Download the formation of giant molecular clouds

THE FORMATION OF
GIANT MOLECULAR CLOUDS
Charles Forbes Gammie
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF
ASTROPHYSICAL SCIENCES
June 1992
Acknowledgments
It is a pleasure to thank my advisor, Jeremy Goodman, for his advice and guidance. He was
patient and generous with his time, and he made so many suggestions that I now find it difficult to
identify where, or if, they all turned up in this thesis. It was very satisfying to work with someone
who always saw things so clearly. The numerical work presented here would have been impossible
without the help of Lars Hernquist and Neal Katz, who provided a version of the TREESPH code in
which they have invested so much time and effort. It is either a credit to Lars and Neal or a reflection
on my programming ability that I found I had introduced almost every bug that I discovered in their
code. Frank Bertoldi, Jill Knapp, Man Hoi Lee, Jerry Ostriker, Dongsu Ryu, David Spergel, and
Tony Stark all shared their wisdom at one time or another. Russell Kulsrud provided sound advice
and some tricky mathematical points were clarified on his blackboard. Bruce Draine made many
detailed and helpful comments on the manuscript, and caught some embarassing errors. Fred Rasio
asked difficult questions about SPH that led to the tests described in Chapter 3. Chapter 4 benefitted
from a discussion with Luke Dones concerning his study of planetary rotation with Scott Tremaine.
Kristin Hoganson read the manuscript and made valuable suggestions for improving the presentation.
Robert Lupton provided beautiful software that vastly simplified the handling of numerical data. The
computations discussed here were run on the Cray Y-MP at the Pittsburgh Supercomputing Center
and on the Convex C-220 at Princeton. Partial financial support was provided by a grant from the
David and Lucille Packard Foundation to Jeremy Goodman.
Less directly, all of the faculty, postdocs, and graduate students at Princeton made it a wonderful
place to study. In particular I owe Jill Knapp an enormous debt; she showed me how Real Observers
work and shared her insight and passion for astronomy. Aloha, Jill. Man Hoi Lee, my friend and
former roommate, cheerfully tolerated my cooking and driving, and my taste in restaurants, music,
movies, and thesis advisors. Steve Thorsett was always there to discuss pulsars and other matters
over deep, cool glasses of beer. The staff and students of Wilson College provided a different sort of
education, and helped excite my interest in the opera. Abe Stone, Karl Krushelnick, Ted Smith, Dave
Coster, and Tom Kundić all provided a place to sleep during the hectic last few weeks of writing.
This thesis is dedicated to my parents, to my brother Jim, and to Kristin.
“Instruct me, for Thou know’st; Thou from the first
Wast present, and with mighty wings outspread
Dove-like sat’st brooding on the vast Abyss
and mad’st it pregnant”
— John Milton, Paradise Lost
ABSTRACT
This thesis considers theoretical issues connected with the formation of giant molecular clouds.
Chapter 1 reviews observations of interstellar gas clouds of mass M > 10 4 M¯ . It critically reviews
theories of cloud formation and discusses what constraints observations place on these theories.
Chapter 2 considers the linear stability and responsiveness of a local model of a mixed star and
gas disk. It is found that (1) the stability properties of the mixed star and gas model are similar to
those of the two fluid model studied by other workers (2) the presence of the gas makes it possible for
a density wave to cross the Lindblad resonance (3) the corotation amplifier can be either inhibited or
enhanced (over that of the stars or gas in isolation) in the mixed system. Applications of the results
to the galaxy and to young galactic disks are discussed.
Chapter 3 considers the nonlinear development of gravitational instability in a local model of a
galaxy disk. An unstable, initially smooth isothermal disk is simulated using the smoothed-particle
hydrodynamics code TREESPH of Hernquist & Katz. The simulations produce objects with mass
7
similar to that of giant molecular clouds in the solar neighborhood on a timescale of <
∼ 10 yr. The
evolution of an initially azimuthal magnetic field is followed in the limit that the field is weak. Collapse
occurs perpendicular to the field lines, so that the magnetic pressure ∝ ρ2 .
Chapter 4 considers the rotation of giant molecular clouds. We calculate the expected rotation
frequency in several models of cloud formation. Gravitational instability produces rapid rotation, and
this can be understood in terms of the conservation of potential vorticity. Magnetic fields torque the
cloud in a retrograde sense and can de-spin a cloud in a fraction of an epicyclic time. The rotation of
giant molecular clouds is unlikely to provide a strong constraint on cloud formation theories until the
magnetic field can be self-consistently included.
CONTENTS
I. Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Interpretation of CO Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Cloud Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Nearby Molecular Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Galactic Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The Solar Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3. Theories of Cloud Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Thermal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Parker Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Supershells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 Spiral Arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4. Constraints on Cloud Formation Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
II. Mixed Star and Gas Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2. Calculation of Linear Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
2.1 Linear Gaseous Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Linear Stellar Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
3. Wave Packet Propagation and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Wave Packet Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Radial Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4. Amplifier Gain and Response to a Point Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1 Gain in the Two-Fluid Corotation Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Gain in the Gas and Star Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
III. Formation of Molecular Clouds by Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2. Boundary Conditions, Initial Conditions, and Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . 92
3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1 Cloud Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3 Cloud Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Appendix: SPH Noise and Shear Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
IV. Rotation of Molecular Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
2. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3. Cloud Formation Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135
3.1 Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.2 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4. Gravitational Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5. Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Chapter 1: Review
This introduction reviews observations of interstellar gas clouds of mass M > 10 4 M¯ . It then
critically reviews theories of cloud formation and discusses what constraints observations place on
these theories.
1. Introduction
The existence of dense clouds of interstellar material has long been apparent from the dark,
starless patches that can be seen with the naked eye on the night sky. A wide array of tools is now
used to study interstellar clouds, ranging in photon energy from sensitive pulsar timing systems to
orbiting gamma-ray observatories. Observations with these tools reveal a large population of massive
clouds in our galaxy (“giant molecular clouds”) that are the sole birthplace of massive stars. Because
of the central role of star formation in the formation, evolution, and appearance of galaxies, the
mechanisms that govern the formation of these clouds are of considerable scientific interest.
Most of what is known about the largest clouds comes from observations of the J = 1→0 transition
of the carbon monoxide molecule. This is because circumstances conspire to make 12 CO easy to observe
and other species (including molecular hydrogen) difficult to observe, both in practice and in principle.
Since we are forced to view the molecular galaxy through this narrow window, the physics of the line
and its relation to interesting physical parameters of the clouds are important, and are briefly reviewed
in §2.1.
The masses of giant molecular clouds (by which we mean all clouds of mass M > 10 4 M¯ ) can
be estimated by several different techniques. Diffuse γ-ray emission, for example, can be used to infer
the mass of a cloud once its distance and γ-ray emissivity are known (e.g. Bloemen, 1985a). The
accuracy of available techniques varies widely, but none of the methods we review in §2.2 give results
that are accurate to better than about 50%.
Some of the most important results from observations through the CO window are contained in
the large-scale surveys of the galactic plane carried out by several groups in the last decade (Sanders,
Scoville, & Solomon, 1985; Dame et al., 1987 and Bronfman et al., 1987; Stark et al., 1988). The
conclusions drawn from these surveys differ in important respects, but there is consensus on some
features of the galactic distribution of molecular clouds. The Milky Way contains ∼ 5000 molecular
5
clouds of mass M >
∼ 10 M¯ (Scoville & Sanders, 1987) whose surface density peaks at a galactocentric
radius ∼ R0 /2 (e.g. Bronfman et al., 1987) and that are strongly correlated with spiral arms (Stark,
1979). Inside the solar circle, the surface density of molecular gas is comparable to that of atomic
gas (e.g. Bronfman et al., 1987). Some of the other important results from observations of molecular
clouds are reviewed in §2.3 and §2.4.
It is natural to ask if other galaxies are like our own. Unfortunately, if the Milky Way were
observed from an external galaxy with one of the single-dish millimeter-wave telescopes now commonly
in use, most of the clouds in our galaxy would be unresolved and would fill only a small fraction of the
beam. Single dish observations of external galaxies are therefore time consuming and yield only the
gross distribution of CO emissivity, while millimeter wave interferometry is still in its infancy. Because
of this observational difficulty, and interpretive difficulties with high resolution data once obtained,
this thesis heavily weights observations of our own galaxy and the solar neighborhood in particular.
In §2.5, we discuss in detail observations of the solar neighborhood.
Numerous theories have been proposed for the formation of interstellar clouds. The problem
remains unresolved largely because the interstellar medium is a complex, inhomogeneous system and
–2–
many physical processes probably play a role in organizing the distribution of gas in the galaxy.
Nevertheless, drastic idealizations of the interstellar medium like those we shall perpetrate later in
this thesis can be conceptually useful and, occasionally, predictive. In §3 we review several of the
mechanisms that have been proposed to form giant molecular clouds.
In §4 we consider what constraints the observations offer for theories of giant molecular cloud
formation. Finally, in §5 we sum up and indicate how some of the problems stated here are addressed
in the other chapters in this thesis.
2. Observations
In sufficiently dense clouds most hydrogen is molecular. H2 is unobservable at temperatures
typical of dense clouds in our galaxy (5 − 25 K) because the only available transitions are weak electric
quadrupole transitions between widely spaced rotational levels (Field, Somerville, & Dressler, 1966).
However, the next most abundant molecule, CO, is roughly coextensive with molecular hydrogen
and is readily observed in emission in its rotational lines. Most of what we know about the largescale distribution of molecular hydrogen is therefore derived from observations of CO and its isotopic
varieties.
2.1 Interpretation of CO emission
12
CO has rotational transitions at frequencies
νJ+1→J ∼
= 1.152719374 × 1011 (J + 1) − 7.3427 × 105 (J + 1)3
Hz,
(1)
(Lovas & Krupenie, 1974) so that the first rotational level is at energy E1 /k = 5.532 K above the
ground state. The opacity of the earth’s atmosphere at the 1→0 transition ranges from τ ' 0.05 at
good sites on good days to τ >
∼ 1, with most of the opacity from water vapor and O2 lines at 60 GHz
and 118 GHz (e.g. Liebe, 1985), so that the J = 1→0 transition is observable from the ground. The
Einstein A coefficients for the rotational levels are
AJ+1→J =
64π 4 ν 3 2 (J + 1)
µ
3hc3
(2J + 3)
s−1
(2)
where µ = 1.0982×10−19 e.s.u.− cm is the dipole moment, giving A1→0 = 7.166×10−8 s−1 (Chackerian
and Tipping, 1983). The first vibrational level is at an energy Evib /k ' 3080 K above the ground state,
so its population is typically negligible. The cross section for deexcitation from the first rotational
level to the ground state from collisions with (para) molecular hydrogen at 10 K is
γ01 = 2.3 × 10−11 cm3 s−1
(3)
(Flower and Launay, 1985). Thus the critical density above which we expect the first rotational level
to be thermalized (ignoring collisional excitation by helium and ortho-H 2 ) is nc = A1→0 /γ1→0 =
3.1 × 103 cm−3 . Radiative trapping of 12 CO line photons substantially enhances emission from low
density regions (Goldreich & Kwan, 1974).
The abundance of CO is dependent on the elemental abundance of C, O, and their isotopic
varieties, the local radiation field, density, temperature, cosmic ray ionization rate ζ, and the history
of an individual parcel of gas. The solar abundance of C and O are 3.6 × 10 −4 and 8.5 × 10−4
–3–
respectively, by number relative to hydrogen (Trimble, 1991). Chemical models suggest that most of
the gas phase carbon is processed into CO in clouds that are well shielded from ultraviolet radiation
on a timescale ∼ 106 yr (e.g. Graedel, Langer, and Frerking, 1982). This may be compared with the
timescale for formation of molecular hydrogen on grains
τH 2 '
³ n ´−1
1
H
' 1.6 × 107
yr
RnH
100
(4)
where R is the product of the collision frequency of hydrogen with grains and the probability that the
hydrogen sticks to the grain (Spitzer, 1978). Van Dishoeck and Black (1987) report that the evidence
is consistent with ∼ 80% of the carbon being depleted into grains in dark clouds, implying, for solar
metallicity, an abundance of CO of 7.2 × 10−5 by number with respect to molecular hydrogen.
We now calculate the opacity in the J→J +1 line through a mythical uniform cloud of thickness L,
density nH , and velocity dispersion σ, assuming local thermodynamic equilibrium at temperature T .
We also assume that the velocities are gaussian distributed with one dimensional velocity dispersion
σ. The line center optical depth is
¶µ
¶
¶µ
µ ¶³
´µ
nH
1 km s−1
δ
fC
L
×
τJ→J+1,0 =13
pc
100 cm−3
3.6 × 10−4
0.2
σ
¢
¡
,
(5)
e−5.53J(J+1)/2T 1 − e−5.53(J+1)/T
(J + 1)
Z(T )
where Z is the partition function, fC the abundance by number of carbon with respect to hydrogen,
and δ is the fraction of carbon in 12 CO . If T = 10K, Z(10 K) = 3.969, and in the J = 1→0 line
µ
µ ¶³
¶µ
¶µ
¶
fC
L
nH ´
δ
1 km s−1
τJ=0→1,0 = 1.4
(6)
pc
100cm−3
3.6 × 10−4
0.2
σ
Thus 12 CO is optically thick for clouds of modest size and density. The optical depth in the 13 CO
line is, modulo abundance differences due to variations in shielding and chemical fractionation, down
by the abundance of 13 C with respect to 12 C. The ratio f12 C /f13 C ≈ 89 for solar abundance (Geiss,
1988), but is measured to be lower, ≈ 67, in Orion (Langer & Penzias, 1990). These authors also
observe a gradient in f12 C /f13 C with galactocentric radius. If the rotational levels are not thermalized
because the density is below the critical density, then the population of the ground state is larger than
in LTE, and the optical depth is correspondingly higher in the J = 1→0 line.
Chemical models of clouds are useful in interpreting observations of dense clouds (Dalgarno, 1986).
Time dependent chemical models show that equilibration times are of order 10 6 yr (e.g. Graedel,
Langer, & Frerking, 1982) in dark clouds. Steady-state cloud models show the importance of a layer
of thickness AV ≈ 1 mag required to shield CO from dissociating UV flux (van Dishoeck & Black,
1988). The accuracy of chemical models is limited by uncertainties in physical input parameters such
as: the cosmic ray ionization rate ζ; unknown reaction rates (particularly at low temperatures and
densities that are difficult to achieve in the laboratory); physics of the gas-grain interface (molecules
should be depleted onto grains in ∼ 3 × 109 /n yr; Williams, 1986); radiative transfer that is not
well represented by the usual plane-parallel models (inhomogeneities may allow UV light deep into a
cloud); mixing of material from outer layers into the cloud interior (Prasad, Heere, & Tarafdar, 1991;
Chièze & Pineau des Forets, 1987); and the interstellar radiation field.
2.2 Cloud Masses
–4–
Methods for determining molecular hydrogen column densities and masses of clouds containing
molecular hydrogen have been critically reviewed by Van Dishoeck and Black (1987) and more recently
by Maloney (1990b). There are seven distinct methods, and we shall proceed from the most direct to
the least.
1. UV Absorption Lines. Absorption in the Lyman and Werner bands of H2 have been used to
directly measure N(H2 ) along lines of sight toward bright, lightly obscured stars (e.g. Savage et al.,
1977; Jenkins et al., 1989). Molecular hydrogen column densities can be obtained in this way to an
accuracy of about 10%.
2. Star Counts. The visual extinction Av is estimated by placing a grid on the sky and counting
the number of stars in each grid cell down to some limiting magnitude. A v is related to the number of
stars counted through the distribution of apparent visual magnitudes. The standard ratio A v /EB−V '
3.1 (Rieke & Lebovsky, 1985) then gives EB−V , which is related to the total column density through the
ratio N (H)/EB−V ' 5.9 × 1021 mag−1 cm−2 , which is calibrated in turn by method 1 (e.g. Frerking,
Langer, & Wilson, 1982; Langer et al., 1989; Dickman & Herbst, 1990). Errors are introduced by shot
noise in the star counts, errors in the original calibration of Av /EB−V and N (H)/EB−V , and real
variation in these ratios.
3. Gamma Ray Emission. Cosmic ray protons and electrons interact with hydrogen and helium in
the interstellar medium to produce observable diffuse γ-ray emission. The most important interactions
are production of π 0 mesons in proton-proton collisions and later decay of the π 0 into gamma rays,
inverse Compton scattering, and bremsstrahlung (Hillier, 1984). Since the ISM is transparent to
gamma rays, the observed gamma ray flux can be used to measure the mass of an interstellar cloud,
given the distance to the cloud and the cosmic ray energy spectrum at the cloud.
The original variant of this method, proposed by Black and Fazio (1973), assumes that the cosmic
ray spectrum at the cloud is equal to that observed at the earth. This appealingly direct method suffers
from our ignorance of the local cosmic ray spectrum at E < 1 GeV because of scattering of cosmic
rays by Alfvén waves in the solar wind. In fact, using an independent estimate of the cloud mass,
this method has been used to infer the cosmic ray spectrum (LeBrun and Paul, 1983). Furthermore,
the gamma ray data are not completely consistent with the assumption that the cosmic ray flux is
spatially invariant (Ramana Murthy and Wolfendale, 1986).
The second variant of this method (e.g. Bloemen et al., 1985b) determines the emissivity of the
ISM empirically by fitting the coefficients A, B, C of the relation
Iγ = A N(HI) + B W12 CO + C.
(7)
Here Iγ is the observed γ-ray flux in some band, W12 CO is the integrated intensity in the J = 1→0 line,
and C (the dominant term on the right hand side) is instrumental noise and extragalactic background.
The column density of HI is readily determined from 21 cm line observations, so the coefficient A gives
the emissivity, while the coefficient B can be used to calibrate the presumed proportionality between
WCO and N (H2 ) (see method 6, below).
The γ-ray method has several difficulties. First, it is not possible, given the coarse angular
resolution of the data (COS-B: >
∼ 1 deg), to test the hypothesis that there exist discrete γ-ray sources
whose distribution varies with the 12 CO intensity, although it must be said that there is no evidence
for such sources locally. It is possible to test if there are sources of CR electrons distributed like the
CO emission, since most of the contribution to the γ-ray flux at high energy comes from interactions
between CR protons and the ISM, while at lower energies bremsstrahlung is increasingly important.
–5–
Indeed, there is some evidence that the CR electron distribution varies with galactocentric radius
(Ramana Murthy and Wolfendale, 1986).
4. 13 CO LTE Column Densities. Here it is assumed that 1) the 13 CO line is optically thin in
the J = 1→0 transition, and 2) that the level populations are described by an excitation temperature
derived from 12 CO observations. N(13 CO )/N(H2 ) is then calibrated by star counts (e.g. Frerking,
Langer, and Wilson, 1982; Dickman and Herbst, 1990). Both assumptions may be tested, and a
multitransition study shows that they are true almost nowhere over a large segment of the Orion A
molecular cloud (Castets et al., 1990). The evidence for significant variations in the abundance of 13 C
(Langer and Penzias, 1989) is also disquieting. New focal plane receiver arrays will decrease the time
required for mapping and make the approach of Castets et al. more widely applicable, but even with
arbitrarily accurate 13 CO column densities one must still know the abundance of 13 CO relative to H2
to obtain a total column density.
5. Virial Theorem. This method assumes that the the gravitational potential and kinetic terms
dominate the virial relation:
¶
Z µ
Z
D2 I
1
B2
r·dS
(8)
=
2T
+
3Π
+
W
+
p
+
(r·B)B·dS
−
Dt2
4π S
8π
where T is the total internal kinetic energy within the (comoving) surface S, Π the integral of the
pressure (thermal, cosmic ray, and magnetic) over the volume, and
I≡
Z
ρr2 dV
W ≡−
Z
ρr·∇φdV.
(9)
(Spitzer, 1978). Then using the velocity dispersion ∆V and some measure of the cloud’s angular
radius Θcl D, where D is the distance,
M =α
∆V 2 Θcl D
,
G
(10)
where α is a constant that depends on assumptions made about the cloud’s internal structure. This
method has been refined both observationally and theoretically by Elmegreen (1989a), Maloney (1988),
Solomon et al. (1987), Langer et al. (1989) and others. It seems likely that for nearby clouds (Magnani,
Blitz, & Mundy, 1985; Pound, Bania, & Wilson, 1990), with masses of ∼ 100 M ¯ , surface terms in
the virial relation become increasingly important, as is likely the case for individual clumps within
molecular clouds (Bertoldi & McKee, 1991). Furthermore, a careful study of Barnard 5 (Langer et
al., 1989; the estimated mass is ∼ 103 M¯ ) suggests that the use of 12 CO line widths in estimating
the virial mass overestimates the cloud mass by a factor of ∼ 2 because the lines are saturated, while
13
CO line widths imply a mass more nearly in agreement with other estimates (method 4). Note that,
unlike all the other mass estimates considered here, virial masses scale as D, not as D 2 .
6. 12 CO Intensities. This method is based on the empirical result that molecular clouds in our
galaxy have a roughly constant ratio of mass to 12 CO luminosity. The oft-used conversion factor is
X ≡ N (H2 )/ICO ' 3 × 1020 . cm−2 K−1 km−1 s
(11)
Calibrations of this relationship against γ-ray emission have been made by Bloemen et al., (1985) for
the Goddard-Columbia CO survey, and by virial theorem analyses. Recently Solomon et al. (1987)
4/5
have suggested that M ∼ LCO , not LCO . Quasi-theoretical derivations of this result (e.g. Elmegreen,
–6–
1989) give dependences on radiation field, metallicity, external pressure and magnetic field strength
that we have no theoretical reason to suspect are invariant with location in the galaxy or from galaxy
to galaxy.
7. Dust. Dust column densities may be obtained from infrared photometry, and are then related
to gas column densities via a “standard” dust to gas ratio. This method has been reviewed by Draine
(1989), who demonstrates that the results are sensitive to the assumed temperature distribution unless
the measurements are taken on the Rayleigh-Jeans tail of the dust emission at λ >
∼ 300µ. In practice,
Langer et al. (1989) find their direct estimate of the dust mass in Barnard 5, using IRAS data at 60
and 100µ, to be an order of magnitude too low. Nevertheless, recent work (Laureijs, Clark, & Prusti,
1991) shows that certain combinations of IRAS fluxes are closely correlated with column densities
measured using method 4.
To summarize, the measurement of molecular column densities is a difficult observational problem,
and the most commonly used methods give results that are only accurate to (optimistically) ±0.3 in
the logarithm. The prospects for improvement are good. Data already taken by COBE promises to
better characterize emission from galactic dust at wavelengths from the near infrared to the millimeter,
although with coarse angular resolution. The development of focal plane receiver arrays will make
mapping a large sample of clouds in several transitions a more practical task, yielding more accurate
CO column densities. Data now being taken with EGRET on the Gamma Ray Observatory will
give a higher resolution (∼ 0.5 deg) and higher sensitivity (∼ 20× COS-B) picture of diffuse γ-ray
emission from the galactic plane (Hunter & Kanbach, 1990). Finally, planned observations in the CI
fine structure line at 609 µm (Stark, 1991) promise to provide an interesting view of the boundary
between atomic and molecular gas.
2.3 Nearby Molecular Clouds
The most intensively studied nearby giant molecular clouds are in Orion. Observations of the
region as of 1982 have been summarized by Goudis (1982). A schematic picture of the region,
reproduced from Kutner et al. (1977), is shown in Figure 1, together with maps of 12 CO and 13 CO
surface brightness (Figure 2a and 2b). It is difficult to say to what extent the Orion clouds are typical
of those in the galaxy as a whole, but because of their proximity they are better understood than
almost any other clouds and we shall describe them in some detail.
The schematic picture (Figure 1) shows two molecular clouds, Orion A and B, each estimated
to have a mass of about 105 M¯ (Blitz, 1991). They are associated with four distinct subgroups of
massive stars, the Orion Ia, b, c, and d subgroups of age 1.2 × 107 , 8 × 106 , 6 × 106 , and 2 × 106 yr,
respectively (Blaauw, 1964). Summing over the 56 stars earlier than B2 (in all the subgroups) gives
a total mass of ∼ 103 M¯ , implying for a Salpeter initial mass function f (M ) ∼ M −2.3 M¯ −1 with
a cutoff at some fraction of a solar mass, a total mass in young stars of several ×10 3 M¯ . Orion
I is a remarkably rich OB association in comparison to others within a kiloparsec of the sun. The
hashed boundaries show the location of Barnard’s Loop, which is observed in both Hα and HI (21 cm)
emission and has a mass of ≈ 5 × 104 M¯ (Goudis, 1982). The radius of Barnard’s Loop is ≈ 50 pc
(assuming a distance of 450 pc), and its expansion velocity is ∼ 20 km s −1 . Both Orion A and Orion
B contain numerous embedded infrared sources and CS cores (Lada, 1990; see also Bally et al., 1991).
Orion is not the only giant molecular cloud in the solar neighborhood. Figure 3 shows the location
of nearby molecular clouds, reproduced from Dame et al., 1987. The ρ Oph cloud, for example, has
also been studied in detail (e.g. de Geus, 1988; Loren, 1989).
–7–
2.4 Galactic Surveys
Three major large scale surveys of CO emission in the galactic plane have been conducted to
date: the Columbia survey (Dame et al., 1987), the FCRAO survey (Sanders, Scoville, and Solomon,
1985), and the Bell Labs survey (Stark et al, 1988). A partial description of these surveys is given in
Table 1.
The Columbia survey has the largest sky coverage, and has been calibrated against γ-ray results
(Bloemen et al, 1985), so derived mass estimates are slightly more secure. The FCRAO survey has
a much smaller beam, but is significantly undersampled. No results are yet available from the Bell
Labs survey except for maps of the Orion and galactic center region (Bally et al., 1987a; Bally et al.,
1987b).
The surface density of H2 obtained by Bronfman et al. (1987) is reproduced in Figure 4, together
with the midplane CO emissivity (in units of K km s−1 kpc−1 ), the scale height, and the location of the
midplane derived from the Columbia survey, using a conversion factor X = 2.8×10 20 cm−2 K−1 km−1 s.
Figure 5 shows the location of a selection of clouds found in the FCRAO survey, as reported by Solomon
et al., 1987. The masses are derived from the virial theorem and most of the distances are kinematic.
A value of R0 = 10.0 kpc has been assumed; a more recent best guess for R0 is 7.7 kpc (Reid, 1988),
decreasing the masses of clouds with kinematic distances and 12 CO masses by a factor of 0.59.
Obviously the distribution of atomic material is also important in studying the formation of
molecular clouds. We shall shortly estimate (§2.5) that the local surface density of atomic hydrogen
−2
is <
∼ 8 M¯ pc . Gordon & Burton (1976) give the surface density of atomic hydrogen as about
−2
3 M¯ pc locally (uncorrected for helium) and declining almost linearly toward zero at the galactic
center (with no correction for optical depth). Most of the atomic hydrogen in the galaxy is evidently
outside the solar circle (Henderson, Jackson, & Kerr, 1982).
2.5 The Solar Neighborhood
Later in this thesis we shall make use of a local model of the galaxy that idealizes a patch of the
disk as a plane-parallel system with a homogeneous, isothermal gaseous component and an isothermal
stellar component. This local model is described by five dimensionless parameters: a temperature for
the gas expressed in terms of Qg ≡ cκ/πGΣg , where c is the sound speed, κ the epicyclic frequency,
and Σg the gaseous surface density; the Toomre stability parameter Qs ≡ σr κ/πGΣs for the stars,
where σr is the radial velocity dispersion and Σs the stellar surface density; a fractional surface density
for the gas fg = Σg /(Σs + Σg ); the logarithmic derivative of the rotation curve β ≡ d ln |vc |/d ln r;
and the ratio of the vertical velocity dispersion of the stars to their radial velocity dispersion. Two
dimensional parameters are required to set the scale of the simulation: κ and Σ ≡ Σ g + Σs . It will be
useful to have values for all these parameters that are appropriate to the solar neighborhood.
Hydrogen is found between the stars in three forms: molecular, atomic, and ionized. Current
opinion (e.g. Kulkarni & Heiles, 1988, hereafter KH) divides the atomic and ionized components into
four phases: the cold neutral medium (T ∼ 80 K), the warm neutral medium (T ∼ 3000 K), the warm
6
ionized medium (T <
∼ 8000 K), and the hot ionized medium (T ∼ 10 K). The four phases are nearly in
pressure equilibrium with each other, although there are probably large fluctuations. Most molecular
hydrogen is found in cold, self-gravitating clouds and can be at higher pressure than the neutral and
atomic components. Molecular, atomic, and ionized gas are all threaded by a dynamically significant
magnetic field: pulsar dispersion measures imply an ordered component of ≈ 1.7 µG (Manchester
–8–
& Taylor, 1977). Cosmic rays also have a non-negligible energy density. In short, it is a gross
oversimplification to treat the interstellar medium as a uniform, isothermal fluid. Stars in the galactic
disk can also be subdivided into into distinct populations; each population has a different velocity
dispersions, and the velocity dispersion increases away from the galactic plane (see Binney & Tremaine,
1987). Thus the stellar component is neither uniform nor isothermal. Nevertheless, since we shall
later adopt a uniform, isothermal model in both linear theory and numerical experiments, we require
model parameters that bring the model as close as possible to reproducing the dynamical behavior of
the disk in the solar neighborhood.
