4. MEASURING POSITION AND INCLINATION ANGLES

Several, more or less independent, ways of measuring the position and inclination angles, of galaxies have been proposed and used so far:

i) Large-scale kinematics. Assuming that the emitting material is in a thin planar disk and is in circular motion around the center of the galaxy, one can minimize, e.g. by least squares, the departures from such a flow (Warner et al. 1973). This method is well adapted to HI velocity fields, provided the galaxy is not too perturbed, and has been widely used in such studies (e.g. Bosma 1981). If the above mentioned assumptions are met, it is the safest of all methods, at least as far as the P.A. is concerned, since it involves data from all the surface of the galaxy and it is not too hindered by noise. This method has been lately (e.g. Pence et al. 1990) applied to optical data which present a sufficiently good coverage of the surface of the galaxy.

ii) From photometry the position and inclination angles are measured traditionally by fitting ellipses to the outermost isophotes which are not too perturbed by noise and background (e.g. Boroson, 1981, etc).

iii) Grosbol (1985, hereafter G85) used a variant of the above method. He applied one dimensional Fourier transforms to the intensity distribution in the outer parts and adopted the deprojection angles that minimised the bisymmetric Fourier component.

iv) Kent (1985) used the photometric data all through the galaxy and did not limit himself to the outer parts. He assumed that the galaxy consists of two components, a bulge and a disk, the former being considered spherical, and fitted the intensity profiles on the major and minor axes.

v) Iye et al. (1982) calculated the projection angles of NGC 4254 by maximizing the power in the axisymmetric component of the deprojected intensity distribution.

vi) Considére & Athanassoula (1988, hereafter CA88) used the m = 2 spectrum to ensure that there was no contribution from a wrongly deprojected disk and that the signal of the spiral stood out well.

vii) Another method related to the above one consists of plotting the positions of the HII regions in a log(r) - plane and fitting a straight line to the arms, thus ensuring that the spirals are logarithmic.

viii) Danver (1942, hereafter D42) obtained visual estimates for 202 galaxies by rotating their images with the help of a special display table, until the galaxy image was circular.

The methods of Iye et al., Considére & Athanassoula, and Danver, i.e. methods v, vi and viii, can be suitably modified so as to apply to HII region distributions. We have refrained from applying method viii, because it is highly subjective, although in the discussion of the individual galaxies that follows, in a few cases, when discordant values are found by the various methods, we will make note of which values gave more reasonable looking results. Method vi necessitates a clear spiral structure, giving a clear signal-to-noise ratio in the m = 2 spectrum. This is often the case when the whole image of the galaxy is being analysed, but seldom in an HII region distribution. Considére & Athanassoula (1982) have already applied it to the HII region distribution of M51 and given a discussion on the results. We have not used this method since very few galaxies in our sample have a spiral structure as clearly delineated as that of M51.

Finally method v cannot be applied as such but rather as described in the previous section. It will be referred to hereafter as our first method. It can of course be applied only to galaxies whose HII region distribution delineates the axisymmetric component. If the HII regions are found only in the spiral arms, i.e. their distribution delineates mainly some m > 0 component, another method should be applied (see next paragraph). Thus all galaxies marked by a Y in Column 6 of Table 1 were found difficult, or impossible to analyse by our first method. Its main disadvantage is that, depending on whether one uses the density, mass or any other form r (r, ), one gives more or less weight to the inner compared to the outer parts of the distribution. Giving weight to the inner parts should give wrong results in the case of B or AB galaxies (respectively 8 and 28 galaxies in our sample according to the RC2), and could falsify the results for galaxies with a slight oval component, if this was reflected in the HII region distribution. Favouring the outer parts does not seem to introduce any obvious biases, but could introduce errors in galaxies with asymmetric outer parts.

We have thus deviced another method, which we will hereafter refer to as our second method, which does not present the above problem. If we cut an axially symmetric galaxy into N equal sectors, like parts of a cake, then each part should have a roughly equal number of HII regions, but these sectors are not equal in a projected galaxy, and the size of each sector depends on the values of PA and IA. For given values of PA and IA we can deproject the HII region distribution, cut the surface into sectors and then count the HII regions falling in each sector. If the deprojection angles chosen are the correct values, the number in each division must be roughly equal and the dispersion around the mean minimal. Thus the correct PA and IA should minimize the dispersion of the number of HII regions in each angular slice from this mean value. Of course this method will not work for the case of strongly barred galaxies and will present more than one minimum in the (PA, IA) plane in the case of ovals, spirals or other structures. Mercifully, most of the time, these minima are relatively wide apart, so that visual inspection of the deprojected distribution or a very rough knowledge of the deprojection angles are enough to tell the relevant minimum. We have also tried to eliminate the effect of asymmetries by adding the contents of two sectors which are symmetric with respect to the center. An important advantage of this method is that the calculations require very little computing time compared to the first method.

We have tested our second method with the help of many random number realisations of axisymmetric inclined disks. We found that this method, like most others, works better with more inclined galaxies than with less inclined ones since for the former the PA stands out better. Of course the larger the number of points the better the method works. It also works less well for very centrally concentrated distributions or distributions having most regions in the outer parts, i.e. the method favours more uniform coverages.

