Lecture 25 Magnetic Fields, Stability,
and Core Collapse
1. Magnetic Fields and Stability
2. Singular Isothermal Sphere
3. Observational Signatures of Infall
References
Stahler, Chs. 9
Shu ApJ 214 488 1977
Evans, ARAA 37 311 1999
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) and that overall they are
gravitationally bound.
More detailed observations show that they contain
small (~ 0.1 pc), cold (~ 10 K), dense (103-105 cm-3),
and often turbulent regions called cores, that are also
gravitationally bound.
But the cores may also contain newly formed stars and thus
manifest both stability and instability to gravitational collapse.
In order to understand how cores can be both stable on one
scale (their size) and unstable on another (presumably the
scale that makes stars), we further examine the stability of
molecular clouds and especially the role of magnetic fields.
The First Stability Results
Without magnetic fields, we found that the conditions for
gravitational instability (collapse) can be given in terms of
certain quantities like the Jeans length, time, and mass.
They are determined solely by two parameters: the dispersion
in the random velocity that determines the effective gas
pressure and the average mass density of the cloud that
determines the effects of gravity:
kT
2
2
c =
+ s turb
m
and, for no turbulence,
1/ 2
p
Ê 10K ˆ
lJ = c
= 0.189 pc Á
˜
Gr
Ë T ¯
3
M J = lJ r = 4.79 M sun
Ê 10K ˆ
Á
˜
Ë T ¯
3/ 2
Ê 1 ˆ
ÁÁ
˜˜
Ë n( H 2 ) ¯
Ê 1 ˆ
ÁÁ
˜˜
Ë n( H 2 ) ¯
-1 / 2
-1 / 2
Stable and Unstable Virial Equilibrium
P
B (stable)
(unstable)A
Rcr
R
For a given mass and temperature, there is a maximum pressure
for stability. Alternatively, given a value of the external pressure,
there is a maximum stable mass. The solution of the Lane-Emden
equation by Bonnor (MNRAS 116 351 1956) & Ebert (Zs. f. Ap 37
222 1955) gives the critical mass as
c4
M cr @ 1.2 3 / 2
= O( M J )
G Pext
Virial Analysis for a Magnetized Cloud
Return to the virial equation for a static (equilibrium) cloud
2
1
1
Pext V - · PÒ V = EK + W + M
3
3
3
Ignore rotation & turbulence (EK=0) to focus on the last
(magnetic) term. In the spirit of previous approximations for
the pressure and gravitational terms, we assume that:
1. the magnetization of the cloud is uniform
2. outside the cloud, the field is dipolar, B(r) ~ (Z/r)-3
3. magnetic flux is conserved: _ = B !R2 ≈ const., i.e, the
cloud is a good enough conductor for flux freezing
2
3
GM
1
4pR 3 P = 3c 2 M + R3B 2
5 R
3
2
2
3
GM
1
F
= 3c 2 M + 2
5 R
3p R
Virial Analysis for a Magnetized Cloud
3 Mc 2
3 GM 2
1 F2
P=
+
3
4
4p R
20p R
12p 3 R 4
The magnetic and gravitational energies vary with R in the
same way.
Since they stand in a constant ratio, once a cloud starts
collapsing the frozen-in magnetic field cannot stop it.
Magnetically dominated clouds are stable, i.e, clouds
whose mass is less than the value:
1/ 2
1Ê 5 ˆ
MF ≡ Á
˜ F
p Ë 9G ¯
More careful analysis, originally due to Mouschovias & Spitzer
(ApJ 210 326 1974) leads to the slightly more accurate formula,
M F ª 0.13 G -1/ 2 F = 1.0 M sun
Ê B ˆÊ R ˆ
˜˜
ÁÁ
˜˜ÁÁ
Ë 20 mG ¯Ë 0.1 pc ¯
2
Magnetically Critical Mass
This analysis leads to the terminology:
Magnetically Sub-Critical Mass (M < M_) – Magnetism
stronger than gravity. The external pressure can maintain
the cloud in equilibrium, but not collapse it
Magnetically Super-Critical Mass (M > M_) - Magnetism
weaker than gravity; external pressure can collapse the cloud.
