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Fig. 80-Sections and projections of homogeneous oblate spheroidal shells.

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Fig. 81-A homogeneous oblate spheroidal shell with a peripheral equatorial ring of absorbing matter of such density as to transmit but 0.2 of the light in the equatorial plane. The intensity curves along the major and minor axes, and the projected images, are shown for inclinations of the equatorial plane to the line of sight of 0°, 15°, 30°, 45°, and 60° respectively.

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Fig. 82-A homogeneous, oblate spheroidal shell, the thickness of the equatorial zone being but a small fraction of that at the poles. The intensity curves and the projected images are shown for inclinations of the equatorial plane to the line of sight of 0°, 30°, 45°, 60°, and 90°, respectively.

secured by Mr. Wright pointing to variations in the condition or in the distribution within the planetary of the different kinds of matter composing it. From the photographic results alone it is evident that we have to do with structures of extraordinary complexity, and the puzzles of the mechanics of the planetary nebulae seem three-fold as bewildering when the attempt is made to fit in as well the almost inexplicable pecularities shown by the spectrographic results. The aberrant wisps and striae and other minor formal irregularities in such complex structures as the Ring Nebula in Lyra, N.G.C. 7009, N.G.C. 7026, and others, would seem to defy all attempts to analyze the details, whatever hypothesis may be adopted regarding the general form of the structure as a whole. Excluding from consideration such minor and apparently haphazard irregularities, and investigating solely the larger details of the forms assumed, is it possible to postulate any general form or forms, which shall be mechanically plausible, and to which the planetaries, or a considerable proportion of the planetaries, will more or less closely conform? In details, each planetary may well be a law unto itself, but even a casual inspection of the forms of the seventy-nine planetaries of which illustrations have been given would seem to indicate certain general similarities of structure.

Of the forms thus far considered, it has been pointed out that the ring hypothesis fails to account for the lack of highly elliptical or edgewise forms; the hypothesis of an ellipsoidal shell of uniform thickness is inadequate in that it does not explain the relatively very faint central regions nor the frequently very marked obliteration at the ends of the major axes; the hypothesis of an ellipsoidal shell surrounded by an equatorial ring of occulting matter must be rejected because no planetaries are observed which are asymmetrical with regard to the dark lane along the major axis.

Two modifications of the ellipsoidal shell hypothesis may be considered. In the first case, let us assume a homogeneous ellipsoidal or spheroidal shell of varying thickness. Possibly as a result of the combination of radiation pressure or gas pressure effects with that of gravitation, we may postulate an actual thinning out of the equatorial zones of the shell because of the higher velocity in and near to the equatorial plane, with a corresponding accretion of nebular matter at the two poles. Figure 82 illustrates this form, showing a homogeneous oblate spheroidal shell, the thickness of the equatorial zone being but a small fraction of the thickness at the poles. The intensity curves are shown for the major and minor axes of the projected forms, and the resulting projected images as well, for inclinations of the equatorial plane to the line of sight of 0°, 30°, 45°, 60°, and 90°, respectively.

These projections show the following general agreements with observed planetary forms: (a) The intensity of the portions near the major axis may be small compared with that at the ends of the minor axis, thus obviating one difficulty in the hypothesis of an ellipsoidal shell of uniform thickness. Compare 6818 (57).

(b) From any position in space (except when the equatorial plane makes an angle of nearly 90° with the line of sight, noted below) the projected image will be symmetrical with regard to a major axis, and will be fainter along this axis.

. (c) Projections at angles of 0° to 30° represent very closely the large group of planetaries classified under group (E) above, showing fainter along the major axis. Compare 6058 (26), 6818 (57), and 7354 (77).

(d) Projections at angles from 45° to 60° correspond approximately to such forms as the Dumb-bell Nebula, 6853 (60), and also to the apparently truncated forms classified in group (D); compare 40 (1).

(e) Projections where the equatorial plane makes an angle of nearly 90° with the line of sight will show a nearly uniform disk, fading out at the edges, similar to those classified in group (F); compare II 3568 (25).

(f) Spectrographic observations made with the slit along the fainter major axis of planetaries of this general shape have frequently shown evidence of rotation. So also in

spectra taken with the slit placed on the lobes of 7026 (71) parallel to the major axis. Against this hypothesis it may be urged that it gives no explanation of the true ring forms with relatively very faint central portions; compare 6369 (31).

An alternative modification of the ellipsoidal shell hypothesis is illustrated in Figure 83. This may be best described as a homogeneous truncated spheroidal shell; the vertical line in the cross-section represents the trace of the equatorial plane. As drawn, the ring-shell is truncated, with no matter in the polar zones, but the same general effects would be produced on the assumption that the thickness of the matter in the polar zones is but a small fraction of that in the equatorial zone. It is perhaps simpler to regard the form as a wide ring in rotation about the central star, under the combined action of gravitation and radiation pressure. The polar zones have fallen in, or are in the process of falling in, toward the central star, lacking the centrifugal force which is assumed to keep the equatorial, rapidly moving zones in position. The intensity curves along the principal axes, and the projected forms, are shown for inclinations of the equatorial plane to the line of sight of 90°, 70°, 45°, 30°, and 0°, respectively.