For most of our test, we used an N(r) = r e^-r profile which, as will be shown in a forthcoming paper, is a realistic representation of the HII region number distribution in a fair fraction of galaxies. Realisations with a relatively large number of points have a minimum in the (PA, IA) plane clearly showing the correct values of the deprojection angles. For a low number of points, i.e. less than 200, one often gets more than one minimum. Since many of our galaxies have less than 200 catalogued HII regions, we have made tests to assess the possibilities of the method in such cases. Fifty different realisations of 150 points were made and projected with PA = 120° and IA = 60°. Our second method was then applied to each of them. In 37 realisations the primary minimum was within ± 5° from the right position, and in 47 within ± 10°. The mean for the PA values for the primary minimum is 120° ± 6°, and for the inclination angle 59° ± 5°. We repeated this for an inclination of 45° which, as discussed above, is less favourable to this method. We found a mean of 122° ± 13° and 43° ± 7° for the primary minima.

We applied the same test to our first method, but now with forty realisations of each distribution since this method is more CPU intensive. In all cases there was only one minimum, however it corresponded to values of the PA and IA further away, in the mean, from the correct values than those predicted by the second method. When the correct values were PA = 120° and IA = 60°, it gave PA = 118° ± 9° and IA = 59° + 6°. Similarly for PA = 120° and IA = 45° it gave PA = 122° ± 20° and IA = 46° ± 9°.

These tests gave an estimate of the expected errors in the case of an axisymmetric distribution of points. However we expect bigger errors in the cases where a considerable number of points are in a spiral. Furthermore the influence of the spiral structure on the errors might be different for the two methods. In order to test this we made 10 different realisations of two "spiral" galaxies. In the first case 100 points were drawn from an axisymmetric distribution and 50 were in spiral component. The numbers were inversed in the second case. Initially we tried drawing numbers from a spiral with a cos 2 component and a reasonable radial distribution. However this gave a spiral structure which was too broad and not at all reminiscent of the distribution shown e.g. in Figure 3. We thus discarded this as unrealistic and preferred to generate the spiral as follows. We placed the points initially equidistantly on a logarithmic spiral with pitch angle i = 15° and then gave to each of them two random nudges, in x and y, between 0 and 0.1 times the maximum radius of the axisymmetric distribution. The result of this ad hoc method looks much more realistic than that of the previous one. Since the amplitude of the spiral does not decrease with radius in these examples, we expected, and got, an important Stocke's effect (Stocke 1955). However, as there are distributions of HII regions with this characteristic and since these were often amidst the most difficult to treat, we have thought it realistic to leave this effect in the random distribution as well. We then applied the same test as above to these two test distributions and found, as expected, bigger deviations in the mean from the correct value than before.

Figure 3. Deprojected HII region distributions of all galaxies in our sample.

Our results are summarized in Table 3. The name of the galaxy is given in Column 1. An I in Column 2 means that the HII region distribution is irregular, and a II that it is very irregular. Columns 3 and 4 give the PA and IA obtained by our first method (similar to Iye et al.'s method) and Column 6 the step in degrees (PA, IA) used in the partitioning and scanning of the (PA, IA) plane when searching for the minimum. Column 5 gives the weight we have assigned to this result. The values of the PA and IA obtained from our second method are given in Columns 7 and 8 and the weight we have assigned to this result in Column 9. A 20, or 40 in Column 10 means that the galaxies were divided into 20, respectively 40, equal sectors. An * was put in Column 11 when the adopted minimum was not the deepest but a secondary one. For this method the angles PA and IA which we used when looking for the minima were always 5°. Thus the error bars of the values we have determined are at least half this step, or half the number given in Column 6 for our first method. However they can be considerably higher if certain structures or associations of points interfere with our methods. In Columns 12 and 13 we give the weighted means of the PAs and IAs obtained by means of our two methods.

Columns 14e thus given only the most trustworthy or most noteworthy values, in our opinion, trying at the same time to include determinations by different methods. Since some galaxies in our sample have been very little studied, we have included for them values less safe than for the best studied cases. The weights were assigned according to the following rules. We consider a priori that the kinematical method based on a complete velocity field is the most accurate one (see also Section 6), so the corresponding values got a weight of 5 (or 3 if the data were of not too high quality). For the photometric values, in the case were the authors presented a convincing isophotal map, the values took a weight of 3, in all the others cases they took a value of 2. The rest of the values, coming from different methods, took a value of 1. For the methods based on the HII regions the weights were ascribed on a subjective basis. The criteria that entered into consideration included the uniqueness of the minimum, the plausibility of the solution (whether it was in agreement with the literature values), the number of HII regions, the regularity of the distribution and whether the HII regions were mostly in the axisymmetric background or rather, delineated a structure like spiral arms or a bar etc.

Finally the adopted PA and IA values are given in Columns 18 and 20 and the corresponding dispersion in Columns 19 and 21. They have been obtained as the weighted means of the values obtained by the different methods. These values have been used to obtain deprojected images, shown in Figure 3, and all other properties of the HII region distributions which will be discussed elsewhere. In figure 3 we show first galaxies with a large number of HII regions in their catalogues and then the ones with fewer, in a smaller format.