are excluded from dense clouds, is difficult to estimate
quantitatively, and is therefore neglected here.

Eq. (BI0) may greatly overestimate the cosmic-ray ionization rate

in dense magnetic clouds. If the magnetic field within a contracting

cloud becomes tangled (e.g. by turbulence), cosmic rays, constrained to

move along the magnetic lines of force, random walk through the cloud

and must traverse more matter to reach the central regions of the cloud.

Nakano and Tadema-u (1972) have estimated the importance of this effect,
3 -3
and have concluded that for densities n>10 cn cosmic rays are

effectively screened, i.e., SCRKCk so that C=-k

When cosmic rays stream along magnetic fields in a collisionless

plasma faster than the Alfy6n velocity, they generate hydromagnetic waves

which in turn scatter the cosmic rays (Wentzel 1974, and references cited

therein). Indeed, Skilling and Strong (1976) have shown that the incoming

cosmic-ray flux in a dense cloud may be substantially reduced by this

mechanism. However, the damping effect of ion-neutral collisions may

inhibit the generation of such waves by the cosmic rays themselves

(Kulsrud and Pearce 1969).

As pointed out by Nakano and Tademaru (1972), cosmic rays, although

not unstable to the generation of hydromagnetic waves, are strongly

influenced by the presence of such waves generated by other mechanisms

(e.g. by magnetic braking during cloud collapse). Cosmic rays are

scattered by these waves and can not freely stream along the open magnetic

field lines. Nakano and Tademaru have shown that effective screening of

the cosmic rays occurs if

2 101 B
<6B > > 1 (B12)
2 1

where <6B >/8rr is the energy density of the hydromagnetic waves having

amplitude 6B, B is the strength of the cloud's large-scale uniform field,

and p=3m/4nR3 is the mass density of the cloud material. By eliminating
-2
p in favor of in and R, and by defining n-R/R with B=B n (see Section III

and IV of the text for a discussion of the rate of growth of the magnetic

field in a contracting protostar), with subscripts denoting initial values,

this expression becomes

8x10" 'R B
<6B2 > oo (B13)
(m/mo)n
12
From Table 1 in the main text, we see that R B = constant- 9x1012 cm gauss
oo
for all cloud masses. Thus, if

-11
<6B2 7x10 (B14)
(m/me)n

cosmic rays are effectively screened.

The energy lost from a magnetically braked, collapsing cloud is just

the difference between the cloud's rotational kinetic energy given by

angular momentum conservation, and its magnetically-braked rotational

energy. For a cloud which is constrained to co-rotate with its sur-

roundings, this energy is given by

2 (8)
E lost 0.2Rn2 R R (BI5)

Assuming that this energy is carried away by the hydromagnetic waves which

are generated by the braking process, we can write

<6B2 >- 2x101 n (B16)

23
were we have taken initial values from Table 1, noting that o R /(m/m )
00 0
Constant= 10 cm s -2. Actually, depending on the mass of the cloud,

<6B2> may be an order of magnitude smaller or larger, so that Eq. (B16)

represents an approximate mean value. A comparison of Eqs. (B14) and

(B16) shows that cosmic rays are effectively excluded from a contracting

cloud very soon after the collapse begins.

It is for these reasons that we believe the ionization in dense

magnetic clouds is determined by the 40K ionization rate, i.e., ,=k .

This being the case, the magnetic flux linking a contracting cloud to

its surroundings, uncouples at relatively low gas densities. This may

explain the absence of large magnetic fields in some dust clouds

(Cnitche.r et al. 1975).

Even if the cosmic rays are not magnetically scattered, there will

be a reduction in the flux of cosmic rays in a magnetic cloud. The

magnetic field lines in the neighborhood of a contracting cloud diverge

outward from the cloud so that charged particles streaming along the field

lines into the cloud will be reflected by the 'magnetic mirror effect.'

Fermi (1949, 1954) proposed that such a magnetic reflection mechanism

might explain the origin of the galactic cosmic rays. For slow variations

of the magnetic field in time and space, the diamagnetic moment of a

charged particle is an adiabatic invariant. Let 6 be the angle between

the direction of the line of force and the direction of motion of the

spiraling particle, viz., the pitch angle. Assuming an isotropic distri-

bution of particle velocities in a region where the field strength is B,

one can easily show (cf. Spitzer 1962) that the velocities must fall

within a solid angle defined by the pitch angle such that

6 = sin- (B/B )2 (B17)

when the field strength increases to B In our case, B is taken to be

the (uniform) galactic field far from a contracting cloud, and B is the

field strength at the cloud surface. Assuming that the particle density

in a given region is proportional to the size of the solid angle given

by Eq. (B17), it follows (cf. Kaplan and Pikelner 1970) that the
fractional decrease in cosmic-ray flux is

f = 1-cose (B18)

lhe cosmic-ray ionization rate is then given by

CR CR (19)

where (CR is given by Eq. (B10). Eq. (B19) provides a workable upper-

limit to the ionization rate in dense magnetic clouds, the lower-limit

determined by Ck being the most likely for reasons already discussed.

Eq. (B17) is valid only if the Lannor radius

T 3.1310 I? [+ Ek ?_1
rL I+ 9--)2-1 a (B20)

of a cosmic ray having a kinetic energy Ek(MeV), and moving in a magnetic

field B microgausss), is much less than the radius of the cloud. One

finds rL(2MeV)=3.4xl010 cm and rL(10 GeV)=6.1>:1012 cm, so that in fact,
I dB -l
rL<
the Larmor period. It is easily demonstrated that this condition is

equivalent to the dynamic time scale being much larger than the Larmor

period, or
1.4(E +938)
tf >> B -s (B21)

Since collapse times are typically 1013-1014 s, this condition is easily

satisfied. Finally, Eq. (B17) neglects collisions among the cosmic rays

which have the effect of randomizing the pitch angle 0. We require

that the time between collisions be much greater than the Larmor period,

or equivalently, that the density of material (primarily molecular

hydrogen) in a cloud satisfies:

-14
5.lxlO 1B Ek 2
n << [(1+ 93) -l] (B22)

where o is the collision cross section. Low-energy cosmic rays (Ek=2)

interact with the cloud material primarily by ionizing molecular

hydrogen. Classgold and Langer (1973a) give oc(2MeV)=l.89xlO17 cm2.

High-energy cosmic rays (Ekl103) interact mainly by p-p scattering and
-26 2
pion-production reactions with a cross section o(10Ge\V)=210 6 cm (see

Nakano and Tademaru 1972 for references). Thus, from Eq. (B22),

n(2MceV)<<4x104 B and n(10GeV)<<2x104 B Since initial values for B

range from 3 up to 103 (see Table 1) and B increases as n2/3 during

gravitational collapse (see Section IV of the text), these conditions are

easily satisfied.

APPENDIX C

MNmENT OF INERTIA FOP. DIFFERENTIALLY
ROTATING MAIN-SEQUENCE STARS

The angular momentum of a rotating spherical body having a radius R,

a radial density distribution p(r), and an angular velocity field u(r,O)

is

J = o- 2w(r,e)p(r)r4sin6d6dr (C1)
3 0 J0

Assuming w(r,0)=w(r), this becomes

J = 8- R L(r)p(rr4dr (C2)
3 J0

The actual distribution of mass throughout chemically homogeneous stars

which are not completely convective often approximates that in the

'standard model' of Eddington (1926), which is just a polytrope of n=3.

For a rigidly-rotating star, w(r)=constant=uR, the angular velocity at

the surface (r=R). The integral in Eq. (C2) is then most easily

evaluated with the aid of the Emden solutions for an n=3 polytrope

(see, for example, Chapter 23 of Cox and Giuli, 1968). Writing the

angular momentum in terms of the moment of inertia I=
gyration constant, we have

2 R (C3)

so that for the case of rigid rotation, we find

K 0.08 (C4)

The pre-main-sequence evolutionary models of Bodenheimer and

Ostriker (1970) for rapidly rotating massive stars predict a marked

differential rotation, with the central angular velocity we being a

factor of ten greater than wR. Their differentially rotating confi-

gurations are stable, according to the criterion developed by Goldreich

and Schubert (1967). From Figure 5 of Bodenheimer and Ostriker (1970)

we approximate the angular velocity as

-2.3xa
w(x) = 10 WRae

where xr/R, and
(CS)
a = 1.4 for 0 < x < 0.3

= 1.1 for 0.3 < x 5 1.0

The mass distribution for a polytrope of n=3 can be approximated as

-bxc
p(x) = pop e

where
(C6)
b = 20 and c = 2 for 0 < x 0.3

b = 11 and c = 1.5 for 0.3 5 x 5 1.0

Here, p = 3m/47R and p (n=3) = 54.18 g on3. Substituting the above

expressions for w(x) and p(x) into Eq. (C2), and evaluating the integral

using Simpson's Rule, we find

K = 0.28 .

(C7)

86

The gyration constant (and therefore, the angular momentum) of the

differentially rotating configuration is thus three times that of a

rigidly-rotating body.

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Alfv'n, H. 1942, Ark. f. Mat., Astr. och Fysik, 2SA, No. 6.

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Alfvyn, H., and Arrhenius, G. 1976, Evolution of the Solar System
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Allen, C.W. 1973, Astrophysical Quantities (3rd ed.; London: Athlone Press).

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Appcnzeller, I. 1971, Astr. Ap., 12, 313.

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Bates, D.R., and Griffing, G. 1953, Proc. Phys. Soc. London, A, 66, 961.

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Berkhvijsen, E.M. 1974, Astr. Ap., 35, 429.

Biermann, P., Kippenhaln, R., Tscharnuter, W., and Yorke, H. 1972,
Astr. Ap., 19, 113.

Blaauw, A. 1961, Bull. Astron. Inst. Neth., 15, 265.

Black, D.C., and Bodenheimer, P. 1976, Ap. J., 206, 138.

Bodenheimer, P., and Ostriker, J.P. 1970, Ap. J., 161, 1101.

Brandt, J.C. 1966, Ap. J., 144, 1221.

Bridle, A.H., and Kesteven, M.J.L. 1976, A.J., 75, 902.

Brosche, P. 1962, Astr. Nachr., 286, 241.

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Cameron, A.G.W. 1962, Icarus, 1, 13.

Cameron, A.G.I'., and Truran, J.W. 1977, Icarus, 30, 447.

Carruthers, G.R. 1970, Ap. J. (Letters), 161, L81.

Chandrasekhar, S. 1961, Hydrodynamic and Hydromagnetic Stability (Oxford:
Oxford University Press).

Clark, F.O., and Johnson, D.R. 1974, Ap. J. (Letters), 191, L87.

Cohcn, N., and Kuhi, L.V. 1976, Ap. J., 210, 365.

Cox, J.P., and Giuli, R.T. 1968, Principles of Stellar Structure
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Crutcher, R.M., Evans, N.J., Troland, T., and Heiles, C. 1975, Ap. J.,
198, 91.

Dalgarno, A., de Jong, T., Oppenheimer, M., and Black, J.H. 1974,
Ap. J. (Letters), 192, L37.

Dalgarno, A., and McCray, R. 1972, Ann. Rev. Astr. and Ap., 10, 375.

Dalgarno, A., and Oppenheimer, M. 1974, Ap. J., 192, 597.

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Dicke, R.H. 1964, Nature, 202, 432.

Dickman, R.L; 1975, Ph.D. dissertation, Columbia University.

S1976, in preparation.

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Drobyshevski, E.M. 1974, Astr. Ap., 36, 409.

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Full Text

PAGE 1

MGNETIC BRAKING DURING STAR FORMATION By ROBERT GVJy.ES FLECK, JR. A DISSERTATION PRESENTED TO TIE GRADUATE COUNCIL OF THE UNIWillSlTY OF FLORIDA IN PAliriAL FULFILLNft-AT OF TrlE T^QUIRBENTS FOR IKE DEGREI: OF DOCTOR OF PHILOSOPHY UI^mERSIlT OF FLORIDA 1977

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.^NIVERSITY OF FLORIDA aJiiifli

PAGE 3

To Sherr>' . . , and my Parents

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We are Stardust . . . Joni Mitchell, V\x)odstock

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ACKNOiVLEDGMENl^S Sincei-e thanks go to ray advisor, Professor James H. Hunter, Jr., who first introduced me to the topic of star formation, and has since guided me through my MS thesis as well as the present dissertation. His friendship and scientific counsel are most deeply appreciated. Thanks also to the other committee members, Professors Edward E. Carroll, Jr., JCwan-Y. Chen, and Charles F. Hooper, Jr., and Drs. Jean-Robert Buchler and Hugh D. Campbell, whose comments throughout the course of this work greatly improved both its content and style. Special thante to Dr. Jean-Robert Buchler for teaching me the "physics" of astrophysics, and for sponsoring my tenure as a graduate research student during the 1975-3976 academic year. I thank Drs. Frederick W. Fallon, Robert B. Dickman, Stephen T. Gottesman, Robert B. Loren, and Telemachos Ch. Mouschovias for privately communicating to me their thoughts on some of the topics treated here. I wish to express my appreciation to Professor Heinrich K. EicMiomvon \Vurmb for "looking out" for me while I was a gi-aduate student at the IMiversity of South Florida, and to Professor Frank Bradshaw Wood for doing the same during my tenure at the University of Florida. Their friendship and advice shall always bo remembered. Special thanks to Professor Wood, wlio along with Professor Kwan-Y. Chen, provided me \vlth a research assistantship for part of the 1974-1975 and 197C-1977 academic years.

PAGE 6

I thank the State of Florida for sponsoring me as a graduate teaching assistant at the University of South Florida from 1972-1974, and at the University of Florida during my final two quarters of residence. Computer time was donated by the Northeast Regional Data Center of the State University System of Florida and is gratjsfully acknowledged. I Vkdsh to thank the Nato Advanced Study Institute Program for their most generous support which enabled me to attend the conference on the Origin of the Solar System held during the spring of 1976 at the University of Newcastle upon Tyne, England. Discussions with other participants lielped to refine some' of the ideas presented in this work. I thank my v/ife, -Sherry, for typing the various drafts of this thesis. Thanks also to Beth Beville for her diligent and accurate typing of the final draft. Most importantly, I express my deepest appreciation to my wife, Sherry, and to my parents for their much needed support and encouragement over t}\e years. Thank God for the many super weekends spent camping on the beach and surfing in Cocoa: without the welcome diversions from my work provided by family, freinds, and The Sea, I might not have lasted. Finally, thank God for collapsing interstellar clouds, from which we originated. Thank God it's over.

