6.6. Other Theoretical Ideas

Magnetic theories already published are based on different assumptions and link various magnetic field directions to compressible turbulences, to energy balance, to fragmentation, self-gravity and star formation, and to other physical processes (chaotic behaviour, fractals, etc).

6.6.1. Magnetism - dominant or not ?

Scalo (1987) proposed that turbulences can explain many scaling relations, such as the velocity dispersion in clumps being proportional to the inverse square root of the gas density. In this picture, turbulence (not magnetic fields) is the dominant mechanism in clouds. The magnetic field may be important for the support and line broadening in some clouds, but it seems likely that such a dominance is restricted.

Theoretical studies have hinted that a small magnetically supported cloudlet should have an oblate core (Lizano and Shu, 1989), whereas some cloudlets show a prolate core, requiring another physical process, such as a rotationally supported cloud (Bonnell & Bastien, 1991).

In contrast, Myers and Goodman (1990) proposed that magnetic fields are dominant in cloudlets, requiring triple equipartition between magnetic energy, gravitational energy, and kinetic energy. Subtracting the thermal part of the line width (proportional to the cloudlet temperature), the remainder (nonthermal part) is attributed entirely to magnetic origins.

In this picture, hydromagnetic waves are expected to be significant. Myers and Khersonsky (1995) proposed that MHD waves, chaotic motions, and clumps are probably more pervasive in clouds than would be expected from cosmic-ray ionization alone, implying a very long time scale for ambipolar diffusion ~ 10⁸ years. This suggests that ambipolar diffusion alone may be too slow a process for low-mass star formation in nearby dark clouds. They also pointed out that the assumption of dominant magnetic support in cores may not be real, since for low mass cores (i) the thermal motions inside the cloud appear observationally energetically sufficient to support the cores against gravity (Fuller & Myers, 1993) and (ii) the required field strength for magnetic support (~ 40 µGauss in TMC-1C) is 10 times larger than the 1 rms observed OH Zeeman upper limit (~ 4 µGauss; Goodman 1992).

Starting from the virial equation for an axisymmetric oblate magnetic cloud of mass M and semimajor axis R and semiminor axis Z embedded in an external medium of pressure P₀ and external magnetic field B₀ with the radius R₀ being the radius far from the cloud of the magnetic tube penetrating the cloud with flux freezing (B₀ R₀² = BR²), one gets (Equ. 1 in Nakano, 1998):

where I is the generalized moment of inertia of the cloud, t is the time, k is Boltzmann constant, T is the mean cloud temperature, µ is the mean molecular weight of the gas, m_H is the hydrogen atom mass, V_turb is the mean turbulent velocity, B is the mean cloud magnetic field, a and b are dimensionless coeficients of order unity. Solutions of this equation have been worked out for different boundary conditions and geometrical assumptions. The same equation can be used for cores, clumps, or cloudlets, provided that the external parameters B₀, R₀, P₀ are those of the cloud or interclump medium.

Nakano (1998) points out that there are no cloud cores which have been confirmed to be magnetically subcritical, defined as having enough or more magnetic field than needed to prevent cloud contraction due to gravity. Also, published lists of magnetic fields in molecular clouds and cores contain mainly upper limits to the field strengths, and these observational upper limits are below the values needed to have magnetic support of the clumps and clouds (e.g., Bertoldi & McKee 1992; Nakano 1998).

Crutcher et al. (1992) made OH Zeeman observations towards 12 cloud cores, and obtained a definite detection in only 1 out of 12; the other 11 gave only upper limits, all several times smaller than required for magnetic control; even the one detection (the B1 core) out of 12 gave a magnetic field value below that required for magnetic control (Heiles et al. 1993; Nakano 1998). Crutcher et al. (1996) also obtained observational upper limits for CN Zeeman observations in 2 cloud cores, still way below that required for magnetic control.

Here we can estimate the amount of magnetic support against the gravitational energy in some clumps. For total magnetic support against gravity, one derives the equation (Equ. 25 in Myers & Goodman, 1988; Equ. 1 in Myers & Goodman, 1990):

The clumps in the cloud M17-SW have been well studied observationally, and from Extreme-IR continuum emission at 800 µ m, Vallée and Bastien (1996) estimated a clump radius 0.17 pc and a clump density 3 × 10⁵ cm^-3, while from ¹²CO line data Bergin et al. (1994) estimated a line width 4 km/s, giving a nonthermal line width ~ 3.7 km/s. Entering these data in the above equation for magnetic control gives B_control = 1.2 mG.