The atomic component is readily observed at 21 cm, and the column density of hydrogen atoms
may be directly obtained from the antenna temperature if the emitting gas is optically thin:
Z
T (V )dV cm−2 K−1 km−1 s.
(12)
NHI = 1.823 × 1018
For the effective sound speed of the gas, we should choose a number that is a compromise between
the velocity dispersion of molecular clouds, the velocity dispersion of the atomic clouds, and the
sound speed in the dynamically stiff hot component. We adopt an effective sound speed of 6 km s −1 ,
consistent with the measured one-dimensional velocity dispersion of HI near the sun (Kulkarni & Fich,
1985). The surface density of atomic material can be obtained more or less directly from observations.
Using data from the Bell Laboratories HI survey (Stark et al., 1991), we have averaged the observed
column density over galactic longitude and fit the coefficients c1 and c2 of hN (b)i = c1 + c2 cscb
separately at b < −10 deg and b > 10 deg. This functional form is appropriate if the sun is sitting in
the middle of a constant column density hole in the HI layer. In the north we find c 1 = −110 K km s−1 ,
c2 = 197 K km s−1 and c1 = −62 K km s−1 , c2 = 196 K km s−1 in the south. Multiplying by 1.36 to
correct for helium, the total mass surface density of the disk in atomic hydrogen is 7.8 M ¯ pc−2 . The
surface density of ionized material is estimated to be 2 M¯ pc−2 from pulsar dispersion measurements
(KH), while the surface density of molecular material is ≈ 3 M¯ pc−2 (Bronfman et al., 1987), for a
grand total of ≈ 13 M¯ pc−2 .
We adopt the values of Kuijken & Gilmore (1989) to describe the stellar component in the solar
neighborhood: Σs ≈ 35 M¯ pc−2 , consistent with our estimate for the gaseous surface density, the
dynamical total surface density of Kuijken and Gilmore, and no dark matter. The mass-weighted
radial velocity dispersion averaged over z is 40 km s−1 , and the ratio of vertical to radial velocity
dispersion is about 2 (Wielen, 1977).
The rotation constants A, B, Ω, and κ are not independent. We derive them from three observed
quantities: the distance to the galactic center, the slope of the rotation curve, and 2AR 0 . We adopt
R0 = 8.5 kpc, which is consistent with most measurements (see Mihalas & Binney, 1981) although a
more recent best guess value is 7.7 kpc (Reid, 1988). The rotation curve is taken to be flat (β = 0)
with 2AR0 = 220 km s−1 (Gunn et al., 1979).
Our standard solar neighborhood parameters are gathered in Table 1; the inferred values of the
Oort constants, the epicyclic frequency, and other derived quantities are shown in Table 2. The
dimensionless parameters for our model of the solar neighborhood are Qg = 1.2, Qs = 2.8, fg = 0.27,
and β = 0, and the scale of the model is set by κ = 37 km s−1 kpc−1 and Σ = 48 M¯ pc−2 . How
malleable are these numbers? As an example, we shall try to reduce Qs . Caldwell & Ostriker (1981)
find Σ = 82 ± 12, Vc = 243 ± 20, and R0 = 9.1 ± 0.6. If we take Σ = 94, Vc = 220, and R0 = 9.7, and
not varying any of the other input parameters, we find Qs = 0.74. Clearly neither Qs nor the other
model parameters are well constrained.
–9–
3. Theories of GMC Formation
Here we review theories of cloud formation. The topic has also been reviewed by Elmegreen
(1990, 1991, 1992).
Several physical mechanisms may play a role, and we consider each in turn. First, collisions
between clouds are highly dissipative, so clouds tend to stick together. This process can be modeled
with the coagulation equation (Field and Saslaw, 1965). Various instabilities may also play a part:
thermal instability (Field, 1965; Cowie 1980); gravitational instability (Goldreich and Lynden-Bell,
1965; Elmegreen, 1989b); magnetic Rayleigh-Taylor instability (Parker, 1966); or a combination of all
these (Elmegreen, 1990). Correlated supernova explosions may form “supershells” (Heiles, 1976) which
become gravitationally unstable and fragment to form GMCs (e.g. McCray & Kafatos, 1987). Finally,
spiral arms may assemble complexes of smaller clouds into larger objects through orbit crowding (Kwan
& Valdes, 1987; Roberts and Stewart, 1987).
3.1 Coagulation
The idea that interstellar clouds form by a coagulation process may be traced to Oort (1954). This
was later quantified by Field & Saslaw (1965), and refined by Kwan (1979) and others. In simplest
form this model considers an interstellar medium populated by clouds of some fundamental mass m 0 .
These clouds are allowed to collide and stick, forming larger clouds. Once a cloud mass exceeds a
critical mass m1 , it forms stars and fragments back into clouds of mass m0 . The conservation of mass
is then expressed by
m1
X
∂n(m, t)
= −n(m)
n(m0 )hσ(m, m0 )v(m, m0 )i
∂t
0
m =m0
+
m−m
X0
1
n(m0 )n(m − m0 )hσ(m0 , m − m0 )v(m0 , m − m0 )i
2 m=m
0
+²
m1
X
δ(m, m0 )
2m0 p=m
0
0
m1
X
(13)
n(m0 )n(m1 + p − m0 )×
m0 =p
hσ(m , m1 + p − m0 )v(m0 , m1 + p − m0 )i.
where ω(m, m0 ) = n(m0 )hσ(m, m0 )v(m, m0 )i is the collision frequency for clouds of mass m with clouds
of mass m0 . This equation can be solved analytically for certain functional forms of the collision
frequency and for certain values of 1 − ², the “star formation efficiency”. Wetherill (1990) discusses
several exact solutions in the case ² = 0, appropriate for the growth of planetesimals in the solar
nebula; Field and Saslaw (1965) consider the case ω = const., and obtain N (m) ∝ m −1.5 in the limit
of large mass.
A more general argument, given by Kwan (1979), is as follows. Consider the number of clouds
of mass m0 when the system is in a steady state. Assume that σ(m) ∝ ma , hv(m)i ∝ mb , and
N (m) ∝ m² , and compute the collision frequency between clouds of mass m and m0 using the larger
of the two velocities and the larger of the two cross sections. After some manipulation, the gain and
loss terms in eqtn.[13] can be written explicitly as power laws in the maximum mass m 1 with constant
coefficients. If the terms are to balance in the limit m1 → ∞, then the power laws must have the
same exponent, leading to the condition
² = −(a + b + 3)/2
(14)
– 10 –
The result of Field and Saslaw (a = b = 0) is contained as a special case. Elmegreen (1989d) has
argued against a coagulation model because the observed values of b ∼ 0 and constant column density
imply ² ∼ −2. There are at least three answers to this objection: first, it is not yet clear that ² ∼ −2
is excluded by observations; second, since the cloud distribution is nearly a monolayer (the typical
cloud radius is only slightly less than the scale height of the cloud layer), it may be more appropriate
to use σ ∝ R ∝ M 1/2 ; third, the observed cloud radius may not be equal to the cross-section for
coalescence.
Effects not included in the simple coagulation model can radically alter the mass spectrum. First,
gravitational focusing and collective effects can enhance the collision cross section. Second, moderately
supersonic collisions can produce fragmentation rather than coalescence; the mass spectrum is then
sensitive to the “reinjection” spectrum, i.e. the mass spectrum of the fragments (Casoli and Combes,
1982). Third, variations in the surface density of molecular material from arm to interarm regions alter
the collision frequency. Treatments accounting for some of these effects include Casoli and Combes
(1982), and Elmegreen (1989). Finally, the topology of the “clouds” at lower masses is not known and
they may be better described as sheets or filaments.
3.2 Thermal Instability
The original analytic treatment of the thermal instability was by Field (1965), whose exhaustive
treatment includes the effects of thermal conduction, magnetic fields, rotation, and zeroth order density
and velocity variations. In the absence of these complicating effects, the instability criterion can be
written in terms of the heating and cooling functions Γ, Λ erg g −1 s−1 as
µ
¶
∂(Γ − Λ)
> 0,
(15)
∂T
p
that is, the fluid is unstable if at fixed pressure a slightly cooler region cools further. (There can also
be a second, overstable mode corresponding to amplifying sound waves). For example, if the heating
function Γ is but a weak function of density and temperature, and, as for many cooling processes,
Λ ∼ ρ, then the condition given by eqtn.[15] reduces to
µ
¶
∂ ln Λ
< 1.
(16)
∂ ln T ρ
If in addition ∂ ln Λ/∂ ln T > 0, as it usually is in the interstellar medium, then in the limit of large
wavelength (λ À cτcool , and τcool = p/Λρ is the cooling time) the growth rate of the instability
∼ c/λ. Hence fastest growing modes have wavelengths <
∼ cτcool . Schwarz, McCray, & Stein (1972)
have studied the role of the thermal instability in forming clouds in a uniformly cooling medium (in
which case thermal instability sets in if (∂ ln Λ/∂ ln T )ρ < 2). These authors find that the favored
wavelengths are λ ∼ cτcool ∼ 1 pc.
A macroscopic thermal instability might occur if the cloud ensemble is treated as a fluid (e.g.,
Cowie, 1980). This possibility has been considered in more detail by Struck-Marcell & Scalo (1984)
and Scalo & Struck-Marcell (1984).
3.3 Gravitational Instability
Gravitational instability has been proposed as a means of assembling 10 5 M¯ clouds rapidly in
the disk (e.g. Jog & Solomon, 1984; Balbus & Cowie, 1985; Elmegreen, 1989). This possibility
– 11 –
is discussed in detail in Chapters 2 and 3. The linear stability analysis given in Chapter 2 makes
several unjustifiable approximations: that the disk is thin, uniform, and isothermal, and that the
most unstable wavelength is small compared to the size of the galaxy. We discuss ways to relax some
of these assumptions here.
The classic works on gravitational instability in a disk are by Toomre (1964), Goldreich & LyndenBell (1965, hereafter GL), and Safronov (1972). The local stability properties of thin stellar disks may
be summarized by Toomre’s result that
Qs ≡
σr κ
>1
3.358GΣ
(17)
is a necessary and sufficient condition for local stability of a two dimensional (“thin”) stellar disk,
where σr is the stellar radial velocity dispersion, κ the epicyclic frequency, and Σ the surface density.
If we consider the stars alone we find that Qs = 2.8 in the solar neighborhood (see Table 2). For a
thin gas disk, 3.358 → π in eqtn.[17]. Then for the gas alone, we find that Qgas ≈ 1.2 in the solar
neighborhood.
One complication that has not been considered fully in the literature is the effect of mixing stars
and gas in a single system. Since the gravitational acceleration of the two components add, but the
pressure support does not, the combined system is less stable than the individual systems. Jog &
Solomon (1984) have considered the stability of a two-dimensional sheet with two fluid components,
one of which represents the stars and one of which represents the gas. The fluids interact only through
gravity. This system is analytically simple to treat and, as we show in Chapter 2, it closely reproduces
the stability properties of a mixed system of stars and gas. We also treat the nonaxisymmetric
response of a mixed system in Chapter 2, and show that it is much more responsive to forcing than
either system in isolation.
A further complication is the finite thickness of the disk. The stability of a local model of a finite
thickness, isothermal gaseous disk has been considered by GL. They find that, as is also the case for
a zero-thickness disk, only radial (ring) modes need to be considered because the nonaxisymmetric
modes are not formally unstable. Also, it is sufficient to consider the static (ω → 0) limit, because
ω 2 is real and is a continuous function of the model parameters, so ω 2 must pass through 0 if there
is to be an instability. Thus the stability test is equivalent to asking if it is possible to wrinkle the
disk infinitesimally slowly, gradually transforming to a new equilibrium where the pressure gradient
parallel to the disk is balanced by gravitational acceleration. If such a series of equilibria exists, then
the disk is marginally stable or unstable. The condition required of an unstable wavenumber k reduces
to the condition that the energy of the mode vanish:
µ 2
¶ Z ∞
κ
2
δΣ
+
c
+
dz ρ0 δψ1 = 0.
(18)
k2
−∞
Here δΣ is the amplitude of the surface density perturbation, c the sound speed, and δψ 1 (z) the
perturbed potential. If the system is two-dimensional, we retrieve the criterion
Qg ≡
cκ
>1
πGΣ
⇔
STABLE
(19)
as a special case. If the disk has finite thickness, then the evaluation of the integral term in eqtn.[18]
is non-trivial. For an isothermal disk, GL find that eqtn.[18] becomes
1 + m + (1/2)m2 ζ(2, 1 + m/2)
πGρc
=
,
κ2
4m(1 − m2 )
(20)
– 12 –
where m ≡ k/kc , kc ≡ 2πGρc /c2 and ζ(a, b) is the generalized Riemann Zeta-function (e.g. Whittaker
& Watson, 1950: note that GL say ζ(a, s) when they mean ζ(s, 1 + a)). Equation [20] implies that
Qg > Qcr = 0.676 is a necessary and sufficient condition for the stability of the finite thickness
isothermal disk.
Linear theory predicts that the disk will become unstable at a characteristic wavelength, and
hence at a characteristic mass scale:
Mchar ' 4π 4 Q4
G2 Σ3
4c4
=
.
κ4
G2 Σ
(21)
For the standard solar neighborhood parameters in Tables 2 and 3, this characteristic mass is about
2 × 107 M¯ . As we shall see in Chapter 3, eqtn.[21] overestimate the mass of the objects that form in
an unstable disk by a factor of 5 − 10. The growth rate of the instability goes from 0 to ∞ as Q g goes
from 1 to 0, but for Qg = 0.9 × Qcr the growth rate is ≈ κ/2.
In practice the ring instability is irrelevant to galaxies because of the “swing amplification” of
nonaxisymmetric modes (GL, Julian & Toomre, 1966). The swing amplifier can increase the amplitude
of a nonaxisymmetric perturbation in the surface density by a factor of several hundred on a timescale
of ∼ 1/κ. The (temporary) exponential growth of these modes is more rapid than the growth of
the axisymmetric ring modes, so unless δΣ/Σ0 ¿ 0.01, the nonaxisymmetric response of the disk
will dominate, quickly bringing small amplitude nonaxisymmetric perturbations into the nonlinear
regime, where they begin to collapse on a free-fall timescale. As an example of a typical surface
density contrast observed in our own galaxy, the sun is in the middle of a “hole” in the surface density
with δΣ/Σ0 ∼ 0.5. Brinks and Bajaja’s (1986) study of M31 shows that such holes are common in at
least one other galaxy.
The presence of an azimuthal magnetic field does not significantly change the local stability
criterion– it is easy to see that the sound speed in the Q criterion is simply replaced by the
magnetosonic speed. It does, however, significantly complicate the linear analysis of nonaxisymmetric
modes (Elmegreen, 1987). The obviously relevant problem of magnetized disks is only briefly discussed
in this thesis because the tools do not yet exist to study the nonlinear evolution of self-gravitating,
magnetized fluids.
Finally, there is the additional complication of boundary conditions. Events may conspire to
introduce a boundary into the disk, either at the center or the edge, that causes a global instability
in a locally stable system (e.g. Narayan, Goldreich, & Goodman, 1987). The growth time of these
modes is proportional to the size of the disk divided by the group velocity of the density waves, which
8
is always less than the stellar velocity dispersion. The growth time is therefore >
∼ 10 yr, which is long
compared to the time required to disperse a giant molecular cloud by internal star formation, and
hence is not directly relevant to the formation of giant molecular clouds.
3.4 Parker Instability
The Parker instability is one of a class of instabilities that develops when a light fluid pushes on a
heavy one. Other members of this class include the Rayleigh-Taylor instability, convective instability,
and the two-fluid instability of Wardle (1990). The original treatment of the problem was by Newcomb
(1961). The first application to galactic equilibria was by Parker (1966); later refinements were given
by Shu (1974) and Zweibel & Kulsrud (1975) and an application to GMC formation was made by
Blitz & Shu (1980).
– 13 –
Consider a compressible, highly conductive and inviscid fluid with adiabatic index γ containing
a horizontal magnetic field B in equilibrium with a vertical gravitational acceleration g. The stability
criterion is
dρ
ρ2 g
−
>
⇔
STABLE,
(22)
dz
γP
identical to the familiar Schwarzschild convection criterion (Newcomb, 1961). The mode whose
stability is decided by this criterion is characterized by translation of fluid along field lines as it
descends into the valleys in the field, feeding off the gravitational potential. The most unstable
wavelength is ≈ 2πH, where H ≡ (d ln ρ/dz)−1 is the density scale height. The growth time for
the mode is of order H/c, where c is the sound speed, multiplied by a dimensionless function of the
fractional departure of the density gradient from the critical value. For the galaxy H/c ≈ 3 × 10 7 yr.
A second mode with wavevector in the plane but perpendicular to the magnetic field, and more closely
akin to the classical convective instability, sets in when −dρ/dz < ρ2 g/(γP +2PB ), where PB ≡ B 2 /8π
is the magnetic pressure. The growth time for this mode also scales with H/c, and it is characterized
by the vertical interchange of flux tubes.
The effects of rotation, cosmic ray pressure, self gravity, and a cloudy medium have been
incorporated into the treatment of Parker’s instability (see the review of Zweibel, 1987). Rotation
slows the growth rate, while self-gravity increases it. Cosmic ray pressure destabilizes, since at fixed
gas pressure and density it increases the scale height by providing additional support. It is as yet
unclear if the galactic field is coherent enough on horizontal scales of ≈ 1 kpc for Parker’s instability
to develop. Finally, estimates of the growth time including rotation give ≈ 10 8 yr, which may be too
long for the assembly of giant molecular clouds.
3.5 Supershells
McCray & Kafatos (1987) have suggested that the fragmentation of swept-up shells
(“supershells”) of gas around OB associations may form clouds of mass ∼ 10 5 M¯ . This characteristic
mass is obtained by considering the expansion of a shell in a uniform medium and calculating the
radius, solid angle, and time at which a “disk” in the shell has a gravitational potential energy
exceeding its thermal energy. Supershells are observed in our galaxy (Heiles, 1979) and in other
galaxies (e.g. M31; Brinks & Bajaja 1986), where they appear as holes in the disk, occasionally
centered on a visible OB association.
Gravity may enhance the development of supershells. In a marginally stable (Q g ≈ 1) disk
underdense regions produce a gravitational wake, just as overdense regions do (Julian & Toomre,
1966). If supershells can remove material from the plane of the disk, for example by ejection into the
galactic halo, then they will make holes in the disk surface density that form negative surface density
wakes.
3.6 Spiral Arms
Roberts and Stewart (1987) and Kwan and Valdes (1987) have shown that, even in the absence
of collective effects, a spiral potential can produce a high density contrast between arm and interarm
regions. Roberts and Stewart suggest that some of the large inner-galaxy molecular clouds may not
be gravitationally bound at all, or only marginally bound. We shall have little more to say about this
theory except to note that almost all the processes already considered proceed more rapidly when the
– 14 –
surface density increases. This does raise the question, however, of what it means for a cloud to be
bound, and what is the binding energy of a molecular cloud.
For a particle orbiting in a disk on a nearly circular orbit there is an energy-like integral analogous
to the Jacobi constant:
1
Γ = v 2 + φ + 2AΩ(R − R0 )2
(23)
2
where A > 0 is Oort’s constant A, Ω < 0 is the rotation frequency, v is the velocity in the rotating
frame, and φ is potential, with the axisymmetric galactic potential subtracted. The last term comes
from an expansion of the effective galactic potential around R0 . The zero velocity surfaces of Γ near
a point mass in the plane of the disk are shown in Figure 6. A particle near a point mass M c is bound
if Γ < Γcr = 1.5(G2 Mc2 /4AΩ)1/3 .
Now consider the binding energy of a molecular cloud. Suppose a cloud is observed to be selfsimilar between an outer scale L0 (which we assume is approximately the size of the cloud) and an
inner scale L1 . The cloud has an angular correlation function of column densities
w(θ) ∼ θ −n
(24)
between θ = L1 /d and θ = L0 /d. Assuming the structure within the cloud is isotropic, this implies
that the three-dimensional dimensionless autocorrelation function ζ(r) ∼ r −n−1 and that the power
spectrum of the density is hρ2k i ∼ k n−2 . Poisson’s equation relates the potential to the density
−k 2 φ̃ = 4πGρ̃, where tilde denotes a fourier transform. The gravitational binding energy is then
õ ¶
!
Z
Z 2π/L1
1−n
1
1
L
GM 2
1
−4πGρ̃2 dk ∼
W ≡
(25)
ρφd3 V ∼
−1
2
1−n
L0
L0
2π/L0
Thus if n < 1 the binding energy is dominated by the large scale structure, but if n > 1 it is dominated
by the smallest scale structure. The configuration with the minimum binding energy is a homogeneous
cloud– as clumping progresses the binding energy increases.
Alternatively, consider a molecular cloud that is not self-similar but has dense clumps orbiting
ballistically in a tenuous interclump medium (e.g. Bertoldi & McKee, 1991). If one assumes the
clumps have constant density and a mass spectrum N (M ) ∼ M −α , then the distribution of binding
energy is W ∼ N (M )M 5/3 ∼ M 5/3−α . Most measurements suggest α ∼ 1.5, so most of the internal
binding energy is in the largest clumps– as is, not coincidentally, most of the star formation (Lada,
1990). How does the binding energy of the clumps compare to the overall binding energy of the
clouds? For a cloud model composed of equal mass, constant density spherical clumps orbiting in an
interclump medium of negligible mass, the sum of the binding energy of the individual clumps will
always exceed the total clump-clump binding energy. If the clumps are not homogeneous then they
contain an even larger fraction of the binding energy.
4. Constraints on Cloud Formation Theories
The observations can now be distilled into a list of constraints on theories of giant molecular
cloud formation.
First, any theory should be able to produce objects of the same mass as nearby, carefully observed
clouds such as Orion and the Rosette, i.e. a few ×105 M¯ .
Second, it is observed that the number of clouds is a decreasing function of the mass. It is often
stated that the mass spectrum is N (M ) ∼ M −1.5 but observational studies (Liszt, Xiang, and Burton,
– 15 –
1981; Drapatz & Zinnecker, 1983; Casoli, Combes, and Gerin, 1984; Terebey et al., 1986; Solomon et
al., 1987; Solomon and Rivolo, 1989; Leisawitz, 1990) do not strongly constrain the functional form
of the mass spectrum. This is because the studies are complete over only a small range in mass,
and because velocity crowding may be producing objects in the inner galaxy that appear to be very
massive but are not. For example, some of the most massive clouds in the survey of of Dame et al.
(1987) were found to break up into much smaller clouds at higher angular resolution (Issa, MacLaren,
& Wolfendale, 1990), and none of their clouds more massive than 106 M¯ are closer than 2.3 kpc.
In short, the only statement we can make with confidence is that the mass spectrum is a decreasing
function of the mass.
Our third constraint comes from the observation that clouds are not rapidly spinning (Blitz, 1991).
The clouds listed by Blitz as having the largest velocity gradients are all spinning in a retrograde
sense (if they are spinning at all– the velocity gradients can also be explained by uniform expansion
or contraction of the cloud). Hence we must either form objects that are slowly rotating or explain
how to de-spin them fast enough so that rapidly rotating clouds are never observed in CO.
Fourth, clouds should form quickly, on timescales of a few ×107 yr. This constraint is controversial
and not as well established as the preceding ones, so we shall discuss the evidence in detail. One argues
that because massive stars require only a few ×107 yr to destroy a giant molecular cloud, and because
almost all giant molecular clouds in the solar neighborhood are forming massive stars, cloud lifetimes
7
must be <
∼ 2 × 10 yr, and so the time required for cloud formation must be of the same order.
Unfortunately, the various methods used to estimate cloud lifetimes, which we shall describe below,
do not all measure the same timescale.
One means of measuring cloud lifetimes uses the abundances of molecules as “chemical clocks”
(Stahler, 1984), and evidently awaits more complete understanding of the internal chemistry and
dynamics of interstellar clouds. The timescale measured by this method is the duration of a parcel
of gas in a particular physical state. If we suppose (undoubtedly incorrectly) that the flow inside
a molecular cloud of size L can be described as homogeneous, isotropic turbulence with outer scale
λ and velocity dispersion (the velocity of the largest eddies) = σ, then the timescale for the a fluid
element to change its physical state by moving across the cloud is ∼ (λ/σ)(L/λ) 2 .
A second method measures the timescale over which clouds are detectable in CO by observing
that if there is an excess of molecular clouds in spiral arms compared to what would be expected from
the continuity equation, then the arm transit time is comparable to the cloud lifetime (Bash & Peters,
1976; Dame et al., 1986; in M51, Vogel, Kulkarni, & Scoville, 1988). If a fraction y of the gas (HI +
H2 ) is in the spiral arms at some fixed galactocentric radius, then conservation of mass implies that
the arm transit time is
2πy
τarm ≈
(26)
m(Ω − Ωp )
where m is the arm multiplicity, Ω the rotational frequency, and Ωp the pattern frequency. If the 3 kpc
expanding arm is close to the inner Lindblad resonance, as has been suggested by Yuan (1984), then
Ω − Ωp ≈ κ ≈ 2κ¯ ≈ 2.4 × 10−15 s−1 . If m = 2, then τarm = y × 4.2 × 107 yr. It is not entirely clear
that there is an excess of clouds in spiral arms in our galaxy above that expected from the continuity
equation, although there is strong evidence that the arm-interarm contrast is high (Stark, 1979).
A third method measures the time required to clear the gas from young clusters using nuclear
−2
7
clocks. The lifetime of a star with M >
∼ 7 M¯ is ≈ 10 (M/15 M¯ ) , where we have normalized to a
B0 star of 15 M¯ and luminosity 2.5×104 L¯ (from Mihalas & Binney, 1981). By selecting associations
of massive stars, which form only in giant molecular clouds (e.g. Blitz, 1991, and references therein)
– 16 –
and searching for nearby molecular material, one obtains an upper limit on the time required to
7
remove the molecular gas from the neighborhood of the cluster of <
∼ 2 × 10 yr (e.g. Bash, Green, and
Peters, 1978; Leisawitz, 1990). Since only one local giant molecular cloud is devoid of internal star
formation (Maddalena’s cloud– see Blitz, 1991), one concludes that the interval over which a giant
7
molecular cloud is detectable in CO is <
∼ 2 × 10 yr.
The lifetime of an interstellar cloud as a GMC recognizable in CO emission is not necessarily
the same as the interval over which the cloud is gravitationally bound. There might, for example, be
a population of massive, long-lived atomic clouds that become molecular only near the end of their
lives. Nevertheless, the dense objects with large visual extinction that are observed in CO must form
7
very rapidly, on a timescale of <
∼ 10 yr, to satisfy the constraint discussed in the last paragraph. We
therefore favor models that form dense cloud rapidly.
A fifth constraint comes from the observations of Kennicutt (1989) on the relation between the
star formation rate (as measured by Hα surface brightness), gaseous surface density, and epicyclic
frequency in a sample of late-type spirals. Kennicutt finds that: (1) There is a sharp cutoff in the star
formation rate at the edges of galactic disks where the Qg parameter rises above a critical value (the
gaseous velocity dispersion is assumed to be 6 km s−1 for all the galaxies in Kennicutt’s sample). (2)
Qg is approximately constant throughout the star forming region of galactic disks. (3) The molecular
surface density (as measured by 12 CO intensity) is poorly correlated with the star formation rate and
there is no correlation between Qg and the transition point from an atomic to a molecular medium.
A natural interpretation of the first two results is that the Qg star formation threshold is really a
threshold for the formation of massive gas clouds by gravitational instability, and that the massive
stars that are observed in Hα are in turn formed solely within these clouds, as is observed in our galaxy.
Kennicutt’s third result seems to contradict this, but only if we assume that massive, self-gravitating
clouds are well traced by 12 CO surface brightness. 12 CO emission may trace different populations of
interstellar clouds in different galaxies, depending on the metallicity, temperature and density, so we
are probably safe in ignoring this apparent contradiction. Nevertheless, if we adopt the hypothesis
that massive stars form only in massive, self-gravitating clouds, as is observed in our own galaxy, then
Kennicutt’s observations imply that there should be a threshold Qg for the formation of these clouds.
As our fifth constraint, then, we require that the growth rate of giant molecular clouds be a sensitive
function of Qg .
5. Conclusions
To summarize, we have distilled five constraints on cloud formation theories from the observations.
Cloud formation theories must: (1) produce objects of mass similar to well observed local clouds such
as Orion, i.e. about 105 M¯ ; (2) make more small clouds than large clouds; (3) form clouds quickly, on
7
a timescale of <
∼ 10 yr; (4) make objects that are slowly rotating or rotating in a retrograde fashion;
(5) have a formation rate that is a sensitive function of Qg .
We are encouraged by the observational result of Kennicutt (1989) (that according to the Q g criterion the gas in disk galaxies is only marginally stable in regions where there is active star
formation) to investigate models of cloud formation that incorporate only the physics that enters into
the Qg -criterion: pressure, self-gravity, and rotation. The rest of this thesis is devoted to investigating
two such models.
In the first model we imagine that clouds form by gravitational instability from a disk that is
initially smooth and isothermal. It is natural then to first examine the linear theory of galactic disks.
– 17 –
This topic has received exhaustive attention (Toomre, 1964; Julian & Toomre 1966; Lynden-Bell &
Kalnajs 1972; Lin, Yuan, & Shu 1969; Lin & Shu 1966; Bertin et al. 1989, to name but a few), but
one aspect that has not been treated completely is the behavior of mixed systems of stars and gas.