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TABLE OF CONTENl'S PAGE AQOsmiEDGMEm'S iv LIST OF TABLES viii LIST OF FiaJRES ix ABSTRACT x QiAFTER I . INTRODUCTION 1 Angular Momentum Problem 1 Present Work 8 II MAGNETIC BRAKING 10 Magnetic and Velocity Fields . . .^ 10 Torque Equation * 12 Toroidal Magnetic Field 14 Rotational Deceleration 19 III STAR FORfÂ«i\TION ., 23 Shockinduced Star Formation 23 Themial Instabilities 25 Physical Conditions in Dark Clouds 27 Initial Conditions for Collapse 33 IV MAGNETIC BRAKING OF COLLAPSING PROTOSTARS 38 V DISCUSSION OF RESULTS: CO^PARISON WITH OBSERVATIONS 49 Specific Angular Momenta of Single and Binary Stars .... 49 ' Rotation of Mainsequence Stars 53 Angular Momentun of tlie Protosun 58 VT CONCLUDING RH-l^RKS 60 APPENDIX A: HEATING AND COOLING RATF-S IN DENSE CLOUDS 62 Heating ^^ Molecular Cooling 63 Molecular Hydrogoi 67

PAGE 8

PAGE Hydrogen Deuteride 68 Carbon Monoxide 69 Grain Cooling , 70 Cloud Temperature 72 APPENUIX B: FRACTIONAJ. IONIZATION IN DENSE. MAGNETIC CLOUDS 76 /^PEMDIX C: VaTXT OF INEFITIA FOR DIFFERENTIALLY ROTATING MAIN-SEQUENCE STARS 84 LIST OF REFERENCES 87 BI(>3R>\PHirAL SKETOI 97

PAGE 9

LIST OF TABLES ,\BLE ' PAGE -3 1 Initial values of particle density n (era ) , cloud radius Rq (cm) , surface magnetic field strength 'Rq/Bq , and angular velocity Uq (s'l) for various cloud masses m/m_ marginally unstable to gravitational collapse. Numbers in parentheses are decimal exponents ^^ 7 -1 2 Cloud radius Ry (cm), angular momentum J^ (g an"^ s ), and poloidal surface magnetic field strength \i (yG or mG) at the uncoupling epoch for ionization by ^^K only (E. ) and ionization by cosmic rays (?c]^) Â• Numbers in parentheses are decimal exponents ^^ 3 Predicted equatorial rotational velocities v^ (km s ) for uniformly rotating (k=0.08) and differentially rotating (k=0.28) main-sequence stars for the two limiting ionization rates, Ck ^nd ^q-^. Tne last column, taken from Allen (1973) , gives mean values for obseived stars

PAGE 10

LIST OF FIGURES HGUT^ ' PAGE 1 Observed magnetic field strength B (gauss) in interstellar clouds and OH maser sources as a function of their particle density n (cnr^) . Bq=3 microgauss is the strength of the large-scale magnetic field of the Galaxy 31 2 -1 2 Specific angular momenta j (cm s ) for binar>' systems (visual, spectroscopic, and eclipsing), single main sequence stars, and the solar system (SS) . Also shoun is tlie specific angular momentum predicted for the two limiting cases of 40x (j^) and cosmic-ray (jcr) ionization, as vrell as j for the case of angular momentum conservation 45 3 Specific angular momentum (cm s ) for the K ionization rate j^^, and threshold angular momentum necessar)' for fission, designated by R and BO. Also shcv.Ti are the specific angular momenta for a number of eclipsing binary systems. The location of ESS represents the specific angular momentimi of the early solar system, and SS designates its present value 52 4 Predicted equatorial rotational velocities v^ (km s ) for uniformly (k'-O.OS) and differentially (k=6.28) rotating main-sequence stars for the cosmic-ray ionization rate, ^Qj^. Also sho^vn for ccanparison are mean values of v* for , obseived stars 57 Al Cosmic-ray (Fj-j^) and compressional (r^-) heating rates (erg s-1 cm" 3) ^ arid molecular (Aj^^ and grain (Ao) cooling rates (erg s"l cm"3) as a function of kinetic gas^tenperature T (K) for gas densities (a) n=105, (b) n=106, and (c) n^lO^ cm~3. For A^\, the solid line represents the molecular cooling rate for a one-solar -mass cloud, and the broken line is the cooling rate for a 40 solar -mass cloud 74

PAGE 11

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfilljnent of the Requirements for the Degree of Doctor of Philosophy MAGNETIC BRAKING DURING STPSi FORMATION By Robert Charles Fleck, Jr. August 1977 Chairman: James H. Hunter, Jr. Major Department: Astronomy Angular maiientum is prima facie a formidable obstacle in the theoiy of star formation: without rotational braking during star formation, stars would rotate with speeds close to that of light. The present investigation suggests that magnetic torques acting on a rotating, contracting, cool interstellar cloud wtiich is permeated by a frozenin magnetic field coupling the cloud to its surroundings, rotationally decelerate a cloud, constraining it to corotate with the background median. Angulaimomentum is thus efficiently transferred from a collapsing cloud to its surroundings. We examine angular momentum transfer from cool, rotating, stellar-mass condensations, collapsing isotherrnally ora magnetically-diluted dynamic time scale. Some mechanisms are discussed for forming gravitationallybound protostellar condensations within cool, dense, molecular clouds. Rotation induces a toroidal magnetic field and the accompanying magnetic stresses generate a set of Alfven waves which pixjpagate into the background medium, thereby transporting angular momentum fran a cloud to its surroundings. A modified virial approach is employed to calculate timedependent quantities of interest at the cloud's surface in order to Â• estimate the braking efficiency of the magnetic torques.

PAGE 12

It is found that so long as a cloud remains magnetically coupled to its surroundings, the magnetic torques constrain a cloud to corotate with the backgroiond medium. Centrifugal forces are al\\:ays kept well below gravity. The one single factor most important in detenmining the angular momentum of a protostar is the ionization rate in dense magnetic clouds: the degree of ionization controls the coupling of a cloud to the galactic magnetic field. The fractional ionization in dense magnetic clouds is therefore discussed in some detail. The angular momentum of magnetically-braked protostars is shoun to be consistent with the observed angular momenta of close binary systems and single earlytype main-sequence stars, llie hy^pothesis of magnetic braking offers support to the fission theory for the formation of close binary systems, and is able to account for the relative paucity of single stars. The calculations also suggest a common mode of formation for (close) binary and planetary systems. IMs investigation shows that magnetic fields do indeed play an inportant, if not dominant, role during the early stages of star formation. Detailed numerical liydrodynamic collapse models have, as yet, ignored the possible effects that magnetic fields may have on the structure and evolution of a protostar. Such models are therefore highly suspect and probably not physically realistic.

PAGE 13

SECTION I INTRODUCTION Angular Momentuni Problem Traditionally, magnetic fields and angular momentum have presented foimidable problems to the theory of star formation ^cf. Mestel 1965) . Due to the high conduct ivit)"of the interstellar mediun, the frictional coupling between plasma and neutral gas is sufficient to cause the largescale galactic magnetic field to become 'frozen' into the fluid and dragged along with it. Accordingly, the magnetic energy density of a collapsing interstellar cloud (or fragment) increases as the cloud contracts, and the collapse is retarded aiid subsequent fragmentation may be prevented. Condensations in the interstellar medium will also possess angular mcmentum by virtue of local turbulence or galactic rotation. A simple calculation shows that a mainsequence star would rotate with an equatorial speed close to that of light if it were formed by isotropic compression from the interstellar gas, conserving angular momentum during contraction. Of course it is doubtful that stars could ever form under such conditions since centrifugal forces at the equator will increase fastpr than the gravitational forces, ultimately resulting in a rotational instability. Mestel and Spitzer (1956) and Nakano and Tademaru (1972) have shown that the 'magnetic field problem' is only temporar>'. Ambipolar

PAGE 14

* diffu'^ion allows the field to tjncouple from the gas when the fractional ionization is reduced. Furthermore,. Mouschovias (1976a, 1976b) has sho^vn that, at least for relatively low gas densities, some material may stream preferentially down the magnetic field lines, thereby increasing the ratio of gravitational to magnetic energy within a condensation. Radio -frequency observations of molecular clouds do not show any clouds rotating much faster than the Galaxy (e.g. Heiles 1970; Heiles and Katz 1976; Bridle and Kesteven 1976; Kutner, et al . 1976; Loren 1977, and private communication; Lada, et al. 1974). Main-sequence stars are observed to rotate with equatorial velocities ranging from a few hundred kilom.eters per second for the early-type stars to just a few kilometers per second for stars later than spectral type F5 (Struve 1930; Abt and Hunter 1962). Evidently, nature has found a solution to the angular momentum problem. A variety of mechanisms have been proposed to reduce the angular momentum of collapsing clouds and protostars. Hoyle (1945) and McCrea (1960, 1961) have suggested that condensation may take place in regions where the local turbulence is abnormally small. However, each object is still likely to have somewhat more angular momentum than is found in single main-sequence stars. Preferential mass flow along the rotational axis would increase the gas density at constant angular velocity. However, tliis process is not without its own difficulties (Mestel 1965; Spitzer 1968a) Nfore 'attention has been, given to the possibility of transforming the Ambipolar diffusion ordinarily 'refers to the process of chargedparticle diffusion due to a balance betiveen a space-charge electric field and density gradients (cf. Krall and Trivelpiece 1973) . In the astropliysical literature, ambipolar diffusion refers to the drift of a weakly ionized plasma across a magnetic field.

PAGE 15

intrinsic (spin) angular momentiun of a single niassive protostar into the orbital angular momentum of a multiple star system (Larson 1972a; Black and Bodenheimer 1976). However, as Mouschovias (1977) points out, the angular momentum of such a hypothetical system is still some t\v'o orders of magnitude greater than that observed for the long -period (visual) binaries. Dicke's (1964) claim that the interior of the Sun is in rapid (differential) rotation suggests that single stars may store a large amount of angular momentum beneath their surface. Although not accounting for the possible stabilizing effect of toroida] magnetic fields, Goldreich and Schubert (1967) have shovm that a necessary condition for stability in differentially rotating stai's of homogeneous chemical composition is that tlie specific angular momentum (i.e. angular momentum per un5.t mass) should increase with increasing distance from the rotational axis. Thus it appears unlikely that a differentially rotatiTig mainsequence star can have an atigular monentimi much in excess of a uniformly rotating star. Furthermore, convective mixing and poloidal magnetic fields redistribute angular momentum in the direction of rigid-body rotation. It has often been suggested that the angular momentum of a contracting cloud or protostar may not be conserved. That is, angular momentum may be transferred in sane manner to the surrounding interstellar material. Weizsacker (1947) has argued that a rapidly rotating star will be rotationally decelerated as angular momentum is transferred from the star to its surroundings by turbulent viscosity. Ter Haar (1949) subsequently showed that Weizsacker 's purely hydrod)Tiamic mechanism for angular momentum transport is probably not very efficient. Recently, Sakurai (1976) has calculated the braking torque on a Jacobian ellipsoid

PAGE 16

by a tidal acoustic wave which is generated in the surrounding medium by the rotating configuration. However, as Sakurai points out, the effectiveness of the braking for pre -mainsequence stars is uncertain because the braking time is of the same order of magnitude as the time scale of evolution. Ffy-dromagnetic braking appears to be more efficient in disposing of angiilar momentiim. In an attempt to account for the sun's observed slow rotation, Alfven (1942) suggested that the interaction of the sun's dipole magnetic field with the surrounding 'ion cloud' would produce a torque on the sun tending to brake its rotation. Ter Hear (1949) generalized this concept to include all stars magnetically coupled to HII regions. Liist and Schliiter (1955) examined in sane detail, particularly for the special case of torquefree magnetic fields, the transport of angular momentum, by magnetic stresses acting on a rotating star. Hoyle (1960) has proposed three stages of development for star formation: (1) the initial stage when a condensation is magnetically coupled to its surroundings by the frozenin galactic magnetic field. Angular mamentum is efficiently transferred from tlie contracting condensation to tlie surrounding medium with the condensation being constrained to co-rotate with the surroundings; (2) a subsequent phase when the fractional ionization becomes low enough so that the condensation uncouples from the galactic magnetic field via ambipolar diffusion, aftei^which angular momentum is effectively conserved; and (3) a recoupling with the galactic field during tlie final stage of slow contraction to the main sequence. Hoyle was able to explain tlie anomolous distribution of angular momentum within the solar system (981 of the total angular momentum of the solar system is concentrated in the planets which comprise less than 1% of the total

PAGE 17

mass of the system) as being the result of a hydromagnetic transfer of angular monicntum from the primitive solar nebula to the planetary material. Hoyle's calculations were confirmed in a more quantitative fashion by Dallaporta and Secco (1975) . Following Hoyle's (1960) paper on the origin of tlie solar system (for a review of this and other theories of solar system formation, see Williams and Cremin 1968), it was generally believed (McNally 1965; lluang 1973, and references cited therein) that single main-sequence stars of spectral type F5 and later were likely to liave planetar>^ systems, and that their observed slow rotation was thus explained ipso facto. Scliatzman (1962) pointed out that the transition between stars with deep envelopes in radiative equilibrium and those with well-developed subsurface hydrogen convection zones occurred among the F types . He introduced a theor>' in which the gas emitted by jets and flares associated with the active chromospheres of the later-type stars (those stars having subphotospheric convective zones) is magnetically constrained to co-rotate with the star out to very large distances where it carries au-ay a large amount of angular momentum per unit mass. This theory is consistent with observational evidence. T Tauri stars undergoing pre-main-sequence contraction are ejecting matter (Herbig 1962; Kuhi 1964, 1966; see, however, Ulrich 1976). The observations of Wilson (1966) and Kraft (1967) shov; a connection between the rotation of stars and their age as determined by chromospheric activity (measured by the presence of H and K emission lines of Call) which is usually associated with the hydrogen convection zone. Dicke (1964) , Brandt (1966) , Modisette (1967) , and Weber and Davis (1967) have calculated the solar-wind induced torque on the Sun. They

PAGE 18

conclude that the torque is sufficient to halve the sun's rotation on a cosmological time scale. Elaborating on the ideas of Schatzman, Mestel (1968) lias fomiulated a theory of magnetic braking by a stellar wind. Using Mestel 's results, Schwartz and Schubert (1969) have shot-m tiiat the Sun may )iave lost a considerable amount of angular momentum if it passed tlirough an active T Tauri stage. Assusning that stars in the pre-mainsequence stage are wholly convective (Hayashi 1961) , Olcamoto (1969, 1970) has shown that solartype stars may lose almost all cf their angular momentum via a Schatzmantype braking mechanism during pre-main-sequence contraction. Tli8 Schatzman -type magnetic braking mechanism would apply only to those stars having appreciable subsurface convection zones and therefore enlianced surface activity (e.g. m.ass loss). This may account for the break in stellar rotation on the main sequence at spectral type F5, although as mentioned earlier, it may be that in some cases, angular momentum has been transferred to a surrounding planetary system. It is not clear that earlytype stars ever develop a fully convective structure during pre-main-sequence contraction (Larson 1969, 1972b). Accordingly, for these stars in particular, we must examine the possibility of rotational braking during the early pre-opaque stages of star formation. Ebert, et al . (1960; see also Spitzer 1968b and Rose 1973), in a pioneering study have investigated the transfer of angular momentum from a contracting interstellar cloud which is magnetically linked to the suiTOunding interstellar m.edium by the frozenin galactic magnetic field. Kinks in the field lines introduced by the rotation of the cloud propagate into the surrounding medium in the form of magnetohydrodynamic (M-ID) waves (in this case, the transverse Alfv^n mode is excited), thereby rotational]y

PAGE 19

decelerating the cloud. This mechanism is expected to be operative only so long as the cloud remains magnetically coupled to the background. As the collapse proceeds to higher densities, ambipolar diffusion (Mestel and Spitzer 1956; Nakano and Tademaru 1972), MHD instabilities (Mestal 1965) , or perhaps intense Ohmic dissipation (Nfestel and Strittmatter 1967) m>ay act to uncouple the cloud's field from the surrounding medium. Altliough their results remain scme^vtiat tentative due to the uncertainties in the formulation of the problem (e.g. assumed cylindrical s>'mmetry) , it appears that the magnetic torques may be sufficient to brake the cloud's rotation so that Hoyle's (1960) argument for efficient angular momentum transfer during the initial stage (Hoyle's stage 1) of star formation is supported. In a more detailed general analysis, Gillis et al. (1974) find, in one particular application of their sanewliat artificial timeindependent pseudo-problem, that the magnetic braking is "embarrassingly efficient" although they admit that their mathematical approximations introduce some degree of uncertainty. Kulsrud (1971) has calculated the rate of emission of energy in the form of N5ID waves (specifically, the fast magnetosonic mode) for a rotating, timedependent, point magnetic dipole. Kulsrud has sho\vn that stars with very large magnetic fields (e.g. magnetic A stars) and initially small rotation may be decelerated to very long periods. Indeed, this mechanism may explain the anticorrelation of rotational velocity and surface magnetic field strength observed for tlie magnetic stars (Landstreet et al, 1975; Hartoog 1977). Tlie magnetic accretion theory of Havnes and Conti (1971) and the centrifugal wind theory of Strittmatter and Norris (1971) have also been proposed to account for the long -period /^ stars. Nakano and Tademaru (1972), Fleck (1974), and Fleck and

