EFFECTS OF ORIENTATION

Fig. 1.—Frequency distribution of apparent magnitudes among nebulae in Holetschek’s list.

Fig. 2.—Relation between luminosity and diameter among nebulae at the beginning of the sequence of types—E0 and E1 nebulae.

The plot is shown inFigure 7, in which the two MagellanicClouds have been included in order to strengthen the bright end of the curve which would otherwise be unduly influenced by the singleobject, M 31. The magnitudes +0.5 and +1.5, which were assigned to the Clouds, are estimates based upon published descriptions.

Fig. 3.—Relation between luminosity and diameter among nebulae at the middle of the sequence of types—E7, Sa, and SBa nebulae.

Fig. 4.—Relation between luminosity and diameter among nebulae at the end of the sequence of types—Sc and SBc nebulae.

The correlation of the data is very closely represented by the formula

This falls between the two regression curves derived from least-square solutions and could be obtained exactly by assigning appropriate weights to the two methods of grouping. The nature of the data is such that a closer agreement can scarcely be expected. No correction to the assumed value of the slope appears to be required. The material extends over a range of 12 mag., and the few cases which have been investigated indicate that the correlation can be extended another 3 mag., to the limit at which nebulae can be classified with certainty on photographs made with the 100-inch reflector. The relation may therefore be considered to hold throughout the entire range of observations.

Fig. 5.—Relation between luminosity and diameter among the irregular nebulae. The Magellanic Clouds are included. N.G.C. 4656 is an exceptional case in that it shows a narrow, greatly elongated image in which absorption effects are very conspicuous; hence the maximum diameter is exceptionally large for its apparent luminosity.

The residuals without regard to sign average 0.87 mag., and there appears to be no systematic effect due either to type or luminosity. The scatter, however, is much greater for the spirals, especially in the later types, than for the elliptical nebulae. The limiting cases are explained by peculiar structural features. The nebulae which fall well above the line usually have bright stellar nuclei, and those which fall lowest are spirals seen edge-on in which belts of absorption are conspicuous.

TABLE VII

* C =mT+ 5logd.

† logd= 0.2 (C—mT);mT= 10.0.

The effect of the orientation is appreciable among the spirals in general. In order to illustrate this feature, they have been divided into three groups consisting of those whose images are round or nearly round, elliptical, and edge-on, or nearly so. The mean values ofmT+ 5 logdwere then computed and compared with the theoretical value, 13.0. The residuals are negative when the nebulae are too bright for their diameters and positive when they are too faint. The results are given inTable IX, where mean residuals are followed by the numbers of nebulae, in parentheses, which are represented by the means.

The numbers of the barred spirals are too limited to inspire confidence in the results, but among the normal spirals there is conclusive evidence that the highly tilted and edge-on nebulae are fainter for a given diameter than those seen in the round. A study of the individual images indicates that the effect is due very largely to dark absorption clouds, which become more conspicuous when the nebulae are highly tilted. These clouds are generally, but not universally, peripheral features. An extensive investigation will be necessary before any residual effect due to absorption by luminous nebulosity can be established with certainty. Even should such exist, it clearly cannot be excessive.

Fig. 6.—Progressive characteristics in the sequence of types. The upper curve represents the progression in total magnitude with type for nebulae having maximum diameters of one minute of arc. The elliptical nebulae and the normal spirals are included as representing the normal sequence, but the barred spirals and the irregular nebulae are omitted. The figures give the number of objects observed in each type. Among the later elliptical nebulae the numbers are so small that means of adjacent types have been plotted. The lower curve represents the progression in diameter along the normal sequence for nebulae of the tenth magnitude.

TABLE VIII

Fig.7.—Relation between luminosity and diameter among extra-galactic nebulae. The nebulae have been reduced to a standard type, S0, which, being the mean of E7, Sa, and SBa, represents a hypothetical transition point between elliptical nebulae and spirals. The Magellanic Clouds have been included in order to strengthen the brighter end of the plot.

