Star Formation1
Kohji Tomisaka
National Astronomical Observatory Japan
August 13, 2021

1

https://astrophys.sakura.ne.jp/Lecture Notes/StarFormation/7.pdf

Copyright ©2001-2021 by Tomisaka, K.
version 0.5: August 5, 2001
version 1.0: August 30, 2001
version 2.0: November 8, 2003
version 3.0: June 1, 2004
version 4.0: Oct. 29, 2007
version 5.0: Dec. 9, 2009version 6.0: Oct. 3 – , 2012
version 7.0: August 13, 2021
version 7.1: May 3, 2022
If you find any problems in this note, I would appreciate very much if you let me know it by e-mail:kohji_tomisaka@yahoo.co.jp. Download from https://astrophys.sakura.ne.jp/Lecture_Notes/
StarFormation/7.pdf This was compiled at 10:31 of 4th May, 2022.

Contents
1 Introduction
1.1 Interstellar Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Case Study — Taurus Molecular Cloud . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 T Tauri Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Disk Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Spectral Energy Distribution (SED) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Protostars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 B335 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Infall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Outflow Motion and a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 L1551 IRS 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Infall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optical Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 L 1544: Pre-protostellar Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Rotation of Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Mechanism to Accelerate Outflows . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Serpens-South Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.2 Prestellar Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.3 Cores with Protostars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bok Globules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.4 Magnetic Field ∼1000AU Scale . . . . . . . . . . . . . . . . . . . . . . . . . . .
T Tauri Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.1 Mass Accretion onto Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10.2 Edge Effect of Cylindrical Object . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12 Line Width - Size Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Physical Background39
2.1 Basic Equations of Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 The Poisson Equation of the Self-Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Free-fall Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43i

ii

CONTENTS
2.4.1 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Sound Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Jeans Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Gravitational Instability of Thin Disk . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Convective Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Super- and Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 Flow in the Laval Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.2 Steady State Flow under an Influence of External Fields . . . . . . . . . . . . .
2.8.3 Stellar Wind — Parker Wind Theory . . . . . . . . . . . . . . . . . . . . . . .
2.9 Virial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1 Virial Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.1 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.2 Einstein’s Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Induced Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Collisional Excitation and Deexcitation . . . . . . . . . . . . . . . . . . . . . .
2.10.3 Relation of Einstein’s Coefficients to Absorption and Emissivity . . . . . . . . .
Relation between Einstein’s Coefficients∗ . . . . . . . . . . . . . . . . . . . . .
2.10.4 Excitation Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.5 Critical Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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63

3 Galactic Scale Star Formation
65
3.1 Schmidt Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.1 Global Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.2 Local Star Formation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Star Formation Seen with Akari . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Distributions of Atomic and Molecular Hydrogen . . . . . . . . . . . . . . . . . . . . . 69
3.3 Gravitational Instability of Rotating Thin Disk . . . . . . . . . . . . . . . . . . . . . . 72
3.3.1 Tightly Wound Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Toomre’s Q Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Spiral Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Density Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.1 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Galactic Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 Local Star Formation Process85
4.1 Hydrostatic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Bonnor-Ebert Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.2 Equilibria of Cylindrical Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Virial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 Magnetohydrostatic Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Equilibrium of Filament . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.3 Subcritical vs Supercritical Filaments . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.4 Observation of Magnetic Field Strength . . . . . . . . . . . . . . . . . . . . . . 94

CONTENTS

iii

Zeeman Splitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Davis–Chandrasekhar-Fermi Method . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.5 Observation of Mass-to-Flux Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.1 Ionization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.2 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.3 Dissipation of Magnetic Field by Ambipolar Diffusion, Hall Effect, and Ohmic
Dissipation∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4 Dynamical Collapse of Spherical Isothermal Clouds . . . . . . . . . . . . . . . . . . . . 104
4.4.1 Inside-out Collapse Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4.2 Protostellar Evolution of Supercritical Clouds . . . . . . . . . . . . . . . . . . . 109
4.5 Star Formation Time Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.6 Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.7 Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.7.1 Magneto-driven Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Angular Momentum Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Centrifugal Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.7.2 Entrainment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.8 Evolution to Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.9 Example of Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.9.1 Angular Momentum Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.10 Evolution in the H-R diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.10.1 Accreting vs Non-accreting Stars . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.10.2 Main Accretion Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Before Deuterium Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
After Deuterium Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.10.3 Pre-main-sequence Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.10.4 Pre-main-sequence Evolution of Massive Stars . . . . . . . . . . . . . . . . . . . 134
5 Addendum
137
5.1 Aligned vs. Misaligned Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Polarization due to Magnetically Aligned Dust Grains . . . . . . . . . . . . . . . . . . 144
5.2.1 Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2.2 Polarization of Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Appendices

149

A Basic Astrophysical Quantities

149

B Basic Equation of Hydrodynamics
151
B.1 What is fluid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.3 Lagrangian and Eulerian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.4 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.4.1 Expression for Momentum Density . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.5 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.5.1 Polytropic Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.5.2 Energy Equation from the First Law of Theromodynamics . . . . . . . . . . . . 155

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CONTENTS
B.6 Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.6.1 Rankine-Hugoniot Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.6.2 Isothermal shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.7 Derivation of Self-similar Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

C Basic Equations of Magnetohydrodynamics
159
C.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.1.1 Flux Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.1.2 Basic Equations of Ideal MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.1.3 Axisymmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
D Hydrostatic Equilibrium
163
D.1 Polytrope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
D.2 Magnetohydrostatic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
E Basic Equations for Radiative Hydrodynamics
167
E.1 Radiative Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
F Polarization
169
F.1 Formalism to Calculate Observed Stokes Parameter . . . . . . . . . . . . . . . . . . . . 169
F.2 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
G Random Velocity

175

H X-factor

177

I

Abundance
179
I.1 Solar system abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
I.2 Isotope ratio (Isotopic abundance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
I.3 Molecular Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

J Gravity in Filament

181

K Magnetic Braking
185
Aligned Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Perpendicular Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Bibliography

187

Chapter 1

Introduction
1.1

Interstellar Matter
8.0
7.0"Coronal" Gas

Stellar Wind

6.0

Supernova

5.0
log T (K)Star

Interaction with Supernova

Dissociation of H2

HII Regions

Second Collapse

4.0
Mass Loss from Red GiantIntercloud Gas
3.0
2.0

Photo-dissociation

Diffuse Cloud

1.0
0.0
-4.0

Warm or Hot Core

Quasi-static Contraction

Isothermal Collapse

Molecular Cloud
-2.0

0.0

2.0

First Stellar Core

Globule
4.0

6.0

8.0

10.0 12.0
log n (cm-3)

14.0

16.0

Second Stellar Core
18.0

20.0

22.0

Figure 1.1: Multiphases of the interstellar medium. The temperature and number density of gaseous
objects of the interstellar medium in our Galaxy are summarized. Originally made by
Myers (1978), reconstructed by Saigo (2000).

Figure 1.1 shows the temperature and number density of gaseous objects in our Galaxy. Coldinterstellar medium forms molecular clouds (T ∼ 10 K) and diffuse clouds (T ∼ 100 K). Warminterstellar medium 103 K ≲ T ≲ 104 K are thought to be pervasive (wide-spread). Hii regions are
ionized by the Ly continuum photons from the nearby early-type stars. There are coronal (hot but
tenuous) gases with T ∼ 106 K in the Galaxy, which are heated by the shock fronts of supernova
remnants. Pressures of these gases are in the range of 102 K cm−3 ≲ p ≲ 104 K cm−3 , except for theHii regions. This may suggest that gases are in the pressure equilibrium.
In this figure, a theoretical path from the molecular cloud core to the star is also shown. We will
1

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CHAPTER 1. INTRODUCTION

see the evolution more closely in Chapter 4.

Figure 1.2: (Left:) Temperature-density relation of interstellar gas in which thermal equilibrium isachieved between two effects of heating and cooling. Above the line, cooling due toradiative loss is dominated, while below it the heating is dominated over the cooling.
(Right:) The same as the left one, but for pressure-density relation. Figures are takenfrom Figs. 1.15 and 1.16 of Hayashi et al. (1978) (see also Draine 2011).

From theoretical point of view, in Figure 1.2, we plot the temperature-density (left) and pressuredensity (right) relations for a gas in which the thermal equilibrium is achieved between two effects
to heat the gas and to cool the gas. If the interstellar pressure p is higher than p ≳ 10−13 dyn cm−2 ,Figure 1.2 (right) indicates that interstellar gas density is higher than ρ ≳ 10−22 g cm−3 . On theother hand, if the interstellar pressure p is lower than p ≲ 10−14 dyn cm−2 , low-density interstellar
matter exists ρ ≲ 10−26 g cm−3 . Between these two pressures, there are three states coexisting in theinterstellar space. Since the middle state is unstable (because increasing the temperature leads toincreasing the heating rate or decreasing the cooling rate), there exist two stable phases in pressure
equilibrium. This is a picture of two-phase interstellar medium (Field et al. 1969; Spitzer 1978).
Intercloud gas and diffuse cloud gas are explained by this mechanism. Historically, the discovery
of hot coronal gas with T ≳ several × 105 K stimulated an idea that the interstellar gas is globally
divided into three phases (Cox & Smith 1974; McKee & Ostriker 1977).
Globally, the molecular form of hydrogen, H2 , is much abundant inside the Solar circle, while theatomic hydrogen Hi is more abundant than molecular H2 in the outer Galaxy. In Figure 1.3 (left),the radial distributions of molecular and atomic gases are shown. The right panel shows similar
distributions for four typical external galaxies (M51, M101, NGC6946, and IC342). This indicatesthese distributions are similar with each other. Hi is distributed uniformly, while H2 density increases

1.2. CASE STUDY — TAURUS MOLECULAR CLOUD

3

greatly reaching the galaxy center. In other words, only in the region where the total (Hi+H2 ) density
exceeds some critical value, H2 molecules are distributed.

Figure 1.3: Radial distributions of H2 (solid line) and Hi (dashed line) gas densities in galaxies.
(Left:) our Galaxy. Converting from CO antenna temperature to H2 column density,n(CO)/n(H2 ) = 6 × 10−5 is assumed. Taken from Gordon & Burton (1976). (Right:)
Radial distribution of H2 and Hi gas for external galaxies. The conversion factor isassumed constant X(H2 /CO) = 3 × 1020 H2 /K km s−1 . Taken from Honma et al. (1995).
About the X-factor, refer to Appendix H.

1.2

Case Study — Taurus Molecular Cloud

Figure 1.4 (left) shows the 13 CO total column density map of the Taurus molecular cloud (Mizuno
et al. 1995) whose distance is 140 pc far from the Sun. Since 13 CO contains 13 C, a rare isotope ofC, the abundance of 13 CO is much smaller than that of 12 CO (for isotopic abundances, see TableI.2). Owing to its low abundance, the emission lines of 13 CO are relatively optically thinner thanthat of 12 CO. Using 13 CO line, we can see deep inside of the molecular cloud. The distributions of
T Tauri stars and 13 CO column density coincide with each other. Since T Tauri stars are young
pre-main-sequence stars with M ∼ 1 M⊙ , which are in the Kelvin-Helmholtz contraction stage anddo not reach the main-sequence hydrogen burning stage, it is shown that these stars are newly formedin molecular clouds.
Since 18 O is much more rare isotope (18 O/16 O ≪ 13 C/12 C, see Table I.2), the distribution of muchhigher-density gases is explored using C18 O lines. Figure 1.4 (right) shows C18 O map of Heiles cloud
2 in the Taurus molecular cloud by Onishi et al. (1996). This shows us that there are many molecular
cloud cores which have much higher density than the average. Many of these molecular cloud cores
are associated with IRAS infrared sources and T Tauri stars. It is shown that star formation occurs in
the molecular cloud cores in the molecular cloud. They found 40 such cores in the Taurus molecular

4

CHAPTER 1. INTRODUCTION

Figure 1.4: (Left:) 13 CO total column density map of the Taurus molecular cloud (Mizuno
et al. 1995). Taken from their home page with URL of http://www.a.phys.nagoyau.ac.jp/nanten/taurus.html (in Japanese). T Tauri stars, which are thought to be premain-sequence stars in the Kelvin-Helmholtz contraction stage, are indicated by brightspots. (Right:) C18 O map of Heiles cloud 2 region in the Taurus molecular cloud (Onishi
et al. 1996). This shows clearly that the cloud is composed of a number of high-density
regions.

cloud. Typical size of the core is ∼ 0.1 pc and the average density of the core is as large as ∼ 104 cm−3 .
The mass of the C18 O cores is estimated as ∼ 1 − 80 M⊙ .
H13 CO+ ions are excited only after the density is much higher than the density at which CO
molecules are excited1 . H13 CO+ ions are used to explore the region with higher density than thatobserved by C18 O. Figure 1.5 shows the map of cores observed by H13 CO+ ions. The cores shown in
the lower panels are accompanied with infrared sources. The energy source of the stellar IR radiationis thought to be maintained by the accretion energy. Here we estimate the accretion luminosity Lacc(energy released by accretion in a unit time). Since the gravitational potential energy per unit massat the surface of a protostar with a radius R∗ and a mass M∗ is equal to
Φ=−

GM∗
,
R∗

(1.1)

the kinetic energy of the gas accreting on the stellar surface is approximately equal to ∼ GM∗ /R∗ .
The energy inflow rate owing to the accretion is
∼

GM∗
c3
GM∗
× Ṁ ∼
×A s,
R∗
R∗
G

(1.2)

where Ṁ = Ac3s /G is the mass accretion rate (cs represents the sound speed of accreting gas). Thisrate was estimated as follows:
1. A mass of gravitationally bounded object (the Jeans mass) is given as MJ ∼ ρ × c3s t3ﬀ , where tﬀ
represents the gravitational free-fall time tﬀ ∼ (Gρ)−1/2 [eq. (2.26)].
2. Thus, the accretion rate is estimated as Ṁ ∼ MJ /tﬀ ∼ ρ × c3s t2ﬀ ∼ c3s /G.
3. Thus, the “natural” accretion rate is proportional to the sound speed cs cubed and the inverseof the gravitational constant G. (See also §§ 4.4.1 and 4.4.2.).
In § 2.10.5 of the next chapter (p.63), we will touch the critical density, above which the upper state levels are well
excited by collision with H2 molecules. In Table 2.1 we show the values of critical densities of CO and HCO + .
1

1.3. T TAURI STARS

5

Figure 1.5: Pre-protostellar versus protostellar cores (H13 CO+ map). Panels of the upper row show
the C18 O cores without associated IRAS infrared sources. Lower ones show the cores with
IRAS sources. Taken from Fig. 1 of Mizuno et al. (1994)

Accretion rate ofṀ ∼ 1.6 × 10−6 M⊙ yr−1



3
cs
190 m s−1

(1.3)

gives the accretion luminosity of
Lacc ∼ 5L⊙

M∗
1M⊙



10R⊙
R∗



3
cs
.
190 m s−1

(1.4)

In the upper panels of Figure 1.5, the cores without IR sources are shown. This core does not show
accretion but collapse. That is, before a protostar is formed, the core itself contract owing to thegravity, which is explained closely in Chapter 4.
In Figure 1.5, H13 CO+ total column density maps of the C18 O cores are shown. Cores in thelower panels have associated IRAS sources, while the cores in the upper panels have no IRAS sources.
Since the IRAS sources are thought to be protostars or objects in later stage, the core seems to evolvefrom that without an IRAS sources to that with an IRAS source. From this, the core with an IRAS
source is called a protostellar core, which means that the cores contain protostars. On the other
hand, the core without IRAS source is called pre-protostellar core or, in short, pre-stellar core.
Figure 1.5 shows that the prestellar cores are less dense and more extended than the protostellar
core. This seems to suggest that the density distribution around the density peak changes between
before and after the protostar formation.

1.3

T Tauri Stars

T Tauri stars are observationally late-type stars with strong emission lines and irregular light variations
associated with dark or bright nebulosities. T Tauri stars are thought to be low-mass pre-main-sequencestars, which are younger than the main-sequence stars. Since these stars are connecting between protostars and main-sequence stars, they attract attention today. More massive counterparts are called
Herbig Ae-Be stars. They are doing the Kelvin-Helmholtz contraction in which the own gravitationalenergies released as it contracts gradually and this is the energy source of the luminosity.

6

CHAPTER 1. INTRODUCTION

Figure 1.6: HR diagram of T Tauri stars. Many emission lines are found in the spectra of T Tauristars. WTTS (Weak Emission T Tauri Stars) and CTTS (Classical T Tauri Stars) areclassified by the equivalent widths of emission lines. That is, the objects with EW of Hαemission < 1 nm are usually termed WTTS. Taken from Fig.1.2 of Hartmann (1998).

1.3. T TAURI STARS

7

The gravitational energy of a star with mass of M∗ and radius of R∗ is given as
W = −α

GM∗2
,
R∗

(1.5)

where α is a numerical factor of order of unity which depends on the radial density distribution ρ(r).
From the virial theorem [eq. (2.120)], total energy of E = W/2 = −U for monoatomic gas with γ =5/3. When the star loses its energy by radiation in a rate of L∗ = −dE/dt, the gravitational energy
decreases as dW/dt = −2L∗ (in other words, absolute value of the gravitational energy increases).
Since dW/dt = αGM∗2 /R∗2 · dR∗ /dt = −2L∗ , the star is contracting in a rate ofdR∗
2L∗ R∗2
=−
.
dt
αGM∗2
This process is called Kelvin-Helmholtz contraction. In short, the gravitational energy is used tocompensate the luminosity. When the system loses its energy, although the potential energy W
decreases, its thermal energy U increases. Thus, the central temperature increases and such a starreaches to the main-sequence star in which the nuclear fusion reaction of 1 H to create 4 He produces
energy.
Problem 1
Obtain the Kelvin-Helmholtz time of our Sun. The Kelvin-Helmholtz time is given
τKH ∼

GM∗2
.
R∗ L ∗(1.6)

Use the physical values of the Sun shown in Appendix A.
Many emission lines are observed in the spectra of T Tauri stars. WTTS (Weak emission T
Tauri Stars or Weak line T Tauri Stars) and CTTS (Classical T Tauri Stars) are classified bytheir equivalent widths (EWs) of emission lines. That is, the objects with an EW of Hα emission< 1 nm = 10 Å are usually termed WTTS. Figure 1.6 is an HR diagram (Teﬀ − Lbol ) of T Tauristars in Taurus-Auriga region (Kenyon & Hartman 1995). WTTS distribute near the main-sequence
while CTTS are found even far from the main-sequence. A number of theoretical evolution tracks forpre-main-sequence stars with M ∼ 0.1 M⊙ − 2.5 M⊙ are shown in a solid line, while the isochorones
for ages of 105 yr, 106 yr, and 107 yr are plotted in a dashed line. Vertical evolutionary paths are
the Hayashi convective track. Since D=2 H has a much lower critical temperature (and density) for afusion nuclear reaction to make He than 1 H, deuterium begins to burn before reaching the zero-agemain sequence. This occurs near the isochrone for the age of 105 yr and some activities related to theignition of deuterium seem to make the central star visible (Stahler 1983).
Disk Frequency
Infrared studies of T Tauri stars in star-forming regions have suggested that initial disk frequency israther high and that the disk lifetimes are relatively short as 3 − 15 Myr. From JHKL photometry,
Haish, Lada, & Lada (2001) obtained the fraction of disk-bearing stars for 6 star formation regions.
L band excess emission indicates an accompanied disk. The fraction decreases with the age of theclusters as in Figure 1.7. This figure shows clearly that the disk fraction is initially very high (≳ 80 %)and rapidly decreases with increasing cluster age. In 3 Myr a half of the disk stars lose their disks.
Overall disk lifetime is estimated as ∼ 6 Myr.



8

CHAPTER 1. INTRODUCTION

Figure 1.7: Fractions of IR excess sources in respective clusters are plotted against the age of theclusters. Taken from Haish, Lada, & Lada (2001).

1.4

Spectral Energy Distribution (SED)

A tool to know the process of star formation is provided by the spectral energy distribution (SED)
mainly in the near- and mid-infrared light. T Tauri stars and protostars have typical respective
SEDs. IR SEDs of T Tauri stars and protostars were classified into three as Class I, Class II, andClass III, from a stand-point of relative importance of the radiation from a dust disk to the stellar
black-body radiation. Today, the classification is extended to the younger protostars, which have no
indication of near- and mid-IR emissions, and they are called Class 0 objects. (Unfortunately, there
is no zero in Roman numerals.) The classification uses the slope in near- and mid-IR spectrum as
α ≡ |d log λFλ /d log λ|λ=2.2µm−25µm . In Figure 1.8, typical SEDs and models of emission regions areshown.
1. Class III is well fitted by a black-body spectrum, which shows the energy mainly comes from acentral star. This SED is observed in the weak-line T Tauri stars (WTTS). Although T Tauristars show emission lines of such as the hydrogen Balmer sequence, the weak-line T Tauri stars
do not show prominent emission lines, which indicates the amount of gas just outside the star
(this seems to be supplied by the accretion process) is small. At this stage, a disk has been
disappeared or only an extremely less-massive disk is still alive. α < −2.
2. Class II SED is fitted by a single-temperature black-body plus excess IR emission. This showsthat there is a dust disk around a pre-main sequence star and it is heated by the radiationfrom a central star. The width of the spectrum of the disk component is much wider than that
expected from a single-temperature black-body radiation. Thus, the disk has a temperature
gradient which decreases with increasing the distance from the central star. In this stage, thedust disk is more massive than that of Class III sources. Classical T Tauri stars (CTTS) havesuch SEDs. −2 < α < 0.
3. In Class I SED, the mid infrared radiation which seems to come from the dust envelope ispredominant over the stellar black-body radiation. Since the stellar black-body radiation seems
to escape at least partially from the dust envelope, a relatively large solid angle is expected for1.4. SPECTRAL ENERGY DISTRIBUTION (SED)

9

Figure 1.8: Spectral Energy Distributions (SED) of young stellar objects (YSOs) 2 and their models.
(Left:) ν − νFν plot taken from Lada (1999). (Right:) λ − λFλ plot taken from André
(1994)

a region where the dust envelope does not intervene. α > 0. Class I SED is expected fromprotostars.
4. Class 0 SED seems to be emitted by an isothermal dust with ∼ 30K. The protostar seems to
be completely covered by gas and dust and is obscured with a large optical depth by the dustenvelope. No contribution can be reached from the stellar-black body radiation.
The reason why the emission from the disk becomes wide in the spectral range is understood(Fig. 1.9) as follows: Temperature of the disk is determined by a balance of heating and cooling.
Assuming the disk is geometrically thin but optically thick, the cooling per unit area is given by
the equation of the black-body Planck radiation. Therefore, the temperature is determined by theheating predominantly by viscous heating and extra heating by the radiation from the central star.
The flux density emitted by the disk is given byZνFν ∼ νπBν [T (r)]2πrdr ∼ r(T or ν)2 νBν .
(1.7)
2

Young stellar objects represent protostars and premain-sequence stars.

10

CHAPTER 1. INTRODUCTION

Figure 1.9: Explanation for the spectral index of the emission from a geometrically thin but opticallythick disk. Taken from Fig. 16 of Lada (1999).

Assuming the radial distribution of temperature as −qR
T = T0,R0(1.8)(q = 3/4 for the standard accretion disk) and taking notice that each temperature in the disk radiates
at a characteristic frequency ν ∝ T (the Wien’s law for black-body radiation)
νFν ∼ r2 νBν ∝ ν 4 T −2/q ∝ ν 4−2/q ,(1.9)

where we used the fact that the peak value of Bν ∝ ν 3 . Therefore, it is shown thatνFν ∝ ν n ;2
n=4− .
q(1.10)As shown in the previous section, we have no young stellar objects found by IR before a protostar
is formed. These kinds of objects (pre-protostellar core) are sometimes called Class −1. The classification was originally based on the SED and did not exactly mean an evolution sequence. However,today YSOs are considered to evolve as the sequence of the classes: Class −1 → Class 0 → Class I→ Class II → Class III → main-sequence star.

1.5

Protostars

1.5.1

B335

Infall Motion
B335 is a dark cloud (Fig. 1.10) with a distance of D ≃ 250pc. Inside the dark cloud, a Class 0 IR
source is found. The object is famous for the discovery of gas infall motion. In Figure 1.11, the line
profiles of CS J = 2 − 1 line emissions are shown (Zhou et al. 1993). The relative position of the1.5. PROTOSTARS

Figure 1.10: Near infrared images of B335, which is Class 0 source. Left is H band image and rightis K band image.

Figure 1.11: Line profile of CS J = 2 − 1 line radio emission. Model spectra illustrated in a dashedline (Zhou 1995) are overlaid on to the observed spectra in a solid line (Zhoug et al.
1993).

11

12

CHAPTER 1. INTRODUCTION

Figure 1.12: Explanation of blue-red asymmetry when we observe a spherical symmetric inflow motion. An isovelocity curve for the red-shifted gas is plotted in a solid line. That for theblue-shifted gas is plotted in a dashed line. Taken from Fig. 14 of Lada (1999).

profiles correspond to the position of the beam. Expression (9,9) represents the offset of (9 ′′ , 9′′ ) fromthe center. At the center (0,0), the spectrum shows two peaks and the blue-shifted peak is brighter
than the red-shifted one. This is believed to be a sign of gas infall motion. The blue-red (or more
specifically “blue-skewed”) asymmetry is explained as follows:
1. Considering a spherical symmetric inflow of gas, whose inflow velocity vr increases with reaching
the center (a decreasing function of r)
2. Considering a gas element at r moving at a speed of vr (r) < 0, the velocity projected on aline-of-sight is equal tovline−of−sight = vsystemic + vr cos θ,(1.11)
where vsystemic represents the systemic velocity of the cloud (line-of-sight velocity of the cloudcenter) and θ is the angle between the line-of-sight and the position vector of the gas element.
The isovector lines, the line which connects the positions whose procession/recession velocities
are the same, become like an ellipse shown in Fig. 1.12.
3. An isovelocity curve for the red-shifted gas is plotted in a solid line. That for the blue-shifted
gas is plotted in a dashed line. If the gas is optically thin, the blue-shifted and red-shifted gases
contribute equally to the observed spectrum and the intensities of the blue- and red-shiftedpeaks of the emission line should be the same.
4. In the case that the gas has a finite optical depth, for the red-shifted emission line a cold gas
in the foreground side absorbs effectively the emission coming from the hot interior. On theother hand, for the blue-shifted emission line, the emission made by the hot interior gas escapesfrom the cloud without being absorbed by the cold gas (there is no cold counter part for theblue-shifted gas).
5. As a result, the blue peak of the emission line becomes more prominent than that of the redshifted emission. This is the explanation of the blue-red asymmetry.

1.5. PROTOSTARS

13

Figure 1.13: C18 O total column density map (left) and H13 CO+ channel map (right) of B335 alongwith the position-velocity maps along the major and minor axes. Taken from Fig. 3 of
Saito et al. (1999).

In Figure 1.11, model spectra calculated with the Sobolev approximation or LVG (large-velocity
gradient; Zhou 1995) are shown. These show the blue-red (blue-skewed) asymmetry (the blue line >the red line).
Outﬂow Motion and a Disk
Early observation of star forming regions have revealed fast molecular outflow are often ejected fromthe central protostar with ≳ 10 km s−1 . B335 is also a typical outflow source. In Figure 1.13,
distributions of high density gases traced by the C18 O and H13 CO+ lines are shown as well as the
bipolar outflow whose outline is indicated by a shadow (Hirano et al. 1988). Comparing left and rightpanels, it is shown that the distribution of C18 O gas is more extended than that of H13 CO+ which
traces higher-density gas. And the distribution of the H13 CO+ is more compact and the projectedsurface density seems to show that the actual distribution is ellipsoid. And the molecular outflow
seems to be ejected in the direction of the minor axis of the high-density gas. It may suggest that (1) amolecular outflow is focused or collimated by the effect of density distribution or that (2) collimation
is made by the magnetic fields which run preferentially perpendicularly to the gas disk. This gas disk
is observed by these high density tracers.
Combining the C18 O and H13 CO+ distributions, the surface density distribution along the major
axis is obtained by Saito et al. (1999). From the lower panel of Figure 1.14, the column density
distribution is well fitted in the range from 7,000 to 42,000 AU in radius, r −0.95
Σ(r) = 6.3 × 1021 cm−2,
(1.12)
104 AU
where they omitted the data of r ≲ 7000 AU since the beam size is not negligible. Similar power-law

14

CHAPTER 1. INTRODUCTION

Figure 1.14: Column density distribution NH (r) derived from the H13 CO+ and C18 O data taken by
the Nobeyama 45 m telescope. Taken from Fig. 9 of Saito et al. (1999).

density distributions are found in the far IR thermal dust emissions.

1.5.2

L1551 IRS 5

Figure 1.15: (Left:) 13 CO column density distribution. The contour lines represent the distributionof 13 CO column density. 2.2 µm infra-red reflection nebula is shown in grey scale which
was observed by Hoddap (1994). (Right:) Schematic view of L1551 IRS 5 region.

L1551 IRS 5 is one of the best studied protostellar objects. This has an infra-red emission
nebulosity (Fig. 1.15). It is believed that there is a hole perpendicular to the high-density disk andthe emission from the central star escapes through the hole and irradiate the nebulosity. In this sense
this is a reflection nebula. L1551 IRS 5 has an elongated structure of dense gas similar to that observedin B335. High-density gas is extending in the direction from north-west to south-east [Fig. 1.15 (left)].
Since the opposite side of the nebulosity is not observed, the opposite side of nebulosity seems to belocated beyond the high-density disk and be obscured by the disk. This is possible if we see the south
surface of the high-density disk as in Figure 1.15 (right).

1.5. PROTOSTARS

15

Figure 1.16: Isovelocity contours measured by the 13 CO J = 1 − 0 line. It should be noticed thatthe isovelocity lines run parallelly to the major axis. The north-eastern side shows a
red-shift and the south-western side shows a blue-shift.

Infall Motion
Inflow motion is actually measured. Figure 1.16 shows the isovelocity contours measured by the
13 CO J = 1 − 0 observation (Ohashi et al. 1996). It should be noticed that the isovelocity lines run
parallelly to the major axis. The north-eastern side shows red-shift and the south-western side shows
blue-shift. Considering the configuration of the gas disk shown in Figure 1.15 (right), this pattern ofisovelocity contours indicates not outflow but inflow. That is, the north-east side is the near side ofthe disk and the south-west side is the far side. Since a red-shifted motion is observed in the nearside and a blue-shifted motion is observed in the far side, it should be concluded that the gas disk of
the L1551 IRS5 is now infalling.
Optical Jet
HST found two optical jets emanating from L1551 IRS5. This has been observed by SUBARU
telescope, which found jet emission is dominated by [Feii] lines in the J- and H-bands. The jet
extents to the south-western direction and disappears at ∼ 10′′ ≃ 1400 AU from the IRS5. The
width-to-length ratio is very small ≲ 1/10 or less, while the bipolar molecular outflow shows a lesscollimated flow. As for the origin of the two jets, these two jets might be ejected from a single source.
However, since there are at least two radio continuum sources in IRS5 within the mutual separationof ∼ 0.5′′ [see Fig. 1.17 (right)], these jets seem to be ejected from the two sources independently.

16

CHAPTER 1. INTRODUCTION

Figure 1.17: (Left:)
Infrared image (J- and K-band) of the IR reflection nebulaaround L1551 IRS 5 by SUBARU telescope.
Taken from Fig. 1 ofItoh et al.
(2000).
(The jpeg file is available from the following url:
http://SubaruTelescope.org/Science/press release/9908/L1551.jpg). (Right:)
Central 100 AU region map of L1551 IRS 5. This is taken by the λ = 2.7 cm radio
continuum observation. Deconvolved map (lower-left) shows clearly that IRS 5 consistsof two sources. Taken from Looney et al. (1997).

1.6. L 1544: PRE-PROTOSTELLAR CORES

17

Figure 1.18: A mosaic image of HH 111 based on HST NICMOS images (left) and WFPC2 images(right). Taken from Fig. 1 of Reipurth et al. (1999) and rotated clockwise.

Although the lengths of these jets are restricted to 10′′ , Herbig-Haro jets, which are much larger
than the jets in L1551 IRS5, have been found. HH30 has a ∼ 500 AU-scale jet whose emission ismainly from the shock-excited emission lines. One of the largest ones is HH111, which is a member of
the Orion star forming region and whose distance is as large as D ∼ 400pc, and a jet with a length of∼ 4pc is observed. Source of HH111 system is thought to consist of at least binary stars or possibly
triple stars [Reipurth et al. (1999)]. Star A, which coincides with a λ = 3.6 cm radio continuum
source (VLA 1), shows an elongation in the VLA map whose direction is parallel to the axis of thejet. Therefore, star A is considered to be a source of the jet. Since the VLA map of star A shows
another elongated structure perpendicular to the jet axis, star A may be a binary composed by two
outflow sources.

1.6

L 1544: Pre-protostellar Cores

L1544 is known as a pre-protostellar core (Taffala et al. 1998). That shows an infall motion but thiscontains no IR protostars. In Figure 1.19 (left), CCS total column density map is shown, which shows
an elongated structure. Ohashi et al. (1999) have found both rotation and infall motion in the cloud.
Position-velocity (PV) diagram along the minor axis shows the infall motion. That along the major
axis indicates a rotational motion, which is shown by a velocity gradient. Due to a finite size of thebeam, a contraction motion is also shown in the PV diagram along the major axis.

1.7

Rotation of Outﬂow

Inspired from the report of rotation of jet (Bacciotti et al. 2002), examples of rotating molecular
outflows are found (a list is summarized in Matsushita et al. 2021).
Figures 1.20 and 1.21 show that the outflow found in globule CB 26 is rotating (Launhardt etal. 2009). CB 26 is a Bok globule with bipolar NIR reflection nebulae, where a very young T Tauri
star and ∼ 200AU-scale high-density disk are observed (Fig. 1.20). Figure 1.21 shows the map of18

CHAPTER 1. INTRODUCTION

Figure 1.19: CCS image of prestellar core L1544. (Left:) Total intensity map. (Right:) PV diagrams
along the minor axis (left) and along the major axis (right).

intensity-weighted velocity (1st moment2 ) asR
T (v)vdv,
Vave = R
T (v)dv(1.13)

where T (v) represents the brightness temperature for line-of-sight velocity v. Left panel indicatesclearly that the right part gas is departing from us while the left part is approaching. This means
there exists a global velocity gradient perpendicular to the flow axis, or in other words, the rotation
motion in which the rotation axis coincides with the symmetric axis. The rotation is toward thesame direction of the high-density disk observed by 13 CO (Launhardt & Sargent 2001). Right panel
is an expecting intensity-weighted velocity distribution for a simple model of 12 CO (J = 2 − 1) line
as follows: r
Vr (r) = v0,
(1.14)r0 −1ϖVrot = VKep,
(1.15)ϖ0 −3rn(r) = n0,(1.16)r0 −qr
TK (r) = T0,
(1.17)
r0
in which we assume (i) the outflow is conical, (ii) gas element at (ϖ, z) in cylindrical coordinate waslaunched from a Keplerlian disk at ϖ0 conserving angular momentum, (iii) the radial expanding speed
Vr (r) is simply proportional to the distance from the central star r [eq. (1.14)]. The assumption (ii)2

Total intensity map is often called 0th moment. In the 2nd moment, the velocity variance around the intensityweighted velocity is plotted.

1.7. ROTATION OF OUTFLOW

19

Figure 1.20: Globule CB 26 has an outflow which seems to be seen edge-on. Grey-scale indicates
the K-band image of the bipolar reflection nebula. Red contours which show SMA 1.1
mm dust continuum emission indicate a high-density disk exists between two lobes ofthe bipolar nebula. Green contours represent the 12 CO(J = 2 − 1) integrated intensity.
Taken from Fig. 1 of Launhardt et al. (2019).

Figure 1.21: Observed intensity weighted velocity is shown in the left panel. Right one is a model.
Taken from Fig. 4 of Launhardt et al. (2019).

leads to [eq. (1.15)] as the rotation speed is inversely proportional to the distance from the rotation
axis ϖ = (r2 −z 2 )1/2 . (iv) Density and temperature distributions are assumed as the density decreases
with the distance r in proportion to r−3 [eq. (1.16)], and the kinetic temperature decreases with the
distance r in proportion to r−q [eq. (1.16)]. Comparing with a simple model (right panel), such global
rotation motion is seen evidently only when the outflow is observed from edge-on.

1.7.1

Mechanism to Accelerate Outﬂows

As is already noticed when the molecular outflows were found (Pudritz & Norman 1986), the momentum of outflow is not explained by the luminosity of central stars. Figure 1.22 shows the correlation
between the bolometric luminosity of central stars, Lbol (x-axis), and the momentum injection rate ofmolecular outflows, Ṁ V (y-axis). Using the relation between momentum p and energy E of photons,
E = pc, the maximum amount of momentum P converted from the photon’s energy satisfies the20

CHAPTER 1. INTRODUCTION

Figure 1.22: Correlation between the bolometric luminosity of central protostars (x-axis) and injection
rate of linear momentum or “outflow force” (y-axis). Triangle and square represent
respectively less-massive and massive star forming regions. This is taken from Fig. 7 of
Wu et al. (2004).

relation

dP
Lbol
≃
,
dt
c

(1.18)

where dP/dt ≡ F ≃ Ṁ V represents the force necessary to accelerate gas to the speed of V .
As shown in Figure 1.22, observed “outflow force” is much larger than that explained by theradiation pressure. The figure shows that the mechanism to accelerate the gas is not the radiationpressure. This is valid not only for the low luminosity central stars (triangles), but also the massive
luminous central stars (squares). Since the correlation relation is smoothly connected between massive
and less-massive central stars, the mechanism to form molecular outflows is common irrespective ofcentral star masses. Today, we believe magnetic field plays an important role in making the molecular
outflows (see next §1.8 and §4.7 of p.115).

1.8. MAGNETIC FIELDS

1.8

21

Magnetic Fields

Directions of magnetic field are studied by (1) measuring the polarization of light which is sufferedfrom interstellar absorption. In this case the direction of magnetic field is parallel to the polarization
vector. The reason is explained in Figure 1.23 (upper-left). In the magnetic fields, the dusts arealigned in a way that their major axes are perpendicular to the magnetic field lines. Such aligned
dusts absorb selectively the radiation whose E-vector is parallel to their major axes. As a result, thedetected light has a linear polarization parallel to the magnetic field lines.
However, the polarization measurement in the optical and near IR wavelengths limited to theregion with low gas density, because background stars suffer severe absorption and becomes hard tobe observed if we want to measure the polarization of the high-density region. More direct method
is (2) the measurement of the linear polarization of the thermal emission from dusts in the mm (orsub-mm) wavelengths; in this case the direction of magnetic field is perpendicular to the polarization
vector. The mechanism is explained in Figure 1.23 (upper-right). The aligned dusts, whose majoraxes are perpendicular to the magnetic field lines, emit the radiation whose E-vector is parallel to themajor axes3 . Since the absorption does not have a severe effect in this mm wavelengths, this gives
information about the magnetic fields deep inside the clouds.

Figure 1.23: Explanation how the polarized radiation forms. Taken from Weintraub et al. (2000).
3

Polarization direction is deﬁned as that of E-vector of the electromagnetic radiation.

22

CHAPTER 1. INTRODUCTION

Figure 1.24: H band (left) and Ks band (right) polarizations in the Serpens-South Cloud. Bars indicate the direction of polarization E-vector, which represents the direction of interstellar
B-vector. Colors represents λ = 1.1 mm dust continuum intensity by ASTE/AzTEX.
Distance to this object is D = 260 pc and this image covers ∼ 1.5 pc × 1.5 pc (Sugitani
et al. 2010).

1.8.1

Serpens-South Cloud

Figure 1.24 shows the structure of the magnetic field measured with H and Ks band polarimetory(Sugitani et al. 2010). Serpens south cluster is located at the center of an elongated filament. At thesame time the location corresponds to the intersection of several minor filaments. This figure shows
clearly that magnetic field runs in the perpendicular direction to the axis of the filamentary cloud.
Although the configuration of magnetic field lines in the filament is not explored sufficiently by thenear IR polarimetory, in the envelope the magnetic field direction is perpendicular to the filament aswell as outside of the filament.

1.8.2

Prestellar Core

Figure 1.25 illustrates the polarization maps of three prestellar cores (L1544, L183, and L43) donein the 850 µm band by JCMT-SCUBA. In L1544 and L183 the mean magnetic fields are at an angleof 30 deg to the minor axes of the cores. L43 is not a simple object; there is a T Tauri star locatedin the second core which extends to south-western side of the core (an edge of this core is seen near
the western SCUBA frame boundary). And a molecular outflow from the source seems to affectthe core. The magnetic field as well as the gas are swept by the molecular outflow. L43 seems an
exception. The fact that the mean magnetic fields are parallel to the minor axis of the high-density
gas distribution seems to mean gas contracts preferentially in the direction parallel to the magnetic
fields.
Figure 1.26 (left) shows the optical image of dark cloud B68. Emission from background stars
suffers extinction due to the dust in the dark cloud. Figure 1.33 (right) gives the distribution ofvisual extinction AV against the distance from the center. The distribution is well fitted to that of
the isothermal cloud in a hydrostatic equilibrium. Thus, the amount of extinction gives an estimateof the column density of interstellar dusts in the dark cloud. Since the absorption by the aligned

1.8. MAGNETIC FIELDS

Figure 1.25: Directions of B-Field are shown from the linear polarization observation of 850 µm thermal emission from dusts by JCMT-SCUBA. L1544 (upper-left) and L183 (upper-right),
the magnetic field and the minor axis of the molecular gas distribution coincide with eachother within ∼ 30deg. Configuration of magnetic field in L43 (lower ) is not simple, which
may be affected by the molecular outflow. Taken from Ward-Thompson et al. (2000).

Figure 1.26: (left:) Optical image of B68 (Alves et al. 2001) and (right:) H band image and polarization E-vector derived for background stars (Kandori et al. 2009). The region shown in(b) corresponds to the region shown by the square in (a).

23

24

CHAPTER 1. INTRODUCTION

dust grains induces the polarization, the direction of magnetic field is obtained from the E vector of
the polarized emission. Figure 1.26 (right) indicates the polarization vector and thus the direction ofinterstellar magnetic field in B68. Figure indicates a magnetic field line like a “hour-glass.”

