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Lecture 23

Magnetic Fields, Core Stability and Collapse

1. Magnetic Fields and Stability

2. Singular Isothermal Sphere

3. Observational Signatures of Infall

ReferencesMcKee, Lec22-06 & OSSP (Crete 1999)

Stahler, Chs. 9 & 10

Shu, ApJ 214 488 1977

Troland & Crutcher (2008)

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1. Magnetic Fields and Stability

• Observations show that GMCs are large (10-100 pc),

cold (~ 20 K), fairly dense (~ 102 - 103 cm-3), andturbulent ( v ~ 1 km s-1) gravitationally bound overall.

• More detailed observations show that they contain

small (~ 0.1 pc), cold (~ 10 K), dense (~ 103 - 105 cm-3)

and often turbulent regions called cloud cores that are

also bound, but perhaps not always gravitationally.

• But the cores may also contain newly formed stars and

thus manifest both stability and instability although on

different spatial scales. To explore this situation, we

further examine the stability of molecular clouds and

especially the role of magnetic fields.

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Virial Analysis of a Magnetized Cloud

Apply the virial theorem from Lec22 for an equilibrium cloud

Ignore rotation and turbulence (EK - 0) and focus on the

last (magnetic) term. In the spirit of the approximations in

Lec22, assume

1. The magnetization is uniform,

2. The field outside is dipolar3. Magnetic flux is conserved, R2B, I.e., the cloud

is a sufficiently good conductor to allow flux-freezing

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NB In some papers, 0.13 is replaced by 1/2 in the last formula.

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Magnetically Critical Mass

This analysis leads to the following terminology:

• Magnetically Sub-Critical Mass: M < M - Magnetism

stronger than gravity. The external pressure maintains the

cloud in equilibrium but does not collapse it.

• Magnetically Super-Critical Mass: M > M - Magnetism

weaker than gravity. The external pressure aids collapse.

The virial equation can be rewritten in terms of M :

P =3

4

Mc 2

R33

20

G

R4(M 2 M 2)

Measurements of magnetic fields in cores (Troland and

Crutcher (2008; slides 10 & 11) Indicate that they are often

on the margin between sub and super critical.

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The Fate of Sub-Critical Cores

• Sub-critical cores may collapse on long timescales via

ambipolar diffusion (see Shu II, Ch. 27).

• Gravity acts on all of the components of the interstellar

cloud, ions as well as neutrals (the dominant component).

• But the ions are tied to the magnetic field; they tend to

force the neutrals to follow them by microscopic ion-neutral

interactions.

• Over time, however, gravity drags the neutrals through

the ions and feeds the center of the collapse.

• The drift speed varies inversely with the electron fraction,

so ambipolar diffusion is faster inside a core compared to

its surface

• Alternatively, gas dynamic forces in a turbulent cloud

may drive a cloud core into collapse

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Ambipolar Diffusion Timescale

A crude estimate, based on replacing the LHS by B2/4 L,

and using B ~ 20 G, K ~ 10-9 cm3 s-1, H2 as the neutral,

and other typical core properties yields w << 1 km s-1.

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Core Magnetic Field Measurements

Troland & Crutcher 2008 astro-ph 0802.2253

• OH Zeeman measurements

are considered the most definitive,

but they are notoriously difficult.

Until recently, only a few results

were reliable.

• Troland and Crutcher reported 9

new “probable” detections at the2.5 level after observing 34 dark

clouds for 500 hours with Arecibo• Error bars are 1 !

critical

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OH Spectra for Troland-Crutcher Detections

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Discussion of Troland & Crutcher 2008

• Arecibo (3’) observations of OH 1665, 1667 line shapes,added and subtracted to give Stokes I( ) and dI( )/d for

Blos for 34 dark cloud cores and 9 “probable” new detections.

• Requires geometrical correction of ~ 3 to get total B.

