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

scale for free-fall collapse is

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

cad

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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