1.8.3

Cores with Protostars

Bok Globules
Bok globules are isolated dark clouds. Wolf, Launhardt, & Henning (2003) have studied the relation
between the magnetic field and outflow directions. Figure 1.27 displays the direction of the magnetic
field obtained with JCMT-SCUBA polarization observation at 850µm as well as the outflow foundby 13 CO observation of Chandler & Sargent (1993) and 12 CO observation by Yun & Clemens (1994).
In B335, CB230, and CB244, outflows are oriented in the direction perpendicular to the major axisof the globules. The magnetic field is running parallel to the outflow but perpendicular to the majoraxis. This means that a disk is formed with a gas flowing along the magnetic field. Further, theoutflow is generated by a twisted magnetic field which is an outcome of a rotating disk. In globule
CB54, the alignment is not perfect, that is, magnetic field direction is slightly aligned with the outflow(Wolf, Launhardt, & Henning 2003). This seems to strengthen the magnetic origin of the outflow
and magnetically guided disk formation.
It is clearly shown that the polarization anti-correlates with the intensity of the thermal emission.
This might be due to the observational error in the low intensity region (or low S/N region). Thismay be related to physical processes such as (1) the alignment owing to the magnetic field becomesinefficient in high density region or (2) the magnetic field is tangled in the dense region and thepolarization due to the aligned dust is canceled.

1.8.4

Magnetic Field ∼1000AU Scale

Single dish observation with JCMT can reveal a structure larger than 5000–10000AU (Fig. 1.27).
Smaller-scale configuration of magnetic field is only able to be reached with interferometric observationsuch as SMA. Figure 1.28 indicates a 1000 AU-scale configuration of magnetic field for a low-massstar forming region NGC 1333 IRAS 4A (Girart, Rao, & Marrone 2006). Although this is a complex
region, a core extends from the upper-left to the lower-right. The magnetic field [Fig. 1.28 (right)]
runs from the upper-right to the lower-left and this clearly shows “hour-glass” shape. This meansthat the magnetic field is squeezed by the effect of contraction in the star formation process.
They fitted the magnetic field line by a polynomial as
y = g(1 + Cx2 ),(1.19)

where x and y, distances along the symmetric axis and along the major axis of the disk, respectively.
They obtained C = 0.12 ± 0.06 sec−1 .
It should be noted that the polarized intensity (color in the middle panel) is weak in the major
axis direction of specific intensity I distribution (contour lines in the same panel). This is understoodby the depolarization effect (Kataoka et al. 2012 and Addendum).
SMA observation for G31.41+0.31 (Fig. 1.29) also reveals that the magnetic field configuration of
10000AU scale for high-mass star forming region is “hour-glass” shape (Girart et al. 2009). This may
indicate the magnetic field is dragged by the infall motion. However, this can not be fitted with theconfiguration of the magnetic field line obtained with single dish observations, since Fourier component
with the small wave-number k is not sufficiently picked up in the interferometric observation. In other1.8. MAGNETIC FIELDS

Figure 1.27: Directions of magnetic field are shown from the linear polarization observation of 850 µm
thermal emission from dusts by JCMT-SCUBA. These four objects are known as Bok
globules. The bars with γ represent the average directions of polarization (E-vector),
which are perpendicular to the magnetic field. This is taken from Wolf, Launhardt, &Henning (2003).

Figure 1.28: SMA observation low-mass star formation region (NGC1333 IRAS 4A). (Middle:) The
total intensity of 879 µm thermal dust emission (contour lines), that of polarized intensity(false color) and polarization E-vector (red bars) are shown. (Right:) The polarization
E-vector of the dust thermal emissions is perpendicular to the direction of magnetic field(Fig. 1.23). The direction of magnetic field (red bars) is overlaid on the total intensity
distribution (false color). 1′′ = 300AU. This clearly shows that the magnetic field looks
like a hour-glass within ∼ 1000 AU scale. This is taken from Girart, Rao, & Marrone(2006).

25

26

CHAPTER 1. INTRODUCTION

Figure 1.29: SMA observation high-mass (G31.41+0.31) star formation region. (Left:) The total
thermal dust emission at 879 µm is shown with contour lines, while the intensity ofpolarized intensity is shown with false-color. 1′′ = 7000AU. Directions of magnetic field
are illustrated with solid thick bars. (Middle:) Directions of magnetic field is overlaid
on the flux-weighted velocity map of the CH3 OH 147 -156 line, which indicates the core
has a systematic velocity gradient. Taken from Girart et al. (2009).

words, large-scale (more or less uniform) magnetic field is underestimated in the interferometricobservations.
T Tauri Disks
Magnetic field at the position of protostars and T Tauri stars are measured for IRAS 16293-2422,
L1551 IRS5, NGC1333 IRAS 4A, and HL Tau (Tamura et al. 1995). Although HL Tau is a T Tauri
star, it has a gas disk. Thus this is a Class I source. The others are believed to be in protostellar
phase (Class 0 sources). It is known that IRAS 16293-2422, L1551 IRS5, and HL Tau have diskswith the radii of 1500-4000 AU from radio observations of molecular lines. Further, near infraredobservations have shown that these objects have 300-1000 AU dust disks. Figure 1.30 shows the
E-vector of polarized light. If this is the dust thermal radiation, the direction of magnetic fields isperpendicular to the polarization E-vector. Figure shows the magnetic fields run almost perpendicular
to the elongation of the gas disk. Global directions of magnetic field outside the gas disk and the
direction of CO outflows are also shown in the figure by arrows. It is noteworthy that the directionsof local magnetic fields, global magnetic fields, and outflows coincide with each other within ∼ 30
deg.

1.8. MAGNETIC FIELDS

Figure 1.30: Polarization of the radio continuum λ = 1mm, λ = 0.8mm. IRAS 16293-2422 (upper-left:
Class 0), L1551 IRS5 (upper-right: Class 0), NGC1333 IRAS 4A (lower-left: Class 0),
and HL Tau (lower-right: Class I). Taken from Tamura et al. (1995).

27

28

1.9

CHAPTER 1. INTRODUCTION

Density Distribution

Motte & André (2001) made λ =1.3 mm continuum mapping survey of the embedded young stellar
objects (YSOs) in the Taurus molecular cloud. Their maps include several isolated Bok globules,
as well as protostellar objects in the Perseus cluster. For the protostellar envelopes mapped in
Taurus, the results are roughly consistent with the predictions of the self-similar inside-out collapse
model of Shu and collaborators (§4.4.1). The envelopes observed in Bok globules are also qualitatively
consistent with these predictions, providing the effects of magnetic pressure are included in the model.
By contrast, the envelopes of Class 0 protostars in Perseus have finite radii ≲ 10000 AU and are afactor of 3 to 12 denser than is predicted by the standard model.
Another method to measure the density distribution is to use the near IR extinction. From
(H − K) colors of background stars, the local value of AV in a dark cloud can be obtained using astandard reddening law,
AV = 15.87E(H − K)
(1.20)
if the intrinsic colors of background stars are known. Here, the color excess is defined as the difference
between the observed color and the intrinsic color: E(H − K) ≡ (H − K)obs − (H − K)intrinsic . We
can convert the extinction to the column density assuming the gas/dust ratio is constant


AV
21
−2
.
(1.21)
N (H + H2 ) = 2 × 10 cm
1 mag
This is a standard method to obtain the local column density of the dark cloud using the near IR
photometry.
See Figure 1.31. Assume the density distribution is expressed as −αrρ(r) = ρ0,(1.22)
r0
where r is a physical distance from the center. The column density distribution against the projecteddistance p of the line-of-sight from the center of the cloud is given asNρ (p) = 2

Z (R2 −p2 )1/2 hiρ (s2 + p2 )1/2 ds,(1.23)

0where R represents the outer radius of the cloud. Using equation (1.21), this yields AV distribution
AV (p) = 10−23

ρ0 r0αZ (R2 −p2 )1/2

(s2 + p2 )−α/2 ds.

(1.24)

0If background stars are uniformly distributed, the number of stars with AV |obs is proportional to thearea which satisfies AV |obs = AV (p). That is, if we plot AV (p) against 2πpdp, this gives the number
distribution of background stars with AV . Figure 1.33 shows the result of L977 dark cloud by Alveset al. (1998).
On the other hand, Alves et al. (2001) derived directly the radial distribution of NH by comparing
the NH (p) model distribution for B68 (see also Kandori et al. 2009). They obtained a distribution iswell fitted by the Bonnor-Ebert sphere (Ebert 1955; Bonnor 1956), in which the hydrostatic balance
between the self-gravity and the pressure force is achieved (lower panels of Fig. 1.33) (see section
4.1.1).

1.9. DENSITY DISTRIBUTION

Figure 1.31: Schematic view to explain an AV distribution.

Figure 1.32: (Left:) Radial intensity profiles of the environment of 7 embedded YSOs (a-g) and 1starless core (h). (Right:) Same as left panel but for 4 isolated globules (a-d ) and 4
Perseus protostars (e-h). Taken from Motte & André (2001).

29

30

CHAPTER 1. INTRODUCTION

Figure 1.33: Density distribution of L977 (top) and B68 (bottom) dark clouds. (Top-left:) L977
dark cloud dust extinction map derived from the infrared (H-K) observations. Taken
from Alves et al. (1998). (Top-right:) Observed frequency distribution of extinction
measurements for L977 and the predictions from clouds models with density structures
ρ(r) ∝ r−α having α = 1 (dashed line), 2 (solid line), 3 (dotted line), and 4 (dash-dottedline). (Bottom-left:) B68 images (false color images made from B, V, and I images(upper ), and B, I, and K images (lower )). (Bottom-right:) Spatial distribution of thecolumn density. Taken from Alves et al. (2001).

1.10. FILAMENTS

31

Figure 1.34: Column density map for Aquila molecular cloud obtained from the thermal emission
from dusts using Herschel satellite. The image shows that the molecular cloud is madefrom many filaments whose typical width is approximately equal to ≃ 10000 AU. Class 0
protostars (green stars) and bound prestellar cores (blue triangle) are distributed along
the filament with high column density. Taken from André et al. (2010).

1.10

Filaments

In this field, we should pay attention to the density distribution in cylindrical clouds. As seen in the
Taurus molecular cloud, there are many filamentary structures in a molecular cloud. Gould belt survey
by Herschel satellite has found filamentary structure in observed molecular clouds: Polaris cloud,Aquila cloud, IC 5146, Vela C, and the Rosette cloud. André et al. (2010) found a positive correlation
between the line-mass of the filament (mass per unit length) and star forming activity assessed by
the number density of protostars. Figure 1.34 shows that a typical star forming molecular cloud(Aquila molecular cloud) is composed of many filaments. If the observed filaments have typical widthD ≃ 104 AU, typical column density of 1022 H2 cm−2 corresponds to 15M⊙ /pc (NH2 /2 × 1022 cm−2 ) ×(D/104 pc) ≃ 2cs2 /G. The line mass of 2c2s /G is the maximum line mass which can be supported by
the isothermal gas pressure against the self-gravity, which is often called the critical line mass (see§4.1). Since signature of star formation is found in the part with Σ ≳ 2 × 1022 H2 cm−2 , Figure 1.34
seems to indicate that the portion whose line mass is larger than this maximum line mass begins
contraction and star formation.
In §4.1, we will give the density distribution for a hydrostatic spherically symmetric and that ofa cylindrical cloud. The former is proportional to ρ ∝ r−2 and the latter is ρ ∝ r−4 . Therefore,
the distribution ρ ∝ r−4 was expected for cylindrical cloud. From near IR extinctions observation(Alves et al. 1998), even if a cloud is rather elongated [Fig. 1.33 (top-left)], the power of the density
distribution is equal to not −4 but ≃ −2. Using Herschel observation, Arzoumanian et al. (2010)
obtained the power which prefers ∝ r−2 rather than ∝ r−4 for filaments in IC 5146. They found the32

CHAPTER 1. INTRODUCTION

Figure 1.35: Average radial distribution of density observed in the filaments of IC 5146 (black solidline). They fitted observed Σ(r) by Plummer-like function of Eq. (1.26). Fitting is muchbetter for p ∼ 2 (red dotted line) rather than p ∼ 4 (blue dashed line). Taken from
Fig. 4 (c) of Arzoumanian et al. (2010).

radial distribution of column density Σ(r) is fitted by the Plummer-like functionρcρ(r) = h
ip/2
1 + (r/Rf )2
ρc R fΣ(r) = A h

1 + (r/Rf )2i(p−1)/2(1.25)(1.26)and p ∼ 2 and Rf ∼ 0.1 pc (see Fig. 1.35). Fiege, & Pudritz (2001) proposed an idea that a toroidal
component of the magnetic field, Bϕ , plays an important role in the hydrostatic balance of thecylindrical cloud.
As already mentioned in §1.8 (Fig.1.24), major filaments are extending perpendicular to theinterstellar magnetic field (Sugitani et al. 2011; Palmeirim et al. 2013; Planck Collaboration 2016),
although less dense filaments called “striation” are often extending parallel to the interstellar magnetic
field.

Problem 2
Consider a cylindrical symmetric density distribution given by equation (1.25). Integrate thisand obtain the column density distribution of equation (1.26).


1.10.1Mass Accretion onto Filaments

To consider the formation mechanism, it is important to see how the mass accretes onto the filament.
In a well studied filamentary cloud, Taurus molecular cloud, Shimajiri et al. (2019) have found massaccretion flow to the B211/213 filament.
The filament is extending from the west-northwest direction to the east-southeast direction.
Northern side gas exterior to the filament has positive line-of-sight velocity, while southern side indi-1.10. FILAMENTS

33

Figure 1.36: (left and middle:) Relationship between Hi supershell and Taurus filamentary cloudB211/213 (taken from Fig. A6 of Shimajiri et al. (2019)). 857 GHz dust emission (greyin the left and middle panels) represents the distribution of dense Hi gas. Hα emissions(color in the middle panel) distributes centered on the Per OB2 association. B211/213
filament is placed on the front side southern hemisphere of the Hi supershell. (right:)
Schematic view of accretion flow on to B211/213 filament (their Fig. 9). Red-shifted
accretion flow distributes in the northern region relative to the filament. In contrast,
blue-shifted flow is seen in the southern region

cates negative velocity. This is systematically observed for several pc along the filament of B211/213.
As shown in Figure 1.36(left and middle), B211/213 filament is overlapped with Per Hi supershell.
The supershell is seen as a hole of 857 GHz dust emission (left), which indicates low-density gas fills
a spherical large volume which seems to originate from stellar wind or supernovae occurred in Per
OB2 association. The supershell sweeps the gas and forms a high-density shell, and on the shell massaccretion flow is realized as shown in the right panel.

1.10.2

Edge Eﬀect of Cylindrical Object

Filaments and cylindrical dense clouds often exhibit indication of star formation at the end of theobjects. The most well known object is M16, in which three spectacular pillars (cylindrical darkclouds) are standing and there is a young cluster on top of one of the pillars (Hester et al. 1996).
There are many examples similar to M16, in which star formation preferentially occurs at the tipof the filament. Theoretical explanation of this phenomenon has been studied by many researches(Bastien 1983, Hanawa, Yamamoto, & Hirahara 1994, Pon et al. 2011, 2012). Filament shrink timescale is the fastest at the edge of the filament (see Appendix J). Thus, the density increases near theedge first. Whether the density enhancement moves to the filament center or not is dependent on theratio of the gravitational to thermal energies (Bastien 1983).

34

CHAPTER 1. INTRODUCTION

Figure 1.37: (Left:) dN/dM for ρ Oph prestellar cores measured by λ = 1.3 mm continuum using
IRAM 30 m telescope and MPIfR bolometer array. (Right:) Cumulative mass distribution of the 70 pre-stellar condensations of NGC 2068/2071. The dotted and dashed lines
are power-laws corresponding to the mass spectrum of CO clumps (Kramer et al. 1996)
and to the IMF of Salpeter (1955), respectively. Taken from Fig. 3 of Motte et al. (2001).

1.11

Mass Spectrum

We have seen that a molecular cloud consists in many molecular cloud cores. For many years, there
are attempts to determine the mass spectrum of the cores.
From a radio molecular line survey, a mass of each cloud core is determined. Plotting a histogramof number of cores against the mass, we have found that a mass spectrum can be fitted by a powerlaw asdN= Mn(1.27)
dM
where dN/dM represents the number of cores per unit mass interval. Many observations indicatethat n ∼ −1.5 as Table 1.1.
Figure 1.37 (left) (André, Ward-Thomson, & Barsony 2000) shows a mass spectrum function
dN/dM for 59 ρ Oph cloud prestellar cores obtained at the IRAM 30-m telescope with the MPIfR
19-channel bolometer array. Prestellar cores with a mass ∼ M⊙ has a spectrum of dN/dM ∝ M −2.5
(n = −2.5) for M ≳ 0.5M⊙ . While the right panel (Motte et al. 2001) shows the cumulative mass
spectrum (N (> M ) vs. M ) of the 70 starless condensations identified in NGC 2068/2071. The mass
spectrum for the 30 condensations of the NGC 2068 sub-region is very similar in shape. The best-fit
power-law is N (> M ) ∝ M n+1 ∝ M −1.1 above M ≳ 0.8M⊙ . That is, n = −2.1. This power derived
from the dust thermal emission is different from that derived by the radio molecular emission lines.
Table 1.1 summarizes the observations to calculate the power index of core mass function.
The mass function of newborn stars is called as initial mass function (IMF). IMF for field stars inthe solar neighborhood has been obtained as shown in Figure 1.39. The most famous one is Salpeter’s
IMF as dN∗ /dM∗ ∝ M∗−2.35 (Salpeter 1955). The low-mass end is flatter than that of M∗ ≳ M⊙ as
dN∗ /dM∗ ∝ M∗−1.2 for 0.1M⊙ ≲ M∗ ≲ 1M⊙ while dN∗ /dM∗ ∝ M∗−2.7 for 1M⊙ ≲ M∗ ≲ 10M⊙ (Meyer
et al. 2000). The powers of stellar (−2.7 ∼ −2.35) and prestellar (−2.5 ∼ −2.1) mass functions are
similar. If one prestellar core forms one star, the stellar mass M∗ is proportional to the prestellar
core mass as M∗ = f Mcore and f ≲ 1, the mass spectrum of prestellar cores completely determines
the IMF.

1.11. MASS SPECTRUM

35

Table 1.1: Mass spectrum indices derived from surveys using various molecular lines and dust continuum emission.

Paper
Loren (1989)
Stutzki &Guesten (1990)

n−1.1−1.7 ± 0.15

Regionρ Oph
M17 SW

Lada et al.
(1991)
Nozawa et al.
(1991)
Tatematsu et al.
(1993)
Dobashi et al.
(1996)
Onish et al.
(1996)
Motte, André, &Neri (1998)

−1.6

L1630

C18 O
(J = 2 − 1),
S34 S(J = 2 − 1,
J = 3 − 2)
CS (J = 2 − 1)

−1.7

ρ Oph North

13 CO−1.6 ± 0.3

Orion A(J = 1 − 0)
CS (J = 1 − 0)

M ≳ 50M⊙−1.6

Cygnus13 CO

M ≳ 100M⊙−0.9 ± 0.2

Taurus−1.5ρOph(J = 1 − 0)
C18 O
(J = 1 − 0)
λ = 1.3mm

−2.5
−2.1

Sarpens

λ = 3mmM ≳ 0.5M⊙
1M⊙ ≲ M ≲ 30M⊙

−1.6 ∼ −1.8

L1457 etc∗

12 CO, 13 CO,

10−4 M⊙ ≲ M ≲ 104 M⊙

Testi & Sargent(1998)
Kramer et al.
(1998)

Heithausen et al.
(1998)−1.84

Johnstone et al.
(2000)−2 ∼ −2.5

MCLD 123.5+ 24.9, Polaris
Flare
ρOph

−1.5
−2.5 ∼ −3

Johnstone et al.
(2001)
Onishi et al.
(2002)
Ikeda, Sunada, &Kitamura (2007)
Ikeda &
Kitamura (2009)

Observation

C18 O(J = 1 − 0,
2 − 1)
12 CO(J = 1−0
and 2 − 1)

Mass range
10M⊙ ≲ M ≲ 300M⊙
a few M⊙ ≲ M ≲ a few 103 M⊙

M ≳ 20M⊙
3M⊙ ≲ M ≲ 160M⊙

3M⊙ ≲ M ≲ 80M⊙
M ≲ 0.5M⊙

MJ ≲ M ≲ 10M⊙

λ = 850µm

M ≳ 0.6M⊙

Ori B

λ = 850µm

M ≲ 0.6M⊙
M ≳ M⊙−2.5

Taurus−2.3 ± 0.1

Ori A

H13 CO+(J = 1 − 0)
H13 CO+(J = 1 − 0)−0.4 ± 0.1−2.3 ± 0.3

Ori A

C18 O
(J = 1 − 0)

3.5M⊙ ≲ M ≲ 20.1M⊙
M ≳ 9M⊙M ≲ 9M⊙M ≳ 5M⊙∗ MCLD126.6+24.5, NGC 1499 SW, Orion B South, S140, M17 SW, and NGC 7538

36

CHAPTER 1. INTRODUCTION

Figure 1.38 plots the power-law indices against the typical gas densities of respective observations,in which the critical density is taken as a typical density for molecular line studies. The powerlaw index is an increasing function of the typical gas density ngas (Ikeda 2007) and the cores withngas ≳ 105 cm−3 have the same power-law index as the IMF. This may indicates these cores are thesites of star formation or direct parents of new-born stars4 .

1.12Line Width - Size Relation

Larson (1981) compiled the observations for molecular cloud complexes, molecular cloud and molecular clumps published in 1974-1979. He obtained an empirical relation that the size of a structure iswell correlated to the random velocity in the structure which is measured by the width of the emission
line (see Appendix G). Figure 1.40 (left) shows this correlation and this is well expressed as
L 0.38−1,(1.28)σ ≃ 1.10 km s
1 pc
where σ and L represent respectively the three-dimensional random speed of gas and the size of thestructure. A similar correlation is found only for giant molecular clouds (Sanders, Scoville, & Solomon1985) as

1/2


3
L 0.62−1σ=∆VFWHM = (0.535 ± 0.16) km s,
(1.29)
23 ln 2
1 pc
for GMCs whose sizes are larger than 10 pc (be careful the typos in their abstract: their power was−0.62). Larson (1981) also found another correlation between the mass M and the random velocity
like Figure 1.40 (right), which is expressed as
M 0.20−1
σ ≃ 0.43 km s.
(1.30)
1M⊙
These correlations are often called Larson’s law.
In the next chapter (§2.9), we will see the virial relation, that is, for an isolated system to achievea mechanical equilibrium the gravitational to thermal energy ratio has to be equal to 2 : 1 for γ = 5/3
gas. The ratio of the gravitational energy ∼ (3/5)GM 2 /(L/2) to the thermal energy M σ 2 /2 is also
fitted as
12 GM
L 0.14≃ 1.1,(1.31)
5 σ2L
1 pc
which is weakly dependent of the size or the mass. This seems to mean the ratio is nearly constantirrespective of the mass or size of the clouds.
Since there is a mutual relation between mass, size, and the velocity dispersion to achieve amechanical equilibrium (the Virial relation), there is only one independent correlation in the above
two correlations [eqs. (1.28) and (1.30)]. Although several reasons to explain the correlation areproposed, we have no consensus yet.

4
However, the same authors (Ikeda and Kitamura 2009) obtained n ≃ −2.3 ± 0.3 even for C18 O cores (its criticaldensity ∼ 103 cm−3 ) in OMC-1 region in the Orion A cloud. Thus, the relationship between the core mass function and
IMF has not been cleared yet.

1.12. LINE WIDTH - SIZE RELATION

Figure 1.38: The power-law index of n = γ of core mass function dN (M )/dM ∝ M −n . n is plottedagainst typical densities of observed molecular cores. In molecular line observations, thetypical densities are assumed equal to the critical densities of the transitions of respective
molecular line observations. For mm and sub-mm dust emission, the density is assumed∼ 106 cm−3 . Taken from Fig. 8.1 of PhD Thesis of Ikeda (2007).

37

38

CHAPTER 1. INTRODUCTION

Figure 1.39: Comparison of initial mass functions for field stars in the solar neighborhood. Respective
symbols represent S55: Salpeter (1955), MS: Miller & Scalo (1979), Scalo 86: Scalo (1986)
and KTG93: Kroupa, Tout, & Gilmore 1993. Taken from Fig. 1 of Meyer et al. (2000).

Figure 1.40: The left shows the relation between cloud size (horizontal axis) and the three-dimensional
internal velocity (vertical axis). The right shows a similar correlation between mass
(horizontal axis) and the random velocity.

Chapter 2

Physical Background2.1

Basic Equations of Hydrodynamics

The basic equation of hydrodynamics are (1) the continuity equation of the density [eq. (B.11)],∂ρ+ div(ρv) = 0,∂t(2) the equation of motion [eq. (B.7)]∂v
ρ
+ (v · ∇) v = −∇p + ρg,
∂t

(2.1)

(2.2)

and (3) the equation of energy [eq. (B.23)]∂ϵ+ div(ϵ + p)v = ρv · g,
∂t

(2.3)

where ϵ = (1/2)ρ|v|2 + ε and ε represents the thermal energy per volume. Occasionally barotropic
relation p = P (ρ) substitutes the energy equation (2.3). Especially polytropic relation p = KρΓ isoften used on behalf of the energy equation. In the case that the gas is isothermal or isentropic, thepolytropic relations of
p = c2is ρ (isothermal)
(2.4)
and
p = Kργ (isentropic)(2.5)

are substitution to equation (2.3). [We can replace equation (2.3) with equations (2.4) and (2.5).]

2.2

The Poisson Equation of the Self-Gravity

In this section, we will show the basic equation describing how the gravity works. First, compare
the gravity and the static electric force. Consider the electric field formed by a point charge Q at a
distance r from the point source as
1 Q
E=
,
(2.6)
4πϵ0 r2
where ϵ0 is the electric permittivity of the vacuum. On the other hand, the gravitational accelerationby the point mass of M at the distance r from the point mass is written down asM
g = −G 2 ,
r
39

(2.7)

40

CHAPTER 2. PHYSICAL BACKGROUND

where G = 6.67 × 10−8 dyn cm2 g−2 is the gravitational constant. Comparing these two, replacing Q
with M and at the same time 1/4πϵ0 to −G these equations (2.6) and (2.7) are identical with eachother.
The Gauss theorem for electrostatic field asρe
div E = ,(2.8)
ϵ0
and another expression using the electrostatic potential ϕe as
ρe
,
ϵ0

(2.9)

div g = −4πGρ,(2.10)∇2 ϕ = 4πGρ,(2.11)∇2 ϕ e = −lead to the equations for the gravity as

andwhere ρe and ρ represent the electric charge density and the mass density. Equation (2.11) is called
the Poisson equation for the gravitational potential and describes how the potential ϕ is determinedfrom the mass density distribution ρ, after the boundary condition for ϕ is given.
Problem 3
Consider a spherically symmetric density distribution. Using the Poisson equation, obtain thepotential (ϕ) and the gravitational acceleration (g) for a density distribution shown below.
= ρ0 for r < Rρ= 0 for r ≥ R
Hint: The Poisson equation (2.11) for the spherically symmetric system is


1 ∂
2 ∂ϕ
r
= 4πGρ.
r2 ∂r
∂r


2.3



Free-fall TimeEven if the pressure force can be neglected in the equation of motion (B.1), the gravitational force
remains. Assuming the spherical symmetry, consider the gravity gr (r) at the position of radial
distance from the center being equal to r. Using the Gauss’ theorem, gr is related to the mass
inside of r, which is expressed by the equation
Z r
ρ4πr2 dr,(2.12)
Mr =0

and gr is written asGMr.
(2.13)
r2
This leads to the equation motion for a cold gas under a control of the self-gravity is written
gr (r) = −

d2 r
GMr
=− 2 .
dt2
r(2.14)

2.3. FREE-FALL TIME

41

Analyzing the equation is straightforward, multiplying dr/dt gives
dv 2 /2
d GMr
=+
,
dt
dt
r

(2.15)

which leads to the conservation of mechanical energy as 
1 dr 2 GMr
−
= E,
2 dt
r

(2.16)

in which E represents the total energy of the pressureless gas element and it is obtained from theinitial condition. If the gas is static initially at the distance R, the total energy is negative asE=−

GMr (R),R(2.17)

because at t = 0, r = R and dr/dt = 0.
The solutions of equation (2.16) are well known as follows:
1. the case of negative energy E < 0. Considering the case that the gas sphere is inflowing v < 0,equation (2.16) becomes


1
1 1/2
dr
= − [2GMr (R)]1/2−,(2.18)
dt
r R
where we assumed initially dr/dt = 0 at r = R. Using a parameter η(t), the radius of the gas
element at some epoch t is writtenr = R cos2 η.
(2.19)In this case, equation (2.18) reduces to
dη
cos η
=
dt



2

This gives us the expression of t ast=R3
2GMr (R)

GMr (R)
2R3

1/2 

1/2.

sin 2η
η+2(2.20).

This corresponds to the closed universe in the cosmic expansion (Ω0 > 1).
2. if the energy is equal to zero, the solution of equation (2.16) is written as
2/3  9GM (R) 1/3

r
3/2
3/2
r −R
=
t2/3 ,
2(2.21)(2.22)

where R = r(t = 0). This corresponds to the Einstein - de Sitter Universe with Ω0 = 1.
3. If E > 0, the expansion law is given by!−1  2 1/2
sinh 2ηṘ2 RR−η(2.23)t=
2E
2GMr (R)
2


and
x=

E
GMr (R)r = sinh2 η,(2.24)

where E represents the total energy
E=Ṙ2 GMr (R)
−
> 0.
2
R

(2.25)

42

CHAPTER 2. PHYSICAL BACKGROUND
1
0.8
0.6
0.4
0.2
0.25

0.5

0.75

1

1.25

1.5

Figure 2.1: Free-fall. The x- and y-axes represent η + sin 2η/2 and cos2 η, respectively.
Problem 4Solve equation (2.16) for E = 0 and obtain (2.22).
Problem 5Solve equation (2.16) for E > 0 and obtain equations (2.23) and (2.24).


In the present case, at t = 0, since dr/dt = 0 the energy is negative. Equation (2.19) shows us
r becomes equal to zero (the gas collapses) if η = π/2 as well as η = 0 at t = 0. Equation (2.21)
indicates it occurs at the epoch oft = tﬀ

1/2
R3
π
=
,
2GMr (R)2
3π 1/2=,32Gρ̄(2.26)

where ρ̄ represents the average density inside of r, that is Mr /(4πr3 /3). This is called “free-falltime” of the gas. This gives the time-scale for the gas with density ρ̄ to collapse. In the actualinterstellar space, the gas pressure is not negligible. However, tﬀ gives a typical time-scale for a gas
cloud to collapse and to form stars in it.

2.3.1

Accretion Rate

Equation (2.26) indicates that the gas shell with a large ρ̄ reaches the center earlier than that witha small ρ̄. Immediately, this means a spherical cloud with a uniform density ρ0 contracts uniformly
and all the mass reaches the center at t = tﬀ = (3π/32Gρ0 )1/2 . In this case, the mass accretion
rate to a central source becomes infinity at an epoch t = tﬀ . In contrast, consider a cloud whose
density gradually decreases outwardly. In this case, the outer mass shell has smaller ρ̄ than the innermass shell. Therefore even when the inner mass shell collapses and reaches the center, the outermass shell is still contracting and does not reach the center. This gives a smaller mass accretion rate
than a uniform cloud. If the gas pressure is neglected, the accretion rate is determined by the initial
spatial distribution of the density. We will compare the accretion rate derived here with results ofhydrodynamical calculation in §4.6 (p.112).

2.4. GRAVITATIONAL INSTABILITY

2.4

43

Gravitational Instability

Here, we will study a typical size where the self-gravity plays an important role and forms density
inhomogeneities with the Jean wavelength.

2.4.1

Linear Analysis

Consider a uniform gas with density ρ0 and pressure p0 without motion u0 = 0. In this uniform gas
distribution, we assume small perturbations. As a result, the distributions of the density, the pressure
and the velocity are perturbed from the uniform ones asρ = ρ0 + δρ,(2.27)

p = p0 + δp,(2.28)

u = u0 + δu = δu,(2.29)

andwhere the amplitudes of perturbations are assumed much small, that is, |δρ|/ρ0 ≪ 1, |δp|/p0 ≪ 1
and |δu|/cs ≪ 1. We assume the variables changes only in the x-direction. In this case the basic
equations for isothermal gas are∂ρ ∂ρu+= 0,(2.30)
∂t
∂x


∂u
∂u
∂p
ρ
+u
=−
+ ρgx ,
(2.31)∂t∂x∂xandp = c2is ρ,(2.32)

where u and gx represent the x-component of the velocity and that of the gravity, respectively.
Using equations (2.27), (2.28), and (2.29), equation (2.30) becomes∂∂(ρ0 + δρ) +[(ρ0 + δρ)(u0 + δu)] = 0.
∂t∂x(2.33)

Noticing that the amplitudes of variables with and without δ are completely different, two equations
are obtained from equation (2.33) as∂ρ0 ∂ρ0 u0
+
∂t
∂x

= 0,

(2.34)∂δρ∂+(ρ0 δu + δρu0 ) = 0,∂t∂x(2.35)

where the upper is the equation for unperturbed state and the lower describes the relation between thequantities with δ. Equation (2.34) is automatically satisfied by the assumption of uniform distribution.
In equation (2.35) the last term is equal to zero. Equation of motion


∂
∂
∂
∂
(ρ0 + δρ)(u0 + δu) + (u0 + δu) (u0 + δu) = − (p0 + δp) + (ρ0 + δρ) (ϕ0 + δϕ), (2.36)
∂t
∂x
∂x
∂x
gives the relationship between the terms containing only one variable with δ as follows:
ρ0

∂δu
∂δp
∂δϕ
=−
− ρ0
.
∂t
∂x
∂x

(2.37)

44

CHAPTER 2. PHYSICAL BACKGROUND

The perturbations of pressure and density are related with each other as follows: for the isothermal
gas
 
δp
∂p
p0
=
=
= c2is ,(2.38)
δρ∂ρ T
ρ0and for isentropic gas
δp
=
δρ

2.4.2



∂p
∂ρ


=γ
ad

p0
= c2s .
ρ0(2.39)

Sound Wave

If the self-gravity is ignorable, equations (2.35) and (2.37) reduce respectively to
∂δρ
∂δu
+ ρ0
= 0,
∂t
∂x

(2.40)

and

∂δu
∂δρ
= −c2is,(2.41)∂t∂x
where we assumed the gas is isothermal. These two equations describe the propagation and growthof perturbations. If the gas acts adiabatically, replace cis with cs .
Making ∂/∂x×(2.40) and ∂/∂t×(2.41) vanishes δρ and we obtainρ02∂ 2 δu2 ∂ δu
−
c
= 0.
is
∂t2
∂x2

(2.42)

Since this leads to
∂δu
∂δu
− cis
∂t
∂x
∂δu
∂δu
+ cis
∂t
∂x

= 0,

(2.43)= 0,(2.44)

equation (2.42) has a solution that a wave propagates with a phase velocity of ±cs (sound speed).
Since the displacement (∝ δu) is parallel to the propagation direction x, and the restoring force comesfrom the pressure, this seems the sound wave.
Problem 6Interstellar gas contains mainly hydrogen and helium, whose number ratio is approximately 10 : 1.
Obtain the value of average molecular weight for the fully ionized interstellar gas with temperature
T = 104 K (components are ionized H+ (Hii) and He+2 (Heiii) and electron e− ). How about themolecular gas (T = 10 K) containing molecular H2 , neutral He (Hei) and no electron?


2.5



Jeans Instability

Sound wave seems to be modified in the medium where the self-gravity is important. Beside thecontinuity equation (2.35)
∂δρ
∂δu
+ ρ0
= 0,
(2.45)∂t∂x
and the equation of motion (2.37)
ρ0

∂δρ
∂δϕ
∂δu
= −c2is
− ρ0
,
∂t
∂x
∂x

(2.46)

2.5. JEANS INSTABILITY

45

Figure 2.2: Dispersion relations.

the linearized Poisson equation∂ 2 δϕ= 4πGδρ,∂x2

should be included. ∂/∂x × (2.46) gives
 2 
∂ δu
∂ 2 δρ
ρ0= −c2is− 4πGρ0 δρ.
∂x∂t
∂x2

(2.47)(2.48)

where we used equation (2.47) to eliminate δϕ. This yields
2∂ 2 δρ
2 ∂ δρ
=c+ 4πGρ0 δρ.
is
∂t2
∂x2

(2.49)

where we used ∂/∂t × (2.45).
This is the equation which characterizes the growth of density perturbation owing to the selfgravity. Here we consider the perturbation are expressed by the linear combination of plane wavesasX
δρ(x, t) =A(ω, k) exp(iωt − ikx),(2.50)
where k and ω represent the wavenumber and the angular frequency of the wave. Picking up a plane
wave and putting into equation (2.49), we obtain the dispersion relation for the gravitational
instability asω 2 = c2is k 2 − 4πGρ0 .
(2.51)
Reducing the density to zero, the equation gives us the same dispersion relation as that of the soundwave as ω/k = cis . For short waves (k ≫ kJ = (4πGρ0 )1/2 /cis ), since ω 2 > 0 the wave is ordinary
oscillatory wave. Increasing the wavelength (decreasing the wavenumber), ω 2 becomes negative fork < kJ = (4πGρ0 )1/2 /cis . For negative ω 2 , ω can be written ω = ±iα using a positive real α. In thiscase, since exp(iωt) = exp(∓αt), the wave which has ω = −iα increases its amplitude exponentially.
This means that even if there are density inhomogeneities only with small amplitudes, they grow ina time scale of 1/α and form density inhomogeneities with large amplitudes.
The critical wavenumber
kJ = (4πGρ0 )1/2 /cis
(2.52)

46

CHAPTER 2. PHYSICAL BACKGROUND

Figure 2.3: Thin disk.

corresponds to the wavelength
2π
=
λJ =
kJ



πc2is
Gρ0

1/2
,

(2.53)

which is called the Jeans wavelength. Ignoring a numerical factor of the order of unity, it is shownthat the Jeans wavelength is approximately equal to the free-fall time scale (eq.[2.26]) times the soundspeed. The short wave with λ ≪ λJ does not suffer from the self-gravity. For such a scale, the analysis
we did in the preceding section is valid.
Typical values in molecular clouds, such as cis = 200 m s−1 , ρ0 = 2 × 10−20 g cm−3 , give us theJeans wavelength as λJ = 1.7 × 1018 cm = 0.56 pc. The mass contained in a sphere with a radiusr = λJ /2 is often called Jeans mass, which gives a typical mass scale above which the gas collapses.
Typical value of the Jeans mass is as follows
4π
MJ ≃
ρ0
3



λJ
2

3

π
=
6



π
Gρ0

3/2
c3is ρ0 .

(2.54)

Using again the above typical values in the molecular clouds, cis = 200 m s−1 , ρ0 = 2 × 10−20 g cm−3 ,the Jeans mass of this gas is equal to MJ ≃ 27 M⊙ .

2.6

Gravitational Instability of Thin Disk

Disks are common in the Universe. Spiral and barred spiral galaxies have disks where stars areformed. In more small scale, gas and dust disks are often found around protostars. Moreover, sucha disk may become a proto-planetary disk. It is valuable to study how the self-gravity works in suchthin structures.RHere, we assume a thin disk extending in x- and y-directions whose surface density∞
is equal to σ = −∞ ρdz, in other word the density is written using the Dirac’s delta function δ(z) as
ρ(x, y, z) = σ(x, y)δ(z).

(2.55)

Integrating the basic equations (2.45) and (2.46) along the z-direction the linearized basic equations
for the thin disk are as follows:
∂δσ
∂δu
+ σ0
= 0,
(2.56)
∂t
∂x
∂δσ
∂δϕ
∂δu
= −c2is− σ0,(2.57)
σ0∂t∂x∂x

2.6. GRAVITATIONAL INSTABILITY OF THIN DISK47

∂ 2 δϕ ∂ 2 δϕ+
= 4πGδσδ(z),(2.58)∂x2∂z 2
where we assumed σ = σ0 + δσ, u = δu, ϕ = ϕ0 + δϕ and took the first order terms (those contain
only one δ).
Outside the disk, the rhs of equation (2.58) is equal to zero. It reduces to the Laplace equation∂ 2 δϕ ∂ 2 δϕ
+
= 0.
∂x2
∂z 2

(2.59)

δX(x, t) = δA exp(iωt − ikx),(2.60)Taking a plane wave of
equation (2.59) is reduced to∂ 2 δϕ− k 2 δϕ = 0.
∂z 2This has a solution which does not diverge at the infinity z = ±∞ asδϕ = δϕ(z = 0) exp(−k|z|).

(2.61)(2.62)On the other hand, integrating equation (2.58) from z = −0 to z = +0 or in other word, applying theGauss’ theorem to the region containing the z = 0 surface, it is shown that the gravity δgz = −∂δϕ/∂z
has a jump crossing the z = 0 surface as
∂δϕ
∂δϕ
−
= 4πGδσ.
∂z z=+0∂z z=−0(2.63)

Equations (2.62) and (2.63) lead a final form of the potential asδϕ = −2πGδσexp(−k|z|).
k(2.64)

Considering δϕ at z = 0, the perturbed potential is written as
δϕ = −

2πG
δσ.
k(2.65)

Putting this to equation (2.57), and using equation (2.56), we obtain the dispersion relation for
the gravitational instability in a thin disk asω 2 = c2is k 2 − 2πGσ0 k.

(2.66)

This reduces to the dispersion relation of the sound wave for the short wave k ≫ 2πGσ0 /c2is . While fora longer wave than λcr = c2is /Gσ0 , an exponential growth of δσ is expected. The dispersion relationis also shown in Figure 2.2.

2.6.1

Rotating Disk

In the case of a rotating disk with an angular rotation speed Ω = (0, 0, Ω), basic equations are

σ0

∂δu
∂δσ
+ σ0
= 0,
∂t
∂x

(2.67)∂δu= −c2is ∇δσ − σ0 ∇δϕ − 2σ0 Ω × δu,∂t(2.68)

48

CHAPTER 2. PHYSICAL BACKGROUND∂ 2 δϕ− k 2 δϕ = 4πGδσδ(z),(2.69)∂z 2
where the third term of r.h.s. of equation (2.68) represents the Colioris force. Equation (2.69) gives
a solution identical to equation (2.64) as
δϕ = −

2πGδσ
exp [i(ωt − kx) − k|z|] .
k(2.70)

Choosing a direction in which the wavenumber vector can be expressed as k = (k, 0, 0), we can reduce
equations (2.67) and (2.68) toiωδσ − ikσ0 δux = 0,(2.71)iωδux = −c2is (−ik)δσ− 2πiGδσ + 2Ωδuy ,σ0iωδuy = −2Ωδux .

(2.72)(2.73)

These three equations together with equation (2.70) bring us a dispersion relation asω 2 = c2is k 2 − 2πGσ0 k + 4Ω2 .