We attempt to remedy this defect in Chapter 2. The nonlinear theory of waves in disks continues to
excite interest, despite early work by Roberts (1969), more recent work by Lubow, Balbus, & Cowie
(1986), and considerable attention to nonlinear density waves in planetary rings (e.g. Shu, Yuan, &
Lissauer, 1985). In Chapter 3, for the first time, we are able to investigate numerically the nonlinear
development of gravitational instabilities in a disk model whose vertical structure is fully resolved and
for which the dynamical effects of the stellar component are self-consistently included. After selecting
clouds from the simulations by a density threshold criterion, we shall find that gravitational instability
can produce objects that look very much like giant molecular clouds in the solar neighborhood.
In the second model one imagines that small dense clouds nucleate the formation of giant
molecular clouds and grow by accretion. Self-gravity can enhance the rate of growth, but generally
speaking accretion in a disk is a slow process because it is inhibited by tidal forces. In chapter 4
we examine whether this model (and the gravitational instability model) can produce objects with
rotation rates comparable to those observed in nearby giant molecular clouds.
– 18 –
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– 22 –
Figure Captions
1. A schematic picture of the Orion region, reproduced from Kutner et al., 1978.
2a. 12 CO brightness contours in the Orion regions, from Kutner et al., 1978.
2b. 13 CO brightness contours in the Orion regions, from Kutner et al., 1978.
3. Position of nearby molecular clouds projected onto the galactic plane, reproduced from
Dame et al., 1987.
Figure 4. Surface density, scale height, and midplane density and location, from Bronfman et al.,
1987.
Figure 5. Location of clouds from the catalogue of Solomon et al., 1987, projected onto the galactic
plane.
Figure 6. Zero-velocity surfaces for motion in the local model of the galactic disk near a point mass
of unit mass.
Figure
Figure
Figure
Figure
TABLE 1
Observed Solar Neighborhood Parameters
symbol
value
comments
Σg
Σs
R0
2AR0
d ln Vc /d ln R
σg,r
σs,r
z0,g
z0,s
σs,z /σs,r
13 M¯ pc−2
35 M¯ pc−2
8.5 kpc
220 km s−1
0
6 km s−1
40 km s−1
100 pc
250 pc
0.5
gaseous surface density
stellar surface density
distance to GC
flat rotation curve
gas radial velocity dispersion
lower in plane
differs with component
TABLE 2
Inferred Solar Neighborhood Parameters
symbol
value
comments
Vc
κ
220 km s−1
37 km s−1 kpc−1
1.2 × 10−15 s−1
12.9 km s−1 kpc−1
−12.9 km s−1 kpc−1
−25.9 km s−1 kpc−1
−8.4 × 10−16 s−1
1.2
2.8
154 pc
1.14 × 106 M¯
2.7 × 107 yr
5.6 km s−1
circular velocity
epicyclic frequency
A
B
Ω
Qg
Qs
L ≡ GΣ/κ2
M ≡ ΣL2
T ≡ 1/κ
V ≡ GΣ/κ
Oort’s constant
Oort’s constant
Vc /R0
σg κ/πGΣg
σs κ/3.358GΣs
length unit
mass unit
time unit
velocity unit
TABLE 3
Galactic CO Surveys
beama
coverage
spacing
line
sensitivity
∆v
Columbiab
0.5 degc
all l,
−10 < b < 10,
−20 < b < 20f
fully
sampled
12
CO J = 1→0
0.1 − 0.3 K
0.65 km s−1 (No.),
1.3 km s−1 (So.)
FCRAOd
4400
−1.05 < b < 1.00,
8 < l < 90,
18 < l < 55
12
CO J = 1→0
0.4 K
1 km s−1
12
CO J = 1→0
CO J = 1→0
0.15 K
0.3 K
0.68 km s−1
0.65 km s−1
BTLe
a
10000
−5 < l < 122,
−1 < b < 1
60
30
30 , 0.750f
Full Width at Half Maximum.
Dame et al., 1987.
c
“Superbeam” mode. The instrumental FWHM is 8.70 .
d
Sanders, Scoville, & Solomon, 1984.
e
Stark et al., 1988.
f
Partial coverage.
b
13
Chapter 2: Linear Theory of Mixed Star and Gas Disks
This chapter considers the stability and responsiveness of a local, two-dimensional model of a disk
composed of stars and gas. The model is described by four parameters: the ratio of the stellar surface
density to gas surface density, a temperature for each component, and the logarithmic derivative of
the rotation curve.
The stability of the local model to radial (ring) modes is examined, and it is shown that the
surface of neutral stability in parameter space is similar to that for a two-fluid model that represents
the stellar component as a hot fluid. The amplification of nonaxisymmetric modes is also investigated,
both by analytic means in the tightly wound (WKBJ) limit, and by direct numerical integration of
the coupled linearized Boltzmann and fluid equations.
We find that (1) the star and gas system is always less stable than the individual components
in isolation, so that there are regions of parameter space where the stars and the gas would be
stable in isolation, but when combined are not. (2) The combined star and gas system is much more
responsive to nonaxisymmetric perturbations than the isolated components. (3) The phase transition
discussed by Bertin & Romeo is an artifact of the restriction of their treatment to radial modes.
When nonaxisymmetric modes are considered, the transition from a star-dominated response to a
gas-dominated response is smoothed out. (4) The presence of gas makes it possible for a density wave
to cross the Lindblad resonance. (5) The stars and gas respond at different characteristic wavevectors;
for parameters appropriate to the the solar neighborhood the stars respond at a longer wavelength
and with a more open (more nearly azimuthal) wavevector than the gas.
We then consider a local model of the solar neighborhood and of a young galactic disk. The
characteristic wavelength for the linear response of a mixed star and gas system with parameters
appropriate to the solar neighborhood is ≈ 2 kpc, similar to the local spacing between spiral arms.
Young galactic disks are difficult to stabilize, and may actually become more unstable by converting
stars into gas.
1. Introduction
Recent theoretical studies of galactic disks have incorporated both gas and stars as dynamically
significant components (Bertin et al., 1989; Bertin and Romeo, 1988; Balbus, 1988; Lubow, 1986;
Lubow, Balbus, and Cowie, 1986; Jog and Solomon, 1986, hereafter JS). Earlier work explored some
aspects of disks in which both stars and gas participated in density waves: Lin, Yuan and Shu (1969)
considered linear, tightly wound waves in a mixed star and gas disk; calculations by Roberts (1969)
and collaborators (Shu et al., 1972) incorporated a nonlinear gaseous response to an imposed spiral
potential. More recently Lubow (1986) has considered the linear theory of tightly wound waves in
a star and gas disk. However, there is no linear analysis of the nonaxisymmetric response of mixed
disk systems in the literature that does not use the tight-winding approximation. The purpose of this
chapter is to remedy this defect by analysing a mixed star and gas system in the simplest possible
context, the local model of Goldreich and Lynden-Bell (1965, hereafter GL) and Julian and Toomre
(1966, hereafter JT).
Several lines of observational evidence support the idea that, for some galaxies, the self-gravity of
the gas is at least as important as the self-gravity of the stars. Kennicutt (1989) has shown that, for a
sample of spiral galaxies, the gaseous component is hovering on the verge of gravitational instability
(the Q parameter for the gas is ≈ 1). Observations of the interstellar medium in the solar neighborhood
– 24 –
also indicate marginal stability (Chapter 1). Kuijken and Gilmore’s (1989) dynamical estimate of the
surface density of the disk in the solar neighborhood suggests a somewhat lower value for the surface
density of stars (35 M¯ pc−2 ) than earlier estimates that, coupled with recent lower estimates for the
distance to the galactic center and a correspondingly higher epicyclic frequency, implies that locally
the stellar component is more stable than previously thought (Toomre, 1974; perhaps the strongest
conclusion that can be drawn from Kuijken and Gilmore’s work is that the stability parameter is poorly
constrained). That spiral structure is rarely observed in galaxies without a Population I component
(for two exceptions, see Strom, Jensen, and Strom, 1976) also argues for the importance of self-gravity
in the gas.
Theoretical prejudice also supports the idea that the gaseous component should be close to
gravitational instability. The cooling timescale for most of the mass in the interstellar medium is
short compared to the dynamical time (Spitzer, 1978) so the ISM must be regularly stirred and heated.
That this stirring and heating is regulated by a feedback loop involving star formation is an appealing
notion that is consistent with all observational evidence. On the other hand, many mechanisms are
available to heat the stellar component (Spitzer & Schwarzschild, 1953; Lacey & Ostriker, 1986; all
reviewed by Binney & Tremaine, 1987) while none are available to cool it except rapid conversion of
gas into stars of low velocity dispersion. Numerical simulations of disks (e.g. Sellwood & Carlberg,
1984) confirm that it is exceptionally difficult to make cool stellar disks because nonaxisymmetric
disturbances are vigorously amplified.
The disk model used here to investigate mixed star and gas disks is sometimes known as
the shearing sheet. It has been exhaustively studied as a general model for differentially rotating
astrophysical fluids (Narayan, Goldreich, & Goodman, 1987; JT; GL). In simplest form it considers
the disk in the neighborhood of some fiducial point and idealizes it as an infinite two-dimensional
sheet with uniform surface density and shear flow. The local model has the virtue that it is an
extremely simple system with a small number of free parameters, and that it captures almost all of
the important physics. However, the principal approximation of the local model, that the size of
the region considered is negligible in comparison to the size of the system as a whole, is marginal in
applications to disk galaxies.
The local approximation requires that some characteristic wavelength for the disk, λ c , be much
smaller than the galactocentric radius of the local patch of disk being considered in the model. If
the stars and gas are lumped together and treated as a fluid with a single velocity dispersion c,
then λc = 2π 2 Q2 GΣ/κ2 is an appropriate characteristic wavelength (Q ≡ cκ/πGΣ > 1 implies local
stability of a thin [two dimensional] gaseous disk, where κ is the epicyclic frequency and Σ is the total
surface density; we assume that Q ∼ 1 throughout). Then the local approximation requires that
λc = 2π 2 Q2 GΣ/κ2 ¿ R0 .
(1)
Estimates of λc for spiral galaxy disks imply λc /R0 ∼ 1. If the surface density and Q for the gas alone
are used to estimate λc then λc is reduced by the ratio of the gaseous to total surface density (≈ 4 in
the solar neighborhood), and the local approximation is more attractive.
A second approximation that will made here is that the disk is two-dimensional, i.e. that the
scale height z0 is much less than the characteristic wavelength λc :
1
πQ2 GΣ/κ2 ¿ λc ,
(2)
2
the expression for z0 being valid for a self-gravitating isothermal sheet. But λc /z0 = 4π. Corrections
must then be made for thickness effects, the most important being a reduction in the self-gravity of the
z0 =
– 25 –
disk that modifies the stability criterion for a self-gravitating isothermal sheet in vertical equilibrium
to Q > 0.68 (GL).
Two further idealizations are made to reduce the problem to tractable form. The interstellar
medium is treated as a uniform, isothermal fluid and the density fluctuations are taken to be linear
perturbations around this steady state. It hardly needs to be said that the interstellar medium is not
well represented by a uniform fluid: the sound speed and density vary by orders of magnitude over
scales comparable to those considered here. And while the stellar component may well experience
changes in surface density that are substantially less than unity during passage through a spiral arm,
the arm-interarm surface density contrast in the gas in our galaxy and in similar galaxies is probably
> 1. Nevertheless, it is of interest to fully understand the behaviour of this simple model before
∼
turning to more difficult and complicated ones.
Finally, the infinite shearing sheet model used here does not capture global instabilities that are
known to set in, even in a locally stable system, when radial boundaries are introduced (e.g. Narayan,
Goldreich, and Goodman, 1987). A single boundary creates a resonant cavity between the boundary
and a forbidden region around the mode’s corotation radius. The mode, which has negative angular
momentum (if the boundary is inside corotation), grows because angular momentum leaks through the
corotation barrier. The boundary might be identified with disk edges, if the surface density declines
sharply enough (see mode D of Toomre & Zang in Toomre, 1981) or disk centers, if the velocity
dispersion and rotation frequency are arranged to vary in the right way (Mark, 1976c).
This chapter is organized as follows. §2 outlines the procedure for the calculation of the linear
response of the combined system. The discussion in §2 is rather technical, while later sections
bear more directly on possible observational implications. §3 derives the dispersion relation for
axisymmetric modes and discuss the implications for local stability and the propagation of wave
packets in the disk. §4 discusses issues related to the amplification of nonaxisymmetric waves. §5
summarizes the results of the previous sections and discusses applications.
2. Calculation of Linear Response
To establish a local model of the disk, we select a fiducial point in the disk and erect a cartesian
coordinate system whose x-axis points outward in radius and whose y-axis points in the direction of
increasing φ. This cartesian coordinate system rotates with the fiducial point about the center of the
system, so it is always oriented in the same fashion with respect to r and φ. Next we neglect terms
in all the relevant equations of more than linear order in the ratio of departures from this fiducial
point to the galactocentric radius. This is essentially the epicyclic approximation, and the equations
of motion for a point mass within this system are sometimes known as Hill’s equations (Hill, 1878).
One is left with an infinite sheet in which differential rotation appears as a linear shear v y = 2Ax,
where A is Oort’s constant A.
Nonaxisymmetric modes have the general form eik·x , where k is oriented at some non-zero angle
with respect to the radius vector. Differential rotation will advect the wave crests with it, so that
the magnitude and orientation of k change. Ultimately k becomes very large and nearly radial. If
τ ≡ 2A(t − t0 ), then
kx = −τ ky
ky = const.,
(3)
so that k makes an angle φ ≡ tan−1 (τ ) with a curve of constant radius. At τ = 0, then, the
wave is azimuthally directed. Waves at negative and positive τ are described as leading and trailing,
respectively.
– 26 –
The Euler and continuity equations can be combined into a single equation which clearly exhibits
the destabilizing effect of the shear and the stabilizing effect of rotation (Hunter, 1964). This equation
can be directly linearized and rewritten as a second-order ordinary differential equation for the
evolution of the mode amplitude in either the potential or the surface density, which are connected
by the Poisson equation. In the potential, this equation has the simple form
d2 ψ̃g
+ S(τ )ψ̃g = f (τ )ψ̃i .
dτ 2
where ψ̃g is the potential amplitude of the shearing wave, and ψ̃i is the amplitude of a shearing fourier
component of the imposed potential.
The function S(τ ) is positive at late times for any choice of parameters for the disk, even those
unstable to radial modes, and the solution becomes oscillatory. There are two kinds of minima in S(τ ):
the first is associated with the self-gravity of the gas and causes an amplification of the wave whose
magnitude has been calculated analytically in the tight-winding limit both for stars (Mark, 1976b)
and for gas (Goldreich & Tremaine, 1978; hereafter GT). The second minimum is purely a result of
differential rotation and results in an amplification of the wave even in the absence of self-gravity
(Narayan, Goldreich, and Goodman, 1987). When these two minima nearly coincide one obtains the
large amplifications observed in the numerical integrations of GL and JT.
The situation for the stars is more complicated. The evolution of the distribution function
for the stars is governed by the collisionless Boltzmann equation. When expressed in action-angle
variables, the Boltzmann equation reduces to a particularly simple form that can be integrated along
the unperturbed orbits to obtain the distribution function at later times. Given a functional form for
the zeroth-order distribution function, one can recover the surface density by performing an integration
over velocity space at fixed position. The result is a Volterra integral equation for the evolution of
the wave amplitude.
There is confusion over the definition of the Oort constants because the galaxy’s angular
momentum vector is more nearly anti-parallel to the Earth’s than parallel, implying the rotation
frequency Ω < 0. We adopt the convention of GT and GL,
A=
r dΩ
.
2 dr
(4)
Then if Ω < 0, A > 0, A + Ω = B, and 4BΩ = κ2 , where κ is the epicyclic frequency. In terms of
β ≡ d ln |vc |/d ln r, the logarithmic derivative of the rotation curve,
1−β
A
=p
κ
8(1 + β)
B
=−
κ
r
1+β
8
−2
Ω
=p
κ
8(1 + β)
A natural choice of units is G = κ = Σ = 1, where Σ is the total surface density. The fractional gas
and stellar surface densities are denoted fg ≡ Σg /Σ and fs ≡ Σs /Σ. The unperturbed velocity field
for the gas is ẋ = 0, ẏ = 2Ax, identical to the unperturbed mean velocity field for the stars (the
mean velocities of stars and gas must be equal in the local model). The x and y components of the
perturbed velocity field are denoted u and v respectively. In general perturbed quantities are in lower
case, while zeroth-order quantities are subscripted with a 0. With this choice of units a mixed star
and gas system is described by four parameters: β; a dynamical temperature for the stars expressed
in terms of the dimensionless parameter Qs ≡ σr κ/3.358GΣg (σr is the radial velocity dispersion of
– 27 –
the stars); a dynamical temperature for the gas expressed in terms of the dimensionless parameter
Qg ≡ cκ/πGΣg (c is a velocity dispersion or effective sound speed); and the fractional surface density
of the gas.
2.1 Linear Gaseous Response
The equations governing the evolution of the gas are the two dimensional continuity equation
the Euler equations
D ln Σg
= −∇·V ,
Dt
(5)
DV
= −∇η
Dt
(6)
∇2 Φ = 4πG(Σg + Σs )δ(z).
(7)
and the Poisson equation
Here D/Dt is the total (or convective) derivative, V is the velocity in an inertial frame, η ≡ Φ + Φ 0 +
Φi + c2 ln Σg , Φ0 is a fixed axisymmetric potential, Φ is the rest of the potential associated with the
disk, and Φi is an imposed external potential.
A useful constraint is provided by Kelvin’s theorem
I
Γ = V ·dl = const.
(8)
restricted to two-dimensional fluid flow, where it becomes the conservation of “potential vorticity”
(e.g. Pedloskey, 1987)
µ ¶
Dξ
ω
D
=
=0
(9)
Dt
Dt Σg
where ω ≡ ∇ × V is the vorticity, and V is the velocity in an inertial frame (we shall use ω to denote
a frequency later on, but the meaning should be clear from the context). Eqtn.[9] holds in the absence
of magnetic fields and viscosity, provided that the pressure acceleration can be written as the gradient
of a scalar field (e.g. for an isentropic or isothermal fluid). For a shearing sheet with constant surface
density and velocity field V = 2Axŷ, ξ = 2B/fg , where B is Oort’s B constant, since B = Ω + A.
Now take the divergence of the Euler equations and D/Dt of the continuity equation, then
eliminate DV /Dt to obtain
∂Vj ∂Vi
D2 ln Σg
.
(10)
= −∇2 η +
Dt2
∂xi ∂xj
The final term can be rewritten in terms of the vorticity ω and the rate-of-strain tensor e ij =
(∂Vi /∂xj + ∂Vj /∂xi )/2 as
∂Vj ∂Vi
1
= eij eij − ω 2 ,
(11)
∂xi ∂xj
2
so that
1
D2 ln Σg
= −∇2 η + eij eij − ω 2
(12)
2
Dt
2
(Hunter, 1964; GL). In this equation one clearly sees the stabilizing effect of the vorticity (the term
involving the vorticity is negative definite) and the destabilizing effect of the shear (the strain term is
positive definite).
Equation [12] is readily evaluated in the rotating frame of the local model; since rotation produces
no contribution to the rate-of-strain tensor, rotation enters only in the vorticity term. Now linearize
– 28 –
eqtn.[12] in the rotating frame, sending Σg →fg + σg , η→η0 + η1 , V →Ω × r + 2Axŷ + ux̂ + v ŷ, and
ω→ω0 + ω1 to obtain
¶
µ
1 D 2 σg
∂u ∂v
2
− ω 0 ω1 .
(13)
= −∇ η1 + 2A
+
fg Dt2
∂y
∂x
The perturbed vorticity ω1 can be rewritten using the conservation of potential vorticity: ω1 =
ξ0 σg + ξ1 fg , where ξ1 , a free function, is the perturbed potential vorticity and ξ0 = 2B/fg is the
original potential vorticity. Next, decompose the perturbation into a sum of shearing waves of the
form exp(iky (−τ x + y)) where τ = 2A(t − t0 ). Terms with this dependence removed are marked by a
tilde. Then ∂/∂x→ − iky τ, ∂/∂y→iky , and D/Dt→2Ad/dτ . Making these substitutions in eqtn.[13],
eqtn.[9], and eqtn.[5], we obtain, after some manipulation,
!
µ
¶Ã
fg ky2 (1 + τ 2 )
2τ dσ̃g
8AB
1
ξ˜1
σ̃g
d2 σ̃g
=
−
η˜1 − 1 +
,
(14)
+
2
2
2
2
dτ
1 + τ dτ
4A
1+τ
fg
ξ0 4A2
From now on, for the sake of simplicity, we shall set the perturbed potential vorticity to 0. (In
obtaining eqtn.[14], we have used the identities B = Ω + A and 4BΩ = κ2 = 1). Using the definition
of η1 and the Poisson equation, η̃1 = (c2 /fg − 2π/|k|)σ̃ + ψ̃s + ψ̃i , where k ≡ ky (1 + τ 2 ) and ψ̃s is the
stellar potential and ψ̃i is the imposed external potential. Then
µ
¶
´
d2 σ̃g
2τ dσ̃g
1
fg k 2 ³
8AB
2 2
=
−
+
1
+
c
k
−
2πf
|k|
σ̃
−
(15)
ψ̃
+
ψ̃
g
i
s
dτ 2
1 + τ 2 dτ
4A2 1 + τ 2
4A2
In numerical work we shall need to add a bulk viscosity term of the form ²∇(∇·v) to the left hand
side of eqtn.[6]. Potential vorticity is still conserved to linear order, and −²k 2 (dσ̃g /dτ )/2A is added
to the right hand side of eqtn.[15].
Ignoring viscosity for now and eliminating the surface density in favor of the potential in eqtn.[15],
µ
¶
1
12A2
8ΩA
d2 ψ̃g
2 2
ψ̃g =
+
1
−
2πf
k
+
k
c
+
+
g
dτ 2 4A2
(1 + τ 2 )2
(1 + τ 2 )
(16)
´
πfg k ³
ψ̃s + ψ̃i ,
2A2
which is of the form promised above.
An equation for the evolution of radial modes may be derived from eqtns.[11] on the assumption
that the solution is of the form exp(ikx x), or from eqtn.[16] by taking the limit ky →0 at fixed kx .
Then
³
´
¢
d2 σ̃g ¡
(17)
+ 1 + kx2 c2 − 2π|kx |fg σ̃g = −kx2 fg ψ̃s + ψ̃i .
2
dt
These equations are identical to those obtained by GT except for notation, and a misprint in GT,
eqtn.[35a].
2.2 Linear Stellar Response
The fundamental equation governing the response of the stars to an imposed perturbation is the
collisionless Boltzmann equation
∂F
∂F
+ α̇i
= 0,
(18)
∂t
∂αi
where F is the distribution function and αi are arbitrary phase-space coordinates. A natural choice
of coordinates is actions and action-angles (e.g. Lynden-Bell and Kalnajs, 1972). If one defines
Jr ≡
1 2
κa
2 r
Jφ ≡ 2Bxg
(19a)
– 29 –
where ar is the epicyclic amplitude of an individual star and xg is the x-coordinate of its guiding
center, then Jr is essentially the epicyclic energy, while Jφ is a linear function of the z component of
the star’s angular momentum. Jr and Jφ are canonically conjugate momenta to the angles Θr and
Θφ , which are the epicyclic phase and the y-coordinate of the star’s guiding center. Thus
x=
p
2Jr cos Θr +
y = −2Ω
The Hamiltonian is
p
1
Jφ
2B
(20a)
2Jr sin Θr + Θφ .
(20b)
A 2
J ,
2B φ
(21)
H = Jr +
and the equations of motion are, of course,
J̇ = −
∂H
∂Θ
Θ̇ =
∂H
.
∂J
(22)
An equilibrium distribution function depends only on the actions Jr and Jφ .
The collisionless Boltzmann equation is now
∂F
∂F
∂F
+ Θ̇
+ J̇
= 0.
∂t
∂Θ
∂J
Writing F = F0 + f , H = H0 + ψ, where ψ is the perturbing potential, and linearizing,
¶
µ
∂H ∂
∂F0 ∂ψ
∂
+
.
f=
∂t
∂J ∂Θ
∂J ∂Θ
(23)
(24)
The derivative on the left is the total (convective) derivative along the unperturbed orbit. Rewriting
as such and integrating,
Z
∂ψ
∂F0 t
dt0
f=
.
(25)
∂J −∞
∂Θ
If F0 depends only on Jr (the sheet is initially uniform) then
f=
∂F0
∂Jr
Z
t
dt0
−∞
∂ψ
.
∂Θr
(26)
In this form the linearized Boltzmann equation admits a simple physical interpretation. The integral
is proportional to the potential drop along an epicycle. If the potential is decreasing along the epicycle
(the epicyclic motion is being forced) and the distribution function is a decreasing function of radial
action, then the action of each star is increased and at fixed action, phase space density increases
(f > 0).
To obtain the perturbed surface density from the perturbed distribution function, one must
perform an integral over velocity space at fixed position. This is done by fixing x and y in eqtn.[20],
eliminating two of the coordinates J , Θ in terms of the other two, then performing an integration
over the remaining phase space coordinates against the velocity space volume element. We choose to
eliminate Jφ and Θφ , so that the volume element is dJr dΘr |∂(vx , vy )/∂(Jr , Θr )| = |2Ω|dJr dΘr and
σ(x, y) = |2Ω|
Z
∞
dJr
0
Z
2π
dΘr f (Jr , Θr , x, y).
0
(27)
– 30 –
With this program in mind, we assume the perturbing potential has the form
ψ(t1 ) = ψ̃(t1 ) exp {iky (−2At1 x + y)} ,
(28)
where we have defined 2At1 ≡ τ = 2A(t − t0 ) = 2At + τ0 to simplify the calculation. Reexpressing ψ
in actions and action-angles,
n
o
p
p
ψ(t01 ) = ψ̃(t01 ) exp iky (−2At01 [ 2Jr cos Θ0r + Jφ0 /2B] + [−2Ω 2Jr sin Θ0r + Θ0φ ]) ,
(29)
and
´
³
p
p
∂ψ
0
0
0
]
+
[−2Ω
]
ψ(t01 ),
2J
sin
Θ
2J
cos
Θ
[−
−2At
=
ik
r
r
y
r
r
1
∂Θ0r
(30)
where the 0 indicates a quantity to be evaluated at time t01 . At fixed x and y,
Θ0φ = y + 2Ω
so that
σ̃(t1 ) =|2Ω|
where
Z
p
Jr0 = Jr ,
(31a)
Θ0r = t01 − t1 + Θr ,
p
Jφ0 = 2B(x − 2Jr cos Θr ),
(31b)
2Jr sin Θr + 2A(t01 − t1 )(x −
t1
−∞
dt01 ψ̃(t01 )
Z
∞
dJr
0
Z
2π
dΘr
0
np
exp i Jr (γ cos Θr + δ sin Θr )
(31c)
p
2Jr cos Θr ),
∂F0 p
Jr (α cos Θr + λ sin Θr )×
∂Jr
,
o
√
α ≡ i 2ky [2At01 sin(t01 − t1 ) − 2Ω cos(t01 − t1 )],
√
λ ≡ i 2ky [2At01 cos(t01 − t1 ) + 2Ω sin(t01 − t1 )],
√
γ ≡ 2ky [−2At01 cos(t01 − t1 ) − 2Ω sin(t01 − t1 ) + 2At1 ],
√
δ ≡ 2ky [2At01 sin(t0 − t) − 2Ω cos(t01 − t1 ) + 2Ω],
(31d)
(32)
(33a)
(33b)
(33c)
(33d)
and
σ(t1 ) = σ̃(t1 )eiky (−2At1 x+y) .
(34)
The integral over Θr can be done using the identity
Z
π
dx cos xeia cos x = iπJ1 (a),
(35)
0
where J1 is a Bessel function of the first kind (Gradshteyn and Ryzhik, 1980, 3.915.2), so that
σ̃(t1 ) = 2πi|2Ω|
Z
t1
−∞
dt01 ψ̃(t01 )
Z
∞
dJr
0
∂F0
∂Jr
√
p
Jr (αγ + λδ)
p
J1 ( Jr (γ 2 + δ 2 )).
γ 2 + δ2
(36)
If it is assumed that the initial distribution function is of the Schwarzschild form,
F0 (Jr ) =
2
fs
e−Jr /σu ,
2
−4πσu Ω
(37)
– 31 –
then the integral over Jr can be done using the identity
Z ∞
2
b −b2
dxx2 e−ax J1 (bx) = 2 e 4a
4a
0
(38)
(Gradshteyn and Ryzhik, 1980, 6.631.4). The result is
σ̃(τ ) =
where
Ks (τ, τ 0 ) ≡
Z
τ
dτ 0 Ks (τ, τ 0 ) (σ̃s + σ̃g + σ̃i ) ,
(39)
−∞
2
2
2
πifs
√
(αγ + λδ)e−σu (γ +δ )/4
02
2Aky 1 + τ
(40)
and we have used the Poisson equation
ψ̃ = −
2πσ̃
√
ky 1 + τ 2
(41)
to eliminate the potential in favor of the density.