3 Mc 2
3 G
2
2
P=
M
M
F
3
4
4p R
20p R
(
)
Measurements of magnetic field in cores (Crutcher et al.,
(1999) indicate that they are on the margin between
sub-critical and super-critical: M ≈M_
Fate of Sub-Critical Cores
Sub-critical cores may collapse on long time scales
via ambipolar diffusion (see Shu 2 Ch. 27).
Gravity on pulls on all of the matter, mainly neutral
hydrogen & helium. The ions are tightly tied to the
magnetic field and they lock the neutrals in place by
microscopic ion-neutral interactions. But on long time
scales, gravity will drag the neutrals through the ions
and feed the center of the core. This process is
estimated to take several Myr.
The ambipolar diffusion time scale depends inversely
on the electron fraction, so it occurs slowly on the
outside of the core and more rapidly in the inside.
Estimating the Ambipolar Diffusion Timescale
Assuming ideal MHD, the force per unit volume is balanced
by the drag force,
r r
r
mi mn
1
(— ¥ B) ¥ B =
K ni nn w
4p
mi + mn
where i and n stand for ions and neutrals, K is the rate
coefficient for ion-neutral momentum-changing collisions,
and w is the drift velocity of interest. One can easily check
the dimensions of the right side, and that w is sensitive to:
B, nH2 and xe.
A crude estimate based on replacing the RHS by B2/4!L,
B ~ 20 _G, K~ 10-9 cm3 s-1, H2 as the neutral, and
typical core properties yields w << 1 km s-1.
Observations of Core Magnetic Fields
• Crutcher et al. ApJ 407 175 1993 – one 3-_ detection
out of 12 dark clouds: B1, B ≈ 19.1 +/- 3.9 _G
• Crutcher ApJ 520 705 2000 – estimates of M/M_ for 27
molecular clouds (about half cores)
• Crutcher & Troland (ApJ 537 L139 2000) 3-_ detections
of two out of 8 dark clouds, including L1544, B ≈ 11_G, for
which an infall signature has been observed.
• Crutcher et al. ApJ 600 279 2004 non-Zeeman measurements of 3 pre-stellar cores: 100-200 _G
Summary: Magnetic Fields in Dense Cores
• Measurements show that magnetic fields play can play
an important role in shaping the structure and
dynamics of molecular clouds.
• Polarization maps reveal that the magnetic fields
threading molecular clouds are ordered over large
scales
• The Zeeman measurements show that the
mass-to-flux ratios are close to critical.
• Gravity amplifies the anisotropic structure associated
with magnetic fields, e.g., clouds collapse preferentially
along the direction of the average field.
MHD Core Formation Simulation in 3-d
Li et al. ApJ 605 800 2004
• 5123 element periodic box
• super-critcal field
• strong supersonic turbulence
• Mcloud ~ 64 MJ, L ~ 4 _J,
• followed for 3 free-fall times
• no microscopic physics
readily forms prolate and triaxial super-critical cores &
later centrifugally supported
massive disks
column density map with disks
2. The Singular Isothermal Sphere
We go beyond the virial analysis and try to understand
how a core can appear to be both stable and collapsing by
studying the Lane-Emden equation for an isothermal sphere:
1 d Ê 2 df ˆ
Ár
˜ = 4pG
2
r dr Ë dr ¯
dp
df
df
2 dr
—p = - r—f Æ
= -r
Æ
= -c
dr
dr
dr
dr
using p = rc 2 . Substituting the 2nd into the 1st gives
— 2f = 4pG
Æ
1 d Ê 2 1 dr ˆ
4pGr
Á
˜
r
=
.