The correlations thus far derived are between total luminosities and maximum diameters. In the most general sense, therefore, theyexpress laws of mean surface brightness. The value,K= 5.0, informula (1)indicates that the surface brightness is constant for each separate type. The variations inCindicate a progressive diminution in the surface brightness from class to class throughout the entire sequence. The consistency of the results amply justifies the sequence as a basis of classification, since a progression in physical dimensionsis indicated, which accompanies the progression in structural form. Although the correlations do not necessarily establish any generic relation among the observed classes, they support in a very evident manner the hypothesis that the various stages in the sequence represent different phases of a single fundamental type of astronomical body. Moreover, the quantitative variation inCis consistent with this interpretation, as is apparent from the following considerations.

TABLE IXResiduals inmT+ 5logdas a Function of Orientation

Among the elliptical nebulae it is observed that the nuclei are sharp and distinct and that the color distribution is uniform over the images. This indicates that there is no appreciable absorption, either general or selective, and hence that the luminosity of the projected image represents the total luminosity of the nebula, regardless of the orientation. If the observed classes were pure, that is, if the apparent ellipticities were the actual ellipticities,formula (1)could be written

wherebis the minor diameter in minutes of arc andeis the ellipticity. The termmT+ 5 logbis observed to be constant for a given type. If it were constant for all elliptical nebulae, then the termCe+ 5 log (1 –e)would be constant also. On this assumption,

whereC0is the value ofCfor the pure class E0. Hence

a relation which can be tested by the observations. An analysis of the material indicates that this is actually the case, and hence that among the elliptical nebulae in general, the minor diameter determines the total luminosity, at least to a first approximation.18

The observed values ofCvary with the class, as is seen inTable VIIandFigure 6, but, excepting that for E7, they are too large because of the mixture of later types of nebulae among those of a given observed class. It is possible, however, to calculate the values ofCe–C0for the pure classes and then to make approximate corrections for the observed mixtures on the assumption that the nebulae of any given actual ellipticity are oriented at random. In this manner, mean theoretical values can be compared with the observed values. The comparisons are shown inTable XIIin the formC7–Ce, because E7 is the only observed class that can be considered as pure. The significance of the table will be discussed later.

The following method has been used to determine the relative frequencies with which nebulae of a given actual ellipticity, oriented at random, will be observed as having various apparent ellipticities.

InFigure 8, let the co-ordinate axesOXandOYcoincide with the major and minor axes,aandb, of a meridian section of an ellipsoid of revolution. LetOO′ be the line of sight to the observer, making an angleiwithOX, and letORbe perpendicular toOO′. LetPP′ be atangent to the ellipse, parallel to and at a distance fromOO′. Letx0andy0be the intercepts of the tangent on theX- andY-axis, respectively. The apparent ellipticity is determined bybx, which, for various values of the anglei, ranges frombtoa. The problem is to determine the relative areas on the surface of a sphere whose center isO, within which the radiusOYmust pass in order that the values ofb1, and hence of the apparent ellipticity,e1may fall within certain designated limits. This requires that the angleibe expressed in terms ofb1.

Fig. 8

From the equation of the tangent,PP′,y equals minus x tangent i plus StartRoot a squared tangent squared i plus b squared EndRooty 0 equals StartRoot a squared tangent squared i plus b squared EndRootSinceb 1 equals y 0 cosine ib 1 squared equals a squared sine squared i plus b squared cosine squared i periodLeta= 1, thencosine squared i equals StartFraction 1 minus b 1 squared Over 1 minus b squared EndFraction commawhereb 1 equals 1 minus e 1 comma b equals i minus e period

From these equations, the values ofican be determined for all possible values ofe1. The limits for the observed classes E0 to E7 were chosen midway between the consecutive tenths, E0 rangingfrome= 0 toe= 0.05; E1, frome= 0.05 toe= 0.15; E7, frome= 0.65 toe= 0.75. The relative frequencies of the various observed classes are then proportional to the differences in sinicorresponding to the two limiting values ofe1. These frequencies must be calculated separately for nebulae of different actual ellipticities.