As in the previous case, we may sum up the advantages and difficulties of this hypothesis as follows:

(a) When the equatorial plane makes an angle of 90° to 70° with the line of sight, this form will exhibit true ring effects; the central portions of such rings will generally be very faint, or entirely blank. The rings will generally be approximately circular, but may be somewhat elliptical. Compare 2610 (21), and 6369 (31).

(b) At inclinations of from 80° to 60° the form of the resulting projected image will depend mainly upon the ratio of the width of the shell to its outer equatorial diameter. When the ring is narrower than the outer equatorial diameter, as shown in Figure 83, the projection will be a ring narrower and brighter at the ends of the major axis, and wider and fainter at the ends of the minor axis, a form which does not occur. When the width of the shell is greater than the equatorial diameter (i.e., a prolate spheroidal shell, somewhat "barrel-shaped") the projection will assume the commonly observed form of a ring wider and fainter at the ends of the major axis, and narrower and brighter at the ends of the minor axis. The rings will be somewhat elliptical, but no high degree of ellipticity will be observed. The central regions will be quite faint relatively to the intensity observed in the ring itself. Compare 6720 (46).

(c) At inclinations from 45° to 0° the projections show considerable resemblance to the large class of planetaries tabulated in groups (D) and (E). Compare 3587 (23), 6058 (26), 6818 (57), 6853 (60), and 7354 (77).

(d) Such a ring-shell, if wider than its outer equatorial diameter, will show for all inclinations of the equatorial plane to the line of sight (except for those in the neighborhood of 90°) projected images which will be symmetrical with regard to a fainter major axis region, and brightest at the ends of the minor axis.

Objections:

(a) This construction shows no disk forms like those classified in Group (F); compare II 3568 (25). It would be necessary to assume solid spheroidal or ellipsoidal masses of nebular matter to account for such forms.

(b) The spectrographic evidence offers most support to the form of Figure 82, although some differences have been found in radial velocities at opposite ends of the apparent minor axis in certain planetaries.

It would seem entirely possible that both the forms illustrated in figures 82 and 83 may exist; taken together, they would explain practically all the forms observed.

It has already been mentioned that spectrographic observations made with the slit placed along or parallel to the major axis of certain planetaries have indicated rotation. It is very unfortunate that it is practically impossible, because of the intrinsic faintness of the nebular matter, to make a series of radial velocity determinations at various points near the ends of the minor and major axes of such a planetary as the Dumb-bell Nebula; such observations would be of great value in determining the real mechanism of the planetaries.

A problem of a far more difficult nature is presented by the remarkable bowed, doubled, and distorted spectral lines observed by Messrs. Campbell and Moore in N.G.C. 3242, 7662, and other planetaries. Absorption effects, or manifestations of the Zeeman or Stark effects, have been suggested to explain these doubled lines, but no theory thus far advanced is entirely satisfactory. It seems equally impossible to postulate any conceivable system of particles in orbital motion which shall show radial velocity effects of this nature. It may be of interest, however, to illustrate one empirical hypothesis which will give spectral lines approximating to the forms actually observed, employing only the known facts of spectral displacements due to motion in the line of sight.

Assume a relatively wide spheroidal ring similar to that illustrated in Figure 83, thickest in the equatorial plane of its rotation about the central star. It would seem that the matter at the poles of such a structure must be falling in toward the central star, lacking the centrifugal force which is assumed to keep the equatorial zones in position. The total number of matter thus descending toward the central star from the polar zones need not be very great; the process might even be quasi-continuous, fed by accretions from the edges of the equatorial ring. Such a structure is shown in cross-section in the first square in Figure 84, where the dotted lines represent the matter falling in from the polar regions; these lines have been drawn as straight for convenience, but might be much more complicated, under the combined action of gravitation, radiation pressure, and the residual motions of rotation if fed in from the edges of the ring, than the parabolas or highly elongated ellipses to be expected in a simple case. The cross-section only is shown for an inclination of the equatorial plane to the line of sight of 0° (the projected form for this inclination is illustrated in the last form in Figure 83). Following this are shown the projected images for inclinations of the equatorial plane to the line of sight of 30°, 60°, 70°, 80°, and 90°, respectively. In each square is shown also the form assumed by the spectral line when the slit is placed along each principal axis of the projected image, and at 45° to these axes: the upper part of the ring in the form for inclination 0° is supposed to be coming toward the observer, and the line of sight in each case is taken perpendicular to the plane of the paper. The details of the form of the resulting spectral lines have been purposely greatly exaggerated. Instead of finely divided or atomic matter falling inward toward the central star, matter being ejected from the polar regions of the central star (compare the polar rays of the solar corona) would give the same effects. A small degree of absorption in this finely divided matter would make that "bow" of the doubled line the fainter which was produced by the radially moving matter on the farther side of the planetary from us. In the cases thus far observed, the red component of the doubled line appears the stronger, thus supporting the hypothesis that descending rather than ascending matter is involved in the phenomenon, if, indeed, the phenomenon is due to moving luminous matter. In objection to this admittedly purely empirical hypothesis, it may be argued that the bowed lines should show very wide at their central portions, due to the combination of the low radial speeds of the outermost particles with the greatly increased velocities as these particles approach the nucleus. The objection is a valid one, but not insuperable. It may be removed either by supposing that the falling particles, fed in from the edges of the ring, form a hollow cone, instead of a solid cone, as drawn in the figure, or by the not

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