Magnetostatic Configurations

The freezing of field lines during cloud contraction means that the investigation of magnetic equilibria can proceed in a manner similar to that for rotating, nonmagnetized structures. Thus, the internal distribution of magnetic flux is inherited from an earlier, more rarefied environment, just as the run of specific angular momentum was previously. Suppose we now picture the cloud condensing from a state where the magnetic field Ba is everywhere straight and parallel. For convenience, we will employ a coordinate system whose origin is at the cloud center and whose z-axis lies in this direction. During contraction, matter can slide freely along field lines, until it is halted by the buildup of thermal pressure. Gas moving in the orthogonal direction, toward the axis, tugs on the field and is retarded by pressure (both thermal and magnetic), as well as by magnetic tension, i. e., bending of the field lines. Thus, within the final equilibrium state, there is an additional outward force roughly similar to that arising from rotation (Figure 9.11).

In more detail, however, the magnetic-rotational analogy breaks down in a significant way. The centrifugal force in a rotating, equilibrium cloud points exactly in the ro-direction, implying that the specific angular momentum is constant along cylinders. If the analogy were exact, the equilibrium magnetic flux would also be constant within cylinders, which is not the case. As we have noted, field lines are curved by the inward tug of the gas. The new equation for force balance is

where we again specify an isothermal equation of state. If the cloud does not rotate while contracting, the equilibrium B-vector is poloidal, i. e., it lies within the ro-z plane. Ampere's law then implies that the current j is toroidal, pointing in the ^-direction. It follows that the magnetic force per unit volume, j x B/c, must be a poloidal vector with both ro- and z-components, unlike the centrifugal force. In summary, the fact that B can remain distorted in the equilibrium state precludes a magnetic analogue to the Poincare-Wavre theorem and renders the actual calculation of equilibria more difficult.

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