(2.74)

Comparing with equation(2.66), this indicates rotation works to stabilize the system.
Equation (2.74) is rewritten as(πGσ0 )2 4Ω2 c2is+−1=
(πGσ0 )2c2is
πGσ0 2 (πGσ0 )22
2= cis k − 2
+
Q
−
1
,
cis
c2is


ω

2

c2is

πGσ0
k− 2
cis

2

where we defined
Q≡

2Ωcis
.
πGσ0

(2.75)(2.76)(2.77)

We can see that ω 2 > 0 for all wavenumbers if Q > 1 and that if Q < 1 for some range of wavenumberω 2 becomes negative. A rotating disk with Q < 1 is unstable for some range of wavenumber. This
number is called Toomre’s Q parameter. This is useful to see whether a galactic disk is stable ornot. For the galactic disk, Toomre’s Q must be modified as
Q≡

κcis
πGσ0

(2.78)

where κ represents the epicyclic frequency as

κ≡

dΩ2
R
+ 4Ω2
dR

1/2
.

(2.79)

This reduces to κ = 2Ω for rigid body rotation (Ω =const). See section 3.5 for the galactic disk.
Problem 7

Derive the dispersion relation of gravitational instability in a rotating thin disk, equation (2.74).


2.7. CONVECTIVE INSTABILITY

49zρ∗(z+∆z)p*(z+∆z)

∆z

ρ∗(z)
p*(z)

Figure 2.4: Convection.

2.7

Convective Instability

If water is heated from the bottom and temperature difference between the top and the bottom
exceeds a limit, convection is driven. Water heated from the bottom climbs and cool water on
the top descends. This transfers the thermal energy from the bottom to the top. In this section, we
describe the condition in which convection is driven.
Consider a hydrostatic balanced atmosphere in which the hydrostatic balance equation is satisfied:
dp
dϕ
dp
−ρ
= − + ρg = 0,(2.80)dz
dz
dz
where we assumed v = 0 in equation (2.2) and the gravity is working downwards in z-direction(g < 0). The pressure and density of the atmosphere are p(z) and ρ(z). We consider a gas element(hatched region in Fig. 2.4), whose density ρ∗ and pressure p∗ are equal to those of the atmosphereρ(z) and p(z) asρ∗ (z) = ρ(z), p∗ (z) = p(z).
(2.81)−

Further, we assume this gas element to move from z to z + ∆z adiabatically, that is,p∗ (z + ∆z)p∗ (z)=.
[ρ∗ (z + ∆z)]γ[ρ∗ (z)]γ(2.82)

Pressure balance is required between the gas element at z+∆z, p∗ (z+∆z), and that of the atmospherep(z + ∆z). If the density of the gas element at z + ∆z, ρ∗ (z + ∆z), is smaller than that of theatmosphere, the gravity force in equation (2.80) is weaker than the pressure force and the element
keeps climbing. Thus the condition for the convective instability is written asρ∗ (z + ∆z) < ρ(z + ∆z).

(2.83)Usingdpdρ∆z and ρ(z + ∆z) = ρ(z) + ∆z,(2.84)dz
dz
we can rewrite equation (2.83) into the relation in the variables of the atmosphere (variables without*) asd ln pd ln ρ
<γ(2.85)dz
dz
p(z + ∆z) = p(z) +50

CHAPTER 2. PHYSICAL BACKGROUND

Figure 2.5: (Left:) Explanation of Laval nozzle. (Right:) The relation between the cross-section S(x)and the flow velocity vx .

ord ln(p/ργ )<0(2.86)dz
This means the specific entropy s = ln(p/ργ ) + K decreases upwardly. Thus, if we consider the
adiabatic process, the atmosphere in which a speciﬁc entropy decreases upwardly is unstable for theconvection.
Problem 8
Obtain equation (2.85) from equation (2.83) using equation (2.84).


2.8



Super- and Subsonic Flow

Flow whose velocity is faster than the sound speed is called supersonic, while that slower than thesound speed is called subsonic. The subsonic and supersonic flows are completely different.

2.8.1

Flow in the Laval Nozzle

Consider a tube whose cross-section, S(x), changes gradually, which is called Laval nozzle. Assuming
the flow is steady ∂/∂t = 0 and is essentially one-dimensional, the continuity equation (2.1) isrewritten as
ρuS = constant,(2.87)
or

1 ∂ρ
1 ∂u
1 ∂S
+
+
= 0.
ρ ∂x u ∂x S ∂x

(2.88)

1 ∂p
c2 ∂ρ
∂u
=−
=− s
,
∂x
ρ ∂x
ρ ∂x

(2.89)

Equation of motion (2.2) becomesuwhere we used the relationship of
∂p
=
∂x



∂p
∂ρ



∂ρ
∂ρ
= c2s .
∂xs ∂x(2.90)

2.8. SUPER- AND SUBSONIC FLOW

51

When the flow is isothermal, use the isothermal sound speed c2is instead of the adiabatic one. Fromequations (2.88) and (2.89), we obtain 2 1 ∂uu
1 ∂u
1 ∂S= M2 − 1=,(2.91)
−1
2
cs
u ∂x
u ∂x
S ∂x
where the factor M = u/cs is called the Mach number. For supersonic flow M > 1, while M < 1
for subsonic flow.
In the supersonic regime M > 1, the factor in the parenthesis of the lhs of equation (2.91) is
positive. This leads to the fact that the velocity increases (du/dx > 0) as long as the cross-section
increases (dS/dx > 0). On the other hand, in the subsonic regime, the velocity decreases (du/dx < 0)while the cross-section increases (dS/dx > 0). See right panel of Figure 2.5.
If M = 1 at the point of minimum cross-section (throat), two curves for M < 1 and M > 1 havean intersection. In this case, gas can be accelerated through the Laval nozzle. First, a subsonic flowis accelerated to the sonic speed at the throat of the nozzle. After passing the throat, the gas follows
the path of a supersonic flow, where the velocity is accelerated as long as the cross-section increases.

2.8.2

Steady State Flow under an Inﬂuence of External Fields

Consider a flow under a force exerted on the gas whose strength varies spatially. Let g(x) represent
the force working per unit mass. Assuming the cross-section is constantρu = constant,(2.92)

immediately we have
1 ∂u
1 ∂ρ
+
= 0.
ρ ∂x u ∂x
On the other hand, the equation motion is
u

∂u
c2 ∂ρ
=− s
+ g(x),∂xρ ∂x

From equations (2.93) and (2.94), we obtain
 2

u
1 ∂u
g(x)
−1
= 2 .
2
cs
u ∂x
cs

(2.93)(2.94)(2.95)

Consider an external field whose potential is shown in Figure 2.6(left). (1) For subsonic flow, thefactor in the parenthesis is negative. Before the potential minimum, since g(x) > 0, u is decelerated.
On the other hand, after the potential minimum, u is accelerated owing to g(x) < 0. Using equation(2.92), this leads to a density distribution in which density peaks near the potential minimum. (2)For supersonic flow, the factor is positive. In the region of g(x) > 0, u is accelerated. After passing
the potential minimum, u is decelerated. The velocity and the density distribution is shown in Figure
2.6(lower-right).
The density distribution of the subsonic ﬂow in an external potential is similar to that of hydrostatic one. That is, considering the hydrostatic state in an external potential, the gas density peaks at
the potential minimum. On the other hand, The density distribution of the supersonic ﬂow looks likethat made by ballistic particles which are moving freely in the potential. Owing to the conservationof the total energy (kinetic + potential energies), the velocity peaks at the potential minimum. And
the condition of mass conservation leads to the distribution in which the density decreases near the
potential minimum.

52

CHAPTER 2. PHYSICAL BACKGROUND

Figure 2.6: (Left:) External potential field. (Right:) Velocity and density variations. Gas flows in
the external field whose potential is shown in the left panel. The upper panel represents
a subsonic flow. The lower panel does a supersonic flow.

2.8. SUPER- AND SUBSONIC FLOW

2.8.3

53

Stellar Wind — Parker Wind Theory

Stellar winds are observed around various type of stars. Early type (massive) stars have large luminosities; the photon absorbed by the bound-bound transition transfers its outward momentum tothe gas. This line-driven mechanism seems to work around the early type stars. On the other hand,acceleration mechanism of less massive stars are thought to be related to the coronal activity or dustdriven mechanism (dusts absorb the emission and obtain outward momentum from the emission).
Here, we will see the identical mechanism in §2.8.1 and §2.8.2 works to accelerate the wind from astar. Consider a steady state and ignore ∂/∂t = 0. Assume spherical symmetric flow. The continuity
equation (2.1) gives
r2 ρu = const,(2.96)
where u represents the radial velocity component of v and we useddiv (ρv) =1 ∂r2 ρu .
2r ∂r

(2.97)

This leads to

2 1 ∂ρ 1 ∂u++
= 0.
r r ∂r u ∂r
The equation of motion (2.2) is as follows:u(2.98)c2 ∂ρ GM∗
∂u
=− s
− 2 ,
∂r
ρ ∂r
r

(2.99)

where we used g = −GM∗ /r2 (M∗ means the mass of the central star). From these two equations(2.98) and (2.99) we obtain 1 ∂u2 GM∗ 1M2 − 1= − 2 2,(2.100)u ∂rr
cs r
where M ≡ u/cs represents the Mach number of the radial velocity. Take notice that this is similar toequations (2.91) and (2.95). That is, the fact that the rhs of equation (2.100) is positive corresponds
to increasing the cross-section dS/dx > 0. On the contrary, when the rhs is negative, the fluid acts
as the cross-section S is decreasing.
For simplicity, we assume the gas is isothermal. The rhs of equation (2.100) varies shown in
Figure 2.7. Therefore, near to the star, the flow acts as the cross-section of nozzle is decreasing andfar from the star it does as the cross-section is increasing. This is the same situation that the gas
flows through the Laval nozzle.
Using a normalized distance x ≡ r/(GM∗ /2c2is ), equation (2.100) becomesM2 − 1This is rewritten asddx 1 ∂M22= − 2.
M ∂x
x x

M2
2
− log M − 2 log x −2x(2.101)= 0,(2.102)

we obtain the solution of equation (2.101) as
M2 − 2 log M = 4 log x +

4
+ C.
x

(2.103)

54

CHAPTER 2. PHYSICAL BACKGROUND

Figure 2.7: Right-hand side of equation (2.100) is plotted against the distance from the center r.
fHML
2

gHxL
2

1.75

1.75

1.5

1.5

1.25

1.25

1

1

0.75

0.75

0.5

0.5

0.25

0.25
0.5

1

1.5

2

M

0.5

1

1.5

2

x

Figure 2.8: f (M) (left) and g(x) for C = −3 (right)

This gives how the Mach number M varies changing x. To explore this, we define two functions:f (M) = 2M2 − 2 log M(2.104)is a function only depending on M andg(x) = 4 log x +4+Cx(2.105)is a function only depending on x.
Since the minima of f (M) and g(x) are respectively 1 and C + 4, the permitted region in (x, M)changes for the values of C.
1. If C = −3, for all values of x > 0 there exist M which satisfies f (M) = g(x). This corresponds
to the two curves which pass through a critical X-point of (x, M) = (1, 1) in Figure 2.9.
2. If C < −3, the minimum of g(x) is smaller than that of f (M). In this case, for x whereg(x) < 1 = min(f (M)), there is no solution. Thus, f (M) = g(x) has solutions for x < x1

2.8. SUPER- AND SUBSONIC FLOW

55

Figure 2.9: M vs x (the x-axis is the normalized distance from the center x ≡ r/(GM∗ /2c2is ) and they-axis is the Mach number M.)

and x > x2 , where x1 < x2 , and g(x1 ) = g(x2 ) = 1. This corresponds to the curves running
perpendicularly in Figure 2.9.
3. If C > −3, the minimum of f (M) is smaller than that of g(x). In this case, f (M) = g(x) has
solutions for M < M1 and M > M2 , where M1 < M2 , and f (M1 ) = f (M2 ) = C + 4. This
corresponds to curves running horizontally in Figure 2.9.
Out of the two solutions of C = −3, a wind solution is one with increasing M while departingfrom the star. This shows us the outflow speed is slow near the star but it is accelerated and asupersonic wind blows after passing the critical point. Since the equations are unchanged even if we
replace u with −u, the above solution is valid for an accreting flow u < 0. Considering such a flow,
the solution running from the lower-right corner to the upper-left corner represents the accretion flow,in which the inflow velocity is accelerated reaching the star and finally accretes on the star surface
with a supersonic speed.

56

2.9

CHAPTER 2. PHYSICAL BACKGROUND

Virial Analysis

For a system to achieve a mechanical equilibrium, a relation must be satisfied between energies
such as potential, thermal and kinetic energies. This is called the virial relation. For example, aharmonic oscillator md2 x/dt2 = −kx has a potential energy of WH = (1/2)kx2 and a kinetic energy
KH = (1/2)mẋ2 . Averaging these two energies over one period, both energies give the same absolutevalue proportional to the oscillation amplitude a squared as ⟨WH ⟩ = (1/4)ka2 and ⟨KH ⟩ = (1/4)ka2 .
Another example is a Kepler problem. For simplicity, consider mass m running on a circular orbit
with a radius a from a body with a mass M . The gravitational and kinetic energies are equal to
WK = −GmM/a and KK = (1/2)mv 2 = (1/2)GmM/a, where we used the centrifugal balance as
mv 2 /a = GM m/a2 . As a result, for the harmonic oscillator ⟨WH ⟩ : ⟨KH ⟩ = 1 : 1 while for thecircular Kepler problem WK : KK = −2 : 1. This ratio is known to be related to the power n of
the potential as ϕ ∝ xn . Important nature of the self-gravity is understood only with this relation
without solving the hydrostatic balance equations. In the following, we describe the virial relation
satisfied with isolated systems such as stars.
Hydrodynamic equation of motion using the Lagrangean time derivative [eq.(B.3)] isρdv= −∇p − ρ∇Φ.
dt

(2.106)For simplicity, consider a spherical symmetric configuration. The equation of motion is expressed as
ρ

∂p
∂Φ
dv
=−
−ρ
.
dt
∂r
∂r

(2.107)

Multiplying radius r to the equation and integrating by the volume dV = 4πr2 dr over a volume fromr = 0 to r = R, we obtain the virial relation as
1 d2 I− 2T = 3(γ − 1)U + W,
2 dt2
where

Z
I=

(2.108)Z2

ρr dV =

r2 dM,(2.109)is an inertia of this body, and T and U are, respectively, the kinetic and thermal energies as
Z
Z
1 2
1 2
T =
ρv dV =
v dM,
(2.110)2
2ZU=
Further,

Z

p
dV.
γ−1
Z

GMr
dM,
r
Rr
Rr
is a gravitational energy, where dM = 4πr2 ρdr and Mr = 0 dM = 0 4πr2 ρdrSince
1 d2 r 2
dv=− u2 ,
r
dt
2 dt2
W =−

GMr
ρ
dV = −
r(2.111)(2.112)(2.113)

2.9. VIRIAL ANALYSIS

57

the lhs of equation (2.107) is rewritten asZ R 2 2
Z R
1d r
dv
2
2
ρr 4πr dr =− u ρ4πr2 dr,
2
dt
2
dt
0
0
Z R
Z R
1 d2
2
=
r
dM
−
v 2 dM
2 dt2 00
1 d2 I
=
− 2T,
2 dt2

(2.114)using equations (2.109) and (2.110).
On the other hand, the first term of the rhs of equation (2.107) becomes


Z R
Z R


dp
2
3 R
2
4πr pdr ,−r 4πr dr = − 4πr p 0 − 3
dr
0
0
Z R
= 3
pdV,
0

= 3(γ − 1)U.

(2.115)

To derive this equation, we have assumed the pressure diminishes at a radius r = R and the surfacepressure term does not appear in the final expression. This is valid for an isolated system such as astar.
The last term of the rhs of equation (2.107) is written as
Z R
Z R
dΦ
GMr
2
−
rρ 4πr dr = −ρ 2 4πr2 dr,(2.116)drr
00where we used equation (2.13). This is rewritten as
Z R
Z R
GMr
GMr
2
−
ρ 2 4πr dr =
−
dM
r
r
0
0
= W.

(2.117)The energy +GMr /r per unit mass is necessary for a gas element to move from the radius r, insidewhich mass Mr is contained, to the infinity. Adding the energy −(GMr /r) dM for whole gas sphere,
the potential energy is obtained. In the case of a star composed of uniform density ρ0 ,W =−3 GM 2,5 R(2.118)where M = 4πR3 ρ0 /3.
To obtain a condition for the mechanical equilibrium, we assume d2 I/dt2 = 0. Equation (2.108)
becomes
2T + 3(γ − 1)U + W = 0.
(2.119)Assuming the system is static v = 0, the above equation reduces to3(γ − 1)U + W = 0.

(2.120)In the case of γ = 5/3 this reduces to 2U + W = 0. The total energy E = U + W is expressed asE= (4 − 3γ)U
4 − 3γ=−W3(γ − 1)= −U (for γ = 5/3)
1
= W (for γ = 5/3).
2(2.121)

58

CHAPTER 2. PHYSICAL BACKGROUND

E

E

W

W

∆E
∆U

U

U
∆W

Figure 2.10: Virial relation for stars composed of γ = 5/3 gas.

For the gas with γ > 4/3, equation (2.121) gives a negative total energy E < 0 and the system is in
a confined state. However, if γ < 4/3, the gravity can not confine the gas.
For γ = 5/3 > 4/3, equation (2.120) gives
2U + W = 0.

(2.122)This shows us a strange nature of the self-gravitating gas (see Fig. 2.10). That is, if the heat flux
flows outward, the total energy decreases ∆E < 0. The system must contract and equation (2.121)
indicates ∆W = 2∆E < 0 (the gravitational energy decreases: the absolute value of the gravitationalenergy increases). However, for the thermal energy, equation (2.122) indicates that ∆U = −∆E =−(1/2)∆W > 0 for this system to be static. This shows that if the heat flux flows out from thesystem the thermal energy increases in the self-gravitating system. This comes from the contraction
due to the self-gravity.

2.9.1

Virial Mass

Consider γ = 5/3 gas, the gravitational energy is equal to twice of the kinetic+thermal energy[eq.(2.122)] in the equilibrium state. This gives definition of the virial mass. The kinetic energy is
obtained from the 3D velocity dispersion σ3D as
1
2
U = M σ3D2

(2.123)for globular clusters and elliptical galaxies, where M is the mass of the object. For molecular cloud
this is estimated from the width of the emission line. Equation (G.7) of Appendix G gives
2
σ3D
=

3 2
v
.
2 ln 2 x,HWHM(2.124)

2.10. RADIATIVE TRANSFER

59

where vx,HWHM is the half width of the half maximum of the emission line (if the radiation is opticallythin; see Fig. G.1). Since the gravitational energy is proportional to M 2 as
W = −ϵGM 2,
R(2.125)the virial theorem gives
2
Rσ3D
ϵG
if the surface term is ignored. This gives an estimation of mass of astronomical object.

M=

2.10

Radiative Transfer

2.10.1

Radiative Transfer Equation(2.126)

Specific intensity of radiation fields I(x, n, ν) is defined by the radiation energy dE transferred byphotons with wavelength ν through a unit surface placed at x whose normal is directed to n per unit
time per unit wavelength, and per unit steradian. Average intensity of radiation J(x, ν) is defined asZdΩJ(x, ν) = I(x, n, ν) ,(2.127)4π
where J is obtained by averaging I for the solid angle Ω. This is related to the energy density ofradiation u(x, ν) as
4πu(x, ν) =J(x, ν).
(2.128)c
If the radiation is absorbed in the displacement δs as δI(x, n, ν), δI must be proportional to δs
and I(x, n, ν) as
δI(x, n, ν) = −κI(x, n, ν)δs,(2.129)where κ is a coefficient and called volume absorption coefficient. The dimension of κ is cm−1 . We
can rewrite the above to the differential equation as
dI(x, n, ν)= −κI(x, n, ν) = −κρ ρI(x, n, ν),ds(2.130)where we used mass absorption coefficient κρ which represents the absorption per mass. Equation(2.129) is reduced todIdI(x, n, ν)≡= −I(x, n, ν),(2.131)
κds
dτ
where
Z s
τ=

κds,

(2.132)0

is called the optical depth. This means that τ is a measure for absorption as the intensity decreases
at a factor of exp(−1) from τ = 0 to τ = 1.
If the ray runs δs crossing a volume whose volume emissivity equal to ϵ the intensity increases
δI(x, n, ν) = +ϵ(x, ν)δs.

(2.133)

60

CHAPTER 2. PHYSICAL BACKGROUND
Level

Energynu

Enuhν=Enu-En

nEn

hν=En-End
ndEnd

Figure 2.11: Energy levels of a hypothetical 3-level atom.

The volume emissivity ϵ(x, ν) is the energy emitted by a unit volume at a position x per unit time
per unit solid angle per unit wavelength. From equations (2.129) and (2.133), the radiation transfer
is written as
dI(x, n, ν)= −κI(x, n, ν) + ϵ(x, ν).
(2.134)ds
Rs
Using the optical depth τ = 0 κds, this givesdI(ν)= −I(ν) + S(ν),
dτ(2.135)where S(ν) is called the source function and is defined as S(ν) = ϵ(ν)/κ(ν). Assuming the specificintensity I = I0 at the point of τ = 0, that at the point τ on the same ray is given
Z τ (x)I = I0 exp (−τ ) +S(s(τ ′ )) exp τ ′ − τ (s) dτ ′ .
(2.136)0

In the case of constant source function S =const it reduces to
I = I0 exp (−τ ) + S (1 − exp (−τ )) .
Equation (2.136) gives


I≃

S ...
τ ≫ 1,
τ S + I0 (1 − τ ) . . . τ ≪ 1.

(2.137)(2.138)This indicates that if we see an optically thick cloud τ ≫ 1 the specific intensity reaches us represents
S, while if we see an transparent cloud τ ≪ 1, I represents that of background.
Problem 9Show that equations (2.136) and (2.137) are solutions of equation (2.135).


2.10.2

Einstein’s Coeﬃcients

In this section, we describe the absorption coefficient κ and the emissivity ϵ for the line emission andabsorption. Here, consider a hypothetical atom which has only three levels named n = 0, 1, and 2.
and assume the energy of each level is, respectively, E0 , E1 , and E2 . When the atom changes its
state from n = 2 to n = 1, an photon with energy of hν2→1 = E2 − E1 is emitted, if this transitionis radiative. Besides the radiative process, atom changes its state exchanging its energy with otheratoms, molecules and electrons.

2.10. RADIATIVE TRANSFER

61

Spontaneous Emission
Atoms in an excited state n = 2 changes its state from n = 2 to n = 1 spontaneously. The numberof transitions is proportional to the number of atoms in the n = 2 state N2 as
A2→1 N2 ,(2.139)where A2→1 represents the probability of transition from n = 2 to n = 1 in a unit time. Thiscoefficient is called Einstein’s A coeﬃcient. The lifetime of n = 2 level is related to this coefficientas1
τlife (n = 2) =,(2.140)
A2→1 + A2→0
for this three-level atom.
Absorption
Photon with energy E2 − E1 is absorbed by the atom in the state of n = 1. At the same time theatom is excited from the state n = 1 to n = 2. The number of this transition is proportional to thenumber of atoms in the state n = 1, N1 , the number of photons passing and a cross-section of thisabsorption. This is written as
B1→2 Jν=(E2 −E1 )/h N1 ,(2.141)

where Jν=(E2 −E1 )/h represents the average specific intensity of radiation with the corresponding wavelength. This coefficient is called the Einstein’s B coeﬃcient for absorption.
Induced Emission
When atoms in the state n = 2 are in the radiation field with the average intensity of Jν=(E2 −E1 )/h ,downward transition from n = 2 to n = 1 is also induced and atoms emit photons with energyhνn=2→1 . This is called the induced emission or stimulated emission. This rate is expressed as
B2→1 Jν=(E2 −E1 )/h N2 .

(2.142)Collisional Excitation and Deexcitation
Beside radiative processes, energy states of atoms are changed by collisions with other atoms andmolecules, which are called collisional excitation and deexcitation. Since such transitions areproportional to the density of atoms or molecules, the number of transitions is written as follows: for
excitation from n = 1 to n = 2
C1→2 N1 nH ,(2.143)

C2→1 N2 nH ,(2.144)and deexcitation from n = 2 to n = 1

where the coefficients C are determined by laboratory experiments and quantum mechanics calculations.

62

CHAPTER 2. PHYSICAL BACKGROUND

2.10.3

Relation of Einstein’s Coeﬃcients to Absorption and Emissivity

Emissivity and absorption coefficients shown in §2.10.1 are related to the Einstein’s coefficients in§2.10.2. Net absorption is defined as the absorption minus the induced emission. Energy absorbed inunit volume is expressed as follows: integrating equation (2.129) over the solid angle, the first termof the rhs becomesZ
κ(ν)I(ν)dνdΩ = (E2 − E1 ) (B1→2 N1 − B2→1 N2 ) Jν=(E2 −E1 )/h .
(2.145)
Assuming isotropic radiation field, sinceZIν dΩ = 4πJν ,(2.146)the absorption coefficient integrated over the frequency range of one transition is written asZ1
κ(ν)dν =(E2 − E1 ) (B1→2 N1 − B2→1 N2 ) .
(2.147)4π
As for the emission, the second term of equation (2.129) becomesZ Zϵ(ν)dνdΩ = (E2 − E1 )A2→1 N2 .
Thus,Z
ϵ(ν)dν =E2 − E1
A2→1 N2 .
4π(2.148)(2.149)Relation between Einstein’s Coeﬃcients∗
Analysis using the Einstein’s coefficients is applicable to the state in which the thermal equilibrium is
not achieved. However, if this is applied to the thermal equilibrium, we can obtain relations between
these coefficients.
Assume the detailed balance is achieved between the spontaneous emission, the induced emission
and the absorption asN2 A2→1 + B2→1 Jν=(E2 −E1 )/h = N1 B1→2 Jν=(E2 −E1 )/h .
(2.150)In the thermal equilibrium, the population of atoms is given by the Boltzmann distribution as


N2
g2
E2 − E1=
exp −,(2.151)
N1
g1
kT
where gn represents the statistical weight for the state n.
Equation (2.150) gives
N2 A2→1
N2= Jν=(E2 −E1 )/h B1→2 −B2→1 .
N1
N1(2.152)In the case of thermal equilibrium, the average specific intensity agrees with the Planck function
Bν (T ) as2hν 31Jν = Bν (T ) = 2,(2.153)c exp (hν/kT ) − 1

2.10. RADIATIVE TRANSFER

63

Using equations (2.151) and (2.153), equation (2.150) reduces to2→1
exp (hν/kT ) − gg21 B
g2
2hν 3
B1→2
A2→1 = 2 B1→2.
g1c
exp (hν/kT ) − 1(2.154)

Lefthand-side of the equation is not dependent on the temperature. For this equation to be valid forvarious temperature,g2 B2→1= 1,(2.155)
g1 B1→2and

2.10.42hν 3
g2
A2→1 = 2 B1→2 .
g1
c(2.156)

Excitation Temperature

When the Boltzman distribution is achieved, the level population of 1 and 2 levels are written asN2g2=
exp(−E21 /kT ),N1
g1(2.157)

where g1 and g2 represent the statistical weight of 1 and 2 levels, respectively. In the case that the
Boltzman distribution is not achieved, the temperature that satisfies equation (2.157) is called theexcitation temperature and is expressed as
Tex = −

2.10.5

E21 /k.
loge (g1 N2 /g2 N1 )(2.158)

Critical Density

Consider a two-level atom (a hypothetical atom which has only two levels), in which the spontaneous
downward transitions and collisional excitations and deexcitations are in balance as(A21 + C21 n) N2 = C12 nN1 .

(2.159)C12 g1 = C21 g2 exp [−(E2 − E1 )/kT ],(2.160)Using
the number fraction of upper level is written
g2
C12
exp [−(E2 − E1 )/kT ]
N2
g1
C21
=
=
,
ncr g2A21
C12
N1 + N2
exp [−(E2 − E1 )/kT ]
1+
+
1+
+
n
g1
C21 n C21
where
ncr ≡

A21
,
C21

(2.161)(2.162)is called critical density. When n ≫ ncr , the second term of the denominator is small and the level
population is given by the Boltzmann distribution. As long as n ≪ ncr , the number of upper-level
population is much smaller than that expected for the Boltzmann distribution. Such a low-density
gas emits only weakly.

64

CHAPTER 2. PHYSICAL BACKGROUND
Table 2.1: A and C coefficients and critical densities for rotational transition of typical molecules(s−1 )A10
C10 (cm−3 s−1 )ncr 10 (cm−3 )A54 (s−1 )C54 (cm−3 s−1 )ncr 54 (cm−3 )CO
7.2 × 10−8
2.8 × 10−11
2.7 × 103
1.3 × 10−5
9.3 × 10−11
1.4 × 105CS
1.8 × 10−6
3.7 × 10−11
5 × 104
3.0 × 10−4
3.3 × 10−11
9 × 106

HCO+
4.3 × 10−5
2.6 × 10−10
1.6 × 105
7.2 × 10−3
4.2 × 10−10
1.7 × 107

Since B-coefficients, which has a meaning of the cross-section for the radiation, is proportional
to the electric dipole moment of the molecule, A-coefficients are also large for molecules with large
electric dipole moment [eq. (2.156)]. In the case of rotational levels, A-coefficients increase ∝ (J + 1)3
and thus the critical density increases for higher transition. In Table 2.1, the critical densities forrotational transitions of typical molecules are shown as well as A and C coefficients. Comparing
J = 1 − 0 transitions of CO, CS, and HCO+ , CS and HCO+ trace higher-density gas than CO. And
higher transition J = 5 − 4 lines trace higher-density gas than lower transition J = 1 − 0 lines.
In the discussion above, we ignored the effect of transition induced by absorption. The above
critical density is defined for optical thin case. For a gas element with a finite optical depth, photons
are effectively trapped in the gas element (photon trapping). If we use the probability for a photon
to escape from the gas element, βν , the critical density is reduced to
ncr ∼

A21
βν .
C21

(2.163)The most abundant CO, 12 C16 O, is sometimes optically thick, while rare molecules 13 C16 O and12 C18 O, which have almost the same Einstein’s coefficients as 12 C16 O, are optically thin. In such a
cloud, the critical density of 12 C16 O is smaller than that of 13 C16 O and 12 C18 O.

Chapter 3

Galactic Scale Star Formation
3.1

Schmidt Law

Schmidt (1959) speculated that the star formation rate is proportional to a power of the surface
density of the interstellar mediumΣ̇SF ∝ Σngas ,(3.1)
where the power n seems between 1 and 2 around the solar vicinity. If n = 2, the star formation
rate is thought to be determined by the collision rate of interstellar clouds. At that time Schmidt
showed us n ≃ 2. On the other hand, if the gas passing through the galactic arms forms stars, thestar formation rate seems proportional to the gas surface density and the arm-to-arm period. Thus
this predicts n = 1. This relation is called Schmidt law of star formation.

3.1.1

Global Star Formation

The star formation rate is estimated by the intensity of Hα emission (Kennicutt, Tamblyn, & Congdon
1994) as
L(Hα)
SFR(M⊙ yr−1 ) =,(3.2)
1.26 × 1041 erg s−1
which is used for normal galaxies. While in the starburst galaxies which show much larger starformation rate than the normal galaxies, FIR luminosity seems a better indicator of star formation
rate
L(FIR)
L(FIR)SFR(M⊙ yr−1 ) =
=
.
(3.3)
2.2 × 1043 erg s−1
5.8 × 109 L⊙
Kennicutt (1998) summarized the relation between SFRs and the surface gas densities [Fig. 3.1 (left)]for 61 normal spiral and 36 infrared-selected starburst galaxies. As seen in Figure 3.1, the star formation rate averaged over a galaxy (ΣSFR (M⊙ yr−1 kpc−2 )), which is called the global star formation
rate, is well correlated to the average gas surface density Σgas (M⊙ pc−2 ). He gave the power of the
global Schmidt law as n = 1.4 ± 0.15. This type of correlation is often called Kennicutt-Schmidt
law. That is,1.4±0.15
Σgas−4ΣSFR ≃ (1.5 ± 0.7) × 10M⊙ yr−1 kpc−2 .
(3.4)
1M⊙ pc−2
The fact that the power is not far from 3/2 seems to be explained as follows: Star formation
rate should be proportional to the gas density (Σgas ) but it should also be inversely proportional to
the time scale of forming stars in respective gas clouds, which is essentially the free-fall time scale.
65

66

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Figure 3.1: Taken from Figs. 6 and 7 of Kennicutt (1998). (Left:) The x-axis means the total (Hi+H2 )gas density and the y-axis does the global star formation rate. (Right:) The x-axis means
the total (Hi+H2 ) gas density divided by the orbital time-scale. The y-axis is the same.

Remember the fact that the free-fall time given in equation (2.26) is proportional to τﬀ ∝ 1/(Gρ)1/2 .
Therefore
ρSFR ∝ ρgas × (Gρgas )1/2 ∝ ρ3/2
(3.5)
gas ,
where ρgas and ρSFR are the volume densities of gas and star formation.
He found another correlation between a quantity of gas surface density divided by the orbital
period of galactic rotation and the star formation rate [Fig. 3.1 (right)]. Although the actual slope isequal to 0.9 instead of 1, the correlation in Fig. 3.1 (right) is well expressed as
ΣSFR ≃ 0.017 Σgas Ωgas = 0.21

Σgas
τarm−to−arm

,

(3.6)

where Ωgas represents the angular speed of galactic rotation. This means that 21 % of the gas mass isprocessed to form stars per orbit. These two correlations [eqs. (3.4) and (3.6)] implicitly ask another
1/2
relation of Ωgas ∝ Σgas .

3.1.2

Local Star Formation Rate

In Figure 3.2 (left), the correlation between star formation rate and gas density is plotted for specificgalaxies (NGC4254 and NGC2841). This shows us that Hα surface brightness (star formation rate)

3.1. SCHMIDT LAW

67

Figure 3.2: Distributions of ΣSF and Σgas . (Left:) Relation between ΣSF and Σgas for an Sc galaxy
NGC4254 and an Sb galaxy NGC2841. (Right:) Relation between ΣSF and Σgas forvarious galaxies. These are taken from Figs.7 and 8 of Kennicutt (1989).

and the gas column density are well correlated each other. Figure 3.2 (left) also indicates that there
seems a critical gas density below which star formation is not observed. The value of this thresholdcolumn density is approximately ≃ 4M⊙ pc−2 for both galaxies in Figure 3.2 (left). The same correlation is seen in other spiral galaxies [Fig.3.2(right)]. Fitting the correlation with a power-law, heobtained
ΣSFR ∝ Σ1.3±0.3
,
(3.7)
gas
for the region active in star formation. Take notice that this power is very close to that of the
global Schmidt law [eq.(3.4)] The threshold surface gas density ranges from 1 M⊙ pc−2 to 10 M⊙ pc−2(1020 − 1021 H cm−2 ). Therefore, theory of star formation must explain(1) the Schmidt law (clear positive correlation between star formation rate and the gas surface
density) above the threshold column density and(2) the fact that there is no evidence for star formation in the gas deficient region below the thresholdcolumn density.

Star Formation Seen with Akari
Figure 3.3 represents the far-IR observation of M101 with Akari (Suzuki et al. 2007). Using 4 bands of
Akari FIS and 3 bands of ISO, the SED of this galaxy is fitted with two components of dust emissions+4
with temperatures Tw = 55+9
−25 K and Tc = 18−9 K. Spiral galaxies (not classified with either AGN

68

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Figure 3.3: Far-IR SED of M101 with AKARI (Suzuki et al. 2007). Dotted and dashed lines represent+4best-fit modified black body spectrum with Tw = 55+9
−25 K and Tc = 18−9 K.

(a)

(b)

Figure 3.4: Akari far-IR observation of M101 (Suzuki et al. 2007). (a:) Distribution of emissions fromcold dust with Tc (left) and warm dust with Tw (right). (b:) The warm-to-cold intensity
ratio is plotted.

3.2. DISTRIBUTIONS OF ATOMIC AND MOLECULAR HYDROGEN

69

or starburst galaxy) have a far-IR temperature with 20 K–50 K as a whole from Revised ShapleyAmes galaxy survey by IRAS (De Jong et al. 1984). As shown in Figure 3.4 (a), cold component
correlates well with that of gas distribution, while warm one has a relation to star formation regions.
Figure 3.4(b) plots the warm-to-cold ratio of intensities, which seems to show the distribution of starformation efficiency ≡ star formation rate/gas mass. Some peaks in this plot correspond to famous
Hii regions. Center of the galaxy and the end of the galactic bar indicate peaks in this map. Sincebarred galaxies have larger star formation rate than spiral galaxies without bars, the central peaks
are regarded as a region where gas is accumulated by the effect of a bar potential. Equations (3.4)
and (3.7) indicates that the star formation efficiency
0.3−0.4
ΣSFR /Σgas ∝ Σgas
.

(3.8)

M101 galaxy do not seem to be explained by a simple law as the star formation efficiency is anincreasing function of gas density.

3.2

Distributions of Atomic and Molecular Hydrogen

We believe that stars form in molecular clouds, in which major component of hydrogen is “molecule.”
MHi is larger than MH2 and Hi gas distributes rather uniformly than H2 gas. Figure 3.5(left) shows
distributions integrated intensities of Hi (red) and CO (blue) for Large Magellanic Cloud (Fukui etal. 2009). CO gas is found as GMCs with a mass M ∼ 105−6 M⊙ , while Hi gas is distributed more
uniform than that of CO gas.
Figure 3.5(right) shows that line width of Hi is much larger than that of CO. Comparison in thephase-space (position vs. line-of-sight velocity) indicates us that the Hi gas coexisting with H2 gas
was overestimated when the comparison is made for the column density σ. H2 gas associated with
GMCs may be a small fraction of Hi gas.
Figure 3.6 displays the distributions of Hi and CO gas. CO and Hi data are divided into pixels
whose 3D size is 40 pc × 40 pc × 0.17 km s−1 . The number of pixels where Hi (> 7.2 K) and CO(> 0.21 K) are both detected are plotted by a histogram with dark color. While that of Tb (Hi) > 7.2 K
but Tb (CO) < 0.21 K (light-shade histogram) shows a different distribution. The ratio of pixels
associated with CO increases with Tb (Hi) [Fig. 3.6 (bottom)]. This may indicates an accretion of Hi
gas to form GMCs.
Another important finding of NANTEN survey is a classification of GMCs from a stand point of
massive star formation. Kawamura et al. (2009) indicate that “Type I shows no signature of massive
star formation, Type II is associated with relatively small HII region(s), and Type III with both HII
region(s) and young stellar cluster(s),” as shown in Figure 3.6(right). The lifetime of a GMC throughType I to Type III is estimated to be a few 107 yr. The average Hi intensity for Types I, II, and III
are estimated as 34 ± 16 K, 47 ± 17 K, and 56 ± 19 K. The Hi intensity increases with the sequence ofstar formation.
The lifetime of GMCs is studied using ALMA also for other galaxies (Chevance et al. 2020).
Nine disk (star-forming) galaxies are chosen from high-resolution multi-wavelength coverage ALMA
survey (PHANG). Since the target galaxies are nearby (D ≲ 17 Mpc) and nearly face-on (i ≲ 75◦ ),distributions of CO and Hα can be compared for the spatial scale up to 35-162 pc. Figure 3.7(left)
shows the variation of gas-to star formation rate when changing the averaging aperture size. The
x-axis represents the aperture size in pc and the y-axis shows the CO-to-Hα ratio normalized withthe value averaged over the galactic scale. We can see the separation between two gas-to-SFR ratios
for small scales: (blue) when we observe the ratio centered at the molecular gas peaks and (red) that

70

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Figure 3.5: (Left:) Distributions integrated intensities of Hi (red) and CO (blue) for LMC. The Hi
integrated intensity is obtained from combination of aperture synthesis and single-dishobservations. (Kim et al. 2003). (Right:) Spectra of Hi and CO for a direction are shown.

Figure 3.6: (Upper-left:) Number of pixels are plotted against the brightness temperature of Hi. CO
and Hi data are divided into pixels whose 3D size is 40 pc× 40 pc×0.17 km s−1 . (Bottomleft:) Number of pixels associated with CO emission increases with higher Hi brightness
temperature (Fukui et al. 2009). (Right:) Classification of GMCs. Type I GMCs areobserved without any indication of massive star formation; Type II’s are GMCs with Hii
regions. However, young clusters are embedded in the GMCs and not observed. Type
III’s are associated with Hii regions and young clusters (Kawamura et al. 2009)

3.2. DISTRIBUTIONS OF ATOMIC AND MOLECULAR HYDROGEN

71

Figure 3.7: (Left:) Gas-to-SFR (CO-to-Hα) ratio (y-axis) is plotted against the aperture size in pc(x-axis; 20 pc - 20 kpc). The value is normalized with the galactic average Blue points
are calculated by focusing on the molecular gas peaks, while red ones are for ionized gas
peaks. In small scales, these two curves are separate. λ = 113+22−14 pc indicates the spatial
resolution scale below which we can distinguish the phase of the system (CO without Hα,
CO with Hα, and Hα without CO). With larger spatial scale > λ objects belonging tothese three phases can not be specified but are mixed. This is a part of Fig. 2 in Chevanceet al. (2010). (Right:) Durations of respective phases of GMC evolution. Duration inwhich CO gas is observed is shown with orange bar. Blue part is the phase in which Hα
is observed. Overlapped part indicates GMCs observed associated with Hα emissions,which corresponds to type II to III based on the classification by Kawamura et al (2009).

when we observe centered at the Hα peaks. This is believed that GMC and young star system in
different phases are observed when looking the system locally (the aperture is smaller than < λ).
Thus, observing with finer spatial resolution < λ, the ratio of durations of GMC and of young cluster
is predicted.
Figure 3.7 (right) shows duration timescales of GMCs (CO gas: orange) and young massive starclusters (stars: blue). GMCs with Hα are distributed in the overlapped region with brown color.
In this phase, stellar feedback is destroying the parent GMC; the duration of this is as short as 15 Myr. Before that, GMCs are inactive in star formation and GMCs spend 75-90 % of their lifetimes,
10-30 Myr in this inactive phase.