PAGE 20

Hunter (1976) have adapted Kulsrud's result (even though Kulsrud's formulae are strictly applicable only to a periodically time -varying dipolar field) in order to estimate the efficiency of braking for collapsing interstellar clouds. The results of Fleck and Hunter are in good agreement with observations of molecular clouds and stellar rotation on the main sequence. Prentice and ter Haar (1971; see also Kraut Schneider 1977) have suggested that a collapsing grain-cloud may lose angular momentum to the neutral gas component. However, this mechanism assumes that the grains are electrostatically neutral, and it ultimately relies on a hydromagnetic transfer of angular momentum to the outside. Â« Present Work The purpose of the present investigation is to show that magnetic fields do indeed play an important, if not dominant, role during the early stages of star formation. V/e examine angular momentum transfer from a cool, rotating, magnetic cloud, magnetically coupled to its surroundings prior to the epoch of ambipolar diffusion, and undergoing essentially pressure-free collapse on a magnetically-diluted dynamic time scale. Rotation induces a toroidal magnetic field in the neighborhood of the cloud and the accompanying magnetic stresses produce a net torque acting on the cloud tending to keep the cloud in a state of corotation witli its surroundings. We do not attempt a detailed solution of the coupled hydrodynamic and electrodynamic equations as to do so would require a sophisticated ccmputer code to handle the problem numerically. Such a formidable (if not impossible) task is liardly justifiable in view of our lack of understanding of many of the details of the star formation process. Instead, we employ a modified virial approach to

PAGE 21

calculate time -dependent quantities oÂ£ interest at the surface of a cloud in order to estimate the efficiency of the magnetic torques in de-spinning the cloud. We coinpare our results vÂ«/it}\ observed properties of molecular clouds , the specific angular momenta of (close) binary systems, the angular momentum of the protosun, and with stellar rotation on the main sequence. Uncertainties in some of the physical processes of star formation and complexities in the mathematical formulation of the problem do, of course, necessitate some degree of approximation and simplification in order that the problen remain tractable. We cannot hope to improve on the approximate nature of any theoretical study of star formation until we better understand the observations that are just now becoming available.

PAGE 22

SECTION II MGNETIC BRAKING Mag netic and Velocity Fields It has been established (Heiles 1976, and references cited therein) that a large-scale magnetic field pervades the Calaxy. Due to the high conductivity of the interstellar medium, this field is 'frozen' into the fluid (Mestel and Spitzer 1956). Consider a unifoim, spherical, interstellar cloud with radius R which is contracting isotropically.* Strict flux conservation implies that the magnetic field strengtli B within a radially contracting cloud increases according to B = B^CR^R)^ , (1) where the subscripts denote initial values. As a consequence of fluxfreezing during an isotropic collapse, the initially uniform (galactic) field lines are draivn out from the cloud into a nearly radial structure (Mestel 1966). Accordingly, we approximate the magnetic field outside, the cloud by the spherical polar coordinates B = (BÂ„,Bq,B.) where r2 ^r " ^o^"T -"Dcose (2) r * ... Obse-rvations of condensations in the interstellar mediuin spanning a range in mass from the massive molecular cloud complexes down to the stellarmass Bok globules (Zuckeman and Paljner 1974, and references cited therein) indicate an approximate spherical geometiy. Isotropic contraction will be partly justified and partly relaxed in a later section of this paper. Of course, the sLmplif)dng assxjniption that a cloud is uniform and has a v;ell-defined boundary at R is somewiiat artificial although it does simplify the calculations a.nd it is not expected to affect the validity of the results ,

PAGE 23

11 B = -B^sine (3) B = . (4) For rÂ»R , the field becanes uniform and is described by the equation o ^ = B Ccos9,-sine,0) . (5) o o Tlie velocity fields outside a radially contracting, rotating cloud are given by V = (v^,0,v^) , (6) where v^ = R(r/R)" (7) and V. = wrsin9 . (8) 9 In Eq. (7) , R E -5, the collapse velocity at the cloud surface (r=R) , and we set the exponent n=l in accordance with the findings of Gerola and Sofia (1975) and Fallon et al. (1977, and private communication) for the Orion A molecular cloud. However, the exact value of the exponent is somewhat uncertain (cf. Loren et al. 1973; Loren 1975, 1977; Snell and Loren 1977). In Eq. (8), Oi)=cj(r,t)a) is the angular velocity of the material, and we have taken the axis of rotation to be parallel to B , i.e., (o=B =Â£, tlie unit vector along the positive z-axis. The z-axis thus becomes the axis of symsnetry and derivatives with respect to the azimuthal coordinate vanish, i.e., ^r"= . Mouschovias (private communication) is investigating magnetic braking for the case of ujlB , and believes that the braking may be more efficient in this case.

PAGE 24

12 Paris (1971) has sho^vn a tendency for the torques exerted by an undetached magnetic field to rotate the angular momentum vector into parallelism with the overall direction of the field, 'flius, our assuniption that wB =1 probably more closely approximates reality. Velocities at the cloud's surface can be found by setting r=R. The cloud is assumed to rotate rigidly at a uniform rate w(r^R)=w(R). The magnetic viscosity due to the cloud's frozenin magnetic field constrains the cloud to rotate uniformly as long as the travel time of an Alfven wave through the cloud is less than the collapse time. Torque Equation The shear at R due to the cloud's rotation generates a toroidal field B , and the resulting magnetic torques react on the rotation field. The magnetic stresses acting to minimize B generate a set of Alfven waves whicli propagate into the surrounding medium, thereby transporting angular momentum from the cloud to its sun"oundings . As pointed out by Liist and Schliiter (1955; see also Mestel 1959), the magnetic torque exerted on currents witliin a given volume can be described by means of a tensor analogous to the Maxwell stress tensor* r2 B,BÂ„ v.'here e, . and 6,^ are, respectively, the Levi-Civita tensor and Kronecker delta, and x , a=i,3,k , denotes the Cartesian coordinates. We employ the Gaussian system of electromagnetic. units,

PAGE 25

13 If (k,i,j) is a cyclic permutation of Eqs. (9) and (10), then 2 B The kcomponent of the magnetic torque density about the origin is given by so that the magnetic torque acting on a volume V may be transformed into a surface integral: d,dV . D^n^dS (13) vAere n. is the unit normal outward from the surface element dS. If the A* surface S is a sphere centered on the origin, then the total outflow of angular momentum is 1 PkÂ£VS = 4lT (-B )(x.B.-x.B.)dS , (14) rijji 2 where B is the radial component: although the magnetic pressure (B /8tt) can interchange angular momentum bet^veen field streamlines, only the tension along the field. lines B.B./Ati contributes to the flux across the sphere S because the pressure acting normally to eacli surface element has zero moment about the center. The total torque can have only a . z-component since our system is symmetric about the z-axis. Letting B denote the poloidal comjionent of the magnetic field (i.e., B =B ,+BÂ„) , the total torque t is (Liist and Schluter 1955) 4Tr J B,B rsinedS , (IS) ({> P Â•

PAGE 26

14 where the surface element for a sphere of radius r is just 6S = r^sined8dc}> . (16) For a rotation field described by Eq. (8) , it is intuitively cleaV that the toroidal magnetic field vanishes along the z-axis where v is zero, and in the xy-p]ane where B changes sign. Thus jne can write the toroidal field as B.(r,e) = B (r)sin9cosG . (17) Since the poloidal magnetic field outside a collapsing cloud has an" almost purely radial structure, we set B =B so that using Eqs. (16) and (17) in Eq. (15) and carrying out the appropriate integration, the torque at the surface of the cloud becanes T = A r5b^CR)B^(R) (18) wiiere B (R) is the surface poloidal field and is given by Eq. (1) . Toroidal Magnetic Field The time dependence of a frozen-in magnetic field is given by (cf . Jackson 1975) ||=Vx(vxS) . (19) Provided tJiat the ratio lB,/B | does not become large, the temporal behavior of the surface poloidal field should be adequately described by Eq. (1) if R=^R(t) is known. For the v and B fields described by Eqs. (2)i (3), and (6) -(8), the tin^e dependence of the toroidal field, given by Eq. (19) is

PAGE 27

15 which becomes 2 ^=-TrVV-Â«-=Â«(f *^5i . (21) where we have made use of the convective derivative, 3t at ^ 8f r 8r ^ -^ In Eq. (21) , the first terni represents the convection of B, due to v^ v.'hile the second term shows clearly the exj^ected dependence of dB /dt on the shear in the azinaithal velocity field 3a)/3r . For iv lÂ«v^ , where V, =-^(23) is the Alfven speed in a plasma having a mass density p, the convection term is uninportant and the rate of gro;\1:h of B, is determined soley by the rotational sliear 9av3r . We now derive an approximate expression for to(r), and finally, 9a}/8r . The equation of motion in a fixed non-rotating inertia.1 frajne is dv Pelt = -V(Pg+p\$) + \ . (24) fixed Here, P is the thermal gas pressure, \$ denotes the gravitational potential and f^'IM (25) is the magnetic force density, j being the electric current density, and c is the speed of light. The transformation of v between a fixed

PAGE 28

16 frame and a frame rotating with angular velocity w is given by (cf. Marion 1970) dv at dv dt fixed + OJXV (26) rotating Thus, in the reference frame of our rotating cloud (u being the cloud's angular velocity) the equation of motion for the velocity field given by Eqs. (6) -(8) reads dv Pdt -^(P +p'J')+t^-n w"r-pwx(wxr) ^M R (27) rotating The B-coiiiponent of this equation is -^i -V (V +D!|'")+f -f dt e*g ^ ' Me c6 ' (28) v-tiere V.S3/89 , f^n is the 9 -component of the magnetic force density and f^Q = p[wx(wxr)]Q ... = pw rsinScosS (29) is the G-component of tlie centrifugal force density. Tlie first term on the right-hand-side of Eq. (28) vanishes for a spherically synffnetric cloud. The left -handside of the equation vanishes as well since the velocity field given by Eq. (6) assumes v =0 . Thus, Eq. (28) reduces to ^6 cO Â• (30) Using Ampbre's law M = ^ t (31)

PAGE 29

17 in Eq. (27) , the magnetic force density becomes whence the 6 -component B . 8B B. ^ Combining Eqs. (29) and (33) in accordance with Eq. (30), using Eqs. (2) and (3) for B and Bg, respectively, yields the following expression for 0)2 , _1^ [Bo^o^-J +l)+2B^(r)(sin^e-cos^e)] , (34) pr r where we have used Eq. (17) to write out the explicit r and dependence of B . The effect of the second term in brackets is to increase w in the equatorial zones (i.e. the xy-plane) and decrease w near the poles (i.e. along the z-axis) . Averaged over a sphere of radius r which is concentric with the cloud, this tenn vanishes, i.e., /^(sin^e-cos^e)de = -^ =0 , (35) . /; ae so that an approximate (average) atigular velocity for the material surrounding the cloud is u B R R^ , =^-^(-f^ir^ Â• (36) r p* r This procedure, which is equivalent to neglecting currents in the radial direction, is similar to that employed by Alfven 1967; also Alfven and Arrhenius (1976) in deriving his 'law of partial co-rotation' for a magnetized plasma.

PAGE 30

18 The gas density p outside the cloud will be, in general, s(xne function of r. Tlie theoretical collapse models of Hunter (1969) and the observational findings of Loren (1977) for the Mon R2 molecular cloud suggest P = PgCf)"^ (37) v/here p = 3m/4TTR is the density at the surface of a uniform spherical cloud having a mass ra. Using this result in Eq. (36) and differentiating with respect to r yields ,2 Â„2 Â„2 (38) . . r, 1 B R r R , R R 1 3r ^ 3m ' ^2 9w (_o..l)''^. _o (..0,13--^. r r r As expected, 7t^<0 . Using this expression for 3a)/3r in Eq. (21), 2 R taking -8-Â» 1 (whicli should be true as the collapse proceeds and has r the virtue of somewhat underestimating the rate of growth of B initially) , and evaluating the result at the surface of the cloud, r=R, yields This is a linear first-order differential equation which can be cast into the form An integrating factor is R . Making use of the free-fall collapse velocity (in a later section, we modify this equation to take into account pressure gradients within the cloud) *= -[20,(1 -i)]'-' _ (,Â„

PAGE 31

19 G being the Ne\rtonian gravitational constant, and defining n = R/R^j , (42) the solution to Eq. (40) assumes the form B^(R,e) = -rf 2 2 0^1 BR sin8cos9 rn ,^ m 1 n (l-n)^J (43) Evaluating the integral and writing B (R,e) = B (R)sin9cQse in accordance with Eq. (19) gives , 1. 45x10"^ Vr^ .-, .H 7 7C 7 xc; T B.(R) = -n2B,(RJ ^ -^U-^^S^'^ll'^^i e' (|>^ o 2 35 ^ 12^ Jin G (m/m ) ^ o^ " l-(i-n)^' " _l+(i-n)^_ 4ri (44) .33 where m_ = 2.0>10 g is the mass of the Sun. Asymptotically, as n->0, B (R)-> n"^ .

PAGE 32

20 = .u + . (47) The collapse velocity at the cloud surface R is given by Eq. (41), and the magnetic torque acting on the cloud is determined from Eq. (18) using Eq. (44) to evaluate B (R) . Notice that for t=0, the above expression reduces to angular momentum conservation. Angular momentum is transferred from the cloud to its surrounding's so that t is intrinsically negative and the cloud is rotationally decelerated. A discussion of some of the relevant time scales is in order. From Eq. (47) it is apparent that braking will be efficient only if the second term on the right-handside dominates the first. Using Eq. (25) for the Alfv^n speed, one can easily show that this is equivalent to the following condition: v^~> IvrV^I . (48) v^ere vÂ„=R and v,=wR is the cloud's surface rotational velocity in the R

PAGE 33

21 t^ R/v^ (51) in a measure of tlie characteristic hydromagnetic time scale, i.e., the travel time for an Alfven wave traversing the cloud. (Interestingly enough, this order-of -magnitude estimate for the power-loss is, excepting for a constant of order one, just that predicted by the magnetic braking model of Ebert, et al . (I960)). Since P = -tw, Eq. (47) becomes ^ = I (^ . 2R) , (52) whence the condition v^ > Ivj^l (53) in order that braking be efficient. A measure of the characteristic time scale for free-fall collapse is so that the condition for efficient braking becomes t^ < t^ (55) For a marginally unstable cloud collapsing from rest, this condition is satisfied during the initial contraction stage since v. is typically a few kilometers per second in the interstellar mediujn. In fact, Mouschovias (private communication) believes that the magnetic stresses acting on the surface of a contracting cloud will prevent Vp from ever exceeding v. . It is sometimes argued that because the magnetic energ>' of a gravitationallybound condensation can never exceed the gravitational energy, the travel tiiiie of Alfven waves through the condensation is at least equal to, and may well be considerably longer than, the free-fall tijue which is given by

PAGE 34

22 3 ^ ^Â£ = ^W^ Â• (56) However, this is the time required for complete collapse to a zeroradius singularity (cf. Hunter, 1962). It is more appropriate to compare time scales of interest with the 'instantaneous' dynamic time scale as given by Eq. (54). As pointed out by Mestel (196S) and Mouschovisis (1976a, 1976b, 1977), the free-fall time as defined by Eq. (56) may have but an academic significance for clouds with frozenin magnetic fields.

