Lecture 23
Magnetic Fields, Core Stability and Collapse
1. Magnetic Fields and Stability
2. Singular Isothermal Sphere
3. Observational Signatures of Infall
References
McKee, 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), and
turbulent (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 dipolar
3. 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:
3 Mc 2
3 G
2
2
P=
(M
M
)
3
4
4 R
20 R
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 B2/4L,
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
critical
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• 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 the
2.5 level after observing 34 dark
clouds for 500 hours with Arecibo
• Error bars are 1!
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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 Formation
Li, Norman, Mac-Low & Heitsch, ApJ 605 800 2004
an alternate scenario for core formation
• 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)
column density map with disk-like structures
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The solutions were first studied in detail
by Bonnor, Ebert and McCrea.
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Bonnor-Ebert-McCrea Solutions
BEM showed that an isothermal
cloud of radius R subject to an
external pressure Pext is unstable
If its mass exceeds a critical value
c4
1/ 2
M cr = 1.18 3 / 2 Pext
G
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”.
Stable and unstable equilibrium
solutions of the BEM Eq.
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Singular Solutions
Subsitute = A /r 2 into
1 d
4 G A
1 d 2 1 d 4 G
+
r
=
0
to
get
(2r)
+
=0
2
2
2
2
2
r dr
c r
r dr dr c
and find
c2 1
2c 2
=
and M(r) =
r
2
2G r
G
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,
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v = uc,
(x)
=
,
2
4 Gt
c 3t
M(r,t) =
m(x)
G
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Inside-Outside Collapse
density
curves labeled
by time in units
of 30,000 yr
log r
velocity
log r
parameters: M = 0.96 Msun, c = 0.2 km/s, Pext = 1.1x105 K cm-3
large x = r/ct: static solution for r > ct
small x = r/ct: free-fall collapse for r < ct
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The parameter m0 = 0.975
19
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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)
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Schematic for Infall Signature
Lada (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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