A global comparison of the results of the different methods may allow us to assess them. This is particularly important for our two methods for which no other comparison or discussion can be found in the literature. We will consider: values from our second method, from our first method, from global HI kinematical studies, from Grosbol's (G85) photometric study, from Considére & Athanassoula's (CA88) spiral structure analysis and from Danver's (D42) eye estimates. We have plotted in figure 4 the PA obtained with each of these methods versus the values obtained by the other methods. Figure 5 gives the same plots for the cosine of the IAs. We have plotted the cosines rather than the angles since these are more uniformly distributed. The comparison is of course not for all galaxies ever measured by a given method but rather only for the galaxies in the present sample. We have also omitted M51 from these comparisons as the different methods give very very discrepant results (see previous section). The solid line gives the least squares fit to the points and the dashed one the diagonal. The correlation coefficients are given in the first line of each entry in Table 4, and their 90% confidence level in the second line. We have included in this Table two entries per correlation. In nearly all cases the two entries are the same yet we kept them both to facilitate the reading of the table. The independent variable is always in the column and the dependent one in the line. In the third line we find the value of spearman's rank correlation coefficient and in the fourth line the two-sided significance level of its deviation, using a student's distribution. In the last line is the number of galaxies in each of the observations.
Figure 4. Correlations between the values of the PA found by various methods.
Figure 5. Correlations between the values of the cos (IA) found by various methods.
From figures 4 and 5 and Table 4 we note that the PAs are in general much better correlated than the IAs. Indeed the mean of the rank correlation coefficients from the PAs in Table 4 is 0.90 ± 0.07, while the same quantity for the IAs is only 0.76 ± 0.15. (From Figs. 4 and 5 one could have thought that the difference was even larger. However this is partly due to the fact that, since PAs can reach values twice as large as IAs, the scales on the figures are not the same, but even so the difference is significant). It reflects the fact that the PAs are easier to measure and are riddled with less errors than the IAs. Furthermore, for some methods, an error in PA will produce a systematic error in IA. For the case of large scale kinematics, the PAs can be measured fairly accurately, whereas IAs can only be obtained from the measurement of V sin i. In photometry the IAs are influenced by the Stocke effect (Stocke 1955), which is important only for galaxies with high amplitude spirals and whose disk does not extend much beyond the region of the spiral. On the other hand the inclination angles are influenced for all galaxies by the thickness of the disk, or even, in the case of intermediate inclination galaxies with large bulges, by that of the bulge. Finally in Danver's method it is difficult to discuss the errors, since the method is purely subjective. However it seems to us that the eye can measure position angles more accurately than inclination angles. The main conclusion that can be drawn from Table 4 is that there is no bad, inadequate or even second rate method. That is particularly important for the two methods we introduced in this paper, since they had not been tested before. Their correlation coefficients are as good as those of the other methods. Table 4 together with Figs. 4 and 5 show that they are devoid of systematic errors and that their mean errors are not higher than those of the other methods. Thus, provided that a given catalogue does not contain systematic errors and contains a sufficient number of HII regions describing reasonably well the disk, then it can be used with confidence for estimating the position and inclination angles of a galaxy.
|(0.92, 0.97)||(0.92, 0.98)||(0.87, 0.95)||(0.89, 0.99)||(0.90, 0.97)|
|(0.93, 0.97)||(0.84, 0.97)||(0.81, 0.94)||(0.57, 0.97)||(0.72, 0.93)|
|(0.93-0.98)||(0.84, 0.97)||(0.74, 0.95)||(0.93, 0.99)||(0.88, 0.97)|
|(0.87, 0.95)||(0.81, 0.94)||(0.74, 0.95)||(-0.1, 0.97)||(0.59, 0.91)|
|ATHA.||(0.89, 0.99)||(0.57, 0.97)||(0.93, 0.99)||(-0.1, 0.97)||(0.93, 0.99)|
|(0.90, 0.97)||(0.72, 0.93)||(0.88, 0.97)||(0.59, 0.91)||(0.93, 0.99)|
|(0.76, 0.90)||(0.78, 0.95)||(0.66, 0.87)||(0.74, 0.97)||(0.69, 0.90)|
|(0.76, 0.90)||(0.68, 0.93)||(0.49, 0.83)||(0.65, 0.98)||(0.62, 0.90)|
|(0.81, 0.95)||(0.68, 0.93)||(0.68, 0.94)||(0.90, 0.99)||(0.71, 0.93)|
|(0.66, 0.87)||(0.49, 0.83)||(0.68, 0.94)||(0.93, 1)||(0.09, 0.73)|
|ATHA.||(0.75, 0.97)||(0.56, 0.98)||(0.90, 0.99)||(0.93, 1)||(0.57, 0.95)|
|(0.69, 0.91)||(0.62, 0.90)||(0.71, 0.93)||(0.09, 0.73)||(0.57, 0.95)|
One should not try from Table 4 to rank the various methods. The differences between most of the correlation coefficients are not significant as can be seen be careful inspection of Table 4. Furthermore different kinds of perturbations (like ovals, asymmetries in the outer parts, spiral structure, etc) influence the various methods in different ways. Unfortunately these effects can not be found by statistical studies as the present one. Only rather elaborate modelling may help us to untangle them.
We would like to thank the referee, P. Grosbol, for interesting comments on the first version of this paper, A. Bosma for many helpful discussions on the galaxies in our sample and P. Hodge for sending us a copy of "An Atlas of HII regions in 125 Galaxies" (Hodge & Kennicutt 1982) which contains a number of otherwise unpublished catalogues of HII regions. M. Gerbal typed in about a third of these catalogues. Our literature search was based on SIMBAD data retrieval system, database of the Strasbourg, France, astronomical Data Center. C. G.-G. acknowledges his financial support by a grant under the contract by the I.N.S.U. and the I.A.C, for the installation of the Themis telescope.