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THE C 4 H ZEEMAN EFFECT IN TMC­1: UNDERSTANDING LOW­MASS STAR FORMATION
B. E. Turner
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903
and
Carl Heiles
University of California, Berkeley, CA
Received 2005 May 13; accepted 2005 September 27
ABSTRACT
We have searched for the C 4 H Zeeman effect toward the cold dense TMC­1 cyanopolyyne peak (CP) core and
obtained an upper limit of 14:5 # 14 #G for a magnetic field B ¼ jBj cos #, where # is the angle between the field and
the plane of the sky. C 4 H is particularly suited to the detection of small dense cores that result from the evolution
of magnetized clumps undergoing ambipolar diffusion (AD) to smaller denser cores that directly form stars. These
are the so­called quasi­static models. We discuss three distinct types of model for such systems, those of the
Mouschovias, Shu, and Myers groups, respectively. We see no indications of line­broadening or high densities in the
core, such as those predicted to follow the ambipolar diffusion and free­fall contractions, but the dearth of obser­
vational data for B fields in low­mass protostars makes it difficult to form definite conclusions from our results. We
also discuss several models emphasizing recent formulations of large­scale (100 pc) flows of supersonic turbulence
that appear to surmount earlier difficulties such as too large a star formation (SF) rate and the need for a magnetically
subcritical initial cloud. Possible reasons for the null result on the C 4 H Zeeman effect are that (1) in the case of the
quasi­static models, TMC­1 CP has not evolved yet to the onset of dynamic collapse, which follows the much longer
AD phase, and (2) there is a weak jBj field throughout the Taurus complex and it is directed close to the plane of the
sky, whereas the Zeeman effect is sensitive only to line­of­sight (circular) polarization. In the supersonic turbulence
models, the subsonic nature observed for the C 4 H emission lines argues that the energetic turbulence that formed
TMC­1 CP has passed by TMC­1 CP at least a Myr ago (the crossing time for the energetic turbulence to traverse the
clump).
Subject heading: ISM: magnetic fields
1. INTRODUCTION
The role of magnetic fields in the evolution of molecular
clouds and their star­forming properties is widely believed to be
important, particularly in the support of clouds against free­fall
collapse, which if unchecked would produce a star formation rate
some 10 times that observed in the Milky Way. Braking by mag­
netic fields may be key to understanding how clouds shed their
excess angular momentum prior to condensing into stars. More­
over, magnetic fields may dictate the size of the subclumps within
a cloud and thus the masses of the protostellar condensations that
ultimately form. At later stages, a strong field is likely to influence
the dynamics and evolution of any accretion disk, outflow, or H ii
region that may form around a star.
By far the majority of magnetic field results utilizing the Zeeman
effect have involved H i and OH. H i characterizes B fields only in
low­density regions (10 1 --10 2 cm #3 , not relevant to star forma­
tion). OH samples higher density gas, #10 2 --10 3 cm #3 via the
1665/1667 MHz lines, and thus traces a large range of molecular
gas, including ``envelope'' gas around GMCs and around smaller
star­forming clumps in large complexes such as Taurus. These
clumps are signposts of low­mass star­forming regions, but due to
large beam sizes and the ubiquity of OH, little information is avail­
able about magnetic fields in the smaller dense structures (those
most important for modeling protostar formation).
To probe higher density condensations and thus later stages of
the star formation process, the molecular Zeeman species require
four attributes: a chemistry that produces a large abundance (only)
at high density; large g factors, high critical densities, and a large
T #
R /T sys ratio. Other than OH, the only molecular species that ex­
hibit the Zeeman effect are CH, CN, SO, CCS, and C 4 H. The spe­
cies CH and SO have relatively weak transitions., Searches with
SO in warm massive star­forming clouds have been unsuccessful
(Uchida et al. 2001), as have two attempts at using CCS to ob­
serve cold dark clouds (Levin et al. 2001; Uchida et al. 2001).
Apart from H i and OH, only CN has so far yielded detections of
magnetic fields (Crutcher et al. 1996, 1999), detecting strong fields
of #0.19, #0.33, and #0.45 mG in OMC­1, M17 SW, and DR 21
OH. No cold dark clouds were attempted. The problem is that
these highly complex, massive star­forming regions, with poorly
characterized physical conditions, are not amenable to compari­
son with detailed models. It appears that these warm, more mas­
sive star­forming clouds are magnetically supercritical, and the
cores are in approximate virial equilibrium, supported primarily
by internal motions rather than static magnetic fields. The motions
may be approximately Alfve ’nic, suggesting that MHDwaves pro­
duce the supersonic motions. Understanding the role of jBj re­
quires detection of fields in smaller quiescent clumps, which can
be compared at small­scale sizes with the many detailed models
now available (cf. Ciolek & Mouschovias 1994). At present, B
fields have been detected in only one small star--forming clump
embedded in a cold dark cloud (L1544).
C 4 H satisfies the criteria of a good high­density magnetometer
with a reasonable critical density, which happens to be roughly
the geometric mean of those for OH and CN. Curiously, C 4 H has
strong lines and a large abundance only in TMC­1. The Green
Bank 140 foot (42.67 m) telescope was used in 1999 (B. E. Turner
and C. Heiles) to survey #30 molecular clouds in the Galactic
388
The Astrophysical Journal Supplement Series, 162:388--400, 2006 February
# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

plane; it yielded only three additional sources of C 4 H, none strong
enough to warrant a Zeeman search. In 1999 we searched unsuc­
cessfully for the C 4 HZeeman effect toward TMC­1CP, the cyano­
polyyne peak. With an imprecise value for the angular size of
TMC­1 CP, it seemed desirable to repeat the C 4 H observation
using the Green Bank Telescope (GBT), whose effective area is
#7 times that of the 140 foot telescope. The TMC­1 CP clump
remains an ideal object for comparison with existing models of
magnetized clump evolution.
2. THE ZEEMAN EFFECT IN C 4 H
The Zeeman splitting is given in terms of the Lande g factor as
EH (M F ) ¼ #g F # I HM F ;
where g F depends on the molecular angular momenta that in­
teract with the magnetic field. C 4 H is a 2 # molecule with dou­
blet splitting arising from coupling of the unpaired electron
spin of the H atom with the orbital angular momentum of the
H atom (J ¼ N × S ). The nuclear spin I of the proton couples
with J (F ¼ J × I ). Since the latter coupling is the weaker one,
we can write
g F ¼ g J # J × g I # I ;
where
# J ¼ ½F(F × 1) × J (J × 1) # I(I × 1)#=2F(F × 1);
# I ¼ ½F(F × 1) × I (I × 1) # J (J × 1)#=2F(F × 1);
g J ¼ ½J (J × 1) × S(S × 1) # N (N × 1)#=J (J × 1);
g I ( H ) ¼ 2:79:
Table 1 lists the resulting g factors and the theoretical line
strengths, the product of which is proportional to the strength of
the Zeeman signals.
3. OBSERVATIONS OF THE C 4 H ZEEMAN
EFFECT IN TMC­1
During 2003 January, August, and September we used the
GBT to observe the 9493.061 and 9497.616 MHz hyperfine
transitions of C 4 H. Specifically, we observed the circular polar­
ization (Stokes V ) and total intensity (Stokes I ) with the intent of
observing the line­of­sight magnetic field strength. The GBT re­
ceiver covering these lines has native dual circular polarization.
This is only approximate, however. We used the calibration tech­
nique of Heiles et al. (2001) to derive the Mueller matrix transfer
function of the feed/electronics combination and the associated
correction technique to obtain pure Stokes V. The calibration
results are given by Heiles et al. (2003).
We used the Green Bank Spectral Processor, which is a
Fourier­transform type spectrometer, in cross­multiply mode.
We observed both lines simultaneously by splitting the spec­
trometer into two parts, each having 512 channels, total band­
width 625 kHz, and providing all four Stokes parameters
simultaneously.
We observed the Taurus­CP clump at # ¼ 04 h 38 m 38 s ;
# ¼ 25 # 35 0 45 00 B1950:0
Ï ÷ [or # ¼ 04 h 41 m 42 s ; # ¼ 25 # 41 0 27 00
J2000:0
Ï ÷]. We used in­band frequency switching with a total
span of 15/32 of the total bandwidth. We accumulated 34.7 hr of
integration time.
3.1. Results: Magnetic Field
We have also formed weighted averages of the spectra at
9493.061 and 9497.616 MHz, respectively, shown in Figures 1
and 2. Figure 1 shows the 9493.061 MHz line, and Figure 2 the
9497.616 MHz line. In each figure the top panel shows the
Stokes I spectrum, which is the sum of two orthogonal polariza­
tions (not the average) and is exhibited with no spectral smooth­
ing. The bottom panel shows the Stokes V spectrum, which is the
difference between orthogonal circular polarizations and is boxcar­
smoothed by three channels to reduce the visual noise. We derive
the line­of­sight B field strength, B los , by least­squares fitting the
Stokes V profile to two terms: (1) the frequency derivative of the
Stokes I profile and (2) the Stokes I profile itself; term 1 is propor­
tional to B los and term 2 is proportional to the instrumentally
caused leakage of Stokes I into Stokes V caused by small errors in
the Mueller matrix. The derived B los is written between the top and
bottom panels. The signs of B los are arbitrary because we did not
determine the sense of the derived circular polarization.