PAGE 35

SECTION III STAR FORMATION Shockinduced St ar Fonrtat ion The fact that young stars are frequently found in clusters suggests that stars are formed by a fragmentation process which occurs during the gravitational collapse of large interstellar clouds. According to the Jeans (1928) instabilit>^ criterion, the minimum unstable mass is -3 related to the gas temperature T (K) and particle density n (cm ) througli the relation !^ > 10 (I-)^/2 . . (57) e Rie to the isothermal behavior of the interstellar medium at relatively low gas densities (Gaustad 1963; Gould 1964; Hayashi and Nakano 1965; Hattori et al. 1969; see also Appendix A of this paper), the minuiium unstable mass decreases as the collapse proceeds to higher gas densities so that a large collapsing cloud is expected to fragment into a number of smaller stellar-mass condensations. However, rather compelling theoretical arguments and observational evidence have been presented suggesting tliat gravitationally-bound stellar-mass condensations (i.e. protostars) may form directly out of the interstellar medium without recourse to fragmentation. Ebert (1955) and McCrea (1957; see also the discussion in Mestel 1965) have sho^vn that external pressures of the order 10 to 10 cm K can reduce tlie minimum

PAGE 36

24 unstable mass to stellar order. Such extreme pressure variations are kno;m to exist in the interstellar mediun (Jura 1975) . Shock waves propagating in the interstellar medium can increase the gas density uj^ to two orders of magnitude, and thus reduce the Jeans mass by a factor of ten. Because of the cooling efficiency of the interstellar medium at low densities, the cooling time behind a shock in an HI region is typically two ordei's of magnitude less than the dynamic time scale (Field et al. 1968; Aanestad 1973), so that the shock propagates Â•iso thermally. The jump in density across an isotliennal shock front is approximately (Kaplan 1966) n, y ^ 5r= ( 1-j-)^ , " (58) "l 2 km :Â• vAere v is the shock velocity, (measured in km s ) and may be as large as 20 km s for a strong shock. Various mechanisms have been proposed for producing and maintaining interstellar shocks, and the possibility of shocktriggered star foraiation has been examined under a variety of physical conditions. The hydrodynamical models of Stone (1970) indicate that star formation may be enhanced by shocks generated during collisions between interstellar clouds. Indeed, Loren (1976) believes that ongoing star foimation in the NGC 1333 molecular cloud is the result of such a cloud-cloud collision. Roberts (1969), Shu et al. (1972), and Biennann et al. (1972) have suggested that shock waves associated with density waves in spiral galaxies may induce the gravitational collapse of gas clouds thus leading to star formation. Giant lUI regions associated with young earlytype stars often line up 'like beads on a string' along the spiral arms of our Galaxy, and there is recent evidence for star formation by density wave shocks in M33 as well (Uubout-Crillon 1977).

PAGE 37

25 The shock front associated with the advancing ionization front of an HII region may trigger the collapse of stellar-mass condensations (l>>'Son 1968) . Large OB associations may be caused by a sequential burst of HII regions in a dense cloud (Elmegreen and Lada 1977) , or perhaps by a supernova cascade process (Ogolman and Maran 1976) . Observations of the Origem Loop supernova remnant (Berklivijsen 1974) and the expansion of the Gum Nebula (Schwartz 1977) suggest that the strong shock from a superriova explosion may trigger star formation. Cameron and Truran (1977) explain various isotopic ananalies and traces of extinct ratioactivities in solar system material as being the result of a nearby Type II supernova that triggered the collapse of a cloud which led eventually to the formation of the solar system. The detailed t^vodimensional numerical hydrod>Tiamic calculations of Woodward (1976) danonstrate the validity of the shockinduced mechanism of star formation, particularly when the effects of self -gravitation, thermal instabilities, and dynamical instabilities of the Kelvin-Helmholtz and Rayleigh-Taylor t>'pe (cjf. Chandrasekar 1961) which are triggered by the shock, are taken into account. Thermal Instabilities Theimal instabilities in the interstellar medium can result in the fcfciiation of non-gravitational condensations of higher density and lower temperature than are found in the surrounding medium (Field 1965) . 2 Basically, this is because cooling rates at low densities vary as n while heating rates vary only as n. Following a thermal instability, ds the density increases and tlie temperature drops (pressure equilibrium obtaining for short-wavelength perturbations) , the critical Jeans mass given by Eq. (57) decreases rapidly. Theoretical studies by Hunter

PAGE 38

26 (1966, 1969) and Stein and McCray (1972) have shown that self -gravitating, primary, stellar-mass condensations can form out of the medium directly, without the occurrence of fragmentation, by the two-step process of thermal instability at pressure equilibrium followed by gravitational collapse. Observationally, isolated primary condensations having stellar masses are laio\v'n to exist (Aveni and Hunter 1967, 1969, 1972; Herbig 1970, 1976). Replacing the usual assumption of isothennal compression with the condition of energy balance, Kegel and Traving (1976) have generalized the Jeans criterion for gravitational instability, and they find that the dÂ£nP 2/2 minimum unstable mass is reduced by a factor (-^ ) <1, v/ith P and p o being the pressure and density at energ)' equilibrium. Thermal -chemical instabilities may also lov;er the Jeans mass. The formation of hydrogen molecules on grain surfaces in interstellar clouds may result in pressure instabilities leading to the formation of protostars (Schatzman 1958; Reddish 1975). Because the cooling efficiency is greater for CO than for CI I, the conversion of CI I to CO during the evolution of dense interstellar clouds (cf. Herbst and Klemperer 1973; Allen and Robinson 1977) may lead to instabilities (Oppenheimer and Dalgamo 1975; Glassgold and Langer 1976). Generalizing Field's (1965) work to include chemical effects, Glassgold and Langer find unstable masses of stellar order. Oppenheimer (1977) has demonstrated that the interstellar gas may be unstable to the isentropic gro\rth of linear perturbations in dense, opticallythick regions where the molecular transitions governing the cooling of the gas are thermal ized, and where strong heat sources are present. Such instabilities may also lead to the formation of protostars. The criterion for thermal instability beccmes modified in the presence of magnetic fields (Field 1965) . Just as in the case of

PAGE 39

2V shockinduced density grovvHh in a magnetized plasma (^cf. KaplaJi 1966), magnetic pressures inhibit conipression of the gas in directions perpendicular to the field lines. Even so, Mufson (1975) has shown for a wide variety of physical conditions, that the post-shocked gas is likely to become tliennally unstable and that condensation m.odes can grow across magnetic field lines. High resolution radio observations of the supernova remnant IC 443 by Duin and van der Laan (1975) give evidence for condensation perpendicular to field lines. Physical Conditions in Dark Clouds Young stars (e.g. T Tauri stars, Herbig Ae and Be stars, and HerbigHaro objects) and pre-stellar objects (e.g. IR and maser sources) are frequently associated with dense molecular clouds (cf. Strom et ai . 1975) . The apparent location of newly formed stars and HIT regions on the outsides of dense, massive clouds and not at their centers (Zuckerman and Palmer 1974; Kutner et al. 1976; Elmegreen and Lada 1977; Vrba 1977) suggests a starformat ion scenario wherein a shock-driven inplosion at the boundary of a cloud initiates a thermal -gravitational instability, ultimately resulting in gravitationally-bound condensations. Theoretical studies by Solomon and Wickramasinghe (1969) and Hollenback et al. (1971), and tlie dense cloud chemical models of Herbst and Klemperer (1973) and Allen and Robinson (1977) , indicate that hydrogen is pre3 -3 doiiinantly molecular in dense (n ^10 cm ) clouds. Rocket observations by Carruthers (1970) and Copernicus satellite observations by Spitzer et al. (1973) support this conclusion. Other major chemical constituents include he, CO, ML, H, HD, OH, HXO, and H-O. A representative mean molecular weight for dense cloud material v.'ould be u=2.5. Dark clouds typically

PAGE 40

28 3 -3 have particle densities i^n,, ~10 an and kinetic gas temperatures "2 T=10K (Heiles 1969; Penzias et al . 1972; Zuckerman and Palmer 1974, and references cited therein, see also Appendix A of this paper for a detailed calculation of cloud thermod>T\ajnics) . Observations do not show any interstellar clouds rotating much faster thaji the Galaxy {cf . Heiles 1970; Heiles and Katz 1976; Lada et al. 1974; Kutner ct al . 1976; Bridle and Kesteven 1976). In the solar -15 -] neighborhood, the Galax)'' rotates v/ith an angular velocity Wp=10 s '. Obser\'ed line widths of molecular transitions originating in dense molecular clouds are almost invariably too wide to be explained by thermal motions, and they frequently imply supersonic velocities. The line widths have commonly been attributed to microturbulence (Leung and Liszt 1976) or macroturbulence (Zuckeiinan and Evans 1974) , but difficulties with line profile interpretation (Snell and Loren 1977) and energetic difficulties associated with supersonic turbulence (Dickman 1976, and private comimmication) have led to the supposition that the line widths reflect systematic motions within the clouds, probably large-scale collapse (Goldreich and Kwan 1974; Scoville and Solomon 1974; Liszt et al. 1974; Gerola and Sofia 1975; de Jong et al. 1975; Snell and Loren 1977; Plambeck et al. 1977; Fallon et al . 1977). However, in favor of a turbulent origin, Arons and Max (1975) have suggested that the obsei-ved large line widths may be due to the presence of moderateamplitude hydromagnetic waves in molecular clouds. Such waves may be generated by the magnetic braking process. Magnetic field strengths in interstellar clouds are very uncertain. Measuring Zeeman splitting in the 21 cm line of neutral hydrogen and the 18 cm ai line, Verschuur (1970) has obtained field strengths in a number

PAGE 41

29 ^ -3 of diffuse (n<10' cm ') clouds. Clark and Johnson (1974) have suggested that the apparently anomalous broadening of millimeter-v;avelength SO lines observed in Orion is caused by the Zeeman effect in 6-gauss T.agnetic fields. However, Zuckerman and Paljner (1975) believe that the. large line widths are probably due to kinematic rather than magnetic broadening. Beichman and Cliaisson (1974) find evidence from infrared polarization measurements and OH Zeeman patterns for milligauss fields in the Orion infrared nebula. Ricka'rd et al. (1975) and Lo et al. (1975) have derived milligauss field strengths for a number of CH maser sources. However, well-kno\\Ti observational and theoretical problems in interpreting CH spectra in terms of Zeeman patterns (Zuckerman and PaLner 1975; Heiles 1976) make these results tentative. Magnetic field strengths obtained by 3-3 6-3 Verschuur (n<10 cm ), Beichman and Chaisson (n=10 cm ), and Lo et al. 8-3 (n=10 cm ) , are plotted in Figure 1 as a function of inferred particle density in the magnetic region. We employ Zuckerman and Palmer's estimate of the gas density in the Beichman-Chaisson source, and for the density in the neighborliood of the source discussed by Lo et al. , we take n=10 an as suggested by Mouschovias (1975b) . 2 -3 At low densities (n<10 cm ) , the magnetic field strength reflects the large-scale galactic field B =3uG. The constancy of the field strength for these low-density clouds suggests that material may stream preferentially along the field lines until higlier gas densities are reached. .Anisotropic gas flow along magnetic field lines increases the thermal gas pressure P =nkT, k=l. 38x10' erg deg' being Boltzmann's constant, \s'hile 2 holding the magnetic pressure P,-=B /Sir constant. Neglecting, inertial forces, these two pressures will come into balance when the gas density readies a critical value given by

PAGE 42

0)

PAGE 43

31

PAGE 44

32 n^^ = B^STTkT , . (59) so that for B=Bg=3MG and T=10K, N^^ = 260^ ob"^ , (60) which is in good agreement with Figure 1. Indeed, the Parker (1966) instability (a magnetic Rayleigh-Taylor instability) may provide a mechanism for preferential gas flow along magnetic field lines at low densities, and the observational findings of Appenzeller (1971) and Vrba (1977) support Parker's predictions. Assuming pressure equilibrium maintains for n>n , tlie magnetic field strength should scale with the gas density according to Eq. (59): B = (STrkT)^*^ , (61) so that B ~ n^' (62) for an isothermal compression, in agreement with the detailed equilibriun models of Mouschovias (1976a, 1976b) for selfgravitating magnetic clouds. This result is to be compared with Eq. (1) which obtained for the case of isotropic contraction and strict fliixfreezing: B ~ n^ , (63) v^ere we have used 5 p = vmwi = 3m/4TrR (64) to relate the radius of a spherical cloud to its particle density, nV,=l. 67x10" g being the mass of the hydrogen atom. Because of the

PAGE 45

33 xmcerVibzies in detennining B and n for Figure 1, it is not possible to det^mir.e precisely whether a slope of 1/2 or 2/3 best fits the obserY?jti,sns: neither is inconsistent. However, as Mouschovias (1976b) points Vy*, a slope of 2/3 may be incoinpatible with certain properties (e.g. 5)2/5, density, inferred magnetic fields) of maser sources. Furthcnir/re:, Scale (1977) has shown that the heating of dense interstellar cLods by ambipolar diffusion imposes a constraint on cloud field strengiL^; 3 must not increase faster than n * so that predicted gas tempo ratijTffis do not exceed those observed in dense clouds. Initial Conditions for Collapse i'Tfjr, -csi equation of motion of the form given by Eq. (27) , one can derive (cf, Cox and Giuli 1968) a fairly general form of the virial equation: 'si 2K+3U+M+fi-3P V . (65) fVrr(j, 1 Â« jL_^ ^ where I is the moment of inertia of the fluid about the '/^Â•J'Jm of coordinates, K, U, M, and Q. are, respectively, the total kinetic f:i\tify^y of ;sass motion, the thermal energ>', magnetic energy, and gravi^wtlo/iyj er^Tgy within the volume V, P is the hydrostatic pressure on Mici b.nr\ii(^c; defined by V, and y is the ratio of specific heats {y-lfS for Â•'' '^'V/ l';ffi|>frrature gas of diatomic molecules). From what has been said ""'1'/ I'^/jxriing rotation and turbulence, we may safely ignore the mass"*''' \i>u yUieiic energy term. Also, although strong surface pressures may *''l'iÂ« f/<>iit ^Classing shock waves, it is primarily the theimal instability ' '*Â« jjas that drives the condensation of material to the higher densities

PAGE 46

34 required for eventual gravitational collapse. Accordingly, we neglect the P teiTA as well, s The condition for collapse is I<0 . Eq. (65) then becomes |ftI>3U+M . (66) 2 For a unifoiin spherical mass distribution, fi = -f ~frÂ• Dividing Eq. (66) by the volume of the (spherical) cloud V, noting that for a non-relativistic gas the pressure is two-tliirds the energy density, and assuming mechanical equilibritim betiveen the thermal pressure (Pp=nkT) 2 and magnetic pressure (Pw=B /Sir) , the condition for gravitational collapse, Eq. (66), becomes n > 3.75xl0^(m/m )"^ cm"^ , (67) or, equivalent ly, R < 6.73xl0-^^(m/m ) cm , (68) \diere we }iave used Eq. (64) to eliminate n in favor of R, and the subscripts here denote initial (i.e. critical) values for collapse. The initial magnetic field strength at the cloud surface is found from Eqs. (59) -(61) to be B = B^(n /260)^ . (69) o G o ^ 'Galactic rotation (wp=10' s' ) sets a lower limit to the angular velocity of a contracting cloud, and an upper limit is imposed by conservation of angular momentum, provided there are no external torques acting on the cloud. Since the evolution of a condensation up to the time it becomes gravitationally-bound is highly uncertain, we do not

