**B. Model of a nebula whose internal viscosity is very great**

This model is built up of stars, dust, and gases in such fashion that the gravitational interactions between the various components, as well as direct impacts, influence the path of every component mass in a radical way. Many changes in energy and momentum of every component mass will take place during time intervals that are short compared with the time an unperturbed mass of the same initial velocity would consume to traverse the system.

Conditions of motion in this model are analogous to the conditions
of motion of elementary particles in a star. This model of a
nebula, therefore, will rotate like a solid body, regardless of what its
total mass and the distribution of mass over different regions of the
system may be. The conclusion which has sometimes been put
forward, ^{(2)} that constant
angular velocity necessarily implies uniform
distribution of mass, is obviously erroneous. Furthermore, it is again
seen that the rate of rotation of a stellar system has no very direct
bearing on its total mass.

Good mechanical models of actual nebulae may presumably be
constructed by combining the distinctive features of the two limiting
cases described in the preceding sections. Such a combined model
will possess a central, highly viscous core whose relative dimensions
are not negligible but are comparable with the extension of the
whole system. If the outlying, and among themselves little interacting,
components of the nebula had no connection with the central
core, we might, at a given instant, observe average angular velocities
which, as a
function of the distance *r* from the center of
rotation, would be given by

(1) |

where *r*_{0} is the radius of the core. For *r* >
*r*_{0}, the angular velocity
(*r*) would be
essentially arbitrary. In reality, however, the
viscosity will not drop abruptly to zero at *r* =
*r*_{0}. From an inspection
of the distribution of the outlying masses in many nebulae it would
seem that these masses at some previous time must have formed part
of the central core. They may have been ejected from this core
because they acquired high kinetic energy through many close
encounters, or they may be the result of a partial disruption of
the core by tidal actions caused by encounters with other nebulae
(Jeans). At the moment these outlying masses have passed the
boundary of the core, their average tangential velocities must have
been of the order *r*_{0}
_{0}.
Assuming that these masses in regions
*r* > *r*_{0}
are essentially subject only to the gravitational attraction of a
central spherical core and that the interactions among themselves may
be neglected (the internal viscosity of the outlying system is equal
to zero), the average angular velocity
(*r*) outside
the core will be approximately given as

(2) |

This relation simply expresses the fact that a mass *m* which, on
being ejected from the core with a tangential velocity
*v*_{t} relative to
this core, describes an orbit whose angular momentum,

(3) |

is a constant. If the average for many particles leaving the core is zero, the average angular velocity = of all the particles in a given point is obtained from

(4) |

which is the same as (2). Unfortunately, relation (2) is superficially
very similar to the relation obeyed by the angular velocities
_{c}
of a system of circular planetary orbits around a heavy central mass
*M*. For such an orbit we have

(5) |

or

(6) |

where is the
universal gravitational constant. The similarity of
the dependence on *r* in (2) and (6) will in reality become still
greater, since the rate of decrease of
(*r*) with
increasing values of *r* will in
actual nebulae be more gradual than that given by (2), owing to the
fact that the internal viscosity will not vanish abruptly at *r* =
*r*_{0}
but will disappear gradually with increasing *r*. The observed angular
velocities in the outlying regions of nebulae actually show a
dependence on *r* which resembles the relation (6). This relation was,
therefore, sometimes erroneously used for the determination of the
mass *M*. The preceding discussion, however, indicates that the
observed angular velocities may be adequately accounted for on the
basis of the considerations resulting in relation (2) rather than in
relation (6). This again shows clearly that it is not possible to
derive the masses of nebulae from observed rotations without the use of
additional information, since the relation (2) does not contain the
mass *M* at all.