The results are given inTable X, where the actual ellipticities, listed in the first column, are followed across the table by the percentages which, on the assumption of random orientation, will be observed as having the various apparent ellipticities. The bottom row will be seen to show the percentages of apparent ellipticities observed in an assembly of nebulae in which the numbers for each actual ellipticity are equal and all are oriented at random.

TABLE X

From this table and the actual numbers in the observed classes as read from a smoothed curve, the numbers of each actual ellipticity mingled in the observed classes can be determined. For instance, the four nebulae observed as E7 represent 0.187 of the total number of actual E7. The others are distributed among the observed classes E0 to E6 according to the percentages listed inTable X. Six nebulae are observed as E6, but 3.6 of these are actually E7. The remaining 2.4 actual E6 nebulae represent 0.224 of the total number of that actual ellipticity, the others, as before, being scattered among the observed classes E0 to E5.Table XIgives the complete analysis and is similar toTable Xexcept that the percentages in the latter are replaced by the actual numbers indicated by the observations.

Finally, the mean values ofC7–Ceare calculated from the numbers of nebulae in the various columns ofTable XItogether with the values ofC7–Cefor the pure classes as derived fromformula (4). The results are listed in the fourth column ofTable XIIfollowing those for the pure and the observed classes. In determining the observed values, N.G.C. 524 and 3998 are included as E0 and E1, although inTable Ithey are listed as peculiar, because they are obviously much flattened nebulae whose minor axes are close to the line of sight.

TABLE XI

* The totals represent the numbers in the observed classes as read from a smooth curve.

The observed values in general fall between those for the pure classes and those corresponding to random orientation. They are of the same order as the latter, and the discrepancies are perhaps not unaccountably large in view of the nature and the limited extent of the material. There is a systematic difference, however, averaging about 0.2 mag., in the sense that the observed values are too large, and increasing with decreasing ellipticity. One explanation is that the observed classes are purer than is expected on the assumption of random orientation. This view is supported by the relatively small dispersion inC, as may be seen inTable IandFigure 2, among the nebulae of a given class, but it is difficult to account for any such selective effect in the observations. The discrepancies may be largely eliminated by an arbitrary adjustment of the numbers of nebulae with various degrees of actual ellipticity; for instance, the values inthe last column ofTable XII, calculated on the assumption of equal numbers, agree very well with the observed values, although the resulting numbers having the various apparent ellipticities differ slightly from those observed. The observed values, however, can again be accounted for by the inclusion of some flatter nebulae among the classes E6 and E7. Very early Sa or SBa nebulae might easily be mistaken for E nebulae when oriented edge-on, although they would be readily recognized when even slightly tilted. If the numerical results fully represented actual statistical laws, the explanation would be sought in the physical nature of the nebulae. The change from ellipsoidal to lenticular figures, noticeable in the later-type nebulae, would affect the results in the proper direction, as would also a progressive shortening of the polar axis. The discrepancies, however, are second-order effects, and since they may be due to accidental variations from random orientation, a further discussion must await the accumulation of more data.

TABLE XIIDifferential Values ofC

* Read from smooth curve inFig. 6. The small numbers of observed E5 and E6 nebulae justify this procedure. The other values are the means actually observed.

† N.G.C. 524 and 3998 are included as E0 and E1, respectively.