2
2
Á
˜
r dr Ë r dr ¯
c
We seek equilibrium solutions satisfying:
R
r ¢(0) = 0
r ( R) = pext c - 2 M (r ) = 4p Ú dr r 2 r (r )
0
The Bonnor-Ebert-McCrea Equation
Use of dimensionless variables
gives the BEM Equation:
r 4pGr 0
r
y=
, x=
r0
c
1 d Ê 2 1 dy ˆ
ÁÁ x
˜˜ + y = 0
2
x dx Ë y dx ¯
schematic solution
Shu ApJ 214 488 1977
The Singular Isothermal Sphere
Shu (ApJ 214 488 1977) noticed the 1/ r2 envelope of
the BEM solutions and showed that 1/ r2 is a particular
solution of the equation, now called the singular
isothermal sphere (SIS). Substitute _ = A/r2 into
1 d Ê 2 1 dr ˆ 4pGr
1 d
4pG A
Á
˜
(
)
r
+
=
0
to
get
2
r
+
=0
2
2
2
2
2
Á
˜
r dr Ë r dr ¯
c
r dr
c r
to find
c2 1
r=
2pG r 2
2c 2
M (r ) =
r
G
Shu rewrote the full equation as two 1st-order ODEs
with no free parameters when scaled variables are used:
velocity
v=u/c
density
_=4!G_t2
position/time x=r/ct
The Singular Isothermal Sphere
The solution that asymptotes to the SIS
is plotted on the left and has the limits:
at 2
A c2 1
c2
large x : u = -(a - 2)
, r=
, M=A r
2
r
2 2pG r
G
1/ 2
1/ 2
Ê 2m0 a 3t ˆ
1 Ê m0 a 3 ˆ
c 3t
˜˜ , r =
ÁÁ
˜ , M=
small x : u = -ÁÁ
m0
3 ˜
4pG Ë 2tr ¯
G
Ë r ¯
m0 is related to A. A_ 2+ is a limiting
case showing inside-outside collapse:
large x : static solution (SIS) for r > ct
small x : free - fall collapse with
2GM (r )
u=
and r µ r -3 / 2 for r < ct
r
For A_ 2+, m0=0.975.
Singular Isothermal Sphere
Collapse at finite r occurs after the signal that collapse
started at the center reaches r, i.e., in time
Ê r ˆÊ T ˆ
r
˜˜Á
t ª = 5.26 ¥105 yr ÁÁ
˜
c
0
.
1
pc
10
K
¯
Ë
¯Ë
-1 / 2
It is of interest to account for the SIS mass distribution M(r):
c 3t
accreted mass : M (0 ) =
m0
G
+
3
c
t
infalling mass : M(ct) - M (0 + ) =
(2 - m0 )
G
2c 3t r
static envelope mass : M(r) - M (ct ) =
( - 1)
G ct
The most remarkable result is that the accretion rate is
determined only by the signal speed c and G:
M& acc
c3
= m0
G
Final Remarks on the SIS
• The time for the mass with r to collapse is ~ the Jeans
time for the average density within that sphere.
• Rotation will cut off the r-2 envelope beyond
-13
Ê 10 rad s
c
17
RC = @ 1.86 ¥10 cm ÁÁ
W
W
Ë
-1
1/ 2
ˆÊ T ˆ
˜˜Á
˜
¯Ë 10K ¯
• The effects of magnetic fields have been studied in depth
by Shu and collaborators, leading to singular isothermal
toroids (SITs), e.g., Allen et al. ApJ 599 363 2003. Also
see the work of Mouschovias and coworkers.
3. Observations of Collapse
It had been the hope of the early mm radio astronomers
to discover molecular clouds in the process of collapse.
They were rather surprised when, instead of collapse,
outflow was discovered in the CO maps of the region
around the YSO, L1551-IRS5 (next page), and later in
hundreds of other sources.