72

3.3

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Gravitational Instability of Rotating Thin Disk

Here, we will derive the dispersion relation for the gravitational instability of a rotating thin disk. We
will see the spatial variation of Toomre’s Q parameter, which determines the stability of the rotating
disk, explains the nonlinearity of star formation rate, that is, there is a threshold density and no stars
are formed in the low density region.
Use the cylindrical coordinate (R, Z, ϕ) and the basic equations for thin disk in §2.6. In linear
analysis, we assume Σ(R, ϕ) = Σ0 (R) + δΣ(R, ϕ), u(R, ϕ) = 0 + δu(R, ϕ), v(R, ϕ) = v0 (R) + δv(R, ϕ),where u and v represent the radial and azimuthal components of the velocity. Linearized continuity
equation is1 ∂∂δΣ Σ0 ∂δv∂δΣ+(RΣ0 δu) + Ω
+
= 0,
(3.9)
∂t
R ∂R
∂ϕ
R ∂ϕ
where Ω = v0 /R.
Linearized equations of motion are


∂
∂
∂
+Ω
δu − 2Ωδv = −(δΦ + δh),
∂t
∂ϕ
∂R


and

∂
∂
+Ω
∂t
∂ϕ


δv +

κ2
1 ∂
δu = −(δΦ + δh),2Ω
R ∂ϕ

(3.10)(3.11)

where h is a specific enthalpy as dh = dp/Σ andκ=

4Ω2 + R

dΩ2
dR

1/2
(3.12)

is the epicyclic frequency.
We assume any solution of equations (3.9), (3.10), and (3.11) can be written as a sum of terms of
the formδu = ua exp[i(mϕ − ωt)],(3.13)

δv = va exp[i(mϕ − ωt)],(3.14)

δΣ = Σa exp[i(mϕ − ωt)],(3.15)

δh = ha exp[i(mϕ − ωt)],(3.16)

δΦ = Φa exp[i(mϕ − ωt)].

(3.17)

Using the equation of state of p = KΣγ ,ha = c2s Σa /Σ0 .

(3.18)

Using equations (3.13)-(3.17), equations (3.9), (3.10), and (3.11) are rewritten asi(mΩ − ω)Σa +1 ∂Σ0 v a(RΣ0 ua ) + im
= 0,
R ∂R
R


d
(Φa + ha ),ua [κ − (mΩ − ω) ] = −i (mΩ − ω)(Φa + ha ) + 2mΩ
dR
R


2

and

2


κ2 d
(Φa + ha )va [κ − (mΩ − ω) ] =,(Φa + ha ) + m(mΩ − ω)2Ω dR
R

(3.19)(3.20)



2

2(3.21)

3.3. GRAVITATIONAL INSTABILITY OF ROTATING THIN DISK

73

Figure 3.8: Tightly wound (left) vs loosely wound (right) spirals.

3.3.1

Tightly Wound Spirals

We assume the wave driven by the self-gravity has a form of tightly-wound spiral [Fig. 3.8 (left)].
When we move radially, the density δΣ varies rapidly. While, it changes its amplitude slowly in theazimuthal direction. In a mathematical expression, if we write the density perturbation δΣ asδΣ = A(R, t) exp[imϕ + i f (R, t)],(3.22)

where the amplitude of spiral A(R, t) is a slowly varing function of R, a tightly wound spiral means
the shape function varies fast (the radial wavenumber k ≃ df /dR is large enough or Rdf /dR ≫ 1).
We consider the gravitational force from the vicinity of (R0 , ϕ0 ), since the δΣ oscillates and cancels
even if we integrate over large region. Thus,δΣ(R, ϕ, t) ≃ Σa exp[ik(R0 , t)(R − R0 )],(3.23)

Σa = A(R0 , t) exp[imϕ0 + if (R0 , t)].

(3.24)

where
Notice that the density perturbation [eq. (3.23)] is similar to that studied in §2.6. The potential
should be expressed in a similar form to equation (2.64) asδΦ ≃ −

2πGΣaexp[ik(R0 , t)(R − R0 )],|k|(3.25)

2πGΣa
.
|k|

(3.26)

which simply meansΦa = −

If we set R = R0 , we obtain our final result for the potential due to the surface density perturbation
δΦ(R, ϕ, t) ≃ −2πG
A(R, t) exp[imϕ + if (R, t)].
|k|

(3.27)

Differentiating this equation with R and ignoring the term dA(R, t)/dR compared to that of df (R, t)/dR =k, we obtainsign(k) dδΦ(R, ϕ, t),(3.28)
δΣ(R, ϕ, t) = i
2πG
dR

74

CHAPTER 3. GALACTIC SCALE STAR FORMATION
Table 3.1: Epicyclic frequency vs rotation law.

Rotation
Rigid-body rotation Ω =const.
Flat rotation vϕ =const.
Kepler rotation vϕ ∝ r−1/2

κ
2Ω
√
2Ω
Ω

Neglecting the terms ∝ 1/R compared to the terms containing ∂/∂R, equations (3.19), (3.20),
and (3.21) are rewritten asi(mΩ − ω)Σa + ikΣ0 ua = 0,(3.29)
ua [κ2 − (mΩ − ω)2 ] = (mΩ − ω)k(Φa + ha ),(3.30)andκ2k(Φa + ha ),(3.31)
2Ω
Using these equations [(3.29), (3.30), and (3.31)], Φa = −2πGΣa /|k|, and ha = c2s Σa /Σ0 , we obtain
the dispersion relation for the self-gravitating instability of the rotating gaseous thin diskva [κ2 − (mΩ − ω)2 ] = i(mΩ − ω)2 = k 2 c2s − 2πGΣ0 |k| + κ2 .

(3.32)

Generally speaking, the epicyclic frequency depends on the rotation law but is in the range of Ω ≲κ ≲ 2Ω (see Table 3.1 for κ for typical rotation laws). It is shown that the system is stabilized dueto the epicyclic frequency compared with a non-rotating thin disk [eq.(2.66)].

3.3.2

Toomre’s Q Value

Consider the case of m = 0 axisymmetric perturbations. Equation (3.32) becomesπGΣ0 2πGΣ0 222 2
222
ω = k cs − 2πGΣ0 |k| + κ = cs k −
+κ −
.
c2s
cs

(3.33)

If ω 2 > 0 the system is stable against the axisymmetric perturbation, while if ω 2 < 0 the system isunstable. Defining
κcsQ=,(3.34)πGΣ0
the last two terms are represented as (πGΣ0 /cs )2 (Q2 −1). Thus, if Q > 1, ω 2 > 0 for all wavenumbers
k. On the other hand, if Q < 1, ω 2 becomes negative for some range of wavenumbers k1 < k < k2 .
Therefore, the Toomre’s Q number gives us a criterion whether the system is unstable or not for
the axisymmetric perturbation. [Recommendation for a reference book of this section: Binney &Tremaine (1988).]
The condition is expressed as
Σ0 > Σcr =κcsπG(Q < 1).

(3.35)

Kennicutt plotted Σ0 /Σcr against the normalized radius as R/RHii for various galaxies, where RHii
represents the maximum distance of Hii regions from the center (Fig. 3.9). Since Σ0 /Σcr = Q, Figure
3.9 shows that Hii regions are observed mainly in the region with Q < 1 but those are seldom seen

3.4. SPIRAL STRUCTURE

75

Figure 3.9: Σ0 /Σcr vs R/RHii . RHii represents the maximum distance of Hii regions from the center.
The sound speed is assumed constant cs = 6 km s−1 . Taken from Fig. 11 of Kennicutt(1989).

in the outer low-density Q > 1 region. This seems to indicate that the gravitational instability plays
an important role.
The above discussion is for the gaseous disk. The Toomre’s Q value is also defined for stellar
system as
σR κQ=,(3.36)
3.36GΣ0where σR represents the radial velocity dispersion.
For non-axisymmetric waves, even if 1 ≲ Q ≲ 2 the instability grows. To explain this, the swing
ampliﬁcation mechanism is proposed (Toomre 1981). If there is a leading spiral perturbation in thedisk with 1 ≲ Q ≲ 2, the wave unwinds and finally becomes a trailing spiral pattern. At the sametime, the amplitude of the wave (perturbations) is amplified (see Fig. 3.10).

3.4

Spiral Structure

Figure 3.11 shows the B- (left) and I-band (right) images of M51. The B-band light originates fromthe massive early type stars. Although the image taken in B-band shows a number of spiral arms,
that of I-band shows clearly two arms. The I-band light seems to come from mainly less-massive
long-lived stars, while the B-band light is essentially coming from the massive short-lived stars which
are formed in the spiral arm. On the contrary, the less-massive stars are not necessarily born in the
spiral arm. This suggests that there are two kinds of spiral patterns: one is made by stars (mainly
less-massive) and the other is the gaseous spiral arm where massive stars are born and contribute to
the B-band image.
In this section, first, we will briefly describe the density wave theory which explains the former
spiral pattern in the stellar component. You will find the amplitude of the spiral pattern in stellar
component is not so large. However, the response of gaseous components (Hi and H2 gas) to the

76

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Figure 3.10: Numerical simulation of the swing amplification mechanism. The number attached eachpanel shows the time sequence. This is obtained by the time-dependent linear analysis.
First, perturbation with leading spiral pattern is added to the Mestel disk with Q = 1.5.
The leading spiral gradually unwinds and become a trailing spiral. Loosely wound spiralpattern winds gradually and the last panel shows a tightly wound leading spiral pattern.
The final amplitude is ∼ 100 times larger than that of the initial state.

spiral density wave potential with small amplitude is much more nonlinear than that of the stellar
component and a high-contrast spiral pattern appears in the gaseous component .

3.5

Density Wave Theory

We have derived the dispersion relation of the gravitational instability in the rotating thin disk as(mΩ − ω)2 = k 2 c2s − 2πGΣ0 |k| + κ2 ,(3.37)

where m represents the number of spiral arms. Although the stability of the stellar system is a little
different, we assume this is valid for the stellar system after cs is replaced to the velocity dispersion.

3.5. DENSITY WAVE THEORY

77

Figure 3.11: B- (left), and I-band (right) images of M51. Taken from Fig. 3 of Elmegreen et al. (1989).

k/kT

Q=1

4

Q=1.2

3.5

Q=1.5

3
2.5

Q=2

2
1.5
1
0.5
-2

-1

1

2

nu

Figure 3.12: (Left:) A plot of the dispersion relation eq.(2.66). x- and y-axes are ν and k/kT . Respective lines are for Q = 1 (straight lines), Q = 1.2, Q = 1.5, and Q = 2. The trailingpart k > 0 is only plotted. Points ν = −1, ν = 0, and ν = 1 correspond respectively
to inner Lindblad resonance (ILR), corotation resonance (CR), and outer Lindblad resonance (OLR). The relation is symmetric against the x-axis and the curve of k < 0represents the leading wave. (Right:) Leading vs trailing spiral.

78

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Since(mΩ − ω)2 = k 2 c2s − 2πGΣ0 |k| + κ2πGΣ0 2πGΣ0 22
2,= cs k −
+κ −
c2s
cs





πGΣ0 2πGΣ0 2 2
2
Q −1 ,+= cs k −2
cs
cs
we obtain|k| = kTip
2 h
2 (1 − ν 2 ) ,
1
−
Q
1
±
Q2

(3.38)(3.39)

where ΩP = ω/m is a pattern speed, ν = m(ΩP − Ω)/κ is the normalized frequency, kT =
κ2 /2πGΣ0 is the Toomre’s critical wavenumber for a cold (cs = 0) system. ν = ±1, which leads
to |k| = 0, represents the Lindblad resonance and is rewritten as
ΩP =

ω
κ
=Ω± .
m
2

(3.40)

Assuming m = 2, the resonance when ΩP = Ω + κ/2 is called outer Lindblad resonance (OLR)
while that of ΩP = Ω − κ/2 is called inner Lindblad resonance (ILR). ν = 0 means the corotation resonance (CR) ΩP = Ω.
Plotting the wavenumber k against the normalized frequency ν of equation (3.39) as Figure
3.12 (left), it is shown that, in the case of Q = 1, the wavenumber exists for all ν. Since ν = −1, 0,
and +1 correspond to the points of ILR, CR, and OLR and these three resonance points appear inaccordance with the radial distance, the x-axis of Figure 3.12 (left) seems to correspond to the radial
distance from the center. In the case of Q > 1, it is shown that a forbidden region appears around the
co-rotation resonance point. Waves cannot propagate into the region. Figure 3.12 (left) shows thatk/kT has two possible wavenumbers in the permitted region. The waves with larger k and smaller k
are called short waves and long waves, respectively.
Consider a wave expressed by Σ ∝ exp[imϕ + ikr]. If k < 0, moving from a point (R0 , ϕ0 ) in the
direction ∆ϕ > 0 and ∆r > 0 the phase difference between the two points [m(ϕ0 + ∆ϕ) + k(R0 +∆r)] − [mϕ0 + kR0 ] can be equal to zero. That is, in the case of k < 0 the wave is leading. On
the other hand, if k > 0, moving in the direction ∆ϕ < 0 and ∆r > 0 the phase will be unchanged.
In this case, the wave pattern is trailing. Since the dispersion relation is symmetric for k > 0 andk < 0, there are two waves, trailing waves and leading waves. Therefore there are four waves: a short
trailing wave, a long trailing wave, a short leading wave, and a long leading wave.

3.5.1

Group Velocity

The wave transfers the energy with its group velocity. Whether the group velocity is positive or
negative is quite important considering the energy transfer. Using the dispersion relation, equation(3.38), The group velocity|k|c2s − πGΣ0dω
= sign(k).
(3.41)
vg (R) =dk
ω − mΩ
For a region R > RCR , ω − mΩ > 0 or ν > 0. On the other hand, for a region R < RCR , ω − mΩ < 0or ν < 0. Consider first the trailing wave. In the region R > RCR , long-waves propagate inwardly
to the CR, since |k|c2s < πGΣ0 for long-waves and ν > 0. Short-waves propagate outwardly from theCR, since |k|c2s > πGΣ0 for short-waves and ν > 0. In the region R < RCR , long-waves propagate
outwardly to the CR since |k|c2s < πGΣ0 and ν < 0. Short-waves propagate inwardly from the CR,

3.6. GALACTIC SHOCK

79

Figure 3.13: Spiral coordinate. Along a curve of constant η, r ∝ exp(θ tan i).

since |k|c2s > πGΣ0 and ν < 0. As a result, it is concluded that the long-wave propagates toward
the CR and the short-wave does away from the CR. In contrast, for the leading wave, the short-wave
propagates toward the CR and the long-wave does away from the CR.
Assuming that the wave packet is made near the Lindblad resonance points, (1) the long-trailingwaves propagate toward the co-rotation resonance points; (2) they are reflected by the Q-barrier;(3) they change their character to short-waves and propagate away from the co-rotation resonancepoints; and finally (4) the waves are absorbed at the center or propagate away to the infinity. (1) the
short-leading waves propagate toward the co-rotation resonance points; (2) they are reflected by the
Q-barrier; (3) they change their character to long-waves and propagate away from the co-rotation
resonance points; and finally (4) they reach the Lindblad resonance points and the energy may be
converted to the long-trailing waves there.
The wave obtains its energy at the resonance points. The density wave transfer the energy to
the co-rotation points. Therefore, the density wave theory predicts the galactic stellar disk has spiraldensity pattern between the inner Lindblad resonance points and the outer Lindblad resonance point
if Q ≳ 1.

3.6

Galactic ShockIn the preceding section, we have seen that a spiral pattern of density perturbation is made by thegravitational instability in the stellar system with Q ≳ 1. This forms a grand design spiral observedin the I-band images which represent the mass distribution which consists essentially in less-massive
stars. The amplitude of the pattern is smaller than that observed in B-band images. In this section
we will see how the distribution early type stars is explained. [Recommendation for a reference bookof this section: Spitzer (1978).]
Gas flowing through a spiral gravitational potential, even if its amplitude is relatively small, acts
rather nonlinearly. Consider the gravitational potential in the sum of an axisymmetric term, Φ 0 (R)and a spiral term Φ1 (R, ϕ). The spiral gravitational field is rotating with a pattern speed ΩP . We
use a reference frame rotating with ΩP . u = (u, v), where u and v are radial (R) and azimuthal(ϕ) component of the flow velocity. Consider the reference system rotating with the angular rotation

80

CHAPTER 3. GALACTIC SCALE STAR FORMATION

speed ΩP . We assume the variables with suffix 0 represent those without the spiral potential Φ 1
and variables with suffix 1 are used to express the difference between before and after Φ 1 is added.
u = u0 + u1 = 0 + u1 , v = v0 + v1 , and"#
1 ∂Φ0 1/2v0 = R(Ω − ΩP ) = R− ΩP ,(3.42)
R ∂R
where Ω means the circular rotation speed. The equation of motion in the r-direction is∂u v ∂u v12
c2 ∂Σ ∂Φ1
∂u
+u
+
−
=− s
−
+ 2Ωv1 ,
∂t
∂r
r ∂ϕ
r
Σ ∂r
∂r
and that in the ϕ direction is∂v1 v1v ∂v1c2 ∂Σ 1 ∂Φ1
κ2
∂v
+u
+
+
=− s
−
−
u,
∂t
∂r
r
r ∂ϕ
Σr ∂ϕ
r ∂ϕ
2Ω

(3.43)(3.44)and the continuity equation isR∂Σ ∂ΣRu ∂Σv
+
+
= 0.
∂t
∂R
∂ϕ

(3.45)As seen in Figure 3.13, we introduce the spiral coordinate (η, ξ) in which ξ-axis is parallel to thespiral pattern which has a pitch angle i and η-axis is perpendicular to the ξ-axis.
rdη = dr cos i + rdϕ sin i,(3.46)rdξ = −dr sin i + rdϕ cos i.

(3.47)vη = u cos i + v sin i,(3.48)vξ = −u sin i + v cos i(3.49)

Assuming that i ≪ 1 (tightly wound spiral), equations (3.43), (3.44), and (3.45) become∂vη
∂vη
c2 ∂Σ ∂Φ1
+ vη
=− s
−
+ 2Ω(vξ − vξ0 ),
∂t
∂η
Σ ∂η
∂η

(3.50)

∂vξ
∂vξ
κ2
+ vη
= − (vη − vη0 ),
∂t
∂η
2Ω

(3.51)∂Σ ∂Σvη
+
= 0.
∂t
∂η

(3.52)

Similar to §2.8 we look for a steady state solution of these equations. Equation (3.52) becomes
dΣ
Σ dvη
=−
.
dη
vη dη

(3.53)

Using this, equation (3.50) reduces tovη2 − c2s

 dvη
dΦ1
= 2Ω(vξ − vξ0 ) −.
dηdη(3.54)dvξκ2= − (vη − vη0 ).
dη
2Ω(3.55)

Equation (3.51) becomesvη

3.6. GALACTIC SHOCK

81

In these equations we used following quantities:vη = vη0 + vη1 ,(3.56)vξ = vξ0 + vξ1 ,(3.57)vη0 = v0 sin i,(3.58)vξ0 = v0 cos i,(3.59)

v0 is given in equation (3.42).
Equations (3.53), (3.54), and (3.55) are solved under the periodic boundary condition: X(η =right end) = X(η = left end). Since equation (3.54) is similar to the equations in §2.8, you may think
the solution seems like Figure 2.6. However, it contains a term which expresses the effect of Coliois
force 2Ω(vξ − vξ0 ), the flow becomes much complicated.
Taking care of the point that a shock front exists for a range of parameters, the solution of
the above equations are shown in Figure 3.14. Numerical hydrodynamical calculations which solve
equations (3.50), (3.51), and (3.52) was done and steady state solutions are obtained (Woodward1975). It is shown that F ≤ 0.7%, the velocity (vη : velocity perpendicular to the density wave) does
not show any discontinuity. In contrast, for F > 2%, a shock wave appears. Steady state solution is
obtained from the ordinary differential equations (3.53) (3.54), and (3.55) by Shu, Milione, & Roberts(1973). Inside the CR, v0 > 0 (Gas has a faster rotation speed than the spiral pattern). As longas vη > cs there is no shock. Increasing the amplitude of the spiral force F , an amplitude of thevariation in vη increases and finally vη becomes subsonic partially. When the flow changes its naturefrom supersonic to subsonic, it is accompanied with a shock. (An inverse process, that is, changing
from subsonic to supersonic is not accompanied with a shock.)In the outer galaxy (still inside CR) since Ω −ΩP decreases with the the distance from the galactic
center, vη0 = R(Ω − ΩP ) sin i decreases. In this region, vη0 < cis and the flow is subsonic if there isno spiral gravitational force, F = 0. In such a circumstance, increasing F amplifies the variation invη and vη finally reaches the sound speed. Transonic flow shows again a shock.
Summary of this section is:
1. There exists a spiral density pattern of the stellar component if the Toomre’s Q parameter is
1 ≲ Q ≲ 2.
2. This is driven by the self-gravity of the rotating thin disk.
3. If the amplitude of the non-axisymmetric force is as large as ∼ 1% of the axisymmetric one,interstellar gas whose sound speed is as large as cis ∼ 8 km s−1 forms the galactic shock. Thisis observed when the flow is transonic.
4. The amplitude of the gas density fluctuation is much larger than that in the stellar density.
5. If stars are formed preferentially in the postshock region of the spiral arm, we expect clear spiralarms made by early-type stars, which are massive and short-lived, as seen in the B-band imagesof spiral galaxies.

82

CHAPTER 3. GALACTIC SCALE STAR FORMATION

Figure 3.14: Galactic shocks. The velocity perpendicular to the spiral wave front is plotted against
the phase of the spiral wave. Gas flows from the left to the right. The spiral potential
takes its minimum at θ = 90 deg. Parameters are taken as cs = 8.6 km s−1 , i = 6.7 deg,
R = 10 kpc, RΩ = 250 km s−1 , Rκ = 313 km s−1 , RΩP = 135 km s−1 . The amplitudeof spiral gravity force is taken F =0.4%, 0.7%, 2%, and 5% of the axisymmetric force−∂Φ0 /∂R. Taken from Fig. 13.3 of Spitzer (1978) [originally Woodward (1975)].

3.6. GALACTIC SHOCK

Figure 3.15: Galactic shock. (Left:) vη vs vξ plot. Increasing F from 0.5 % to 0.9 %, the variation invη increases. And at F = 0.97 %, the minimum speed reaches the sound speed 8 km s−1 .
Further increasing F , a subsonic region appears (vη < cis for F = 1.05 % and 5%). Whenflow changes its nature from supersonic to subsonic, a shock wave appears. R = 10 kpc,
cs = 8 km s−1 , and i = 6.7 deg are taken. Other parameters are the same as Fig. 3.14.
(Right:) Comparison of two density distributions: there are no jumps but only cusps for
F < 0.97 %, while shock fonts and high compressions in the postshock region appear formodels with F = 5 %. Taken from Shu, Milione, & Roberts (1973).

83

Chapter 4

Local Star Formation Process
4.1

Hydrostatic Balance

Consider a hydrostatic balance of isothermal cloud. By the gas density, ρ, the isothermal soundspeed, cis , and the gravitational potential, Φ, the force balance is written as
c2 dρ dΦ
− is
−
= 0,
ρ dr
dr
and the gravity is calculated from a density distribution asZ
GMr
4πG r 2dΦ−=− 2 =− 2
ρr dr,
dr
r
r
0

(4.1)

(4.2)

for a spherical symmetric cloud, where Mr represents the mass contained inside the radius r. The
expression for a cylindrical cloud is
Z
dΦ
2Gλr
4πG r
−
=−
=−
ρrdr,
(4.3)
dr
r
r
0
where λr represents the mass per unit length within a cylinder of radius being r (line mass).
For the spherical symmetric case, the equation becomes the Lane-Emden equation with the polytropic index of ∞ (see Appendix D.1). This has no analytic solutions. However, the numericalintegration gives us a solution shown in Figure 4.1 (left). Only in a limiting case with the infinite
central density, the solution is analytically given as
ρ(r) =

c2is −2
r .
2πG

(4.4)

Increasing the central density, the solution reaches the above singular isothermal solution (SIS).
On the other hand, a cylindrical cloud has an analytic solution (Stodólkiewicz 1963; Ostriker1964) as
−2r2
ρ(r) = ρc 1 +
,
(4.5)
8H 2
where
H 2 = c2is /4πGρc .
85

(4.6)

86

CHAPTER 4. LOCAL STAR FORMATION PROCESS
1

1’fort.4’’1.1.d’’1.d’’0.9.d’

0.1

0.1

0.01

0.01

0.001

0.001

0.00010.0001

1e-05
0.01

0.1

1

10

100

1000

1e-05
0.01

0.1

1

10

100

1000

Figure 4.1: Radial density distribution. A spherical cloud (left) and a cylindrical cloud (right). In theright panel, solutions for polytropic gases with Γ ≡ 1 + 1/n = 1.1 or n = 10 (a relatively
compact curve plotted with “+”) and Γ = 0.9 or n = −10 (relatively extended curve with“*”) are plotted as well as the isothermal one (middle curve with “×”).

Far from the cloud symmetric axis, the distribution of equation (4.5) gives
ρ(r) ∝ r−4 ,(4.7)

ρ(r) ∝ r−2(4.8)

while the spherical symmetric cloud has

distribution.

Problem 10

Show that the SIS is a solution of equations (4.1) and (4.2).
Problem 11Show that the density distribution of equation (4.5) is a solution of equations (4.1) and (4.3).


4.1.1



Bonnor-Ebert Mass

In the preceding section [Fig. 4.1 (left)], we have seen the radial density distribution of a hydrostatic
configuration of the isothermal gas. Consider a circumstance that such kind of cloud is immersed
in a low-density medium with a pressure p0 . To establish the pressure equilibrium, the pressure
at the surface c2is ρ(R) must equal to p0 . This means that the density at the surface is constantρs = ρ(R) = p0 /c2is .
Figure 4.2 (left) shows three models of density distribution, ρc = ρ(r = 0) = 10ρs , 102 ρs , and3
10 ρs . Comparing these three models, it should be noticed that the cloud size (radius) decreases withincreasing the central density ρc . The mass of the cloud is obtained by integrating the distribution,
which is illustrated against the central-to-surface density√ ratio ρc /ρs in Figure 4.2 (right). The y-axis
represents a normalized mass as m = M/[4πρs (cis / 4πGρs )3 ]. The maximum value of m = 4.026
means
c2is
.
(4.9)
Mmax ≃ 1.14
1/2
G3/2 p0

4.1. HYDROSTATIC BALANCE

87

This is the maximum mass which is supported against the self-gravity by the thermal pressure with anisothermal sound speed of cis , when the cloud is immersed in the pressure p0 . This is called BonnorEbert mass [Bonnor (1956), Ebert (1955)]. It is to be noticed that the critical state Mcl = Mmax isachieved when the density contrast is rather low ρc ≃ 16ρs ≡ ρcr .
Problem 12
Show that the normalize form of equations (4.1) and (4.2) are represented as
1 df
f dξ

=

m
,
ξ2
Z ξ
f ξ 2 dξ,m =03
where nondimensional variables√ are defined as ξ ≡ r/H, f ≡ ρ/ρs , and m ≡ M/(4πρs H ) using
the scale-length of H = cis / 4πGρs .

Problem 13

Numerically solve the above simultaneous differential equations and obtain the critical mass [eq.
(4.9)] and the central density of the critical state, ρcr . Boundary conditions of the above equations
are: at the center (ξ = 0), m = 0 and f = 1.
It should be noted that a gas cloud with Mcl < Mmax with isothermal sound speed cis andimmersed in the external pressure pext will have a mechanical balance between the pressure and theself-gravity (mechanical equilibrium). However, that with Mcl > Mmax is not supported against the
self-gravity. The former is called “subcritical” and the latter “supercritical.” Later, we will show
the critical mass is increased by rotation (centrifugal force) and magnetic field (Lorentz force).
Another important result from Figure 4.2 (right) is the stability of an isothermal cloud. Even
for a subcritical cloud with Mcl < Mmax , any clouds on the part of (∂Mcl /∂ρc )p0 < 0 are unstable,
whose clouds are distributed on the branch with ρc > ρcr . This is understood as follows: For ahydrostatic cloud, the mass should be expressed with the external pressure and the central density[Fig.4.2 (right)] as
Mcl = Mcl (p0 , ρc ).
(4.10)In this case, a relation between the partial derivatives such as






∂Mcl
∂p0
∂ρc
·
·
= −1,
∂p0 ρc∂ρc Mcl∂Mcl p0(4.11)

is satisfied, unless each term is equal to zero. Figure 4.1 (left) shows that the cloud mass Mcl is adecreasing function of the external pressure p0 = ρs c2is , if the central density is fixed. Since this means


∂Mcl
< 0,
(4.12)∂p0 ρcequation (4.11) gives us



∂p0
∂ρc




·
Mcl

∂ρc
∂Mcl


> 0.

(4.13)

p0

For a cloud with ρc < ρcr ≃ 16p0 /c2is , mass is an increasing function of the central density as


∂Mcl
> 0.
(4.14)∂ρc p0

88

CHAPTER 4. LOCAL STAR FORMATION PROCESS
1000

5’F1000.d’’F100.d’’F10.d’’rhoc-M.d’

4

100

M

rho

3

2
10

1

1
0.001

0
0.01

0.1
x

1

10

1

10

100
rho

1000

10000

Figure 4.2: (Left:) radial density distribution. Each curve has √different ρc . The x-axis denotes theradial distance normalized by a scale-length as c2is / 4πGp0 . It is shown that the radiusincreases with decreasing ρc in this range (ρc = (10, 102 , 103 ) × p0 /c2is ). (Right:) The
relation between mass (Mcl ) and the central density (ρc ) is plotted, under the conditionof constant external pressure. The x-axis represents the central density √normalized byρs ≡ p0 /c2is . The y-axis represents the cloud mass normalized by 4πρs (cis / 4πGρs )3 .

Thus, equation (4.13) leads to the relation


∂ρc
∂p0


> 0,

(4.15)

Mclfor ρ < ρcr . This means that gas cloud contracts (the central density and thus pressure increase),
when the external pressure increases. This is an ordinary reaction of a stable gas.
On the other hand, the cloud on the part of (∂Mcl /∂ρc )p0 < 0 (for 16 ≲ ρc /(p0 /c2is ) ≲ 2000)
behaves


∂ρc
< 0,
(4.16)∂p0 Mcland this represents that an extra external pressure decreases the central density and thus the pressure.
This means the pressure encourages expansion. This reaction is unstable.

4.1.2

Equilibria of Cylindrical Cloud

In Figure 4.1 (right), we plotted the structure for a polytropic cylindrical cloud. Although inner
structure is not dependent of the polytropic exponent, Γ, it is clear the slope of the outer envelope isdependent on Γ.
1. In the spherically symmetric case, consider a polytrope (p ∝ ρΓ ) with Γ < 6/5 (at least theenvelope of Γ = 6/5 cloud extends to r → ∞.), in which gas extends to r → ∞. If ρ ∝ r−p ,the mass inside of r is proportional to Mr ∝ r3−p . Thus, the gravity per unit volume at aradius r, GM ρ/r2 , is proportional to GM ρ/r2 ∝ r1−2p . On the other hand the pressure force is|(∂p/∂r)| = (∂p/∂ρ)|(∂ρ/∂r)| ∝ (r−p )Γ−1 r−p−1 ∝ r−pΓ−1 . These two powers become the same,
only if p = 2/(2 − Γ).
2. In the case of cylindrical cloud with Γ ≤ 1, which is called negative index polytrope N =1/(Γ − 1) < 0 or Γ = 1 + 1/N ≤ 1, the mass per unit length λ ∝ r2−p . The gravity at r,
Gλρ/r ∝ r1−2p . Note that the power is the same as the spherical case. Since the power ofthe pressure force should be the same as the spherical case, the resultant p should be the same

4.2. VIRIAL ANALYSIS

89

p = 2/(2 − Γ). Singular solutions with infinite central density ρc = ∞ are expressed by thepower law as ρ(r) ∝ r−2/(2−Γ) ∝ r−2N/(N −1) (Viala & Horedt 1974a,b). For the isothermal gasof Γ → 1 or N → ∞, it should be noticed that the power index is −2 different from that of theregular solution, −4 [Eq.(4.5)].
For 0 ≤ Γ ≤ 1 or N ≤ −1, polytropic cylinders have infinite radius and line mass. On the other hand,Γ > 1 or N > 0 the density and thus the pressure goes to zero at a finite radius [see, e.g., Toci &Galli (2015a)]. The case of Γ = 0.9, an envelope extending to a large radius indicates the power-law
distribution much shallower than that of the isothermal Γ = 1 one (Fig. 4.1).
As is mentioned earlier, for the isothermal cylindrical hydrostatic equilibrium, which is governedby the basic equations of (4.1) and (4.3), the solution is given analytically by Stodólkiewicz (1963)
and Ostriker (1964) as shown in equation (4.5). Equation (4.5) gives the line mass inside of the radiusr asZ rλ(r) =
ρ(r)2πrdr
0=2c2is r2 /8H 2.
G 1 + r2 /8H 2(4.17)

This is a monotonically increasing function of r and the maximum line mass is given as λcr = 2c2is /G.
When an isothermal cylinder has a line mass larger than 2c2is /G, this cylinder has no equilibrium
configuration and it must begin to contract radially. The cylinder with λ > λcr is called supercritical,
while less massive one with λ < λcr is called subcritical.

4.2

Virial Analysis

Hydrodynamic equation of motion using the Lagrangean time derivative [eq. (B.3)] is dvρ= −∇p − ρ∇Φ.
dt

(4.18)

Multiplying the position vector r and integrating over a volume of a cloud, we obtain the virial
relation as
1 d2 I= 2(T − T0 ) + W,
(4.19)
2 dt2
where
Z
I = ρr2 dV,(4.20)Z 

is the inertia of the cloud,
T =


1 2
3
3
pth + ρv dV = p̄Vcl ,22
2

is a term corresponding to the thermal pressure plus turbulent pressure,Z
3
pth r · ndS = p0 Vcl
T0 =
2
S

(4.21)(4.22)

comes from a surface pressure, p0 , andZW =−ρr · ∇ΦdV = −3 GM 2
5 R(4.23)

90

CHAPTER 4. LOCAL STAR FORMATION PROCESS

is the gravitational energy. To derive the last expression in each equation, we have assumed the cloudis spherical and uniform. Here we use a standard notation as the radius R, the volume Vcl = 4πR3 /3,
the average pressure p̄, and the mass M .
To obtain a condition of mechanical equilibrium, we assume d2 I/dt2 = 0. Equation (4.19) becomes
4π p̄R3 − 4πp0 R3 −3 GM 2= 0.
5 R(4.24)

Assuming gas is isothermal p = c2is ρ, the average pressure is written asp̄ = c2is ρ̄ = c2is

3M
.
4πR3

(4.25)

Using equation (4.25) to eliminate p̄ from equation (4.19), the external pressure is related to the massand the radius as
3c2 M
3GM 2
p0 = is 3 −.
(4.26)
4πR
20πR4
Keeping M constant and increasing R from zero, p0 increases first, but it takes a maximum, p0,max =3.15c8is /(G3 M 2 ), and finally declines. This indicates that the surface pressure must be smaller thanp0,max for a cloud to be in the equilibrium. In other words, keeping p0 and changing R, it is shownthat M has a maximum value to have a solution. The maximum mass is equal toMmax = 1.77

c4is
1/2

G3/2 p0.

(4.27)

Clouds massive than Mmax cannot be supported against the self-gravity. This corresponds to the
Bonnor-Ebert mass [eq. (4.9)], although the numerical factors are slightly different.

4.2.1

Magnetohydrostatic Clouds

Consider here the effect of the magnetic field. In the magnetized medium, the Lorentz force
F =

1
1
1
(∇ × B) × B = − ∇(B · B) +(B · ∇)B
4π
8π
4π(4.28)

works in the ionized medium, where we used vector formula of ∇(A · B) = (B · ∇)A + (A · ∇)B + A ×(∇ × B) + B × (∇ × A). The first term of equation (4.28), which is called the magnetic pressure,has an effect to support the cloud against the self-gravity.
The virial analysis is also applicable to the magnetohydrostatic clouds. The terms related to themagnetic fields are
Z
Z
Z
B2
B2
M =
dV + (r · B)B · ndS −r · ndS
8π
S
S 8π


Z
Φ2B
Φ2
B 2 − B02
1
(4.29)≃
dV ≃ 2
− B ,
8π
6π
R
R0
where ΦB represents a magnetic flux and it is assumed to be conserved if we change the radius, R,
that is ΦB = πB0 R02 = πBR2 . Equation (4.24) becomes
4π p̄R3 − 4πp0 R3 −3 GM 21 Φ2+ 2 B = 0,
5 R
6π R

(4.30)

4.2. VIRIAL ANALYSIS

91
Φ2

where we ignored the term 6π1 2 RB0 . The last two terms are rewritten as3G
M 2 − MΦ2 ,5R
22
where MΦ is defined as 3GMΦ /5 = ΦB /6π 2 .
This shows the effects of the magnetic fields:(4.31)

1. B-fields effectively reduce the gravitational mass as M 2 −MΦ2 = M 2 −5Φ2B /(18π 2 G). This plays
a part to support a cloud.
2. However, even a cloud contracts (decreasing its radius from R0 to R), the ratio of the gravitational to the magnetic terms keeps constant since these two terms are proportional to∝ R−1 . Thus, if the magnetic term does not work initially, the gravitational term continues topredominate over the magnetic term.
If M < MΦ , a sum of last two terms in equation (4.30) is positive. Since the second term of rhs ofequation (4.26) is positive, there is one R which satisfies equation (4.30) irrespective of the externalpressure p0 . While, if M > MΦ , there is a maximum allowable external pressure p0 . Therefore,
M = MΦ gives a criterion whether the magnetic fields work to support the cloud or not. Morerealistic calculation [Mouschovias (1976a, 1976b), Tomisaka et al. (1988)] gives us a criteriondm
G1/2 σ1== 0.17 ≃,(4.32)dΦBB
2π
where σ and B means the column density and the magnetic flux density. A cloud with a massΦB
(4.33)
M>
2πG1/2
is sometimes called magnetically supercritical, while that with
ΦB
M<
(4.34)
2πG1/2is magnetically subcritical.
More precisely speaking, the criterion showed in equations (4.33) and (4.34) should be applied fora cloud which has a much larger mass than the Bonnor-Ebert mass. That is, even without magnetic
fields, the cloud less-massive than the Bonnor-Ebert mass has a hydrostatic configuration shownin Figure 4.2 (left). The cloud with central density of ρc = 10 has a stable density distribution.
Magnetohydrostatic clouds with different magnetic fluxes are calculated by Mouschovias (1976a,
1976b) and Tomisaka et al. (1988)]. Mass of the cloud is obtained against the central density, which
is an increasing function of magnetic flux (Fig. 4.3). The maximum allowable mass (critical mass)
supported by some magnetic flux increases with the magnetic flux. To fit the numerical results,
Tomisaka et al. (1988) obtained an expression for the critical mass when the cloud has a mass-to-flux
ratio dm/dΦB , the isothermal sound speed cis , and the external pressure p0 as(2 )−3/2
1/2πc4s
Mcr = 1.3 1 −.
(4.35)
1/2
G1/2 dm/dΦB |r=0
p0 G3/2
G1/2This shows that the critical mass is a decreasing function of the mass-to-flux ratio or increasing
function of the magnetic flux. And the critical mass becomes much larger than the Bonnor-Ebert
1/2
mass ≃ c4s /(p0 G3/2 ) only when the mass-to-flux ratio at the center of the cloud is reaching 1/2π
at which the term in the curry bracket goes to zero. Hereafter, we call here the cloud/cloud corewith mass larger than the critical mass Mcr a supercritical cloud/cloud core. The cloud/cloud coreless-massive than the critical mass is subcritical.

92

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.3: Cloud mass is plotted against the central density. Respective curves correspond to clouds
with different magnetic flux. Plasma beta of the external medium (β0 ) specifies themagnetic flux. Model with β0 = ∞ corresponds to that shown in Fig. 4.2 (right).

4.2. VIRIAL ANALYSIS

4.2.2

93

Equilibrium of FilamentIn §4.1.2, we have seen the hydrostatic configuration of isothermal cylindrical cloud [eq. (4.17)]. Whenthe magnetic field lines run in the parallel direction to the cylindrical axis, hydrostatic equilibrium
solution is obtained by Stodólkiewicz (1963) for a special model with constant plasma β as β ≡B 2 /8πρc2is = const. In this case, since the magnetic pressure gradient plays a role similar to thethermal pressure gradient, larger mass must be supported against
R r the self-gravity compared withequation (4.17). The line mass contained inside radius r (λ(r) ≡ 0 ρ2πrdr) is expressed as
λmag,// (r) =2
 r2 /8Hmag
2c2is,
1 + β −12G
1 + r2 /8Hmag(4.36)

where Hmag represents the scale-height at the center of the filament as Hmag = cis (1+β −1 )/(4πGρc )1/2
with the central density ρc . Comparing equation (4.36) with equation (4.17), the magnetic filament
is more massive than the non-magnetic model (λmag,// > λ). However, observations (e.g., Fig. 1.24)
indicate that the interstellar magnetic field is not parallel but perpendicular to the filaments. Since
the Lorentz force must play a role in supporting the filament, the line-mass of the filament extending
perpendicular to the interstellar magnetic field, λmag,⊥ , must be larger compared with the nonmagnetic filament, too.
Using similar formulation shown in Appendix D.2, Tomisaka (2014) obtained magnetohydrostatic
configurations of isothermal filaments with perpendicular magnetic field. Figure 4.4 (left) indicates atypical solution of the filament, in which width of the filament perpendicular to the magnetic field iswider than that parallel to the field lines. Comparing the solutions with identical magnetic flux, thelarger the central density ρc the larger the line-mass λ [Fig. 4.4 (middle)]. Although this corresponds
to Figure 4.3, which is the mass-density relation for the axisymmetric isothermal 3D cloud, there is aclear difference in 2D filament: M (Φ, ρc ) takes maxima at a finite ρc (isothermal axisymmetric cloud )while λ(Φ, ρc ) increases monotonically with ρc (isothermal ﬁlament). Maximum line-mass which canbe supported by thermal pressure and the Lorentz force against the self-gravity increases with themagnetic flux [Fig. 4.4 (right)] as
λmax ≃ 0.24

c2is
Φcl
+
1.66
,
G
G1/2

(4.37)

where Φcl represents one half of the magnetic flux threading the filament per unit length. Thisequation is for dimensional quantities of λ, Φcl , and cis . The second term of rhs represents thethermal contribution ≃ 2c2is /G and the first term comes from the magnetic pressure and tension.
As for the stability of the filament, similar discussion to §4.1.1 (see the discussion of p.86) can be
possible. That is, when the line-mass, λ, increases with the central density, ρc , as (∂λ/∂ρc )Φcl ,cis > 0,
the system is stable against the radial contraction/expansion of the filament. Figure 4.4 (middle)
shows that (∂λ/∂ρc )Φcl ,cis > 0 in all the parameter ranges, which is slightly different from the 3D
isothermal cloud. This is a story of the isothermal filament. Considering negative index polytropes
obeying P = Kρ1+1/N = KρΓ with −∞ < N < −1 or 0 < Γ < 1, some models indicate a peakline-mass λ at a finite central density ρc < ∞ (Kashiwagi & Tomisaka 2021).

