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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 re