Credible detections of infall were not made until the
1990s. The subject is reviewed by:
• Evans NJ II, ARAA 37 311 1999
• Myers, OSPS 1999
• Myers, Evans, & Ohashi, PPIV 2000, p. 217
• Evans 2002, astro-ph/0211526
Discovery of Bipolar Molecular Outflows
Snell, Loren & Plambeck, ApJ 239 L17 1980
CO contours on an optical photo of the
L1551-IRS5 region with H-H objects.
Paradigm-setting cartoon for the
outflows from all low-mass YSOs.
Optical Jet
HH1
HH2
YSO and recent jet
Optical photo of HH1-HH2: Hz (green), S II (red) plus
continuum (blue). Stapelfeldt et al,, ApJ 116 372 1998
Signature of Collapse
red
Evans ARAA 37 311 1999
blue
Myers 1999 Schematic Infall Signature
blue and red shifted surfaces of constant
l.o.s velocity assuming v ~ r- -1/2
Optical depth unity occurs for red and blue
in regions with different excitation
Myers Two-Slab Model
Effects of varying optical depth and approach speed
B335: The Simplest Case
Choi et al. ApJ 448 742 1995
H2CO & CS, with
critical densities ~ 106 cm-3
dashed line is model
rounded Bok globule
distance ~ 250 pc
IRAS (> 60 m): L ~ 3 Lsun
many other mm lines
clumpy bipolar outflow
The Case of B335
Choi et al. (ApJ 442 742 1995) modeled the line shapes
(shown on the previous page) using Shu’s SIS by varying
the temperature distribution, the CS & H2CO abundances,
and the beak point between the r-3/2 and r-2 density
distributions. They concluded that the observations
support the inside-out collapse model.
But Wilner et al. (ApJ 544 L69 2000) observed the highfrequency 245 GHz CS(5-4) transition with the PdB
interferometer at much higher angular resolution (3”
instead of 11”, corresponding to ~ 750 AU), and found a
clumpy bipolar outflow & an inner off-center peak. The fit
to the inside-out collapse model is degraded.
IRAM CS Observations of B335
Interferometer channel maps
showing outflow using CS(5-4)
SIS modeling of line shape that
includes fitting map data: dotted line
IRAM Dust Observations of B335
Harvey et al. (ApJ 596 383 2003)
mapped the 1.2 & 3.0 mm continuum
(dust) emission at a spatial resolution
of 0.3” (10 times better than the
previous CS(5-4) maps). They detect a
disk-like structure of 100 AU and model
the dust with an inside-outside density
profile:
-0.4
T = 10K (r/5000 AU )
-p
for r < 5000 AU and 10K for r > 5000 AU
r = r 0 (r / R0 ) for r < R0
-2
and r 0 (r / R0 ) for R0 < r < 25000 AU
They find p=1.55 +\- 0.04, R0=6500 AU, n0(H2)=3.3x104 cm-3,
M(<6,500 AU) = 0.58 Msun & M(<25,000 AU) = 3.06 Msun,
and argue for Shu’s 1977 model.
Starless Core L1544
Tafalla et al. ApJ 504 9000 1998
HCO+
H2CO
CS
C3H2
N2H+
C34S
optically thick lines
opacity increasing upwards
map of CS(2-1) profiles
asymmetry extends beyond N2H+ core
does not fit simple SIS models
Summary of Infall Signature Observations
• There is a basic problem to start: small infall velocity
requires high spatial resolution observations.
• The line shape is affected by other dynamical motions,
e.g., turbulence, rotation & outflow
• The analysis has to use good thermal, chemical &
excitation models and deal with many lines reflecting
different properties; dust also plays a role.
• The results depend on core shape and magnetic field.
• The observations probably do show infall, but the overall
result is ambiguous, e.g., low resolution surveys yield
many red-peaked profiles.
• ALMA is needed to go to the next level.