Part 3

DISTRIBUTION ACCORDING TO SIZE OF THE NEBULAE IN THREE FIELDS

=========+===================+============== | FIELD | DIAMETER +-------------------+ WOLF’S FIELD | I | III | VII | IN PERSEUS ---------+-----+------+------+-------------- 5″ | | | | 3 10 | 7 | 38 | 5 | 34 15 | 27 | 67 | 12 | 52 20 | 25 | 42 | 18 | 26 25 | 6 | 19 | 10 | 6 30 | 5 | 9 | 4 | 2 35 | 3 | 2 | 4 | 1 40 | 4 | 5 | 4 | 2 45 | 1 | 3 | 3 | 50 | 1 | | 3 | 60 | | | 3 | 90 | | | 1 | ---------+-----+------+------+--------------

Exposures of two hours gave all of the nebulae recorded. Five-hour exposures brightened the images somewhat, but revealed no new ones. The conclusion would seem to be that the limits of the clusters had been reached. This, however, is uncertain, for the longer exposures gave relatively few stars which were not on the plates of shorter exposures. As a matter of fact, the 24-inch reflector at this altitude seems to have a maximum working efficiency at slightly over two hours. Save for very exceptionally clear and steady skies, the longer exposures add much labor for negligibly greater results. The diameters of the faintest stars near the center of the field, with the full aperture, fine sky, and an hour’s exposure, which are of about magnitude 17.5, are about 2.0″. Longer exposures are apparently subject to change of focus, differential refraction, and other disturbances which tend to increase the size of the images unduly, and hence to spread the total light over a larger area. The result is that the value of p in the reciprocity equation Itᵖ = iTᵖ does not remain constant throughout the exposure, but varies, beyond a certain value of T, depending on the adjustments of the telescope, the position of the field, and the condition of the sky. However, the effect should be more noticeable on the stars than on nebulae which present surfaces.

I have plotted Wolf’s lists of nebulae, Nos. 3-14, in the same manner, converting estimates of size into seconds of arc according to his table. These lists were made from plates taken with the 16-inch (41 cm) Bruce camera, of focal length 203 cm, of the Heidelberg Observatory. The curves (Fig. 1) take the same form, save that for most the maximum frequency is for diameters between 20″ and 25″. One of his lists was made from plates taken with the 30-inch reflector at Königstuhl. It is of a field in Perseus, α = 3ᵸ12ᵐ, δ = +41° 6′. Diameters of the 124 Wolf nebulae and five others were measured from plates taken with the Yerkes 24-inch reflector. This plot (Fig. 2) gives a maximum for diameters around 15″, and the longer focus of Wolf’s 30-inch apparently does not add to the number of small nebulae distinguishable on the plates made with the shorter telescope.

TABLE V

WOLF’S NEBEL LISTEN NOS. 3-14

=====+================================================= | Diameter List +-----+-----+------+------+------+-------+-------- | 4″ | 6″ | 15″ | 25″ | 60″ | 200″ | >200″ -----+-----+-----+------+------+------+-------+-------- 3 | 205 | 322 | 291 | 280 | 195 | 36 | 6 4 | | 1 | 69 | 153 | 19 | 6 | 1 5 | | 6 | 99 | 106 | 23 | 1 | 3 6 | | 14 | 114 | 72 | 2 | 2 | 1 7 | | 9 | 103 | 156 | 35 | 4 | 12 8 | | 80 | 372 | 243 | 34 | 3 | 9 | | 48 | 174 | 160 | 19 | | 10 | | 3 | 31 | 26 | 2 | | 11 | | | 13 | 60 | 19 | 1 | 12 | | 14 | 61 | 162 | 27 | 10 | 13 | | | 20 | 61 | 26 | 3 | 14 | | 3 | 160 | 296 | 43 | 1 | 1 -----+-----+-----+------+------+------+-------+--------

The evidence, while far from conclusive, appears to indicate the existence of actual clusters of these small nebulae in the sky. If this is true, it is natural to suppose them physically connected, as is the case in star-clusters. It is not possible to form a conception of this state of affairs until some idea of their distance is acquired. Suppose them to be extra-sidereal and perhaps we see clusters of galaxies; suppose them within our system, their nature becomes a mystery.

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

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

α ( v )² (5) ρ = 1.4 × --- × 10¹²(------) in suns per cu. LY, or β ( dα )

α ( v )² = 2.8 × --- × 10⁻⁶(------) in atmospheres β ( dα )

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.