• OH/H2 assumed to be 8x10-8 to obtain H2 column.• Core mass = 2.8mH R2 N(H2)

• Cores are mildly supersonic: vFWHM = 0.7 km/s; other

average properties are:Blos ~ 8 μG, R ~ 0.3 pc, N(H2) ~ 4x1021cm-2

• Main result: M/M ~ 1.5 (with geometrical correction of 3),

and Eturb > Egrav > Emag with each ratio is ~ 2.

• Technical issues: large uncertainties (error bars), cores

barely resolved in OH

• Heterogeneous sample: cores come from a variety of

molecular clouds

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Summary: Magnetic Fields in Dense Cores

• The observations show that magnetic fields play a

significant role in molecular clouds.

• The best but still incomplete OH Zeeman measurements

indicate mass-to-flux ratios the same order as critical, and

even slightly larger.

• Turbulence must also play a role since the linewidths

indicate that the turbulent energy is somewhat larger

than the gravitational.

• Polarization maps reveal that magnetic fields threading

molecular clouds are ordered over large scales.

• Magnetic fields play a role the formation of stars by

shaping the structure of the clouds and cores, e.g., clouds

tend to collapse preferentially along the direction of the

average field.

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3-D MHD Simulation of Core FormationLi, Norman, Mac-Low & Heitsch, ApJ 605 800 2004

an alternate scenario for core formation

column density map with disk-like structures

• 5123 element periodic box

• super critical field

• strong supersonic turbulence

• M = 64 MJ, L - 4 LJ

• followed for 3 free-fall times

• no microscopic physics

readily forms prolate

and tri-axial super critical

cores and centrifugally

supported massive disks

at late times (numbered

in figure)

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The solutions were first studied in detail

by Bonnor, Ebert and McCrea.

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Bonnor-Ebert-McCrea Solutions

Shu (ApJ 214 488 1977) noticed that

the unstable solutions asymptote to a

1/r2 envelope, and he showed that

1/r2 is a particular (singular) solution

of the BEM Eq., which he described

as the “singular isothermal sphere”.

BEM showed that an isothermal

cloud of radius R subject to an

external pressure Pext is unstable

If its mass exceeds a critical value

Mcr =1.18c 4

G3 / 2 Pext1/ 2

Stable and unstable equilibrium

solutions of the BEM Eq.

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Singular Solutions

Subsitute = A /r2 into

1

r2

d

drr2 1 d

dr

+

4 G

c 2 = 0 to get 1

r2

d

dr( 2r) +

4 G

c 2

A

r2 = 0

and find

=c 2

2 G

1r2 and M(r) =

2c 2

Gr

These solutions are self-similar, i.e., they have no intrinsic spatial scale.

Shu took advantage of this property to find time-dependent

solutions for the isothermal sphere using the scaled

variables

x = r /ct, v = uc, =(x)

4 Gt 2 , M(r,t) =c 3t

Gm(x)

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Inside-Outside Collapse

density velocitycurves labeled

by time in units

of 30,000 yr

log r log r

large x = r/ct: static solution for r > ct

small x = r/ct: free-fall collapse for r < ct

parameters: M = 0.96 Msun, c = 0.2 km/s, Pext = 1.1x105 K cm-3

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Final Remarks on the SIS

The final physical picture is a two component cloud core: a collapsing inner region with ~ r-3/2 and v ~ r-1/2

an outer static envelope with ~ r-2 and v ~ constant

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The difficulty in analyzing optically thick lines continue to

make this a controversial topic,

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One of Many Images Confirming The

Snell, Loren, Plambeck Paradigm

HH-30 Disk & Jet (HST optical image)

450 AU

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Schematic for Infall SignatureLada (1999)

Notice the assumption about locations of high/low excitation.

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Alternate Schematic for Infall Signature

Evans ARAA 37 311 1999

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Another Schematic for Infall Signature

Evans ARAA 37 311 1999

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Myers Two-Slab Model

Effects of varying the optical depth and collapse speed.

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