The equation for the evolution of the radial modes is obtained by taking the limit k y →0 at fixed
kx of eqtn.[39], giving
Z t
dt0 Ks,r (t, t0 ) (σ̃s + σ̃g + σ̃i ) ,
(42)
σ̃s =
−∞
where
ª
©
Ks,r (t, t0 ) ≡ −2πfs |kx | sin(t0 − t) exp −kx2 σu2 (1 − cos(t0 − t)) .
(43)
Equations [39],[40],[42], and [43] are all, except for differences in notation, identical to those
derived by JT, although they have been derived by different means.
3. Wave Packet Propagation and Stability
The dispersion relation for radial modes describes both the propagation of wave packets and the
local stability properties of the disk. It is obtained from eqtn.[43] and eqtn.[42] on the assumption
that the mode has time dependence exp(−iωt) and it has the form
1=
2π
|k|
µ
fs k 2
fg k 2
+
2
2
2
−ω + 1 + c k
sin πω
Z
π
dτ sin τ sin ωτ e−k
0
2
2
σu
(cos τ +1)
¶
,
(44)
identical to the dispersion relation obtained by Lin, Yuan, and Shu (1969) for tightly wound (nearly
radial) waves. For comparison, the dispersion relation for radial modes in the two-fluid model of JS is
¡
−ω 2 + 1 + k 2 c2g − 2πfg |k|
¢¡
¢
−ω 2 + 1 + k 2 c2s − 2πfs |k| = 4π 2 k 2 fg fs
(45)
where cg and cs are the “sound” speeds in the gas and stars respectively.
Figure 1 shows the locii of solutions to the dispersion relation for stars and gas (solid line) and for
the stars (dashed line) and gas (dotted line) in isolation, in units appropriate to the combined system.
The model parameters are set to the solar neighborhood values estimated in Chapter 1: Q g = 1.2,
Qs = 2.8, and fg = 0.27 (the dispersion relation is independent of β, which characterizes the local shear
rate). At any given wavenumber the frequency is multiple-valued. As we shall see below, frequencies
between 0 and ±1 correspond, for tightly wound modes (ky ¿ 1, |kx /ky | ¿ 1 when Qg , Qs ∼ 1), to
– 32 –
locations between corotation and the Lindblad resonances. Modes at higher frequencies are trapped
between the higher order Lindblad resonances (ω/κ = ±2, ±3, . . .), which usually do not exist in
global galaxy models. The minimum of the dispersion relation is at |k x,min | = 0.8, corresponding to
a wavelength of 1.2 kpc. This may be regarded as a characteristic wavelength for the response of the
model to perturbations, while ω(kx,min ) is an indicator of the sensitivity of the model to perturbations–
the lower the frequency, the more responsive the disk. The location of the minimum in the star-gas
system is nearly identical to that for the gas alone; the stars are hot and unresponsive enough that at
their most responsive wavenumber there is only a dimple in the star-gas dispersion curve.
How sensitive is the dispersion relation to the model parameters? Figure 2 shows the dispersion
relation after each of the parameters has been turned up and down. Increasing f g (at fixed total
surface density) increases the wavelength of the minimum in the dispersion relation and decreases the
frequency at the minimum. The dispersion relation is very sensitive to Q g , and lowering Qg by a
small amount sharply decreases the minimum frequency. The stellar component contributes little to
the response and so the dispersion relation is relatively insensitive to Q s .
Figure 3 shows the ratio of the perturbed gaseous surface density to the perturbed stellar surface
density as a function of wavenumber, again for the solar neighborhood parameters. Three curves
are shown: one for the mode with |ω| < 1, one for the mode with 1 < |ω| < 2, and one for the
mode with 2 < |ω| < 3 (see Figure 1). The peaks are the points where ω, k satisfy both the star-gas
dispersion relation and the dispersion relation for the gas alone. For the principal mode (|ω| < 1) the
response is dominated by the gas at long wavelengths, but as the wavelength decreases the gas becomes
increasingly important, for the simple physical reason that the stars have a difficult time responding
at wavelengths shorter than an epicyclic amplitude. At even shorter wavelength the response is again
dominated by the stars because the stellar density wave piles up in the vicinity of the Lindblad
resonances (|ω| = 1).
Finally, how well does the two-fluid dispersion relation reproduce the star-gas dispersion relation?
Figure 4 shows the dispersion relation for both the two-fluid model (solid line) and the star-gas model
(dashed line). Obviously the two curves track one another fairly closely inside the Lindblad resonance
(|ω| < 1). Near and outside the first Lindblad resonance the dispersion relation for tightly wound
modes in the two-fluid model is not applicable to star and gas disks.
3.1 Wave Packet Propagation
It is instructive to consider the evolution of a wave packet in the local model in order to make
a connection between the evolution of the shearing waves discussed so far and events in the spatial
domain. The prescription for evolving a wave packet is to first choose a perturbing potential that
generates a disturbance localized in both the fourier and spatial domains. Then a fourier transform
gives the potential as a function of (ky , τ, τ0 ), so that the disturbance it creates can be evolved using
eqtns.[39], [42], [16], and [17]. An inverse fourier transform then gives the evolved wave packet as a
function of (x, y, t).
GT carried out this prescription analytically for the gaseous sheet in the tight-winding limit. If
the original disturbance has azimuthal wavenumber ky and is steady in time, then a wave packet
localized near some radius x sees a frequency ω(x) = 2Axky as the shear causes it to sweep through
the pattern. Clearly x = 0 is the corotation radius for the disturbance, and x = ±1/2Ak y gives the
location of the Lindblad resonances. As the packet evolves it is wound up by differential rotation so
that kx (t) = −2Aky t − ky τ0 . These relations for ω(x) and kx (t), together with the dispersion relation
– 33 –
ω(k), imply a relation x(t) that describes the position of the wave packet as a function of time. Proving
that the x(t) obtained this way describes the evolution of the wave packet is the substance of GT’s
calculation. Since there is a one-to-one correspondence between ω and x, and between k x and t, the
path of the wave packet can be read off the gaseous single-component dispersion relation shown in
Figure 1.
Consider the dispersion relation for a gaseous disk with Qg = 1.2 shown in Figure 5. A leading
packet excited outside the Lindblad resonance (near point A in the Figure, with k x > 0) moves inward
across the Lindblad resonance (x = 1/2Aky ) and toward corotation, is reflected at the “corotation
barrier” (point B; x = (Q2g − 1)/2Aky Q2g ) and moves back toward the Lindblad resonance as a long
leading wave. At point B, when the wave packet is closest to corotation, it experiences a small
amplification for reasons to be discussed in the next section. As kx approaches 0 and the packet
approaches the Lindblad resonance (point C), the tight winding approximation fails. By solving
eqtn.[16] exactly in the vicinity of τ = 0 and invoking conservation of wave action flux, GT show that
the packet is totally reflected at the Lindblad resonance. It then travels back toward corotation (point
B) as a long trailing wave, is reflected, and finally moves away from corotation as a short trailing wave
that is asymptotically a sound wave.
The obvious analogous calculation can be carried out for the two-fluid model, although we do not
do so here. Again the path of the wave packet can be read off the radial mode dispersion relation.
Figure 6 shows an extreme case where both the stars and the gas are marginally stable: we have set
fg = 0.1, Qg = 1.25, and Qs = 1.25. The two-fluid dispersion relation is shown as a solid line, while
the star-gas dispersion relation is a dashed line. Starting as a leading wave at point A, with k x > 0,
the two-fluid wave packet makes two approaches to corotation at points C and D before approaching
the Lindblad resonance at point E.
Similarly, one expects the radial mode dispersion relation to describe the path followed by a
wave packet in the star-gas model. The major difference from the two-fluid model is at the Lindblad
resonances. Near point B in Figure 6 in the star-gas model (but not in the two-fluid model) the mode
attempts to cross the Lindblad resonance but cannot; the group velocity vanishes. However, there is
no real resonance in the gaseous component. As GT have shown, the gas is only very responsive in
the neighborhood of the Lindblad resonance. The group velocity of a wave packet in a gaseous disk
carries it undisturbed across the resonance. However, the |ω| > 1 mode gets close to a second mode at
|ω| < 1 that hovers just inside the Lindblad resonance. In the limit that the stellar content of the disk
is negligible (fg − 1 ¿ 1) there are still formally two modes, but they approach so closely that there
is a mode conversion of almost unit efficiency and the wave packet crosses the Lindblad resonance
uninhibited. In the limit that the gas content of the disk is negligible the mode conversion efficiency
is 0 and the wave packet is entirely absorbed at the Lindblad resonance (see the review of Toomre,
1977). The other difference between the two dispersion relations in Figure 6 is that the star-gas model
is less responsive at high wavenumber than the two-fluid model, because the hot fluid that represents
the stellar component in the two-fluid model is more capable of responding at high wavenumber than
the stars themselves.
If the mode conversion efficiency is a continuous function of fg , disks with intermediate gas
fraction (0 < fg < 1) ought to have intermediate values of the mode conversion efficiency. An
immediate consequence is that disks that contain gas have a means of transporting a density wave
across the inner Lindblad resonance. Some of the energy carried by the density wave is absorbed by
the stellar component (increasing the velocity dispersion of the stars), while some is carried across
– 34 –
by the gas. It may be possible to calculate the conversion efficiency analytically in the tight-winding
limit, but this appears to be technically rather challenging. Once a density wave can cross the inner
Lindblad resonance it may be able to propagate through the center of the disk and emerge as a leading
wave on the opposite side (see the discussion of Binney & Tremaine, 1987). This provides a means
of closing the feedback loop on the swing amplifier and generating a genuine nonaxisymmetric, global
instability.
3.2 Radial Stability
Our results here hopefully clarify the work of Lin and Shu (1966) and extends the work of Jog
and Solomon (1986) and Bertin and Romeo (1988) to star and gas models.
To demonstrate that the disk is locally stable we need only show that the radial modes are stable,
because there are no local nonaxisymmetric instabilities. This is a consequence of the evolution
of the radial wavenumber of the modes– the radial wavenumber is always large at early and late
times, and for high enough wavenumber the disk is always stable in both stars and gas. This alone
does not guarantee stability, since at fixed wavenumber the amplitude could grow exponentially (the
distinction is the same as that between uniform and pointwise convergence); it must also be shown
the disk responds weakly to short wavelength forcing. A proof of the stability of nonaxisymmetric
modes can be constructed using arguments similar to those suggested by JT, but is not given here.
If ω 2 is real, a condition for local stability is obtained by studying modes in the neighborhood of
ω 2 = 0, since the modes must pass through ω 2 = 0 to become unstable. A sufficient condition for ω 2
to be real is that f (0) be a monotonically decreasing function of Jr , and a proof of this is given in the
Appendix based on the instability test of Goodman (1987). On taking the limit ω → 0 of eqtn.[43],
1=
2π
|k|
µ
¶
´
2
fg k 2
fs ³
−k2 σu
2 2
)
+
1
−
e
I
(k
σ
0
u
σu2
1 + k 2 c2
(46)
where I0 is a modified Bessel function of the first kind (see, e.g. Binney & Tremaine, 1987, Appendix
6A). We refer to the boundary of the region in parameter space where solutions to this dispersion
relation exist as the neutral stability surface.
Figure 7 shows the neutral stability curves as a function of Qg and Qs for various values of the
gas fraction fg . A comparison of these curves with those for the two-fluid model of JS shows close
agreement, the sign of the difference being such that some regions of parameter space are unstable
in the two-fluid model but stable in the star-gas model. These differences may be due as much to
the parametrization we have used as to any physical effect, and for all practical purposes the curves
are identical. Note that there is a region of parameter space where the mixed system is unstable but
either component would be stable in isolation (the lower left corner of Figure 7). The stars can under
no circumstances stabilize the gas. (This is true locally but not globally, since the addition of a hot
stellar component can increase κ and hence Qg . In the local model κ is fixed.)
Star-gas models where the gas fraction fg ≈ 0.5 are the least stable in the sense that it takes
the largest Q for each component to stabilize them. (At fixed fg , however, there is always a point on
the neutral stability curve for the two-fluid system where the surface energy density is equal to that
required to stabilize a one-component system. However, this point is a minimum for the surface energy
density along the neutral stability curve; all other points have higher surface energy density.) This
raises the possibility that a young disk system may actually destabilize itself by converting a significant
fraction of its gas into stars. Whether or not this actually happens depends upon the trajectory of
– 35 –
the galaxy in parameter space, which depends in turn upon the velocity distribution of newborn stars,
how fast the gas is being turned into stars, how fast the stars are being dynamically heated, and how
the gas is being heated by the stars. If the stars are born with a velocity dispersion smaller than the
velocity dispersion of the gas (as might happen if the stars are formed primarily in massive clouds
with a low velocity dispersion) then it is possible that when fg >
∼ 0.5 star formation decreases the gas
fraction while Qg and Qs remain fixed, moving the galaxy toward the neutral stability surface and
increasing its responsiveness.
The dispersion relation also gives the natural scale for structure in the disk. In general the star-gas
model is always stable at very long wavelengths (because of rotation) and at very short wavelengths
(because of pressure and stellar streaming). When the disk is unstable, the growth rate is strongly
peaked at one or two characteristic scales. One does not therefore expect gravitational instability in
an initially uniform disk to produce object with a spectrum of masses: instead one expects objects of
a characteristic mass, given by Mc ' Σg λ2c , where λc is the scale with maximum growth rate. The
scale λc changes suddenly when the instability goes from being gas-dominated to star-dominated.
Figure 8 shows two dispersion relations, both with fg = 0.1. The curve marked A is unstable near the
characteristic wavelength of the gas, and has Qg = 1.19 and Qs = 1.226. The curve marked B has
Qg = 1.205 and Qs = 1.21, and is unstable at the characteristic scale for the stars, which is about 7
times the characteristic scale for the gas. A small change in parameters thus brings about a dramatic
change in the scale of the instability. This is the “phase transition” discussed by Bertin & Romeo
(1988) in the context of the two-fluid model. However, there is usually enough noise in galactic disks
that the swing amplification of nonaxisymmetric modes (which can cause the density amplitude of the
wave to grow by a factor of several ×102 , and which have larger instantaneous growth rates than the
axisymmetric modes) dominates over the axisymmetric instability. The gain of the swing amplifier is
less dramatically dependent on the model parameters and the transition between the gas-dominated
and star-dominated regimes is smoothed out (§4).
4. Amplifier Gain and Response to a Point Mass
This section discusses the responsiveness of the disk to nonaxisymmetric perturbations. First,
we consider the gain of the corotation amplifier in the two-fluid model in the tight-winding limit and
calculate the lowest order (in fg ) correction to the gain from the introduction of a cool gaseous
component. Second, we consider the response of the star-gas model to an imposed point-mass
potential.
4.1 Gain in the two-fluid corotation amplifier
To make this calculation somewhat less opaque, it may be helpful to first describe it in words.
We model the disk as two thin fluid sheets coupled only by gravity, since fluid sheets are easier to
treat analytically than collisionless sheets. Earlier work on the corotation amplifier in the stars (Mark,
1976a) and the gas (GT) shows nearly identical amplification, so we expect the hot fluid representation
for the stars to be adequate. A tightly wound (ky ¿ 1, |kx /ky | ¿ 1) wave packet propagating toward
its corotation radius in the two-fluid model will follow the path described in §3, (see Figure 6) with
ω ∼ x and k ∼ t, where ω and k are related by the dispersion relation, eqtn.[45]. When f g ¿ 1 and
both components are marginally stable, the wave packet makes two approaches to corotation, and is
reflected at each (again, see Figure 6). At each approach part of the wave packet tunnels through
corotation and excites a wave packet on the opposite side (with frequency of opposite sign). The less
– 36 –
stable the sheet, the greater the amplitude of the wave excited on the opposite side. Because density
waves inside corotation have negative angular momentum density while those outside have positive
angular momentum density, the exchange of angular momentum across corotation amplifies the both
modes and while conserving total angular momentum flux. The magnitude of the amplification has
been calculated analytically for a fluid sheet and a stellar disk (GT) and for a star-gas combination
(Mark, 1976a. Mark considers the amplification at only one of the approaches to corotation evident
in Figure 6).
The evolution of the amplitude of a mode in the two-fluid model is governed by the coupled single
fluid equations:
´
πf1 k ³
¨
ψ̃ 1 + S1 ψ̃1 =
ψ̃
+
ψ̃
(47a)
2
i
2A2
´
πf2 k ³
¨
ψ̃
+
ψ̃
(47b)
ψ̃ 2 + S2 ψ̃2 =
1
i
2A2
where
µ
¶
12A2
1
8ΩA
2 2
Si ≡
1 − 2πfi k + k ci +
.
(48)
+
4A2
(1 + τ 2 )2
(1 + τ 2 )
We adopt the convention that component 1 represents the gas, with low surface density and low
velocity dispersion, while component 2 represents the stars. ψ̃i is the external perturbations in the
potential not due to stars or gas. Equations [47] can be rewritten as
Ψ̈ + W Ψ = Ψi
(49)
where Ψ is a vector with components ψ̃1 and ψ̃2 , W is a matrix defined in the
q obvious way, and
Ψi ≡ (f1 , f2 ) × πk ψ̃i /2A2 . W can be symmetrized by the transformation ψ̃2 → ff21 ψ̃20 , so that it is
clear that there is always a basis in which W is instantaneously diagonal.
Now consider the case ky ¿ 1 − (πfi /ci )2 = (Q2i − 1)/Q2i , so that Si2 /Ṡi À 1, that is, the Si
change on a timescale long compared with 1/Si . Away from τ = 0, the last two terms in Si (the last
of which is responsible for the swing amplifier) may be neglected and eqtns.[47] is formally analogous
to a slowly rotating two dimensional harmonic oscillator. The first order WKB solutions are
µ
¶
√
R
êa (τ ) 2iν ωa êb (τ ) ±i τ dτ 0 ωa (τ 0 )
±
(50a)
Ψa,± = √
e
ωa
ωa2 − ωb2
µ
¶
√
R
êb (τ ) 2iν ωb êa (τ ) ±i τ dτ 0 ωb (τ 0 )
±
Ψb,± = √
(50b)
e
2
ωb
ωb − ωa2
ωa2 (τ ) and ωb2 (τ ) are the instantaneous eigenvalues of W , while êa (τ ),êb (τ ) are the instantaneous
normalized eigenvectors. Here ν ≡ (d/dτ )(tan−1 (êa ·ê2 /êa ·ê1 )) is the rate of rotation of the
eigenvectors with respect to the physical basis. I adopt the convention that ω a2 → S1 and ωb2 → S2 at
late times, so that Ψa± asymptotes to a sound wave in the cold fluid (gas), while Ψb± asymptotes to a
sound wave in the hot fluid (“stars”). It is possible to approximately calculate the fourier transform
of these modes, (e.g. GT) to recover Ψ (x, t), and it can be shown that the fast mode exists only
outside the Lindblad resonances, while the slow mode penetrates almost to corotation. Again, note
that the sign of the frequency indicates the position of the mode with respect to corotation; modes
with frequencies of opposite sign are found on opposite sides of corotation.
Now consider a time τm when the frequency ωa is a minimum. Near this time,
2
ωa2 (τ ) ' ωa,m
+ µτ 2
2
ωb2 (τ ) ' ωb,m
+ γτ
(51)
– 37 –
m
êa (τ ) ' êm
a + ντ êb
m
êb (τ ) ' −êm
a ντ + êb
(52)
where m indicates a quantity to be evaluated at time τm , and because of the tight-winding
approximation α,µ and ν are small in comparison to ωa and ωb . Suppose that ωa,m becomes
comparably small because we lower the Q of either the cool component or the hot component. A
wave train arriving from τ < τm now suffers a slight nonconservation of action near τ = τm and at
τ > τm two wave trains emerge, one with positive frequency and one with negative frequency.
If we assume that the form of the solution to the coupled two-fluid equations in the neighborhood
of τm , the point of closest approach to corotation, is
m
Ψ = Aêm
a + Bêb
(53)
where |B| ¿ |A| (this may be verified a posteriori), then the equation describing the amplitude A is
Ã
!
2
2
− νm
d2 A
1 02 ωa,m
+
τ +
A=0
(54)
√
dτ 02
4
2 µ
where τ 0 = (4µ)1/4 (τ − τm ). This is the parabolic cylinder equation. Standard asymptotic matching
techniques determine the amplitudes of the outgoing WKB solutions (e.g. Bender & Orszag, 1978);
connection formulae for parabolic cylinder functions are provided by Abramowitz and Stegun (1972).
√
√
The final amplitude j ≡ A/ ω of the original wave train Ψa,± is increased by a factor 1 + e−2πb while
the amplitude of Ψa,∓ (which lies on the opposite side of corotation) is e−πb , where the amplification
parameter b is
2
− ν2)
(ωa,m
b≡
,
(55)
√
2 µ
so that when b is reduced the amplification is enhanced.
Assuming that Q1 ∼ Q2 ∼ 1 and f1 ¿ 1, we expand the eigenvalue equation of W near τm in
powers of f1 . Near the minimum in the dispersion relation where the gaseous response dominates
(e.g. point C in Figure 6) k ≈ 1/πf1 Q21 (to lowest order in ky ) and
¶
µ
1
4f1 Q21
2
2
−1
(56)
ωa ≈
Q1 −
4A2 Q21
Q22
so that Q1 > 1 + 2f1 /Q22 is required for stability. Next expanding b to lowest order in f1 , we have
√
ν ≈ 2πQ41 ky f12 /Q22 , µ ≈ π 2 Q21 ky2 f12 / 2A2 , and
µ
¶
4f1 Q21
1
2
Q1 −
−1 .
(57)
b≈
8πAf1 Q31 ky
Q22
The amplification of the density wave in the gas is thus always enhanced by the presence of the stars.
Near the minimum in the dispersion relation where the stellar response dominates (e.g. point D
in Figure 6) k ≈ 1/πf2 Q22 (again, to lowest order in ky ) and
ωa2 ≈
¡ 2
¢
1
Q2 − 4f1 − 1
2
2
4A Q2
(58)
so that Q2 > 1 + 2f1 is required for stability, in close agreement with the neutral stability curve
for fg = 0.1 shown in Figure 7. Expanding to lowest order in f1 , we have ν ≈ 2πQ22 ky f1 , µ ≈
√
π 2 Q22 ky2 (1 − 10f1 )/ 2A2 , and
b≈
¡ 2
¢
1
Q1 + f1 (5Q22 − 9) − 1 .
3
8πAQ2 ky
(59)
– 38 –
The amplification of the wave train near the star-dominated approach to corotation can be inhibited
p
if Q2 > 9/5 = 1.34. This is because there are two corrections to b: one to the frequency (which
is lowered by the presence of the gas, increasing the amplification) and one to the curvature of the
dispersion relation (which is lowered by the presence of the gas, decreasing the amplification. When
the wave packet makes a rapid approach and retreat from corotation the transmission across corotation
p
is more efficient). For Q2 = 9/5 the two effects balance exactly. Mark (1976) has calculated the
amplification parameter b at the star-dominated approach to corotation for a mixed star and gas disk
with small gaseous surface density, and obtains no dependence on Q s ≡ Q2 in the correction due to
the presence of the gas. It is not as yet clear if this is because of a genuine qualitative difference
between stellar and gaseous disks at corotation.
4.2 Gain in the Gas and Star Amplifier
Here we consider the nonaxisymmetric response of the mixed star and gas system without the
tight-winding approximation. This requires numerical integration of eqtns.[16], [17], [39], [42]. First,
however, we need a quantitative measure of responsiveness. For the gas alone, the natural definition is
the ratio of wave action at early and late times. For the stars alone a similar definition is not possible
because phase mixing washes out shearing waves at early and late times (large inclinations). One
possibility is to measure the amplitude of the maximum of the response to an imposed point-mass
potential in fourier space, that is, to find the nonaxisymmetric wave with the largest amplitude. This
works well for the stars because they converge rapidly to a steady response to the imposed potential.
In a mixed system neither approach works very well. The ratio of ingoing and outgoing wave actions
in the gas is well defined at early and late times because the stellar contribution is negligible. In
practice this method is difficult to implement because it requires integrating the coupled fluid and
Boltzmann equations for long times and the computational expense of the integration ∝ t 3 . The
ratio of wave actions also may not properly measure the responsiveness of a disk with a cool stellar
component and a warm gaseous component. Taking the maximum of the response to an imposed
point-mass potential does not work in the star-gas sheet because the maximum is ill-defined, since the
amplitude of an individual wave in the gaseous component increases as t 1/2 as t→∞. The approach we
shall take below is to calculate the response of a star-gas model with our standard solar neighborhood
parameters, then ask what is the effect of changing each parameter in turn upon the shape and
amplitude of the response.
The coupled coupled star and gas equations (eqtns.[21] and [41]) were integrated numerically.
First, the integral equation for the stars was stepped forward by direct integration using the trapezoidal
rule. This can be done without knowledge of the gas surface density at the new time since K(τ, τ ) = 0
(see eqtn.[39]). Then the gas surface density was integrated forward over the stellar timestep using a
standard Runge-Kutta method (Press et al.,1988) and linear interpolation to obtain the stellar surface
density at intermediate times. This method proved to be stable and to converge rapidly.
Figures 9, 10, and 11 show “typical” time evolutions for a single fourier component in the gas,
the stars, and the mixed star-gas system respectively. In each case the evolution was started with the
impulsive application of a sinusoidal potential at the indicated inclination variable τ .
Figure 9 shows the evolution of the the wave amplitude in the surface density (σ k ) and the
evolution of the wave action
J. We have set ky = 0.08 and Qg = 1.0. The wave action J is defined so
Rτ
p
i
ω(τ 0 )dτ 0
that ψk ≈ (J/ (ω))e
, and is directly related to the energy and angular momentum carried
by the wave. When ω̇/ω 2 ¿ 1, J is constant, as is the case at large negative and positive τ . In
– 39 –
addition to the impressive increase in the amplitude of the mode, there are several points to notice
here. First, the surface density amplitude increases algebraically, like |τ | 1/2 , as |τ |→∞. Second, the
growth of the action J is approximately exponential over the interval −5 < τ < 5. The bumps evident
in the log(J) curve occur when ω 2 (τ ) changes sign. Third, the action converges rapidly to a constant
value at early and late times.
Figure 10 shows the evolution of the wave amplitude in a stellar disk, with identical initial
conditions except that the wave was stimulated at τ = −6.2 rather than τ = −20. The difference in
amplitude of the star (Figure 10) and the gas (Figure 9) should not be interpreted as a difference in
responsiveness, but rather differing initial conditions. Note the rapid exponential decay of the stellar
surface density at late times.
Figure 11 shows the evolution of the combined star and gas disk with f g = 0.5, Qg = 1.5,
Qs = 1.5, β = 0.0, and ky = 0.09. The total surface density response is the solid line, the dashed line
is the stars, and the dotted line the gas. The gaseous and stellar responses track one another closely
in the region of exponential growth, but afterwards the stellar response decays rapidly and moves out
of phase with the gaseous response.
Now let us consider what happens when a point mass is introduced at the origin. The response
of the disk to this imposed potential is of interest because it is a kernel that can be integrated against
an arbitrary imposed potential to obtain the surface density response of the disk, and because there
exist dense, almost point-like perturbers (molecular clouds) in the disk. To calculate σ(x, y) one
first calculates the evolution of the fourier components under the influence of the perturbing potential
(ψ̃ = −M/|k|) and then takes a fourier transform. To guard against the possibility of numerical errors,
two tests were made of the code that calculates the linear response. First, the frequencies of radial
modes evolved by the the code were checked against those predicted by the dispersion relation. Second,
to test the evolution of nonaxisymmetric modes, the response to a point mass was calculated in the
limit β→1, i.e. nearly solid body rotation. For solid body rotation axisymmetric and nonaxisymmetric
modes are not distinguished because there is no shear, so the response should be (and was) nearly
circularly symmetric.
To establish a point of comparison, we first calculate the response in the limit f g →0 and fg →1,
i.e. for the purely stellar and purely gaseous model. Figure 12 shows the surface density response for
the stars and the gas in both the fourier and spatial domain at time 5/κ after the point mass was
introduced. The arrows indicate the direction of shear. The gaseous component has Q g = 1.2, while
the stellar component has Qs = 2.8. A small viscosity has been added to accelerate the convergence
of the gaseous surface density; in the form of the viscosity discussed in §2, we have set ² = 0.2. We
have also multiplied the fourier transform by a Gaussian of the form exp(−0.5k 2 /1.32 ) to reduce the
ringing that occurs because the gaseous response is non-zero at the boundaries of the calculation in
k-space. The response has not converged at t/κ = 5; we shall discuss convergence shortly. The time
t = 5/κ was chosen to evaluate the response because molecular clouds, the likely perturbers, do not
persist over much longer timescales. If one accepts the estimate for the lifetime of molecular clouds
discussed in Chapter 1, then 5/κ ≈ 1.3 × 108 yr in the solar neighborhood is already somewhat longer
than the lifetime of an individual cloud.
What is learned from these images of “spiral streaks”? First, the shape of the gaseous and stellar
response is different. The stellar response tends to be smoother (have less power at high wavenumber);
this is true for all values of the model parameters, and is a consequence of the damping of the stellar
waves by phase mixing already discussed, and the persistence of tightly wound modes in the gas.
– 40 –
The gas response in fourier space peaks at kx,max = −0.40, ky,max = 0.22 and has an amplitude
of 8.44, while the stellar response peaks at kx,max = −0.10, ky,max = 0.06 and has amplitude 0.55.