Meanwhile, it is evident that, to a first approximation at least, the polar diameters alone determine the total luminosities of all elliptical nebulae, and the entire series can be represented by the various configurations of an originally globular mass expanding equatorially. A single formula represents the relation, in which the value ofCis that corresponding to the pure type E0. FromTable XII,this is found to be 2.62 mag. less than the value ofC7The latter is observed to be 12.75, hence

If this relation held for the spirals as well, the polar diameters could be calculated from the measured magnitudes. Unfortunately, it has not been possible to measure accurately the polar diameters directly, and hence to test the question, but they have been computed for the mean magnitudes of the Sa, Sb, and Sc nebulae as given inTable III, and the ratios of the axes have been derived by a comparison of these hypothetical values with the means of the measured maximum diameters. The results, 1 to 4.4, 1 to 5.7, and 1 to 7.3, respectively, although of the right order, appear to be somewhat too high. An examination of the photographs indicates values of the order of 1 to 5.5, 1 to 8, and 1 to 10, but the material is meager and may not be representative. The comparison emphasizes, however, the homogeneity and the progressive nature of the entire sequence of nebulae and lends some additional color to the assumption that it represents various aspects of the same fundamental type of system.

From the dynamical point of view, the empirical results are consistent with the general order of events in Jeans’s theory. Thus interpreted, the series is one of expansion, and the scale of types becomes the time scale in the evolutionary history of nebulae. In two respects this scale is not entirely arbitrary. Among the elliptical nebulae the successive types differ by equal increments in the ellipticity or the degree of flattening, and among the spirals the intermediate stage is midway between the two end-stages in the structural features as well as in the luminosity relations.

One other feature of the curves may be discussed from the point of view of Jeans’s theory before returning to the strictly empirical attitude. The close agreement of the diameters for the stages E7 and Sa suggests that the transition from the lenticular nebula to the normal spiral form is not cataclysmic. If the transition were gradual, however, we should expect to observe occasional objects in the very process, but among the thousand or so nebulae whose images have been inspected, not one clear case of a transition form has been detected. The observations jump suddenly from lenticular nebulaewith no trace of structure to spirals in which the arms are fully developed.

If the numerical data could be fully trusted, the SBa forms would fill the gap. Among these nebulae, the transition from the lenticular to the spiral with arms is gradual and complete. It is tempting to suppose that the barred spirals do not form an independent series parallel with that of the normal spirals, but that all or most spirals begin life with the bar, although only a few maintain it conspicuously throughout their history. This would also account for the fact that the relative numbers of the SBa nebulae are intermediate to those of the lenticular and of the Sa. The normal spirals become more numerous as the sequence progresses, while the numbers of barred spirals, on the contrary, actually decrease with advancing type.

Visual magnitudes have been determined by Hopmann for the nuclei of 37 of the nebulae included in the present discussion. These data, together with types and diameters of the nebulae, are listed inTable XIII. When the magnitudes are plotted directly against the logarithms of the diameters, they show little or no correlation. When, however, the nebulae are reduced to the standard type (by applying corrections for differences in diameter along the sequence), a decided correlation is found whose coefficient is 0.76. This is shown inFigure 9The simple mean of the two regression curves is

where the slope differs by about 1 per cent from that informula (2). The list contains 16 elliptical nebulae, 15 normal, and 6 barred spirals. The nebulae are fairly representative, except that few late-type spirals are included. This is an effect of selection due to the fact that nuclei become less and less conspicuous as the sequence progresses.

The same result can be derived from a study of the differences,mn–mT,for the individual nebulae. The mean value is 1.55 ± 0.08, and the average residual is 0.60 mag. Means for the separate types are to be found inTable XIV.

TABLE XIIIDiameters and Nuclear Magnitudes

The low value for Sa-SBa is due to N.G.C. 5866, for which themagnitude difference of 0.06 is certainly in error, and the high value for Sc and SBc, to M 51, for which the difference of 3.98 mag. is not representative. The latter is accounted for in part by the fact that themTrefers to the combined magnitude of the main spiral and the outlying mass, N.G.C. 5195. When these two cases are discarded, the final mean becomes 1.52 ± 0.05, and the average residual, 0.52 mag., is consistent with the probable errors of the magnitude determinations.The small numbers of objects within each class are insufficient for reliable conclusions concerning slight variations along the sequence. From the constancy ofmn–mT,the relation expressed byformula (6)necessarily follows, the small difference in the constant being accounted for by the different methods of handling the data.

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