4.2.3

Subcritical vs Supercritical Filaments

We have seen that there is a critical mass above which the cloud has no (magneto)hydrostatic configuration but below which the cloud has at least one equilibrium state. This gives us an idea that
there are two kind of clouds/cloud cores: that with a mass larger than the critical mass which has to94

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.4: (Left:) Typical solution of isothermal filaments. Magnetic field lines run in the y-direction
in large scale and the filament extends in z-direction. Density contour (solid line) andmagnetic field lines (dashed line) are plotted for the cross-section of the filaments. Parameters of this model are R0 = 2, β0 = 1, and ρc = 100. (Middle:) Relation between theline-mass λ and the central density ρc . In the cylindrical geometry, λ looks to increase
monotonically toward λmax against ρc for the isothermal equation of state. Attachednumbers represent (R0 , β0 ). (Right:) Maximum line-mass λmax is plotted against the
1/2
magnetic flux per unit length Φcl ≡ R0 /β0 . Numerically obtained λmax (asterisks andcrosses) is well fitted by λmax = 4.3Φcl + 20.8 with use of non-dimensional quantities.
Taken from Tomisaka (2014).

collapse dynamically and that with a mass smaller than the critical mass which is in an equilibrium
state. In the density range of 104 cm−3 ≲ n ≲ 1010 cm−3 , the interstellar gas is essentially isothermal.
In this region a major cooling agent is dusts; that is, the dust is heated by the collision of molecules.
The excess energy liberated at the collision increases the dust temperature. Finally the thermal
emission from the dust cools down the dust again. By this process, the thermal energy of the gas
is reduced. Therefore, we consider the cloud/cloud core is isothermal and study the collapse of theisothermal cloud.
Since the supercritical cloud has no hydrostatic configuration, it must evolve in a dynamical way.
Radial contraction of such massive non-magnetic filaments is studied by Kawachi & Hanawa (1998).
They obtained semi-analytic self-similar solutions and compared with the hydrodynamical simulations
for polytropic exponent of 0 < Γ < 1. In the limit of Γ → 1 (Γ = 0.999), solutions realized before
the protostar formation have special character such as: ρ = ρc (1 + Ar2 )−2 -type density distribution
for the central part extends and reduces to the contracting envelope of ρ ∝ r−2 . This is a cylindricalcounter-part of Larson-Penston solution appeared in spherical system (see p.104).
On the other hand, since the subcritical cloud is in a static state, it evolves in much longer timescale of the free-fall time. Such cloud evolves by the effect called the ambipolar diffusion, a kind ofmagnetic diffusion, with which the magnetic flux escapes from the central part of the gas cloud.

4.2.4

Observation of Magnetic Field Strength

To determine whether a cloud is magnetically subcritical or supercritical, we have to measure thestrength of the magnetic field.

4.2. VIRIAL ANALYSIS

95

Figure 4.5: Magnetic field strength is plotted against cloud gas density. The magnetic field strengthof the parallel component to the line-of-sight is measured using the Zeeman splitting.

Zeeman Splitting Method
If the atoms and molecules are permeated by magnetic field B, in the normal Zeeman splitting, thetransition splits into three lines
ν1 = ν0 − β|B|,(4.38)

ν2 = ν0 ,(4.39)

ν3 = ν0 + β|B|,(4.40)

where β = eℏ/2me = 1.3996 Hz µG−1 is the Bohr magneton constant. This comes from the fact thatthe upper and lower levels are splitted into Um = ml βB using the magnetic quantum number ml .
Hi λ = 21cm, OH (maser emissions and thermally excited emissions), H2 O maser emissions are used
to measure the magnetic field strength. Figure 4.5 represents the correlation between the magnetic
field strength and gas density compiled by Fiebig & Guesten (1989), which indicates that Blos isapproximately proportional to ρ1/2 .
Davis–Chandrasekhar-Fermi Method
Davis (1951) and Chandrasekhar & Fermi (1953) proposed an idea to obtain the strength of interstellar
magnetic field using the character of the Alfvén wave. Applying the same method to the molecular
cloud, Crutcher et al. (2004) obtained the magnetic field strength in the plane of the sky in threeprestellar clouds as ≃ 80 µG (L183), ≃ 140 µG (L1544), and ≃ 160 µG (L43). The method is asfollows: Using the phase velocity of transverse Alfvén wave, VA = B/(4πρ)1/2 , the wave is describedasy = A cos [k(x − VA t)] ,(4.41)
where y is a displacement in the plane of the sky and perpendicular to the mean magnetic field. Thisgives
 2  2
∂y
∂y
VA2
=
,
(4.42)∂x∂t
where X represents the time average. The rhs of the equation, which is the one-dimensional velocity
dispersion in y direction, seems to be equal to the velocity dispersion in the line-of-sight ∆V as 2∂y= ∆V 2 .
(4.43)∂t

96

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.6: Mass-to-flux ratio (λc ≡ µc ) is plotted against the column density (NH2 ). The y-axis represents the mass-to-flux ratio normalized with the critical one. Dots are obtained from the
Zeeman splitting measurements (Blos ) and stars are based on the Davis–ChandrasekharFermi method (Bpos ).

4.3. AMBIPOLAR DIFFUSION

97

The lhs of the equation is given with a dispersion of polarization direction ∆ϕ (rad)


∂y
∂x

2
= ∆ϕ2 ,

(4.44)

in which the lhs is measured by the wave-pattern of the Alfvén wave. Therefore, the magnetic fieldstrength on the plane of the sky is given as
Bpos = (4πρ)1/2

4.2.5

∆V
∆ϕ

(4.45)

Observation of Mass-to-Flux Ratio

Figure 4.6 shows the mass-to-flux ratio normalized with the critical value
µc ≡
where



M
Φ

(M/Φ)obs(M/Φ)crit


=
crit

1
2πG1/2(4.46)(4.47)

is the critical mass-to-flux ratio which is obtained numerically from equation (4.32). µc > 1 represents
a supercritical cloud while µc < 1 represents a subcritical cloud.
Figure 4.6 indicates cloud cores are more or less found near the critical mass-to-flux ratio. They
are not distributed either in the regions µc ≫ 1 (very supercritical) or µc ≪ 1 (very subcritical).

4.3

Ambipolar Diﬀusion

4.3.1

Ionization Rate

In the dense clouds, the ionization fraction is low. Since the uv/optical radiations from stars can notreach the cloud center, potential ionization comes from the cosmic ray particles. In this case the rateof ionization per volume is given as ζnn , where ζ represents the ionization rate due to the cosmic
ray and typically we take ζ ∼ 10−17 s−1 . In Figure 4.7, the ionization fraction for various density isshown (Nakano, Nishi, & Umebayashi 2002). This clearly shows that the fraction of ions decreases−1/2
approximately in proportion to nH
for nH ≲ 108 cm−3 . This is understood as follows: Equilibriumbalance between one kind of ion m+ and its neutral m0 ,m0 ↔ m+ + e− ,(4.48)

is considered. The recombination (reaction from the right to left) rate per unit volume is expressed
αne nm+ , while the ionization rate (left to right) per unit volume is ζnm0 , where α means the recombination rate coefficient. Nakano (1984) obtained in the range of 102 cm−3 ≲ n ≲ 108 cm−3
ρi = Cρ1/2
n ,where the numerical factor C = 4.46 × 10−16 g1/2 cm−3/2 .

(4.49)

98

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.7: Ionization fractions X/nH are plotted against the number density nH . The ionization rateof an H2 molecule by cosmic rays outside the cloud is taken ζ0 = 1 × 10−17 s−1 . 20% of C
and O and 2% of metallic elements are assumed to remain in the gas phase and the rest in
grains. M+ represents the metallic ions such as Mg + , Si+ , and Fe+ , while m+ represents+
molecular ions other than H+
3 , e.g., HCO . Taken from Nakano, Nishi, & Umebayashi(2002).

4.3. AMBIPOLAR DIFFUSION

4.3.2

99

Ambipolar Diﬀusion

The ionization fraction of the gas in the cloud is low. Neutral gas and the ions are coupled via
mutual collisions and ions are coupled with the magnetic field. Thus, the neutral molecules, a major
component of the gas, are coupled with the magnetic field indirectly via ionized ions. Assuming
ρn ≫ ρi and ρ = ρn , equation of motion for the molecular gas isρdvn= −c2s ∇ρ − ρ∇ψ + ρrω 2 er + F ,dt(4.50)

where the forces appeared in the rhs represent, respectively, the pressure force, the self-gravity, thecentrifugal force, and the force exerted on the neutral component per unit volume through two-bodycollision with ions. The friction force has a form
F = ρi ρn γ(vi − vn ).

(4.51)We have assumed ρ ≃ ρn since the mass density of the charged component is much smaller than thatof the neutral one. On the other hand, that for ions is
ρi

dvi
1
=
(∇ × B) × B − F ,dt
4π(4.52)

where we ignored the pressure and self-gravity forces compared with the Lorentz force. If the inertiaof the ions are ignored [lhs of eq. (4.52)=0], equations (4.50) and (4.52) give the equation of motion
similar to that of the one-fluid as follows:ρdv1= −c2s ∇ρ − ρ∇ψ + ρrω 2 er +(∇ × B) × B,
dt
4π

(4.53)

where we used the fluid velocity v ≡ (ρn vn + ρi vi )/(ρn + ρi ) ≃ vn .
Ignoring the inertia of the ions, equation (4.52) indicates that the friction force should be balancedwith the Lorentz force as1
F =(∇ × B) × B.
(4.54)
4π
Using equations (4.49) and (4.51), this gives the drift velocity of ions against the neutrals as
vD ≡ vi − vn ,F=,
ρi ρn γ1=(∇ × B) × B,
4πρi ρn γ
1(∇ × B) × B.
=
3/2
4πCγρn(4.55)The term (∇ × B) × B/4π represents the Lorentz force. Equation (4.55) indicates the drift velocity
vD = vi − vn is toward the direction of the Lorentz force. In magnetized cloud, the Lorentz force
works outwardly, the drift velocity vD also points outwardly. Viewing from the ions and magnetic
field lines, vn − vi = −vD points inwardly. Thus, the mass in a flux tube varies with time; that of thecloud center increases while that of the cloud surface decreases. The ambipolar diffusion is a process
to change the mass distribution against the magnetic flux tube.

100

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.8: Relation central density ρc against mass-to-flux ratio M/Φ. Plane-parallel and axisymmetric models are illustrated in the left and right panels, respectively.

It is useful to derive the timescale of ambipolar diffusion. The characteristic timescale of magnetic
flux loss is given as τA = L/|vD |, where L represents the size of the cloud. This gives
4πCγρ3/2 L|(∇ × B) × B|
3/2 
−2 
2
n
B
L
6
≃ 3 × 10 yr.
104 H2 cm−330 µG
0.1 pcτA ≃(4.56)(4.57)

This ambipolar diffusion timescale is several times longer than the dynamical contraction timescale[eq. (2.26)] or free-fall timescale as
3π 1/2tﬀ =(4.58)
32Gρ̄

−1/2
n
6
≃ 0.5 × 10 yr.
(4.59)
104 H2 cm−3
Thus, the magnetic flux of the cloud is not be lost during the dynamical contraction (≃ free-fall).
How is the structure affected by the increase of the mass-to-flux ratio at the cloud center? Here,we will derive the relation between the mass-to-flux ratio and central density and answer the question. Numerical calculation is necessary for the axisymmetric (realistic) cloud. However, this can beobtained for the case of a plane-parallel disk in which the magnetic field is running parallel to the
disk. For the plane-parallel disk the hydrostatic balance is achieved as


∂
B2
∂ψ
p+
= −ρ
= −2πGρσ,(4.60)∂z8π∂zRz
where σ = −z ρdz. Equation (4.60) is integrated to obtain the relation between the variables at thecenter z = 0 as


Bs2
πG 2
Bc2
− ps +=pc +σ ,(4.61)
8π
8π2 s

4.3. AMBIPOLAR DIFFUSION

101

where we used the relation dσ/dz = 2ρ. Quantities with subscript c and s represent those at thecenter and at the surface, respectively. This is rewritten as
 
 
 
ρc
1 Bc 2 ρs ρc 2
1 Bs 2 ρs
πGσs2
+
−
=
+ 1.
(4.62)
ρs 8π ρcc2s ρs8π ρsc2s2ρs c2s
1/2

In the second term of lhs, yc ≡ (ρc /Bc )(cs /ρs ) is a nondimensional mass flux ratio at the center andthis is related to the center-to-surface density ratio ρ′c ≡ ρc /ρs . Using these nondimensional variables,
equation (4.62) is expressed as1(ρ′ )2= ρ̄′c ,(4.63)
ρ′c + c 2 −
8πyc
8πys2
where ρ̄′c ≡ πGσs2 /(2ρs c2s ) + 1 represents the central density necessary for the disk to be supported
without magnetic field (y → ∞). Figure 4.8 (left) illustrates this relation. This shows that if thecentral mass-to-flux ratio increases (moving upward), the central density increases. In low density,
ρ′c ≪ ρ̄′c , equation (4.63) indicates that ρ′c ∝ yc ∝ ρc /Bc . This means that on the low density regime,
the density increases but the strength of magnetic field does not increase. Equation (4.63) also
indicates that yc increase much as yc → ρ̄′c ys when ρ′c → ρ̄′c . This relation between mass-to-flux ratio
and the central density is well fitted to the numerical calculation of plane-parallel disk cloud driven
by the ambipolar diffusion by Mouschovias, Paleologou, & Fiedler (1985). Does this indicate that the
mass-to-flux ratio increases much when the central density increases, for example, from 10 4 cm−3 to106 cm−3 ?
From this point, quasistatic evolution of the magnetized cloud was first studied by Nakano(1979) adopting a method seeking for magnetohydrostatic configuration (Mouschovias 1976a, 1976b).
Nakano (1979) obtained that the mass to magnetic flux ratio (µ) near the center of the cloud does
not increase much before the central density reaches a critical density above which magnetohydrostatic configurations no more exist. Paleologou & Mouschovias (1983), Shu (1983), and Mouschovias,
Paleologou, & Fiedler (1985) gave a completely different result. That is, the mass to magnetic flux
ratio near the center increases much when the density becomes larger than n ≳ 106 cm−3 . Critical
discussion has been done between them1 . However, we have now realized that the difference comesfrom the geometry assumed.
Relation of the mass-to-flux ratio to the central density for the axisymmetric realistic (not planeparallel disk) cloud is plotted in Figure 4.8 (right) (Tomisaka, Ikeuchi, & Nakamura 1990). The
central mass-to-flux ratio increases by the ambipolar diffusion. This increases the central density.
Lines with arrows indicate the quasistatic evolutionary paths driven by the ambipolar diffusion. It is
to be noticed that the relation is completely different from that of the plane-parallel disk. Increase of
the mass-flux ratio is small although the central density increases much. This indicates that in thequasistatic evolution driven by the ambipolar diffusion the mass-flux ratio does not increase much incontrast to the plane-parallel case.
In fully ionized plasma, the magnetic field is coupled to the matter. However, in the low ionizedgas the magnetic field is coupled only to the ions. Thus the induction equation of magnetic fieldshould be written∂B+ ∇ × (vi × B) = 0.
(4.64)∂t1

One of the reasons why large increase in the mass/ﬂux ratio was favored is understood as follows: There has
been a long-standing “magnetic ﬂux problem of stars” in which the magnetic ﬂux of, say, 1 M⊙ main-sequencestar Φ ∼ 5 × 1026 G cm2 (R∗ /2R⊙ )2 (B0 /3 × 104 G) is much smaller than that of the parent cloud Φ ∼ 3 ×1031 G cm2 (Rcl /0.1 pc)2 (B0 /100 µG). The magnetic ﬂux must be reduced in the star formation process. If the mass-ﬂux
ratio increases much in these density ranges, this might resolve the magnetic ﬂux problem of stars.

102

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.9: (Left:) Hall parameters for ions and electrons. Two boundaries come from the conditionof ions’ Hall parameter βi = 1 and that of electrons’ βe = 1, both of which are given by
B/ρ =const. Generally, magnetic field strength B increases in proportion to ρ1/2 in thecontraction. In consequence, contraction proceeds from ambipolar diffusion region to theohmic dissipation region. Taken from Wardle (2007). (Right:) Direction of drift velocity
for various Hall parameter. Taken from Wardle (2004).

This leads to the final expression as∂B+ ∇ × (vn × B) + ∇ ×
∂t

(

)

1
3/2

4πCγρn[(∇ × B) × B] × B= 0,(4.65)

where we used the expression of the drift velocity [eq. (4.55)]. In the isothermal cloud, there isa maximum mass supported by the magnetic field. Since in the subcritical cloud M < Mcr themagnetic flux escapes from the center, the critical mass decreases with time. After the critical mass
becomes smaller than the actual mass (Mcl > Mcr ), there is no hydrostatic configuration. The cloud
evolves into the supercritical cloud region and experiences dynamical collapse.

4.3.3

Dissipation of Magnetic Field by Ambipolar Diﬀusion, Hall Eﬀect, andOhmic Dissipation∗

Beside the ambipolar diffusion process, there are several processes which induce dissipation of magnetic field in the low ionization plasma. These processes break the ideal MHD condition and flux
freezing character (see C.1.1). Magnetic dissipation is by the resistivity of the plasma. Measuring
the resistivity, we begin with the drift of charged particles against the neutrals. Drift is controlled
by the balance between the electromagnetic force and momentum exchange with neutral molecules
via mutual collisions (Wardle 2007). Considering ionized species named j (ions and electrons), thebalance is expressed asvj
Zj eE ′ + Zj e × B − mj γj ρvj = 0,(4.66)cwhere E ′ and vj represent the electric field in the frame of neutral molecules, velocity of j-th charged
species (mass mj and charge Zj e) relative to the neutral molecules (drift velocity), respectively.
Momentum exchange rate between j-th charged species and neutral molecules are given using
γj = ⟨σv⟩j /(mj + m).

(4.67)In deriving equation (4.66), we assumed that the balance is achieved in much shorter timescale thanthe macroscopic evolution timescale. Drift direction is controlled by the ratio of the second to the4.3. AMBIPOLAR DIFFUSION

Figure 4.10:

103

′
′
(Left:)
Definition of E//
and E⊥.
(Right:)Electric current J is divided into three parts: current parallel to B is equal to σ0 E// , that flowing in the direction of electric field perpendicular to B as σ1 E⊥ , and thatflowing perpendicular to both E⊥ and B as σ2 [(B/|B|) × E⊥ ].
Taken fromhttp://wdc.kugi.kyoto-u.ac.jp/ionocond/exp/icexp.html.

third terms, which is called Hall parameter asβj =|Zj |eB 1.
mj c γj ρ(4.68)It should noticed that the Hall parameter is dependent on magnetic field strength and the density,
B/ρ. When βj ≫ 1, for j-th ionized species, magnetic force is dominated over the collision withneutrals. These charged particles are coupled with the magnetic field. On the other hand, even forcharged particles, j-th species are decoupled from the magnetic field if βj ≪ 1. In Figure 4.9 (left),the Hall parameters for ions and electrons are plotted in the B − ρ plane (Wardle 2007).
Ambipolar diﬀusion region: When both βi and βe are much larger than unity (1 ≪ βi ≪ βe ),ions and electrons are coupled with magnetic field.
Hall region: When βi ≪ 1 ≪ βe , electrons are coupled with magnetic field, but ions are not.
Ohmic resistive region: When βi ≪ βe ≪ 1, not only ions but also electrons are decoupled fromthe magnetic field.
The boundary between ambipolar diffusion and Hall regions is given by the condition of ion’s Hallparameter equal to 1. The other boundary between the Hall region and the Ohmic resistive region isfrom βe = 1. From equation (4.68), both boundaries are expressed by B ∝ ρ. Since in the contraction
by the self-gravity magnetic field strength B is amplified approximately in proportion to ρ1/2 , themajor process of magnetic dissipation changes from the ambipolar diffusion region to the Ohmic
resistive region, during the contraction.
P
P
From equation (4.66), the current made by the drift velocity, J =j Jj =
j Zj enj vj , is′obtained depending on E as′
+ σHJ = σO E//

B
′
′
× E⊥
+ σ P E⊥|B|

(4.69)′ and E ′ represent components of E ′ parallel and perpendicular(generalized Ohm’s law), where E//⊥
to the magnetic field [see Fig. 4.10 (left)]. The first term represents the current flowing parallel to B,
in which the magnetic field does not play a role. Thus, this conductivity is equal to that measured
for no magnetic field (the Ohmic conductivity, σO ). The second term represents the current flowing

104

CHAPTER 4. LOCAL STAR FORMATION PROCESS′ and B, which comes from the charged particles drifting in the direction of
perpendicular both to E⊥′ × B) (the Hall effect). Thus, the coefficient to the E ′ is called the Hall conductivity (σ ). The±(E⊥
H⊥′ and its coefficient is called the Pedersen
third represents the current flowing in the direction of E⊥
conductivity (σP ). Although the induction equation is written as ∂B/∂t = ∇ × (v × B) in the ideal
MHD condition, in this case an additional term should be added as∂B= ∇ × (v × B) − c∇ × E ′ .
∂t(4.70)c∇ × B,
Assuming only the Ohmic dissipation, E ′ = σ1O J = 4πσO∂B= ∇ × (v × B) − ∇ × (ηO ∇ × B) ,∂t

(4.71)

where we used Ampere’s law J = (c/4π)∇ × B and the magnetic diffusibility is given by the conductivity as ηO = c2 /4πσO . The last term for the Hall resistivity and the ambipolar diffusion are written
ashi−∇ × ηH (∇ × B) × B̂ ,n hio
∇ × ηA (∇ × B) × B̂ × B̂

(4.72)(4.73)(Wurster 2016, Tsukamoto et al. 2017), where B̂ ≡ B/|B|.

4.4

Dynamical Collapse of Spherical Isothermal Clouds

In 1969, Larson (1969) and Penston (1969) found a self-similar solution which is suited for the
dynamical contraction. Figure 4.11 is a radial density distribution for a spherical collapse of an
isothermal cloud, where the cloud has a four-times larger mass than that of the Bonnor-Ebert mass.
Although the figure is taken from a recent numerical study by Ogino et al. (1999), a similar solution
was obtained in Larson (1969). We can see that the solution has several characteristic points asfollows:
1. The cloud evolves in a self-similar way. That is, the spatial distribution of the density [Fig. 4.11
(left)] at t = 0.326 is well fitted by that at t = 0.293 after shifting in the −x and the +ydirections. As for the infall velocity spatial distribution (right panel), only a shift in the −x
direction is needed.
2. The density distribution in the envelope, which is fitted by ∝ r−2 , is almost unchanged. Only
the central part of the cloud (high-density region) contracts.
3. The time before the core formation epoch (the core formation time t0 is defined as the time atwhich the central density increases greatly) is a good indicator to know how high the central
density is. That is, reading from the figure, at t0 − t = 0.003 (t = 0.326) the central density
reaches ρc ≃ 104 and at t0 − t = 0.036 (t = 0.293) the density is equal to ρc ≃ 102 . This shows
the maximum (central) density is approximately proportional to (t0 − t)−2 , which is reasonablefrom the description of the free-fall time ∝ ρ−1/2 (see p.40 of §2.3).

4.4. DYNAMICAL COLLAPSE OF SPHERICAL ISOTHERMAL CLOUDS

105

Figure 4.11: Evolution of isothermal clouds massive than the Bonnor-Ebert mass. Density (left) andvelocity (right) distributions are illustrated. Solid lines show the cores of the preprotostellar phase (prestellar core) and dashed lines show those of protostellar phase (protostellarcore). The evolution of the protostellar phase is studied by the sink-cell method, where
we assume the gas that entered in the sink-cells is removed from finite-difference grids
and add to the point mass sitting at the center of the sink-cells which corresponds to a
protostar. Taken from Ogino, Tomisaka, & Nakamura (1999).

The basic equations of spherically symmetric isothermal flow are

∂ρ
1 ∂
+ 2
r2 ρv = 0,
∂t
r ∂r
∂v
∂v c2is ∂ρ GM
+v
+
+ 2
= 0,
∂t
∂r
ρ ∂r
r

(4.74)(4.75)Z r

M (r, t) = M (0, t) +4πr′2 ρ(r′ , t)dr′ ,(4.76)

0where M (r, t) represents the mass included in the radius r and M (0, t) denotes the mass of the
protostar. A self-similar solution which has a formΩ(ξ),
4πG(t − t0 )2v(r, t) = cis V (ξ),
c3 |t − t0 |M (r, t) = is
m(ξ),
G
r
ξ =
,
cis |t − t0 |ρ(r, t) =(4.77)(4.78)(4.79)(4.80)

should be found, where Ω and V are functions only dependent on ξ. For example, equation (4.77)
asks the shape of the density distribution is the same after resizing of equation (4.80) r → r/cis |t − t0 |and re-normalizing ρ → ρ · 4πG(t − t0 )2 as Ω(ξ). After calculation as shown in Appendix B.7, the106

CHAPTER 4. LOCAL STAR FORMATION PROCESS

basic equations for the spherical symmetric model yieldm = (ξ − V )ξ 2 Ω,



 dV
2
2
= Ω(ξ − V ) −(ξ − V ),(ξ − V ) − 1dξξ(ξ − V )2 − 1 dΩ
2
= Ω − (ξ − V ) (ξ − V ),Ωdξ
ξ

(4.81)(4.82)(4.83)[eqs. (2.9)-(2.14) of Whitworth & Summers 1985; Hanawa 1999]. Solve a problem of Appendix B.7(p.157). Equations (4.82) and (4.83) have a singular point at which (ξ − V )2 − 1 = 0 or V = ξ ± 1.
Since the point of ξ =const moves with cis , the flow velocity relative to this ξ =const is equal tov − cis ξ = (V − ξ)cis . Thus the singular point ξ∗ at which V = ξ ± 1 corresponds to a sonic point.
Therefore, since the flow has to pass the sonic point smoothly, the rhs of equations (4.82) and (4.83)
have to be equal to zero at the singular point ξ∗ . This gives at the sonic point ±ξ∗ > 0,Ω−22(ξ − V )= Ω(ξ − V ) − = 0,ξξ(4.84)

which leads toV∗ = ξ∗ ∓ 1,2Ω∗ = ± .
ξ∗These equations (4.81), (4.82) and (4.83) have an analytic solution
V (ξ) = 0, Ω =2, m = 2ξ, −∞ < ξ < ∞.
ξ2

(4.85)(4.86)(4.87)

This is a solution which agrees with the Chandrasekhar’s SIS [eq. (4.4)]. Generally, solutions are
obtained only by numerical integration. For ξ → ∞, the solution has to converge to an asymptotic
form ofA − 2 V∞ 4V∞ + (A − 2)(A − 6)
V (ξ) = V∞ −+ 2 ++ O(ξ −4 ),(4.88)ξξ
6ξ 3
AΩ∞ (A − 2)Ω(ξ) =−+ O(ξ −6 ).
(4.89)2
ξ2ξ 4
This shows that for sufficiently large radius the gas flows with a constant inflow velocity V∞ cis .
This has a solution in which the density and the infall velocity should be regular with reaching
the center (ξ ≪ 1). Such kind of solution is plotted in Figure 4.12 (solid lines represent the solutionsof τ = t − t0 < 0). This time evolution is expected from the self-similar solution. This shows that≃ ρc(in the central region),ρ(4.90)∝ r−2 (in the outer envelope),∝r(in the central region),v(4.91)≃ 3.28cis (in the outer envelope).
Reaching the outer boundary, the numerical solution (Fig. 4.11) differs from the self-similar solution(Fig. 4.12). For example, v is reduced to zero in the numerical simulations, while it reaches a finite
value 3.28 in the self-similar solution. And as for the density distribution, ρ drops near the outerboundary in the numerical simulations while it decreases proportional to ∝ r−2 . However, in the region except for the vicinity of the outer boundary the self-similar solution expresses well the dynamicalcollapse of the spherical isothermal cloud. This solution gives the evolution of a pre-protostellar core
formed in a supercritical cloud/cloud core.

4.4. DYNAMICAL COLLAPSE OF SPHERICAL ISOTHERMAL CLOUDS

16

107

3000

t=0y

14

+104
12 -10vr [m s-1]

log n [m-3]

2000
4

+105
10

-105
+106

8

+104 +105 +106
1000
t=0y

-106

-104
6
13

14

15
16
log r [m]

17

0
13

14

-105

-106

15
16
log r [m]17

Figure 4.12: A self-similar solution indicating a dynamical collapse of isothermal spherical cloud(Larson-Penston solution). Spatial distributions of the density and inflow velocity which
are expected from the self-similar solution are plotted. Dashed lines show the evolutionof prestellar core phase and solid lines show that of protostellar core phase. Attachednumbers represents time τ counted from the epoch (t0 ) when the central density reaches
infinity as τ = t − t0 . Taken from Hanawa (1999).

108

CHAPTER 4. LOCAL STAR FORMATION PROCESS

4.4.1

Inside-out Collapse Solution

In 1977, Shu found another self-similar solution which is realized after a central protostar with
infinitesimal mass is formed in the singular isothermal sphere (SIS) solution. The gas begins to accrete
to the protostar. Outside the region where the accretion occurs, the initial SIS is kept unchanged,
since the SIS is a hydrostatic solution. And the front of accretion expands radially outward in time.
Since the inflow region expand outwardly, he called it the inside-out collapse solution. In Figure
4.13, the evolution is shown for density and inflow velocity. This solution gives the evolution of a
protostellar core formed in a subcritical cloud/cloud core.
Using equations (4.88) and (4.89) and assuming the inflow velocity should reduce at large radius,
we obtain
A − 2 (A − 2)(A − 6)−
+ ··· ,
(4.92)V (ξ) = −ξ6ξ 3Ω(ξ) =A
A(A − 2)−
+ ··· .
2
ξ2ξ 4

(4.93)

Since ξ → ∞ means t → t0 (if r is finite), Ω(ξ) → A/ξ 2 means thatρ(r, t0 ) =

Ac2is
.
4πGr2

(4.94)

Comparing with the SIS, when A = 2 this gives the SIS and when A > 2 this gives a density
distribution in which the pressure is inefficient and the cloud is contracting. The solution with A > 2
is obtained by a procedure as (1) at a sufficiently large radius ξ1 , calculate V (ξ1 ) and Ω(ξ1 ). (2) fromthese values, integrate equations (4.82) and (4.83) inwardly. Figure 4.13 show the solution of thistype. The solution with A > 2 inflow speed is accelerated towards the center. Decreasing A (A → 2),
it is shown that an outer part ξ ≳ 1 reaches V → 0. For A = 2+ , the solution reaches the singularline V = −ξ + 1 at ξ = 1 (V = 0) 2 . Since V = 0 and Ω = 2 at ξ = 1, this solution with A = 2+
converges to the SIS at ξ = 1. This means that if there is an infinitesimally small amount of excess
mass at the center of SIS, the accretion begins from the center while outside a radius the cloud is left
static. The inner part of the solution ξ ≲ 1, V and Ω are well expressed as V ∝ ξ 1/2 and Ω ∝ ξ 3/2 .
The power-law distributions of V ∝ ξ 1/2 and Ω ∝ ξ 3/2 are explained as follows: Conservation of
the total energy of an inflowing gas shell is expressed as
v 2 GMr
GMr0
−
=−
,
2
r
r0

(4.95)

where r0 denotes the initial radius of a gas shell and Mr0 = Mr represents the mass inside the gas
shell. Neglecting GMr0 /r0 compared with the term GMr /r (r0 ≫ r), this gives

v≃

2GMr
r

1/2

∝ r−1/2 ,(4.96)

where Zwe assumed a major part of Mr comes from the mass of a protostar M∗ , that is, Mr =r

ρ4πr2 dr ≃ M∗ . Since the average density inside the radius r is written as ρ̄(< r) =
M∗ +
0
Z r
Z r
2
4πr2 dr = 3ρ(r) for SIS distribution, the time necessary for a gas shell to reach theρ4πr dr/00center is proportional to ∝ ρ̄(< r)−1/2 ∝ ρ(r)−1/2 ∝ r1 ∝ (t − t0 )1 , where we used a fact that the2This means A = 2 + ϵ where ϵ > 0 and ϵ → 0

4.4. DYNAMICAL COLLAPSE OF SPHERICAL ISOTHERMAL CLOUDS

109

Figure 4.13: Self-similar solution which shows the inside-out collapse. (Left:) Infall velocity (−V =−v/cis ) is plotted against the similarity variable ξ ≡ r/cis (t−t0 ). Three curves correspondto models A = 2.4, 2.1, and 2.01. (Right:) Density (Ω = 4πGρ(t−t0 )2 ) is plotted against
the similarity variable ξ ≡ r/cis (t − t0 ).

front of accretion expands with a constant sound speed cis . This means that the time necessary forthe gas shell to travel from the radius of the accretion wave front to the center is proportional tot − t0 . Since the mass of the shell which begins accretion in a unit time is equal to ρ(r)4πr2 cis andis constant irrespective of t − t0 . These two facts indicate that the mass accretion rate is constant intime. That is,Ṁ = 4πr2 ρv = const.
(4.97)
Since the mass swept by the sound wave per unit time is equal to Ṁsw = 4πr2 ρSIS cs = 2cs3 /G, theaccretion rate is proportional to c3s /G. Shu (1977) obtained the accretion ratec3
Ṁ = 0.975 s ,G(4.98)

for SIS A = 2+ . Using this equation, equations (4.96) and (4.97) indicate that the spatial density
distribution is expressed by a power-law as
ρ(r) ∝ r−3/2 ,(4.99)

which is valid for the region except for the vicinity of the front of the accretion.

4.4.2

Protostellar Evolution of Supercritical Clouds

What is a protostellar core formed in a supercritical cloud/cloud core? Is this different from theinside-out solution of Shu (1977)? A solution which corresponds to the protostellar core is obtainedby Hunter (1977) and Whitworth and Summers (1985). This is a solution with t > t0 in equation(4.80). The asymptotic behaviors of the density and infall velocity reaching the center are different

110

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Table 4.1: Ages of stellar population observed for respective star forming regions (Hartmann 2009).

4.5. STAR FORMATION TIME SCALE
from that of the Larson-Penston self-similar solution for a prestellar collapse. That is,→ finite(LP)Ω→ infinite (Inside − out)→ finite(LP)V→ infinite (Inside − out)

111

(4.100)(4.101)

Using the boundary conditions suitable for the inside-out type solution, another self-similar solution
is obtained. In Figure 4.12 (p.107), such kind of solution is also plotted with solid line for τ > 0.
Take notice that the solutions of t < t0 (τ < 0; prestellar) and t > t0 (τ > 0; protostellar) agreewith each other at t = t0 (τ = 0). Even if the boundary conditions at the center for the similarity
variables, V and Ω, are completely different, the difference between the two is small in the physicalvariables v and ρ at the epoch t = t0 (τ = 0). Therefore, the evolution of a supercritical core isthought to be expressed by the Larson-Penston self-similar solution extended to the protostellar corephase by Hunter (1977) and Whitworth and Summers (1985).
Assume that we observe a protostellar core and obtain their density and infall velocity spatial
distributions. Can we distinguish which solution is appropriate for the Shu’s inside-out solution or theextended Larson-Penston solution? This seems hard, because the structure of density and velocity
distributions are similar after the protostar is formed: the density and velocity show almost similar
power-law as ρ ∝ r−3/2 and v ∝ r−1/2 irrespective of the inside-out solution or the extended LarsonPenston solution. The region where the infall velocity is accelerated toward the center (accretiondominated region) is expanding after the protostar is formed. Therefore, to distinguish between thetwo solutions becomes harder and harder after the protostar is formed. The difference would be large
and we would have a definite answer which solution is appropriate to describe the cloud collapse, if we
can observe a very young protostellar core or a preprotostellar core which shows dynamical collapse.
However, since the time-scale of such a phase is much shorter than the evolved protostellar phaseor a younger preprotostellar core, the number of such kind of objects would be small (τﬀ ∝ ρ−1/2 ).
Therefore, we are looking for such objects just before or after the protostar formation. First core(§4.8 in p.121) is attracting attention, which is regarded as a candidate of earliest stage of a protostar.

4.5

Star Formation Time Scale

There are long-standing controversy on how fast stars are formed in the molecular clouds. One idea isthat the interstellar cloud is in a hydrostatic balance in which magnetic field plays an important rolein supporting against the self-gravity. That is, the cloud is magnetically subcritical. In such a cloud,the magnetic field escapes from the center in a timescale of τA ∼ 10 × tﬀ (see §4.3). After the massto-flux ratio exceeds a critical value, dynamical contraction begins. The timescale of cloud lifetime isequal to the time scale of ambipolar diffusion, τA ∼ 3 × 107 yr (ρ/102 H2 cm−3 )−1/2 . Long lifetime ofmolecular clouds is necessary to explain giant molecular clouds are formed by agglomeration processof small clouds. The necessary lifetime is estimated 107 yr – 108 yr.
Another idea is that the star formation timescale is much shorter than this timescale. Table 4.1(Hartmann 2009) shows the ages of stellar population observed for respective star forming regions.
Stellar clusters and associations accompanied with molecular clouds have ages younger than 5 Myr.
And clusters and associations without accompanied molecular clouds are older than 5 Myr. Thismeans that a molecular cloud will disappear after 5 Myr since star formation begins in the cloud.
Since this timescale is shorter than τA , the clouds seem to continue to be formed and the turbulence
supports the cloud, the cloud core begins dynamical contraction after the turbulence decays away
within the sound crossing timescale ∼ Myr.

112

CHAPTER 4. LOCAL STAR FORMATION PROCESS

4.6

Accretion Rate

Using equation (2.26), the necessary time for a mass-shell at R to reach the center (free-fall time) isexpressed as

1/2
R3
π
T (R) ≡
(4.102)
2GM (R)
2[for detail of this section see Ogino, Tomisaka, & Nakamura (1999)].
Consider two shells whose initial radii are R and R + ∆R. The time difference for these two shells
to reach the center ∆T (R) can be written down using equation (4.102) as


πR1/2
3
R dM (R)∆T (R) = 3/2−∆R.
(4.103)2 (GM (R))1/2 2 2M (R) dR
Mass in the shell between R and R + ∆R, ∆M ≡ M (R + ∆R) − M (R) = (dM/dR)∆R, accretes on
the central object in ∆T (R). Thus, mass accretion rate for this pressure-free cloud is expressed as∆M/∆T . This leads to the expression asdM (R)dR.
dM (R)
2M (R) dR

R

dM
23/2 G1/2 M (R)3/2 M (R)
(R) =
3
dT
π
R3/2
− R
2

(4.104)This gives time variation of the accretion rate.
Case 1: Consider two clouds with the same density distribution ∂ log ρ/∂r but different absolutevalue. Since these two clouds have the same ∂ log M (R)/∂ log R and thus the last factor of
equation (4.104) is the same, the mass accretion rate depends only on M (R)/R, and is expressedasdM(R) ∝ M (R)3/2 .
(4.105)dTThis indicates that the accretion rate is proportional to ρ3/2 , while the time scale is to (d log M/dt)−1 ∝ρ−1/2 . This is confirmed by hydrodynamical simulations of spherical symmetric isothermal
clouds (Ogino et al. 1999). In Figure 4.14 we plot the mass accretion rate against the clouddensity. α represents the cloud density relative to that of a hydrostatic Bonnor-Ebert sphere.
This shows clearly that the mass accretion rate is proportional to α3/2 for massive clouds α > 4.
This is natural since the assumption of pressure-less is valid only for a massive cloud in which
the gravity force is predominant against the pressure force.
Case 2: When the initial density distribution is the SIS as ρ ∝ r−2 , the mass included inside R is
proportional to radius M (R) ∝ R. In this case, equation (4.104) gives a constant accretion rate
in time.
Similar discussion has been done by Henriksen, André, & Bontemps (1997) to explain a decline
in the accretion rate from Class 0 to Class I IR objects. They assumed initial density distributionof(r ≤ rN ), = ρN .. . 
ρ(4.106)r −2/D1
 = ρN. . . (r > rN ),rN
as shown in Figure 4.15 (left). Expected accretion rate is plotted against the elapsed time in
Figure 4.15 (right). Since the free-fall-time of the gas contained in the inner core r ≤ rN is4.6. ACCRETION RATE

Figure 4.14: Mass accretion rate against the typical density of the cloud. α represents a factor at which
the initial density distribution is more massive than the hydrostatic solution (BonnorEbert sphere). γ is the polytropic ratio using the polytropic index as γ ≡ 1 + 1/n. The
steeper slope shown in the models of γ ≥ 1.05 indicates that the highest mass accretion
rate in time is approximately proportional to α3/2 . Taken from Fig. 8 of Ogino et al.
(1999).

113

114

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.15: A model proposed to explain time variation in accretion rate by Henriksen, André, &Bontemps (1997). The density distribution ρ(r) at t = 0 (left) and expected accretion
rate (right).

the same, such gas reaches the center simultaneously. It makes a very large accretion rate at
3 δ(t − (3π/32Gρ )1/2 ). If D = 1, ρ ∝ r −2 for r ≳ r .
t = (3π/32GρN )1/2 as Ṁ = (4π/3)ρN rN
1
NN
1
Since M ∝ R and tﬀ ∝ R, equation (4.104) predicts Ṁ ∝ R0 ∝ t0 . A constant accretion rateis expected for this power-law and the accretion rate is converged to a constant value after the
3 . Ifaccreted mass is much larger than that was contained in rN , M∗ ≫ M (rN ) ≡ (4π/3)ρN rN−1
2
1/2
D1 = 2, ρ ∝ r for r ≳ rN . Since for this power M ∝ R and tﬀ ∝ R , equation (4.104)
predicts Ṁ ∝ R3/2 ∝ t3 . They gave Ṁ ∝ t3D1 −3 for D1 > 2/3.
Problem 14

Show that Ṁ ∝ t3D1 −4/3D1 if ρ ∝ r−2/D1 and D1 < 2/3 [Henriksen, André, & Bontemps (1997)].



4.7. OUTFLOW

115

2

1.5

1

0.5

0
0

0.5

1

1.5

2

Figure 4.16: Magneto-centrifugal wind model. Figure shows the effective potential for a gas parcel
with conserving the angular rotation speed ω0 of the disk [eq. (4.108)]. The gas parcel
is assumed to located originally at (r0 , 0). The x- and y-axes mean normalized distancefrom the central star as r/r0 and z/r0 . The darker color represents the deeper potential.

