Fig. 2.—Distribution of size of nebulae in Wolf’s Perseus
and three Yerkes fields
The question of nebular distances is of first importance, for it is in terms of this quantity that the various dimensions may be expressed. The dark nebulosities, by their very nature, and the great diffuse clouds, some obviously connected with even naked-eye stars, may safely be considered as galactic, and this view is in accord with their low radial velocities with reference to our system.
The planetaries have repeatedly been measured for proper motion, with negligible results. Taking a value of 40 km/sec. for the average radial velocity, and an assumed lower limit of 0.02″ for the average annual proper motion, a tentative lower limit for the average distance of the largest, and hence probably the nearest, is found to be about 2000 light-years. There is thus no reason on this ground for placing them outside our system, especially in view of their decidedly systematic galactic distribution.
Rotation of these nebulae, as detected by the spectroscope, furnishes a means of relating mass and average density with distance. Assume an axis perpendicular to the line of sight, the mass as concentrated in the nucleus and the individual distant particles as rotating in equilibrium; let α be the radius, P the period of rotation, M the mass, and suppose the rotation to be circular. Then
α³/P²M = C, a constant.
Let the unit of distance be the light-year (LY); of time, the year; of mass, that of the sun (S), then C, as computed from the earth-sun system, is about 4 × 10⁻¹⁵.
Let
- 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 dαv²
- (2) ρ = 1.4 × 10⁻⁶ (v/dα)²
- (3) P = 9.15 dα/v