Thus the characteristic inclination of the responses is identical, and would correspond to a “pitch
angle” φ ≡ arctan ky,max /kx,max = 29 deg. The maximum of the gaseous response is well defined
in this case because of viscous damping. The wavenumber of the response peak corresponds to a
wavelength λpk = 52.9→6.0 kpc in the stars and λpk = 13.7→570 pc in the gas, using the standard
solar neighborhood values for Σ and κ (see Chapter 1).
Now consider response to a point mass when the stars and gas are coupled. Figure 13 shows
the response of the mixed star-gas model evaluated at t = 5/κ. In this case the stars respond
rather differently. They are essentially driven by the gas and respond at a wavelength closer to the
natural wavelength of the gas. Nevertheless, the stars are extremely reluctant to respond at such
short wavelengths (the epicyclic amplitude for a star with the rms epicyclic amplitude is 6.9), and
make an effort to escape the clutches of the gaseous wake– thus the peculiar flaring at the ends of the
stellar wake. The peak of the response in the gas is located at kx,max = −1.29, ky,max = 0.67 and has
amplitude 9.8, while the peak response of the stars is located at kx,max = −0.26, ky,max = 0.40 and
has amplitude 1.2. These peak wavenumbers correspond to wavelengths of 670 pc and 2.0 kpc for the
gas and stars, respectively. This characteristic wavelength for the stellar response is comparable to
the spacing between spiral arms in the solar neighborhood.
Next we consider variations in the parameters about these standard solar neighborhood
parameters. Figure 14 shows the response for fg = 0.17 in the top two panels, while the bottom
two panels show fg = 0.37. Increasing the surface density increases the amplitude of the response and
decreases the characteristic scales; the compromise between the stars and the gas is tilted in favor of
the gas. Reducing the surface density makes the reluctance of the stars to respond even more evident
in the upper right hand panel and decreases the amplitude of the response, particularly in the stars.
Figure 15 shows the response for Qg = 1.0 (top two panels) and Qg = 1.4 (bottom two panels). When
Qg = 1.0 the model is unstable to radial modes; the response is radically increased in this case. Figure
16 shows the response for Qs = 2.0 and Qs = 3.6. Here increasing Qs makes self-gravity even less
important for the stars, increasing the extent of the flaring wings on the stellar response. Decreasing
Qs to 2.0 makes the disk unstable to radial modes and sharply increases the amplitude of the response.
Variation of β is not shown, but increasing the shear rate (decreasing β) makes the wake more tightly
wound.
We have promised to consider the issue of convergence. Figure 17 shows the time evolution of
the solar neighborhood star-gas model. As the wake evolves, it grows, winds up, and its amplitude
increases. Even at t = 9/κ the response has not yet converged; for the standard model with ² = 0.2
viscosity, convergence occurs at t ≈ 15/κ or ≈ 4 × 108 yr; smaller values of the viscosity require longer
convergence times. Giant molecular clouds probably do not persist for so long.
Finally, we are obliged to give some sense of how the responsiveness of the disk changes with the
model parameters, even if our definition of responsiveness is not completely satisfactory. Therefore we
have evaluated the response of the disk to a point mass at t = 5/κ and found the local maximum of
the response of the gas in fourier space with |k| closest to 1/πfg Q2g , which is the expected wavenumber
of the maximum in the gaseous response. The amplitude of the gaseous response at this maximum
is an indicator of the responsiveness of the disk; Figure 18 shows this amplitude as a function of
Qg and Qs for four different values of fg . The gas is most responsive at small values of the gas
fraction where Qg can get very close to 1 while the disk remains stable. Notice that there is no
– 41 –
sharp change in amplitude from regions where the response is dominated by the gas (low Q g , high
Qs ) to regions where the response is dominated by the stars (low Qs , high Qg ). Hence the phase
transition discussed by Bertin & Romeo (1988) exists only for axisymmetric modes because those
modes must either grow or oscillate, so a small shift in parameter space can radically alter the scale
of the instability. Nonaxisymmetric modes are all formally stable, but experience large temporary
amplification. The magnitude of the amplification is a smooth function of the model parameters and
changes only gradually from the gas-dominated regime to the star-dominated regime.
5. Conclusions
1. The stability properties of the star and gas disk are more complicated than those for the single
component disks, but they are well reproduced by the two-fluid model.
2. As expected, the nonaxisymmetric response of the mixed system is not well represented by
the two fluid model near the Lindblad resonances.
3. With the addition of a gaseous component, density waves can propagate across the Lindblad
resonances. While we have not calculated the transmission coefficient for this process (this is
technically rather challenging), it is likely that the efficiency of the mode conversion that carries
the wave across the resonance is a monotonically increasing function of gas fraction.
4. As in the two-fluid model, there are two most unstable wavelengths. For model parameters
appropriate to the solar neighborhood, the most pronounced nonaxisymmetric response is at the
smaller of these wavelengths, which is approximately the most unstable wavelength in the gas.
5. The phase transition discussed by Bertin & Romeo (1988) is an artifact of the restriction of
their treatment to radial modes. The nonaxisymmetric response of the disk changes smoothly from
the gas-dominated to the star-dominated regime.
6. Application to the solar neighborhood. If Σg ≈ 13 M¯ pc−2 , κ ≈ 1.2×10−15 s−1 , σg ≈ 6 km s−1 ,
then Qg ≈ 1.2. If Σst ≈ 35 M¯ pc−2 , and σst ≈ 40 km s−1 , then Qst = 2.8, fg = Σg /(Σg + Σst ) = 0.27,
and the disk is stable locally, even before corrections are made for thickness effects. Lowering Q s by
raising the surface density of stars makes the disk less stable. Lowering Q s to 2.0 makes the disk
locally unstable. The characteristic scale of the nonaxisymmetric response in the gas is λ g = 670 pc,
corresponding to a characteristic mass of 5.8 × 106 M¯ , substantially larger than a typical molecular
cloud in the solar neighborhood. The characteristic scale for the nonaxisymmetric response of the
stars is 2.0 kpc, comparable to the spacing of spiral arms in the solar neighborhood.
We have not incorporated thickness corrections in our stability analysis, as might be done by using
a softened gravitational kernel. The sign of the effect of properly including the vertical structure of the
disk is not obvious. Finite thickness tends to increase stability, and for an isothermal self-gravitating
gaseous sheet Qg = 0.68 is all that is required for stability (GL). Nevertheless, the galactic disk is
not isothermal– the velocity dispersion of every component decreases toward the midplane, decreasing
stability. For example, the column-averaged radial velocity dispersion of the stars is 40 km s −1 , but
the midplane radial velocity dispersion is 30 km s−1 (see Mihalas & Binney, 1981).
What sort of errors are introduced by using a uniform, isothermal model of the interstellar
medium? One effect that could change the dynamics significantly is the variation of velocity dispersion
with length scale (see Issa, Maclaren, & Wolfendale, 1988; Stark, 1988, as suggested by Larson, 1988).
If, for example, σ ∼ L0.5 , then on small scales the disk is dynamically cold and less stable.
7. Application to young galactic disks. If the galactic disk in the solar neighborhood once had
a gaseous surface density exceeding the current surface density in gas (13 M ¯ pc−2 ) by a factor of
– 42 –
−1
≈ 2, it would require a velocity dispersion >
to maintain Qg = cκ/πGΣg > 1. Such a
∼ 10 km s
high velocity dispersion is difficult to maintain in view of the sharp rise in the cooling curve of atomic
gas at T = 104 K. (If one views the interstellar medium as an ensemble of discrete clouds, then
it is difficult to maintain the velocity dispersion above the sound speed because the average energy
dissipated in cloud-cloud collisions rises sharply when the collisions are supersonic, and because of
the onset of instabilities at the cloud surface that shred the cloud.) Furthermore, the presence of a
young stellar disk may force the required velocity dispersion even higher, since (if the local rotation
curve remains fixed) the presence of the background stellar component always destabilizes the gas.
How destabilizing the stellar component is depends upon the initial velocity dispersion of young stars
and how rapidly they are dynamically heated. That young galactic disks are unstable is consistent
the galaxy formation models of Katz (1989), who finds that a rapid burst of star formation occurs in
the disks of newborn spirals, and with the numerical results of Hernquist (private communication),
who finds that global disk models with gas are very difficult to stabilize above a critical value of the
gas fraction. Finally, if damped Lyman-α systems (reviewed by Wolfe, 1988) are indeed rotationally
supported, they may be unstable because their large size implies a low rotation frequency and hence
low epicyclic frequency. Thus for a typical surface density of 10 M¯ pc−2 , a velocity dispersion of
8 km s−1 , and an assumed rotation frequency of half the solar value (26/2 km s−1 kpc−1 ), Qg = 1 and
the cloud is only marginally stable.
In summary, we have studied the linear response of a local model of a mixed star and gas disk.
The stability of the system has been analyzed and comparisons made to earlier work. Calculation
of the nonaxisymmetric response of the local model to a point-mass on a circular orbit in the disk
give typical scales for the response and indicate how the response changes with the model parameters,
which are the gas fraction, the Q parameter for the stellar and for the gaseous component, and the
logarithmic derivative of the rotation curve. Young galactic disks are hard to stabilize, both because
they require velocities dispersions above 10 km s−1 and because the stellar component is destabilizing.
– 43 –
Appendix
Here we show that it is necessary and sufficient for stability of radial modes that there be no
solutions to the dispersion relation when ω 2 = 0. First we show that ω 2 is real provided that the
condition ∂f0 /∂Jr < 0 is satisfied, then we show that if an instability exists then there exists a mode
with ω 2 = 0. We consider an infinite thin sheet with a collisionless and a gaseous component, each
with an initially uniform surface density.
Consider the operator
µ
¶
∇2
Mψ ≡ R −
ψ
(60)
4πG
Here R is the linear response operator, defined so that σ = Rψ is the linear surface density response
to an imposed potential ψ. If ψ is a mode, Mψ ≡ 0. The linear response operator may be decomposed
into a gaseous and stellar part, R = Rg + Rs . Goodman (1988) has shown that the linear response
operator for purely radial modes in a stellar system like the shearing sheet (assuming a time dependence
eiωt ) may be written
)
(∞
Z
X (nκ)2 ∂f0 /∂Jr
inΘ
ψn e
(61)
δΣs = Rs ψ = |∂vr /∂Jr |dJr
−ω 2 + (nκ)2
n=1
where
Z
dΘein·Θ ψ
hψ, φi =
ψ ∗ φd3 r
ψn ≡
(62)
is the decomposition of the perturbing potential into fourier components in the epicyclic phase Θ =
Θ(r, vr ). The linear response operator for the gas is easily derived from the continuity and momentum
equations,
Z
fg k 2 ψk eikx
δΣg = Rg ψ = dk 2
.
(63)
ω − κ2 − c 2 k 2
where ψk is the k-th component of the fourier transform of the perturbing potential.
Now define the inner product
Z
(64)
where the integral is carried out over all of phase space for the stars and over configuration space for
the gas. For a mode ψ, hMψ, ψi = 0. Since h∇2 ψ/4πG, ψi is positive definite, the imaginary part of
the product hRψ, ψi must vanish if ψ is to be a mode. The imaginary part is
!
̰
(Z
2
2
X
(nκ)
∂f
/∂J
|ψ
|
0
r
n
−
Im(hRψ, ψi) = Im(ω 2 )
2πdJr
(−Re(ω 2 ) + (nκ)2 )2 + Im(ω 2 )2
n=1
)
(65)
Z
σ0 k 2 |ψk |2
dk
.
(Re(ω 2 ) − κ2 − c2 k 2 )2 + Im(ω 2 )2
Since the term in curly braces is negative definite if ∂f0 /∂Jr < 0, the left hand side vanishes if and
only if Im(ω 2 ) = 0. Therefore ω 2 is real.
This is not sufficient to justify the use of the static limit in studying the stability of the mixed
star and gas system, since there might be a mode with ω 2 < 0 even if there is no solution at ω 2 = 0.
Goodman (1988) has shown that there is an unstable mode with ω 2 < ω12 (ω12 < 0) if for some test
function with time dependence eiω1 t we have
hM(ω1 )ψ, ψi < 0.
(66)
– 44 –
Goodman’s proof is valid only for stellar systems, but the generalization to a mixed system is
straightforward. Since
d
hMψ, ψi = ω 2
dω 2
Z
dk
(Z
2πdJr
Ã
∞
X
(nκ)2 ∂f0 /∂Jr |ψn |2
(−ω 2 + (nκ)2 )2
n=1
)
σ0 k 2 |ψk |2
(ω 2 − κ2 − c2 k 2 )2
!
−
(67)
>0
if ω 2 < 0, it follows that modes with ω 2 < 0 exist if and only if there is a solution to the dispersion
relation with ω 2 = 0.
– 45 –
References
Abramowitz, M., and Stegun, I., 1972, Handbook of Mathematical Functions, (New York: Dover).
Balbus, S.A., 1988, Ap. J., 324, 60.
Bender, C.M., & Orszag, S.A., 1978, Advanced Mathematical Methods for Scientists & Engineers,
(New York: McGraw-Hill).
Bertin, G., Lin, C.C., Lowe, S.A., and Thurstans, R.P., 1989, Ap. J., 338, 78.
Bertin, G., and Romeo, A.B., 1988, Astr. Ap., 195, 105.
Binney, J., and Tremaine, S., 1987, Galactic Dynamics, (Princeton: Princeton University Press).
Goldreich, P., and Lynden-Bell, D., 1965, M.N.R.A.S., 130, 125.
Goldreich, P., and Tremaine, S., 1978, Ap. J., 222, 850.
Goodman, J., 1988, Ap. J., 329, 612.
Gradshteyn, I.S., and Ryzhik, I.M., 1980, Table of Integral, Series, and Products, (New York:
Academic Press).
Graham, R., 1967, M.N.R.A.S., 137, 25.
Hill, G.W., 1878, Am. J. Math., 1, 5.
Hunter, C., 1964, Ap. J., 139, 570.
Issa, M., Maclaren, I., & Wolfendale, A.W., 1990, Ap. J., 352, 132.
Jog, C.J., and Solomon, P.M., 1984, Ap. J., 276, 114.
Julian, W.H., and Toomre, A., 1966, Ap. J., 146, 810.
Katz, N., 1989, Ph.D. Thesis, Princeton Univ.
Kennicutt, R., 1989, Ap. J., 344, 685.
Kuijken, K., & Gilmore, G., 1989, M.N.R.A.S., 239, 605.
Larson, R.B., 1988, in Galactic & Extragalactic Star Formation, ed. R.E. Pudritz & M. Fich, (Boston:
Kluwer) p. 459.
Lin, C.C., Yuan, C., and Shu, F.H., 1969, Ap. J., 155, 721.
Lin, C.C., & Shu, F.H., 1966, Proc. Nat. Acad. Sci., 55, 229.
Lindblad, B., 1927, Medd. Astron. Obs. Uppsala, 19, 1.
Lubow, S.H., 1986, Ap. J., 307, L39.
Lubow, S.H., Balbus, S.A., and Cowie, L.L., 1986, Ap. J., 309, 496.
Lynden-Bell, D., and Kalnajs, A., 1972, M.N.R.A.S., 157, 1.
Mark, J.W.-K., 1976a, Ap. J., 203, 81.
Mark, J.W.-K., 1976b, Ap. J., 205, 363.
Mark, J.W.-K., 1976c, Ap. J., 206, 418.
Narayan, R., Goldreich, P., and Goodman, J., 1987, M.N.R.A.S., 228, 1.
Pedloskey, J., 1987, Geophysical Fluid Dynamics, (New York: Springer-Verlag).
Press, W.H., et al., 1988, Numerical Recipes in C, (New York: Cambridge Univ. Press).
Roberts, W.W., 1969, Ap. J., 158, 123.
Sellwood, J.A., and Carlberg, R.G., 1984, Ap. J., 282, 61.
Shu, F.H., et al., 1972, Ap. J., 173, 557.
Spitzer, L., and Schwarzschild, M., 1953, Ap. J., 118, 106.
Spitzer, L., 1978, Physical Processes in the Interstellar Medium, (New York: Wiley).
Strom, S.E., Jensen, E.B., & Strom, K.M., 1976, Ap. J., 206, L11.
Toomre, A., 1974, in Highlights of Astronomy, 3, (Boston: Reidel), p. 457.
Toomre, A., 1977, Ann. Rev. Astr. Ap., 15, 437.
– 46 –
Toomre, A., 1981, in The Structure & Evolution of Normal Galaxies, (New York: Cambridge), p. 111.
Wolfe, A.W., 1988, in QSO Absorption Lines: Probing the Universe, (New York: Cambridge), p.297.
– 47 –
Figure Captions
Figure 1. The dispersion relation for the mixed star-gas model with model parameters appropriate
to the solar neighborhood.
Figure 2. The dispersion relation for the mixed star-gas model after varying the model parameters
about those appropriate to the solar neighborhood. The top left panel shows the effect of
changing the gas fraction fg ≡ Σg /(Σs + Σg ). The solid line is the standard model shown in
Figure 1. The top right panel the effect of varying Qg , while the lower left panels shows the
effect of varying Qs . The dispersion relation for radial modes does not depend on the fourth
model parameter β ≡ d ln Vc /d ln R.
Figure 3.
The ratio of gaseous to stellar surface density response along the dispersion curve. At
each |k| the dispersion relation is multiple-valued, and the curves correspond to the various
branches of the dispersion relation shown in Figure 1. The principle branch is that inside the
Lindblad resonance, i.e. with |ω| < 1. For this branch the stellar response dominates at long
wavelength, the gas response at intermediate wavelength, and the stellar response at very
short wavelength, when the density wave piles up in the vicinity of the Lindblad resonance
(|ω| = 1).
Figure 4. A comparison of the dispersion relation for the two-fluid and star-gas dispersion relation
for the solar neighborhood parameters.
Figure 5. The single-fluid dispersion relation.
Figure 6. The two-fluid dispersion relation (dashed line) and the star-gas dispersion relation (solid
line) for fg = 0.1, Qs = Qg = 1.2. The dispersion relation describes the path of the wave
packet in the shearing sheet; two approaches are made to corotation in the mixed models.
Figure 7. Neutral stability curves in Qg and Qs for different values of the gas fraction fg . That part
of the parameter space to the lower left of the neutral stability curve is unstable, while that
to the upper right is stable. The heavy lines mark the stability curve for f g = 0.1, 0.3, 0.5, 0.7,
and 0.9. The light lines are spaced at intervals of ∆fg = 0.05.
Figure 8. The dispersion relation in the star-gas model for two closely spaced points in parameter
space. The ordinate shows the frequency of the mode (if ω 2 > 0) and the growth rate (if
ω 2 < 0). A small change in the model parameters causes a dramatic change in the scale of
the response.
Figure 9. Evolution of a single nonaxisymmetric wave in the gaseous surface density after being
perturbed by an impulsive, sinusoidal potential at τ = −20. The left two panels show the
evolution of the surface density amplitude, while the right two panels show the evolution of
the wave action J (defined in text). Qg = 1.0
Figure 10. Evolution of a single nonaxisymmetric wave in the stellar surface density being perturbed
by an impulsive, sinusoidal potential at τ = −6.2. Qs = 1.0.
Figure 11. Evolution of a single nonaxisymmetric wave in the mixed star-gas model after being
perturbed by an impulsive, sinusoidal potential at τ = −5.8. The solid line is the sum of the
stars and the gas, the dashed line is the gas, and the dotted line is the stars. We have set
fg = 0.5, Qs = 2.0, Qg = 2.0, and β = 0.0.
Figure 12.
The response of the gas (Qg = 1.2, left two panels) and the stars (Qs = 2.8, right
two panels) to an imposed point-mass potential. The top two panels show the respective
surface density response in physical space, while the bottom two panels show the response in
fourier space. The arrows indicate the direction of shear; were this the solar neighborhood
– 48 –
Figure
Figure
Figure
Figure
Figure
Figure
Figure
the galactic center would be located at the far left. The contour intervals are set to be 1/30
of the total range of the response and are different for each panel.
13. The response of the mixed star-gas model for the solar neighborhood. The arrangement
of panels is the same as in Figure 12.
14. A comparison of the response of the stellar surface density (right two panels) and the
gaseous surface density (left two panels) for different values of the fractional gaseous surface
density.
15. A comparison of the response of the stellar surface density (right two panels) and the
gaseous surface density (left two panels) for different values of Qg .
16. A comparison of the response of the stellar surface density (right two panels) and the
gaseous surface density (left two panels) for different values of Qs .
17. Evolution of the response for the mixed star-gas model of the solar neighborhood in the
gas.
18. Evolution of the response for the mixed star-gas model of the solar neighborhood in the
stars.
19. Responsiveness of the mixed star and gas system as a function of the model parameters.
The responsiveness is defined here as the maximum amplitude of the gaseous response in
fourier space (near |k| = 1/2πfg Q2g ) evaluated at 5/κ after the introduction of a point mass
with unit mass at the origin.
Chapter 3: Formation of Molecular Clouds by Gravitational Instability
This chapter considers the nonlinear development of gravitational instability in a local model of
a galaxy disk. Simulations of an unstable, isothermal disk are carried out using the smoothed-particle
hydrodynamics code TREESPH of Hernquist & Katz. The vertical structure of the disk is fully
resolved.
7
In the simulations (one of which incorporates a stellar component) dense lumps form in <
∼ 10 yr,
with a characteristic mass about an order of magnitude smaller than that predicted by naive linear
theory.
The evolution of an initially azimuthal magnetic field is followed in the limit that the field is
weak. Collapse occurs roughly perpendicular to the field lines so that the magnetic pressure ∝ ρ 2 .
For reasonable values of the initial magnetic field, magnetic pressure will eventually halt the collapse.
Tests of the TREESPH code, including measurement of an effective shear viscosity, are described
in an appendix. It is shown that terms in the SPH estimates of gradients of fluid variables that are
proportional to the gradient of the smoothing length h are of order h2 and should be neglected.
1. Introduction
5
Giant molecular clouds are massive (>
∼ 10 M¯ ) clouds of molecular hydrogen with mean diameters
≈ 100 pc and mean separation ≈ 500 pc (e.g. Blitz, 1991). They are closely correlated with massive
star formation (e.g. Shu, Adams, & Lizano, 1987) and with spiral arms (Stark, 1979). Because
giant molecular clouds are intimately connected with star formation, and hence the evolution and
appearance of spiral galaxies, the mechanisms that govern their formation are of considerable scientific
interest.
Various processes have been proposed to assemble giant molecular clouds, including gravitational
instability, coagulation, accretion, the Parker instability, fragmentation of supershells, and temporary
assembly in spiral arms (see Chapter 1 or Elmegreen, 1991, for a review). The observation that most
7
nearby giant molecular clouds are forming massive stars that will destroy them in <
∼ 10 yr (e.g. Bash,
Green, & Peters, 1978) places a demanding time constraint on cloud formation theories. Coagulation,
accretion, and the Parker instability all probably take too long to make giant molecular clouds. Of
the remaining theories, gravitational instability is particularly promising in view of the observation
of Kennicutt (1989) that Qg ≡ cκ/πGΣg is approximately 1 throughout the star-forming regions of
galactic disks. Here c is the gaseous velocity dispersion, κ is the epicyclic frequency, and Σ g is the
gaseous surface density. The Qg parameter is an indicator of the importance of self-gravity in the disk;
for Qg < 1 a thin, gaseous disk is gravitationally unstable. A plausible explanation of the constancy
of Qg is regulation by a feedback loop involving gravitational instability and star formation. But the
7
mass scale typical of gravitational instability in a disk is large, >
∼ 10 M¯ in the solar neighborhood,
so that gravitational instability is unlikely to produce individual stars. Rather it seems likely that
instability should produce massive gas clouds, like the giant molecular clouds observed in our galaxy,
that in turn give birth to many individual stars. We are encouraged, therefore, to investigate a theory
of giant molecular cloud formation that incorporates only self-gravity, rotation, and pressure, which
are the physical ingredients that enter the Qg criterion.
We investigate the possibility that molecular clouds form by gravitational collapse in the context
of a local model of the galactic disk (or “shearing sheet”). The local model is constructed by choosing
some point in the disk and expanding the equations of motion to first order in the ratio λ c /R0 , where
– 50 –
λc is a characteristic scale for disturbances in the disk and R0 is the distance to the galactic center.
This approximation is usually marginal for the stellar component in galactic disks and usually better
for the gaseous component. The local model is a useful tool because it captures the most important
aspects of motion in the disk (rotation and shear) and because it is simple.
A second approximation made in our numerical models of the galactic disk is that the disk is
initially uniform and isothermal. Clearly one is compelled to treat this relatively simple and well-posed
problem before turning to the more difficult issues of inhomogeneity, cooling, and star formation. We
shall not attempt to defend our model as anything other than a zeroth-order approximation to the
galactic disk.
The local model is cast in dimensionless form by setting G = Σ = κ = 1, where Σ is the total
surface density and κ is the epicyclic frequency. In the event that the gas and stars are uniform
and isothermal, the model is fully described by five dimensionless parameters: Q g ≡ cκ/πGΣg , the
Toomre Q-parameter for the gas layer; Qs ≡ σr κ/3.36GΣs , the Q-parameter for the stellar component
(it is assumed that the stars have a Schwarzschild velocity distribution) ; f g ≡ Σg /Σ, the fractional
surface density of gas; β ≡ d ln |Vc |/d ln R, the logarithmic derivative of the rotation curve, which
we shall always take to be 0 (i.e. the rotation curve is flat); and σz /σr , the ratio of the vertical
velocity dispersion in the stars to the radial velocity dispersion, which we always take to be 0.6. In
the solar neighborhood, Σg ≈ 13 M¯ pc−2 , Σs ≈ 35 M¯ pc−2 , κ ≈ 37 km s−1 kpc−1 , σr ≈ 40 km s−1 ,
and c ≈ 6 km s−1 (the origin of these numbers is explained in Chapter 1). The time unit is then
2.7 × 107 yr. When the stellar component is neglected, the length unit is 42 pc, the velocity unit
1.5 km s−1 , and the mass unit 2.3 × 104 M¯ . When the stellar component is included, the length unit
is 154 pc, the velocity unit 5.6 km s−1 , and the mass unit 1.1 × 106 M¯ . All dimensional estimates that
follow are made by reference to these solar neighborhood units.
2. Boundary Conditions, Initial Conditions, and Numerical Techniques
A local model of the galactic disk was evolved using the three-dimensional, self-gravitating
hydrodynamics code TREESPH developed by Hernquist and Katz. TREESPH combines the tree
method for calculating the gravitational acceleration (Barnes & Hut, 1986) with the smoothed particle
hydrodynamics (SPH) method for evolving the fluid (e.g. Gingold & Monaghan, 1977). A complete
description of the original TREESPH code is given in Hernquist & Katz (1989, hereafter HK). The
code is fully vectorizable and was run on the Cray Y-MP at the Pittsburgh Supercomputing Center
and a Convex C-220 at Princeton.
TREESPH has been tested on a number of exactly solvable problems, including three dimensional
shock tubes (for the hydrodynamic portion of the code) and the collapse of an oblate spheroid, as
described by Lin, Mestel, & Shu (1965; for the gravitational portion of the code). A shock tube
calculation is shown in Figure 1a, which displays particle positions, and Figure 1b, which shows fluid
variables along the length of the tube. Initially the fluid at z < 0 has density 0.085 and thermal
energy 10.0 (per unit mass) ; the fluid at z > 0 has density 0.7 and thermal energy 10.0. We have
set γ = 1.4. The tube is periodic in all coordinates, so that as soon as the simulation is started a
planar shock moves left from z = 0 and right from z = −4. The shock tube contains 5, 000 particles
and required 2.5 hrs to run on a Convex C-220. The initial conditions were prepared using the same
constrained Poisson sampling scheme used to prepare all our other initial conditions (to be described
shortly), and received no other special preparation. The code parameters are identical to those used in
our production runs, except that Nsm = 32, rather than 64 (Nsm controls variations in the smoothing
– 51 –
length. The smoothing length is always set so that there are Nsm SPH particles within two smoothing
lengths of each particle). The solid lines drawn in the Figure are the expected values, while the boxes
are an average over bins of width 0.1 in z. The particle positions are plotted as faint dots. The
agreement is satisfactory, although there is some post-shock oscillation in the velocity and thermal
energy. The Lin-Mestel-Shu collapse of an oblate spheroid produced a disk with the expected radius
at the expected collapse time, with a root mean square separation of the particles from the disk of
less than the gravitational softening length. Energy was conserved to better than 0.5%. Additional
tests to determine the noise level in SPH estimates of fluid variables and effective shear viscosity were
also performed and are described in the Appendix. Further tests of TREESPH will be described in a
(real) forthcoming paper (Gammie, Hernquist & Macmillan, 1992).
Modifications were made to the original TREESPH code to adapt it to the local model. First,
however, it is best to explain how the local model is constructed. Consider the galactic disk in the
neighborhood of some fiducial point that orbits the galactic center at radius R 0 with frequency Ω0 .