PAGE 47

35 attempt to calculate the efficiency of magnetic braking during this stage. Therefore, the angular velocity of a marginally-unstable cloud cannot be determined a priori. It is pos^ble that a condensation may derive its rotation from (subsonic) turbulence which may be generated by the dynamical instabilities, particularly the Kelvin-Helmholtz modes (cf. Woodward 1976) , following the passage of a shock. Because of the strong dissipation of supersonic turbulence (Heisenburg 1948) , turbulent velocities must not exceed the sound speed c^ = (yicr/m^)^' (70) -1 7 v^ich is about 0.3 km s for T=10K, y=2.5, and Y= t . If the correlation length of the turbulence is of the order of the cloud's diameter, then % cy2R . (71) A turbulent origin for the angular momentum of protostars has the attractive feature of explaining (1) the random orientation of rotational axes of early-type stars (Huamg and Struve 1954) and field Ap stars (Abt et al. 1972), (2) the lack of a dependence of inclinations in visual binary systems on galactic latitude (Finsen 1933) , and (3) tlie lack of evidence (Huang and Wade 1966) for preferred galactic distribution of orientations of orbital planes of eclipsing binaries. If stars and stellar systems acquired their angular momenta directly from galactic rotation, angular momentum vectors would generally be aligned perpendicular to the galactic plane. Initial values of n^, R^, B^, and w appear in Table 1 for a range of protostellar masses from 1 m up to 40 m . The noninteger masses correspond to main-sequence spectral tyj^es for which main-sequence

PAGE 48

36 Table 1. Initial values of particle density n (an ), cloud radius R (cni) , surface magnetic field strength B /Bp , and angular velocity w (s ) for various cloud masses m/m marginally unstable to gravitational collapse. Numbers in parentheses are decimal exponents. m/m^

PAGE 49

37 rotation data have been accumulated. Spectral types later than early G will not be considered since these stars are expected to lose most of their primordial angular momentum during pre-mainsequence contraction and ma insequence nuclear burning (cf. Section I). Interestingly enough, the values of o} are roughly the same as those one v-.'ould calculate assuming conservation of angular momentum for a condensation having initial densities of the order of a few particles per cubic centimeter, and initially rotating with the Galaxy. Clearly then, the adopted values for 0) are probably overestimated. By adopting a possibly exaggerated co^, we require the magnetic braking to be correspondingly, more efficient in decelerating a protostar. (It will turn out tliat the calculations are quite insensitive to a wide range of values of w .) Hie values of n and to for the 1 m and 2.1 m clouds are in agreement (fortuitously) o Â© Â© with the initial conditions assumed by Black and BodenJieimer (1976) in their calculations of rotating protostars. For all mases, the initial 3 2 ratio of gravitational to centrifugal forces at the equator, F /F =Gm/R c^ , g c o o is about ten. Thus, centrifugal forces are not sufficient to stabilize tlie initial configurations, and the assumption that 2KÂ«r2 appears reasonable.* * Goldreich and Lynden-Bell (1965) and Toomre (1964) have shown by a generalization of the Jeans stability criterion to include the stabilizing effect of rotation, that whereas in the classical Jeans case with pressure effects stabilizing short -wavelength perturbations, long waves are stabilized by rotation.

PAGE 50

SECTION IV MAGNHTIC BRAKING OF COLLAPSING PROTOSTARS We now proceed to calculate the rotational deceleration of a magnetically braked, contracting protostar. The value of the rotational deceleration is given by df "R" 2 Â• (^^^ With the initial conditions given in Table 1, a)^=w(R) is obtained by nuiTierical integration of Eq. (47) together witli R, t, and B (R) given by Eqs. (41), (18) , and (44), respectively. A fourth-order Runga-Kutta integration scheme with variable stepsize was employed. The accuracy of the numerical code was tested by setting t=0 in Eq. (47) and checking conservation of angular momentuni (J=0) . The value of the cloud radias at each step in the integration was computed by a subroutine which solved Kepler's equation for a collision orbit (i.e., a degenerate ellipse with eccentricity e=l) by a standard iterative procedure. This was found to be easier than integrating Eq. (41) and solving the resulting transcendental equation for R at each step. In Older to keep the problem tractable, a number of simplifying, although reasonable, assumptions have been made. To simplify the geometry, the collapse is assumed to be isotropic and homologous, which implies furtlier, that the surface poloidal magnetic field increases according to B^(R) = B^(R^/R)2 , (72)

PAGE 51

39 vdth B taken from Table 1. Furthemore, it is assumed that the magnetic o stresses within a cloud will constrain the cloud to rotate uniformly as a rigid body. It may be argued that the assumption of homologous collapse may be somewhat artificial in view of the numerical collapse models of Larson (1969, 1972b) and blunter (1969). However, there is little or no observational evidence for 'Larson-type' d>Tia]nical evolution (Cohen and Kuhi 1976), and Disney' p.976) believes the impressed boundaiy' conditions in the Larson approximation are probably not realistic. A nonlaomologous collapse would have the effect of lowering the moment of inertia of a contracting cloud, as well as increasing the gravitational (binding) energy somewhat. Hie initial value of the aziinnthal magnetic field Baj(R ) can not be deteimned a priori. We adopt B (R )=0, v;ith the understanding that the braking efficiency will be sonewhat underestimated during the initial collapse phase since t ==0. Because the magnetic field of a cloud remains frozenin during the collapse, the free-fall time given by Eq. (S6) which is valid only for a pressure-free collapse, will underestimate the true collapse time. From the equilibrium virial theorem [1=0 in Eq, (65)] , one can define an effective Nevvtonian gravitational constant G. = G (lg|-2K-M-5U3 ^ (733 Since tlie collapse proceeds isothermally at the relatively low densities 8 -"? (n<10 cm ) considered here (Gaustad 1963; Gould 1964; Hayashi and Nakano 1965; Hattori et al. 1969; see also Appendix A of this paper), the term 3U representing thermal gas pressure is not expected to contribute significantly to Eq. (73). Furtheiroore, provided a cloud loses angular

PAGE 52

40 momentum during collapse, the term 2K can be neglected Csee discussion in Section III). Thus, the collapse of a magnetic protcstar proceeds on a magnetically-diluted time scale given by 3tt ^^ t^=(3^)^ , (74) where G' G(]b| -M)/|n| . Since \Q. \ 2l{ (see Eq. (66) and discussion leading to Eq. (67)), and both lfi| and M grow lite -R for an isotropic collapse with flux conservation, G'G/2 . The magnetic braking mechanism is operative only so long as a cloud remains magnetically linked to its surroundings. Mestel (1966) and Nfestel and Strittmatter (1967) have argued that, as a cloud contracts, the almost oppositely directed field lines at the equatorial plane give rise to strong 'pinching' forces that dissipate flux and reconnect field lines, so that the magnetic field of a cloud is effectively detached from that of the background.* However, as Mestel and Strittmatter themselves point out, tlie time scale for this process is so long that this process may not be efficient. Furthermore, the equilibrium models of Mouschovias (1976b) do not show any tendency for equatorial pinching. The expulsion of a cloud's magnetic field, by ambipolar diffusion provides a more efficient mechanism for detaching the cloud's field from * Kulsrud's (1971) mechanism for the rotational deceleration of a rotating dipole applies in tliis case. Kulsrud finds t-RSb^Bq. However, because the magnetic torques are then proprotional to the magnetic field in the suiTounding medium instead of the aziiirathal field, the braking efficiency is reduced considerably. Kulsixid's formulae, derived for a harmonically time-var>''ing dipole, may not be strictly applicable to a contracting protostar, the radius of which decreases secularly with time.

PAGE 53

41 that of its surroundings (Mestel and Spitzer 1956; Nakano and Tademaru 1972) . l"he time scale for ambipolar diffusion is (Nakano and Tademaru 1972) 2-2 tÂ„ = SttR mij nn B D "2 e = 8.26x10^-^ (n^/n) , (75) v-herc we have used Eqs. (64), (67) -(69), and (72) to \Nrrite the second -9 "S -1 equality, taking =2xlO cm""s (Osterbrock 1961). The electron density n is calculated from Eq. (B7) which appears in Appendix B (see Appendix B for a discussion on the fractional ionization in dense magnetic clouds) . Because the ionozation rate in dense clouds is soinewhat uncertain (cf. AppendLx B) , we consider the two limiting cases: weak ionization by radioactive K nuclei only, at a rate given by Cj^ (Eq. (B9)), and ionization by both K and (magnetically screened) cosmic rays at a rate determined essentially by C^n (Eq. (B19)), bearing 40 in mind that the K rate is probably the more realistic of the two (^cf. Appendix B) . As pointed out by Mouschovias (1977), if there is tension in the field lines, Eq. (75) overestimates the diffusion time scale. However, the field inside the cloud is assumed uniform so that t^ is probably not much less than that given here by Eq. (75). The magnetic field becomes essentially uncoupled when t^ =^ t (Mestel and Spitzer 1956; Nakano and Tademaru 1972). Af ter\>:ards , a cloud contracts conserving angular momentum. Thus, the ajigular momentum of a protostar is established at the uncoupling epoch. The integrations were therefore terminated when this condition was met. For a unifoim, spherical, rigidly-rotating protostar, the angular momentum

PAGE 54

42 xz the uncoupling epoch is simply J^ 0.4 mR^co^ . (76) ii2sults of th.e calculations appear in Table 2. In all c.ises considered here, the magnetic torques rotationally :iecelerate the clouds, constraining them to co-rotate with their -15 -1 Jirj^oundings at an angular velocity w =a) =10 s . Physically, the U (j /_^ven speed iust outside a cloud is always greater than the collapse J^locity at the cloud surface, so that the torques are able to transmit liiijular momentum from a collapsing cloud on a time scale which is less iian the (mai^iietically-diluted) free-fall time. Tlius, the supposition liiat mignetxi: braking is efficient all the way down to the breakdown of fliix-freezimr (e.g. Hoyle I960; Mouschovias 1977) appears to be vindicated. The angular momentum of an initial condensation is reduced some three T.r four orders of magnitude for the case of ionization by K and cosmic 40 riys, respectively. Tne angular momentim of the K clouds at the mnnupling epoch is an order of magnitude greater than that of the cosmic ny ionized clouds because uncoupling occurs earlier in the collapse sequence (n^^lO cm"^) for the weaker K ionization rate. Clouds ionized 7 -3 iroinarily b)' cosmic rays uncouple from their surroundings later (n =10 cm ) ^ice the time scale for ambipolar diffusion is longer for these clouds. The surface magnetic field strengths at the uncoupling epoch fall itcely within the range of those observed in dense clouds (see discussion JT Section 111 leading up to Figure 1). Tlie ratio |B,/B | at imcoupling f^r. foimd to be near unity for all clouds. This is consistent with the nndings of tUllis et al. (1974) who predict a similar ratio in their time.ndopcndent inngnetic braking model. If lB,/B \ become much larger than

PAGE 55

43 o

PAGE 56

o Â•s PJ U P. o O W VI u w < o owe: Cuco -H w ^-^ -t-J Â•H E E r-i CD -H rt +-Â• rH P (/) OT X P Â•H to r< > *-Â• ^^ ji 03 O E O +-> CJ I/) Â•M fi cn (1) C >.X <+-! (Â« +-< rt cd O Â•H Â•> o ^-> r>j -^ (D +-> 1 U C C (D 10 Q) fc P O e o to !-i a rH Â•n-H 3 rt e S c o O i-f u E W>;H Q C H-l w u c3 p, p-4 rÂ— . CO C -H X CO to +-> O -H 10 Â•H i-l -H m o Â•H to u a> 4-1 to V) a o 01 Â•H O Pi u X

PAGE 57

45 23 rr T CsJ o 3 *f.?-^^5 0.0 0.4 0.8 LOG (m/m^l 1.2 1.6

PAGE 58

46 assoiing Â• rigid rotation, and an upper bound is estimated by adopting Bodenheimer and Ostriker's (1970) differential rotation law for the earlytype stains (see Appendix C) . For rapidly rotating stars, gravity darkening may lead to an underestimate of the observed equatorial velocity by as much as 40 percent (Hardorp and Strittmatter 1968) , so that tlic transition of j between the early-type main-sequence stars and the eclipsing binaries is relatively smooth. The decline in j for stars later tlian spectral type early-F (m/m < 2) is thought to be the result of angular momentum transfer during pre-mainsequence contraction (Schatzman 1962; Mestel 1968; Schwartz and Schubert 1969; Okamoto 1969, 1970) aid mainsequence nuclear burning (Dicke 1964; Brandt 1966; ^fodisette 1967; Weber and Davis 1967), or perhaps an indication of the presence of planetary systems (Hoyle 1960; McNally 1965; Huang 1973, and references cited thei'ein) . ' Tarafdar and Vardya (1971) accoimt for this discrepancy in j by presiming a rapidly rotating interior for the later-type stars and/or a slowly rotating interior for the earlytyj'je stars. Hie important point illusti'ated by Figure 2 is that if angular moTientura is conserved during star formation, then the angular momentUTii of a protostellar condensation, being almost two orders of magnitude greater than that of the widest separated visual binaries, will be inconsistent with the observed angular momenta of stellar systems. ITie two-dimensional numerical hydrod)Tiamic models of Larson (1972a) and Black and Bodenheimer (1976) for rotating, collapsing protostars are thus highly suspect and probably not physically realistic since they assume strict angular momentuiTi conservation throughout the collapse. Even if the toroidal figures predicted by their models (for which there is no

PAGE 59

47 observational evidence) arc unstable to nonaxis>niimetric breakui:), as suggested by Wong's (1974) stability analysis of equilibrium toroids, the angular momentum of the system remains undianged, and one is hard pressed to find a mechanism to dispose of angular momentum. On the other hand, the calculations presented here for magnetic braking during star formation are consistent with the observations presented in Figure 2. Single stars are rare (Blaauw 1961; Heintz 1969; Abt and Le\'"/ 1976) . It is therefore not surprising that the most likely 40 ionization rate in dense clouds, namely, ionization by K radioactive nuclei (_cf. Appendix B) , predicts angular momenta corresponding to that obseived for close binaries, while the much less likely situation of ionization by cosmic rays accounts nicely for the angular momenta of (rare) single stars. The minimum angular momenta of single main-sequence stars is in excellent agreement with the minimum angular monentujTi of a contracting protostar, jÂ™ . Single mainsequence stars with rapidly rotating cores and/or large equatorial velocities apparently form in regions of lower cosmic ray flux. These are the stars in Figure 2 having Apparently, clouds having a specific angular momenta j become unstable ana fragment into a multiple (e.g. binar)-) star system. After the magnetic field is expelled from a contracting cloud, the cloud continues to contract conserving angular momentum since no external (magnetic) torques act on the cloud. Eventually, as the rotational kinetic energy of the cloud increases, the ratio of centrifugal forces to gravity exceeds a critical value determined essentially by the distribution of mass and angular momentum within the configuration (see Ostriker 1970 for references) , and the cloud breaks up into two or more condensations

PAGE 60

48 unity, there would be the danger that hydromagnetic instabilities oÂ£ the tv/isted field might interfere with the assumed poloidal magnetic topography.. 3 2 The ratio F /F = Gra/R w indicates the important result that centrifugal forces are kept well below gravity throughout the collapse sequence . .