4.7

Outﬂow

In Chapter 1, we mentioned “outflows” ejected from the protostars and pre-main-sequence stars.
L1551-IRS5 is a typical example which shows a number of outflows are ejected in the course of starformation. One is molecular outflow, whose lobes extends L ∼ 0.5 (D/160 pc) pc in two opposite
directions from the IRS5 (Snell et al. 1980). This is traced by CO J = 1 → 0 emission line. They
estimated an expansion velocity of the CO gas of VCO ∼ 15 km s−1 , a mass of 0.3 M⊙ . The dynamical
age is equal to τdyn = L/VCO ∼ 3 × 104 yr. Inside the CO outflow lobe, several Herbig-Haro objects
are found, which are emission-line nebulosities. Proper motion studies have revealed the motion ofsuch H-H objects. Cudworth & Herbig (1979) reported HH28 and 29 have 0.15′′ yr−1 − 0.20′′ yr−1 ,which corresponds to 150 − 200 km s−1 . In a neighbor of the IR source a stellar jet is found (Mundt& Fried 1983) by optical emission lines. Its size is equal to 17′′ = 4 × 1016 (D/160 pc) cm and the jet
indicates rather well-collimated shape (the opening angle ≃ 10◦ ). The relationship between these two
outflows (massive molecular outflow and less massive optical jet) are not clear yet.

4.7.1

Magneto-driven Model

To accelerate gas till supersonic speed, the magnetic force is considered to be an important player.
Blandford & Payne (1982) have proposed the “magneto-centrifugal wind model” for the accelerationin two-dimensional axisymmetric configuration.
Figure 4.16 explains the magneto-centrifugal wind mechanism, which corresponds to Fig. 1 ofBlandford & Payne (1982). Consider a disk under the gravity of a central star with mass M∗ .
Assume that gas in the disk at the radius r0 rotates with a Kepler speed ofGM∗ 1/2vϕ (r0 ) =.
(4.107)r0

116

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Consider a gas parcel whose original position was (r = r0 , z = 0). The effective potential for theparticle rotating with an angular rotation speed of ω0 = vϕ (r0 )/r0 is written as
"
  #
1 2 2
GM∗
r0
1 r 2
GM∗
− r ω0 = −+,(4.108)ϕ(r, z) = −r0(r2 + z 2 )1/2 2(r2 + z 2 )1/2 2 r0
3/2

where ω0 = (GM )1/2 /r0 . Be careful that the ordinary effective potential is calculated for constantangular momentum. The second term is ordinarily written as (1/2)(j 2 /r2 ) using the specific angularmomentum j ≡ rv = r2 ω. However, equation (4.108) is for constant angular rotation speed. The gas
will co-rotate with the same angular rotation speed as the disk, if the magnetic field is sufficiently
strong. Equation (4.108) represents the effective potential for the gas co-rotating with the disk at r0 .
Figure 4.16 plots the isopotential contour lines for the effective potential [eq. (4.108)]. The contour
line passing the point (r0 , 0) represents ϕ = −3GM∗ /2r0 . The angle between the isopotential line of
ϕ = −3GM∗ /2r0 and the r-axis is equal to ±60◦ (see the problem below). Further, it can be obtainedthat the effective potential increases upwardly while it decreases with reaching the central star orradially outwardly.
This leads to an important conclusion that
1. If the angle between the magnetic field line and the r-axis, θmag , is larger than 60◦ (θmag > 60◦ ),gas parcel moving along the field line must climb the effective potential. It needs extra energy
to depart from the disk.
2. If θmag < 60◦ , gas parcel moving along the field line slides down the effective potential. It can
depart from the disk without extra energy.
Be careful that this is valid if the magnetic field is sufficiently strong. If the Kepler disk is threadedwith the magnetic field lines with a shallow angle, and if the angular momentum is sufficiently
transferred along the magnetic field line, in this case, the gas is accelerated by the extra centrifugal
force and finally escapes from the potential well of the central star. This mechanism is called the“magneto-centrifugal wind” mechanism.
Problem 15
Show that the angle between the isopotential line which passes the point (r0 , 0) and the r-axis
is equal to ±60◦ . Calculate the Taylor expansion of the effective potential near the point (r0 , 0)and show that to satisfy ϕ(r0 + ∆r, ∆z) = ϕ(r0 , 0), (∆z/∆r)2 must be equal to 3.


Angular Momentum Transfer
In the axisymmetric case, the poloidal (z, r) and toroidal (ϕ) components of the magnetic field B andcurrent j are decoupled with each other. That is, the poloidal (Bp ) and toroidal (Bϕ ) magnetic fields
are made by the toroidal (jϕ ) and poloidal (jp ) electric currents, respectively. As for the Lorentz force
j × B, the poloidal component comes from jϕ Bp or jp Bϕ , while the toroidal component does fromjp × Bp . Even if there is no toroidal magnetic field (thus no poloidal electric current), there exists thepoloidal component of the Lorentz force, which acts as a pressure to counter-balance the self-gravity(§4.2). On the other hand, the toroidal component of the Lorentz force appears only the case withthe poloidal electric current and thus toroidal component of magnetic field Bϕ . This means thatthe angular momentum is transferred by the magnetic field only when Bϕ exists. Equation (C.14)
explains how the angular momentum density ρvϕ r is transferred. The left-hand side of equation

4.7. OUTFLOW

117

Side View
Pole-on View

Figure 4.17: Pole-on (left) and side (right) views of the magnetic field lines, which drive the outflow.

(C.14) represents the advection of the angular momentum density, while the right-hand side


 
∂Bϕ
Bp
r 1 ∂
(rBϕ )Br +Bz =· ∇(rBϕ ),(4.109)4π r ∂r∂z
4π
represents the torque exerted on the gas parcel.
The induction equation of the magnetic field [eq.(C.17)] shows that Bϕ is generated from poloidal
magnetic field by the effect of rotational motion vϕ . This indicates that the angular momentum istransferred as follows:
1. The toroidal component of magnetic field Bϕ is amplified by the dynamo process [equation(C.17)]. The magnetic field lines run like spiral viewing from the top like Figure 4.17.
2. Negative torque works near region attached to the fast-rotating disk, while positive one appears
in the region connected to the radial magnetic field. The angular momentum is transferredfrom inside to outside along the same field line.
3. Angular momentum received by the gas near disk surface leads to the excess centrifugal force
which accelerate the gas. The disk inside which is affected by the negative torque loses its
angular momentum and insufficient centrifugal force drives further accretion.
This angular momentum transfer on the same magnetic field has a character of angular momentum
redistribution.
Centrifugal Radius
In a diffuse cloud, the centrifugal force does not play an important role. Consider a gas parcel whose
specific angular momentum is equal to j. When the gas element contracts till the distance r fromthe center, the centrifugal force per mass of
vϕ2
r

=

j2
,
r3

(4.110)works. Contraction stops when the gravitational acceleration GM/r2 is balanced by the centrifugal
force. Its radius (centrifugal radius) is expressed as
rc =

j2
.
GM

(4.111)

118

CHAPTER 4. LOCAL STAR FORMATION PROCESSAssuming that the disk is near the hydrostatic balance in z-direction, total column density is related
to the volume density on the z = 0 plane asσ=

2c2s ρcπG

1/2.

(4.112)

We can show that a nondimensional ratio of specific angular momentum to the mass (lhs) is equal to
the ratio of the free-fall time to the rotation period (rhs) as follow:
c s ωc r 2
cs j
ωc
=
=
.
GM
Gσπr2
(2πGρc )1/2

(4.113)Many numerical simulations confirm that the contracting disk in the runaway isothermal contraction
phase has some ‘universality.’ The ratio of the free-fall time to the rotation period is approximately
equal toωc
≃ 0.3,(4.114)(2πGρc )1/2
regardless of the initial conditions (Matsumoto, Nakamura, & Hanawa 1997). Thus, cs j/GM ≃ 0.3.
Finally, we can see the centrifugal radius ofGM
cs j 2GM
rc = 2≃ 0.3 2
(4.115)cs
GMcs
increases with time in proportional to the mass M , because gas element with large j contracts later.
As the centrifugal radius increases with time, the launching point of the outflow also expands with
time.

4.7.2

Entrainment Model

Models that a protostar is driving a jet and the jet transfers its momentum to the ambient material andforms a molecular outflow are called as entrainment models. From hydrodynamical calculations,
De Young (1986) pointed out there are two mechanisms of entrainment. The ejected jet forms a bow
shock and matter hit by the bow shock is accelerated promptly (prompt entrainment). His numericalsimulations predicted that gas within the radius from the jet axis of (1 − 3)× jet radius is entrainedin this mechanism. After the bow shock has passed away, slower mechanism of entrainment works,
in which a turbulent layer is developed around the jet and the Reynolds stress in the layer transfers
the momentum (Stahler 1994). In this mechanism, the ratio of entrained ambient gas to the injectedmass by the jet is estimated as (1/3 − 3) (De Young 1986).

4.8. EVOLUTION TO STAR

119

Figure 4.18: (Left:) Relation between density and temperature at the center of the cloud (Masunaga& Inutsuka 2000). (Right:) Cooling and heating rates, which are calculated for the cloudcenter are plotted against the density.

4.8

Evolution to Star

In the preceding sections we have seen the evolution from gas cloud to stars. However, we wererestricted to the isothermal gas. To understand the change in the temperature, we have to consider theradiation which keeps the gas isothermal. Spherical symmetric radiative hydrodynamical simulations
have been studied to understand the evolutionary path from interstellar cloud to star. Figure 4.18
(left) illustrates the relation between ρ and T obtained by Masunaga & Inutsuka (2000). Several
characteristic power-laws are seen in the figure. Figure 4.19 plots the density and temperature
distribution.
Isothermal Phase
Below ρ ≲ 10−13 g cm−3 , gas is essentially isothermal T ∝ ρ0 . This corresponds to points number 1-3
of Figure 4.19. Since internal energy of gas is transferred to the thermal energy of dust grains bycollisions, the main coolant in this regime is the dust thermal radiation. The cooling rate per mass isΛ = 4κp σT 4 ,(4.116)

where κp (10 K) ∼ 0.01 cm2 g−1 and σ represent Planck mean absorption coefficient and the StephanBoltzmann constant. Main heating process for ρ ≲ 10−14 g cm−3 is cosmic-ray heating: ϵCR ∼2 × 10−4 erg s−1 g−1 (Goldsmith & Langer 1978). Balance between these two majors asks that temperature is constant as1/4 
1/2
κpϵCRT ∼ 3K.
(4.117)2 × 10−4 erg s−1 g−1
0.01 cm2 g−1
Heating rate due to dynamical compression,d(1/ρ)≃ c2s (4πGρ)1/2 ,(4.118)dt
increases according to the contraction and it balances with the above cooling at the density ρA ∼10−14 g cm−3 (Masunaga, Miyama, & Inutsuka 1998).
The cloud in this phase experiences the dynamical contraction as described in § 4.4, which is calledrunaway collapse. Structure of ρ(r) and vr (r) is well represented by the Larson-Penston self-similarsolution.
Γ = −p

120

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.19: Dynamical evolution of a spherical cloud studied with RHD simulation (Masunaga &Inutsuka 2000). Density (left) and temperature (right) distributions are plotted against
the radius.

4.8. EVOLUTION TO STAR

121

First Core
After ρ ≳ ρA , gas is no more isothermal and becomes adiabatic. Between 10−13 g cm−3 ≲ ρ ≲10−9 g cm−3 , gas obeys the γ = 5/3 polytropes. Above the density ρ ≳ ρA , the optical depth for thethermal radiation exceeds unity τ ≳ 1 and radiative cooling can not compensate the compressional
heating. As long as the temperature is low as T ≲ 100 K, neither rotation nor vibration is excited forH2 molecules. Therefore, even H2 gas behaves like single-atom molecule. Thus γ ≃ 5/3.
Between 10−9 g cm−3 ≲ ρ ≲ 10−7.5 g cm−3 , the exponent becomes γ = 7/5, which characterizes
that the gas consists of two-atom molecule H2 .
In this phase, relatively large gas pressure supports against the gravity and the cloud becomeshydrostatic (points number 4-6 of Fig. 4.19). This is called as “ﬁrst core” made by the molecularhydrogen. The density structure of the first core is well represented by a polytrope sphere with thespecific heat ratio of γ = 7/5 or the polytropic index n = 2 12 . From equation (D.11) in Appendix
D.1, such a polytrope has a mass-density relation as
Mc1 ∝ ρ1/10,
c(4.119)where Mc1 and ρc represent, respectively, the mass of the first core and the central density. At the
beginning, the core mass is equal to Mc1 ≃ 0.01M⊙ . As long as the mass increases a factor 3, thecentral density increases 5 orders of magnitude.
Second Collapse
For 10−7.5 g cm−3 ≲ ρ ≲ 10−2.5 g cm−3 , the exponent becomes γ = 1.1. In this density range ofρ ≳ 10−7.5 g cm−3 the temperature exceeds 103 K. Due to this temperature, the hydrogen molecules
begin to dissociate and the dissociation process absorbs the energy. By this, gas becomes much softer(γ → 1) than the first core. This introduces another dynamical contraction, the second collapse(points number 6-7 of Fig. 4.19).
Second Core
After the dissociation is completed, the exponent becomes γ = 5/3 again. This gas forms a second
core composed of atomic hydrogen, which is called the second core (points number 7-8 of Fig. 4.19).
Since the accretion rate [eq. (4.98)] is proportional to c3s or T 3/2 , the accretion rates onto the second
core Ṁ2nd is larger than that of the first core Ṁ1st as
3.5 K 3/2
T
/10
Ṁ2nd
2nd
≃ 5 × 103.
Ṁ1st
(T1st /10 K)3/2(4.120)Thus, the first core disappears quickly and after that the gas begins to accrete onto the second
core, which will be a protostar. However, this is the case of non-rotating, spherical symmetric cloud
collapse. As previously seen, the angular momentum plays a crucial role and forms a disk. The
evolution must be different completely for such a case.

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CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.20: Explanation of nested grid method.

4.9

Example of Numerical Simulation

We have described the cloud runaway collapse and succeeding accretion process. In the latter phase,
there is a possibility that the outflow is driven by the effect of magnetic Lorentz force. A restricted
number of numerical simulation can simulate such evolution throughout. The size of the molecularcores ∼ 0.1 pc is more than 3×104 times as large as the typical size of the first core ∼ 1 AU. Therefore
numerical scheme to calculate such process must have a large spatial dynamic-range. In the finitedifference scheme, the spatial dynamic range is restricted by the number of the cells. Although at
least 3 × 105 × 3 × 105 grid points (in 2D) seem necessary to resolve a factor of 3 × 104 , to solve
the MHD equation with the Poisson equation of the self-gravity has not been done. This is done by
the Eulerean nested-grid simulation. This method uses a finite number of grid systems withdifferent spatial resolutions simultaneously. Coarse grid covers whole the cloud and is to see a global
structure. Fine grid covers only the central part of the cloud and is to see the fine-scale structureappeared near the center. Figure 4.20 explains grid cells of the nested grid method. The size of thecells of the n-th level grid (Ln) is taken equal to a half of those of the Ln − 1 (Tomisaka 1998, 2000,
2002).
A cylindrical cloud with coaxial magnetic field in hydrostatic balance is studied. Rotation vectorΩ and the magnetic field B are parallel to the symmetric axis of the cylinder (z-axis). For theenergy equation, a double-polytropic equation of state is adopted, since the radiation hydrodynamical
calculation (for example Masunaga & Inutsuka 2000) predicts ρ − T relations shown in Figure 4.18,
which consists of a number of power-laws for some density ranges.
(
p=

c2s ρ
c2s ρA



ρ
ρA

7/5

(for ρ < ρA ),(for ρ ≥ ρA ),(4.121)

where we take ρA = 1010 H2 cm−3 . Figure 4.21 shows the model of Ωc = 10−13 rad s−1 , Bc =10 µG, ρc = 104 H2 cm−3 , cs = 190 (T /10K)1/2 m s−1 , and scale-length H = 0.341(cs /190 m s−1 )(ρc /104 H2 cm−3 )−1/2 pc. This is a snapshot at t = 0.6066 τﬀ = 1.061 Myr. The cylindrical cloud4.9. EXAMPLE OF NUMERICAL SIMULATION

(a)

123

(b)

(c)

Figure 4.21: Ωc = 10−13 rad s−1 , Bc = 10 µG, ρc = 104 H2 cm−3 , and cs = 190 (T /10K)1/2 m s−1 .
Snapshots at the time of 0.6066τﬀ = 1.061 Myr represented in different levels are shown:
L1 (upper-left), L5 (upper-right), and L10 (bottom). The actual size of the frames ofL5 (upper-right) and L10 (bottom) are, respectively, 1/16 and 1/512 smaller than thatof L1 (upper-left). Magnetic field lines (red solid lines) and isodensity contours (blacksolid lines) are presented as well as the velocity vectors by arrows. Taken from Fig. 1 ofTomisaka (2002).

124

CHAPTER 4. LOCAL STAR FORMATION PROCESS
(a)

(b)

(c)
P3

P2

P1

Figure 4.22: Continuation of the evolution shown in Fig. 4.21. Snapshots at the time of 0.6069 τﬀ =1.066 Myr just ∼ 500 yr after from Fig. 4.21. Panel (a) plots the structure of L10 which
is similar to Fig. 4.21 (c). Spherical region inside ∼ 1.2 × 10−3 H has been swept by theoutflow. Panel (b) is a snapshot of L12 grid which has 4-times finer spatial resolution than
L10 (a). In panel (c), we plotted magnetic force dominated region with ‘+’, centrifugal
force dominated with ‘*’, and thermal pressure force dominated with blank ’ ’. Taken
from Fig. 3 of Tomisaka (2002).

breaks into pieces with a length in z-direction equal to the most unstable wavelength of the gravitational instability (Matsumoto, Nakamura, & Hanawa 1994) as
λmax ≃

2π(1 + α/2 + β)1/2

cs
.
i1/2
(4πGρc )1/2
0.72 (1 + α/2 + β)1/3 − 0.6
h(4.122)The magnetic fields prevent gas motions perpendicular to the field lines B. Further, the centrifugal
force works in the perpendicular direction to Ω. A disk forms which running in the perpendicular
direction to B and Ω and thus to the z-axis. Figure 4.21 illustrates a snapshot at the time of
0.6066 τﬀ = 1.061 Myr. Level 1 grid (upper-left) captures the global structure of the contracting disk
perpendicular to the z-axis. Close up view is shown in L5, which has 16 times finer spatial resolution(upper-right). Since speed, vz , of infalling gas is larger than the sound speed, a number of shock
fronts are formed near z ≃ 0.02H and z ≲ 0.01H. Further inner structure is represented in L10
grid (bottom) which has 32 times finer resolution than L5. Another discontinuity in density is now
forming near z ∼ 0.0002H. Since gas inside of this discontinuity has the density larger than thecritical density ρ > ρA , in this phase the first core (p.121) begins to be formed. In the isothermal
collapse phase, a characteristic power-law for the Larson-Penston self-similar solution is seen in theradial distributions of density as ρ(r, z = 0) ∝ r−2 , and Bz (r, z = 0) ∝ r−1 .

4.9. EXAMPLE OF NUMERICAL SIMULATION

125

After t > 0.6067τﬀ the first core is formed. Radial infall motion is accelerated toward the surfaceof the first core and the rotational motion vϕ also increases. This amplification promotes the toroidal
magnetic field, Bϕ . Since the torque is proportional to (∇ × Bϕ eϕ ) × Bp , the amplitude of the torqueincreases after the first core formation.
Structure just after ∼500 yr has passed from Figure 4.21 is shown in Figure 4.22. In τ ∼ 500
yr from the epoch of Figure 4.21, flow is completely changed and outflow is launched. Panel (b),which corresponds to a four-times close-up of panel (a), shows clearly that outflow is ejected froma region near r ≃ 1.5 × 10−4 H and z ≃ 0.5 × 10−4 H. And the outflow vectors and the disk has anangle of approximately ≃ 45◦ . The magneto-centrifugal wind model favors a small angle between
the magnetic field and the disk, θmag < 60◦ . This model predicts that the angle between the flow
and the disk is also smaller than ≲ 60◦ . Numerical results is not inconsistent with this prediction ofmagneto-centrifugal wind model.
To see which force is driving such an outflow, amplitudes of three forces are compared: the Lorentz
force Fm = (∇ × B) × B/4π, the centrifugal force Fc = (ρvϕ2 /r)er , and the thermal pressure gradientFp = −∇p. The parallel components to the poloidal magnetic field of the three forces are as
Fm ·

Bp
|Bp |

(∇ × B) × B Bp
·
,
4π
|Bp |
1 Bp
= −
· ∇(rBϕ )2
8πr2 |Bp |=(4.123)(Ustyugova et al. 1999)
Fc ·

Bp
Br
= ρvϕ2 r ·
,
|Bp |
|Bp |

(4.124)

Fp ·

Bp
Bp
= −∇p ·
.
|Bp |
|Bp |

(4.125)andFigure 4.22 illustrates the largest force at each grid point:
Region C: Centrifugal force dominated region (region C), which is shown by asterisk (*), overlaps
the outflow region around point P1 (z = 10−4 H, r = 2.5 × 10−4 H).
Region M: Just radially exterior to this region C, there is another region near point P2 (z =2 × 10−4 H, r = 3.5 × 10−4 H) filled with plus signs where the magnetic force is dominated(region M).
Comparing with the flow vectors in the panel (b), the strong outflow coincides with this region M andthe above region C. Therefore, the outflow is driven by the centrifugal force and the Lorentz force(pressure gradient by toroidal magnetic fields).
Difference between models with different magnetic field strength but the same rotation speed isillustrated in Figure 4.23. In this comparison, we assume virtually a hard equation of state such asp ∝ ρ2 for ρ ≥ ρA gas in equation (4.121). Magnetic to thermal pressure ratio α is taken 1 [panels(a), and (d )], 0.1 [panels (b), and (e)], and 0.01 [panels (c), and (f )]. Panels (a), (b), (c) in theleft column are snapshots when the central density reaches ρA while panels (d ), (e), (f ) in the rightcolumn show the structure after τ = 4.5 × 10−3 τﬀ = 8000 yr passed. In the model with α = 1, theoutflow gas flows through a region whose shape resembles a capital letter U. This is similar to thatobserved in the model with α = 1 and Ω = 5 shown in Figure 4.22. Decreasing the magnetic fieldstrength from α = 1 to 0.01, the shape of disk changes from a flat disk to a round ellipsoid. Thisis reasonable, since the model with neither magnetic field nor rotation shows a spherical contracting

126

CHAPTER 4. LOCAL STAR FORMATION PROCESS

(a)

(d )

(b)

(e)

(c)

(f )

Figure 4.23: Difference in models with the same rotation speed Ω = 2 × 10−14 rad s−1 but different
magnetic field strength ((a) and (d ): Bc = 10 µG, (b) and (e): 3 µG, and (c) and (f ):1 µG). Taken from Fig. 7 of Tomisaka (2002).

4.9. EXAMPLE OF NUMERICAL SIMULATION

127

core. Outflow is also affected by decrease in magnetic field strength. Globally folded magnetic fieldlines are seen in the outflow region in panel (e). Smaller-scale folding structure of the magnetic fieldlines develops in the outflow region in panel (f ). The structure seen in the poloidal magnetic field
is well correlated to the distribution of toroidal magnetic field strength. That is, a packet of strong
Bϕ pinch the gas and poloidal magnetic field lines radially inwardly. This means that the toroidal
magnetic field is amplified by the rotation motion and becomes predominant over the poloidal one in
the case with weak initial magnetic field. Thus, the whoop stress, which is expressed by the firstterm of the Lorentz force of equation (C.13) and comes from the tension of toroidal magnetic field,
pinches the plasma contained inside of the loop of Bϕ . For this case, the magnetic force dominantregion is widely spread compared with the centrifugal force dominant region, which is far from themodel with strong magnetic field (α = 1). In this case, the magnetic pressure gradient plays animportant role. Gas of outflow is ejected perpendicularly from the disk and forms flow similar to thecapital letter I. In conclusion, U-type and I-type outflows are generated for strong magnetic modeland weak magnetic model, respectively.

4.9.1

Angular Momentum Distribution

Figure 4.24: Evolution of specific angular momentum contained in accumulated mass from the center.
Open circles, filled squares, and triangles denote, respectively, the beginning and the endof the runaway collapse phase and the accretion phase. Taken from Tomisaka (2000).

During this evolution, magnetic torque is working to the outflowing gas and let the gas rotate.
As the reaction, the magnetic torque is working the disk gas to reduce the angular momentum. To
see the time evolution of angular momentum distribution, we plotted angular momentum spectrum 3
in Figure 4.24 (Tomisaka 2000). Vertical axis is the specific angular momentum and the horizontal
Angular momentum spectrum means literally that mass ∆M with speciﬁc angular momentum of j − j + ∆j (y-axis)
is plotted against j (x-axis). However, to consider the anglar momentum transfer, i.e., extraction and deposition, we
regard the accumulated mass from the center as the independent variable and its speciﬁc angular momentum as thedependent variable.
3

128

CHAPTER 4. LOCAL STAR FORMATION PROCESS

axis is the accumulated mass from the center. In a spherical symmetric calculation, the accumulated
mass is well defined as
Z rMr =ρ4πr2 dr.
(4.126)0

However, in a 2D axisymmetric calculation this is not apparently defined. Since the gas density
increases monotonically reaching the center, the mass shell with higher density is considered to belocated nearer to the center. Total gas mass whose density is higher than ρ1 is calculatedZ
M (> ρ1 ) =ρdV,(4.127)
ρ>ρ1which contains the total angular momentum ofZJ(> ρ1 ) =ρrvϕ dV.

(4.128)

ρ>ρ1

Specific angular momentum contained in M (> ρ1 ) is calculated asj(> ρ1 ) =J(> ρ1 ).
M (> ρ1 )(4.129)In Figure 4.24, we plotted j(> ρ1 ) as a function of M (> ρ1 ).
Before the first hydrostatic core is made, the angular momentum distribution evolves from the linewith open circles to that with filled squares. The central ≃ 0.5 (cis /190 m s−1 )3 (nc /104 H2 cm−3 )−1/2 M⊙
reduces its angular momentum to 1/3 before the core is formed.
Since we found no outflows (the outflow extracts angular momentum from the central part) inthis phase, this reduction seems to be explained by the magnetic braking (Appendix K). This is
explained also by considering that gas with smaller angular momentum is preferentially accumulated
during the contraction and constructs the high density region (Tomisaka 2000).
Angular momentum distribution changes greatly after the protostar formation. In ≃ 7 × 103 yr(nc /104 H2 cm−3 )−1/2 , the distribution evolves from the line with filled squares to that of filled triangles in Figure 4.24. The central ≃ 0.17 (cis /190 m s−1 )3 (nc /104 H2 cm−3 )−1/2 M⊙ has been reduced toa factor ≲ 10−4 of the initial value. The angular momentum transfer at this stage is mainly occurredby the wind. That is, outward flowing gas brings a finite angular momentum and as a result angularmomentum is transferred outwardly.
In conclusion, the outflow process in molecular cores both with magnetic field and rotation plays
a role in segregating angular momentum and mass. That is, about 10 % of the accreting mass isejected by the magnetic wind mechanism. At the same time, the outflow brings almost all theangular momentum of the originally accreting gas. From the rest 90 % of the gas mass, the angularmomentum is extracted and the gas continues to accrete to the central protostar. The specific angularmomentum of the protostar is reduced to a factor of 10−4 of the initial value. Today, this reduction
process of angular momentum is essential to form a star with avoiding the centrifugal barrier.

4.10. EVOLUTION IN THE H-R DIAGRAM

129

Figure 4.25: Typical stellar evolution in HR diagram with mass accretion. Thin lines represent theevolution without mass accretion. Attached numbers represents the evolutions of constant mass stars of 0.2, 1, 3, and 15 M⊙ . In contrast, 4 models with the accretion rates,
10−6 (dash-dotted line), 10−5 (dotted line), 10−4 (dashed line), and 10−3 M⊙ yr−1 (solidline) are shown. Ṁ∗ = 10−3 M⊙ yr−1 stars accumulate 13.4 M⊙ in 13.4 kyr. Ṁ∗ = 10−4 ,10−5 , and 10−6 M⊙ yr−1 models reach main sequence stars of M∗ = 9.4 M⊙ (94 kyr),6.9 M⊙ (690 kyr), and M∗ = 3.3 M⊙ (3.3 Myr), respectively. The initial condition of theaccretion models is taken as M∗ = 0.1 M⊙ and R∗ = 1 R⊙ . Taken from Fig. 1 of Yorke& Sonnhalter (2002).

4.10

Evolution in the H-R diagram

4.10.1

Accreting vs Non-accreting Stars

Here, we summarize the pre-main-sequence evolution briefly. In Figure 4.25, pre-main-sequence stellar
evolution is calculated (Yorke & Sonnhalter 2002). Accreting stars (Ṁ = 10−6 −10−3 M⊙ yr−1 ) evolvefrom the lower-right to the upper-left in the HR diagram. In contrast, less massive non-accretingstars (Ṁ∗ = 0 and M ≲ 3 M⊙ ) evolve vertically downward (Hayashi’s convective track). Contrarily,
massive counterparts with M ≳ 3 M⊙ evolve horizontally to the left (Henyey’s radiative track). From
this Figure, the less-massive pre-main-sequence star seems to change the evolution direction, after
the accretion stops. If the evolutionary paths of accreting stars are confined in H-R diagram, thepre-main-sequence stars appear below the accreting star’s evolutionary paths.

130

CHAPTER 4. LOCAL STAR FORMATION PROCESS

4.10.2Main Accretion Phase

Before Deuterium Ignition
Gas accreting onto the protostar generates energy as follows

Lacc =GM∗ Ṁ∗,R∗(4.130)where M∗ , R∗ and Ṁ∗ represent the mass and radius of the protostar and the mass accretion rate.
Temperature at the stellar center increases with time. Finally thermonuclear fusion reaction of
deuterium 2 H(p, γ)3 He begins4 . Before deuterium burning begins, the protostar is radiative, that is,
the energy is transported radiatively.
When a fresh gas of ∆m accretes, thermal energy of ∆U = GM∗ ∆m/R∗ increases. Virial theorem[eq. (2.122)] requires the potential energy must decrease (increase in the absolute value) as∆W = −2∆U ≃ −GM∗ ∆m,R∗ /2(4.131)to achieve mechanical equilibrium. As a result, the fresh gas must contract to about half the radiusat which the gas first joined the core (Stahler, Shu, & Taam 1980).
When the free-falling fresh gas accretes on a static star, an accretion shock forms. Since the gas
temperature increases after passing the shock front, temperature of the postshock gas, Tg , is muchhigher than that of the radiation, Tr . Thus, the postshock gas cools very effectively. The postshock
region with Tg > Tr is called radiative relaxation region. The outgoing luminosity at the accretion
shock is much larger than that inside of the radiative relaxation region.
In the case of less mass stars, the Kelvin-Helmholtz contraction time tKH ≃ GM∗2 /(R∗ L) (Lrepresents the luminosity at the base of the radiative relaxation region) is much longer than theaccretion time scale tacc ≃ M∗ /Ṁ∗ . This gives Lacc ≃ GM∗ Ṁ∗ /R∗ is much larger than L, whichis consistent with the above statement that in the radiative relaxation region a large amount ofaccretion luminosity is radiated away. Since tKH ≫ tacc , the specific entropy inside the relaxation
region is essentially frozen to that when the gas obtained passing through the relaxation region.
Figure 4.26 taken from Stahler, Shu, & Taam (1980) shows the distribution of the specific entropy
against the accumulated mass Mr . The bottom curve corresponds to the state before the nuclear
burning begins when no entropy generation occurs. The temperature increases with mass, because
the star must be compressed to support an extra mass. After the temperature becomes high enoughfor the deuterium burning reaction 2 H(p, γ)3 He as T ∼ 106 K, an extra energy is liberated by the
nuclear fusion reaction. This increases the specific entropy mainly in the off-center region. Figure
4.26 clearly shows that shell deuterium burning occurs at Mr ∼ 0.025M⊙ (shown in the dashed line,
which indicates the specific entropy distribution if no convection is assumed). In reality, convective
instability grows in the region of ds/dr < 0 (see §2.7), which flattens the specific entropy distribution,and s(r) shown in the upper-most solid line is realized.
4

Notation to express the following reaction: 2 H + p → 3 He + γ.

4.10. EVOLUTION IN THE H-R DIAGRAM

131

Figure 4.26: Specific entropy distribution is plotted against the accumulated mass (Stahler, Shu, &Taam 1982). The bottom curve shows the distribution at t = 2.3 × 104 yr (M∗ =0.23 M⊙ ). The top curve is for t = 3.4 × 104 yr (M∗ = 0.34 M⊙ ). The dashed curve
indicates the entropy distribution if no convection is generated. While, the solid curves
represent that of new distribution achieved with convection.

After Deuterium Ignition
One gram of interstellar matter generates energy by the reaction of 2 H(p, γ)3 He as
 

D
XQ
δ≡
H
mH
 


X
Q
[D/H]
14
erg g−1 ,= 0.92 × 10
2.5 × 10−5
0.7
5.5 MeV(4.132)where [D/H] is the deuterium number fraction in H, X is the mass abundance of hydrogen, Q is theenergy rereased by one reaction, and mH is the mass of the hydrogen atom. If this energy is containedin the gas thermal energy, kTd /mH = δ yields

 

[D/H]
X
Q
6
Td = 1.1 × 10
K,
(4.133)
2.5 × 10−5
0.7
5.5 MeVwhich is considerably close to the deutrium ignition temparature ∼ 106 K (Stahler 1988). This gives
an estimate of the deuterium-produced luminosity as!
Ṁ
LD = Ṁ δ = 15 L⊙,(4.134)10−5 M⊙ yr−1
for a star with the accretion rate of Ṁ ≃ 10−5 M⊙ yr−1 .
Entropy inversion ds/dr < 0 is unstable for the convection (see §2.7 in p.49). Convection is
generated in the region of ds/dMr < 0. Convective motion transfers the energy and the entropy.
The specific entropy peak generated by the deuterium burning is smoothed out by the convection.
The convection induced with the deuterium burning brings a fresh gas containing deuterium into the
hot deuterium burning region. Thus, although the abundance of the deuterium is small and thus
the deuterium burning could not continue long, deuterium burning continues actually as long as theconvection can mix the outer deuterium rich fresh gas into the burning region by the convection.

132

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.27: Luminosities in protostars accreting at Ṁ∗ = 10−5 M⊙ yr−1 (left) and Ṁ∗ =10−4 M⊙ yr−1 (right) (Palla & Staler 1993). Accretion luminosity Lacc , deuterium burning luminosity LD , maximum radiatively transferable luminosity Lrad , and luminosity at
the surface Lsurf are plotted.

In Figure 4.27, luminosities in protostars accreting at Ṁ∗ = 10−5 M⊙ yr−1 (left) and Ṁ∗ =−1 (right) are plotted (Palla & Stahler 1993). L⊙ yr
rad represents the maximum luminosity
transferable radiatively for a star with mass M∗ and radius R∗ as
10−4 M
Lrad ∼ 1 L⊙M∗
1M⊙

11/2 

R∗
R⊙

−1/2
,

(4.135)where the Krammers opacity κ ∝ ρT −3.5 is applied for stars with M ≲ 5M⊙ . Actual luminosity at
the surface of the star4
Lsurf = 4πR∗2 σTeﬀ,
(4.136)is larger than Lrad for low mass stars (M∗ < M1 = 2.4 M⊙ for Ṁ∗ = 10−5 M⊙ yr−1 and M1 = 3.4 M⊙
for Ṁ∗ = 10−4 M⊙ yr−1 ). Excess energy is transferred by the convection, Lsurf = Lrad + Lcon . This
figure also shows that except for extremely low-mass object the accretion luminosity is the majorenergy source over deuterium burning Lacc > LD . Since massive objects with M∗ > M3 satisfies
Lrad ≃ Lsurf > Lacc , interior of such stars are fully radiative (energy is transferred by the radiation).
M3 = 3.9 M⊙ for Ṁ∗ = 10−5 M⊙ yr−1 and M3 = 5.9 M⊙ for Ṁ∗ = 10−5 M⊙ yr−1 .
Figure 4.28 (left) illustrates how the deuterium burning in protostars evolves. Accumulating infalling gas and increasing M∗ , the structure of protostars is changed. (a) in low-mass stars, convection
feeds the deuterium of the accreting matter to the burning center. (b) as increasing the mass and thus
the temperature, the opacity of the stellar interior decreases. Finally a radiative zone appears andit prevents from mixing deuterium. (c) as a consequence, central deuterium is depleted and energy
generation rate is reduced. Thus the central region becomes radiative. (d) For higher mass stars,
deuterium ignites as a shell source just outside the radiative central region. Due to this deuterium
shell burning, which is similar to the hydrogen shell burning in the red giants, radius of the protostar expands. Figure 4.28 (right) plots the mass-radius relation of a protostar with accretion rate

4.10. EVOLUTION IN THE H-R DIAGRAM

133

Figure 4.28: Schematic view of deuterium burning in protostars (left). Evolution of the radius of a
protostar is plotted against its mass (right). Open circle before the first swell represents
the ignition of deuterium burning at the center. The second open circle before the larger
swell shows the beginning of deuterium shell burning (M∗ ≃ 2.5M⊙ ). Both figures aretaken from Palla & Stahler (1990).

of Ṁ∗ = 10−5 M⊙ yr−1 . The left open circle represents the beginning of central deuterium burning(state a). On the second circle, radiative barrier appears (state b) and deuterium shell burning begins.
Beyond the mass, the radius swells dramatically. Further increase in the mass increases the gravity
and the radius shrinks. At the third open circle, hydrogen (1 H) begins to burn at the center of the
protostar.
In the HR diagram, the mass-radius relation for the protostars shown in Figure 4.28 (right)
corresponds to the dotted line of Figure 4.29. Observed pre-main-sequence stars: T Tauri stars(M∗ ≲ 2M⊙ ) and Herbig Ae/Be stars (2M⊙ ≲ M∗ ≲ 5M⊙ ) distribute below the dotted line. Thus,
this line is often called the birth line for stars. In Figure 4.29, evolutionary tracks without massaccretion Ṁ = 0 are plotted from the birth line. Modulation of the birth line is related to theswell and shrink of radius shown in Figure 4.28. That is, increase of luminosity around M ≃ 1M⊙
corresponds to the first swell after deuterium burning begins (between the first and second open
circles). Another increase around M ≃ 3.5M⊙ is related to the deuterium shell burning (after thesecond open circle).

4.10.3Pre-main-sequence Evolution

Pre-main-sequence evolutionary path is understood as follows: Consider a situation that a star withM∗ is left when accretion stops. Figure 4.27 indicates that a less-massive star with M∗ < M1 isfully convective. Thus, this evolves along the convective Hayashi track (Hayashi 1961) down to themain-sequence. In contrast, a massive star with M∗ > M3 (Fig. 4.27) is fully radiative and evolves

134

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.29: Pre-main-sequence evolutionary path in the H-R diagram (Palla & Staler 1993). Evolution of stars with different masses from 0.6M⊙ to 6M⊙ are calculated from the birthline (dotted line).

along the radiative Henyey track (Henyey, LeLevier, & Levee 1955) to the main-sequence. These
differences are clearly indicated in Figure 4.29. We can see vertical Hayashi track is seen essentiallyfor stars with M ≲ 1.5M⊙ . Massive stars with M ≳ 2.5M⊙ evolve to the upper-left direction along
the Henyey track.

4.10.4Pre-main-sequence Evolution of Massive Stars

Since mass accretion rate Lacc is large in the massive star formation process, entropy brought fromaccreting material to the stellar interior has more important effect in the pre-main-sequence evolutionof massive stars. There are several ideas how the accreting gas falls onto stellar surface (Fig. 4.30):
Spherical accretion: If the angular momentum of accreting gas does not play an important role, gas
accretes in a spherically symmetric manner [Fig. 4.30 (left)]. Accretion speed seems supersonicand gas falls on to the stellar surface after passing the accretion shock, a shock front facing
outwardly. Since the kinetic energy of the falling gas is changed to the thermal one throughthe shock, the specific entropy brought to the stellar interior is higher than the disk accretion
below.
Disk accretion: If gas cools efficiently and forms the accretion disk in the plane perpendicular to
the angular momentum vector J . When the heat liberated by the friction in the accretion disk
is efficiently radiated away from the disk, the disk keeps relatively cool [Fig. 4.30 (right)]. The
specific entropy brought to the stellar interior seems smaller than the spherical accretion.
Considering the adiabatic constant K ≡ p/ργ ∝ exp s, and the approximations of density and pressureof stellar interior as ρ ∼ M∗ /R3 and p ∼ GM∗2 /R∗4 , the stellar mass-radius relation is writtenR∗3γ−4 ∼ KM∗γ−2 ,(4.137)

4.10. EVOLUTION IN THE H-R DIAGRAM

135

Figure 4.30: Comparison between spherical accretion (left) and disk accretion (right).
−1/3

which reduces to R∗ ∼ KM∗
when the specific heat ratio is γ = 5/3. Since the specific entropy(s = log K) introduced to the stellar interior is much larger for the spherical accretion, K should
be much larger for the spherical accretion as Kspherical accretion ≫ Kdisk accretion . Thus as long as
the specific entropy is mainly determined by the temperature of accreting gas, the stellar radiusof spherical accretion is expected much larger than that of disk accretion. Figure 4.31 shows thedifference between M∗ − R∗ relations of respective accretion models.
The time evolution of pre-main-sequence star with large mass accretion rate is divided into thefollowing four phases [see Fig. 4.31 from Hosokawa, Yorke, & Omukai 2010)].
(I) Convective transport phase (M ≲ 6 M⊙ ): Before the phase of swell (we call the phase inwhich radius expands around M∗ ∼ 10 M⊙ as “swell”), the stellar radius expected for the
spherical accretion is approximately 10 times larger than the disk accretion. In this phase thestellar entropy is transferred by the convection.
(II) Swelling (6 M⊙ ≲ M ≲ 10 M⊙ ): As the temperature rises, the opacity decreases (see Krammer’s opacity of κ ∝ ρT −3.5 ). As a consequence, the high entropy is transferred from theinterior to the outer part radiatively, which brings the surface region (and thus R∗ ) to expand.
(III) KH contraction (10 M⊙ ≲ M ≲ 30 M⊙ ): The stellar envelope no longer receives the entropy,
after the opacity is reduced whole the stellar interior. The phase when the radius decreases withtime/mass is governed by the Kelvin-Helmholtz contraction.
(IV) Main-sequence (M ≳ 30 M⊙ ): The last phase in which radius increases with mass is Hburning phase in the main-sequence star. In the central part, H-burning is beginning. Thus
this is main-sequence stage.

136

CHAPTER 4. LOCAL STAR FORMATION PROCESS

Figure 4.31: (Left:) Spherical accretion. (Right:) Disk accretion. For comparison, M − R relation forthe spherical accretion model is also shown in the thin line. Difference in the radius before
the swell is clear. Decrease in the radius with time (and mass) is due to Kelvin-Helmholtz
contraction. When H burning begins at the center, the star reaches the main-sequence.
These are taken from Figs. 2 and 4 of Hosokawa, Yorke, & Omukai (2010).