TABLE 1
The N ¼ 1 0 Transition of C 4 H
J u J F u F
Frequency
( MHz) Strength g­Factor Product
3/2 .......... 1/2 0 1 9493.061 0.1389 #2.457 #3.413
3/2 .......... 1/2 1 2 9497.616 0.4150 0.897 3.726
3/2 .......... 1/2 1 1 9508.005 0.1102 0.197 0.022
1/2 .......... 1/2 0 1 9547.961 0.1111 0.0 0.0
1/2 .......... 1/2 1 0 9551.717 0.0840 1.562 0.131
1/2 .......... 1/2 1 1 9562.904 0.1409 1.728 0.243
Fig. 1.---Spectrum of Stokes I (top) and Stokes V (bottom) of the line at
9493.061 MHz of C 4 H toward TMC­1 CP.
C 4 H ZEEMAN EFFECT IN TMC­1 389

We have formed a weighted average of the first two figures
(not shown). The Stokes I spectrum is a straight average. The
Stokes V spectrum is a weighted average, with the weights pro­
portional to fg, where f is the theoretical optically thin line inten­
sity and g is the Lande g factor. The g factors for the two lines are
of opposite signs, so the two V spectra contribute with opposite
signs (``subtract'') instead of the same sign. This weighted aver­
age of the other two individual spectra produces B los ¼ #2:7 #
12:9. However, this value is relatively meaningless except for
visual purposes. Our final value for the B field is derived from a
weighted average of the other two individual spectra; i.e., from a
simultaneous fit to both individual spectra. It is not derived from
a fit to the weighted average. The simultaneous fit is statistically
optimal and provides what is, in principle for Gaussian noise
statistics, the best estimate of the B field and the most realistic
estimate of the uncertainty.
Our adopted procedure gives the optimal result because it fits
only three parameters: the magnetic field, which is proportional
to the frequency derivative as in (1) above, and two leakage pa­
rameters, as in (2) above. In contrast, if we instead obtained the
final result from a weighted combination of the separate, indi­
vidual fits in Figures 1 and 2, we would have fit four parameters:
two magnetic fields and two leakage parameters. This is clearly
less optimal. Our final result is a nondetection:
B los
j j ¼ 14:5 × 14:0 #G:
Note again that the sign of the derived field was not calibrated
and is undetermined.
3.2. Results: Gaussian Fits
We fit the C 4 H Stokes I lines in Figures 1 and 2 with two
Gaussian components. The center velocities differ by about
0.2 km s #1 , but this difference is not real: it is a result of uncer­
tainties in local oscillator frequencies in the GBT electronics.
Unfortunately, we cannot recover the central velocities more ac­
curately than a few tenths of a km s #1 . The height ratios are very
close to 3, the expected ratio for optically thin lines, and the de­
rived widths for the two components are equal almost to within
the errors. We conclude that the line is very well represented by
two Gaussians. Table 2 gives the component parameters.
4. MODELS OF THE COLLAPSE OF MAGNETIZED
STAR­FORMING CORES
4.1. The Basis of the Models
Many support mechanisms have been considered against free­
fall collapse of star­forming molecular clumps (on a timescale of
#10 5 yr), and all, including now magnetic fields, are seen to en­
counter difficulties. Magnetic fields are still considered an impor­
tant mechanism in supporting clumps in quasi­equilibrium against
gravity, largely as a result of the theoretical and simulation work of
the Shu and Mouschovias groups over the past two decades. But it
is now recognized that other issues need study, such as the extent
of the role an expanded model of the turbulence can play. The
observational data are still very sparse, especially in regions of
low­mass star formation. The more difficult question is how the
field evolves as the clumps form dense cores just prior to the
onset of star formation. We discuss approaches by Mouschovias
and colleagues, by Shu and coworkers, and, briefly, by Myers &
Goodman. The latter is not a ``theory'' on the level of the other
two, but it considers the consequences of a virial equilibrium
between kinematic, gravitational, and magnetic energies.
It is necessary to define carefully the distinction between
``clouds,'' ``clumps,'' and ``cores.'' In regions of massive star for­
mation, ``cloud'' means GMC, ``clump'' means any of a large
number of secondary objects such as M17 SW in M17, or W51M
in W51; and ``core'' means the smaller, denser object(s) that form
within clumps, via (e.g.) ambipolar diffusion (AD) or other agent
and become the massive protostar(s). In regions of low­mass star
formation the Taurus and Ophiuchus regions are good examples of
a similar hierarchy. The Taurus­Auriga region is the archetypal cold
dark complex exhibiting all of the hierarchical categories of Scalo
(1985; cf. Turner 1997), starting with the Lynds dark clouds (Scalo
category 4) and ending with the cores (category 6). The overall
TMC­1 structure is a ``cloud.'' Whether it is subcritical or super­
critical is unknown. TMC­1 SE and TMC­1 CP (cyanopolyyne
position) are two ``clumps'' within TMC­1, only 0.4 pc apart. 1
Within each clump is, or will form, a highly dense core, and from
Fig. 2.---Spectrum of Stokes I (top) and Stokes V (bottom) of the line at
9497.616 MHz of C 4 H toward TMC­1 CP.
TABLE 2
Gaussian Fits to Figures 1 and 2
N
Frequency
( MHz)
HGT
( K )
V LSR
( km s #1 )
FWHP
( km s #1 )
1............ 9493 0.432 # 0.004 5.899 # 0.001 0.214 # 0.003
9497 1.266 # 0.004 5.706 # 0.001 0.210 # 0.002
2............ 9493 0.088 # 0.003 6.193 # 0.009 0.275 # 0.020
9497 0.266 # 0.006 5.989 # 0.008 0.314 # 0.016
1 We choose to observe TMC­1 CP because Zeeman observations of CCS have
already been made toward TMC­1 SE (Uchida et al. 2001) and because the CP
position favors carbon­chain chemistry, and C 4 H is a longer chain than CCS.
TURNER & HEILES
390 Vol. 162

them stars. Only low­mass stars appear to form in this hierarchy.
All of these categories, as well as the larger overall complex, are
believed to be self­gravitating. By definition, conditions in the
overall cloud differ from those in clumps or cores, the actual sub­
units where the gravitational collapse takes place. The evolution­
ary states of clouds, clumps, and cores are poorly known, although
there is evidence that several massive clumps lie close to the
subcritical­supercritical boundary in massive star­forming regions
(Crutcher et al. 1999). Essentially no such information exists for
low­mass clumps such as those in TMC­1, because no jBj fields
have been detected in them.
It is instructive to review the simple case of gravitational col­
lapse in the absence of AD. In this case clouds that are initially
magnetically supercritical will collapse on relatively short time­
scales (#free fall). If the jBj field is insufficient to stop the initial
collapse, its compression during collapse cannot bring the cloud
into equilibrium and halt the collapse. Conversely, an initially
magnetically subcritical cloud will become unstable in its core
owing to AD, which causes a drift of neutrals into the core with­
out a significant increase in the magnetic flux. Eventually, the
mass­to­flux ratio in the core becomes supercritical, and dynam­
ical collapse and star formation can proceed. The envelope re­
mains essentially in place while the supercritical core collapses.
The connection between the core and the surrounding enve­
lope by B field lines can transfer angular momentum outward
and make it possible for stars to form. The way in which the field
evolves as the clump starts to form a dense core is usually stated
in the form of a power law, jBj # n # . The field B scales as a
power of the density regardless of whether it is important. If the
field is not important energetically or dynamically, and is frozen
in, then isotropic contraction of the core will amplify the field as
B # n # . In the case of spherical isotropic collapse, which occurs
if the field is weak and unable to break the spherical symmetry of
the contraction, then # ¼ 2/3. If the field is energetically im­
portant, the same scaling between B and n holds in the central
magnetic flux tubes, but now # ¼ 1/2. The value of # decreases
because the magnetic field is now strong enough to prohibit
lateral contraction perpendicular to the field, yet unable to stop
flow of matter along the field, since magnetic forces act only
perpendicular to field lines. The field causes an initial spherical
structure to flatten along the field lines while remaining sup­
ported perpendicular to the field lines. If equilibrium is attained
along the direction of the field, it occurs without magnetic forces,
so the equilibrium occurs due to a balance between thermal pres­
sure and gravitational forces. In that case, the gravitational force
scales as the mass column density # 2 , and the thermal pressure
force scales as the density (cf. Ciolek & Basu 2001). This results
in the relation n # # 2 . For a flux­frozen cloud, the mass­to­flux
ratio B/# is a constant for each magnetic flux tube. Hence,
B/n 1/2
¼ constant for a flattened, flux­frozen cloud, and there­
fore B # n 1/2 , or # ¼ 1/2.
More detailed theories are needed if AD is effective. Two
groups have been pursuing detailed static or quasi­static models
for many years, the Mouschovias group (with R. A. Fiedler,
S. Basu, and G. E. Ciolek) and the Shu group (with S. Lizano,
F. Adams, D. Galli, and Z.­Y. Li). These are ongoing programs.
A third effort ( Myers & Goodman 1988) has explored the im­
plications of a simpler approach. We discuss these efforts in the
context of current observations. Finally, we comment on models
of supersonic turbulence driving the evolution of clouds.
4.2. The Shu et al. Models
The classical problem of star formation is to find relevant,
analytic, asymptotic solutions among a large family of self­
similar solutions to the one­dimensional collapse problem for the
equilibrium structure resulting from spherical density perturba­
tions in self­gravitating, isothermal, nonmagnetic, ideal gases.
All solutions to the isothermal collapse problem are members of
a two­parameter family with the Larson­Penston ( LP)­type solu­
tions (collapse of spheres with finite uniform central density) at
one end of parameter space, and at the other end initial condi­
tions that lead analytically to a singular isothermal sphere (SIS),
and in turn to an expansion­wave or ``inside­out'' collapse.
The latter model has been studied initially by Shu (1977), fol­
lowed by the Shu group. Lizano & Shu (1989) introduced mag­
netic fields and hence ambipolar diffusion to these models. Of
interest are solutions in which effects arising from the B field are
a small perturbation to the dynamical collapse of the innermost
regions. The resulting picture is of dense clumps condensing into
cores inside a cloud by gradually losing magnetic (and turbulent)
support via the process of quasi­static AD. As with the nonmag­
netic SIS models, the quasi­static ADmodels inevitably build up
a r #2 density profile that contracts on timescales of order t AD #
10t A . Acting in a magnetically subcritical, isothermal cloud, AD
is supposed to lead eventually to formation of a strict singularity
at the center: N (r ¼ 0) ¼ 1. The cores then approach a state
resembling a singular isothermal sphere, and if they pass the
brink of instability, they ultimately collapse from inside out, build­
ing up a central protostar and a nebular disk surrounded by an in­
falling envelope. The advantage of the Shu­type single singular
static core is that it allows construction of self­similar models. The
gravitational collapse of a singular isothermal sphere has a self­
similar solution taking the form of a spherical expansion wave
propagating outward at the sound speed, which can be found
semianalytically, thereby providing a convenient starting point for
further analysis.