PAGE 61

SECTION V DISCUSSION OF RESULTS: COMPARISON WITH OBSERVATIONS Tlie brightest flashes in the world of thought are incomplete until they have been proven to have their counterparts in the world of fact. John Tyndall (British Physicist 1820-1893) Tlie hypothesis of magnetic braking during star formation offers a plausible explanation for tlie observational fact that interstellar clouds, in general, do not rotate much faster than the Galaxy (Heiles 1970; feiles and Katz 1976; Bridle and Kesteven 1976; Kutner et al . 1976; l^xen 1977 and private coinmunication; Lada et al. 1974). Further obser'Vational evidence for angular momentum transfer during star foniiat ion is afforded by a consideration of the angular momenta of binaiy systems fe'id single main-sequence stars, stellar rotation on the main sequence, ^Ad the angular momentum of the protosun. Specific Angular Momenta of Single and Binary Stars The specific angular momenta j (angular momentum per unit mass) for t^lnary systems (visual, spectroscopic, and eclipsing) , single main-sequence stars, and the solar system (SS) , is illustrated in Figure 2. In calculating j for the main-sequence stars, we have assimed that the stars htsve a mass distribution given by Eddington's standard model (polytropic i/idex n = 3) . Taking the observed mean equatorial rotational velocities f.OT main-sequence stars (Allen, 1973), a lower bound for j is obtained

PAGE 62

50 vdth most of the angular momentum of the original cloud going into orbital motion of the fission fragments. The details of the 'fission' Â•process have yet to be worked out. That a cloud having a specific angular momentum ir, is expected to fission can be seen most easily in Figure 3. Here we plot the specific angular momenta for a number of eclipsing binary systems for which absolute orbital dimensions have been determined. Most of the data are taken from Kopal (1959) . Roxburgh (1966) and Bodenheimer and Ostriker (1970) have calculated the threshold (i.e. minimum) angular manentum necessary for fission to occur; Roxburgh (R) for the W Ursae Majoris systems, and Bodenheimer and Ostriker (BO) for earlytype close binary systems. Their results, reproduced in Figure 3, are shown to be in good agreement witli tlie specific angular momenta of close binary systems, and more importantly here, indicate that our primary condensations, having a specific angular momentum j^, will be unstable to bifurcation since j,, is greater than the threshold j for all masses. According to our theor>' of magnetic braking, jj, is an upper limit to the specific angular momentum of a cloud of given mass. How then are the wide (i.e. long-period spectroscopic and visual) binaries formed? Evidently, an independent mode of formation exists for these systems. Indeed, there has been increasing evidence for tivo separate modes of binary formation. Contrary to the earlier suggestions of Kuiper (1955) , Van Albada (1968a) finds that the division of earlytype binaries into close (spectroscopic) and wide (visual) pairs is probably real and not due to the obvious selection effects. l\liatever the period distribution, smooth or bimodal, Huang (private communication to H.A. Abt and S.G. Levy, 1976) feels that no single formation process will produce binaries with such a

PAGE 64

52 CVJ 00 O d in A. . o Â« o o B O -J O CO O (|S ^no) I 901

PAGE 65

53 g vdde range of periods (1 day< P< 10 days). From a statistical analysis of the frequency of binary' secondaiy masses, Abt and Lev)^ (1976) conclude that there are tv-c* types of binaries:, those with the shorter periods are fission systems in which a single protostar subdivided because of excessive angular momentum, whereas the longer periods represent pairs of protostars tliat contracted separately but as a common gravitationallybound system. This 'neighboringcondensation' or 'earlycapture' mechanism for the origin of long-period binaries is supported by the ninnerical calculations of Van Albada (1968b) and Amy and Weissman (1973) , as well as by the findings of the recent two-body tidal capture theory of Fabian et al. (1975; see also Press and Teukolsky 1977). It is not likely that a wide binar>^ system can evolve into a close system by disposing angular momentum, thereby making unnecessary a separate formation mechanism for close binaiy systems. Binary stellar winds (Mestel 1968,; Siscoe and Heinemann 1974) can operate only in the later-t>i3e, low-mass stars which have outer convection zones. Ant^lar momentum loss via gravitational radiation is efficient only for low-mass systems already in near contact (Webbink 1976) , and the disposal of angular momentum by mass -loss is not expected to occur until the late main-sequence evolutionar>' stages of already close binaiy systems (Webbink 1976). Furthermore, Webbink (1977) believes that most W Ursae Majoris systems have always existed in a contact state, and that fission is the only obvious formation mechanism satisfying tliis requirement. Rotation of Main-Sequence Stars The equatorial rotational velocity of a mainsequence star having a specific angular momentum j is v^ = jAR* (77)

PAGE 66

54 where k is the effective gyration constant for the star having a mainsequence radius 11^. Assuming angular momentum is conserved after tlie uncoupling epoch, j is given by jj, or jÂ™. Since stars later than early-F may lose large quantities of angular momentum during pre-maijisequence contraction, we consider only the eaily-t>'pe (m/m^ > 2) stars. The mass distribution within such stars is given to a good approximation by the Eddington standard model, cliaracterized by a polytropic index n=3. For rigid rotation k=0.08, and for the differentially rotating models of Bodenheimer and Ostriker (1970), ic=0.28 (see Appendix C for details of these calculations). Using this information together with the angular momentum data in Table 2, rotational velocities are calculated from Eq. (77) and the results appear in Table 3, Independent of the assumed rotation 40 law, v^, for the K ionization rate, far exceeds the critical equatorial breakup velocities for all spectral types (Slettebak 1966) ; a fortiori these stars are expected to fission into a close pair. Within the frame, work of our present theor)' of magnetic braking, single stars are believed to acquire an amount of angular momentum determined by the cosmic ray ionization rate. Figure 4 illustrates the reasonable agreement particularly for the case of uniform rotation (k=0.08), of predicted rotational velocities with those observed for single main-sequence stars (Allen 1973). Rigid rotation may result from the actions of circulation currents, convective mixing in the core, or magnetic viscosity (Roxburgh and Strittmatter 1966), so that the case k:=0.08 may in fact repiresent the most plausible sitiiation. The anomalously high rotational velocities predicted for the lower-mass stars on the assumption of angular momentum conservation during contraction to the main sequence, is evidence for further. rotational braking during the later pre-main-sequcnce. evolutionary stages of these stars.

PAGE 67

55 Table 3. Predicted equatorial rotational velocities v^. (km s ) for unifonnly rotating (k==0.08) and differentially rotating (k:=0.28) main-sequence stars for the two liiniting ionization rates, Cir and E,Â„^. llie last column, taken from Allen (1973), gives mean values for observed stars. m/m^

PAGE 68

?w r-l m (U e ^H H W C5J ,o M Â•H 4-> O u rt Â§? :3 O /-> cr u tn 1 o * box > C w Â•H +J +-> O Â•H U U O ^-^ rH 00 > Â• O i-t I! c o CO f-. o Â•H +J cS N Â•H O to o o o u Â•H O U I U Â•H f= t/^ O o 0) X! K <4-l O r-l > o a Â•H PL,

PAGE 69

57 UJ a. >H < ho UJ Ol CO CO C\J o CO o E o o -J d o d (|-S K^M) A

PAGE 70

58 Angular Momentina of the Protosiin Hoyle .(I960, 1963) has estimated the angular momentum of the early 51 2 -1 solar system J^g^ 4^10 g cm s by augmenting the planets up to nonnal solar composition. This value is in good agreement v;ith our 51 2 -1 predicted J^ 3x10 g cm s (see Table 2) for a one-solar-mass cloud ionized primarily by K. The specific angular momentum of the early solar system (ESS) is ccmpared with that of the present solar system (SS) in Figure 3. The fact that jggg, lying well above the threshold j necessary for fission, is comparable to the specific angular momenta of close binaiy systems, suggests that both (close) binary and planetary systems may be formed by a similar process {cf. van den Heuvel 1966; Fleck 1977) involving the rotational instability of a single primary condensation.* Drobyshevski (1974) has suggested a mechanism for close-binary formation wherein the convective outer layers of a rapidly rotating protostar are throv^i off fonning a ring in the star's equatorial plane which becomes unstable and forms a second component. This is very similar to the generally accepted Kant-Laplace nebular hypothesis for solar-system fonnation. The criteria for determining whether the end-product of such an instability will be planetary of stellar would be of interest. Circumstellar disks commonly * ,, In the past (cf. Brosche 1962; McNally 1965), the angular momentum presently in the solar system (SS) had always been compared with that of single main-sequence stars. Such a comparison is probably fortuitous because, as can be seen from Figure 3, a condensation having a speci^fic. angular momentum j^c is not expected to become rotationally unstable*. One must then look for a mechanism, other than the popular nebular h>'pothesis wherein a planetaiy system fonns in the equatorial plane of a rotationally unstable protostar, to account for the formation of a planetar)' system.

PAGE 71

59 associated with Be stars may be the result of a rotational instability, particularly since these stars rotate vdth near-breakup velocities (Slettebak 1966).

PAGE 72

SECTION VI CONCLUDING REMARKS Without rotational braking during star formation, single stars would rotate with speeds close to that of light. 'ITiis statement is actually a reductio ad absurdum', centrifugal forces will halt the collapse perpendicular to the rotation axis long before such speeds are attained. However, even if the resulting highly flattened system fragments with most of the angular momentiOT going into orbital motion of the fragments about their center of mass, multiple star systems (e.g. binar>" stars) would liave periods two to three orders of magnitude longer than the periods typically observed for even the widest pairs. The present investigation suggests that magnetic torques acting on a rotating, contracting cloud which is permeated by a frozenin magnetic field coupling the cloud to its surroundings, rotationally decelerate a cloud, constraining it to co-rotate with the background medium. Centrifugal forces are always kept well below gravity. Die angular momentum of magnetically braked clouds is consistent with the observed angular momenta of close binary systems and single earlyt^'pe main-sequence stars. The hypothesis of magnetic braking offers support to the fission theory for the formation of close binary systems, and is able to account for the relative paucity of single stars. The calculations also suggest a common mode of formation for (close) binary, and plcinetary systems. Throughout this investigation, some simplifying assumptions (most of which are physically justifiable) have been made in order to keep the

PAGE 73

61 ,..u\o \i\ even' instance, we have deliberately underestimated tiafc t-ffc-ciw.V Pl" ma^'^etic braking. Even so, the braking is still able Â•33.^ r^jroÂ»-'v* the cK^i'^s to co-rotate vith their surroundings; a fortiori tr-T.. Â» .^Â»,A .viAlvsis will reach thtcsame conclusion. The one single ^Tjj^^jjj^^jy. ^Â„^Â» t-HAvvitti^At -i" determining the angular momentum of a protostar .... . . ice in dense magnetic clouds: the degree of ionization csmrraic 'V v-ct"--'!'^'^ of ^ cloud to th^ galactic magnetic field. For this r2:ai?Â»."i7i thj^wtt'rri-*!discussed in Apper.dix B concerning the fractional xcni-i*i {Â«trj v^ >K'r.>^^ clouds is in need sf further study, particularly the laecnmivUas t^r tvcluding cosmic rays from magnetic clouds. Â•^ \ CJC:x;I-t---'-'"S suggest that the toroidal configurations obtained iit tn^wuisrncxl collapse models of Larson (1972a) and Black and BodsijX^ ;ftrr 15^" v wherein lingular nonentum is conserved throughout the COtIst->*j ^ t, ' : ippear until the latsr pre-nain-sequence evolution of a prctc-s-C';-. -r Tormation is a complicated process and the (simplifying) as-^J-rooCTi-st .raseeniv made that magnetic fields (and therefore their role in mai'p'^f-' ~ Â• 'in be ignored may n-.t be physically realistic. Indeed, it .nay w* Â«^"^ "n ignore effects of i-.tation en the evolution of a il/rfytr,r.ri> Tjrk of Larson and Bla-,;
PAGE 74

APPENDIX A HEATING AND COOLING RATES IN DENSE CLOUDS Heating Cosmicray heating and compressional heat generated by the collapse are the dominant heat uig mechanisms in dense molecular clouds. Heating by photo-dissociation of H2 (Stephens and Dalgamo 1973) , by photoelectrons ejected from grains (Spitzer 1948), by photoionization of the gas (Takayanagi and Nishimura 1960), and by chemical reactions (Dalgamo and Oppenlieiiner 1974) is unimportant in dense clouds because the ultraviolet photons are mostly screened out. Heat of formation released by newly fomed H2 molecular (Spitzer and Cochran 1973) is unimportant because of the low neutral hydrogen abundmice in dense clouds. Scalo (1977) has suggested that the action of ambipolar diffusion may generate an appreciable amount of heat if the magnetic field in a contracting cloud k 3 grows like B~n where k > -f. This mechanism is probably unimportant for our clouds which are characterized by k = y during the initial compression 2 stages and k == -I for gravitational collapse. Heating by the dissipation of (supersonic) turbulence is ignored. The cosmicray heating rate per unit volume is The cosmic-ray ionization rate per H molecule K^^ is computed in AppendLx B and is given by Eq. (B19) . Glassgold and Langer (1973a) give the mean energy gain per ionization ^-17 eV. The heating rate by freely

PAGE 75

63 propagating cosmic rays (Â£=1 in Eq. (B19)) is then r^^j^ = 2.72x10"^^ n exp[1.58x10' '^(m/ng n'] erg s""'" on'^ . (A2) A measure of the heat generated by the collapse r is the ratio of the thermal energy density of a cloud at temperature T, j nkT Ck=l. 38x10 erg deg' , is Boltzmaim's constant), to its free-fall time t^ defined by Eq. (56). Thus r ^ 2x10' ^V2' T erg s~^ m'^ . (A3) Nfolecular Cooling Inelastic gas-grain collisions, and rotational transitions among the more abundant molecular species, H2, CO, and HD, will cool the gas. Other perspective molecular coolants such as H2CO (lliaddeus 1972) , HCl (Dalgamo et al . 1974) , and CS and SiO (Goldreich and Kv%'an 1974) are probably not important compared to CO and HD. Atomic coolants such as CI, CII, and 01 (cf. Penston 1970) are unimportant in dense clouds: due to the attenuation of ionizing ultraviolet radiation, carbon is mostly neutial when n>10 an (Werner 1970) , and atomic carbon and oxygen are depleted by chemical reactions in dense clouds (cf. Allen and Robinson 1977). Furthermore, the cross section for collisional excitation of the CI and CII fine -structure levels by H2 is much lower than for atomic hydrogen. The energy radiated per ujiit volume per second in a transition between states u and I is n E .A Â„P . , (A4) u uÂ£ uJl uil '

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64 _3 where n is the nijmber of r.Â»lecules an in state u, E Â„ is the energy difference between u and Â£, A Â„ is the Einstein A-coefficient (i.e. the transition probabilit>' per unit time for spontaneous emission) , and is the plioton escape probability. The optical depth in a line having a rest frequency v, and a thermal Doppler width (Mihalas 1970) is given by (Penzias 1975) CA7) T _^\^ulh r ^" SiJg^vhv 1 exp(~r) In these equations, A is the molecule's atomic weight, g^ arid g^ are, respectively, the statistical weights of levels u and i, N^=n^R/2 is the column density of molecules in state I through a mean path length R/2, R being the cloud radius, c=3.0xl0 cm s' is the speed of light, and h=6. 626x10 org s is Planck's constant. In Eq. (A7) , the teim in brackets is the correction for stimulated emission. Under steadystate conditions, the relative population of levels is given by where "i

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65 is the collisional de-excitation rate, o . being the collisional de-excitation cross section, and B is the molecule's rotational constant. The cross sections are averaged over Max\\'ellian velocity distributions to obtain rate constants for rotational excitation. For simplicity, we assume that a = o (^a) , and furthermore, that the rotational levels are excited (and de-excited) by collisions with H, , the most abundant molecular species in dense clouds. Thus, the term in brackets in the denominator of Eq. (A8)_ is a measure of the deviation from thermal equilibrium: at higher gas densities, collisional de-excitation dominates spontaneous emission so that the levels become thermalized, and Eq. (A8) reduces to the Boltzmaim distribution. Mien level populations become 2 thermalized, the cooling rate utiieh is proportional to n at lower densities, goes like n [see Eq. (M)] . The cooling efficiency is thus reduced at high densities, and the cooling is said to be 'collisionally quenched. ' The rotational levels of a diatomic molecule are indexed by the rotational angular momentum quantum number J. The level J has enei'gy Ej = hBJ(J+l) , (AlO) and the level degeneracy is gj = 2J-^1 . (All) The energy separating levels J and J-1 (corresponding to an electric dipole transition governed by the selection rule AJ=Â±1) is For thermalized levels, the population of the Jth level is, from Eq. (A8) ,