Chapter 5

Addendum
5.1

Aligned vs. Misaligned Rotation

In preceding sections, we studied the evolution of molecular cloud whose angular momentum J andmagnetic field B are initially parallel each other and their direction coincides with the axisymmetricaxis. As shown in Appendix K, in the parallel configuration, the torsional Alfvén wave brings theangular momentum away from the cloud and the wave propagates before the cloud rotation is reduced
to 1/2 of the initial one asρclzswept ≃ h
(5.1)
ρext
where h, ρcl , and ρext represent the height of the cloud, the cloud density, and that of the external
medium, respectively [eq. (K.7)]. In contrast, when J ⊥ Ω, the wave front travels in the same foldingtimeρcl 1/4
rswept ≃ R
,
(5.2)
ρext
where R represents the cloud radius [eq. (K.6)]. Considering high contrast between ρcl and ρext ,we expect rswept ≪ zswept . Thus, we expect the perpendicular rotator J ⊥ Ω looses its angularmomentum more rapidly than the aligned rotator J //Ω (see Mouschovias & Paleologou 1979 and1980).
Using MHD simulations, Matsumoto & Tomisaka (2004) studied the evolution of misaligned
rotating cloud (J ∦ Ω). The initial density distribution is similar to the Bonnor-Ebert sphere (Ebert
1955, Bonnor 1956) but its density is enhanced by a factor f . The cloud has the mass of Mcl =6.130 M⊙ , radius Rcl = 0.178 pc, the central density ρc = 1 × 10−19 g cm−3 , and the rotation rate
Ωcl = 7.11 × 10−7 yr−1 . In commonly used non-dimensional parameters, the thermal to gravitationalenergy ratio is α0 ≡ Eth /|Egrav | = 0.5 and the rotational to gravitational energy ratio is β0 ≡Emag /|Egrav | = 0.02. Magnetic field and rotation are initially uniform as



sin θ
B = B0  0  ,
cos θ




0
Ω =  0 ,
Ωcl

(5.3)

where θ represents the angle between B and the z-axis (and thus Ωcl ). Respective models with θ = 0◦and θ = 90◦ correspond to the parallel and perpendicular rotators. Similar to equation (4.121), the
137

138

CHAPTER 5. ADDENDUM

Figure 5.1: Time evolution of the directions of (1) the central magnetic field Bc (thin solid line),(2) the central angular momentum Jc (dotted line), (3) normal vector of the central disk
structure nd (dashed line), along with (4) the magnetic field in the large scale (thick solidline). Diamonds denote the epoch when the adiabatic core formation ρc = ρcr . This shows
three vectors (1), (2), and (3) converge each other. Taken from Fig. 5 of Matsumoto &Tomisaka (2004).

5.1. ALIGNED VS. MISALIGNED ROTATION

139

Figure 5.2: Final stage of the simulation. (Left:) Cross-section along the plane which contains Bc
and z-axis (Plane A of the right panel). (Middle:) Cross-section along the plane whichcontains Bc and x-axis (Plane B of the right panel). Grey scale represents the density.
Velocity component lying in the plane is shown by the arrow and that out of the plane isshown with red and blue contour lines. (Right:) Explanation of Planes A and B. Taken
from Fig. 1 (d ) - (f ) of Matsumoto & Tomisaka (2004).

double-polytropic equation of state is assumed for the gas pressure( 2(ρ < ρcr ),cis ρ 5/3p=c2is ρcr ρρcr(ρ ≥ ρcr ),where ρcr = 2 × 10−11 g cm−3 and cis = 0.19 km s−1 .
Magnetic field and angular momentum at the center of the cloud are defined as
R
BdV
ρ>ρ
,
Bc = R 0.1max
ρ>ρ0.1max dVZJc =ρr × vdV,

(5.4)

(5.5)

ρ>ρ0.1max

where we average the quantities for the central gas with higher density than 1/10 of the maximum
density (ρ > 0.1ρmax ). From the moment of inertia Iij averaged for the same area, we obtain the
shortest principal axis, which gives the direction of the disk-like structure (pseudo-disk), nd .
We begin with the case B0 = 18.6 µG, which corresponds to 1/4 of critical magnetic flux density(mass to magnetic flux ratio µ = 4) and magnetic to gravitational energy ratio is equal to γ0 ≡Emag /Egrav = 0.721. Figure 5.1 √shows the√ evolution of Bc , Jc and nd . Initially Bc is parallel to theglobal magnetic field B = t (1/ 2, 0, 1/ 2)B0 of equation (5.3), which points to (x, y) = (45◦ , 0◦ )in the figure. The large-scale B (thick solid line) rotates counter-clockwise as the cloud rotates asa whole. Although Bc (thin solid line) rotates also counter-clockwise, rotation speed is much fasterthan the large-scale’s. Just after the simulation begins, a pseudo-disk is formed in the directionnd ≃ (30◦ , 0◦ ) (dashed line). This vector rotates also counter-clockwise. Although the angularmomentum Jc (dotted line) is set in the z-direction initially, the direction of Jc begins to approach
those of Bc and nd . After the core formation epoch (open diamond), the outflow begins to flow. It
is shown that directions of the three vectors converge to the point ≃ (16◦ , 33◦ ). It should be noticed
that this direction agrees neither with the initial direction of magnetic field nor that of the angularmomentum. In conclusion, the magnetic field, rotation, and the disk normal direction converge eachother finally.

140

CHAPTER 5. ADDENDUM

Figure 5.3: Jc /Mc2 represents the rotation speed normalized by the central free-fall timescale. Solidline represents Jc,// /Mc2 , in which Jc,// is the component of angular momentum parallel tothe local magnetic field, Bc . Jc,⊥ /Mc2 (dotted line) is calculated from the perpendicularcomponent of Jc,⊥ . MF00, MF45, MF90 are the models with θ = 0◦ , 45◦ , and 90◦ . Other
parameters are the same. NM represents the result of nonmagnetic model B0 = 0. Taken
from Fig. 4 of Matsumoto & Tomisaka (2004).

Figure 5.2 shows the gas flow around the protostar at the final stage of this simulation or at theendpoint of Figure 5.1. Left and middle panels show 2D cross-cuts, in which density and velocity
field are displayed. Panel (d ; left) corresponds to the plane determined by Bc and the z-axis (plane
A of panel (f )) and panel (e; middle) corresponds to that by Bc and the x-axis (plane B of panel(f )). Both panels capture the so-called U-type outflow (see p.127). Thus, we can conclude that evenfrom the misaligned rotator θ ≲ 45◦ , outflow is ejected, in which flow is almost axisymmetric.
R From the angular momentum of2 the central part Jc [eq. (5.5)] and the corresponding mass Mc =
ρ>0.1ρmax ρdV , the quantity Jc /Mc indicates the normalized central rotation rate, which gives theangular rotation speed ωc normalized with the central free-fall timescale (4πGρc )−1/2 as shown inequation (4.113). We divide Jc into the components, of which J// and J⊥ are the angular momentum
parallel and perpendicular to the local magnetic field as
Jc,// = |Jc · Bc |/|Bc |,Jc,⊥ = |Jc × Bc |/|Bc |.

(5.6)

In Figure 5.3, we plotted Jc,// /Mc2 (solid line) and Jc,⊥ /Mc2 (dotted line) as a function of the maximum
density ρmax , which is substituted for the elapse time t, since ρmax is a monotonically increasing
function of t. Figure 5.3 shows following points:
1. Comparing the solid and dotted lines of MF45 model (θ = 45◦ ), Jc,⊥ decreases faster than Jc,// .
2. Comparing MF00 model (θ = 0◦ ) and MF90 model (θ = 90◦ ), the perpendicular rotator(θ = 90◦ ) loses its angular momentum more rapidly than the aligned rotator (θ = 0◦ ).
3. In NM model (dashed line), Jc /Mc2 increases with ρmax before the adiabatic core formation(open diamond). This is explained by the fact that gas which was originally located far fromthe center and had larger angular momentum falls to the center later. Comparing models NM
and MF00, magnetic braking works in MF00 and reduce the normalized rotation rate to ≃ 1/5.

5.1. ALIGNED VS. MISALIGNED ROTATION

141

Figure 5.4: Specific angular momentum j(ρ > 10−12 g cm−3 ) plotted against the central density ρc .
Model with α0 = 0.2 (left) and that of α0 = 0.4 (right:) are compared. In each α0 , 3
models of θ are shown: θ = 0◦ (aligned rotator; solid line), θ = 45◦ (misaligned rotator;
dashed line), and θ = 90◦ (perpendicular rotator; dotted line). Taken from Fig. 10 ofTsukamoto et al. (2018).

Matsumoto & Tomisaka (2004) concluded that the perpendicular rotator (θ = 90◦ ) looses its angularmomentum faster than the aligned rotator (θ = 0◦ ).
Joos, Hennebelle, & Ciardi (2012) revisited the same problem. Their setup is a bit different from
Matsumoto & Tomisaka (2004): the density distribution is taken as ρ = ρ0 /[1 + (r/r0 )2 ], the thermal
to gravitational energy ratio α0 = 0.25, the rotational to gravitational energy ratio β0 = 0.03, mass
to flux ratio ratio are in the range from µ = 2 to µ = 17. They obtained the central specific angularmomentum by the formulaR
ρ>ρ1 ρr × vdV
R
j(ρ > ρ1 ) =(5.7)
ρ>ρ1 ρdV
where ρ1 is chosen as 108 cm−3 , 109 cm−3 , and 1010 cm−3 . From their Figure 4, for µ = 3 and5, and for ρ1 = 109 cm−3 and ρ1 = 1010 cm−3 , the specific angular momentum j(ρ > ρ1 ) of theperpendicular rotator (θ = 90◦ ) is larger than that of the aligned rotator (θ = 0◦ ). They concluded
that the perpendicular rotator (θ = 90◦ ) looses its angular momentum slower than the aligned rotator(θ = 0◦ ), which contradicts with the conclusion of Matsumoto & Tomisaka (2004).
The contradiction is now solved by paying attention to the value of the thermal to gravitationalenergy ratio, α0 (Tsukamoto et al. 2018, see also Hirano et al. 2020). They compared their models
with α0 = 0.4 similar to Matsumoto & Tomisaka (2014) and that of α0 = 0.2 similar to Jooset al. (2012). When α0 = 0.2, j(θ = 90◦ ) is larger than j(θ = 0◦ ) until ρc reaches 10−10 g cm−3
(Fig. 5.4 (left)). In contrast, in the model of α0 = 0.4, j(θ = 90◦ ) < j(θ = 0◦ ) until the end of thecalculation (Fig. 5.4 (right)). The cloud with small α0 contracts faster than that with large α0 . Thus,
the duration of the isothermal contraction in which the magnetic braking is working is longer forlarger α0 .
Beside the effect of α0 , they found a mechanism that weakens the magnetic braking in the modelwith θ = 90◦ and strengthens that in θ = 0◦ . This selective accretion mechanism is as follows:θ = 0◦ : In the model with θ = 0◦ , gas flows mainly in the z-direction and gas initially distributednear the z-axis accretes and form a protostar. Since such gas has slight angular momentum,
angular momentum of the pseudo-disk is thus small (strengthens the magnetic braking).
θ = 90◦ : In the model with θ = 90◦ , gas initially distributed far from the z-axis accretes. Since suchgas has relatively large angular momentum, angular momentum of the pseudo-disk is thus large(weakens the magnetic braking).

142

CHAPTER 5. ADDENDUM

Figure 5.5: Outflow and disks at the epoch of 5000 yr after the protostar formation. Model parameters
are identical in all models. Initial B runs in z-direction in all models but the angularmomentum vector J is inclined at an angle θ from B. Panels (a), (b), (c), and (d )correspond to the model of θ = 0◦ , 10◦ , 45◦ , and 90◦ . Box sizes are taken as 3000 AU(left), 400 AU (middle), and 100 AU (right). A disk is displayed as face-on view in theright box. Taken from Fig. 4 of Hirano et al. (2020).

5.1. ALIGNED VS. MISALIGNED ROTATION

143

Figure 5.6: Outflows and disk with different (α0 , θ). Comparison with models different α0 clearlyshows that angular momentum brought to the central region increases with decreasing
α0 . This is understood the magnetic braking is effective in larger α0 . Taken from Fig. 10of Hirano et al. (2020).

As a result, the angular momentum jc is affected by the duration of magnetic braking and the selective
accretion mechanism.
The former mechanism mainly works during the isothermal phase before the protostar formation.
The latter works even in the adiabatic core stage after the protostar formation. How does themisalignment work in the later evolution? As accretion proceeds onto the protostar, the selective
accretion mechanism seems more important in the later phase. Figure 5.5 shows the outflow expectedfor θ = 0◦ , 10◦ , 45◦ , and 90◦ (Hirano et al. 2010). Initial condition of Figure 5.5 is relatively compact
compared with the previous studies: nc = 106 cm−3 , Mcl = 1 M⊙ , Rcl = 5900 AU, B0 = 430µG,Ωcl = 4.73 × 10−6 , which corresponds to α0 = 0.4, β0 = 0.02, and µ0 = 3. Owing to the sink cellmethod, their simulation can reach ≃ 5000 yr after the core formation.
Figure 5.5 shows that a prominent outflow is observed in (a) and (b), that is, the angle between B
and J is θ ≲ 10◦ . Outflow becomes weak with increasing θ. This seems to show that the componentof magnetic field parallel to the angular momentum is necessary to drive a prominent outflow. In
contrast, the rightmost box shows that the radius of the disk increases with increasing θ. This isunderstood owing to the selective accretion mechanism; specific angular momentum of accreting
gas increases as θ increases up to 90◦ .
Effect of α0 is explored by comparison of models with different α0 . Figure 5.6 plots the outflow and disk at the epoch of 5000 yr after the core formation. Models of Figure 5.6 have different (α0 , θ) but the same β0 = 0.02, µ = 3, and Rcl = 6200 AU. Thus B0 , Ωcl , and Mcl arechosen as: (B0 , Ωcl , Mcl ) = (970 µG, 6.63 × 10−6 yr, 2.5 M⊙ ) for models α0 = 0.2, (B0 , Ωcl , Mcl ) =(480 µG, 4.73×10−6 yr, 1.2 M⊙ ) for models α0 = 0.4, and (B0 , Ωcl , Mcl ) = (260 µG, 3.79×10−6 yr, 0.82 M⊙ )for models α0 = 0.6.
The effect of the angle θ is seen similar to α0 = 0.4 (Fig. 5.5) even for α0 = 0.2 and α0 = 0.6.

144

CHAPTER 5. ADDENDUM

That is, the smaller θ, the more prominent the outflow; the larger θ, the larger the disk. Comparing
three columns, models α0 = 0.2 have a larger outflow in θ = 0 case and a larger disk in θ = 90◦
case. On the other hand, models α0 = 0.6 have a smaller outflow in θ = 0 case and a smaller disk inθ = 90◦ case. This variation is understood if we pay attention to the duration of magnetic braking
in the isothermal collapse phase is longer in α0 = 0.6 models compared with those of α0 = 0.2. This
indicates that a long timescale evolution is affected by (α0 , θ) and that observation of outflows anddisks can specify the initial condition of the star formation.

5.2

Polarization due to Magnetically Aligned Dust Grains

NGC1333 IRAS 4A (Fig.1.28) and G31.41+0.31 (Fig 1.29) are regarded as typical examples whichshow magnetic field in the star forming region seems like a hourglass shown in MHD simulations suchas Figure 4.21. However, this is too intuitive. Direct comparison of MHD simulation and polarization
observation is necessary. If (non-spherically symmetric) dust grains are aligned with magnetic field,
emitted thermal dust emission in mm and sub-mm range is polarized. As is shown in Appendix F.1,
polarization is expressed using Stokes parameter. Polarization degree P and polarization angle χare calculated as equations (F.13) and (F.14),α(q 2 + u2 )1/2,
Σ − αΣ2q,
cos 2χ = 2(q + u2 )1/2
usin 2χ = 2.
(q + u2 )1/2
P =Here, we use the relative Stokes parameter shown in equation (F.11),Zq = ρ cos 2ψ cos2 γds,Zu = ρ sin 2ψ cos2 γds,Z
Σ = ρds,Z  2
cos γ 1
Σ2 = ρ
−
ds,
2
3

(5.8)

(5.9)

where α represents the ratio of polarization cross section to extinction cross section [eq. (F.12)] andα ∼ 0.1. The meaning of the angles γ and ψ are shown in Figure F.1 (right).
Since γ represents the angle between B and the celestial plane, when γ = π/2, i.e., magnetic
field is parallel to the line of sight, contribution is zero; while when γ = 0, i.e., magnetic field isperpendicular to the line of sight, contribution becomes maximum. A dust cloud having magnetic
field parallel to the line of sight indicates low polarized intensity and low polarization degree. Thatwith B perpendicular to the line of sight must be observed with high polarization degree.
Summarizing in short, calculating the relative Stokes parameter from ρ and B data gives the
expected polarization for the simulation.

5.2.1

Depolarization

As seen in NGC1333 IRAS 4A and G31.41+0.31 [Figs. 1.28 (middle) and 1.29 (left)], distribution ofpolarized intensity (Q2 + U 2 )1/2 = P I is unique. Polarized intensity P I is weak in the direction of the5.2. POLARIZATION DUE TO MAGNETICALLY ALIGNED DUST GRAINS

element 2

B

B

145

ρ

ρ
element 1

Line of sight

Figure 5.7: Explanation of depolarization. Passing two volume elements which have position angles
of B as ψ1 and ψ1 + π/2 but have the same density ρ and inclination of the magnetic field
γ, contributions are canceled.

major axis of intensity I. Considering depolarization process, this unique distribution is naturally
understood.
As shown in equation (5.9), contribution to (q, u) from an element ds is proportional to ρ cos 2ψ cos2 γ
and ρ sin 2ψ cos2 γ. Even when ρ cos2 γ is spatially constant but magnetic field changes its position
angle, cancellation is expected. As shown in Figure 5.7, when two elements have position angle of ψ1and ψ2 = ψ1 + π/2, then dq2 = −dq1 and du2 = −du1 .
Kataoka, Machida, & Tomisaka (2012) calculated the evolution of a magnetized cloud with misaligned rotation. They showed also a non-rotating case Ω0 = 0 with thermal to gravitational energy
ratio α0 = 0.5, rotational to gravitational energy ratio β0 = 0, magnetic to gravitational energy ratio
γ0 = 0.06, and normalized mass to flux ratio µ0 = 6.6 × (0.53/3π)51/2 /(1/2π) = 5.2.
Figure 5.8 (left) shows the expected polarization structure when the protostar has accumulated
0.5 M⊙ or t = 8.9 × 104 yr has passed. This snapshot is made for the case with the angle between theline of sight and the symmetric axis of the cloud equal to θ = 60◦ . Contour lines indicate Stokes I
and color represents the polarized intensity distribution P I = (Q2 +U 2 )1/2 . Figure 5.8 (left) indicatesclearly that P I is weak in the direction of major axis of the Stokes’ I, of which characteristic point
coincides with polarization observations (Figs. 1.28 (middle) and 1.29 (left)). The polarized intensity
P I is weakened in the major axis direction of I by the depolarization. Figure 5.8 (right) shows themechanism: Contour and color shows the density distribution on the symmetric plane y = 0. Red andblue curved lines represent the magnetic field, which indicate an hourglass shape in 3-dimension andare projected on the celestial plane. Near the major axis of Stokes I, front-side projected field lines(red) and rear-side ones (blue) crosses perpendicularly, each other. In the rest, such perpendicularcrossing is rare. Thus, near the major axis, P I is weakened by the depolarization. If we find thiskind of distribution of P I as well as the two-dimensional (projected) hourglass shape, we expect the
subject has an three-dimensional hourglass shape.

5.2.2

Polarization of Outﬂow

In the preceding section, we have seen that the hourglass-type magnetic field gives a unique distribution of P I, where Ω0 = 0. Coexisting of rotation and magnetic field brings the contracting
gas cloud the outflow (see § 4.9). Expected polarization is calculated for the MHD flow after theoutflow is brown (Fig. 5.9). In Figure 5.9 (top), we plot the gas flow obtained with axisymmetrictwo-dimensional ideal MHD simulations (Tomisaka 2002). Expected polarization is shown in thebottom row. The line of sight is specified with the inclination angle θ from the symmetric z-axis.

146

CHAPTER 5. ADDENDUM

Figure 5.8: (Left) Expected polarized intensity (color) and polarization B-vector (bar). Stokes I
distribution is shown by contour lines. P I is displayed in an arbitrary unit. Inclination
angle of the line of sight is θ = 60◦ . Although this corresponds to Fig. 6 (d ) of Kataoka
et al. (2012), not P but P I is displayed. (Right:) Three-dimensional view of magnetic
field is shown as well as the density distribution of y = 0 plane [Fig. 7 (d ) of Kataoka etal. (2012)]. Red field lines run in the front side of the cloud while the blue ones run in therear side. In the major axis direction, these two field lines are perpendicular each other,which induces the depolarization.

From (Bottom-left) to (Bottom-right), the models with θ = 0◦ (pole-on view), 45◦ , and 90◦ (edge-onview) are shown, respectively.
Looking from the symmetry axis (bottom-left), polarization degree is low. This is understood by
the fact that poloidal magnetic field is largely parallel to the z-axis especially in the high-density
disk, which realizes γ ∼ 90◦ and cos γ ≪ 1 in the integration of relative Stokes parameter (q, u). Not
only radial component of polarization B-vector but also azimuthal component exist in the outer partr ≳ 500 AU in the pole-on view. Since large amplitude of polarized intensity is not expected, however,
it is hard to catch such polarization B-vector field. Disk of r ≲ 500 AU can be seen associated with
ϕ-component of polarization B-vector.
On the other hand, in the model of θ = 90◦ , a disk of ∼ 1000 AU scale width shows high degree ofpolarization (bottom-right). In addition to it, the outflow itself is seen as a low-polarization region.
Hour-glass type polarization pattern is seen in the edge-on view.
Between these two extremes, different symmetry appears in θ = 45◦ model (bottom-middle). That
is, Stokes I (black contour line) has the mirror symmetry with respect to the η-axis (or ξ = 0) andto the ξ-axis (or η = 0). (See Appendix F.2 for the definition of ξ- and η-coordinate.) However,polarization degree P (color) and thus the polarized intensity P I indicates the point symmetry.
To understand this symmetry pattern, a schematic model is prepared in Figure 5.10. In Figure
5.10, we show magnetic field at four points, in which + and − points are symmetric with respect
to z = 0 plane and R and L points are symmetric with respect to x = 0 plane. Here, we assume
magnetic field is composed of Bz = const and Bϕ asBϕ,0(z < 0)Bϕ =(5.10)−Bϕ,0 (z > 0).
Looking from a line of sight with the inclination angle θ = 45◦ , magnetic field vector B is seen like

5.2. POLARIZATION DUE TO MAGNETICALLY ALIGNED DUST GRAINS

Pol(max)=12.6%
Sigma max=25.5 min=22.5

A1O5.L1001.64.phi00/Lev05/th=45= 10.00

Pol(max)=13.4%
Sigma max=25.0 min=22.5

A1O5.L1001.64.phi00/Lev05/th=90

Pol(max)=15.0%
Sigma max=25.5 min=22.0= 10.00

4= 10.00

4

A1O5.L1001.64.phi00/Lev05/th=00

147

4

15

15

1000
22.70

22.704

23
.1

8
23.40
12

23.40

8

8

10
4

12
23.40

4

8

22.70

5

23
.00

-1000

4

24.10
4

0

4

5

η(AU)

48

8

-1000

10

8

P(%)

23.50

η(AU)

0

23
.00

23
.1

10

P(%)

23
12 .7

η(AU)

4

0

22.70

5

-1000
4

4

4

0
ξ(AU)

1000

0
-1000

0
ξ(AU)

1000

4

0
-1000

-1000

0
ξ(AU)

1000

Figure 5.9: Polarization of outflows. (Top:) MHD simulation result. Solid and dashed lines represent
density contour lines and magnetic field line, respectively. Horizontal and vertical axes
mean r and z of the cylindrical coordinate, respectively. (Bottom:) Expected polarization.
Color and the black contour line represent polarization degree P and Stokes I. Short bars
illustrate the direction of polarization B-vector. (left), (middle), and (right) panels show
the models with inclination angle of θ = 0◦ (pole-on), θ = 45◦ , and θ = 90◦ (edge-on),respectively. Taken from Fig. 5 of Tomisaka (2011).

Bz
Bφ

Bz
Bφ

Bz

L+

Bφ

Bz

R+
Bz
Bφ

L-

Bφ

RBφ

Figure 5.10: Explanation of point symmetric polarization pattern. Taken from Fig. 10 of Tomisaka(2011).

0

P(%)

.00
23

1000

4

15

1000

148

CHAPTER 5. ADDENDUM

Figure 5.10. As is seen easily, two kinds of rhombus appear, when performing composition of vectors.
That is, R+ and L− are the same; L+ and R− are the same. Therefore position angle of B, ψ,satisfies the following relation:
ψ(L+) = −ψ(R−),ψ(L−) = −ψ(R+),ψ(L+) ̸= ψ(R+),ψ(L−) ̸= ψ(R−),(5.11)

ψ(L+) ̸= −ψ(L−),ψ(R+) ̸= −ψ(R−).
The first two relations explain the point symmetric nature; The last four relations explain breakdown of the mirror symmetry. Therefore, the point symmetric polarization pattern is coming fromtoroidal magnetic field like Bϕ = −z/|z|Bϕ,0 . As the consequence, weak P I, which originates in the
depolarization between fore- and rear-side of the flow, indicates the same mirror symmetry. Thus,
if polarization pattern shows the mirror symmetry, there exists a toroidal field anti-symmetric withrespect to the major axis of Stokes I.
If this point symmetric polarization pattern is observed, this indicates the coexistence of poloidal
Bp ≡ (Br , Bz ) and toroidal Bϕ components. Since the toroidal magnetic field is made from thepoloidal current jp ≡ (jr , jz ) (Ampère’s law), there exists the azimuthal component of Lorentz forceFϕ eϕ = jp × Bp , which applies the magnetic torque of rFϕ . That is, the point symmetric pattern is
an identifier of the magnetic torque, which transfers angular momentum.
Polarization observation has an ability to explore the magnetic field as well as the density distribution, by which we can reach deeper understanding on the relationship between phenomenon
and its mechanism. This shows the importance of this kind of visualization named “observational
visualization.”

Appendix A

Basic Astrophysical Quantities
M⊙
L⊙
R⊙
mp
me
AU
pc
amu
G
k
h
c
σ

the Solar mass
the Solar luminosity
the Solar radius
proton masselectron mass
the astronomical unit
parsecatomic mass unit
gravitational constantBoltzmann constant
Plank constantlight speed
Stefan-Boltzmann constant

1.989 × 1033 g
3.83 × 1033 erg s−1
6.96 × 1010 cm
1.67262 × 10−24 g
9.10938 × 10−28 g
1.49598 × 1013 cm
3.0857 × 1018 cm
1.66054 × 10−24 g
6.67 × 10−8 dyn cm2 g−2
1.38065 × 10−16 erg K−1
6.62607 × 10−27 erg s
2.99792458 × 1010 cm s−1
5.67040 × 10−5 erg s−1 cm−3 K−4Table A.1: Astrophysical quantities, units, and constants.

149

Appendix B

Basic Equation of Hydrodynamics
B.1

What is ﬂuid?

Gas and liquid change their shape according to the shape of the container. This is a definition of
ﬂuid.

B.2

Equation of Motion

Consider a fluid element with mass m as in section B.1. From the Newton’s second law of mechanics,
the acceleration of an element is written as
dv
F= fm ≡ ,
dt
m

(B.1)

where m, v, F , and f m represent the mass, the velocity of the element, and force working on theelement and the force per mass working on the element, respectively.
In fluid dynamics, using the mass density ρ and the force working on the unit volume f v equation(B.1) is rewritten as
dv
ρ
(B.2)
= f v.
dt
Which kind of force works in a fluid? Pressure force does work in any fluids. Beside this, if thereis the gravity, ρg should be included in f v . If the electric currents is running in the fluid and themagnetic field exists, the Lorentz force j × B should be added.
To write down the expression of the pressure force, consider a fluid element between x and x + ∆x
as shown in Figure B.2. Pressure force exerting on the surface S at x is p(x)S, while that on the

m
V
ρ

m
V’
ρ’

t

t+∆t

Figure B.1: Movement of fluid element

151

152

APPENDIX B. BASIC EQUATION OF HYDRODYNAMICSSp(x)p(x+∆x)x

x+∆x

Figure B.2: Pressure force exerted on an element with volume S∆x

opposite side is −p(x + ∆x)S. The net pressure force working on the volume of S∆x is equal to(p(x) − p(x + ∆x))S, which is approximated as −(∂p/∂x)∆xS + O(∆x2 ). Thus, the pressure force
working on the unit volume is written −∂p/∂x. Equation (B.2) can be rewritten asρdv= −∇p + ρg,
dt

(B.3)

when gravity g is working.

B.3

Lagrangian and Eulerian Equations

The time derivative appearing in equation (B.1) expresses how the velocity of a speciﬁc particle
changes. Therefore, what appears in equation (B.3) means the same, that is, the position of the gas
element concerning in equation (B.3) moves and the positions at t and t + ∆t are generally different.
However, considering the velocity field in the space, the time derivative of the velocity should be
calculated staying at a fixed point x.
These two time derivative are different each other and should be distinguished. The former time
derivative is called Lagrangian time derivative and is expressed using d/dt. On the other hand,
the latter is called Eulerian time derivative and is expressed using ∂/∂t. These two are relatedwith each other. Consider a function F whose independent variables are time t and position x, that is
F (x, t). The difference dF
dt ∆t, using the Lagrangian time derivative of F , represents the the differenceof F (t + ∆t) from F (t) focusing on a specific fluid element, whose positions are different owing to its
motion. The element at the position of x0 at the epoch t0 moves to x0 + v 0 ∆t in time span of ∆t.
Thus the difference is expressed asdF∆t = F (x0 + v 0 ∆t, t0 + ∆t) − F (x0 , t0 ),
dt




∂F
∂F
≃
· v 0 ∆t +∆t,
∂x t x0 ,t0∂t x x0 ,t0(B.4)

where we used the Taylor expansion of F . The difference corresponding to the Eulerian derivative iswritten down as∂F∆t = F (x0 , t0 + ∆t) − F (x0 , t0 ),(B.5)∂t
and this is equal to the second term of the rhs of equation (B.4). Comparing equations (B.4) and(B.5), the Lagrangian derivative contains an extra term besides the Eulerian derivative. That is, theB.4. CONTINUITY EQUATION

153

Lagrangian derivative is expressed by the Eulerian derivative as
d
∂
∂
∂
=
+v·
=
+ v · grad.
dt
∂t
∂x
∂t

(B.6)

Applying the above expression on equation of motion based on the Lagrangian derivative (B.3),
we obtain the Eulerian equation motion:
∂v
ρ
+ v · ∇v = −∇p + ρg.
(B.7)
∂t

B.4

Continuity Equation

Another basic equation comes from the mass conservation. This is often called the continuity equation,
which relates the change of the volume to its density. Consider a fluid element whose volume is equal
to ∆V . The mass contained in the volume is constant. Thus
dρ
d∆V
dρ∆V
=
∆V +
ρ = 0.
dt
dt
dt
The variation of the volume d∆Vdt is rewritten as
Z
Z
d∆V
=
divvdV,v · dS =
dt
∂∆V
∆V

(B.8)

(B.9)

where ∂∆V represents the surface of the fluid element ∆V . From equations (B.8) and (B.9), we
obtain the mass continuity equation for Lagrangian time derivative asdρ+ ρdivv = 0.
dt

(B.10)

Using equation (B.6) this is rewritten to Eulerian form as∂ρ+ div(ρv) = 0.
∂t

(B.11)

Basic equations using the Lagrangian derivative are equations (B.3) and (B.10), while those of
the Eulerian derivative are equations (B.7) and (B.11).

B.4.1

Expression for Momentum Density

It is useful to describe the equation for momentum ρv. Eulerian derivative of momentum density ρv
is rewritten as
∂ρv
∂t

∂ρ
∂v
v+ρ ,
∂t
∂t
= −∇(ρv)v − ρv · ∇v − ∇p + ρg,=(B.12)

where we used equations (B.7) and (B.11). In the Cartesian coordinate (x, y, z) = (xi , x2 , x3 ), thisbecomes

 



 


∂/∂x
p 0 0
gx
ρvx vx ρvx vy ρvx vz
∂/∂x
ρvx
∂ 
ρvy  +  ∂/∂y   ρvy vx ρvy vy ρvy vz  +  ∂/∂y   0 p 0  = ρ  gy  .
∂t
∂/∂z
0 0 p
gz
∂/∂z
ρvz vx ρvz vy ρvz vz
ρvz(B.13)

154

APPENDIX B. BASIC EQUATION OF HYDRODYNAMICS

Using the momentum stress tensor
πi j = ρvi vj + δi j p,

(B.14)∂ρvi ∂πi j+= ρgi ,
∂t∂xj

(B.15)

the above equation is written as

where δi j represents the Kronecker’s delta as δi j = 0 for i ̸= j and δi j = 1 for i = j.
If there is no external force, rhs of equation (B.15) is equal to zero. If the momentum density ρvwere to obey the continuity equation, equation (B.15) would be∂ρvi ∂ρvi vj
+
= 0.
∂t
∂xj

(B.16)

However, this is incorrect, because there exists the pressure force in the fluid and thus the momentumof the fluid element is not conserved.

B.5

Energy Equation

The above basic equations (B.3) and (B.10) or equations (B.7) and (B.11) contain three dependent
variables ρ, p, and v. The number of the variables, 3, is larger than the number of equations, 2.
Therefore, an extra equation is needed to close the basic equations.

B.5.1

Polytropic Relation

If the pressure of the fluid, p, is expressed only by the density, ρ,
p = P (ρ),(B.17)

the number of dependent variables is reduced to two and the above equations are sufficient to describe
the dynamics of the fluid. Occasionally, the pressure is assumed proportional to the power of ρ as
p = KρΓ ,(B.18)

where Γ is a constant. This assumption is called polytropic relation.
The fact should be reminded that the validity of the assumption comes from the physical conditionof the system. In the case that the temperature of the gas is kept constant owing to the cooling andheating process, the gas pressure is proportional to the density
p = c2is ρ,(B.19)

where cis = (kT /µmp )1/2 (k: Boltzmann constant, µ average molecular weight, and mp is the protonmass) represents the isothermal sound speed and is constant.
Another example is the isentropic fluid, in which the entropy is kept constant. In this case thepressure is proportional to ργ , as
p = c2s ργ ,(B.20)
where γ is the specific heat ratio = cp /cv and cs = (γkT /µmp )1/2 represents the adiabatic soundspeed. In these cases the polytropic relations such as equations (B.19) and (B.20) play a role as thethird equation of basic equations of hydrodynamics.

B.6. SHOCK WAVE

B.5.2

155

Energy Equation from the First Law of Theromodynamics

In more general cases, the last equation comes from the first law of the thermal physics,∆Q = dU + pdV.

(B.21)

where U and V represent the internal energy and the volume and ∆Q is the heat added to or
subtracted from the system. Using the total energy per unit volume,1U
ϵ = ρv 2 + ,2V(B.22)

the above equation gives the equation for the total energy per unit volume:∂ϵ+ div(ϵ + p)v = ρv · g.
∂t

(B.23)

Since the total energy per unit volume is expressed using the basic physical quantities as11
ϵ = ρv 2 +p2
γ−1(B.24)

for the ideal gas, equation (B.23) is the final basic equation for hydrodynamics. Equations (B.7),(B.11), and (B.23) are basic equations of hydrodynamics using the Eulerian time derivative.
Problem 16
From the continuity equation (B.11) and the equation of motion (B.7) or the equation for momentum density (B.15), show that the equation for the kinetic energy density is expressed asfollows∂ 12 ρv 2
1 2+∇·
ρv v = ρg · v.
(B.25)∂t2
Then, obtain equation for the total energy (B.23).




B.6

Shock Wave

B.6.1

Rankine-Hugoniot Relation

Passing through a shock front moving with a speed Vs , the physical variables ρ, p, and u change
abruptly. Since the basic equations of hydrodynamics is unchanged after choosing a system moving
Vs (the Galilean transformation), the continuity equation∂ρ ∂ρu+=0∂t∂x(B.26)

gives an equation for a steady state as∂ρu= 0.
(B.27)∂x
Considering a region containing the shock front at xs which extends from x = xs − ∆x to x = xs + ∆x(∆x > 0) and integrating the above equation, we get
Z xs +∆xdρu(B.28)
dx = [ρu]xxss +∆x−∆x .
xs −∆x dx

156

APPENDIX B. BASIC EQUATION OF HYDRODYNAMICS

50

40

30

20

10

0
1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure B.3: p2 /p1 (y-axis) is plotted against ρ2 /ρ1 (x-axis) as shown in equation (B.34). In thisfigure, we assumed γ = 5/3. In the limit of p2 /p1 → ∞, ρ2 /ρ1 = (γ + 1)/(γ − 1) = 4 (seeequation (B.36)).

Thus, we obtain the jump condition concerning the mass conservation as
ρ1 u1 = ρ2 u2 ,(B.29)

where the quantities with suffix 1 are for preshock and those with suffix 2 are for postshock.
Equation of motion for steady state
∂u
∂p
=− ,
∂x
∂x

(B.30)

p1 + ρ1 u21 = p2 + ρ2 u22 ,(B.31)

ρu
gives

where we used equation (B.29).
From the energy equation (B.23) without gravity, the energy conservation equation is rewritten
as

 
  2
xs +∆x
Z xs +∆x
d
1 2
γ
u
γ p
ρu +
p u dx = ρu+.
(B.32)
2γ−1
2
γ − 1 ρ xs −∆x
xs −∆x dx
Using equation (B.29), this gives the third jump condition as
u21
γ p1
u2
γ p2
+
= 2+
.
2
γ − 1 ρ12
γ − 1 ρ2

(B.33)

Using equations (B.29), (B.31), and (B.33), jump conditions in density and temperature are given as
follows:ρ2
u1(γ − 1) + (γ + 1)(p2 /p1 )==,(B.34)
ρ1u2(γ + 1) + (γ − 1)(p2 /p1 )c2
p2 (γ + 1) + (γ − 1)(p2 /p1 )
T2
= s2
×
=
.
2
T1
p1 (γ − 1) + (γ + 1)(p2 /p1 )cs1(B.35)

These relations are called as Rankine-Hugoniot relation. The relation between density jump (ρ2 /ρ1 )and pressure jump (p2 /p1 ) is illustrated in Figure B.3. In the limit where the preshock pressure is
negligible compared with postshock pressure (p2 ≫ p1 ), the density ratio is still finite as
ρ2
γ+1
=
,
ρ1
γ−1

(B.36)

B.7. DERIVATION OF SELF-SIMILAR SOLUTION

157

although the temperature ratio T2 /T1 goes to ∞.
Problem 17We have learned the basic equations of hydrodynamics [eqs. (B.7) and (B.11)] described using
the system S with coordinate (x, y, z). Show that the equations are unchanged when we use thecoordinate system S’ of (x′ , y ′ , z ′ ) moving with a constant velocity (V, 0, 0) relative to S as ′
 x = x − V t,y ′ = y,(B.37)
 ′z = z,(Galilean invariant).
Problem 18

Derive equations (B.34) and (B.35) from equations (B.29), (B.31), and (B.33).


B.6.2



Isothermal shock

In the case of the gas is isothermal p = c2is ρ, equation (B.31) becomes
ρ1 c2is + u21 = ρ2 c2is + u22 .

(B.38)

Eliminating ρ from equations (B.29) and (B.38), we obtainu1 u2 − c2is (u1 − u2 ) = 0,

(B.39)

u1 u2 = c2is .

(B.40)u1
u2
ρ2== 21 .
ρ1u2
cis(B.41)

which means

From equation (B.29),

This indicates that when the preshock Mach number M = u1 /cis ≫ 1, the ratio of the postshockdensity to the preshock density becomes large. When the gas is well approximated by the isothermal
equation of state even when it passes the shock front, the shock is called isothermal shock.