In these models, it is not possible to state when clouds undergo
dynamical collapse and magnetic fields decouple. Dynamical
collapse is initiated while the field is still reasonably well coupled
to the bulk of the gas; the major episode of flux loss (decoupling)
must occur at much higher central densities when self­gravitation
has already overwhelmed the magnetic and thermal means of sup­
port. The reason one cannot follow the evolution during the stage
when the core tries to develop a central cusp (akin to a singular
isothermal sphere) is because the flow velocities along the field
lines approach supersonic values, and the quasi­static assumption
breaks down. An a posteriori calculation of the neutral particle
velocities is used (Lizano &Shu 1989). This gives t 0 ¼ 6 ; 10 6 yr
for the intrinsic diffusion timescale associated with contraction
from the envelope, nearly an order of magnitude longer than the
dynamical time. Nevertheless, the existing results suggest that an
inside­out collapse initiated from a power­law density distribution
is a good description of the dynamical stages of the evolution of
low­mass protostars.
The onset of the dynamical stage marks the time at which ac­
cretion onto the stellar core begins, with zero infall velocity and
acceleration. Before core formation, the density profile n # r #2
for all r (singular isothermal sphere), and the velocity profile is
v ¼ 0 for all r. After core formation, n # r #3/2 for r # c s t,
n # r #2 for r > c s t; the velocity profiles are v # r 1/2 and v # 0
for the same two ranges of r. Finally, the mass accretion rate is
dM /dt ¼ 0:975c 3
s /G. This is significantly below the values de­
rived for LP collapse. In the latter case the entire system is
collapsing dynamically and delivers mass to the center very effi­
ciently, while in the Shu case inward mass transport is relatively
inefficient, as the clump envelope remains at rest until reached
by the rarefaction wave. The density structure of the inside­
out collapse, however, is essentially indistinguishable from the
C 4 H ZEEMAN EFFECT IN TMC­1 391
No. 2, 2006

predictions of dynamical collapse. To differentiate between the
two models requires kinematical data from which the magni­
tude and spatial extent of infall must be determined with high
accuracy.
The next part of the problem is where the (dynamical ) effects
of the B field must be introduced by a dubious ``patching'' tech­
nique in the form of a perturbation, in which the nonmagnetic
collapse of the marginally unstable single isothermal sphere with
infinite central density serves as the zero­order reference state,
which has a known analytic solution. Departures introduced by
an initial uniform field B 0 are developed as a series expansion in
terms of the similarity variables x # c s t, where c s is the effective
sound speed, t is the elapsed time since initiation of dynamical
collapse, and r is the radial distance from the protostar. Thus, the
Shu­type models follow the quasi­static evolution of a single
isolated unstable magnetized cloud core with r #2 density profile,
and threaded with a uniform magnetic field, as AD lowers the
amount of magnetic and ``turbulent'' support in the dense cores.
Their code allows them to follow only the quasi­static stages. By
contrast, the Mouschovias­type models have a cloud­clump­core
hierarchy and a more realistic initial model (see x 4.1).
The advantage of the Shu­type single singular static core is
that it allows construction of self­similar models. The singularity
causes a gravitational collapse to set in at the origin and move
outward at the head of an expansion wave. In the first model of
Galli & Shu (1993a), magnetic effects are assumed to be only a
small perturbation on the expansion wave solution for a singular
isothermal sphere. In the subsequent models of nonrotating mag­
netic clouds (Galli & Shu 1993b), this assumption is relaxed for
a very small region near the axis of symmetry and an infall
problem is solved for this region, using the results of some quasi­
static contraction models of Lizano & Shu (1989) in which the
r #2 profile is built up slowly by ambipolar diffusion. However,
modeling the formation of cores by ambipolar diffusion using a
quasi­static model means that the acceleration of the neutrals is
assumed to be negligible, or equivalently, that there is a balance
between gravitational and magnetic forces at all times. Effec­
tively, this is analogous to the subcritical conditions required in
the early phases of clump evolution in the Mouschovias models.
This assumption breaks down upon the formation of a super­
critical core, whereupon gravitational forces overwhelm restor­
ing magnetic forces and sustained collapse sets in, as shown by
the fully dynamical models of Ciolek & Mouschovias (1995).
Under sufficiently early core contraction, the Shu et al. models
are able to operate under the required magnetostatic equilibrium
and thus to follow the central density to a modest enhancement
of about a factor of 10, that is, from 10 3 to #10 4 cm #3 , at which
point the equilibrium assumption breaks down, and the core evo­
lution suffers a transition to dynamical collapse. In the Lizano
& Shu model it was also found that the velocities remained small
compared with the speed of sound and that the density in their
central region suggested the formation of a profile that indeed
scaled as r #2 . Using the properties of self­similar models, Galli
& Shu (1993a) assume that these two essential features---small
velocities and a density profile that scales as r #2 ---would still be
valid up to the point that a singularity would form at the origin.
Their calculation (Galli & Shu 1993b) follows the evolution
after this event takes place. Now the problem is that the extra­
polation of the Lizano & Shu (1989) results to the initial state
that Galli & Shu (1993a, 1993b) want to use has largely been
excluded by the work that Ciolek, Basu, and Mouschovias have
published. Specifically, the low­velocity or quasi­static be­
havior no longer holds as the core density increases. Instead, the
Mouschovias group finds that the infall becomes increasingly
more dynamical as the central density (and mass­to­flux ratio)
increases. So, the singular, quasi­static initial state that Galli &
Shu (1993a, 1993b) or Li & Shu (1996) have advocated has
never occurred in model simulations. That is, their initial input
assumptions are invalid by the Mouschovias models. Note that
Shu et al. use the same physical equations (actually a restricted
subset) as Mouschovias et al., so if the Shu et al. self­similar
models and states were valid, the Mouschovias et al. models
should have seen them. The Shu et al. states appear never to have
been recovered, largely because they start from a physical state
that is never obtained by the Mouschovias et al. models, which
start from a simpler, earlier, and more diffuse state than that used
by Shu et al. (i.e., a more realistic initial state as suggested by
observations).
Mac Low & Klessen (2004) have summarized the major dif­
ficulties facing SIS­based models. Of all proposed initial config­
urations for protostellar collapse, quasi­static SISs appear to be
the most difficult to realize in nature. Stable equilibria for self­
gravitating, spherical, isothermal gas clouds embedded in an ex­
ternal medium of pressure # e are possible only up to a density
contrast of # c /# e # 14 between cloud center and surface. More
centrally concentrated clouds can reach only unstable equilib­
rium states. Hence, all evolutionary paths that could yield a
central singularity lead through instability so collapse will set in
long before an r #2 density profile is established at small radii r
( Whitworth et al. 1996).
4.3. The Mouschovias et al. Models
In addition to the difficulties encountered in modeling the
evolution of singular isothermal spheres, and in explaining how
they can exist, it is recognized that random external perturba­
tions of any small clump will act to break spherical symmetry in
the innermost region and flatten the overall density profile at small
radii. The resulting behavior in the central region then more
closely resembles the LP description of collapse.
Early numerical simulations of the spherical collapse of ini­
tially uniform­density, isothermal, nonmagnetic spheres by Larson
(1969) and Penston (1969) found that the solution remains iso­
thermal over a wide range of density and spatial scales and that the
approach to the formation of a central density singularity can be
described by an asymptotic similarity solution for an isothermal
sphere. It is characterized by a shrinking­near uniform­density
central region surrounded by a density profile proportional to r #2 .
The establishment of a central singularity merely corresponds to
the formation of a protostar that grows in mass by accretion of the
remaining envelope. When a finite mass begins to develop at
r ¼ 0, the density is everywhere proportional to r #2 . The impor­
tant point is that the initial state has finite density everywhere.
The LP problem is conceptually simpler than the Shu case
because there is no singularity, and hence no restrictions such as
those required in the Shu Models by the perturbative nature of the
solution. One simply solves numerically the system of equations
including two­fluid MHD equations governing AD, Poisson's
equation, and equations describing the ideal gas law, the accelera­
tion of gravity, the terminal ion drift velocity, and the fact that the
magnetic flux is frozen in the ions. The four free parameters in
the model are density, velocity (Alfve ’n velocity in the neutrals),
magnetic field, and time (the neutral­ion collision timescale;
Fiedler & Mouschovias 1993). With the inclusion of a magnetic
field and hence of AD into the model, numerical simulations by
Mouschovias (1976a, 1976b, 1979) show that AD in magnetically
supported clumps results in a self­initiated dynamical LP­type
collapse of the central region where magnetic support is lost, while
the outer part is still held up primarily by the field (and develops a
TURNER & HEILES
392 Vol. 162

r #2 density profile 2 ). This leads to star formation in the deep in­
teriors, where the ionization fraction is lowest, while clump enve­
lopes remain magnetically supported. Mass is fed to the center not
by an outward moving expansion wave, but by AD in the outer
envelope. Mouschovias (1979) finds that the AD timescale is
typically 3--4 orders of magnitude smaller in the core than in the
surrounding envelope. The essence of AD is found to be a redis­
tribution of mass in the clump's central flux tubes, rather than a
reduction in total magnetic flux. Because of this, the assumption
of flux­freezing no longer holds, and the ratio B/# ¼ constant
does not apply. If the AD is moderately effective and the collapse
along the field lines is slow, then in the core # is in the range 1/3 to
1/2 if magnetic and kinematic energy densities are comparable,
and even lower if the magnetic energy dominates. If the AD is
very effective, there should be no growth of B as the column
density (hence density) increases. Thus, the value of B remains
close to fixed, resulting in # ¼ 0 in this case (indeed, no theo­
retical lower limit exists; it must be set by observations of the
cores). Such behavior is seen in Figures 2f and 4f of Ciolek &
Mouschovias (1994), which show # as a function of density in
the central flux tubes of an evolving clump. The ``classical''
Mouschovias­type model starts with a subcritical clump, in which
neutrals gradually contract, diffusing inward through the ions and
field, leaving behind a magnetically supported envelope and on
the slow timescale of AD, forming a supercritical core that under­
goes dynamical collapse. A summary of core properties follows.