In a frame rotating with frequency Ω0 , erect a Cartesian coordinate system with origin at the fiducial
point, x-axis pointing outward in radius (so that x = R − R0 ), y-axis pointing forward in azimuth (so
that y = R0 (φ − φ0 )), and z normal to the disk. One now expands the equations of motion in this
frame to first order in x/R0 . In our version of the local model this means neglecting gradients in the
surface density as well. The undisturbed equilibrium is an infinite plane-parallel sheet with uniform
shear, vy = 2Ax, A ≡ (1/2)dΩ/d ln R > 0 is Oort’s A constant, and Ω < 0 (we use the sign conventions
of Goldreich & Lynden-Bell that give the correct sense of rotation for the galaxy). The equations of
motion for a particle, or the momentum equations for the fluid, take on their usual form except for the
addition of two new accelerations. The first is the usual Coriolis acceleration, −2Ω × v. The second
is a tidal acceleration that arises from an expansion of the effective potential about the fiducial point:
φef f = φgrav + 2AΩx2 (which is second order in x but gives rise to a first order acceleration). This
local cartesian approximation dates back to Hill’s researches on the lunar theory (Hill, 1878), so the
particle equations of motion are sometimes known as Hill’s equations, and it has been widely used
in studies of galactic disks (Lindblad, 1927; Spitzer & Schwarzschild, 1955; Julian & Toomre, 1966;
Goldreich & Lynden-Bell, 1965; Goldreich & Tremaine, 1978) and of planetary rings (e.g. Wisdom &
Tremaine, 1988).
The Coriolis force and tidal acceleration are trivially included in TREESPH, but the boundary
conditions are more problematic. The difficulty arises because one must represent only a small
section of the disk yet maintain the translational invariance of the shearing sheet. In cosmology the
natural choice of boundary conditions under similar circumstances is periodic, but periodic boundary
conditions are excluded in the local model because of the shear flow. Nevertheless, it is possible
to implement boundary conditions that are periodic in a shearing coordinate system. This shearing
periodic system is equivalent to the “sliding bricks” scheme used by Wisdom & Tremaine (1988) to
study planetary rings. Each particle in the simulation may be thought of as having an image at the
grid points of a shearing lattice:
xi,j = x0,0 + iLx̂ + (jL + 2AiLt)ŷ
(1)
where L is the size of the simulation box, t is the current time, A is Oort’s constant, and i and j
are integers. Particle-particle interactions are restricted to r < L/2 so that each particle interacts
only with the nearest image of each other particle. If one focuses attention on the particles within a
box of size L centered at a particular location, the images of this box give the appearance of a brick
– 52 –
wall with each layer of bricks sliding on top of the next. Figure 2 illustrates the boundary conditions
with particle positions from one of our test runs. Every particle in the simulation has an image in
the central box. The dense clump near (-10,5) has images near (35,-15) and (-10,-40). The arrows
indicate the direction of the shear. With our sign conventions, the center of the galaxy is located at
the far left. The simulation is invariant under translations in the plane of the disk.
The sliding-brick boundary conditions are implemented in TREESPH with the aid of a mapping
function MAP(xi , xj ) that returns the position and velocity of the image of particle j nearest to
particle i. Thus whenever a relative distance is required, e.g. for the purposes of calculating the
gravitational acceleration of particle j on particle i, one simply MAPs the position of particle j before
calculating the separation vector. The performance penalty from the MAPping operation is negligible.
Note that relative velocities must also be mapped since different images have different velocities due
to the shear.
The reader familiar with the tree algorithm may wonder how the tree is maintained in the presence
of shear. The trick is to only record the position of the image of each particle that lies within the box
−L/2 < x < L/2, −L/2 < y < L/2. When the recorded position of a particle crosses a box boundary,
a second image enters the box on the opposite boundary; the position of the second image is thereafter
used as the recorded position. Since the particle positions all lie within a box, the tree construction
and descent proceeds as if there were no shear at all, as long as the relative positions of particles and
nodes in the tree are calculated with MAP. No special treatment of the box boundaries during the
tree descent is required.
Because our simulations are periodic on a shearing lattice, the modes are not the continuous set
that characterizes the infinite shearing sheet, but rather a discrete set. At t = 0 the wavenumbers are
multiples of 2π/L, so we use the notation (nx , ny ) to denote a mode that has initial kx = 2πnx /L, ky =
2πny /L.
To prevent the gravitational interaction of particles with more than one image of other particles,
the gravitational acceleration is modified from the usual r −2 to the truncated gravitational kernel

G(², r)


−1/r 2
ar = m p
2

 −(1 − 2r/L)/(L/3)
0
r < 2²
2² ≤ r < L/3
L/3 ≤ r < L/2
r ≥ L/2
where G is the spline kernel given in HK and mp is the particle mass. The truncation of the
gravitational kernel causes very little error in a highly flattened system like the galactic disk, where
most of the acceleration is contributed by nearby particles. To convince oneself of this, consider the
gravitational acceleration at some point in a two dimensional disk due to a single fourier component
in the surface density. The contribution to the acceleration due to all parts of the disk with
R
R
−1/2
r > r 0 ∼ r0
, since ak (r > r0 ) ∼ (1/r)J1 (kr)dr ∼ cos(kr)r −3/2 ∼ r−1/2 × an oscillating
part. Furthermore, at large scales the dynamics is governed by rotation rather than self-gravity,
as may be seen by considering the dispersion relation for radial modes in a gaseous thin disk,
ω 2 = κ2 + c2 k 2 − 2πGΣg |k|. In the limit that k→0, ω 2 →κ2 , and the pressure (c2 k 2 ) and selfgravity (−2πGΣg |k|) are negligible. As long as L À GΣ/κ2 , then, the truncation does not change
the dynamics significantly. In fact, the most important changes in the dynamics are caused by the
gravitational softening, which reduces the importance of self-gravity.
The use of a nonstandard gravitational kernel changes the vertical equilibrium structure of the
disk from the usual sech2 (z) density profile to a new configuration that is the solution to a nonlinear
– 53 –
integral equation. Rather than solve this equation numerically, we introduce a harmonic potential of
the form ψ(z) = (π/L)z 2 , so that at z = L/2 the acceleration is −2π, just as it would be above an
infinite sheet of surface density 1 with ar ∝ r−2 . Equilibria in this potential have a very nearly sech2
density profile. Figures 3a shows particle densities as a function of z from a test run, and Figure 3b
shows the ratio of this density to that expected in a self-gravitating isothermal sheet.
Several tests were made of the boundary conditions. A trivial check is to integrate particle orbits
with epicyclic amplitude larger than L/2. A much more demanding test is to evolve a set of initial
conditions, then translate the initial conditions in a random direction in the x − y plane, evolve them
again, and compare particle positions. Differences in particle positions were small (consistent with
differences in gravitational accelerations caused by shifting the tree structure) and not correlated with
the boundaries of the box.
Initial conditions were prepared by Poisson sampling the density field and then removing particles
with separation less than a fixed fraction of the mean interparticle separation. This minimizes the
energy associated with closely spaced particles that is transformed into random particle motions, and
hence an effective particle pressure, once the simulation is started.
The number of particles used in simulations of the shearing sheet is constrained by two conditions.
First, the SPH smoothing length must be smaller than the disk scale height so that the vertical
structure of the disk is resolved. Second, the simulation size L must be larger than several critical
wavelengths λcr = 2π 2 (Q/Qcr )2 GΣ/κ2 (λcr is the wavelength where ω 2 is a minimum, and Qcr is
the neutrally stable value of Q; Qcr = 0.676 in an isothermal, gaseous, self-gravitating disk with a
standard gravitational kernel), so that structure develops on a scale smaller than the size of the box.
This leads to the requirement
N > 5.2 × 104 (Nsm /64)(L/3λcr )2 ,
(2)
where Nsm is the number of particles required to be within two smoothing lengths and so relates
the smoothing length to the particle number density. N = 5 × 104 is very near the practical limit
with current computing facilities, and we always run with fewer particles, compromising the vertical
structure of the disk slightly.
Yet how can we be sure that the simulation has converged? Defining convergence in TREESPH
simulations is problematic because there are several accuracy parameters all of which must be taken
to the limit of high accuracy in the appropriate order. The accuracy parameters are the softening
length ², the smoothing length parameter Nsm , and the total number of particles N . In test runs
we simulated a box that is 2/3 the size of our production runs with different numbers of particles,
and set Nsm and ² just as we would in setting up a simulation with the same number of particles:
Nsm = 64, and ² ' hhsm i. Figure 4 shows the positions of particles at t = 4/κ in the test simulations.
The runs were made with N = 104 particles, N = 1.3 × 104 (identical to the surface number density
used in our production runs), and N = 2 × 104 . The initial conditions were perturbed by introducing
a gaussian random field in the velocities for the N = 2 × 104 run (using the same power spectrum as
in the production runs), then randomly removing particles from those initial conditions to create the
initial conditions for the other two runs. The outcomes are not identical, but one should not expect
them to be because there are different levels of Poisson noise in the initial conditions for each of the
test runs. The amplitude of fourier component in the surface density from Poisson noise is of order
N −1/2 ∼ 0.01 to 0.007; these modes grow exponentially and significantly affect the outcome. But
there are also qualitative differences. The lower-N simulations have enhanced viscous transport in the
– 54 –
dense regions, giving rounder, denser clumps (but the distribution of densities is similar for all three
test runs). Nevertheless, the runs are similar enough to support the very modest conclusions we shall
attempt to draw from the production runs.
Several integrals of the motion exist that can be used to check the accuracy of the integration
(Wisdom & Tremaine, 1988). The total y-momentum, which is a linear function of the orbital angular
momentum, is conserved (because of the invariance of the Lagrangian under translations in y). The
P
P
initial conditions have i ẏi = 0; at the end of each run | i ẏi /2AL| <
∼ 0.01. A second, energy-like
integral is given by
µ
¶
X
1 2
Γ=
mi
v + φ + 2AΩx2 + c2 ln ρ ,
(3)
2
i
if the fluid is isothermal and the net flux of the associated “energy” density across the boundaries at
|x| = L/2, |y| = L/2 is zero. Typically δΓ/Γ <
∼ 0.01.
3. Experiments
The initial conditions for the two runs described here are summarized in Table 1, along with
the numerical parameters for the runs. In both runs the initial surface density is uniform, and
perturbations are introduced in the velocity field. Run 1 has Qg = 0.9Qcr ' 0.6, implying a midplane
density that is unrealistically large; this low Qg was used because we wanted to simulate an unstable
gaseous component in isolation. Including the stellar background makes the disk more unstable at
fixed Qg and lowers the midplane density to more reasonable values. The initial peculiar velocity field
in Run 1 is a random field with a Gaussian power spectrum, hPv (k)i = (0.013cg )2 exp(−vk2 /σk ), and
random phase. Here σk = 6[2π/L], and there is no power at 0 wavenumber. This power spectrum
was chosen because it is similar to that expected from a random distribution of softened point masses.
In Run 2 a stellar background component is also included, and Qg = 1.0, Qs = 2.0, and fg = 0.3.
The initial peculiar velocity field is again a random field applied to both the stars and the gas,
with hPv (k)i = (0.087cg )2 exp(−vk2 /σk ), where σk = 6[2π/L], and again there is no power at zero
wavenumber. Run 1 is our fiducial run; Run 2 tests the sensitivity of the outcome to the stellar
background component.
Figures 5 and 6 show the positions of 5000 randomly selected particles from runs 1 and 2,
respectively, projected onto the plane of the disk. In Figure 5 the frames are spaced at intervals
of 0.5/κ, beginning at t = 1/κ in the upper left hand corner. In Figure 6, the frames are spaced
at intervals of 0.4/κ, beginning at t = 0.8/κ in the upper left. The star particles from Run 2 are
not shown because the amplitude of the surface density perturbation, even at the end of the run, is
almost imperceptible. In each case the nonaxisymmetric modes grow rapidly and several dense regions
collapse to form clouds. The collapse is limited by gravitational softening.
3.1 Cloud Masses
Linear theory predicts that gravitational instability in a uniform disk should produce objects
of a characteristic mass (Goldreich & Lynden-Bell, 1965; Jog & Solomon, 1984). The scale of the
instability is set by a compromise between pressure support, self-gravity, and rotational support, and
for axisymmetric modes is
λcr = 2π 2 (Qg /Qcr )2 G2 Σ3g /κ4 ,
(4)
– 55 –
where Qcr is the critical value of Qg required for linear stability. If we take the characteristic mass to
be the product of the surface density and the square of the characteristic wavelength, then we obtain
Mchar = Σg λ2cr = 4π 4 (Qg /Qcr )4 G2 Σ3 /κ4 .
(5)
In the solar neighborhood this is ≈ 8.5×106 M¯ (Qg /Qcr )4 (Σg /13 M¯ pc−2 ), too large for gravitational
instability to be responsible for the formation of GMCs with masses similar to the Rosette. Yet, as
we shall see, a succession of factors reduces the efficiency of cloud formation and produces smaller
masses than predicted by eqtn.[5].
CO observations detect gas above a critical density, so it is natural to define clouds in our
simulations in terms of a density threshold. Figure 7 shows the evolution of the density distribution
of particles with time in Run 1. We set the density threshold to the solar neighborhood equivalent of
50 cm−3 ; this is close to densities where one expects the material to become molecular. Only 10.4% of
the particles are at n > 50 cm−3 in the final frame. For the particular set of initial conditions evolved
here, then, gravitational collapse is inefficient at forming dense objects. The clouds picked out by the
density threshold are shown in Figure 8, and their vital statistics are gathered in Table 2. Column 2
of the Table shows the mass of the clouds referred to the standard solar neighborhood model. The
average cloud mass is 19G2 Σ3 /κ2 = 4.3 × 105 M¯ , or about 1/12 of the characteristic mass predicted
by eqtn.[5], since Qg = 0.9Qcr . Raising the density threshold eliminates some of the smallest clouds
altogether but lowers the mass of each cloud; for the three largest clouds in Table 2, the average mass
is 1.3 × 106 M¯ , or Mchar /4.
How rapidly do the clouds form? Figure 9 shows the evolution of Cloud 5 as a function of density
threshold. The mass at density > 80 cm−3 jumps from 0 at t = 108 yr to more than 4 × 105 at
t = 1.06 × 108 yr. It is not particularly surprising that gravitational collapse can change the density
rather rapidly (the dynamical time (Gρ)−1/2 is 1/κ at a density of ≈ 13 cm−3 ), but we note that
unlike most other proposed mechanisms for giant molecular cloud formation, gravitational instability
can change the density of a large mass of interstellar gas from a density at which it is likely to be
7
atomic to a density at which it is likely to be molecular in <
∼ 10 yr.
The reader should be warned that the evolution within the densest regions is completely
compromised by gravitational softening, which controls the linear size of the collapsed clouds (they
are somewhat larger than clouds observed in the solar neighborhood). In fact, the evolution at high
densities is doubtful for several reasons. First, cooling, which we have not included, will cause the
temperature and pressure to drop in regions of high density (pressure is important in supporting the
clouds). Second, even if the mean initial magnetic field is weak it will dominate the dynamics in the
collapsed regions (we show this in the next section). Third, some of the clouds in the final frame
are old enough that star formation would likely have disrupted them already. Therefore the internal
structure of the clouds (their linear size, for example) is set by the softening length.
Why are the clouds that form in Run 1 smaller than the prediction of eqtn.[5]? Consider cloud 5
at t = 4/κ, when its mass = 48G2 Σ3g /κ4 . It, like most of the other clouds in the simulation, is part of
a long, narrow filament. Suppose we model the filament as a narrow, straight, self-gravitating string
supported against collapse by pressure and tidal forces. The string will break in the middle unless its
length l satisfies
µ
¶1/2
2Gµ
l<
,
(6)
A|Ω| sin θ
where µ is the mass per unit length, and θ is the angle that the string makes with curves of constant
radius (the “pitch angle”). To verify this, calculate the gravitational acceleration at position x k from
– 56 –
the center of the string; near the center, the gravitational acceleration varies linearly with x k . The tidal
acceleration is −4AΩxk sin θ. If the tidal acceleration increases more rapidly than the gravitational
acceleration, the string will break. For cloud 5, µ ≈ 6, l ≈ 8, and θ ≈ π/4, so eqtn.[6] requires 8 < 8.2,
and so is consistent with the cloud being tidally limited. (In fact, we can check the accelerations
directly, and the tidal acceleration begins to dominate at the each end of the cloud.) This explains
why the cloud only grows slightly until the end of the simulation. The mass per unit length is the
surface area × the spacing between the clouds, which we write ²1 λcr , × the fraction of the material
between the clouds that falls in, which we write ²2 . From the simulation, we have ²1 ≈ 0.6, and since
µ ≈ 6 and λcr ≈ 16 we have ²2 ≈ 0.63. The mass is then Mc = ²3 µl, where ²3 ≈ 1.0 takes account
of pressure support of the cloud which reduces its length somewhat below the tidally limited value.
Then Mc = 48(Qg /Qcr )3 G2 Σ3g /κ4 , which is about 1/5 of the characteristic mass.
Next we consider the effect of adding an isothermal stellar component. From Figure 6, one sees
that the outcome is qualitatively similar, although the dense regions are better organized into armlike features. This is because the (1, 1) mode dominates the evolution, and the outcome is in part
determined by the limited wavenumber resolution of the simulation and the consequent discreteness of
the most unstable modes. The amplitude of the stellar surface density in the (1, 1) mode is about 6%,
implying a potential amplitude for the mode of 20 km2 s−2 , or very nearly c2 . The stellar component
alone is therefore enough to force a variation in the gaseous surface density of order unity.
The clouds formed in the star and gas run with n > 50 cm−3 are listed in Table 3; their average
mass is 3×106 M¯ . A direct comparison of Run 1 with Run 2 is not possible because all the parameters
for the gas are not the same (the gas would be much more unstable if we set Q g = 0.6, as in Run
1), and we have not evolved the model quite as long. Nevertheless, the masses are about 1/5 of what
would be predicted by naive linear theory.
3.2 Magnetic Fields
For large-scale motion in the galactic disk resistivity and ambipolar diffusion are negligible, and
the magnetic field is transported bodily with the fluid. The field at time t 1 can then be written in
terms of the field at an initial time t0 as
Bi (x1 , t1 )/ρ(x1 , t1 ) = Bj (x0 , t0 )/ρ(x0 , t0 )
∂x1,i
∂x0,j
(7)
(Parker, 1979). In the limit that the initial field is weak the magnetic field pressure and tension are
negligible and the evolution of the fluid is unaffected by the magnetic field. Given an initial magnetic
field configuration, the evolution of the field in our simulations can then be evaluated using eqtn.[7].
A reasonable initial configuration is a purely azimuthal field. The configuration of the field can be
illustrated by drawing lines between particles to represent field lines, and Figure 10 shows an initially
azimuthal field from Run 1 (the particles are selected to lie in a narrow cylinder in the plane of the disk,
then sorted in y and connected). Figure 11 shows the same field at times t = 2.75/κ, 3.25/κ, 3.75/κ,
and 4.25/κ. Collapse occurs perpendicular to the field lines, so that within the collapsing cloud
B ∝ ρ. This differs from the scaling for quasi-static contraction (e.g. Mouschovias, 1976) because
the field geometry is completely different. Concentrating on Cloud 5 at time t = 3.75, we find that
the field inside the cloud is enhanced by a factor of ≈ 12. About 1/2 of this enhancement is from
vertical compression, while the rest is the result of motion in the plane. The magnetic field unit
is G1/2 Σ = 0.7 µG in the solar neighborhood, so for an initial ordered field exceeding 2.1 µG the
– 57 –
magnetic field will dominate at the cloud surface (n = 50 cm−3 ), while for an initial ordered field of
1 µG magnetic pressure dominates at 220 cm−3 . Pulsar rotation measures indicate a large-scale field
of 1.7 µG (Manchester & Taylor, 1977), so it is likely that magnetic field pressure will terminate the
collapse.
In Figure 12, the field is almost always nearly parallel to isodensity contours in dense regions.
A simple argument based on the conservation of potential vorticity shows that this is likely to be
a general feature of large scale collapse in disks, and not just the outcome of the particular initial
conditions evolved here. In a two-dimensional fluid the potential vorticity ξ
ξ=
ẑ·(∇ × v)
Σ
(8)
is constant for each fluid element, provided that there is a one to one correspondence between pressure
and density, the viscous acceleration is negligible (i.e. there are no shocks), and the magnetic field
is negligible (e.g. Pedloskey, 1987; this is simply the conservation of circulation in two dimensions).
Here v is the velocity in an inertial frame; in the shearing sheet eqtn.[8] becomes
ξ=
ẑ·(∇ × vp ) + 2B
Σ
(9)
where B is Oort’s constant B and and vp is the peculiar (i.e. shear subtracted) velocity in the rotating
frame.
Because the fluid is isothermal and the velocity is nearly independent of z in our simulation,
ξ ' 2B for each fluid element as long as it is unshocked. Evidently in dense regions |z·(∇ × v)| should
increase to compensate for the increase in surface density. Consider a dense and as yet unshocked
region in the flow, and suppose that it is elongated along some coordinate x k (as are most of the dense
regions in the simulations). Then v⊥ , the infall velocity, reaches a maximum in the middle of the
elongated region, so that ∂v⊥ /∂xk = 0. Then |z·(∇ × v)| ≈ ∂vk /∂x⊥ ≈ 2B(Σg /Σ0 − 1). For a density
contrast of >
∼ 1, the shear is enough to stretch the field along the cloud on a timescale <
∼ 1/2B.
3.3 Cloud Kinematics
For reasons related to the conservation of potential vorticity discussed in the last section, collapse
in the disk produces rapid prograde rotation, but we shall defer a detailed discussion of cloud rotation
until Chapter 4.
As a check on the plausibility of our simulations, we “observe” the dense regions by choosing
a location in the galactic disk for the center of the simulation and mapping particle positions into
l, b, v space. Figure 13 shows Run 1 at time t = 5.0/κ in projection onto the l, v plane. The center
of the simulation is assumed to be located at l = 34 deg and a distance of 7 kpc from the sun (the
choice is arbitrary). Its galactocentric distance is then 4.8 kpc. Since the density unit is κ 2 /G, and
we assume a flat rotation curve, the densities increase by a factor of (8.5/4.8) 2 = 3.1 over the solar
neighborhood values. To scale the simulation, we set Σg = 20 M¯ pc−2 . Figure 14 shows all points
with density n > ncr = 100 cm−3 . The boxes in the lower left hand corner of the Figure indicate the
resolution of the the Columbia, FCRAO, and Bell Labs CO surveys. The Columbia and BTL survey
are fully sampled on their smallest grid. The FCRAO has 3’ spacing between beams and this spacing
is indicated by the pair of boxes next to the FCRAO label. Most of the objects seen here are similar
in size and velocity dispersion to objects observed in the FCRAO survey.
– 58 –
4. Conclusions
We have investigated gravitational collapse in a simple, local model of the galactic disk. Assuming
the equation of state of the gas is isothermal, and introducing a small amplitude noise in the velocity
field, we find that gravitational instability produces objects of mass comparable to giant molecular
7
clouds in the solar neighborhood (several ×105 M¯ ) on a timescale of <
∼ 10 yr. The typical mass
is about 1/10 of that predicted by naive linear theory based on the most unstable wavelength of
axisymmetric modes. The difference arises because the typical wavelength of the most important
nonaxisymmetric disturbances is smaller than the most unstable axisymmetric mode at the time of
collapse, and because the accretion of material onto the proto-cloud is inefficient.
We follow the evolution of an initially azimuthal magnetic field in the limit that the field is weak,
and find that collapse always occurs perpendicular to the field lines, so that B ∝ ρ. For even modest
magnetic field strength this implies that the magnetic field will dominate the dynamics of the collapsed
object. The magnetic field always has a definite orientation with respect to the cloud and the galaxy,
making an angle of ' 45 deg with respect to the radius vector.
The most robust prediction of the gravitational instability model for the formation of giant
molecular clouds is that their mass should scale as Q4g G2 Σ3g /κ4 ∼ c4 /G2 Σg , with Qg ≈ const. In our
galaxy this implies little variation between the solar neighborhood and the molecular ring. In some
particularly large galaxies, however, κ varies by large factor from near the center to the edge of the
disk. If millimeter interferometers can resolve individual molecular clouds in these galaxies (and if the
clouds are molecular), then there should be a definite correlation of cloud mass with galactocentric
radius.
– 59 –
Appendix: SPH Noise and Shear Viscosity
In this Appendix we describe various tests of smoothed particle hydrodynamics and address two
problems with SPH algorithms: the magnitude of terms that arise because of variable smoothing
length, and unphysical particle clustering.
A1.1 Variable Smoothing Length
In the HK implementation of SPH, the smoothing length h is adjusted continuously so that
approximately Nsm particles are found within two smoothing lengths. This controls the noise level
in estimates of fluid quantities and allows higher density regions to be simulated with higher spatial
resolution. However, time and space variable smoothing lengths introduce terms proportional to dh/dt
and ∇h into estimates of fluid variables (HK; Evrard, 1988). These terms are not incorporated into the
code for algorithmic reasons, and this has been a persistent source of doubt about the performance of
the SPH algorithm. Nevertheless, it can be shown that for an isothermal fluid the terms proportional
to ∇h are second order in h (there are no terms proportional to dh/dt, since no time derivatives need
to be explicitly estimated in an isothermal fluid).
Consider the extra terms that arise in estimates of spatial derivatives because of spatially variable
smoothing lengths in the scatter formalism:
Z
∂W
h∇Ai = ∇r hAi − A
(10)
∇r0 h(r0 )d3 r0
∂h
and in the gather formalism:
h∇Ai = ∇r hAi − ∇r h(r)
Z
A
∂W 3 0
d r
∂h
(11)
(HK, eqtn. 2.31 and 2.33). In both cases ∂W/∂h is integrated against a fluid quantity (∇h ∝ ∇ρ)
that can be expanded in a Taylor series in r 0 . The zero and first order terms in the expansion vanish
when integrated against ∂W/∂h for any normalized, symmetric kernel W . The second order term is
of order h2 , and so may be neglected. Thus the terms proportional to ∇h are of second order.
Attempts to include these second-order terms are likely to introduce serious errors because SPH
estimates of fluid variables are so noisy. Even when SPH tries to estimate 0 (for example the pressure
gradient in a constant density box), it produces noise at some irreducible amplitude that depends on
the number of particles within a smoothing length and the quantity being estimated.
A1.2 Noise Levels
TREESPH adjusts the smoothing length of each particle so that approximately N sm particles are
found within two smoothing lengths. The parameter Nsm determines the noise level in SPH estimates
of fluid variables and must be set with care. We performed a series of controlled experiments in a very
simple system and measured the variation in pressure acceleration, density, and velocity of individual
particles. The result of this investigation, described below, is that there is no optimal value of N sm ,
and an acceptable noise level is ultimately a matter of taste.
When Nsm = 32, the smoothing length is almost exactly the mean interparticle separation. At
lower values the particle mean free path begins to increase and the SPH system is similar to a gas,
with a substantial component of the internal energy contained in the random motion of particles. This
– 60 –
is not usually desirable. Larger values of Nsm produce increasingly quieter estimates of fluid variables,
but rapidly increase TREESPH storage requirements.
The context used to study SPH noise levels was a uniform density cube containing 4×10 3 particles
with velocity field vy = x which we shall refer to as the shearing box. The boundary conditions are
identical to those used for the shearing sheet, except that the z boundaries are periodic. The x
boundaries enforce the shear and prevent it from decaying (they do work on the fluid). We set
c = L = 1, and the equation of state is polytropic with γ = 5/3. While we have not used a polytropic
equation of state elsewhere in this work (the production runs were isothermal), by integrating the
energy equation we can monitor the viscous heating and measure an effective viscosity for the flow.
After two sound crossing times, the rms pressure acceleration was measured. Figure A1 shows the rms
value of each component of the acceleration as a function of Nsm . As Nsm increases, the acceleration
converges to a constant value because there are real sound waves present with wavelength slightly
−3/2
larger than a smoothing length. Each curve is well fit by a function of the form ∼ N sm + const.
Figure A2 shows the rms deviation of the density from the average value. Again, the curve is well fit
−3/2
by a function of the form ∼ Nsm + const., consistent with the presence of a of a real spectrum of
sound waves. The variation in the density here, <
∼ 3%, is significantly better than the variation we
have observed in less idealized systems.
The last source of noise we shall consider is the tree algorithm used to calculate the gravitational
forces. Because the tree algorithm divides the simulation box into a nested set of smaller boxes of
size L2−j , where j is the level of the box in the tree, it introduces errors in the acceleration at fixed
scales. The amplitude of this noise is readily checked and was small compared to the real potential
noise in the initial conditions.
A1.3 Effective Shear Viscosity
We define an effective viscosity by requiring that the measured heating rate in the flow be that
for a viscous fluid described by the Navier-Stokes equations with the effective value of the viscosity.
For an incompressible fluid with viscosity ν, the heating rate is
¶2
µ
Z
νρ
∂vj
∂vi
U̇ =
+
d3 r
.
(12)
2
∂xj
∂xi
Specializing to the shearing cube, the effective viscosity for a cube of volume V with uniform shear
S ≡ ∂vy /∂x is
U̇
.
(13)
νef f =
ρV S 2
In all the test runs ρ = S = V = 1, so νef f = U̇ . Figure A3 shows the evolution of the total thermal
energy with time for different values of Nsm . In each case there is an initial drop in total thermal
energy, followed by relaxation and then a steady rise because of viscous heating. The relaxation
time declines as Nsm increases, and this is clearly a powerful motivation for increasing the value of
Nsm . The effective shear viscosity is shown in Figure A4 as a function of Nsm in units of the mean
interparticle separation times the sound speed. U̇ is measured between t = 1 and t = 2.
It must be stressed (ahem) that the effective SPH viscosity is not a Navier-Stokes viscosity, in
the sense that it is a function of the dimensionless ratio hsm S/c. This may be seen by considering the
viscous acceleration for particle i, which has the form
Ã
!
