=======+==========+=========+=================+==================
_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 _d_α_v_²
α ( _v_ )²
(5) ρ = 1.4 × --- × 10¹²(------) in suns per cu. LY, or
β ( _d_α )
α ( _v_ )²
= 2.8 × --- × 10⁻⁶(------) in atmospheres
β ( _d_α )
α
(6) _P_ = 9.15 _d_ ---
_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.
The observations point to very large values for the rotational
components of velocity, although the necessarily small scale of the
instruments employed in their study renders the measuring difficult.
Pease has determined the velocity of rotation for N.G.C. 4594 with
some degree of accuracy. At 120″ from the nucleus it amounts to 300 km
and varies linearly with the distance outward. V. M. Slipher reports
that for the Andromeda nebula the angular rotation is fastest near the
nucleus, and that this type of rotation promises to be the more common.
Assume a typical spiral 400″ in diameter, with the ratio of the axes
of figure as β/α = 0.1, and with rotation perpendicular to the line of
sight at a velocity of, say, 200 km at the periphery. These figures
are apparently not very different from the average of the two dozen
brighter spirals. Table VII gives the dimensions in terms of distance.
The velocity of escape is 280 km/sec.
At the lower limit of average distance of spirals the typical nebulae
would be fifteen light-years in diameter, forty-five million times
as massive as our sun, and 3 × 10¹³ as dense as our atmosphere. On
the other hand, if the typical spiral nebula is placed at a suitable
distance, its dimensions assume the same order of magnitude as those of
our own stellar system.
The conception of our galaxy set forth by Eddington in his book
_Stellar Movements and the Structure of the Universe_ is that the Milky
Way forms a ring around a central, slightly flattened cluster. This
ring is supposed to rotate in equilibrium so that the stars may remain
concentrated in the configurations they now form. Assuming the ratio
of the two radii as one to five, and using Eddington’s figures of 2000
parsecs for the distance of the Milky Way, and 10⁹ suns as the mass of
the inner cluster, the period and density may be computed and compared
with those of the typical nebula placed at a distance such as will make
the diameter the same as that of our system. The results are given in
Table VIII.