Before core formation (t < 0) the properties of the LP solution
of isothermal collapse are (1) the density profile: # # (r 2
× r 2
0 ) #1
(r 0 ! 0 as t ! 0), flattened isothermal sphere; and (2) the ve­
locity profile: v # r/t as t ! 0; v # #3:3c s , r !1. After core
formation (t > 0) the properties are # # r #3/2 , r ! 0; # # r #2 ,
r !1; velocity profile: v # r #1/2 , r ! 0; v # #3:3, c s , r !1;
accretion rate dM /dt ¼ 47c 3
s /G. Two important predictions are
that (1) supersonic infall velocities extend over the entire collaps­
ing core, and (2) the accretion rate is 48.2 times larger than the rate
for the Shu type models.
The Mouschovias models formulate the problem of the self­
initiated (due to ambipolar diffusion) formation and contraction
of protostellar cores in axisymmetric, isothermal, self­gravitating,
thermally supercritical but magnetically subcritical clouds. The
model clouds are initially in magnetohydrostatic equilibrium and
would remain so indefinitely except for AD. The models require
as input the mass, size, density, and temperature of the clump,
ionization fraction, mass­to­flux ratio (critical value # 1), external
magnetic field (B ref ), and parameters describing the friction of
both neutral and charged particles in a magnetic field. The mass­
to­flux ratio is the key parameter defining to what extent a static
field can support a cloud against gravitational collapse. Early
models used parameters suitable for massive star formation: a typ­
ical clump size of #1--3 pc, and a mass of #50--200
M# . At the
temperatures of interest in the initial clump (#10 K), the Jeans
mass is <10
M# . Since this is less than the #100
M# of the
massive clump, then to be initially in equilibrium, these clumps
must be predominantly supported by magnetic forces, i.e., the
clumps must be magnetically subcritical. Ciolek & Mouschovias
(1994) initially chose a mass­to­flux ratio equal to 0.26 times
the critical mass­to­flux ratio. Given an initial central density of
3(3) cm #3 and a radius of 4.3 pc, then M is fixed, and hence B ref
is defined, at 35 #G in this example. This value is consistent with
observations in regions of massive star formation. Subsequent
observational results suggested a mass­to­flux ratio of #0.5 rather
than 0.26.
The initial magnetic field may be specified either by the factor
by which the envelope is magnetically subcritical (e.g., 3.9) and
the initial central field strength (B 0 ¼ 35 #G; Ciolek 1995) or by
specifying the field in the envelope (B ref ) and the mass­to­flux
ratio, the reciprocal of the above subcritical factor. In a model pa­
rameterized to describe a typical massive star­forming cloud com­
plex [n 0 ¼ 2:6(3) cm #3 , T ¼ 10 K, central field B 0 ¼ 35:3 #G,
radius R 0 ¼ 4:29 pc, M ¼ 98:3
M# ], Ciolek & Mouschovias
(1994, their Figs. 1 and 3a; 1995) have argued that the dimension­
less mass­to­flux ratio is subcritical (<1) in the large­scale cloud
in which the clump forms. This is an important point in comparing
observations of a clump with the models, viz., that the models
actually predict that the clumps should be on the verge of being
critical or have already become supercritical and initiated col­
lapse. It is the larger ``clouds'' in which the clumps reside that are
subcritical. Nevertheless, Nakano (1998) points out that there is
presently no observational evidence for clouds that are signifi­
cantly subcritical. The situation can be rectified by accepting the
argument that most clumps are within a factor of 2 above or below
the critical value for gravitational collapse. TheMouschovias mod­
els require a subcritical envelope for the clumps. However, those
clumps that are supercritical may still be supported through a com­
bined ``static'' field ( large­scale mean field) and a ``fluctuating'' or
turbulent field that may arise from a spectrum of hydromagnetic
waves (Mouschovias & Psaltis 1995). For these supercritical
clumps the Mouschovias models could still be applied if the hydro­
magnetic ``wave turbulence'' decays rapidly, and then AD takes
over. In fact, it has been suggested that it is the damping of HM
waves by AD in magnetically supported molecular clouds, rather
than the growth of perturbations in collapsing clouds, that initiates
fragmentation and star formation (Mouschovias 1987a, 1987b).
After many papers exploring parameter space, the first evolu­
tionary model designed to fit any one specific molecular cloud
was that of the massive star­forming clump B1 (Crutcher et al.
1994). B1 is one of several concentrations in part of an extensive
molecular cloud complex in Perseus. A model of B1 has jBj ¼
30 #G, clump parameters of M # 600
M# , R ¼ 2:9 pc, and
n 0 ¼ 1:3(5) cm #3 , core parameters of 13.4
M# , R ¼ 0:13 pc,
and n 0 ¼ 1:3(5), and mass­to­flux ratio 0.42. Free­fall time is
9.8(5) yr, and total contraction time is t r ¼ 1:5(7) yr. There is
good agreement between observed and model parameters.
Prior to about 1998, these models followed the core evolution
to central densities of #1(9) cm #3 , at which the assumption of
isothermality breaks down because of radiative trapping. Ciolek
&Konigl (1998) first simulated the dynamical collapse of a non­
rotating magnetic cloud core, following the evolution through
the formation of a central point mass and its subsequent growth
into a 1
M# protostar. The epoch of point­mass formation ( PMF)
is studied with a self­consistent extension of previous models.
Prior to PMF, the core is dynamically contracting and is not well
approximated by a quasi­static equilibrium model. AD is unim­
portant during the dynamical pre­PMF collapse phase. However,
the appearance of a central mass, through its effect on the grav­
itational field in the inner core regions, ``revitalizes'' the AD in
the weakly ionized gas surrounding the protostar. This leads to a
decoupling of the field from the matter and results in an outward­
moving hydromagnetic C­type shock of the type predicted by Li
& McKee (1996). For present purposes, this carries us past the
evolutionary phase of TMC­1 (cf. x 5.2). See Ciolek & Konigl
(1998) for a complete account.
In comparing models with observations, note that the central
field B 0 invariably reaches mG levels when the central density n 0
2 The fact that the ionization of any clump is larger in the outer part than in the
inner part, owing to larger extinction toward smaller r, increases the likelihood
that the outer envelope can remain subcritical for a longer period.
C 4 H ZEEMAN EFFECT IN TMC­1 393
No. 2, 2006

reaches 1(9) cm #3 , but the mean field strength in the supercritical
core is usually
# 0:1 pc (for OH), the mean field hBi # 70 #G should be com­
pared with the models. For C 4 H ( beamwidth 5.6 times smaller
than OH beam), hBi # 400 #G. In addition, this value should be
projected along the line of sight.
The OH Zeeman effect is detected in about 15 massive star­
forming regions, and the CNZeeman effect is detected in three of
these. By contrast, next to nothing is known observationally
about jBj fields in low­mass star­forming regions. In the well­
known Taurus and Ophiuchus regions of low­mass star forma­
tion, of #15 small dark clumps examined, the OH Zeeman effect
is seen in only one object, L1544 (Crutcher & Troland 2000).
The L1544 clump has M ¼ 3:2
M# , B ref ¼ 11 #G. Its star­
less core appears to be contracting and may soon form a star.
Well before its jBj field was measured, Tafalla et al. (1998) and
Williams et al. (1999) suggested that the apparent contraction
motions could not be explained by AD. Ciolek & Basu (2000)
emphasized that ADmodels must be chosen with parameters ap­
propriate for a specific core in order to fit the data. Their model
made two predictions: (1) as is the case for all symmetric core
models with gravity balanced by magnetic and thermal pressure
support, the model core is an oblate spheroid with jBj aligned
along the minor axis; and (2) the jBj field strength and its line­of­
sight component as a function of radius are predicted. The mor­
phological prediction was satisfied by subsequent polarization
mapping ( Ward­Thompson et al. 2000); the field is fairly uni­
form in direction, but the direction in the plane of the sky is 29 #
off the minor axis, an apparent discrepancy of the model. The
model mass­to­flux ratio is 0.8 of the critical value, so the dense
core is not as strongly supported by the jBj field as it is for the
massive cores. The model gives jBj ¼ 16 #G at the core center,
13 #G at 0.06 pc from the center, and B ref ¼ 11 #G. The best fit
of the model occurs at 20 # offset of the field direction from the
plane of the sky. The best­fit parameters of L1544 resemble
closely those known for TMC­1 (density, temperature, line width,
core mass, and an assumed mass­to­flux ratio of 0.8). Therefore,
the detected values of jBj in the center (16 #G) and envelope
(13 #G) become the upper limits to the corresponding positions in
TMC­1, but the uncertainties are large and the orientation of the
field relative to the plane of the sky is unknown for TMC­1. With
the sparse data on hand, a few parameters such as mass­to­flux ra­
tio #0.8 and n 0 # 3(3) cm #3 seem suitable for both low and high
massive star­forming clumps, while B 0 , B ref , clump mass, and ra­
dius are source dependent. The status of t AD is not clear, but the
duration of collapse appears to meet the requirements for star
formation in the Galaxy.
4.4. The Myers & Goodman Model
Myers &Goodman (1988) have found that for an ensemble of
clump and cloud sizes of 0.1--100 pc, a virial equilibrium model
of thermal and magnetic support against gravity is found to fit the
observed trends known as the Larson laws relating line widths,
densities, source sizes, and masses, provided the one free param­
eter, magnetic field strength, lies in the range 15--40 #G. Low­
mass dense cores have significantly smaller ratios of kinematic
energy to gravitational potential energy than do larger clouds.
According to the equilibrium model, these cores have substan­
tially less magnetic support against gravity, and substantially
smaller mass­to­flux ratio, than do the larger clouds. This rela­
tively weak magnetic support can arise from AD: for constant jBj
strength, a model cloud with thermal and magnetic balance
against gravity is found to have a critical size for which AD is
fastest, and this size is similar to that of typical low­mass cores
with R # 0:1 pc. Conversely, more massive cores have equilib­
rium jBj values 3--10 times greater than that of their lower
density surroundings.