X
−αµij cij + βµ2ij
mj
ai,visc = −
(∇i W (rij , hi ) + ∇i W (rij , hj )) ,
(14)
2ρij
j
– 61 –
where j runs over nearest neighbors, cij ≡ (ci + cj )/2, ρij = (ρi + ρj )/2, W is the smoothing kernel,
rij is the distance of particle i from j, hp is the smoothing length of particle p, and
µij =
vij ·rij
2
hij (rij /h2ij +
η2 )
(15)
if vij ·rij < 0, and µij = 0 otherwise, hij is the average of the smoothing lengths of particle i and
j, vij is the particle-particle velocity, and η is a black magic parameter that prevents divergences. α
and β are determined by the requirement that shocks similar to those that appear in the simulation
(mach number <
∼ 4) are captured correctly, and in all production runs they are both set to 0.5. From
eqtn.[15] it is clear that the viscous stress increases sharply at hsm S/c >
∼ 1 because of the term ∝ β. In
the shearing box tests 0.06 < hsm S/c < 0.1; in the production runs hsm S/c is typically less than 0.15
in the initial conditions, so the effective viscosity is comparable. In some regions late in the evolution
of the production runs the viscosity is much higher and significantly affects the evolution.
Since particles have small random velocities with respect to the mean flow, there is also a stress
σij = hρvi vj i set up by random particle motions. This stress should be small compared to the SPH
estimates of the pressure if the SPH system is to represent a smooth fluid, although it could be of
interest to simulate a quasi-collisional fluid under some circumstances. In the shearing cube tests the
components of σij are always less than 3% of the physical pressure and the equivalent viscosity due
to the off-diagonal terms was essentially unmeasurable.
A1.4 Particle Clustering
Schüssler and Schmitt (1981) have suggested that under certain conditions SPH particles can
cluster artificially and produce unphysical outcomes. To check this, we calculated the correlation
function of particles in the shearing cube. Figure A5 shows the dimensionless autocorrelation function
ξ ≡ hρ(x0 )ρ(x−x0 )i/hρi2 −1 as a function of particle separation for different values of Nsm (normalized
to the smoothing length and calculated at the end of the run). For Nsm = 32 the smoothing length
is equal to the mean interparticle separation and particles are excluded from r < h sm . At r ≈ hsm
they are strongly correlated, and at larger separations there is a decaying oscillation in the correlation
function that reflects the ordered structure of the particles. For larger values of N sm this structure
becomes less pronounced and more particles are permitted at small separation. Artificial clustering
of particles does not occur, and will not as long as the interparticle force is repulsive.
When self-gravity is included, the interparticle force is attractive for some values of the
interparticle separation. In all the shearing sheet experiments described in the body of this chapter,
the gravitational softening is set to a constant value for all the particles. If the gravitational softening
length is set to too small a value, then the interparticle force is attractive at small separations, and the
SPH system condenses, forming bound clusters of SPH particles. The condition that the interparticle
acceleration ar be repulsive near the origin is dar /dr(r = 0) > 0, which for the gravitational kernel
and smoothing kernel used in TREESPH is sufficient to guarantee that the interparticle force is
repulsive near the origin. This conditions reduces to ² > (h4 2π/9λJ )1/3 in TREESPH, where ² is the
p
gravitational softening length, h the SPH smoothing length, and λJ ≡ c/ (Gρ) is (nearly) the Jeans
length.
– 62 –
References
Barnes, J., & Hut, P., 1986, Nature, 324, 446.
Bash, F.N., Green, E., & Peters III, W.L., 1977, Ap. J., 217, 464.
Blitz, L., 1991, in Physics of Star Formation & Early Stellar Evolution, (Dordrecht: Kluwer), p. 3.
Evrard, A., 1988, M.N.R.A.S., 235, 911.
Elmegreen, B.G., 1991, in Physics of Star Formation & Early Stellar Evolution, (Dordrecht: Kluwer),
p. 35.
Gammie, C.F., Hernquist, L., & MacMillan, S., 1992, in preparation.
Gingold, R.A., & Monaghan, J.J., 1977, M.N.R.A.S., 181, 375.
Goldreich, P., & Lynden-Bell, D., 1965, M.N.R.A.S., 130, 125.
Goldreich, P., & Tremaine, S.D., 1978, Ap. J., 222, 850.
Hernquist, L., & Katz, N., 1988, Ap. J. Suppl., 70, 419.
Hill, G.W., 1878, Am. J. Math., 1, 5.
Jog, C.J., & Solomon, P.M., 1984, Ap. J., 276, 114.
Julian, W.H., & Toomre, A., 1966, Ap. J., 146, 810.
Lin, C.C., & Shu, F.H., 1964, Ap. J., 140, 646.
Lin, C.C., Mestel, L.,& Shu, F.H., 1965, Ap. J., 142, 1431.
Lindblad, B., 1927, Medd. Astron. Obs. Uppsala, 19, 1.
Manchester, R.N., & Taylor, J.H., 1977, Pulsars (San Francisco: Freeman), p. 135.
Mouschovias, T.Ch. 1976, Ap. J., 207, 141.
Parker, E.N., 1979, Cosmical Magnetic Fields (Oxford: Clarendon) p. 36.
Pedloskey, J., 1987, Geophysical Fluid Dynamics, (New York: Springer-Verlag).
Schüssler, M., & Schmitt, D., 1981, Astr. Ap., 97, 373.
Shu, F.H., Adams, F., & Lizano, S., 1987, Ann. Rev. Astr. Ap., 25, 23.
Spitzer, L., and Schwarzschild, M., 1953, Ap. J., 118, 106.
Toomre, A., 1964, Ap. J., 139, 1217.
Wisdom, J., & Tremaine, S., 1988, A. J., 95, 925.
– 63 –
FIGURE CAPTIONS
Figure 1. (a) the positions of particles in a shock tube test of the TREESPH code (b) Fluid variables
along the long axis of the shock tube. The straight solid line is the analytic solution, the
boxes are averages over 0.1 in z. The lighter dots are the fluid variables evaluated at the
particle positions.
Figure 2. Illustration of the shearing periodic boundary conditions. The central box shows particle
positions in a test run. The outer boxes, with fainter dots for particle positions, show images
of particle positions in the central box. Arrows indicate the direction of the shear. With our
sign convention, the galactic center is at the far left.
Figure 3. (a) Particle densities perpendicular to the disk in the initial conditions of a run containing
an isothermal, self-gravitating sheet. The density profile is very nearly ∝ sech 2 (z/2z0 ), as
can be seen in (b), which shows the ratio of the density to ρ0 sech2 (z/2z0 ).
Figure 4. Positions of particles at time t = 4/κ in convergence test runs.
Figure 5. Evolution of an unstable isothermal gaseous sheet with Q g = 0.6. The frames are spaced
at intervals of 0.5/κ, beginning with t = 1/κ in the upper left hand corner. The initial
conditions are described in Table 1.
Figure 6. Evolution of an unstable sheet containing both collisionless particles and an isothermal
fluid, with fractional gas surface density 0.3, gaseous Qg = 1.0, and stellar Qs = 2.0. The
initial conditions are described in Table 1.
Figure 7. The distribution of particle densities at intervals of 1/κ.
Figure 8. Particles with n > 50 cm−3 . The circles and numbers correspond to the clouds listed in
Table 2.
Figure 9. Evolution of the mass of “Cloud 5” as a function of density threshold.
Figure 10. Magnetic field lines in the initial conditions for the purely gaseous sheet.
Figure 11. Magnetic field configuration at intervals of 0.5/κ beginning with t = 2.75/κ in the upper
left hand corner.
Figure 12. Observation of the purely gaseous collapse after placing it at l = 34 deg, d = 7 kpc (these
coordinates were chosen arbitrarily). Particles shown have density n > 100 cm −3 , while it is
assumed that Σg = 20 M¯ pc−2 and that the rotation curve is flat.
Figure A1. Root-mean-square pressure acceleration as a function of the smoothing length parameter
Nsm in the shearing cube test simulations.
Figure A2.
Root-mean-square deviation of the density from the expected value (ρ = 1) in the
shearing cube simulation.
Figure A3. Variation in the thermal energy as a function of the smoothing length parameter N sm .
The slope of this curve at t > 1 is used to measure an effective viscosity for the SPH fluid.
Figure A4. Effective viscosity of the SPH fluid as a function of the smoothing length parameter
Nsm . The viscosity is measured in units of the mean interparticle separation times the sound
1/3
speed, which is proportional to Nsm .
Figure A5.
The dimensionless autocorrelation function ξ for particles in the shearing box test
simulations.
TABLE 1
Simulation Parameters
Qg
Qs
fg
σz /σr
hz 2 i1/2
λcr
L
n0
T
c
Nsph
Nst
Nsm
msph
mst
hhsm i
²
∆T
rcut
a
gas fraction
gas scale height
char. wavelength
midplane density
temperature
sound speed
Num. SPH part.
Num. star part.
SPH particle massa
star particle massa
average smoothing length
grav. softening length
time evolved
grav. cutoff radius
(1) Gas
(2) Star & Gas
0.61
—
1.0
—
1.0
15.6
47
5.5 cm−3
1000 K
2.9 km s−1
3 × 104
0
64
1.7 × 103 M¯
—
0.84
0.7
5/κ
15.6
1.0
2.0
0.3
0.6
0.4
6.2
31
3.1 cm−3
3400 K
5.3 km s−1
3 × 104
2 × 104
64
1.1 × 104 M¯
3.8 × 104 M¯
0.48
0.4
5/κ
10.3
Assuming κ = 37 km s−1 kpc−1 , Σg = 13 M¯ pc−2 , Σs = 35 M¯ pc−2
TABLE 2
Simulation 1 Cloudsa
M [ M¯ ]
1
2
3
4
5
6
7
8
9
10
11
12
a
1.4 × 106
1.2 × 105
9.6 × 104
1.1 × 105
1.4 × 106
1.1 × 105
9.5 × 105
1.5 × 105
1.5 × 105
4.1 × 104
3.1 × 105
2.3 × 105
Assuming κ = 37 km s−1 kpc−1 , Σg = 13 M¯ pc−2
TABLE 3
Simulation 3 Cloudsa
M [ M¯ ]
1
2
3
a
8.4 × 106
2.5 × 105
4.6 × 105
Assuming κ = 37 km s−1 kpc−1 , Σg = 13 M¯ pc−2 , Σs = 35 M¯ pc−2
Chapter 4: Rotation of Molecular Clouds
This chapter considers the rotation of giant molecular clouds. We calculate the expected rotation
frequency for three models of cloud formation: accretion of collisionless clouds onto a seed mass;
accretion of an isothermal fluid onto a seed mass; and gravitational instability in an initially uniform
disk. We then estimate the magnitude of the torque that might be exerted by magnetic fields, cosmic
rays, and other stresses.
Collisionless accretion produces slowly rotating objects because the cloud can draw its mass from
a reservoir of material that has low spin angular momentum with respect to the cloud. The growth
time is ∼ 1/Q2 κ when the cloud has a tidal radius much smaller than the disk scale height and 1/κ
when the cloud’s tidal radius is much larger than the disk scale height.
Fluid accretion produces rapidly rotating objects. The growth rate is similar to that for
collisionless accretion.
Gravitational instability produces objects that are more slowly rotating than in our fluid accretion
model, but with rotation frequencies that are somewhat high compared to observations. Gravitational
torques are sufficiently strong to halt rotation in a few epicyclic times for some of the clouds that form
in our simulations.
Magnetic fields dominate the dynamics for reasonable values of the field strength. They torque
the clouds in a retrograde sense and can de-spin a cloud in a fraction of an epicyclic time. Rotation
is unlikely to provide a strong constraint on cloud formation theories until the magnetic field can be
self-consistently included.
1. Introduction
By now many physical mechanisms have been alleged to be involved in the process of giant
molecular cloud formation (see Chapter 1 or Elmegreen, 1991, for a review), and some test for
distinguishing the most important is required. The rotation of molecular clouds may be such a
test (Blitz, 1991). Insofar as giant molecular clouds form a link in the chain of events leading to
star formation, it is also interesting to examine how the angular momentum problem (posed by, e.g.,
Spitzer, 1978; Mestel, 1965) is resolved at the scale of giant molecular clouds. This chapter considers
the rotation frequency expected in three models of cloud formation and then estimates how effective
torques might be in braking the rotation of a giant molecular cloud.
The kinematic signature of rotation (and uniform expansion or contraction) is a gradient of mean
velocity across the cloud. We define the velocity gradient ∆l and ∆b
∆l ≡ −
1 dvr
d dl
∆b ≡
1 dvr
d db
(1)
where d is the distance to the cloud, l, b the galactic coordinates and vr the radial velocity. By
1/2
symmetry, the average gradient perpendicular to the plane h∆b i = 0. h∆2b i
is a measure of the
contribution of processes not related to galactic rotation to the velocity structure of the cloud.
An observed cloud velocity gradient is contaminated by galactic rotation. Consider a cloud in
solid body rotation with angular velocity perpendicular to the galactic plane. The cloud is centered at
galactic longitude l1 and galactocentric radius R1 , and has circular velocity V1 . The circular velocity
and galactocentric radius of the LSR are denoted V0 , R0 . The separation vector of the center of the
cloud from the sun is r, and the location of a point within the cloud is r + δr. The average velocity
– 65 –
of the cloud with respect to the LSR is v, and the velocity of a point within the cloud is v + δv. The
radial velocity of some point within the cloud is then
vr =
(r + δr)·(v + δv)
|r + δr|
(2)
Expanding to first order in δr and δv, we have
vr = −
(δr·r)(v·r) (δr·v) (δv·r)
+
+
d3
d
d
(3)
where d ≡ |r|. Assuming the cloud is on a circular orbit and rotating with angular frequency ω relative
to the rotating frame of its local standard of rest, we find
∆l = −
V1
q
1 − (R0 /R1 )2 sin2 l1 − V0 cos l1
d
+ ω + Ω1
(4)
where, for our galaxy, Ω1 < 0. When the galaxy is in solid body rotation ∆l vanishes, while for a
galaxy with a flat rotation curve, the velocity gradient in a cloud located at a tangent point is
∆l = ω + Ω1 − Ω0 .
(5)
In a rapidly rotating cloud, the cloud velocity gradient is negative if it is rotating in a prograde sense,
and positive if it is rotating in a retrograde sense.
Observations of cloud velocity gradients have recently been reviewed by Blitz (1991), and the list
of four velocity gradients in Table 1 is taken from that paper. There have been no systematic studies,
and the values quoted in Table 1 are simply the largest of those clouds that have been observed. For
comparison, the rotation frequency in the solar neighborhood is ≈ −0.026 km s −1 kpc−1 , that is, the
Table 1 clouds exhibit rapid retrograde rotation, if they are rotating at all.
We evaluate the timescales relevant to giant molecular cloud rotation by reference to the Orion
A molecular cloud (Bally et al., 1987; Maddalena et al., 1986; Goudis, 1982; Kutner et al., 1977).
At a distance of 450 pc, its mass is ≈ 105 M¯ . The rms clump velocity dispersion is 2 km s−1 , and a
velocity gradient of ω ≈ +100 km s−1 kpc−1 is observed parallel to the galactic plane. A filament in
the northern end of the cloud that is oriented parallel to the plane of length ≈ 9 pc and width 0.5 pc
and has a velocity gradient of ≈ 400 km s−1 kpc−1 . The oldest subgroup in the Orion I OB association
is ≈ 107 yr old (Blaauw, 1964). This should be compared with the rotation period for the whole
cloud (if it is rotating) of ≈ 6 × 107 yr, the rotation period for the filament of ≈ 1.6 × 107 yr, and the
crossing time for a clump in the cloud of ≈ 5 × 106 yr. Since the first OB stars formed in Orion, then,
the filament has had a chance to rotate about one turn (provided that it turns at constant angular
velocity) and the cloud has rotated through ≈ 70 deg. Similar estimates obtain for the other clouds
in Table 1. Hence cloud rotation is a transient phenomenon: clouds have a chance to turn about once
in their lifetimes.
2. Governing Equations
Here we derive an expression for the spin angular momentum of a cloud and an expression for
torques acting on the cloud including tidal torque, gravitational torque, and magnetic torque. Finally,
we derive the potential vorticity constraint.
– 66 –
The total angular momentum J about the center of the galaxy of the gas contained within a
volume V is
Z
J=
r × ρv.
(6)
V
To separate the angular momentum into spin and orbital parts we define
r = R0 + r 0
v = V0 + v 0
(7)
so that R0 is the location and velocity of some point in the cloud, and V0 ≡ Ṙ0 is nearly the cloud’s
circular velocity. Then
J=
Z
ρ(R0 × V0 ) +
V
Z
0
ρ(R0 × v ) +
V
Z
0
ρ(r × V0 ) +
V
Z
ρ(r 0 × v 0 )
(8)
V
which is simplified by defining
M≡
Z
ρ
X≡
V
Z
ρr
0
P ≡
V
Z
ρv
0
S≡
V
Z
ρr 0 × v 0 ,
(9)
V
to
J = S + M (R0 × V0 ) + X × V0 + R0 × P .
(10)
Here S is the spin angular momentum, M is the total mass, X/M is the displacement of the center
of mass of the cloud from R0 , and P /M is the velocity of the center-of-momentum frame of the cloud
with respect to V0 .
As an example, consider the spin angular momentum of a patch of disk in the local model of
Chapters 2 and 3 (the patch has dimension l ¿ R0 , with x ≡ (R − R0 ) and y ≡ R0 (φ − φ0 )). In an
unperturbed state the velocity field is Vrot = 2Axŷ in the rotating frame and V = 2Axŷ + Ω × x in
an inertial frame. The spin angular momentum of the patch is then
S = ẑ
Z
ρ(2Ax2 + Ωx2 + Ωy 2 )
(11)
V
R
where A ≡ (r/2)dΩ/dr is Oort’s constant. For a flat rotation curve Ω = −2A, so L z = ρΩy 2 . Thus a
counter-rotating cloud cannot be assembled in a galaxy with a flat rotation curve if the cloud conserves
its spin angular momentum (as was pointed out by Mestel, 1966). If the rotation curve is declining
then 2A/Ω < 1 and a needle aligned along the radius vector can be assembled into a counter-rotating
cloud, again assuming conservation of spin angular momentum. Spin angular momentum is not in
general conserved, of course, since the galactic tide torques the cloud.
The time derivative of the spin angular momentum is
dS
=
dt
Z
∂ρv 0
+
r ×
∂t
0
V
Z
S
dA·vs0 (r 0 × ρv 0 ),
(12)
where the prime implies that the quantity is evaluated in the noninertial but nonrotating frame r 0 , S
is the surface of the volume V , and vs is the velocity of a surface element. Then
∂ρv 0
= −∇·σ 0 − ρV̇0 ,
∂t
(13)
– 67 –
where the last term enters because the r 0 frame is accelerated. The stress tensor σ 0 is
µ
¶
1
1
0 0
0
2
σij ' ρvi vj + pδij +
gi gj − δij g
4πG
2
µ
¶
Z
1
1
1
−Bi Bj + δij B 2 +
dνdΩI(ν, Ω)Ωi Ωj .
4π
2
c
(14)
The first and second terms are the Reynolds stress and thermal pressure. The third term is
the gravitational stress, and g is the gravitational acceleration (e.g. Lynden-Bell & Kalnajs,
1972; one can verify that this term reduces to a more familiar expression for the gravitational
acceleration by taking its divergence and using Poisson’s equation). The fourth and fifth terms are
the electromagnetic stress, which has been divided into a part with frequency ω ¿ ω p ≡ plasma
frequency= 9 × 103 (ne cm−3 )1/2 Hz (the third term) and a part with frequency ω À ωp (the fourth
term). I(ν, Ω) is the usual specific intensity of radiative transfer theory. Cosmic rays are coupled
to the gas via the magnetic fields, and so are indirectly included here. The approximate equality
signifies that viscous stress and the electric field have been neglected, which is appropriate since both
the Reynolds number and magnetic Reynolds number are large in the interstellar medium. Ambipolar
diffusion is also negligible for the scales and densities in a proto-giant molecular cloud. The energy
density in random motions of the gas, thermal energy, magnetic energy, starlight, and cosmic rays are
all comparable, so none of the terms in eqtn.[14] are negligible.
Substituting eqtn.[13] into eqtn.[12] and rewriting the volume integral as an integral over the
surface of the cloud, we obtain an expression for the spin torque:
dS
=−
dt
Z
dA·(σ 0 × r 0 ) +
S
Z
S
dA·vs0 (r 0 × ρv 0 ) − X × V̇0 .
(15)
This expression will prove useful for estimating the magnetic and gravitational torques but is not
useful for directly evaluating torques from an acceleration field (the sort of data that one obtains from
a simulation).
Returning to eqtn.[12], we rewrite the gravitational torque as
dS
=
dt grav
Z
V
´
³
ρr 0 × −∇φ − V̇0
(16)
where φ is the gravitational potential, which we separate into an axisymmetric part φ 0 and a
nonaxisymmetric part φ1 . Next, −∇φ0 can be expanded in a Taylor series in local cartesian
coordinates about x = y = 0 (x ≡ R − R0 , y ≡ R0 (φ − φ0 ), so that −∇φ0 ≈ g0 + xx̂(∂gr /∂r)|0 . But
∂gr /∂r|0 = −Ω20 (2β − 1), where β ≡ d ln |Vc |/d ln r, and Vc is the circular velocity. Hence
dS
≈
dt grav
Z
V
´
³
ρr 0 × −∇φ1 + g0 − xΩ20 (2β − 1)x̂ − V̇0 .
(17)
The third term in the integrand is the tidal torque. If R0 is moving on a circular orbit then the second
and fourth terms cancel.
The bulk transport of spin angular momentum across the cloud boundary is the sum of the
Reynolds stress and the boundary term in eqtn.[12]:
dS
=
dt bulk
Z
S
dS·(vs0 − v 0 )(r 0 × ρv 0 ),
(18)
– 68 –
which vanishes if the boundary moves with the fluid. Other torques are readily calculated from
eqtn.[12], [13], and [14].
3. Cloud Formation Models
In this section we calculate the cloud rotation frequency expected in three idealized models of
cloud formation. The first two models view cloud formation as an accretion process: in the first model,
accretion of an isothermal fluid; in the second, of a collisionless fluid. This roughly brackets the range
of possibilities for the accreting material, from a cloudy medium with very short mean free path to
a cloudy medium composed of dense lumps with a very long mean free path. The third model views
clouds as the result of gravitational instability in an initially smooth and isothermal differentially
rotating sheet.
The context for numerical tests of accretion and gravitational instability will be the local model
of the galactic disk that has already been described in earlier chapters. Since we won’t consider a
stellar component here, the self-gravitating, isothermal sheet is characterized by two dimensionless
parameters: the stability parameter Qg ≡ cκ/πGΣg , where c is a sound speed in the gas, κ is the
epicyclic frequency, and Σg is the gaseous surface density; and the logarithmic derivative of the
rotation curve, which we shall always assume to be 0, i.e. that the rotation curve is flat. The model
is cast in dimensionless form by setting G = Σ = κ = 1. In the solar neighborhood, we estimate that
(see Chapter 1) Qg ≈ 1.2, κ ≈ 37 km s−1 kpc−1 , and Σg ≈ 13 M¯ pc−2 , implying that the length unit
≈ 42 pc, the mass unit ≈ 2.2×104 M¯ , the time unit ≈ 2.7×107 yr, and the velocity unit ≈ 1.5 km s−1 .
3.1 Accretion
In Chapter 3 we assumed that the interstellar medium could be represented by a smooth
isothermal fluid. Obviously interstellar gas is not smoothly distributed, so we must consider how
inhomogeneity affects cloud growth. Since giant molecular clouds are well separated, dense objects
with a small filling factor, one is immediately tempted to regard their growth as an accretion process.
4
In this section we suppose that molecular clouds of mass Mc <
∼ 10 M¯ (which are numerous– recall
that the mass spectrum is consistent with N (Mc ) ∝ M −3/2 ) nucleate the formation of giant clouds
and grow by accretion of the surrounding (again, smooth and isothermal) gas.
3.1.1 Accretion Rate
Because the accretion model is not discussed elsewhere in this thesis, we first consider the mass
accretion rate. One might hope to estimate the accretion rate of the cloud by reference to classical
Bondi-Hoyle accretion, where it is roughly the product of a density (the density at infinite distance
from the accretor), a velocity (the sound speed), and the surface area of a sphere with a characteristic
radius (the sonic radius, ∼ GM/c2 ). When the accreting object is located in a rotating, self-gravitating
disk, however, the problem is complicated by the presence of not one but four characteristic scales: the
sonic radius rs ≡ GM/c2 , the physical size of the cloud rc , the tidal radius rt ≡ (GM/|4AΩ|)1/3 , and
the scale height of the disk z0 ≡ πQ2g GΣ/2κ2 (for a self-gravitating, isothermal disk). We estimate the
ratios of these length scales by reference to the Orion A molecular cloud. The molecular mass of Orion
A is ≈ 105 M¯ (Maddalena et al., 1986), with probably an equal mass of atomic hydrogen (Chromey
et al., 1989), implying rt ≈ 70 pc. The Orion A cloud is elongated parallel to the galactic plane, with
an apparent size of ≈ 80 × 30 pc in 12 CO (Maddalena et al., 1986). The average radius is therefore
– 69 –
≈ 25 pc; a generous estimate of the physical radius, inside which a small cloud will inevitably coalesce
with the giant cloud, is therefore ≈ 35 pc. The dynamics of an interstellar cloud in the neighborhood
of a GMC is uncertain, and the coalescence radius might be increased by magnetic fields (Clifford &
Elmegreen, 1983) or reduced because most collisions are more likely to produce fragmentation than
coalescence (e.g. Hausman, 1981). The sonic radius ra = GMc /2c2 for isothermal Bondi accretion
in uniform homogeneous medium with sound speed c. Taking the internal velocity dispersion of the
cloud, 2 km s−1 , as a representative velocity we have ra ≈ 50 pc. The (theoretical) scale height of the
disk z0 = πQ2 GΣg /2κ2 is ≈ 90 pc (putting Orion A almost two scale heights out of the disk!). The
observed HI scale height is somewhat larger (e.g. Spitzer, 1978), while the observed scale height of
molecular material is somewhat smaller (Bronfman et al., 1986). To summarize the results from this
round of estimates, the tidal radius rt ≈ 70 pc, the physical or coalescence radius rc = 35 pc, the sonic
radius ra ≈ 50 pc, and the scale height z0 ≈ 90 pc. These scales are nearly identical, so there are
no small dimensionless numbers that might give us an analytical handle on the problem. We must
therefore obtain the growth rate by numerical simulation.
In setting up a numerical simulation of accretion onto a molecular cloud, we must decide how to
represent cloud’s outer boundary and how the effective accretion radius scales with cloud mass. The
growth rate will be insensitive to the cloud boundary condition as long as the boundary is located
inside the sonic point (so it cannot communicate with the rest of the flow, except gravitationally). We
therefore suppose that the boundary is an absorbing, spherical surface of radius r c . This is a liberal
assumption in the sense that it gives an upper limit to the cloud growth rate. Since it will turn out
that the cloud growth time is rather long, we shall consistently make choices that maximize the growth
rate to obtain a lower limit on the growth time. Next, there are two obvious candidates for scalings of
the accretion radius rc with mass: the first assumes that the cloud grows at constant column density,
1/2
1/2
so rc ∼ Mc ; the second assumes that the cloud grows at constant density, so that r c ∼ Mc . The
majority of observations seem to favor the first choice (e.g. Blitz, 1991), although Stark & Blitz (1978)
argued that clouds are tidally limited, implying that they have a constant density. If we calibrate the
scalings with the Orion A molecular cloud and ask how rapidly the cloud grew at earlier times, then
a constant volume density scaling will produce more rapid growth (Mc ∼ t3 , if the accretion rate is
proportional to the cloud surface area) than a constant column density scaling (M c ∼ et/T , where T
is a characteristic time, and again assuming the accretion rate is proportional to the surface area). To
maximize the growth rate, therefore, we choose the constant density scaling. Furthermore, calibrating
the scaling on Orion A, we set rc ' 0.5rt .
By setting the accretion radius proportional to the tidal radius, and by adopting a boundary
condition that forces supersonic flow at the surface of the cloud, we have eliminated two scales from
the problem (both have been tied to the tidal radius). The only length scales left are r t and z0 .
Defining H ≡ z0 /rt , we can now estimate the accretion rate in the limits H À 1 and H ¿ 1.
Neither limit is physical, but, as we shall see, they give nearly the same growth time. When H À 1,
dimensional analysis tells us that the accretion rate must be roughly the produce of the surface area of
the cloud, rt2 , a characteristic velocity, c = πQg GΣg /κ, and a density, taken to be the midplane density
of an isothermal, self-gravitating sheet: ρ0 = κ2 /2πGQ2g . Then dM/dt ∼ Σκ2 rt2 /2Qg , and the mass
grows as t3 . In the limit that H ¿ 1, the flow of material onto the cloud is nearly two-dimensional.
Again, dimensional analysis gives the scaling: the accretion rate is roughly the product of the twodimensional cross-section 2rt , a typical velocity, the shear velocity at the tidal radius = 2Art , and the
surface density Σg . Then dM/dt ∼ 4Art2 Σg , and again the mass grows as t3 .
