In this section we will critically discuss the uncertainties in our approach.
The linearity of the magnitude version of the Bottema relation. We have discussed above that the power-law nature of the Tully-Fisher relation (Eq. (11)) would imply a nonlinear form of the magnitude version of Bottema's relation (Eq. (2)). We have used it as an empirical relation to help (together with Eq. (1)) to estimate the radial velocity dispersions from the observed photometry. One may argue that it is internally consistent to use instead of Eq. (2) a fit of the form
This has only a noticable effect on galaxies with faint absolute disk magnitudes. We have repeated our analysis using such a fit and find no change in our results. To be more definite we repeat the table of the average axis ratio as a function of morphological type that we then obtain.
Choice ofQ. We have adopted a value for Q of 2.0, and this enters directly in our results in the calculation of the vertical velocity dispersion through the value of that follows from this choice. Had we adopted a value of 1.0 for Q, would have been a factor 2 higher and the value for a factor -see Eq. (15)-. Bottema (1993, his Fig. 11) has shown that his observations of the stellar kinematics do not show any evidence for systematic variations in Q among galaxies.
We have used Eq. (19) to argue that Q is likely of order 2 in order to prevent strong barlike (m=2) disturbances. It does not necessarily follow from this that galaxies with more spiral arms should have higher values of Q or that Q should be significantly lower in galaxies with very strong two-armed structure. Spiral structure may arise in a variety of ways; we only argue that disks do not have grossly distorted m=2 shapes and that therefore swing amplification is apparently not operating.
We have assumed that the stellar disk is self-gravitating and ignored the influence of the gas in the evaluation of Q. At one scale length this is probably justified (it is only a small effect in solar neighbourhood), even for late-type disks.
Non-exponential nature of the disks. Often the disk in actual galaxies can be fitted to an exponential only over a limited radial extent. In that case our description is unlikely to hold. However, in our sample the fits can be made reasonably well at one scale length from the center and we believe this not to be a problem.
Vertical structure of the disks. Our results depend on the adoption of a particular form of the vertical mass distribution (namely the sech(z)-form) of Eq. (4). This enters our results through the value of the numerical constant 1.7051 in Eq. (5), and in Eq. (15) it enters into the value for as its square root. Had we assumed the isothermal distribution, then the constant would have been 2.0, while it would have been 1.5 for the exponential distribution. This would have given us values for which are only 6 to 8% higher or lower. As we have shown in de Grijs et al. (1997), for our sample galaxies the vertical luminosity distributions in these disk-dominated galaxies are slightly rounder than or consistent with the exponential model. However, the vertical mass distribution is probably less sharply peaked, and thus expected to be more closely approximated by the sech(z ) model.
Non-constancy of central (face-on) surface brightness. We have assumed in Sect. 2 that for all galaxies the central surface brightness is constant. This is certainly unjustified for so-called "low surface brightness galaxies"; however, our galaxies have brighter surface brightnesses than galaxies that are usually considered to be of this class. But even for galaxies as in our sample it remains true that the (face-on) central surface brightness is in general somewhat lower for smaller systems (van der Kruit 1987; de Jong 1996a). From de Jong's bivariate distribution functions, it can be seen that late-type, low absolute magnitude spirals may have a central surface brightness (in B ) that is up to 1.0 magnitude fainter. Note, however, that for our derived velocity dispersons we have used the actually observed surface brightness for each galaxy.
Effects of colour variations on the mass-to-light ratio. There is a fairly large variation in the colours of the disks in the sample. De Grijs (1998) has argued that this is the result of internal dust extinction. However, we have used the I -band data; de Grijs' Fig. 11 shows that the variation is much less in than in colours involving the B -band.
The colours of the disks do not correlate with morphological type (de Grijs 1998); although such a correlation has been seen in more face-on galaxies (de Jong 1996b), we believe that the fitting procedure has ignored most of the young population and dust absorption near the plane, and that contributions to the scatter as a result of young populations has mostly been avoided; and that the I -band scale lengths determined away from the galactic planes are fairly representative of the stellar mass distributions (de Grijs 1998).