Criticism of this model (Crutcher et al. 1994); Mouschovias
& Psaltis 1995) is based on problems that occur with any virial­
based (equilibrium) model. First, being dependent only on re­
lations between integral quantities, such models provide no
information about the structure of clouds. Second, the concept of
a magnetically subcritical cloud is the key to understanding the
structure and evolution (initiated by ambipolar diffusion) of self­
gravitating clouds, but the magnetically subcritical cloud cannot
be in equilibrium. G. E. Ciolek (2004, private communication)
also notes that if the nonthermal support is due to hydromagnetic
waves, linear or nonlinear, which is consistent with the line
width--size relations (cf. Mouschovias & Psaltis 1995), then the
assumption of virial equilibrium will break down on smaller
scales in the interior of a cloud, where a core is likely to form. In
the interior of a cloud ambipolar diffusion will preferentially
damp hydromagnetic waves and therefore remove the nonther­
mal source of support in these regions, perhaps triggering the
onset of collapse. This contradicts the assumption of virial equi­
librium on these scales. Thus, the model is quantitatively inap­
plicable for objects as small as cores. On larger scales, so long as
there is minimal ambipolar diffusion and reasonable support
against gravity, the model may be adequate.
Finally, we note that low­mass cores have lower kinematic
energy--to--potential energy ratios and less magnetic support
against gravity than larger cores. But this contradicts the
Mouschovias idea that ionization decreases in the centers of
cores because of attenuation of ionizing photons. Low­mass
cores ought to have less attenuation and thus higher ionization,
and thus less ability for AD to allow the field to leak out, as
compared to high­mass cores. ( Mouschovias models point to
3--4 orders of magnitude difference in the AD timescale be­
tween inside and outside cores. Such effects as these cannot be
integrated into the Myers­Goodman model.
It is interesting to apply the simple heuristic model of Basu
(2000) to the data. In this model, B scales as velocity dispersion
times the square root of the density. The model assumes virial
equilibrium along the mean field direction, and flux freezing. We
have but one new point to add to the plot of all data by Heiles &
Crutcher (2005). Current observational data for B strengths are
plotted versus #(V )(density) 1/2 . This relation characterizes the
magnetic field data very well. Given the density sampled by the
C 4 H Zeeman experiment (10 4 cm #3 ) and the quoted line width
(2.1 km s #1 FWHM), this empirical relation predicts a total field
strength of 14 #G. This prediction is remarkably consistent with
the null result quoted in x 3.1.
5. COMPARISON OF MODELS WITH OBSERVATIONS
A successful model should predict the following observables
for both high­ and low­mass clumps. We consider only the low­
mass case here.
5.1. Magnetic Field Strengths
In the Mouschovias et al. models, if B is along the line of
sight, the ratio of jB core j/jB ref j must be large enough to perceive a
larger Zeeman effect toward the core. The quantity jB ref j is the
field in the external envelope. The ``ambient'' field outside the
core, jB ref j, is an important parameter in the Mouschovias et al.
and Myers­Goodman models. Unfortunately, it has not been de­
tected in the Taurus region, and we adopt 14.5 #G, reflecting our
upper limits. For large clumps destined to form massive stars,
TURNER & HEILES
394 Vol. 162

Mouschovias models (Ciolek 1995) predict a modest increase
(factor of #2) in jBj from its envelope value of 35 #G to its
central core value of 67 #G. An estimated core value of 30 #G
should have been detected, although such a small variation along
the line of sight is expected to be difficult to detect even for C 4 H,
and certainly for OH.
We illustrate the behavior of Shu­type models by using the
summary from Galli & Shu (1993a) of one of the cases con­
sidered by Lizano & Shu (1989), applied to low­mass star for­
mation. At dimensionless time t ¼ 0, this model starts as a
somewhat flattened molecular core, of mass 7
M# , and central den­
sity #1(4) cm #3 . After 1(6) yr (t ¼ 0:21), the AD of neutrals has
increased the central density to #3(4) cm #3 . After another 1(5) yr
(from t ¼ 0:21 to 0.23) the central density has increased so much
that the fluid velocity of contraction along field lines begins to
exceed the isothermal sound speed. The quasi­static approxima­
tion breaks down beyond this point, and further evolution of the
core requires use of an MHD code. The transition to dynamical
behavior arises from the evolution of the differential mass­to­
flux ratio dM /d#. As time progresses, AD increases dM /d#
along the central flux tube until at T ¼ 0:23, dM /d# exceeds the
critical value for which a conducting cloud can be stably sup­
ported by embedded magnetic fields against its self­gravitation.
The results of the Myers­Goodman model are summarized in
their Figure 5, which plots the timescale for ambipolar diffusion
versus clump radius for the magnetic equilibrium model (eq. [17]),
assuming the magnetic field scale length equals the radius R. The
plot is for jBj ¼ 5, 30, and 100 #G.Theminima for these fields lie,
respectively, at t AD ¼ 1 ; 10 8 , 2 ; 10 7 , and 5 ; 10 6 yr and at
clump radii 10, 0.1, and 0.05 pc. For the smaller fields, this model
appears to overestimate t AD for clumps appropriate to low­mass
star­formation. Except for very high jBj values, the t AD scales
appear too long to be consistent with the Galactic star formation
rate. The choice of jBj value must be made solely on the basis of
observation.
5.2. Timescales, etc.
Timescales for ambipolar diffusion models have far­reaching
importance: Given that the Galactic molecular clouds are self­
gravitating, then if the total mass of molecular gas, 2 ; 10 9
M# ,
formed stars at the free­fall rate characteristic of the mean den­
sity of #1000 cm #3 of a typical molecular cloud or clump, the
Galactic rate of formation would be 30
M# yr #1 ( Zuckerman &
Palmer 1974), but it is observed to be 3
M# yr #1 . The factor of 10
discrepancy can be explained if the timescale for the quasi­static
ambipolar collapse is of order 10 times that of free fall.
In the Mouschovias et al. models, the total timescale from
clump to fully developed core is t AD ¼ t A /½t ni /t A #, plus the time­
scale for free­fall collapse, producing a total time t R ¼ t A ½1 × ##,
where # # (t A /t ni )# # 10t A is a good approximation for massive
star­forming clumps in general. Here t ni is the collision time
between neutral and ionized particles. The value of # varies from
#5 to 22 among published models, depending on various as­
sumptions about t ni . For an observed ionization fraction X ¼
2(#8) a semianalytical model of Mouschovias (1990) suitable
for TMC­1 yields t AD ¼ 4 ; 10 5 yr, t A ¼ 2:6 ; 10 5 yr. The ob­
served core density of <#1(5) cm #3 implies that free fall has not
onset, in agreement with observed line widths. In a later im­
proved model Ciolek (1995) shows that the central density rises
from 10 3 to 10 9 cm #3 as t R rises from 1 to 17:4t A . For three mod­
els differing in the details of interactions among ions, neutrals
and grains, t R assumes values of 11.8, 16.9, and 17.4 times t ff .
Model core densities start at 2:6 ; 10 3 , reach #10 6 at the start
of dynamic collapse and attain 2:6 ; 10 9 cm #3 at the end of
free fall.
We note that the total collapse time in these models, 1:3 ;
10 7 --1:9 ; 10 7 yr, is within the range needed to explain the rate of
star formation in the Galaxy, although nearly all clumps would
have to be on the verge of forming stars. That is, the AD time­
scale, when compared with the time between formation of the
core and formation of a star, is just the probability of observing the
object in the phase of interest. Satisfying the Galactic star for­
mation rate implies a high probability for a source chosen at ran­
dom to be in an AD phase at an observing time chosen at random.
From the observational viewpoint, the quantities of interest
are (1) the overall field strength (detectability); (2) the difference
in jBj values in the dynamically infalling central region, and in
the envelope; (3) the presence of high core densities; and (4) the
presence of infall or perturbed velocities. It is not clear that such
high densities would be easily detected using just one rotational
transition of C 4 H. However, results from the GBT and 140 foot
telescopes (x 1) will show different intensities if the core angular
size does not exceed the 140 foot beam size. This relation is
satisfied for the model size, which is chosen to equal the core
size. Thus, the lack of an observed intensity difference is incon­
sistent with the predicted core densities. Clearly, many more
Zeeman observations of individual objects are needed to define
the questions and refine the models. Because the signatures of AD
are difficult to recognize, L1544 becomes important as the only
source with a detected (OH) Zeeman effect, which also shows
indications of infall (Ciolek & Basu 2000).
Some 12 objects have been searched for the OHZeeman effect
in low­mass star­forming regions, with one ( L1544) being de­
tected. Given an evolutionary sequence of t AD followed by t ff ,
with t AD # t A , on evolutionary arguments alone we should expect
to see exactly 1 object if the 12 searched objects are of random
ages. This prediction is met.
For the Shu et al. models, once they have established a sem­
ianalytical perturbative solution to the contraction of a nonmag­
netic clump modeled as a singular isothermal sphere, the full
effects of the field can be obtained by solving the MHD equa­
tions that include AD, in terms of the similarity variable x #
r/c st , where c s is the sound speed, (0.35 km s #1 ), t is the time
elapsed since initiation of dynamical collapse, and r is the radial
distance from the center of the protostar. A controversial step is
taken here, whereby a numerical solution of the strongly non­
linear B field of the ``inner'' problem is ``patched'' onto the
semianalytical solution of the intermediate problem by requiring
continuity of all physical variables at an arbitrary interface. Thus,
it is not clear that the time axis is correct. Notwithstanding, a
single model is worked out in detail (Galli & Shu 1993b), with an
ambient B ¼ 30 #G, density 4:3 ; 10 3 cm #3 , and B ref ¼ 15 #G.
For a suitably small value of magnetic to thermal pressure ratio
one gets t(precore) ¼ 1:7 ; 10 5 yr, t(dynamic) ¼ 2:8 ; 10 5 yr,
velocity(outer) ¼ 0:7 km s #1 , velocity(inner) ¼ 2:0 km s #1 . If
the optical depth is moderate, the predicted profile width is much
larger than observed. The small lifetime of the core phase is
probably underestimated, but if not, makes it improbable that a
single object would be observed.