We can also estimate the expected magnetic to be observed from observations of magnetic fields in many other cloudlets and clouds, having obtained the statistical relation (e.g., Fig. 1 in Vallée 1997):

giving again for the clumps in M17-SW a value B_{stat obs} 0.3 mGauss.

Preliminary Zeeman data near clump P4 in M17-SW by Brogan et al. (1998) gave a magnetic field B_Zeeman 0.2 mG.

Thus in the clumps of M17-SW the B_{stat obs} and B_Zeeman give an average of ~ 0.3 mG, while the assumption of magnetic control gives B_control ~ 1.2 mG, hence we have B_{stat obs} / B_control 25% and the statistical and observed magnetic energy density is roughly [0.3 / 1.2]² 6% of the large magnetic energy density needed when assuming magnetic control against gravity.

Hence in addition to the weak magnetic energy density, there is a need for another non-magnetic process to provide the 94% of energy density needed against gravity. The most likely candidates would be turbulences, clump collisions, shocks from travelling clumps, stellar outflows near embedded protostars, convective motions due to IR photon heating near embedded protostars, thermal instabilities due to time-dependent shielding variations near embedded protostars, etc.

6.6.2. Fractals and Waves

Can magnetic field structures be represented with 'fractals' ? There have been some hints that a fractal nature of molecular cloudlets is real, i.e., the number of clouds N of size L is proportional to the size elevated to some fractal power D, like N(L) ~ L^-D where D is near 2.7 (Henriksen 1991; Henriksen 1986). Then it follows that the gas density n varies as n(L) ~ L^-E where E is near 1.0 In a similar vein, it is not yet known if the magnetic field structures is showing a fractal behaviour with size, like B(L) ~ L^-F .

Are there many 'wavy' magnetic field structures in the interstellar medium ? Magnetic field structures are known to display a uniform and a random component. In some departures from this view, wavy transverse vibrations of an otherwise uniform component have been observed (Shuter & Dickman 1990; Moneti et al. 1984). The degree of this magnetic wavyness is not known, but it could be a strong indicator of the existence of Alfvén-type magnetic disturbances, with potential consequences for subsequent star formation (e.g., magnetic braking by an uniform component, see Fig. 10 in Warren-Smith et al. 1987). In turn, these magnetic disturbances would predict increased CO line widths (e.g., Blitz 1991).

Martin et al. (1997) proposed theoretically that small molecular clouds or cloudlets are supported along the magnetic field lines by an Alfvén wave pressure force. The origin of these Alfvén waves would be the orbital motions of clumps (clump size ~ 0.1 pc) within a small cloud (cloud size ~ 1 pc). For a gas density ~ 1000 cm^-3 and a cloud magnetic field ~ 100 µGauss, their model requires a minimum wavelength _Alfven ^. _min (= 4.5 km / s × 0.3 Myr 1.4 pc, after their Equ. 43) and smaller than a maximum equal to the cloud thickness ( 5 pc, after their Equ. 44). Also, their wave model requires the magnetic wave pressure to be strong, about 30 times the isothermal gas pressure, hence B / B₀ 0.6, after their Equ. 69 and 70. Thus their wavelength range is quite limited, and their strong Alfvén wave would imply a nearly random overall magnetic field B₀(  PA B / B₀ = 30°), much larger than what observations of small clouds show. Their model excludes non-magnetic supports, such as (i) clump collisions, (ii) internal cloud hydrodynamic (micro) turbulence, (iii) cloud rotation, (iv) outflows from embedded protostars, (v) shocks from traveling clumps or outflows, (vi) convective motions due to gas heating by IR photons from embedded protostars, and (vii) thermal instabilities due to time-dependent shielding variations of photons from embedded protostars, etc.

Nakano (1998) studied the possibility of having MHD waves travelling inside clouds or clumps. He found that in most cases the waves' dissipation time t_wave is significantly smaller than the free-fall time, typically t_wave is 10% of the free-fall time, yet the whole clump or cloud may last much longer than the free-fall time (because of various supports against gravity). In addition, any such MHD wave in weak B field could be supersonic and super-Alvénic so shock dissipation must then occur in a timescale t_shock much smaller than t_wave .