### 2. MASSES FROM INTERNAL ROTATIONS OF NEBULAE

It has apparently been taken for granted by some astronomers (2) that from observations on the internal rotations good values for the masses MN of nebulae could be derived. Values of the order of MN = 109 M up to MN = 4 × 1010 M were obtained in this way, (2) where M = 2 × 1033 gr is the mass of the sun. A closer scrutiny of the behavior of suitably chosen mechanical models of stellar systems, unfortunately, soon reveals the fact that the masses of such systems, for a given distribution of average angular velocities throughout the system, are highly indeterminate, and vice versa. This conclusion may, for instance, be derived from the consideration of two limiting models of a nebula as a mechanical system.

This model consists of a heavy and small nucleus of mass M0 around which a given number, n, of stars of average mass M8 << M0/n describe planetary orbits. The mutual gravitational interactions between these outlying stars are negligible, and the system may therefore be said to have an internal "viscosity" equal to zero. It is obvious that under these circumstances we may build up models that satisfy almost any specifications in regard to total mass, total luminosity, and internal distribution of luminosity as well as distribution of the average angular velocities. We may, for instance, distribute our n stars over the six-dimensional manifold of all possible planetary orbits (including the epochs or phases) in such fashion that the average angular velocity of the resulting system S0 is zero in every point. Since all these orbits are essentially non-interacting, we may reverse the sense of rotation (direction of stellar motion) in an arbitrary number of these orbits. In this way a system S of specified distribution of average angular velocities may be constructed whose remaining characteristics, such as the mass, the luminosity, and the external shape, are identical with those of S0. Thus, the observed angular velocities in themselves give no clue regarding the mass of the system.

2 E. Hubble, The Realm of Nebulae (New Haven: Yale University Press, 1936), p. 179; also Ap. J., 69, 150, 1929. Back.