The Myers­Goodman model predicts clump densities (2 ;
10 4 cm #3 ), radii (0.14 pc), and line velocity widths (0.19 km s #1 )
for their low­mass star formation case with jBj ¼ 30 #G. These
predictions are in excellent agreement with representative values
of upper limits observed in an ensemble of #120 objects but are
not consistent with our upper limits of 14.5 #G measured in
TMC­1 for C 4 H, and similar limits for OH.
C 4 H ZEEMAN EFFECT IN TMC­1 395
No. 2, 2006

5.3. Other Observations of B Fields in Taurus
Until recently, polarization observations of Taurus and other
complexes have been limited to bright stars at optical wave­
lengths ( Messinger et al. 1997). The polarization mechanism is
the alignment of spinning grains with respect to the ambient B
field. Only linear polarization can be measured, which provides a
map of the field only in the plane of the sky. Line­of­sight mea­
sures are derived via the Faraday rotation of numbers of small
ionized continuum sources with typical sizes of 2.0 pc in di­
ameter, seen associated with the edges of molecular clouds
( Wolleben & Reich 2004). These objects, none of which are
within 5 # of TMC­1, act as Faraday screens. Minima in the polar­
ization are located at the boundary of the molecular cloud.
Analysis includes the effects of background sources (depolar­
ization) and of sources in the foreground of the screen. For a
measured Faraday rotation of #29 rad m #2 , together with a value
of 0.8 cm #3 for the thermal electron density n e , we find a line­of­
sight field of 27 #G. Further observational constraints from H#
measures limit n e to <0.8 cm #3 , and it is concluded that the
regular B field strength parallel to the line of sight exceeds 20 #G
to account for the intrinsic rotation measure ( RM). We may
refine this estimate somewhat. For an extinction of #3 mag, at
one extreme we may assume all electrons come from ionized C,
yielding a thermal electron density of 0.2 cm #3 . The opposite
extreme is when all H atoms as well as the C atoms are ionized,
producing an electron density of 0.8 cm #3 . Chemical models are
used to obtain the C­to­H ratio as a function of depth into the
molecular cloud. With an adopted beam filling factor of #1, the
two extremes imply a range of 15--77 #G for the B field intensity.
It is important to recognize that the TMC­1 clump is completely
dissimilar in its physical properties to the Faraday screen objects.
The latter have E(B # V ) # 0:5 mag, so assuming the usual gas­
to­dust ratio, N (H i × 2H 2 )/E(B # V ) ¼ 5:8 ; 10 21 cm #2 mag #1
and L ¼ 2:0 pc, we find n # 470 cm #3 . By contrast, the TMC­1
clump has n # 10 4 cm #3 , L ¼ 0:4 pc, and extinction immeasur­
ably large. TMC­1 is also located some 5N5 distant from the map
center of the Faraday screen objects. Unfortunately, the lack of the
sign for B los prevents a comparison with the TMC­1 CP and RM
results.
6. NEW THEORIES OF STAR FORMATION
The ``standard'' theory of low­mass star formation (xx 4.2 and
4.3) entails ambipolar diffusion to control quasi­static conden­
sation of dense cores out of magnetically subcritical clumps and
inside­out core collapse leading to star formation. In both, the
formation of the required initial states is unexplained; and in
both, the collapse timescale to form stars is too slow by a factor
of #10 to explain the Galactic SF rate. These deficiencies seem
fundamental and have led to alternative approaches, as follows.
6.1. Turbulence­controlled SF
Mac Low & Klessen (2004) have recently reviewed com­
prehensively the role of turbulence in controlling SF, with B
fields playing a minor role, if any. The importance of this ap­
proach is underscored by our failure to detect a B field in TMC­1
using the C 4 H Zeeman effect as well as that of OH earlier. TMC­1
is among the most quiescent clouds known. Because of the turbu­
lent nature of supersonic motions in molecular clouds, it appears
that dense structures such as filaments and clumps are formed by
shocks in a turbulent flow. The most characteristic consequence of
turbulent fragmentation is that dense postshock gas traces a gas
component with a smaller velocity dispersion than lower density
gas, since shocks correspond to regions of converging flows,
where the kinetic energy of the turbulent motion is dissipated.
Such a relation between the rms velocity centroid and the inte­
grated column density has been found in Perseus, Rosette, and
Taurus clouds (Padoan et al. 2001).
The basis of the new approach is the maintenance of super­
sonic turbulence, which decays in thought to be a few t ff , so there may be continuing energy input
into the clouds, which may come from the same compressive
motions suggested to form the clouds. Molecular clouds do ap­
pear to be driven from large scales, and numerical simulations
suggest a driving scale of #100 pc. Early numerical simulations,
recently confirmed in three dimensions, suggest two conclu­
sions: (1) in the case of global collapse, even in the presence of
B fields, gravitational collapse with freely decaying turbulent
velocity fields does not itself generate sufficient turbulence to
slow further collapse significantly; and (2) in the case of local
collapse in globally stable regions, numerical models show that
continuously driven turbulent support does not prevent collapse
even when the turbulent velocity field carries enough energy to
counterbalance gravitational contraction on global scales.
Supersonic turbulence can completely prevent collapse only
when it can support not only the average density, but also the
shocked, high­density regions. The length scale and strength of
energy injection into the system determine the structure of the
turbulent flow and hence the locations where stars likely form.
Large­scale driving leads to large coherent shock structures. Lo­
cal collapse occurs predominantly in these filaments and layers
of shocked gas. Increasing the driving scale produces larger and
more massive structures that can become gravitationally unsta­
ble, so the SF efficiency increases. Reducing the turbulent ki­
netic energy means that more and larger volumes exceed the
Jeans criterion for gravitational instability. The more massive the
unstable region, the more stars it will form. Dense clusters or
associations of stars build up, either in the complete absence of
energy input, or when small­scale turbulence is too weak to
support large volumes, or when large­scale turbulence sweeps
up large masses of gas that collapse. In all of these cases, SF is
highly efficient and proceeds on a free­fall timescale.
6.2. The New Theory
The key new point is that supersonic turbulence produces
strong density fluctuations, sweeping gas up from large regions
into dense sheets and filaments, even in the presence of magnetic
fields. Supersonic turbulence decays quickly, but so long as it is
maintained by input of energy from some driver, it can support
regions against gravitational collapse. This has a cost: the same
turbulent flows that support the region produce enhancements in
which the Jeans mass drops as M J # # 0:5 , and the magnetic crit­
ical mass above which B fields can no longer support against
that collapse decreases even faster, as M cr # # #2 . For local col­
lapse to actually result in star formation, Jeans­unstable, shock­
generated, density fluctuations must collapse to sufficiently high
density on timescales shorter than the typical time interval be­
tween two successive shock passages. Only then can they de­
couple from the ambient flow and survive subsequent shock
interaction. The shorter the time between shock passages, the
less likely these fluctuations are to survive. Hence, the timescale
and efficiency of protostellar core formation depend strongly on
the wavelength and strength of the driving source, and the accre­
tion histories of individual protostars are strongly time varying.
Global support by supersonic turbulence thus tends to produce
local collapse and low­rate star formation, as seen in low­mass
TURNER & HEILES
396 Vol. 162

SF regions characteristic of the disks of spiral galaxies. Con­
versely, lack of turbulence support results in regions that col­
lapse freely. In gasdynamic simulations, freely collapsing gas
forms a web of density enhancements in which SF can proceed
efficiently, as seen in regions of massive SF. The regulation of the
SF rate then occurs not just at the scale of individual star­forming
through AD balancing magnetostatic support, but rather at all
scales via the dynamic processes that determine whether regions
of gas become unstable to prompt gravitational collapse.
6.3. Applications of the Turbulent Picture
The problem is to identify the role played by turbulence, jBj,
and AD. How many combinations are relevant to TMC­1 and
might explain our failure to detect the B field? Do we need AD
anymore? Or can we adjust it for a given turbulent model to fine­
tune a model for any source?
6.3.1. Dissipation in Compressible MHD Turbulence
(Stone et al. 1998)
Before the rise of the standard models, turbulent motions in
molecular clouds were thought to act as a ``turbulent pressure'' to
support a cloud against self­gravity. Without this, there was a dis­
crepancy between estimated cloud collapse times (#3 ; 10 6 yr,
after which it was presumed that most of the cloud would convert
to stars (violating limits on the Galactic SF rate), and estimated
cloud lifetimes #3 ; 10 7 yr. Moreover, supersonic, sub­Alfve ’nic
turbulence was thought to persist for more than a cloud flow
crossing time over the cloud, because B fields provide a cushion
that reduced dissipation rates. In particular, molecular cloud tur­
bulence would be primarily in Alfve ’nic motions because, for
linear­amplitude waves, no compressions are involved. Numeri­
cally, these notions are now questioned. The present authors
(Stone et al. 1998) evaluate the dissipation rate of supersonic, sub­
Alfve ’nic, three­dimensional, compressible, isothermal turbulence
models with uniform mass­to­flux ratio with no AD and no self­
gravity. The new features are those of driven turbulence and tur­
bulence decaying from saturated initial conditions. The equations
of compressible, ideal MHD are integrated in a cubic box con­
taining a plasma of uniform density threaded by an initially uni­
form B field, and sound speed constant in space and time.
Turbulence is driven by adding random velocity perturbations 9v,
each of which is an independent realization of a Gaussian random
field with prescribed power spectrum. The input parameters are —
E,
the time derivative of the total fluctuation energy (kinetic and
magnetic field); and #, related to the magnetic field as
B
j j ¼ 1:4 #G # #1=2 (T=10) 1=2 N H 2
Ï ÷=100
½ # 1=2 :
For L ¼ 2 pc, N (H 2 ) ¼ 10 3 cm #3 , T ¼ 10 K, then the sound
speed is c s # 0:2 km s #1 , implying the sound crossing time is
#10Myr and driving power —
E ¼ 0:4
L# . Thus, the magnetic fields
are as follows:
1. ``Strong'' field: # ¼ 0:01, B ¼ 44 #G, v A # 2:0 km s #1 ,
t a ' 1 Myr.
