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.
TABLE VI
| d | Diameter | Mass | Period | Density |
|---|---|---|---|---|
| 10 LY | 0.001 LY | 1.2S | 1.5×10² year | 1.8×10⁻⁷ρ |
| 10² | 0.01 | 12. | 1.5×10³ | 1.8×10⁻⁹ |
| 10³ | 0.1 | 120. | 1.5×10⁴ | 1.8×10⁻¹¹ |
| 10⁴ | 1.0 | 1200. | 1.5×10⁵ | 1.8×10⁻¹³ |
For an assumed typical planetary nebula, 20″ in diameter, rotating with a velocity of 6 km at 10″ from the perpendicular axis, [Table VI] has been constructed from formulae (1)-(3), expressing the order of magnitude of dimensions in terms of distance.
The velocity of escape would be about 8.4 km per second, whatever the distance.
Spectroscopic rotation of spirals furnishes an analogous set of formulae, and here the inclination of the axis may be roughly determined from the ratio of the two diameters of the nebulae. Let β be the semi-minor axis, then the formulae will be:
(4) M = 3.4 × 10⁻⁴ dαv²
| (5) ρ = 1.4 × | α | × 10¹² |  | v |  | ² | in suns per cu. LY, or |
| β | dα | ||||||
| = 2.8 × | α | × 10⁻⁶ | | v | | ² | in atmospheres |
| β | dα | ||||||
| (6) P = | 9.15 | dα | |
| v |
The spirals form a continuous series from the great nebula of Andromeda to the limit of resolution, the smaller ones being much the more numerous. Considering them to be scattered at random as regards distance and size, some conception may be formed of their dimensions from the data at hand. The average radial velocity of those so far observed is about 400 km, while the proper motion is negligible. Putting the annual proper motion at 0.05″, the lower limit of the average distance is found to be about 7500 light-years. If they are within our sidereal system, then, as they are most numerous in the direction of its minor axis, the dimensions of our system must be much greater than is commonly supposed.