α = the angular radius in seconds of arc
_d_ = the distance in light-years
ρ = the density in terms of earth’s atmosphere at sea-level
_v_ = the linear velocity of rotation in km/sec.
Then the following relations hold:
(1) _M_ = 3.4 × 10⁻⁴ _d_α_v_²
(2) ρ = 1.4 × 10⁻⁶ (_v_/_d_α)²
(3) _P_ = 9.15 _d_α/_v_
The velocity of escape for the nebula is proportional to α/ρ and
hence to _v_. This follows from the assumption that the particles are
rotating in equilibrium, and therefore the factor of proportionality is
the ratio between parabolic and circular velocity, that is, 1.4, and is
independent of the distance. The value of _v_ for those nebulae so far
observed is small, ranging from 5 to 10 km. Hence, if the assumptions
held only approximately, the velocity of escape would be small and
of the same order as that for the earth. Since these nebulae are
composed of the lightest gases, it follows that at any save very low
temperatures the molecules would escape at a very rapid rate. Certainly
the nebulae would dissipate if the temperatures were of the order of
that of our own atmosphere.