2. ``Moderate'' field: # ¼ 0:1, B ¼ 14 #G, v A # 0:6 km s #1 ,
t a ' 3 Myr.
3. ``Weak'' field: # ¼ 1:0, B ¼ 4:4 #G, v A # 0:2 km s #1 ,
t a ' 10 Myr.
These values have not been chosen with any particular source in
mind. For TMC­1 we have L ¼ 0:4 pc, a factor of 5 lower than
the above model. Values of # and jBj are unaffected, but the rate
of injection of turbulence must increase by the same factor of 5.
In driven turbulent models, the total fluctuation energy
(kinetic+magnetic) rises with time to a saturated value. The
amplitude of E increases monotonically with jBj, hence dissi­
pation decreases as jBj increases. There is a tendency toward
equipartition of kinetic and magnetic energy in all of the models
(the magnetic­to­kinetic ratio varies from 0.3 to 0.6). In the weak
field case, significant amplification of jBj is produced by the tur­
bulence, so that after saturation, the energy in the fluctuations
in jBj is 10 times larger than that in the mean field. In weakly
magnetized models jBj lines are thoroughly tangled; in strong
field models they are relatively well ordered.
In decaying models, starting at saturated conditions, E decays
as a power law in time. Defining the cloud flow crossing time as
the time taken for 50% of the initial (kinetic) energy to be lost,
one finds that decay times are in the range 0.4--0.8 of the crossing
time, comparable to the range of steady state dissipation times.
In summary, compressible MHD turbulence dissipates rapidly---
in less than one flow crossing time at the energy­containing scale---
regardless of jBj and the details of the initial or ongoing energy
input characteristics. The models show that the dissipation time is
always >0.25 Myr when the largest scale is the same size as the
cloud. The energy input rate needed to maintain the turbulence is
not large, #0.4
L# in mechanical power. Over a wide range of
parameters turbulent jBj is found to assume values comfortably
within the ``weak'' field domain relevant to the observed TMC­1
clump.
6.3.2. Supersonic Turbulence Only ( Hartmann et al. 2001)
Formation of molecular clouds and stars is rapid. Observa­
tions of SF regions and young gas­free stellar associations in­
dicate that most nearby clouds form stars only over a short time
span ( less than lateral crossing times of molecular clouds) before
dispersal. 3
Large­scale flows (>100 pc) are involved in forming molec­
ular clouds and require several Myr to accumulate enough mass
to form stars. When column densities reach threshold values
where clouds become molecular, numerical simulations show
that rapid dissipation of turbulence allows gravitational collapse
to occur. Pressure balance constraints and increasing mass­to­
flux ratios imply that clouds are generally both self­gravitating
and magnetically supercritical, hence AD reduction of magnetic
flux is not an essential feature of SF and is in fact omitted. The
picture outlined here is at variance with the standard model of
low­mass SF. The standard picture assumes that stars form in
dark clouds because it is only in such regions that the ionization
decreases to a level where AD can proceed. Instead, dark clouds
are the sites of SF because they represent a stage of evolution
closer to stellar densities than is the atomic phase, partly as a
result of turbulent energy dissipation. Further, the molecular
clouds form in a magnetically supercritical state, and hence ion­
ization effects on AD are not important. Thus, the low rate of
Galactic SF is not the result of slowing by AD of magnetic fields
through dense gas; instead, it is the result of a low efficiency in
converting gas into stars ( Hartmann 1998, p. 33). The rapid
dispersal of clouds may be the main factor determining the ef­
ficiency. It is easier to see how turbulent flows powered by stellar
energy input would form, shape, and disrupt clouds rather than
trying to maintain a quasi­equilibrium that would allow clouds to
survive for long periods; this is why clouds do not have long
lifetimes.
3 Not everyone agrees with this point. There is evidence, in the form of
abundances of some particular elements that are destroyed as stars evolve, that
some T Tauri stars in the Taurus region have ages exceeding 10 7 yr.
C 4 H ZEEMAN EFFECT IN TMC­1 397
No. 2, 2006

In summary, the observational requirement for rapid SF can be
satisfied because the column densities necessary for formation of
molecular gas are comparable to those required for self­gravity
to become important in the solar neighborhood, and because
collapse times can be #1 Myr under these conditions. The pic­
ture requires that B fields, while having important dynamical
effects, do not substantially slow or prevent collapse in at least
some parts of molecular clouds. Both general theoretical argu­
ments and numerical simulations favor this conclusion.
6.3.3. The Turbulent Shock Origin of Proto­Stellar
Cores ( Padoan et al. 2001)
Because of the turbulent nature of supersonic motions in mo­
lecular clouds, dense structures such as filaments and clumps
appear to be formed by shocks in a turbulent flow. The most
characteristic result of turbulent fragmentation is that dense post­
shock gas traces a gas component with a smaller velocity disper­
sion than lower density gas, since shocks correspond to regions
of converging flows, where the kinetic energy of the turbulence
is dissipated. Using synthetic maps of spectra of molecular tran­
sitions, computed from the results of numerical simulations of
supersonic turbulence, one finds that the dependence of velocity
dispersion on gas density generates an observable relation be­
tween the rms velocity centroid and the integrated intensity (col­
umn density), which is indeed found in the observational data.
The theoretical models (maps of synthetic 13 CO spectra) show
excellent agreement with 13 CO maps observed toward the Taurus
and Perseus complexes. Maps of synthetic spectra contain much
information about the structure, kinematics, and thermal proper­
ties of the clouds. The new method of analysis utilizes the fact that
dense gas traces a gas component with a smaller velocity disper­
sion than lower density gas. This occurs because the gas density is
enhanced in regions where the large­scale turbulent flow con­
verges (compresses) and the kinetic energy of the flow is dissi­
pated by shocks. If local compressions are instead due to local
instabilities (e.g., gravitational instability, or gravitational instabil­
ity mediated by AD) and the large­scale motions are only the con­
sequences of local instabilities (as it should be in a self­consistent
picture), then gas density increases with the flow velocity disper­
sion, contrary to the observational evidence presented here.
In summary, this work presents a new statistical method for
analyzing spectral­line maps, which probes directly a very gen­
eral property of supersonic turbulence, that is, the fact that dense
gas traces a gas component with a smaller velocity dispersion
than lower density gas. This property arises because the gas den­
sity is enhanced in regions where the large­scale turbulent flow
converges (compressions) and the kinetic energy of the turbulent
flow is dissipated by shocks. Detailed comparisons between ob­
servational data and models support the idea of the turbulent
origin of the structure and kinematics of molecular clouds. How­
ever, while the Taurus cloud complexes appear to satisfy this
picture of supersonic turbulent fragmentation, in other aspects it
appears much colder and more quiescent.
6.3.4. Magnetically Regulated SF in Turbulent Clouds
( Nakamura & Li 2004)
The combined effects of supersonic turbulence, strong B fields,
and AD on cloud evolution leading to SF are studied numerically.
In clouds that are initially magnetically subcritical, with (e.g.) a
mass­to­flux ratio of 1.2, supersonic turbulence can speed up SF,
through enhanced AD in shocks. The speed­up overcomes one of
the major objections to the ``standard'' scenario of low­mass SF
involving AD, since the diffusion timescale at the average density
of a molecular cloud is typically longer than the cloud life time. At
the same time, the strong B field can prevent the large­scale su­
personic turbulence from converting most of the cloud mass into
stars in one (short) turbulence crossing time and thus alleviate the
high efficiency problem associated with the turbulence controlled
picture for low­mass SF. Thus, relatively rapid but inefficient SF
results from supersonic collisions of somewhat subcritical gas in
strongly magnetized, turbulent clouds.
Clouds that are initially magnetically supercritical, with (e.g.)
a mass­to­flux ratio of 0.8, also form a network of filaments, as
in the subcritical case, but the supercritical filaments are thinner
due to weaker magnetic resistance to shock compression. These
filaments are so supercritical that they can break up gravitation­
ally into a string of dense cores along their length.
In summary, the low SF efficiency is made possible by the
strong B field, which prevents the global cloud collapse in one
turbulence crossing time. The supersonic turbulence, conversely,
speeds up AD in localized regions through shock compression.
The turbulent flow is highly compressive.
7. CONCLUSIONS
The existence of a magnetic field in the Taurus­Auriga mo­
lecular cloud complex seems established by optical polarimetry
(e.g., Messinger et al. 1997) and possibly by Faraday rotation
measures ( Wolleben & Reich 2004). The latter authors believe
the observed Faraday screen is associated with the molecular
gas, although this is not yet generally accepted. If correct, then
the intrinsic rotation measure of ##29 rad m #2 requires an ex­
cessive value for the thermal electron density or an excessive reg­
ular magnetic field component parallel to the line of sight, when
compared to average Galactic values. The total power maps show
no enhanced emission toward the Faraday screen, which con­
strains n e to <0.8 cm #3 . With this limit, a regular magnetic field
strength jBj not less than 20 #G along the line of sight would be
needed to explain the intrinsic rotation measure. The RM results
appear consistent with upper limits of 9 and 13 #G for the OH
Zeeman results (Crutcher 1999) toward two clumps in TMC­1
(including TMC­1 CP), and our present upper limit of 14:5 #
14 #G using C 4 H. The optical results suggest complexity in the
Taurus­Auriga region, data from two sight lines indicating two
distinct cloud regions with differing average magnetic field direc­
tions, one being a sheet of continuous material extending over
much of the Taurus dark cloud, together with an embedded dense
core (or cores) in the lines of sight to the stars. We proceed by
assuming that a combination of errors could result in a lower
estimate for the field derived from the RM results, and/or a higher
value for the upper limit derived from the C 4 H Zeeman effect.
Thus, we require a reduction by a factor of #2 for the field derived
from the RM results.