– 70 –
For molecular clouds in the solar neighborhood, H ∼ 1, so neither limit applies, although one
suspects that a smooth transition between the two limiting regime might give M c ∼ t3 . When Q ≈ 1,
however, the self-gravity of the accreting material is not negligible, and could enhance the accretion
rate. We have simulated the accretion flow with the self-gravitating hydrodynamics code TREESPH
developed by Hernquist & Katz (1988). The algorithm and various tests of the algorithm are described
in Chapter 3, and we shall not redescribe them here. The “shearing bricks” boundary conditions
described in Chapter 3 were used for the outer boundary of the simulation box. An imposed potential
corresponding to the accreting cloud was centered on the origin in the simulation. Particles were
removed from the simulation whenever they stepped across the accretion sphere at r = r c , and the
mass of the imposed potential was incremented. We shall describe a grid of four runs for a standard
set of model parameters (to be described shortly). In the first run, the accreting fluid is isothermal
and self-gravitating. In the second run, the accreting fluid is isothermal but not self-gravitating– the
fluid moves in a potential corresponding to an isothermal disk in vertical equilibrium, together with
the cloud potential. When a particle crosses the accretion sphere the mass of the cloud potential
is incremented. In the third run the accreting material is a self-gravitating collisionless fluid with
an initially Schwarzschild distribution function (in which σz = σr ). In the fourth run the accreting
material is again collisionless but not self-gravitating. In all these runs we have set Q g = 0.68, which
is the marginally stable value in a self- gravitating isothermal sheet. This corresponds to a density
and pressure at the midplane that are unreasonably high, but since the growth rate decreases with
increasing Qg we obtain an upper limit on the growth rate. The box size is L = 20GΣg /κ2 ≈ 830 pc.
The initial mass of the accreting cloud is set to Mc = 4G2 Σ3g /κ4 ≈ 9 × 104 M¯ . The number of
particles used is always 104 .
To give the reader a better sense for the geometry of the simulation, Figure 1a shows particle
positions in the initial conditions projected onto the R − z plane. The solid circle is the tidal radius
of the cloud. There is material inside the accretion sphere in the initial conditions that is removed
at the first timestep. Figure 1b shows the particle positions projected onto the x − y plane late in
the fluid, self- gravitating simulation. The solid lines are the zero-velocity curve for the Jacobi-like
integral ΓJ ≡ v 2 /2 + φ + 2AΩx2 , which is conserved for test particle orbits in the shearing sheet (φ
is the gravitational potential). Most of the material falls in along a band that lies perpendicular to
the density maximum that runs from the third quadrant to the first (i.e. the accreted material came
from the second and fourth quadrants).
Figure 2 shows the mass of the accretor as a function of time for the Grid of Four simulations.
The vertical and horizontal dashed lines show the time and mass referred to the solar neighborhood,
in units of 5 × 107 yr and 2 × 105 M¯ , respectively. First consider the fluid, self-gravitating simulation.
Initially the mass jumps sharply because the accretor swallows all the particles within its accretion
radius. Then the mass grows as t3 until M 1/3 >
∼ 4, when it turns over slightly because it has exhausted
all the mass in its accretion band. One expects this to happen when the mass at radii between x = ±r t
equals the mass of the accretor, i.e. M 1/3 ' L1/2 = 4.47. Turning off the self-gravity of the accreting
particles enhances the accretion rate because the local deficit of fluid caused by accretion is not
reflected in the potential. In the self-gravitating simulation accretion is inhibited by a local maximum
in the potential (local maximum taken in the plane of the disk; it is, of course, really a saddle point)
along the accretion corridor in the second and fourth quadrants of Figure 1b.
The collisionless simulations are similar to the fluid simulations but the growth rate is reduced
(this is mostly because Q = σr κ/3.358GΣ instead of cκ/πGΣ for a collisionless disk, so that at fixed
– 71 –
Q the vertical velocity dispersion is higher, the disk is thicker, and the central density is lower). In all
four simulations the slope of the M 1/3 vs. t curve is within 30% of its value in the fluid, self-gravitating
case:
µ 2 3¶
dMc
G Σ
2/3
' 2.1Mc
= 4.9 × 103 (Mc /105 M¯ )2/3 M¯ Myr−1
(19)
dt
κ3
where the latter equality holds for the standard solar neighborhood model, but with Q g = 0.68. The
straight solid line on top of the fluid, self-gravitating curve in Figure 2 shows this (rather good)
fit. Equation [19] implies Mc = 8, 200((t − t0 )κ)3 M¯ , so that the cloud requires 8 × 107 yr to reach
2 × 105 M¯ , and 1.3 × 108 yr to reach 106 M¯ .
Experiments with self-gravitating fluid sheet with different values of Q g and much larger and
much smaller values of the initial cloud mass also give Mc ∼ t3 , if we adopt the scaling rc ∼ rt . Hence
the mass always grows as t3 , and the growth time for a 2 × 105 M¯ cloud is at least 8 × 107 yr, which
7
is long compared to the cloud destruction timescale of <
∼ 2 × 10 yr. If clouds do indeed have constant
column density, the growth time will be longer yet.
3.1.2 Spin Angular Momentum Accretion
Now we consider the flux of spin angular momentum across the cloud boundary in the case that
the accreting material is an initially uniform isothermal fluid. A simple argument gives an upper limit
to the spin rate of the cloud. For a contour C comoving with the fluid, Kelvin’s Theorem states that
I
D
DΓ
dV ·dl = 0,
(20)
=
Dt
Dt C
where Γ is the circulation, D/Dt is the total, or convective, derivative, and V is the velocity in an
inertial frame. This theorem holds provided that the acceleration of a fluid element can be written
as the gradient of a scalar field. In the interstellar medium circulation will be conserved if some sort
of effective cloud viscosity (which we shall make no attempt to define more precisely) is small, and in
our simulations it will be conserved if the artificial viscosity is negligible, which is generally the case
if there are no shocks. Suppose for the moment that circulation is conserved, and consider a circle
drawn in the plane of the disk very close to the equator of the accretion surface. The azimuth-averaged
rotation frequency around this contour is
I
1
hΩi ≡
V ·dl.
(21)
2πrc2 c
When the mass of the cloud was 0 and the disk was unperturbed, this contour lay at the midplane
(by symmetry) some distance from the nascent accretor. If the material on this contour is unshocked,
then it is steadily deformed by the accretion of material inside it and does not cross itself; the contour
is like a lasso being drawn around the accretor. In the unperturbed state the velocity in an inertial
frame is V = 2Axŷ + Ω × r, so that the circulation around the loop is just Γ = 2(Ω + A)AC = 2BAc ,
where Ac is the area enclosed by the loop. Since the material inside the loop moves in columns, so
that ∂vx /∂z, ∂vy /∂z ≈ 0, (because the timescale in the accretion flow is long compared 1/ the vertical
frequency), the area enclosed by the contour is directly related to the amount of mass M a that has
been accreted by the time the contour reaches the equator of the perturber: Ma ≈ Σg Ac . Hence
hΩi =
BMa
Σc
=B ,
2
πrc Σg
Σg
(22)
– 72 –
where Σc is the average surface density of the cloud. This is similar to the more familiar whirlpool or
tornado, wherein the gathering of vortex lines also produces rotation. Assuming that this azimuthaveraged rotation frequency is nearly the mass-averaged rotation frequency of the material being
accreted by the cloud, eqtn.[22] predicts that Ṡz ' hΩirc2 Ṁc . Since Ṁc ∼ rc2 ∼ M 2/3 , we have
Sz = BMc2 /2π, whence
Sz
BM
ωef f ≡
'
.
(23)
Mc rc2
2πrc2
In the solar neighborhood
ωef f ≈ −7.5 km s−1 kpc−1 Σc /13 M¯ pc−2 ≈ −55 km s−1 kpc−1
(24)
for a cloud with column density N ≈ 1022 cm−2 . As we have already mentioned, this estimate holds
provided there are no shocks in the accretion flow. It turns out that in our simulations there are
always shocks, even for a modest initial accretor mass, so the effective spin frequency always falls
below (but only slightly below) the value predicted by eqtn.[24].
Now suppose that the accreting material is collisionless; what spin angular momentum accretion
rate is expected? In the limit that the tidal radius is much larger than the scale height, the accretion
rate must be proportional to the surface density, but self-gravity will be negligible and the surface
density irrelevant to the dynamics. If the accretion radius scales like the tidal radius, then dimensional
analysis shows that Ṡz ∼ rt4 κ2 Σg , since the only dimensional parameters left are G, Mc , and κ. Then
hΩi ∼ Sz /M ra2 ∼ κM 0 , provided that ra ∼ M 1/3 , and the cloud will spin with nearly constant angular
frequency. We can also estimate the spin angular momentum accretion rate in the limit that the tidal
radius of the cloud is much smaller than the scale height of the disk. In this case the self-gravity of
1/3
1/3
the cloud is negligible, since GMc /ra2 ∼ Mc →0 as Mc →0 (assuming that ra ∼ rt ∼ Mc ). Then
the collisionless particles strike the accretion surface with velocities that are unperturbed by the cloud
potential. The velocity distribution function at the surface of the cloud is the same as it would be
in the absence of the cloud, except that all particles with v·r > 0 have been removed by collisions
(no particles are rising out of the accretion surface). Thus the mean particle velocity of the particles
parallel to the surface (and hence the mean spin angular momentum) is the same as it would be
in the absence of the cloud. The mean velocity in an inertial frame of the collisionless particles is
2Axŷ+Ω×r, so that the mean rotation frequency of particles at the surface is ∼ B. Therefore the spin
angular momentum accretion rate will be Ṡz Brt2 × ρrt2 σ ∼ Brt4 (κ2 /2πGQ2g )(πGΣg Qg /κ) ∼ κ2 rt4 Σg ,
where σ is the particle velocity dispersion. Thus in both limits the spin angular momentum accretion
rate ∼ rt4 κ2 Σg ∼ M 4/3 , and the effective rotation frequency hΩi ∼ Sz /Mc ra2 ∼ κM 0 , so we suspect
that when z0 ∼ rt , the effective rotation frequency will also be approximately constant.
We can measure the spin angular momentum accretion rate in the simulations of cloud growth
discussed in the last subsection by recording the angular momentum of each particle as it accretes.
Figure 3 shows the effective rotation frequency ωef f ≡ Sz /Mc ra2 as a function of time for a simulation
in which the accreting material is self-gravitating and collisionless, and for a simulation in which the
accreting material is self-gravitating and an isothermal fluid. Assuming that the cloud is initially
nonrotating, ωef f is 2/5 of the actual rotation frequency if the cloud has uniform density and is in
solid body rotation, and an even smaller fraction if the cloud is centrally condensed. The rotation
frequency in the fluid case grows slowly, while the rotation frequency in the collisionless case is either
constant or very slowly increasing. The dashed line is ω = (BMc2 /2π − BM02 /2π)Σg /Mc ra2 , which
is the effective value of the rotation frequency predicted by eqtn.[24] and assumes that there are no
– 73 –
shocks in the accretion flow. The difference between the two lines can be entirely accounted for by
the viscous torques that are applied in shocks.
To summarize, fluid accretion produces relatively rapid rotation, while collisionless produces slow
rotation because the cloud can always draw its mass from particles with low spin angular momentum.
3.2 Gravitational Instability
In our simple model for formation of giant molecular clouds by gravitational instability, we imagine
that an initially uniform, isothermal disk cools until it is nearly unstable. Even small disturbances
in the velocity field or surface density in the disk are then swing amplified into large disturbances,
creating dense regions that rapidly collapse. This model is discussed in detail in Chapters 2 and 3;
here we shall focus on the rotation of the collapsed clouds.
First, how much rotation is expected? If the clouds form from an initially uniform disk, then
conservation of circulation again gives an upper limit on the rotation frequency of the collapsed cloud.
Following the argument given in the last subsection, we find that
BMc
ω<
∼ 2A Σ
c g
(25)
where Ac is the projected area of the cloud. For the clouds that formed in a purely gaseous simulation
(Run 1 of Chapter 3), the surface area of the cloud is typically 104 pc2 , so the corresponding rotation
frequency is −57(Mc /5 × 105 M¯ )(104 pc2 /Ac )(13 M¯ pc−2 /Σg ).
Now let us consider in detail the outcome of Run 1 of Chapter 3, which evolved an unstable
isothermal gaseous sheet. At the end of the runs, the simulation contained a number of dense lumps;
the positions of all the lumps (defined as connected regions with n > 50 cm −3 ) is shown in Figure 4.
P
P
We define an effective rotation frequency Ωef f ≡ i Sz,i / i mi ri2 for each of the clouds, where the
sum is taken over all particles i within the cloud, and ri is the distance of particle i from the cloud
center of mass. Table 2 lists the effective rotation frequencies for clouds formed in Run 1. All but
one of these clouds are rotating in a prograde sense, with hΩef f i = −43 km s−1 kpc−1 . The effective
rotation frequencies are generally somewhat smaller than the conservation of circulation argument
predicts, and this is in part due to viscous torques.
What torques are acting on the clouds? Figure 5 shows the specific torques averaged over narrow
cylinders centered on cloud 5 at the end of Run 1. The tidal torque is the most important, and is
acting to spin the cloud in a prograde sense. The next most important torque is the gravitational
torque; the viscous torque is almost negligible.
The total spin angular momentum of the cloud is not well conserved. Figure 6 shows the spin
angular momentum of the particles that wind up in the cloud as a function of time from the beginning
of the simulation. The spin angular momentum changes by more than an order of magnitude because
of tidal torques. That this should be so is more apparent when one considers the evolution of particle
positions as they collapse to become part of the cloud. Figure 7 shows the positions of particles that
are destined to become part of dense clumps at times t = 0/κ, 0.5/κ, . . . , 4.0/κ. The particles start
out in a narrow band with a leading orientation. The spin torque from tides can be obtained from
eqtn.[17]:
Z
dSz
=
ρr 0 × xΩ20 (1 − 2β)x̂
(26)
dt tidal
V
Since the bands start with a leading orientation, the tidal torque is initially positive and tends to spin
down the clouds, which have negative spin angular momentum.
– 74 –
We can now observe the clouds as they might be seen in CO emission by centering the simulation
at some position in the galactic plane and projecting particle positions into the l − v plane. Figure
8 shows the projection of particles in the gaseous collapse simulation after placing the center of the
simulation at a distance of 7 kpc at l = 34, and assuming that the surface density is 20 M ¯ pc−2 . Only
particles with n > 100 cm−3 are shown. The cloud with the largest velocity dispersion, near l = 33,
is rotating so rapidly that a velocity gradient is not readily distinguishable. The next most rapidly
rotating cloud, near l = 36, would be observed to have a velocity gradient of ≈ 0.8 km s −1 pc−1 . This
velocity gradient is uncomfortably high in comparison to the clouds listed in Table 1, and is prograde
rather than retrograde. Unfortunately, we cannot honestly use this result to discriminate against the
gravitational instability model. This is because there are several torques that are effective on the
scale of the molecular cloud that are capable of de-spinning it in of order an epicyclic time. These are
discussed in the next three sections.
4. Gravitational Torques
Gravitational torques are not particularly important for the example cloud discussed in the last
section, but they are important for the most massive cloud (cloud 1) in our simulation of gravitational
collapse in a gaseous sheet (Run 1). Figure 9 shows the azimuth-averaged torques on cloud 1 at
t = 4/κ; the inset shows the particle positions and offset from the center of mass of the cloud,
in parsecs (referred to the standard solar neighborhood model). The total gravitational torque on
material interior to 150 pc is sufficient to remove its angular momentum on a timescale ∼ 1/κ.
Following an argument made by Larson (1984) and Lynden-Bell & Kalnajs (1972), we estimate
the gravitational torque using eqtn.[12]:
dS
=−
dt grav
Z
dA·(σG × r 0 )
(27)
S
where σG is the gravitational part of the stress tensor. Assuming the surface S is an infinite cylinder,
and adopting a cylindrical coordinate system centered on the cloud, we have
dSz
1
=−
dt
2G
Z
dφdz r 2 gφ gr .
(28)
S
Only the nonaxisymmetric part of gr contributes to the integral. If both nonaxisymmetric components
are of order GMc /R2 , as might occur if the mass is arranged in the form of a twisted, trailing bar,
then the timescale for transport of angular momentum out of the cloud is
¯ ¯
¯ Sz ¯ GM Ω
1
¯ ¯∼
¯ Ṡ ¯ R3 Ω Gρ ∼ Ω ,
z
(29)
if the object is rotationally supported. That is, gravitational torques on highly deformed objects like
some of the clouds shown in our simulation are efficient enough to slow rotation within a rotation
period.
5. Magnetic Torques
The transport of angular momentum by magnetic fields away from a rotating mass embedded
in a plasma has been considered by Mestel & Spitzer (1956), Kulsrud (1971), Mouschovias (1977),
– 75 –
Mouschovias & Paleologou (1979), Draine (1983), and others. A characteristic timescale in the problem
is the Alfven crossing time for the cloud, but the torque varies by orders of magnitude depending on
the exact configuration of magnetic fields. Here we discuss an upper limit to the magnetic torque on
a collapsing cloud and then estimate how seriously magnetic tension violates the potential vorticity
constraint.
The magnetic torque on the interstellar matter contained within a spherical surface s 1 of radius
R is
Z
dA·(σB × r 0 ),
(30)
S˙B = −
s1
where σB is the magnetic stress. Taking the orientation of the magnetic field that maximizes the
magnetic torque perpendicular to the plane at every point on the surface,
π
S˙B ·ẑ < hB 2 sin Θ > R3 ,
8
(31)
where Θ is the usual polar coordinate separation from the north pole and the angle brackets indicate
an average taken over the surface of the sphere. Then
¯ ¯
¶Ã 2 2 !
µ ¶µ
2
¯ Sz ¯
M
R
ω
ω R
ρ̄
1
c
τB ≡ ¯¯ ¯¯ ∼ 2 2 ∼
,
2
R
B
R
ω
ρ
V
Ṡz
edge
A,edge
(32)
where edge denotes some average value at the cloud boundary, VA,edge is a typical Alfvén velocity on
the cloud boundary, and ω is a rotation frequency for the cloud. Clearly τB ∼ 1/ω, so that magnetic
torque could rapidly despin the cloud.
Another way of understanding the dynamical importance of the magnetic field is to calculate how
rapidly it changes the potential vorticity. Magnetic fields cause the potential vorticity to decay at a
rate
Dξ
H
=−
(∇Σ × (B·∇)B) ·ẑ,
(33)
Dt
4πΣ3
where we imagine that the shearing sheet is confined between two frictionless, non-conducting plates
separated by a distance H. The potential vorticity of a disk with initially uniform velocity field is
2B/Σ (n.b. B here is Oort’s constant), so the timescale for changing ξ locally is
¯ µ
¯
¶µ
¶ µ ¶µ 2 2¶
¯ ξ ¯
4πΣ3 L2
Ω L
1
¯ ∼ 2B
τξ ≡ ¯¯
∼
Dξ/Dt ¯
Σ
HΣB 2
Ω
VA2
(34)
where L is a typical scale for variation of both the magnetic field and surface density. As might be
expected, the timescale for changing the potential vorticity is about 1/Ω× the ratio of the spin energy
of the proto-cloud to its magnetic energy. Since these are comparable for L ≈ 500 pc, V A ≈ 10 km s−1 ,
and Ω ≈ 26 km s−1 kpc−1 , τξ ∼ 1/Ω.
These two estimates differ from that usually given for the spin-down of a spherical cloud threaded
by a magnetic field (e.g. Spitzer, 1978) in that they consider the instantaneous torque on the cloud.
If the cloud spins several times the magnetic field is wound around the cloud and the magnetic field
strength increases in proportion to the angle through which the cloud has turned. The time required to
bring the cloud to a complete halt is then just the Alfvén crossing time multiplied by a dimensionless
constant that depends on the magnetic field configuration. Because the observations are consistent
with the cloud only having time to rotate through about one turn, the instantaneous torque is more
relevant to the present discussion.
– 76 –
While we cannot dynamically incorporate magnetic fields in our simulations, we can estimate the
magnetic torque in the limit that the field is weak. Assuming that the initial field is nearly azimuthal,
we select particles from the initial conditions that are at nearly constant radius and close to the
midplane of the disk. After ordering this group of particles in azimuth we imagine that all the fluid
elements represented by the SPH particles are threaded by the same field lines. Figure 10 shows lines
connecting the series of particles selected by this procedure from the initial conditions of the gaseous
collapse simulation of Chapter 3.
Figure 11a-d shows the field geometry at t = 2.75, 3.25, 3.75, and 4.25 (at intervals of about
1.3 × 107 yr in the solar neighborhood). We focus attention on the condensation near x = 3, y = 8 in
the frame at t = 3.75. The magnetic field is enhanced by a factor of ' 12 above the initial azimuthal
magnetic field B0 (this is obtained by measuring the reduction in the spacing of field lines parallel to
the plane of the disk of ' 6 and the compression of the fluid perpendicular to the disk of ' 2). If we
define the cloud as all points with n¯nbhd > 50 cm−3 , the torque may be estimated by reference to
eqtn.[30]. The cloud is approximately an ellipsoid of length L = 6, width w = 1, and height h = 1.
The field enters the cloud normal to its surface in one quadrant, runs the length of the cloud, and
exits in the opposite quadrant. The torque is therefore
Ṡz ≈ 2
B2
h
8π
Z
L/2
ldl ≈
0
B2 2
L h ≈ 52B02 [G4 Σ5g /κ6 ],
32π
(35)
where B0 is the initial magnetic field. The total spin angular momentum of the cloud about its center
of mass is −31.7 [G4 Σ5g /κ7 ], so the timescale for spindown by the magnetic fields is
τB ≈ 0.61/B02 [κ−1 ]
(36)
or about 1.1 × 107 (B0 /1.7 µG)−2 yr in the solar neighborhood (the magnetic field unit is
0.7Σg /13 M¯ pc−2 µG), where we have taken the pulsar rotation measure estimate of 1.7 µG for the
ordered component of the magnetic field (Manchester & Taylor, 1977).
For reasonable values of the initial field strength, the magnetic torque on the cloud at the
end of the simulation is sufficiently strong to violate the assumption that the magnetic field has a
negligible effect on the dynamics. It is of course difficult to estimate how the self-consistent inclusion
of the magnetic field would alter the outcome. Nevertheless, the sign of the various effects are clear:
gravitational collapse tends to produce prograde rotation because of conservation of potential vorticity.
A weak magnetic field resists the rotation and torques the cloud in a retrograde sense.
6. Conclusions
Other torques act on a giant molecular cloud. The terms in the stress tensor (eqtn.[23]) that
we have not yet considered are the Reynolds stress, pressure, radiation, and small-scale magnetic
stresses. If the interstellar medium in the neighborhood of a giant molecular cloud is concentrated in
dense clumps with long mean free path, the Reynolds stress will cause a torque to be exerted on the
cloud; this torque can also be described as an effective viscosity in the cloud “fluid”. This torque can
de-spin the cloud on a timescale of 1/κ if the clumps have a random velocity that is comparable to
the cloud rotational velocity. Pressure torques can be exerted in connection with the rocket effect.
Young stars heat gas in a blister on the cloud surface; the gas leaves the cloud with a velocity on
order the sound speed. Momentum is transferred efficiently to part of the cloud. Since young clusters
– 77 –
eventually succeed in removing all molecular gas from their immediate vicinity this mechanism is
undoubtedly capable of exerting a significant torque on the cloud. Radiation pressure is negligible.
Cosmic ray pressure would be negligible if the cosmic ray pressure in the vicinity of the cloud were
equal to the cosmic ray pressure in the ambient interstellar medium. However, because supernovae
occur preferentially in and near molecular clouds, it is not obvious that this is the case.
To summarize, we have investigated the rotation frequencies expected for two simple models of
giant molecular cloud formation. Fluid accretion and gravitational instability both produce rapid
rotation, and this can be understood in light of the conservation of potential vorticity. Collisionless
accretion produces slow rotation because the cloud draws its mass from a reservoir of material with
low spin angular momentum. Unfortunately, these estimates of the rotation frequency do not seriously
constrain cloud formation theories because several mechanisms exist, including gravitational and
magnetic torques, that can remove the cloud’s spin angular momentum within its lifetime.
– 78 –
REFERENCES
Bally, J., et al., 1987, Ap. J., 312, L45.
Blaauw, A., 1964, Ann. Rev. Astr. Ap., 2, 213.
Blitz, L., 1991, in Physics of Star Formation & Early Stellar Evolution, (Dordrecht: Kluwer), p. 3.
Bronfman, L., et al., 1987, Ap. J., 324, 248.
Chromey, F.R., Elmegreen, B.G., & Elmegreen, D.M., 1989, A. J., 98, 2203.
Clifford, P., & Elmegreen, B.G., 1983, M.N.R.A.S., 202, 629.
Draine, B.T., 1983, Ap. J., 270, 519.
Elmegreen, B.G., 1991, in Physics of Star Formation & Early Stellar Evolution, (Dordrecht: Kluwer),
p. 35.
Goudis, C., 1982, The Orion Complex: A Case Study in Interstellar Matter, (Dordrecht: Reidel).
Hausman, M., 1981, Ap. J., 245, 72.
Hernquist, L., & Katz, N., 1988, Ap. J. Suppl., 70, 419.
Kulsrud, R.M., 1971, Ap. J., 163, 567.
Kutner, M.L., et al., 1977, Ap. J., 215, 521.
Larson, R.B., 1988, in Galactic & Extragalactic Star Formation, ed. R.E. Pudritz & M. Fich, (Boston:
Kluwer) p. 459.
Larson, R.B., 1984, M.N.R.A.S., 206, 197.
Lynden-Bell, D., & Kalnajs, A., 1972, M.N.R.A.S., 157, 1.
Maddalena, R.J., Morris, M., Moscowitz, J., & Thaddeus, P., 1986, Ap. J., 303, 375.
Manchester, R.N., & Taylor, J.H., 1977, Pulsars, (San Francisco: Freeman), p. 135.
Mestel, L.,& Spitzer, L., 1956, M.N.R.A.S., 116, 503.
Mestel, L., 1965, Quart. J.R.A.S., 6, 161.
Mestel, L., 1966, M.N.R.A.S., 131, 307.
Mouschovias, T.Ch., 1977, Ap. J., 211, 147.
Mouschovias, T.Ch.,& Paleologou, E.V., 1979, Ap. J., 230, 204.
Spitzer, L., 1978, Physical Processes in the Interstellar Medium, (New York: Wiley).
Stark, A.A., & Blitz, L., 1978, Ap. J., 225, L15.
– 79 –
FIGURE CAPTIONS
Figure 1. (a) The projection of particle positions onto the x − z plane from initial conditions of a
fluid accretion run. (b) The projection of particle positions onto the x − y plane from the
same initial conditions. The solid lines are the zero-velocity curves for the Jacobi integral.
Figure 2. The mass of the accreting object as a function of time in four different accretion runs:
in two of the simulations, the accreting material is collisionless, while in the other two the
accreting material is an isothermal fluid.
Figure 3.
The effective rotation frequency ωef f ≡ Sz /Mc ra2 as a function of time for the selfgravitating collisionless accretion simulation and for the self-gravitating isothermal fluid
accretion simulation.
Figure 4. The location of all particles with n > 50cm−3 at the end of a simulation of gravitational
collapse in an isothermal gaseous sheet. The circles and numbers indicate the location of
clouds whose effective rotation frequencies are listed in Table 2.
Figure 5.
The azimuth-averaged specific torques on “Cloud 5” at the end of the simulation of
gravitational collapse in an isothermal gaseous sheet.
Figure 6. Spin angular momentum of the particles that eventually become part of “Cloud 5” as a
function of time.
Figure 7. Positions of particles that are destined to become part of dense clouds at the end of
the simulation of gravitational collapse in an isothermal gaseous sheet. The frames begin at
t = 0/κ in the upper left hand corner and are spaced at intervals of 0.5/κ.
Figure 8. Observations of the isothermal gaseous collapse simulation assuming that the simulation is
centered at l = 34 deg, d = 7 kpc and that the surface density at that location is 20 M ¯ pc−2 .
Particles with n > 100 cm−3 are projected into the l − v plane.
Figure 9. Azimuth averaged torques on “Cloud 1” in the isothermal gaseous collapse simulation at
t = 4/κ.
Figure 10. The initial magnetic field configuration in the isothermal gaseous collapse simulation.
The field lines were set by selecting particles lying within narrow tubes set along lines of
constant x, sorting the particles within the tube in y, and connecting them.
Figure 11. Magnetic field geometry at intervals of 0.5/κ beginning at t = 2.75/κ in the upper left
hand corner.
TABLE 1
Cloud Velocity Gradients
Cloud
gradient [ km s−1 pc−1 ]
Rosette
L1641 (Orion A)
W3
Monoceros R1
0.18
0.10
0.13
0.2
TABLE 2
Cloud Rotation: Run 1
Cloud No.
I[107 M¯ pc2 ]
Sz [106 M¯ pc km s−1 ]
Ωef f [ km s−1 kpc−1 ]
1
2
3
4
5
6
7
8
9
10
11
12
160
23
23
20
500
13
560
16
66
1.7
44
26
−260
−4.2
−7.2
−5.0
−390
−4.2
−140
4.5
−21
−0.18
−6.5
−31
−160
−19
−32
−25
−77
−33
−24
28
−33
−11
−15
−120