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66 n, = n (2J+l)exp[-J(J+l)hB/kT] , , (A13) *J o where n is the population oÂ£ the ground state. The total number density of a particular molecular species is IV = I n ^ J=0 ^ = n I (2J+l)exp[-J(J+l)hB/kT] . (A14) Â° J=0 For kTÂ»hB , as is the case for all molecules considered here, Hj. = n^ r'(2J+l)exp[-J(J+l)hB/kT]dJ Using this result to eliminate n^ in favor of n in Eq. (A13) , we have o hB nj = n ~ C2J+1) exp[-J(J+l)h5/kT] . (A16) The cooling rate per unit volume for electric dipole transitions is, after Eq. (A4) , ^ = jl^ ^/j,JA,j-iej (A17) with n,, Ej j_^ and g^ defined by Eqs. (A16) , (A12) , and (A5) , respectively, and 4 3 2 4 A = 512^ ^ ^ J fA18) ^'Â•^ ^ 3hc-^(2J+l) where \i is the molecule's electric dipole moment. For lines which are optically thick, cooling occurs from the surface of a cloud at a rate per unit volume given by

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67 3n,R cK) where n,/n is given by Eq. (A16) and the photon spectral energy distribution for tÂ»1 is given by the Planck fuiiction, B^OM " ^ T^ Â• Â• CA20) c e -1 For thennal line broadening, dv is given by Eq. (A6) . Molecular I{>'drogen Â• The lU molecule is a homonuclear diatomic molecule and therefore has no pernianent electric dipole moment. For low temperatures (T<100 K) , hydrogen molecules are predominantly in the ground rotational state, so that cooling occurs mainly via the electric quadrapole J-2-Â»-0 transition for para-H-,. At such low temperatures, para-ortho collisional conversion is very slow. Because of the long radiative lifetimes of the excited 'o -3 levels, the level populations for gas densities n>100 cm , are given -14 by the Boltzmann equation. For the J=^2->0 transition, Â£2^ = 7.07x10 erg so that n^ = Sn^e"^^^/"^ . (A21) The cooling rate by H2, assuming n ^n^ =n is then = 1.04xlO"^\B2e"^^^^^ erg s"""an'^ , . (A22) viiere we have set A ^ = 2.947x10 s""*(Thaddeus 1972). The photon escape probability is related to tJie optical depth, as defined by Eq. CA7) , through Eq. (A5) . For H^,

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3 use-^ i/^>afijjn=3m/47rR to eliminate R in favor of n and m, the ^y^^.^'A^ cooling follows from Eq. (A19) : .n = T .'-///;-9''^^T^ n^ (m/m^)"^ e'^-^^^'^Cl-B,) erg s'^ cm"^. (A24) f th6 1^'^'''"' temperatures characterizing dense clouds, H^ cooling .(ected -'' Â•^ ^^^ efficient. , neuteV/'^-^c i iinorf*^^'^ of HD as a molecular coolant in dense clouds was ' \sted '''/ -Valgamo and Wright (1972) . Duo to vibronic inter-22 i..j ijfj///; i'>y> has a permanent dipole moment y=5.85>-'10 esu cm. MtO K on}'/ ^^^'^ first two excited rotational levels need be considered. > vÂ»olin2 /''^'' per ^^i^ volume is then Â„.iH> and V/rJiiht: give Â£2;^ = 3.54x10-^^ erg, E^q = I'^ll^w'^'^ erg, 41x10''' !' . 3Jid A, ^ = 2.54x10" s' . The levels are thermalized 'Â•tacively J"W densities, and n -ajj, for T<65 K, as can be seen from 'Ilio Hii cooling rate is then Â« x(!in)n(1.35xl0"^^e^e"-^^^/'^+1.28xl0'^^62^"^^"^^^) ^^g ^'^ Â°"'^ > (A26) :i'-n /fi , is the fractional abundance of HD relative to IL. '-"111/ I U 2 '^ -5 Â•Me H:U latio is 20,000:1. Accordingly, we take x(IID) = 5x10 '. Â•^ of Il!):ii2 by Spitzer et al. (1973), and DCN:HCH by Wilson ^3) sug^iest that x(HD) may be two orders of magnitude larger.

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69 However, we follow the ideas of Watson (1973) and assume that these ratios reflect ciiemical fractionation rather than true isotopic abundances. Tlie photon escape probabilities are determined by the optical depths which follow from Eq. (A7) : -1 JL J. T^(m) = 1.39x(Iffi)T ^{m/m^)3 n^ T^(HD) = 2.78x(HD)T"^(m/]Ti )'^n7 e'-^^^/'^ . (A27) c o The surface cooling rate for x(ro) = 5-<10 follows from Eq. (A19) , and is given by 1 r22j-J r^,^ ,' 1-I29n\r,^a W7o.-256/T,, _Â« ,/_ ,-1 i\^ = 3.46x10 "-^T^n^ fm/m^) e "''^^'[(1-3^+796 """''(I-62)] erg s " cm ". (A28) Carbon Monoxide The most abund^mt hea\7molecule in dense interstellar clouds is CO. Because only 5.5 K separates the first excited rotational level from the ground state, CO is a potentially efficient coolant in the relatively cool (T=10 K) environment of molecular clouds. Indeed, Glassgold and Langer (1973b) have shown, neglecting optical depth effects, that the low temperatures [T=10 K; see Zuckennan and Palmer (1974) for references] tyjiically observed in dense molecular clouds can be maintained by CO cooling alone. The CO cooling rate follows from Eq. (A17) : Â•I where the summation is discontinued at the first level (J=J ) where the ^ m collisional de-excitation rate is less than the spontaneous transition rate. The cross section for rotational excitation of CO by His

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70 -15 2 o-lO cm (Green and Thaddeus 1976). Because of its low dipole -19 moment (y =1.12x10 esu an) and high abundaiice, the CO molecule thermalizes at low densities. Using Eqs. (A16) , CA12) , and (A18) , Eq. (A29) becomes A^Q = 7.42x10" ^^x(CO) ~l^ Bj J^ exp[-J(J+l)h^kT] erg s'"^ cm"^ . (A30) 12 -5 We adopt a fractional abundance x( CO) = 3x10 for the main isotopic 13 -7 species, and x( CO) = 3.4x10 for the less abundant species; the adopted 12 isotopic ratio is the terrestrial value 89:1.For CO, 6=57,700 MHz, 13 and for CO, B=55,100 Wz. The optical depth in a line arising from a transition where J^J-1 is, from Eq. (A7) , Tt(CO) = 7.57xl0"^x(CO) JT (m/m )^n^ exp[-J(J+l)hB/kT] Â• J Â•[l-exp(-J(J+l)hB/kT)] . (A31) Surface cooling in the optically thick lines occurs at a rate A^^ = 5.62xl0"^2 ~-^-^ f /(2J+1)C1-6t) ' T2 /Â„,/,Â„ ^T ^ ' THm/m )3 exp[-2J(J+l)hB/kT] l-exp[-J(J+l)h8/kT] erg s cm . (A32) Grain Cooling .Cooling by inelastic collisions of grains with the gas particles (mostly molecular hydrogen) at temperature T occurs at a rate (Dalganio and McCray 1972) A = 2xlO'^Vf2'(T-T )e erg s""^ an'^ , (A33)

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71 vjhere T is the temperature of the grains, and G is the energy accomo modation coefficient, and is a measure of the elasticity of the collision; 6 is unity for a completely inelastic collision, and becoiries zero for elastic collisions. We take 0=1. Because of the strong density dependence, grain cooling is expected to dominate molecular cooling at high densities. The grain temperatures can be determined from the energy balance of a grain (Lo^v and L>niden-Bell 1976) : aT^QpCy ^nk(T-y(-^)G aT^Q^Cy . , (A34) vhere a=5.67xlo" erg cm' deg" s" is the Stefan-Bolt zmann constant, (T) is the Planck mean absorption efficiency, m., is the mass of the hydrogen molecule, and T, =2.7 K is cosmic blackbody radiation temperature. The first term represents the heat loss due to radiation of a dust grain at temperature T . The second term is the collisional energ)^ gain from. the gas, and the last tenn represents heating from an isotropic blackbody radiation of temperature T, . Eq. (A34) assumes that the cloud is shielded from the external stellar radiation field, and that there are no embedded stars within the cloud which might heat the grains to a temperature T >T, in which case the grains heat the gas (Leung 1976) . KelMan and Gaustad (1969) give Q (T) = 4.1xl0'^ T^'^^ for 0.2 micron ice grains. The optical depth through a cloud of radius R for the theimal radiation from the grains is T (T ) = Q (T )a n R . (A35) g^ g^ >' g^ g g By eliminating R in favor of n and m, the mass of a unifonn, spherical cloud, and by taking a n = 6x1 0" n, and Q (T =10) =10' , the optical depth

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72 Tg(Tg=10) lO'^m/m^)"^!!^ . '(A36) For stellar-mass clouds, ^ >i for densities n^lO cm '^, which is in good agreement with the results of Hattori et al. (1969). Since densities encountei-ed during the initial collapse stages are much less than this, the thennal radiation from grains is assumed to pass freely out a contracting cloud. Cloud Temperature Figure Al illustrates the temperature and density dependence of the various heating and cooling rates described in this Appendix. The total molecular cooling rate 7V..=A, '^[jn"*"^pn Â• As expected grain cooling completely dominates for n>10 cm . Even if gas-grain collisions are only weakly inelastic (Â©-0.01 say), grain cooling will still dominate at high densities. Grain temperatures ranged from T =6 K for T=10 K up to T =35 K for T=100 K. Gas temperatures much in excess of 20 K are rarely encountered in dense clouds; temperatures up to 100 K are considered merely to illustrate the rapid rise in the molecular coolijig rates at these relatively high temperatures. Surface cooling for all molecules never amounts to more than 101 of the total molecular cooling rate, and cooling by molecular hydrogen becomes noticeable (-101 of the total Biolecular cooling) only for the highest tonperatures . Carbon monoxide dominates the cooling at T=10 K for all 3 -3 densities. However, except for n=10 cm where \rrr^nr\> hydrogen deuterlde is by far the dominant molecular Coolant for T ^ 20 K, its cooling rate becoming tliree orders of magnitude greater than CO cooling for the higliest temperature. . It appears that cloud temperatures equivalent to those typically observed in dark clouds (T~1D K ; see Zuckerman and Palmer (1974) for

PAGE 86

74

PAGE 87

75 references) are certainly attainable at lo\v gas densities, and should prevail for higher densities as well, provided grain cooling operates with at least a one-percent efficiency. The rapid rise of the molecular cooling rate at. higher tenperatures indicates that a cloud will collapse approximately isothermally near some equilibrium (r=A) temperature 9 -3 T =10-20 K at least for densities n<10 cm . Molecular cooling becomes e " even more efficient if large velocity gradients (e.g. due to cloud collapse) develop in a contracting cloud (Goldreich and Kwan 1974; de Jong et al. 1975). Furthermore, as is sho^vn in Appendix B, the cosmic -ray ionization rate (and therefore, the heating rate) adopted licre is probably greatly overestimated for densities n>10 cm . .-.-^^

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U' 1 1 It I' I'lt'H'^ controls the I1tÂ« Â»lÂ»>frcc of ionvraticsn in dt >icn5Â« mW'V' ,1,1 When the fractional -<Â» Â• > , Â»Â» /n, I* ht^h, t: u.avipnl.u diffusion is Â» . .MitMthnliuor and Dalgamo ^''r . . , (Bl) Â»i e r c g ***Â«",.j -rV) .i^>f^ t,S^ -*. irjs easily r-hovvT) that for lU^iVsities .T. .-^ ^rs in liur faces '!'Â»Â«i:^ate:4 nuliative recomxfi tern on the right -hAr.>d-sieb of F.q. (Bl) can bo tÂ»*if equilibrium, Im. (i'-V, then uives *Â«^ -.-J: . (B2) V n 2 .. ' .. : i . (B3)

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77 The equilibrium rate equation for tlie molecular ion density gives (Oppenlieimer and Dalgamo 1974) Cn ^^^ ) " an +3n(M) (B4) where n(Nf) is the total density of heavy neutral atoms (e.g. Mg, Ca, Na, and Fc) that undergo cliargc transfer with molecular ions, and 3 is the rate coefficient for the charge -transfer process. We neglect associative ionization (Oppenheimer and Dalgamo 1977) which is probably an insignificant source of electrons in cold interstellar clouds. Putting Eqs. (B2) and (B4) into Eq. (B3) gives a quadratic equation for n which has the solution e "e "2^ ^ 2 gn(NO a a gn(M) a +n 1/2 (B5) Following Oppenheimer and Dalgamo (1974), we define a depletion factor 6 by the expression n(M) = 4x10'^ 6 n . (B6) Observations with the Copernicus satellite (Morton et al . 1973; Morton 1974, 1975) suggest an average depletion factor for heav/ metals, 6==0.1. Oppenheimer and Dalgamo give B=10 cm s , a=10 cm s , and -17 3 -1 a =10 an s . Eq. (B6) then becomes 26, n = 2xw\ [U 12_i.]l/2 _ , n 26, (B7) so that for densities nÂ»10 E,, n^.lO^'c (B8)

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78 Ions in dense molecular clouds are supplied primarily by the 40 ionization of molecular hydrogen by cosmic rays and K radioactivity; ulti-aviolet radiation (Werner 1970) and X-rays (Nakano and Tademaru 3 -3 1972) are screened in the peripheral regions of dense (n>10 cm ) clouds. Cameron (1962) has estimated the ionization rate by the g-decay 40 of K radioactive nuclei to be Cj^ = 1.4x10"^^ s'-^ . (B9) Nakano and Tademaru (1972) have calculated the ionization rate of atomic hydrogen by cosmic rays. Adjusting their result for ionization of molecular hydrogen by taking into account the difference in ionization cross sections, aÂ„ =1.65o,, (Bates and Griffing 1953), the cosmic-ray ionization rate in a cloud having a mass m is given by 5^j^ = 10"-^^ exp[-l. 54x10""^ (m/m^)~n'3'] . (BIO) The exponential term reflects the attenuation of cosmic rays due to their interaction with matter as they propagate through a cloud. The total ionization rate C = CcR Cj, . (BID By coTiparing Eqs. (B9) and (BIO), one can see that for stellar-mass clouds 10 12 -3 having densities n>10 -10 cm , cosmic rays are effectively screened and 40 ions are produced primarily by in situ K nuclei. Brown and Marcher (1977) have shoun that ionization of H and H^ in dense clouds may be enhanced by energetic secoiidary electi-ons producGd by knock-on collisions, neutron-decay reactions, and pion-dccay reactions, Tr--Â»-Vi^^*, following the interaction of fast (prunary) cosmic rays with the material in a cloud. This effect, wliich may be significant only wh'?n lowenergy cosmic rays are excluded from dense clouds, is difficiilt to estimate quaiititativcly, and is therefore neglected here.