B.7

Derivation of Self-similar Solution

Equations (4.77)-(4.80) give the relation between self-similar variables (independent variable, ξ anddependent variables, Ω(ξ), V (ξ), m(ξ)). For t < t0 , equation (4.80) is written as ξ = −r/cs (t − t0 ),which gives
 
∂ξ
−1
=
(B.42)∂r t
cs (t − t0 )
 
∂ξ
−r
ξ
.
(B.43)==−
2∂t r
cs (t − t0 )t − t0

158

APPENDIX B. BASIC EQUATION OF HYDRODYNAMICS

When t − t0 < 0, ∂ξ/∂t > 0 means that a typical point of the solution (ξ = const) shrinks with time.
The first term of the continuity equation (4.74) is rewritten as


∂ρ
∂
Ω(ξ)
=
,
∂t
∂t 4πG(t − t0 )2
1
dΩ ∂ξ
−2Ω
+
,
=
4πG(t − t0 )3 4πG(t − t0 )2 dξ ∂t


−1
dΩ
=
2Ω + ξ
,(B.44)
4πG(t − t0 )3dξ
and the second term is

1 ∂
2
r
ρv
=
r2 ∂r
=
=
=

∂ρv 2ρv
+
,
∂r
r

∂
Ω
2Ωcs V
cs V +,
2∂r 4πG(t − t0 )−cs (t − t0 )ξ4πG(t − t0 )2dΩ2Ωcs V−1+cs V,
dξ 4πG(t − t0 )2
cs (t − t0 ) −cs (t − t0 )ξ4πG(t − t0 )2


dΩV
2ΩV
−1
+
.
3
4πG(t − t0 )dξξ(B.45)

Thus, we obtain the continuity equation for similarity variables as
dΩV
2ΩV
dΩ
+
+ξ
+ 2Ω = 0.
dξξdξEquation (4.76) or ∂Mr /∂r = 4πr2 ρ is rewritten using self-similar variables as 3
c (t − t0 )
dξ
Ω
d
− s
m
= 4π [cs (t − t0 )ξ]2
,
dξ
G
dr
4πG(t − t0 )2c3 (t − t0 )
−1
dm
c2
∴− s
= s ξ 2 Ω,G
cs (t − t0 ) dξGdm
= ξ 2 Ω.
∴dξOn the other hand, ∂Mr /∂t = −4πr2 ρv leads to 3
∂cs (t − t0 )Ω−
m = 4π [cs (t − t0 )ξ]2 cs V∂t
G
4πG(t − t0 )2c3
c3 (t − t0 ) dm ∂ξc3∴ − sm− s= s ξ2V Ω
G
G
dξ ∂t
G
dm
= ξ 2 V Ω,
∴ −m + ξdξ(B.46)(B.47)(B.48)

where we used equation (B.43). With help of equation (B.47), this is reduced tom = ξ 2 (ξ − V )Ω,(B.49)

by which equation (4.81) is derived.
Problem 19Referring to the above, derive the equation of motion using the self-similar variables from equation(4.75). Then, show the other two equations (4.82) and (4.83).
Appendix C

Basic Equations ofMagnetohydrodynamics
C.1

Magnetohydrodynamics

Here, we derive the basic equations of magnetohydrodynamics. Differences from hydrodynamics are
the Lorentz force in the equation of motion and the induction equation of magnetic field. The Lorentz
force is written as
1
j×B =(∇ × B) × B,
(C.1)
4π
where we used the Ampère’s law ∇ × B = (4π/c)j and the displacement current is ignored. Thus,
the equation of motion becomes as


∂v
1
ρ
+ (∇ · v)v = −∇p − ρ∇ψ +(∇ × B) × B,
(C.2)
∂t
4π
in the Eulerian form.
The Faraday’s law ∇ × E = −(1/c)(∂B/∂t) and the Ohm’s law j = σ(E + v × B/c) (σ is theelectric conductivity) lead the induction equation of the magnetic field as
∂B
c2
∇×B
= ∇ × (v × B) −
∇×
.
∂t
4π
σ

(C.3)

The factor c2 /4πσ is the coefficient of magnetic diffusibility. When we assume the electric conductivity
σ → ∞, diffusive MHD equation, equation (C.3), reduces to the ideal MHD equation as∂B= ∇ × (v × B).
∂t

C.1.1

(C.4)

Flux Freezing

An important nature of the ideal MHD is derived from equation (C.4), that is, the matter and themagnetic field is coupled with each other. Consider aR gas parcel threaded with the magnetic field B.
The magnetic flux anchored to the gas parcel ΦB ≡ B · dS is changes
Z
Z
Z
dΦB
∂B
d
B · dS ==· dS + B · v × ds,
(C.5)
dt
dt
∂t
159

160

APPENDIX C. BASIC EQUATIONS OF MAGNETOHYDRODYNAMICS

Figure C.1: The change of flux anchored to a surface dS. One part of the change comes from thetemporal change in magnetic flux density. The other comes from the change of theboundary of the surface due to gas motion. Time variation of the area is given by
vdt × ds.

where s = ∂S. The first term of the rhs of the equation comes from a time derivative of the magnetic
flux density, while the last term represents the change of the
R integral regionR due to the fluid motion(see Fig. C.1).
∵
dS
=
v
×
ds.
Using
the
Stokes’theorem∇ × A · dS = A · ds, the last term isRR
reduced to B × v · ds = ∇(B × v) · dS. Finally, we obtain

Z 
dΦB
∂B
=
− ∇(v × B) · dS = 0,
(C.6)
dt
∂t
where we used equation (C.4). This means that the magnetic flux anchored to a gas parcel does notchange in the ideal MHD regime.

C.1.2

Basic Equations of Ideal MHD

Basic equations are as follows: The mass continuity as∂ρ+ ∇ · (ρv) = 0,∂t
the equation of motion as


∂v
1
ρ
+ (∇ · v)v = −∇p − ρ∇ψ +(∇ × B) × B,
∂t
4π

(C.7)

(C.8)

the equation of thermal energy as∂ϵ+ div(ϵ + p)v = ρv · g,
∂t

(C.9)

or some barotropic relation p = P (ρ) and the induction equation as∂B= ∇ × (v × B).
∂t

C.1.3

(C.10)

Axisymmetric Case

The basic equations to be solved are the magnetohydrodynamical equations and the Poisson equation
for the gravitational potential. In cylindrical coordinates (z, r, ϕ) with ∂/∂ϕ = 0, the equations are
expressed as follows:
∂ρ
∂
1 ∂
+
(ρvz ) +(rρvr ) = 0,
(C.11)∂t∂zr ∂r

C.1. MAGNETOHYDRODYNAMICS
∂ρvz
∂t

∂ρvr
∂t

∂
1 ∂
(ρvz vz ) +(rρvz vr ) =∂zr ∂r 

 
∂Bϕ
∂Br
∂ψ
1
∂Bz
2 ∂ρ
− cs
−ρ
+
−
Bϕ −
−
Br ,
∂z
∂z
4π
∂z
∂z
∂r

161

+

∂
1 ∂
(ρvr vz ) +(rρvr vr ) =∂zr ∂r 

 
∂ψ
1
1 ∂
∂Br
∂Bz
2 ∂ρ
−ρ
+
−
(rBϕ )Bϕ +
−
Bz ,
− cs
∂r
∂r
4π
r ∂r
∂z
∂r


∂ρrvϕ
∂Bϕ
∂
1 ∂
r 1 ∂
+
(ρrvϕ vz ) +(rρrvϕ vr ) =(rBϕ )Br +
Bz
∂t
∂z
r ∂r
4π r ∂r∂z(C.12)

+

1 ∂
∂Bz
=
[r(vz Br − vr Bz )],
∂t
r ∂r
∂Br
∂
= − (vz Br − vr Bz ),
∂t
∂z
∂Bϕ
∂
∂
=
(vϕ Bz − vz Bϕ ) −(vr Bϕ − vϕ Br ),
∂t
∂z
∂r


∂ψ
∂2ψ 1 ∂+
r
= 4πGρ,
∂z 2
r ∂r
∂r

(C.13)(C.14)(C.15)(C.16)(C.17)(C.18)

where the variables have their ordinary meanings. Equation (C.11) is the continuity equation; equations (C.12), (C.13) and (C.14) are the equations of motion. The induction equations for the poloidal
magnetic fields are equations (C.15) and (C.16) and for the toroidal magnetic field is equation (C.17).
The last equation (C.18) is the Poisson equation.

Appendix D

Hydrostatic Equilibrium
D.1

Polytrope

If we choose the polytropic equation of state1 ,p = Kρ1+1/n ,

(D.1)

 2 
r dp
= −4πGρ,
ρ dr

(D.2)

the hydrostatic balance is expressed by
1 d
r2 dr

where we used equations (4.1) and (4.2). A hydrostatic gaseous star composed with a polytropic gas
is called polytrope. Normalizing the density, pressure and radius asρ ≡ ρc θ n ,
p ≡

(D.3)

Kρ1+1/nθn+1 ,c(1−n)/nr ≡ Hξ =(n + 1)Kρc
4πG

(D.4)

!1/2
ξ,

(D.5)

we obtain a normalized equation as
1 d
ξ 2 dξ


ξ2

dθ
dξ


= −θn ,

(D.6)

which is called Lane-Emden equation of index n. The boundary condition at the center of polytrope
which is regular at the center (r = 0) should beθ = 1,dθ= 0,dξat ξ = 0. Solution of this equation is plotted for several n in Figure D.1.
1

Reference book for this appendix is chapter 4 of Chandrasekhar (1939).

163

(D.7)
(D.8)

164

APPENDIX D. HYDROSTATIC EQUILIBRIUM
1
0.8
0.6
0.4
0.2

1

2

4

3

5

6

7

Figure D.1: Solutions of eq. (D.6) are plotted for indices of n = 3 or Γ = 4/3 (the most extendedone), n = 5/2 or Γ = 7/5 (the middle), and n = 3/2 or Γ = 5/3 (the most compactone). Although, these three solutions have no zero-point at finite ξ, for n ≥ 5 gas extends
infinitely and the solution has no zero-point.

Mass of the polytrope is written down asZ r
M =4πρr2 dr,
0
Z ξ
3
= 4πH
θn ξ 2 dξ,0Z ξd 2 dθ
ξdξ,= 4πH 3dξ0 dξdθ= −4πH 3 ξ 2
dξ ξ1


(n + 1)K 3/2 (3−n)/2n 2 dθ
ρc
ξ
,
= −4π
4πG
dξ ξ1

(D.9)

where ξ1 represents the zero point of θ or the surface radius normalized by H. For n = 3 or
Γ ≡ 1 + 1/n = 4/3, equation (D.9) reduces to
M = −4π

K
πG

3/2
ξ2

dθ
.
dξ ξ1

(D.10)

Thus, the mass does not depend on the central density for Γ = 4/3 polytrope. For n = 5/2 orΓ = 1 + 1/n = 7/5, M is written down as
M = −4π

7K
8πG

3/2
ρc1/10 ξ 2

dθ
,
dξ ξ1

(D.11)

where ξ1 = 5.35528 and −ξ12 dθ/dξ|ξ1 = 2.18720. Polytrope with Γ = 7/5 gas, the mass-density
relation becomes ρc ∝ M 10 . While, for n = 3/2 of Γ = 1 + 1/n = 5/3, M is written down as
M = −4π

5K
8πG

where ξ1 = 3.65375 and −ξ12 dθ/dξ|ξ1 = 2.71406.

3/2
ρc1/2 ξ 2

dθ
,
dξ ξ1

(D.12)

D.2. MAGNETOHYDROSTATIC CONFIGURATION

D.2

165

Magnetohydrostatic Conﬁguration

In section 4.2, we obtained the maximum mass which is supported against the self-gravity using thevirial analysis. In this section, we will survey result of more realistic calculation.
Formalism was obtained by Mouschovias (1976a,b), which was extended by Tomisaka, Ikeuchi, &Nakamura (1988) to include the effect of rotation. Magnetohydrostatic equilibrium is achieved on abalance between the Lorentz force, gravity, thermal pressure force, and the centrifugal force as
1(∇ × B) × B − ρ∇ψ − ∇p − ρω × (ω × r) = 0.
4π

(D.13)In the axisymmetric case, the poloidal magnetic fields is obtained by the magnetic flux function, Φ,or the ϕ-component of the vector potential as
1 ∂Φ
,
r ∂r
1 ∂Φ
= −
,
r ∂z
bϕ
=
.
r

Bz =
Br
Bϕ

(D.14)

Equation (D.13) leads tobϕ ∂bϕ
∂Φ
1
∆1 Φ
−
2
4πr
∂z
4πr2 ∂zbϕ ∂bϕ
1
∂Φ
−
∆1 Φ
−
2
4πr
∂r
4πr2 ∂r∂bϕ ∂Φ ∂bϕ ∂Φ
−
+
∂r ∂z
∂z ∂r
−

with
∆1 Φ ≡

∂ψ ∂p
+
,
∂z
∂z
∂ψ ∂p
= ρ
+
− ρω 2 r,
∂r
∂r
= ρ

= 0,

∂2Φ
∂
+r
∂z 2
∂r



1 ∂Φ
r ∂r

(D.15)(D.16)(D.17)

.

(D.18)

Equation (D.17) indicates bϕ is a function of Φ as bϕ (Φ), which is constant along one magnetic fieldline. Ferraro’s isorotation law demands, that is, to satisfy the stead-state induction equation ω isconstant along a magnetic field. This means ω is also constant along one magnetic field line, ω(Φ).
From this, the density distribution in one flux tube is written
 


q
1 2 2
2
ρ = 2 exp − ψ − r ω /cs .
(D.19)
cs
2This means q is also constant along one magnetic field line, q(Φ) . Since the forces are expressed by
the derivative of function q
 


1 2 2
2
(∇q) exp − ψ − r ω /cs = ∇p + ρ∇ψ − ρrω 2 − ρr2 ω∇ω,
(D.20)
2
where equation (D.19) is used, equation (D.15) and (D.16) are rewritten as




 
2
∂Φ ∂(bϕ /2)
1 2 2
2
2 ∂ω
2 ∂q
,
∆1 Φ
+
= −4πrexp − ψ − r ω /cs + ρr ω
∂z
∂z
∂z
2
∂z




 
2
∂Φ ∂(bϕ /2)
1 2 2
2
2 ∂ω
2 ∂q
.
∆1 Φ
+
= −4πrexp − ψ − r ω /cs + ρr ω
∂r
∂r
∂r
2
∂r

(D.21)(D.22)

166

APPENDIX D. HYDROSTATIC EQUILIBRIUM

Finally, using the fact that b, q, and ω are functions of Φ, these two equations are reduced to

 



d(b2ϕ /2)
dq
1 2 2
2
2
2 dω
∆1 Φ = −− 4πr
exp − ψ − r ω /cs + ρr ω.
(D.23)dΦdΦ2dΦAnother equation to be coupled is the Poisson equation as
 


1 2 2
q
2
∆ψ = 4πG 2 exp − ψ − r ω /cs .
cs2(D.24)The source terms of equations (D.23) and (D.24) are given by determining the mass δm(Φ) andthe angular momentum δL(Φ) contained in a flux tube Φ–Φ + δΦ. Mass and angular momentum
distribution of
Z zs (Φ) Z r(z,Φ+δΦ)δm(Φ) = 2dz
dr2πrρ(r, z),(D.25)
0r(z,Φ)

Z zs (Φ)δL(Φ) = 2

Z r(z,Φ+δΦ)
dr2πrρ(r, z)r2 ω(Φ)

dz
0(D.26)r(z,Φ)

is chosen arbitrary in nature, where zs (Φ) is the the height of the cloud surface where the magnetic
potential is equal to Φ. For example, m(Φ) and L(Φ) are chosen as a uniformly rotating uniformR r(Φ+δΦ)
density spherical cloud threaded by uniform magnetic field. Since r(Φ)
2πrdr = δΦπ∂r2 /∂Φ Z zs (Φ)
 1
1 2 2
2
2
dz2π ∂r /∂Φ 2 exp − ψ − r ω /cs ,q(Φ) = (dm/dΦ)/
cs
2
0
 


Z zs (Φ)
 1
1 2 2
2
2
2
ω(Φ) = (dL/dΦ)/dz2π ∂r /∂Φ 2 q(Φ)r exp − ψ − r ω /cs ,cs
20(D.27)(D.28)The source terms of PDEs [eqs. (D.23) and (D.24)] are given from equations (D.27) and (D.28). While
the functions q(Φ) and ω(Φ) are determined from the solution of these PDEs after m(Φ) and L(Φ)
are chosen. This can be solved by the self-consistent field method.

Appendix E

Basic Equations for Radiative
Hydrodynamics
E.1

Radiative Hydrodynamics

The basic equation for gas which is affected by the radiation is as follows:dρ+ ρ∇ · v = 0,
dt
dv
χF 0
ρ
= ρ∇ϕ − ∇ψ +
F0 ,
dt
c
dϵ
ρ + p∇ · v = hCR ρ + cχE0 E0 − 4πχP 0 B,
dt
∇2 ψ = 4πGρ,p,ϵ=(γ − 1)ρ

(E.1)
(E.2)
(E.3)
(E.4)
(E.5)

where χ, hCR , and B are the absorption coefficient, the cosmic-ray heating rate per unit mass, and
the Planck function B = σT 4 /π. In equation (E.2), the term χFc 0 F0 represents the acceleration ofgas due to the photon pressure. In equation (E.3), hCR ρ, cχE0 E0 and 4πχP 0 B represent, respectively
the heating due to the CR particles, heating due to the absorption of radiation and cooling die to theemission. The frequency-integrated radiation energy density and radiation flux are defined as
Z ∞Zdν0 dΩI(ν, n),
(E.6)
E0 =
0
Z
Z ∞
(E.7)
F0 =
dν0 dΩI(ν, n)n,0

where I(ν, n) denotes the specific intensity of radiation at frequency ν along the direction vector
n. These equation could be solved if the the radiation transfer is solved. The frequency-averaged
absorption coefficients are defined as follows:Zdνχ(ν)F0 (ν) /F0 ,(E.8)
χ F0 =Zdνχ(ν)E0 (ν) /E0 ,(E.9)
χ E0 =Zdνχ(ν)B(ν) /B,(E.10)
χ P0 =
167

168

APPENDIX E. BASIC EQUATIONS FOR RADIATIVE HYDRODYNAMICS
While the equation for the radiation transfer is basically as follows:
1 ∂I(ν, r, n)+ n · ∇I(ν, r, n) = −χ(ν, r) [I(ν, r, n) − S(ν, r)] ,c∂twhere S(ν, r) represent the source term.

(E.11)

Appendix F

Polarization
F.1

Formalism to Calculate Observed Stokes Parameter
losz’

OblateProlate
B

Cpara

J

β
γ
y’

ψ

Cperp

x

y

x’

Figure F.1: (Left:) Relations between the cross section (C⊥ and C// ) and the shape of spheroid. C⊥
and C// are the cross sections for the electromagnetic waves which have polarizations perpendicular and parallel to the symmetric axes, respectively. Symmetric axis of the oblate
spheroid is the shortest axis, while that of the prolate spheroid is the longest one. (Right:)
Schematic explanation of the coordinate. Distribution of Stokes parameter looking fromthe z-axis is calculated with respect to (x, y). When calculating the contribution of z –z + dz, local Cartesian coordinate (x′ , y ′ , z ′ ) is introduced, although z and z ′ are identical.
Orientation of (x′ , y ′ ) is determined as ex′ = (B/|B|) × ez′ and ey′ = ez′ × ex′ . In thiscoordinate system, vector B lies in the y ′ -z ′ plane. The angle between the vector B andy ′ -axis is γ and that between y- and y ′ -axis is ψ. Spinning direction makes precession
and the angle between angular momentum J and magnetic field B is β.

This section is devoted for the polarization observation of the thermal emission from aligned dust
grains with magnetic field, based on the formalism by Lee & Draine (1985), Wardle & Königl (1990),
and Fiege & Pudritz (2000) (see also Reissl, Wolf, & Brauer 2016).
We consider oblate and prolate spheroidal dust grains and their cross-sections for the electromagnetic wave whose polarization is ⊥ to and // to the symmetry axis of the dust grains. Respective
cross-sections are expressed as C⊥ and C// , respectively (see Fig. F.1 (left)). To derive the radiationtransfer equation, we introduce a local Cartesian coordinate (x′ , y ′ , z ′ ). The z ′ -axis is taken along theline of sight, x′ -axis is perpendicular to the local magnetic field B and the y ′ -axis is perpendicular
to both x′ - and z ′ -axes. In this coordinate system, B lies in the y ′ -z ′ plane (Wardle & Königl 1990;
169

170

APPENDIX F. POLARIZATION

see Fig. F.1 (right)) . That is, B = By′ ey′ + Bz ′ ez ′ . Absorption cross section for the light linearlypolarized along the x′ - and y ′ -axes areC⊥ (for oblate grain),
′
Cx =
1
C
+
C
(for prolate grain),
⊥
//
 2
(F.1)
C⊥ + (C// − C⊥ ) cos2 γ(for oblate grain),
Cy ′ =C⊥ + 21 (C// − C⊥ )(1 − cos2 γ) (for prolate grain),where we assumed the precession angle β = 0 in Figure F.1 and took β = 0 in equations (3.6) and(3.7) of Lee & Draine (1985) and ι = 0 in equations (3.1) and (3.2) of Wardle & Königl (1990).
Radiation transfer for the light polarized along the x′ - and y ′ -directions are written (eqs. [3.3] and[3.4] of Wardle & Königl (1990)) asdIx′ = nd Bν (Td )Cx′ dz ′ ,dIy′ = nd Bν (Td )Cy′ dz ′ ,(F.2)

where Bν (Td ) represents the Planck function for the dust temperature of Td . In this coordinate
system, contribution to the Stokes parameter (I ′ , Q′ , U ′ , V ′ ) from the segment dz ′ is given as
dI ′ = dIx′ + dIy′ ,dQ′ = dIx′ − dIy′ ,dU ′ = 0,(F.3)

dV ′ = 0.
The orientation of the frame of (x′ , y ′ ) is dependent on the direction of B(z). Integrating along theline of sight dz = dz ′ , the orientation changes with z. When we look B(z) from the z-direction, let
the position angle of the vector be ψ(z). The contribution to Stokes parameter based on the spatially
fixed (x, y, z) coordinate (I, Q, U, V ) is obtained from (I ′ , Q′ , U ′ , V ′ ) asdI = dI ′ , dIx′ − dIy′
cos 2ψ − sin 2ψ
dQ=,
sin 2ψ cos 2ψ
dU
0

(F.4)

dV = dV ′ ,Thus,dI= nd Bν (Td )(Cx′ + Cy′ ),dz
dQ= cos(2ψ)nd Bν (Td )(Cx′ − Cy′ ),dzdU= sin(2ψ)nd Bν (Td )(Cx′ − Cy′ ),dz
V =0
From equation (F.1), cross section for the polarized light is given(C⊥ − C// ) cos2 γ(for oblate grain)′′
Cx − Cy =1
2 γ (for prolate grain)
(C
−
C
)
cos
⊥
//
2
≡ Cpol cos2 γ.

(F.5)

(F.6)

F.1. FORMALISM TO CALCULATE OBSERVED STOKES PARAMETER
Using Cpol (eq.[F.6]), the Stokes parameter Q and U are
Z
Q = Cpol Bν (Td ) nd cos 2ψ cos2 γdz,
Z
U = Cpol Bν (Td ) nd sin 2ψ cos2 γdz,

171

(F.7)

where nd represents the dust number density in the interstellar space. As for the Stokes’ I, the crosssection Cext ≡ (Cx′ + Cy′ )/2 is written down [assuming RF = 1 or Φ = cos2 γ in eq. (3.14) of Lee &Draine (1985)] as 2
2C⊥ + C//C x′ + C y ′cos γ 1=
− Cpol
−
2
3
2
3
 2

(F.8)
2C⊥ + C//cos γ 1∴ Cext =− Cpol−.
3
2
3
Using this cross section, Stokes’ I is calculated

 2
Z
Z
2C⊥ + C//cos γ 1
Bν (Td ) nd dz − Cpol Bν (Td ) nd
−
dz.
(F.9)
I=
3
2
3
In summary, Stokes parameters are obtained as
Q = Cpol cd Bν (Td )q,
U = Cpol cd Bν (Td )u,2C⊥ + C//I=
cd Bν (Td )Σ − Cpol cd Bν (Td )Σ2
3
with so called relative Stokes parameter asZ
q = ρ cos 2ψ cos2 γdz,Zu = ρ sin 2ψ cos2 γdz,Z
Σ = ρdz,Z  2
cos γ 1dz
Σ2 = ρ
−2
3(F.10)(F.11)

where cd = nd /ρ represents the ratio of dust number density to the gas mass density. To obtain
the absolute value of Stokes parameter it is necessary to know following factors: the polarization
efficiency Cpol /[(2C⊥ + C// )/3], degree of grain alignment with the magnetic field, and degree of small
scale turbulence in magnetic field. However we can discuss the polarization angle distribution andrelative polarization degree distribution only with the relative Stokes parameter.
Putting the ratio of polarization cross section to the extinction cross section as
Cpol,(2C⊥ + C// )/3(F.12)(Q2 + U 2 )1/2
Iα(q 2 + u2 )1/2=.
Σ − αΣ2(F.13)

α=the polarization degree is written
p≡

172

APPENDIX F. POLARIZATION

Figure F.2: (Left:) Relationship the coordinates of target object (x, y, z) and that of observation (ξ, η).
Filaments extend in z-direction and x − y plane is the cross-cut of the filament. η-axis istaken perpendicular to the line of sight expressed by n and toward the z-axis. ξ-axis istaken perpendicular both to η-axis and to the line of sight n. (Right:) Integration along
the line of sight ∆s corresponds to theq path in the x-y plane ∆ℓ. These two path lengths
are related as ∆s = ∆ℓ/ sin θ = ∆ℓ/ n2x + n2y , where θ represents the angle between theline of sight and the z-axis.

Polarization angle χ is determined fromcos 2χ =sin 2χ =F.2

q(q 2 + u2 )1/2
u.
(q 2 + u2 )1/2

(F.14)

Coordinate System

We introduce a plane onto which observational data are displayed and its Cartesian coordinate (ξ, η)1 .
In Figure F.2, we show the relation of observed target written using 3D Cartesian coordinate (x, y, z)
and the observation coordinate (ξ, η).
ez − (ez · n)n|ez − (ez · n)n|
= eη × n,eη =(F.15)eξ(F.16)

Relative Stokes parameter q and u are obtained with integrating density along the line of sight asZq =ρ cos 2ψ cos2 γds,
(F.17)Zu =ρ sin 2ψ cos2 γds,
(F.18)ZΣ =ρds,(F.19)

Z
cos2 γ 1ds(F.20)
Σ2 =
ρ
−
2
3
1

Greek letters ξ and η correspond respectively to Roman letters of x and y.

F.2. COORDINATE SYSTEM

173

where the integration is performed along the line of sight, ρ is the density, and γ and ψ represent
the angle between the magnetic field and the celestial plane and the angle between the η-axis andthe magnetic field projected on the celestial plane, respectively. From the relative Stokes parameters,
polarization angle χ and polarization degree P are derived ascos 2χ =sin 2χ =Pq(q 2 + u2 )1/2
u,,(q 2 + u2 )1/2
1/2
q 2 + u2.
= p0
Σ − p0 Σ 2(F.21)(F.22)(F.23)The parameter p0 controls the maximum polarization degree and we assume p0 = 0.15 to fit thehighest degree of polarization observed for the interstellar cloud.

Appendix G

Random Velocity
Considering gas in Maxwellian velocity distribution, the distribution function for the velocity is asfollows:
 2
v
f (vx ) = A exp − x .
(G.1)
σ
This gives the one-dimensional random velocity as
 2
2 exp − vx dv
v
x
x
0
σ2
 2
,
R∞
vx
exp
−
dv
x
2
0
σ

R∞
⟨vx2 ⟩ =
=

σ2
.
2

(G.2)

If we observe emissions from such a gas, the emission line is broaden due to the Doppler shift. Using
equation (G.2), the HWHM (half width of half maximum: the line width measured from the thecenter of the emission line to the point of the half intensity; see Fig.G.1) of the emission line is"  #vx,HWHM 2
1
exp −
= ,
(G.3)
2
2⟨vx ⟩
2
which leads tovx,HWHM = (2 ln 2⟨vx2 ⟩)1/2 ,(G.4)

vx,FWHM = (23 ln 2⟨vx2 ⟩)1/2 .

(G.5)

and
Thus, if we assume isotropic distribution, three-dimensional random velocity of gas
2⟨v3D⟩ = ⟨vx2 ⟩ + ⟨vy2 ⟩ + ⟨vz2 ⟩ = 3⟨vx2 ⟩(G.6)

is obtained with the line width as


2 1/2
⟨v3D
⟩

1/2
3
=
vx,FWHM ,23 ln 2
1/2
3=vx,HWHM .
2 ln 2

175

(G.7)

176

APPENDIX G. RANDOM VELOCITY

Figure G.1: HWHM (half width of half maximum: the line width measured from the the center of theemission line to the point of the half intensity) and FWHM (full width of half maximum:
the line width measured between the points of the half intensity).

Appendix H

X-factor
Since the H2 molecule is the homonuclear diatomic molecule, the H2 molecule has no electric dipole.
Thus, the electric dipole radiation is not expected for this molecule. While the second abundant
molecule CO is the heteronuclear diatomic molecules and has electric dipole. Lower levels of rotationaltransition of CO are relatively easily excited even in the cold interstellar medium of T ∼ 10K.
Therefore, the rotational transition of CO molecule is the first possibility to observe cold moleculargas. The X-factor represents the ratio of the column density of H2 molecules N (H2 ) to the integrated
intensity of 12 CO line ICO as
NH2 (cm−2 )
X≡
.
(H.1)
ICO (K km s−1 )There are several empirical but physical estimations of the X-factor: Using γ-rays, X-factor is estimated as X ≃ 2.3 × 1020 (Strong et al. 1988). This is based on the idea that the emissivity of
γ-ray emission is proportional to the cosmic-ray intensity times the number of target nuclei. The
γ-ray intensity is proportional to the column density if the cosmic-ray intensity is uniform. From the
distribution of γ-ray intensity, Strong et al. (1988) estimated the total column density NH2 (cm−2 )using some model. This gives the value of X-factor.
Estimations using the Virial mass of the interstellar cloud and the visual extinction AV .

177

Appendix I

Abundance
I.1

Solar system abundance

Table I.1: Solar system abundance taken from meteorites and the Solar photosphere. Taken from
Table 3.2 of Allen’s Tables (Cox 2000).

179

180

I.2

APPENDIX I. ABUNDANCE

Isotope ratio (Isotopic abundance)

Table I.2: Isotopic abundances for H, C, N, and O for various places. Taken from Table 21.10 ofAllen’s Tables (Cox 2000).

I.3

Molecular Weight

Table I.3: Group abundance ratios. Taken from Table 3.1 of Allen’s Tables (Cox 2000).

Appendix J

Gravity in Filament
Let’s calculate the gravity working in a filament with the line-mass of λ. Consider the point at the
distance d from the edge shown in Figure J.1 (left) and calculate the gravity at the point. Since thegravity due to the gas in the region L − 2d < x < L − d and that of L − d < x < L must be canceled,
we calculate the part of −L ≤ x ≤ L − 2d. In this analysis, we assume the width of the filament in
y- and z-directions, w, is finite but much smaller than the length L. Consider a finite element dx
placed at x (x ∼ x + dx). Gravity by the mass element is approximated as dg = −Gλdx/r2 , where r
represents the distance to the element for r ≫ w. Integrating this expression, we obtain the gravity
exerted from the filament at the point x = L − d. Considering the distance r is equal to r = L − d − x,
we integrate the gravity
Zdm
g0 ≃ − G 2 ,(J.1)
r
Z x=L−2d
dx
= −Gλ
,
(J.2)
(L − d − x)2
x=−L

1
1
= −Gλ −.
(J.3)+
2L − d dExact expression for the gravity by a filament with finite length and uniform density is shown inequation (J.5). In Figure J.1 (right), we plot the gravity (g0 /Gλ) at the point x = L − d as
Plot[1/(2 L - d) - 1/d, {d, 0, L}]}.

Figure J.1: (Left:) Calculating self-gravity at the distance d from the edge of a filament with length
2L. We assume the periodic boundary condition and the same filaments are placed withthe displacement of 2L. (Right:) Gravity working at x = L − d. Equation (J.3) iscalculated. The x- and y-axes represent d and g0 (d)/Gλ and we assume L = 10 here.

181

182

APPENDIX J. GRAVITY IN FILAMENTAs shown in the figure, the gravity from the own filament becomes strongest at d = 0, i.e., the edgeof the filament. The gravity is canceled at the center, e.g., g = 0, at x = 0 or d = L.
Problem 20
Consider a cylinder with a length ℓ, radius R and uniform density ρ̄ placed at −ℓ/2 ≤ z ≤ ℓ/2(Burkert & Hartmann 2004).
1. A disk with a radius R, uniform density ρ̄, and thickness ∆z is located at the origin z = 0.
The gravity working parallel to the axis (z-axis) on the cylinder axis (r = 0, z) is writtenas"#1∆az (r = 0, z) = 2πGρ̄∆z
−1 .
(J.4)
(1 + R2 /z 2 )1/2
Show this.
2. Integrate ∆az , and show the gravity working inside the cylinder (0 ≤ z ≤ ℓ/2) is expressed
as


"

2 #1/2 "2 #1/2 ℓℓ− R2 +
−z
+z
.
(J.5)
az (r = 0, z) = −2πGρ̄ 2z + R2 +22
3. If ℓ ≫ R, at the edge of the cylinder z = ℓ/2, equation (J.5) is reduced toRaz (z = ℓ) ≃ −2πGρ̄R 1 −.
2ℓShow this.
4. Considering ℓ/2 ≫ R and ℓ/2 − z ≫ R, the above equation (J.5) is reduced to
"
−1 
−1 #
ℓ
ℓ
az (r = 0, z) ≃ −πGρ̄R2
−z
−
+z
.
2
2

(J.6)

(J.7)

Show this.
5. Show that equation (J.3) is equivalent to the above equation (J.7).


The free-fall time scale is given as

tﬀ =

L−d
|g0 |

1/2

1/2



L−d
∝  2(L−d) ∝ [d(2L − d)]1/2 .

(J.8)

d(2L−d)Since this goes to zero at d = 0 and d = 2L, gas at the edge looks to contract first. In the actual gas
dynamics, outer gas element can not overtake inner one. Thus, density enhancement is made near
the edge of the filament.
Next, we see the gravity from the external filaments. The gravity from the filament next to theright is given11
g1 = Gλ−.
(J.9)
2L + d 4L + d
Attraction from all the filaments x > L is


1
1
1
1
gR = g1 + g2 + · · · = Gλ−+−
+ ··· ,
(J.10)
2L + d 4L + d 6L + d 8L + d183

Figure J.2: Gravity from the ensemble of external filaments. x- and y-axes represent d and (gR +gL )/Gλ. Although this is positive and becomes strongest at d = 0, the absolute value ofthe external component is much smaller than the internal one. We assumed L = 10 here.

while that from the left x < −L isgL = g−1 + g−2 + · · · = −Gλ



1
1
1
1
−
+
−
+ ···
4L − d 6L − d 8L − d 10L − d.

(J.11)

From equations (J.10) and (J.11), we plot the gravity from external filaments (gR + gL )/Gλ as
Plot[Sum[1/(n L + d) - 1/((n + 2) L + d) - 1/((n + 2) L - d) +1/((n + 4) L - d), {n, 2, 1000, 4}], {d, 0, L}]
In Figure J.2, (gR + gL )/Gλ is plotted against the distance d. The gravity from the external filaments
is much smaller than that from the own filament, g0 /Gλ.

Appendix K

Magnetic Braking
Angular momentum transfer or redistribution is possible in several ways: (for simplicity, we consider
2D axisymmetric system, here)
Advection: Gas with angular momentum ρrvϕ moves with velocity v = (vr , vz ) brings angularmomentum flux density as ρrvϕ v.
Magnetic tension: Since the torque N = r × F changes angular momentum, even in the axisymmetric geometry, the ϕ-component of Lorentz force transfers the angular momentum.
Aligned Rotator
A simple mechanism to reduce the angular momentum of the cloud by magnetic tension is called
magnetic braking (Mouschovias & Paleologou 1980). Shown in Figure K.1 (a), consider a cloud withthe column density Σcl being immersed in the intercloud medium with density ρext and magnetic fieldof B0 . The cloud is rotating with Ωcl but the intercloud gas has no angular momentum. Torsional
Alfvén wave propagates in the both vertical directions (z > 0 and z < 0). Wave fronts propagate with
the Alfvén speed as ±VA = B0 /(4πρext )1/2 . After passing the wave front |z| < VA t, the intercloud
medium begins corotation with the same angular speed of the cloud Ωcl . Since the inertia increases
with a rate of ρext VA , the angular momentum conservation is expressed asΣcl Ωcl (t = 0) = [2ρext VA t + Σcl ] Ωcl (t),(K.1)

which gives the time derivative of Ωcl as
1 dΩclΩcl dtΣcl 2ρext VA(2ρext VA t + Σcl )22ρext VA
∼ −
,
Σcl
= −

(K.2)

where we assumed 2ρext VA t ≪ Σcl to fit the early phase evolution. Thus, the rotation stopping
timescale is given as
Σcl
.
(K.3)
τ∼
2ρext VAIt should be noticed that the timescale is longer for lower intercloud gas density, although the prop−1/2
agating speed is increased as VA ∝ ρext .
185

186

APPENDIX K. MAGNETIC BRAKING

(b)

(a)
ρext
B0
ρext
B0

z

Ωcl
Σcl

Ωcl

2h

z=0
R

Figure K.1: Schematic model of magnetic braking. (a:) Aligned rotator (Ωcl //B0 ). A rotating cloudwith the column density Σcl and angular speed of Ωcl is considered. Torsional Alfvénwave propagating in the intercloud medium with the magnetic field strength of B0 andthe density of ρext . (b:) Perpendicular rotator (Ωcl ⊥ B0 ). A cloud with radius R andhalf height h is immersed in the intercloud medium. Torsional Alfvén wave is assumed topropagate radially only in a restricted region of −h < z < h. Cloud-to-intercloud density
ratio ρcl /ρext controls which rotation is braked more rapidly.

Perpendicular Rotator
In contrast, when the cloud is rotating in the perpendicular direction to the magnetic field, Ω ⊥ B(Fig. K.1 (b)), the moment of inertia of intercloud gas swept by the wave increases
Z
Iext =
r2 dm,
Z r=
r2 2hρext 2πrdr,0= ρext πhr4 ,(K.4)

where h represents the half thickness of the cloud and we assumed the Alfvén wave propagates in theintercloud restricted region of −h < z < h. After sweeping the external medium of r < rswept , Iext
becomes equal to that of the cloud
Icl = ρcl πhR4 ,(K.5)
where ρcl and R represent the cloud density and its radius. This gives
rswept = R

ρcl
ρext

1/4
.

(K.6)

In the aligned rotator, equation (K.3) gives the height where the Alfvén wave has reached as


ρcl
.
(K.7)
zswept = hρextComparing equations (K.6) and (K.7), rswept /zswept = (R/h)/(ρcl /ρext )3/4 ≪ 1, since ρcl ≫ ρext . The
perpendicular rotator seems to stop its rotation much faster than the aligned rotator (Mouschovias& Paleologou 1979).

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Indexabsorption coefficient κ, 59
accretion luminosity, 4
accretion rate, 4, 42
AKARI, 68
Alfvén speed, 185
Alfvén wave, 185aligned rotator, 137
ambipolar diffusion, 99, 102
angular momentum (specific), 128
angular momentum transfer, 116
Aquila cloud, 31
ASTE/AzTEX, 22
average intensity J, 59
B211/213, 32
B335, 10, 11, 13, 24
B68, 22, 30
barotropic relation, 39
blue-skewed asymmetry, 12
Bok globules, 24
Bonnor-Ebert mass, 86, 90
CB230, 24
CB244, 24
centrifugal radius, 117
Class I, 8Class II, 8Class III, 8
co-rotation resonance, 78
color excess, 28
continuity equation, 153
convection, 49, 131
core mass function, 34
corotation resonance (CR), 77
critical density, 63
critical line mass, 31
critical mass, 87
CTTS, 6–8
density wave theory, 75, 76

depolarization, 144
deuterium burning, 130, 131
disk accretion, 134
disk frequency, 7
dispersion relation, 45
dissociation, 121
double-polytropic equation of state, 122, 139
drift velocity, 99
dynamical collapse, 104
Einstein’s A coefficient, 61
Einstein’s B coefficient, 61
Einstein’s coefficient, 60, 62
emissivity (volume) ϵ, 59
energy density of radiation, 59
entrainment, 118
epicyclic frequency, 48, 72, 74
equation of motion, 151
equivalent width, 7
Eulerian time derivative, 152
excitation temperature, 63
filament, 22, 31
first core, 111, 121
first law of thermodynamics, 155
flux freezing, 159
free-fall time, 40
G31.41+0.31, 26, 144
galactic shock, 79
Gauss theorem, 40, 47
grand design spiral, 79
gravitational instability, 43, 72
gravitational instability of thin disk, 46
H-R diagram, 129
Hall effect, 102
Hall parameter, 103
Hayashi track, 7, 129, 133
Henyey track, 129, 134
196

INDEX
Herbig Ae-Be star, 5
Herbig-Haro objects, 115
Herschel satellite, 31high density tracer, 13
hour-glass shape, 24hydrodynamics (basic equation), 151hydrodynamics — continuity equation, 39hydrodynamics — equation of energy, 39hydrodynamics — equation of motion, 39hydrostatic balance, 85
I-tyle outflow, 127
IC5146, 31
ideal magnetohydrodynamics, 159
induced emission, 61
infall motion, 10
infrared (IR) excess, 8
initial mass function, 34
inner Lindblad resonance (ILR), 77
inside-out collapse, 106
interstellar medium — three-phase, 2interstellar medium — two-phase, 2
IRAS, 69
IRAS source, 5isochrone — pre-main-sequence star, 7
isothermal sound speed, 51
isovector line, 12
JCMT-SCUBA, 22, 24
Jeans instability, 44
Jeans mass, 46
Jeans wavelength, 46
Kelvin-Helmholtz contraction, 3, 5
Kelvin-Helmholtz time, 7
Kennicutt-Schmidt law, 65
L1544, 22
L1551-IRS5, 115
L183, 22
L43, 22
L977, 28, 30
Lagrangian time derivative, 152
Lane-Emden equation, 85, 163
Laplace equation, 47
Larson’s law, 36
Larson-Penston solution, 104, 119
Laval nozzle, 50, 53

197
leading spiral, 78
Lindblad resonance, 78
line width size relation, 36
linear analysis, 43long wave, 78
Lorentz force, 90, 99, 159
M101, 68
M51, 75
Mach number, 51, 53
magnetic braking, 128, 141, 185magnetic field, 21
magnetic pressure, 90
magnetic to gravitational energy ratio (γ0 ), 139
magneto-centrifugal wind, 115, 125
magnetocentrifugal wind, 115
magnetohydrodynamics, 159
mass accretion rate, 4, 112
misaligned rotator, 137
NANTEN, 69
nested grid method, 122
NGC1333 IRAS4A, 24, 144
NGC2068/2071, 34
NGC2841, 66
NGC4254, 66
observational visualization, 148
ohmic dissipation, 102
outer Lindblad resonance (OLR), 77
outflow, 115
Parker wind, 53
Plummer-like function, 32
Poisson equation, 39
polarization, 21
polarization (angle), 144
polarization (degree), 144
polytrope, 121, 163
polytropic index, 113, 121
polytropic relation, 39, 154
pre-main-sequence star, 3pre-stellar core, 5
protostar, 8, 10
protostellar core, 5radiative relaxation region, 130
radiative transfer, 59

198
relative Stokes parameter, 144, 171
rho Oph cloud, 34
rotational to gravitational energy ratio (β0 ), 137,
141
runaway collapse, 119, 122
runaway contraction, 118
Salpeter’s initial mass function (IMF), 34
Schmidt law of star formation, 65
second collapse, 121
second core, 121selective accretion mechanism, 141
self-similar solution, 104, 157
Serpens-south cloud, 22
short wave, 78
singular isothermal solution (SIS), 85, 106
singular isothermal sphere (SIS), 106
SMA, 24
sound speed, 44
sound wave, 44
specific intensity I, 59
spectral energy distribution (SED), 8, 9
spherical accretion, 134
spontaneous emission, 61
standard reddening law, 28
star formation rate, 65
starburst galaxy, 65
stelar wind, 53
stellar wind – dust-driven mechanism, 53
stellar wind – line-driven mechanism, 53
Stokes parameter, 144, 169, 171
subcritical, 93, 97
subcritical cloud, 87
subsonic flow, 50
supercritical, 93, 97
supercritical cloud, 87
supersonic flow, 50
swing amplification mechanism, 75
T Tauri star, 4, 5, 8
Taurus molecular cloud, 3, 32
thermal to gravitational energy ratio (α0 ), 137,
141
tightly wound spiral, 73
Toomre’s Q parameter, 48, 72, 74
trailing spiral, 78
U-type outflow, 127

INDEX
virial mass, 58
virial relation, 56, 89
WTTS, 6–8
young stellar objects (YSOs), 9
Zeeman splitting, 95