Let us now consider the angular momentum carried off by the solar wind. Angular momentum loss is a crucially important topic in astrophysics, since only by losing angular momentum can large, diffuse objects, such as interstellar gas clouds, collapse under the influence of gravity to produce small, compact objects, such as stars and proto-stars. Magnetic fields generally play a crucial role in angular momentum loss. This is certainly the case for the solar wind, where the solar magnetic field enforces co-rotation with the Sun out to the Alfvén radius, . Thus, the angular momentum carried away by a particle of mass is , rather than . The angular momentum loss time-scale is, therefore, shorter than the mass loss time-scale by a factor , making the angular momentum loss time-scale comparable to the solar lifetime. It is clear that magnetized stellar winds represent a very important vehicle for angular momentum loss in the Universe. Let us investigate angular momentum loss via stellar winds in more detail.

Under the assumption of spherical symmetry and steady flow, the azimuthal
momentum evolution equation for the solar wind, taking into account the
influence of the interplanetary magnetic field, is written

(770) |

where is the angular momentum per unit mass carried off by the solar wind. In the presence of an azimuthal wind velocity, the magnetic field and velocity components are related by an expression similar to Eq. (761):

The fundamental physics assumption underlying the above expression is the absence of an electric field in the frame of reference co-rotating with the Sun. Using Eq. (772) to eliminate from Eq. (771), we obtain (in the ecliptic plane, where )

where

(774) |

Note that the angular momentum carried off by the solar wind is indeed equivalent to that which would be carried off were coronal plasma to co-rotate with the Sun out to the Alfvén radius, and subsequently outflow at constant angular velocity. Of course, the solar wind does not actually rotate rigidly with the Sun in the region : much of the angular momentum in this region is carried in the form of electromagnetic stresses.

It is easily demonstrated that the quantity
is a constant,
and can, therefore, be evaluated at to give

(777) |

at large distances from the Sun. Recall, from Sect. 5.7, that if the coronal plasma were to simply co-rotate with the Sun out to , and experience no torque beyond this radius, then we would expect

(779) |

The analysis presented above was first incorporated into a quantitative
coronal expansion model by Weber and Davis.^{} The model of Weber and Davis is
very complicated. For instance, the solar wind is required to flow smoothly
through no less than *three* critical points. These are associated
with the sound speed (as in Parker's original model), the radial Alfvén
speed,
, (as described above), and the total
Alfvén speed,
.
Nevertheless, the simplified analysis
outlined above captures most of the essential features of the outflow.
For instance, Fig. 20 shows a comparison between the large- asymptotic
form for the azimuthal flow velocity predicted above [see Eq. (778)] and
that calculated by Weber and Davis, showing the close agreement between
the two.