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79 Eq. (BIO) may greatly overestimate the cosmic-ray ionization rate in dense magnetic clouds. If the magnetic field within a contracting cloud becomes tangled (e.g. by turbulence), cosmic rays, constrained to move along the magnetic lines of force, random walk through the cloud and must traverse more matter to reach the central regions of the cloud. Nakano and Tademaru (1972) have estimated tlie importance of this effect, 3 -3 and have concluded that for densities n>10 cm , cosmic rays are I effectively screened, i.e., Cpn > i^Â—^ , (B12) 2_ 1 m^ p^ 2 where <6B >/8tt is the energy density of the hydromagnetic waves having

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80 an^^litude 6B, B is the strength of the cloud's large-scale miifoTm field, 3 and p=3m/4iTR is the mass density of the cloud material. By eliminating -2 p in favor of m and R, and by defining n-R/R with B=B n (see Section III and IV of the text for a discussion of the rate of groirth of the magnetic field in a contracting protostar) , with subscripts denoting initial values, this expression becomes , 8x10" ^''r B <5BS> Â°-^ . (B13) (m/m^)n 12 FrcMii Table 1 in the main text, we see that R B ~ constant =^ 9^10 cm gauss o o for all cloud masses. Tlius, if (m/m Jn Â© cosmic rays are effectively screened. The energy lost from a magnetically braked, collapsing cloud is just the difference between the cloud's rotational kinetic energy given by angular momentum conservation, and its magnetically -braked rotational energy. For a cloud which is constrained to co-rotate with its surroundings, tliis energy is given by E, ^ =^ 0.2im)VR'^ Â• (B15) lost o o Assuming that this energy is carried away l)y the hydroraagnetic waves which are generated by the braking process, we can write <6B^>=2xlO"ll n'^ , (B16) 2 3 where we have taken initial values from Table 1, noting that u^R^/(m/m^) =: constant -10 cm s . Actually, depending on the mass of the cloud, <6B > may be an order of magnitude smaller or larger, so that Eq. (B16) represents an approximate mean value. A comparison of Eqs. (B14) and

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81 (B16) shows that cosmic rays are effectively excluded fran a contracting cloud very soon after the collapse begins. It is for these reasons that we believe the ionization in dense magnctic clouds is determined by the K ionization rate. I.e., s-si, Â• Tliis being the case, the magnetic flux linking a contracting cloud to its surroundings, uncouples at relatively low gas densities. This may explain the absence of large magnetic fields in some dust clouds (Crutcher et al . 1975) . Even if the cosmic rays are not magnetically scattered, there will be a reduction in the flux of cosmic rays in a magnetic cloud. The magnetic field lines in the neighborhood of a contracting cloud diverge outward from the cloud so. that charged particles streaming along the field lines into the cloud will be reflected by the 'magnetic m.irror effect.' . Fermi (1949, 1954) proposed that such a magnetic reflection mechanism might explain the origin of the galactic cosmic rays. For slow variations of the magnetic field in time and space, the diam.agnetic moment of a cliarged particle is an adiabatic invariiuit. Let be the angle between the direction of the line of force and the direction of moticai of the spiraling particle, viz., the pitch angle. Assuming an isotropic distribution of particle velocities in a region where the field strength is B, one can easily show (cf. Spitzer 1962) that the velocities must fall within a solid angle defined by the pitch angle such tliat e = sin"-^(B/B )^ (B17) vAien the field strengtli increases to B . In our case, B is taken to be the (uniform) galactic field far from a contracting cloud, and B is the' field strength at the cloud surface. Assuming that the particle density

PAGE 94

82 in a given region is proportional to the size of the solid angle given by Eq. (B17) , it follows (_cf . Kaplan and Pikelner 1970) that the fractional decrease in cosmic-ray flux is f = 1-C0S9 . (B18) "ITie cosmic^ ray ionization rate is then given by t where K^^n is given by Eq. (BIO). Eq. (B19) provides a workable upperlimit to the ionization rate in dense magnetic clouds, the lower-limit determined by 5, being the most likely for reasons ali'eady discussed. Eq. (B17) is valid only if the Larmor radius S.lSxlO^"^ E U g3gJ J cm (B20) of a cosmic ray having a kinetic energy E, (MeV) , and iiK)ving in a magnetic field B (microgauss) , is much less than the radius of the cloud. One finds rj^(2MeV)=3.4xlO'^Â° cm and r^^ClO GeV)=6.1xlO^^ cm, so that in fact, rjÂ«R. Also, the quantity {^ IhtI-' i^ust be significantly greater than the Larmor period. It is easily demonstrated that this condition is equivalent to the d>'namic time scale being much larger tlian the Larmor period, or 1.4CE,+938) tf Â» / s . (B21) Since collapse times are typically 10 ""-10 s, this condition is easily satisfied. Finally, Eq.. (B17) neglects collisions among the cosmic rays which have the effect of randomizing the pitch angle 6. We require that the time between collisions be much greater than the Lamior period.

PAGE 95

83 or equivalently, that the density of material (primarily molecular hydrogen) in a cloud satisfies: n Â« ^ [(1+ 93I) -1]' cm"-^ , (B22) v/here a is the collision cross section. Low-energy cosmic rays (E,==2) interact with the cloud material primarily by ionizing molecular hydrogen. Glassgold aid Langer (1973a) give a(2MeV)=1.89xio'-'-'' cm^. 3 High-energy cosmic rays (E. >10 ) interact mainly by p-p scattering and pion-production reactions with a cross section a(10GeV)-2xlo" an (see Nakano and Tademaru 1972 for references). Thus, from Eq. (B22) , 4 4. n(2MeV)Â«4xlO B m\d n(10GeV)Â«2xlO B . Since initial values for B 3 ' 2/3 range from 3 up to 10 (see Table 1) and B increases as n during gravitational collapse (see Section IV of the text), these conditions are easily satisfied.

PAGE 96

APPENDIX C mMEM' OF INERTIA FOR DIFFERENTIAIXY ROTATING MAIN-SEQUE.NCE STARS The angular momentum oÂ£ a rotating spherical body having a radius R, a radial density distribution p(r), and an angular velocity field (jj(r,6) is T _ 8u ^ " T ^/2 4 a)(r,e)p(r)r sinededr . (CI) Assuming w(r,e)=a)(r) , this becomes 8u R a)(r)p(r)r'^dr . (C2) The actual distribution of mass throughout chemically homogeneous stars which are not completely convective often approximates that in the 'standard model' of Eddington (1926), which is just a polytrope of n^3. For a rigidlyrotating star, a)(r)=constant=Wj., the angular velocity at the surface (r=R) . The integral in Eq. (C2) is then most easily evaluated with the aid of the Emden solutions for an n=3 polytrope (see, for example, Chapter 23 of Cox ai\d Giuli, 1968), Writing the 2 angular momentum in terms of the moment of inertia I=kMR , where < is the gyration constant, we have

PAGE 97

85 so that for the case of rigid rotation, we find tc = 0.08 . (C4) The pre -main -sequence evolutionary models of Bodenheimer and Ostriker (1970) for rapidly rotating massive stars predict a marked differential rotation, with the central angular velocity w^ being a factor of ten greater than Wj,. Their differentially rotating configurations are stable, according to the criterion developed by Goldreich and Schubert (1967) . From Figure 5 of Bodenheimer and Ostriker (1970) we approximate the angular velocity as a u)(x) = 10 03 e 2.3x' > where x^r/R, and (C5) a = 1.4 for ^ X 0.3 = 1.1 for 0.3 X 1.0 . The mass distribution for a polytrope of n=3 can be approximated as f ^ -bx^ p(x) = pQp e where (C6) b = 20 and c = 2 for x 0.3 b = 11 and c = 1.5 for 0.3 ^ x 1.0 . Â— 3 -3 Here, p = 3m/47rR and p (n=3) 54.18 g an " . Substituting the above expressions for aj(x) and p(x) into Eq. (C2) , and evaluating the integral using Sijjipson's Rule, we find K = 0.28 . (C7)

PAGE 98

86 Tlie g^Tation constant (and therefore, the angular momentuni) of the differentially rotating configuration is thus three times that of a rigidly-rotating body.

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LIST OF REFEI^NCES Aanestad, P. A. 1973, Ap. vT. Suppl. , 25 , 205. Abt, H.A., Chaffee, F.H., and Suffolk, G. 1972, Ap. J. , 175 , 779. Abt, H,A., and Hunter, J.H. 1962, Ap. J. , 156 , 381. Abt, H.A. , and Levy, S.G. 1976, Ap. J. Suppl. , 30, 273. Alfven, H. 1942, Ark. f. Mat., Astr. och Fysik , 28A , No. 6. . 1967, Icarus , 7_, 387. Alfven, H. , and Arrhenius, G. 1976, Evolution of the Solar System (NASA SP-345). Allen, C.W. 1973, Astrophysical Quantities (3rd ed.; London: Athlone Press) Allen, M. , and Robinson, G.V/. 1977, Ap. J. , 212 , 396. Appenzeller, I. 1971, Astr. Ap. , 12 , 313. Amy, T., and Weissman, P. 1973, A.J. , 78^, 309. Arons, J., and Max, C.E. 1975, Ap. J. (Letters) , 196, L77. Aveni, A., and Hunter, J.H. 1967, A.J. , 77, 1019. . 1969, A.J. , 74, 1021. . 1972, A.J. , 77, 17. Bates, D.R. , and Griff ing, G. 1953, Proc. Phys. Soc. London, A , 66, 961. Beichman, C.A. , and Chaisson, E.J. 1974, Ap. J. (Letters) , 190, L21. Berklwijsen, E.M. 1974, Astr. Ap. , 35, 429. Biermann, P., Kippenhahn, R. , Tschamuter, W. , and Yorke, H. 1972, Astr. Ap. , 19, 113. Blaau\>r, A. 1961, Bull. As t ron. Inst. Neth. , 15, 265. Black, D.C., and Bodenheimer, P. 1976, Ap. J. , 206 , 138. Bodenheimer, P., and Ostriker, J. P. 1970, Ap. J. , 161, 1101.

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88 Brandt, J.C. 1966, Ap. J. , 144 , 1221. Bridle, A.H. , and Kesteven, M.J.L. 1976, A.J. , 75, 902. Brosche, P. 1962, Astr. Nach r., 286 , 241. Brovm, R.L., and Marscher, A. P. 1977, Ap . J . , 212 , 659. Cameron, A.G.W. 1962, Icarus , 1_, 13. Cameron, A.G.W. , and Truran, J.VJ. 1911 ^ Icarus , 50 , 447. Carruthers, G.R. 1970, Ap. J. (Letters) , 161, L81. Chandrasekhar , S. 1961, It/drod>namic and Hydromagnetic Stability (Oxford: Oxford University Press) . Clark, P.O., and Johnson, D.R. 1974, Ap. J. (Letters) , 191, L87. Cohen, M. , and Kuhi, L.V. 1976, Ap. J. , 210, 365. Cox, J. P., and Giuli, R.T. 1968, Principles of Stellar Structure (New York: Gordon and BreachTT Crutcher, R.M. , Evans, N. J. , Troland, T., and Heiles, C. 1975, Ap. J. , 198 , 91. Dalgamo, A., de Jong, T. , Oppenheimer, M. , and Black, J.H. 1974, Ap. J. (Letters) , 192 , L37. Dalgamo, A., and McCray, R. 1972, Ann. R ev. Astr. and Ap. , 10, 375. Dalgamo, A., and Oppenheimer, M. 1974, Ap. J. , 192 , 597. Dalgamo, A., and Wright, E.L. 1972, Ap. J. (L etters) , 174, L49. Dallaporta, N. , and Secco, L. 1975, Ap. Space Sci. , 57 , 335. Dicke, R.H. 1964, Nature , 202 , 432. Dickman, R.L: 1975, Ph.D. dissertation, Columbia University. . 1976, in preparation. Disney, M.J. 1976, M.N.R.A.S. , 175 , 323. Drobyshevski, E.M. 1974, Astr. Ap. , 36, 409. Dubout-Crillon, R. 1977, Astr. Ap. , 56, 293. Duin, R.M. , and van der Laan, H. 1975, Astr. A}J. , 40, 111. Dyson, J.E. 1968, Ap. Space Sci. , 1, 388.

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94 Rickard, L.J., Zuckerman, B., and Palmer, P. 1975, Ap . J . , 200 , 6. Roberts, W.V/. 1969, Ap. J., 158 , 123. Rose, W.K. 1973, Astrophysics (New York: Holt, Rinehart and Winston, Inc.), p. 99. Roxburgh, I.W. 1966, Ap. J. , 143, 111. Roxburgh, I.W. , and Strittmatter, P. A. 1966, M.N. R. A. S. , 155 , 1. Sakurai, T. 1976, Ap. Space Sgi. , 41, 15. Scalo, J.M. 1977, Ap. J. , 215 , 705. Schatzman, E. 1958, Rev. Mod. Phys. , 30, 1012. . 1962, Ann. d'Ap. , 25, 18. Schwartz, K. , and Schubert, G. 1969, Ap. Space Sci. , 5^, 444. Schwartz, R.D. 1977, Ap. J. (Letters) , 212, L25. Scoville, N.Z., and Solomon, P.M. 1974, Ap. J. (Letters) , 187, L67. Shu, F.H. , Vincenzo, M. , Gebel, W. , Yuan, C. , Goldsmith, D.W. , and Roberts, W.W. 3972, Ap. J. , 175 , 557. Siscoe, G.L. , and Heincmann, M.A. 1974, Ap. Space Sci. , 51 , 565. Skilling, J.., and Strong, A.W. 1976, Astr. Ap. , 55, 255. Slettebak, A. 1966, Ap. J. , 145, 126. Snell, R.L., and Loren, R.B. 1977, Ap. J. , 211, 122. Solomon, P.M., and Wickramasinghe, N.C. 1969, Ap. J. , 158 , 449. Spitzer, L. 1948, Ap. J. , 107, 6. . 1962, Physics of Fully Ionized Gases (Neiv York: Interscience) , p. 14, . 1968a, in Stars and Stellar Systems . 7, eds, B.H. Middlehurst and T7H. Aller (Chicago: University of Cliicago Press), p. 1, . 1968b, Diffuse Matter in Spac e (New York: Interscience) Spitzer, L., and Cochran, W.D. 1975, Ap. J. (Letters) , 18_6, L25.^ Spitzer, L. , Drake, J.F., Jenkins, E.B. , Morton, D.C. , Rogerson, J.B., and York, D,G. 1975, Ap. J. (Letters) , 181, L116. Stein, R.F., and McCray, R. 1972, Ap. J. (Letters) , 177 , L125.

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96 Wong, C.-Y. 1974, Ap. J., 190 , 675. Woodward, P.R. 1976, Ap. J. , 207, 484. Zuckennan, B., and Evans, N.J. 1974, Ap. J. (Let ters), 192 , L149. Zuckemian, B., and PaLner, P. 1974, Ann., Rev. Astr. and Ap. , 12 , 279. . 1975, A p. J. (L etters) , 199 , L35.

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BIOGRAPHICAL SKETCH Robert Charles Fleck, Jr., was bom, the first of five children, on New Year's Eve 1949 in Jackson, Michigan. Shortly afterwards, he (was) moved to Holly^vood, Florida, and in 1961 his family moved to Fort Lauderdale, Florida, where he entered Cardinal Gibbons High School in 1963. Entering the University of Florida in 1967 to pursue a career in astrophysics, he took his BS degree (physics) in 1971. He entered graduate school in astronomy at the University of South Florida one year later and received his MS degree in 1974, afterwhich he continued his graduate studies at the University of Florida.

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I certif)' that I have read tliis study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James H. Hunter, Jr. , Chairman Professor of Astronomy I certify that I. have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Edward E. Carroll, Jr. Professor of Nuclear Ejigineering Science I certify that I have read this study and that in my opinion it confomis to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. JTvw ^ W .dU Â•vvy Kwan-Y. Che4^ Professor of Astronomy I certify that I have read this study and tliat in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degi^ee of Doctor of Philosophy. Charles F. Hooper, Jr Professor of Physics Tliis thesis was submitted to the Graduate Faculty of the Department of Astronomy in the College of Arts and Sciences and to the Graduate Council, ajid was accepted as partial fulfillment of the requirements for the degree of Doctor of Pliilosophy. August, 1977 _!:Â— -y^v^
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^ OM3 3 7.7. 3.10 4,