Our C 4 H Zeeman result is not likely to be affected by beam
dilution. Thus, our value, jBj ¼ j14:5j # 14 #G, along with the
OH values of jBj < 14 #G (with hint of a signal at 14 #G;
Crutcher et al. 1996) and upper limits of 9 and 13 #G for TMC­1
CP and another clump, comprise the total molecular Zeeman
data known to date. This paucity is not surprising since the mo­
lecular Zeeman effect has been detected in no other species in
TMC­1 CP nor anywhere else in the Taurus­Ophiuchus region
with the exception of OH in L1544. Based on its reasonably
large Lande g factor and a chemistry that favors high densities,
we argued that C 4 H is superior to OH as a magnetometer for
detecting weak fields in low­mass cold dense cores. We also
based our search on model predictions of high core densities and
modestly amplified magnetic fields.
In the following we compare observed parameters of TMC­1
CP with Mouschovias­type models, which firmly link predicted
TURNER & HEILES
398 Vol. 162

parameters with a time line well defined in terms of the source
evolution. To assess the probability that TMC­1 CP is in a suit­
able evolutionary phase to observe a magnetic field, we divide
the 10 7 yr timescale for the magnetic evolution into 10 equal time
intervals for the quasi­static AD phase, and one additional in­
terval for the final stage of dynamic collapse. In a typical model,
the central density increases from #10 3 to 10 6 cm #3 from the
first to the 9th intervals inclusively, from 10 5 to 10 6 cm #3 for the
10th interval, and from 10 6 to 10 9 cm #3 over the final 11th in­
terval. A 10 6 cm #3 core would probably be marginally detectable
if a more sensitive map of the C 4 H line intensities were available.
The 10 9 cm #3 core would be only somewhat easier to detect than
the 10 6 cm #3 core. Therefore, we are only able to see the core
during the last 2 of the 11 intervals, or 1:8 ; 10 6 yr of the
magnetic­field epoch, after which a star rapidly forms. Similar
arguments apply using velocities and accelerations as a diagnos­
tic. These quantities also show prominent effects only in the last
two time intervals of the models, so we would expect to detect
only 2 sources out of every 11 in any of these parameters.
7.1. A Summary of Possibilities for Quasi­static Models
1. If all parameters (e.g., alignments) are optimized, and a
suitable B field exists, then as a result of evolution there is only a
0.18 chance of detecting both an enhanced jBj and a dense core
per source observed. One source was seen out of 13 searched
( L1544).
2. No B field, even oriented along the line of sight, is observ­
able toward TMC­1 CP at the present time because the source
has not evolved yet to the onset of dynamic collapse, which
follows the quasi­static contraction under AD, a phase that oc­
curs over #10 7 yr. The core density is predicted to be as high as
#10 6 cm #3 when free fall begins, but from other observations
the actual core density is #10 5 cm #3 . Further, if there were any
free fall, the line widths would be increased, but this is not seen.
A density of 10 5 cm #3 is higher than the canonical 10 3 cm #3 of a
typical clump, so that most likely TMC­1 CP is in the earlier
stages of the AD phase, and should be subcritical with an am­
bient field of #15 #G. Even if we view the field along the line of
sight, at an early phase of the AD the intensity jBj will be am­
plified over B ref by only a small factor, #1.4, so that jBj # B ref <
B ref and we should expect to see only B ref , and no sign of C 4 H
line intensity enhancement.
3. As seen by RM and optical recombination lines, a B field
exists 5N5 away from TMC­1 CP, but the field might be absent
toward TMC­1 CP, or be parallel to the plane of the sky. Since
both RM and the Zeeman effect are sensitive to a line­of­sight
field, either the field exists at the RM position and not at TMC­1
CP, or the field curves sharply so as to veer away from the line­
of­sight field at the RM position. The Taurus region is highly
complex, as shown also by the polarization work of Messinger
et al. (1997).
4. There is a B field in and around TMC­1 CP, and in fact
throughout much of the Taurus­Ophiuchus complex as observed
by optical polarimetry, but that field is directed nearly in the
plane of the sky, preventing detection of the Zeeman effect. The
simplest explanation for a field seen along the sight line toward
the RM objects but not toward TMC­1 CP 5N5 away is that at
least the parallel field varies strongly with position. The optical
maps do show a ``hole'' toward TMC­1 CP in the perpendicular
field. This hole probably exists for the parallel field toward
TMC­1 CP as well, because otherwise the line of sight to the
TMC­1 CP core would present a high­density source if evolved
to the dynamic phase (with an 0.18 probability). A core density
of 10 7 --10 9 cm #3 should be visible via its effect on the intensity
of the C 4 H J ¼ 1 0 lines as seen with the 140 foot and GBT
telescopes. Line broadening but not a velocity shift may be
expected.
5. There is no field at all in the vicinity of TMC­1 CP strong
enough to establish a subcritical initial condition. Thus, the clump
must be in the free­fall stage, but this is unlikely since no line
broadening or velocity shifts are seen.
6. In early Mouschovias models ( Mouschovias 1987b) the B
fields were treated as one­dimensional. Together with AD time­
scales, which were much smaller than in the outer regions where
the density is high, the result was that gravity was found to be too
weak, leading to some B field leakage out of the core and into a
region where the ionization is high enough to prevent AD. In this
case one should not expect strong fields or maximum flux in
cores, but rather in regions surrounding cores. The C 4 H source
distribution on the sky, which samples the densest regions, will
be less well coupled to the telescope beam. However, as multi­
dimensional models of the field came into use, the calculated
gravitational fields became stronger and a redistribution of the
mass in the inner flux tubes resulted. Now there is no longer any
loss of central core flux that samples the densest regions that are
now well­coupled to the telescope.
7.2. The Controlled Turbulence Models and TMC­1 CP
TMC­1 CP is a clump­sized object located at the southeast
extreme of the TMC­1 elongated cloud. Its mass may be roughly
determined from the results of Pratap et al. (1997), who derive a
density of 2 ; 10 4 cm #3 and a mass of 0.48
M# . The cloud ap­
pears static judging from the line widths given in Table 1. The to­
tal observed line width 9v 2
obs ¼ 9v 2
T ×9v 2
NT ¼ 0:375, and since
9v T ¼ 0:32 km s #1 , we find 9vNT ¼ 0:195 km s #1 , i.e., sub­
sonic. At the present epoch the turbulence has negligible effect
on TMC­1 CP but could have dominated the formation of the
clump. Indeed, the present filamentary character of the TMC­1
object, of dimensions 11 ; 2 0 , suggests formation via super­
turbulent compression, which features filamentary structures. If
so, the turbulence has by now dissipated almost completely from
all of the half­dozen or so clumps embedded in the overall TMC­1
object. Nomagnetic fields are detected anywhere in TMC­1, there
is insufficient energy in the turbulence to have any effect on the
clumps, and the narrow line widths indicate no infall or other ma­
terial motions within the clumps. In short, their future depends on
small but immeasurable departures from equilibrium within the
clumps---infall may be proceeding, or the onset of a dynamical
collapse phase. We cannot classify the clumps (e.g., sub­ or su­
percritical) in their present state, however. Unfortunately, there­
fore, they cannot act as test objects for the effects of controlled
turbulence models.
An important question for all models is how well the model
mass­to­flux ratio agrees with the value actually observed. It is
customary to measure the mass­to­flux ratio in units of the quan­
tity
c# /
## t
p (g) (Crutcher 1999) or (2#G 1/2 ) #1 (Shu et al. 1999) in
terms of which M
/c# ¼ 1 at the critical point. For the only low­
mass SF object with a detected molecular Zeeman effect (OH in
L1544), Crutcher estimated the observed quantity M
/# B from
the observed column depth N(H 2 ) and magnetic field jBj, and
obtained a measured value of #8. Two projection effects (see
below) need to be applied, which reduces M
/# B to 2. This is still
firmly in the supercritical domain. We have used the Shu et al.
form to calculate M
/# B for C 4 H in TMC­1 CP. This ratio is given
by the projected surface mass per unit area mN(H 2 ), where
m ¼ 2:4mH (mass of an H atom), divided by the projected flux
per unit area, B k , in units of (2#G 1/2 ). The two projection effects
mentioned above are, first, that jBj is on average twice as large as
C 4 H ZEEMAN EFFECT IN TMC­1 399
No. 2, 2006

a component of it, B k , seen along some random direction; and
second, if molecular clouds are flattened, then measuring the
column density along some random (slant) path will typically
give a value mN (H 2 )) twice the true surface density. Using the
result B k ¼ 14:5 # 14:0 #G, we calculate the mass­to­flux ratio
as
c#=B ¼ 0:25mN (H 2 )=B k 2#G 1=2
# # #1 ;
where 0.25 includes the two projection effects. For B k # 14:5 #G
(our 1 ¼ # detection), we find c#/B k > 1:05 for our C 4 H result,
in good agreement with the value >0.892 for the average lower
limit over 12 low­mass sources not detected in OH (Shu et al.
1999). The detected OH Zeeman pattern in L1544 gives c#/B #
2:0, somewhat higher than the average value for the undetected
sources (Crutcher & Troland 2000).
What if anything can we conclude? Aside from our upper limit
of 14.5 #G, we have only the following facts to start with: (1)
there is a B field in the Taurus cloud; and (2) there is negligible
supersonic turbulence in the Taurus cloud. But fact 1 proves
nothing because the field might be patchy and hence too weak
toward our observed line of sight, and it might also lie orthog­
onal to the line of sight. Fact 2 describes the present condition
but suitable turbulence may have existed within the last Myr (the
crossing time for the TMC­1 CP clump). So in effect nothing is
known of the initial conditions and history of the supersonic
turbulence fundamental to the so­called dynamic class of mod­
els. The failure to observe supersonic turbulence leaves some
hope for the static models if they can overcome the subcritical
requirement, and in the Shu­type models, explain the origin of
the singular isothermal initial state. At present, the best prospect
for understanding low­mass SF and the role of B fields appears
to be that of Nakamura & Li (2004), who combine the main
attributes of supersonic turbulence and of AD. The need for ob­
servational results is paramount, especially in the case of low­
mass SF.
We thank Glenn Ciolek and Zhi­Yun Li for many very helpful
discussions. This work was supported in part by NSF grant AST
04­06987.
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