The colour of the fitted disks in our sample has in the range 2 to 4. The latter is red, even for an old population and may well be caused by excessive internal extinction, but we see no strong evidence for a substantial systematic correction in our velocity dispersions from this.
Effects of metallicity on the mass-to-light ratio. De Jong (1996b) has drawn attention to the non-negligible effects of metallicity on the mass-to-light ratio. From his compilation of models, in particular his W94 (Worthey 1994) models with ages of 12 Gyr, we estimate that the effect in the I -band amounts to 10 to 20% over the range of relevant metallicities. The effect on the derived velocity dispersions is the square root of this.
Effects of radial colour variations. Since we use an empirical relation to derive the radial velocity dispersion at one B -scale length from the center, we have to consider the effect of using the I -band. We have used the latter as the proper scale length to use for the mass density distribution. The correct one to use here would be the one measured in the K -band, which is 1.15 0.19 times smaller for this sample. The scale length in the B -band is 1.64 0.41 times longer than in the K -band (values quoted here from de Grijs 1998). We deduce from this that we may have underestimated the scale length to use by a factor 1.43 and systematically overestimated the vertical velocity dispersion by about 20%.
The effect of radial metallicity variations on the scale length is probably very small; these gradients in the older stellar populations are in any case expected to be significantly less than in the interstellair medium. This is so, because in models for galactic chemical evolution the mean stellar metallicity of the stars approximates the (effective) yield, while that in the gas grows to much large values in most models (van der Kruit 1990, p. 322).
Non-flatness of rotation curves. This may be an effect for small, late-type galaxies that have slowly rising rotation curves. It enters however in our analysis only in the derivation of the Bottema relations and this holds empirically to rather small rotation velocities ( 100 km s-1).
The slope of the Tully-Fisher relation. Although we use a relation between the luminosity of the disk alone and the rotation velocity, it remains true that our "slope" of is steeper than the usually derived slopes of Tully-Fisher relations, which would indicate (e.g. Giovanelli et al.'s 1997 "template relation" yields an exponent of 3.07 0.05). If this slope were to be put into Eq. (9), Eq. (1) would have , which is very significantly in disagreement with Bottema's observations. The same holds for the other Bottema relation, Eq. (2).
The analysis of the present sample (see de Grijs & Peletier 1999) has resulted in slopes in the Tully-Fisher relation of 3.20 0.07 in the I -band and 3.24 0.21 in the K -band. On the other hand, Verheijen (1997), in his extensive study of about 40 galaxies in the Ursa Major cluster, finds a slope of 4.1 0.2 in the K -band.
Effects of the gas on the value ofQ. In our calculations we have already made crude allowance for the effects of the gas on the gravitational field. But in our derivations we have not taken into account the effect of the HI on the effective velocity dispersion to be used in the evaluation of the Q-parameter. The effect of the HI is a decrease of the effective velocity dispersion and therefore in Q. This means that the assumed value should in reality be decreased on average, but beyond this numerical effect, it does not affect our results.
The effects just discussed can produce errors in the estimated velocity dispersions of the order of 10 to 20% each. The final result of the dispersion ratios in Fig. 3 may therefore be wrong by a few tenths, which is comparable to the scatter in that figure. However, we have no cause to suspect that we have introduced serious systematic effects that would be strong functions of the rotation velocity or the morphological type and the lack of correlation of the axis ratio with these properties is unlikely to be an artifact of our analysis.
We conclude that it is in principle possible to infer information on the axis ratio of the velocity ellipsoid from a sample of edge-on galaxies for which both the radial scale length as well as the vertical scale height have been measured. The result, however shows much scatter, most of which is a result of the necessary assumptions. There is one significant improvement that can be made and that is the direct observation of the stellar velocity dispersion in these disks. That this is feasible in practice for edge-on systems has been shown by Bottema et al. (1987, 1991). The observed velocity profiles can be corrected for the line-of-sight effects, giving the tangential velocity dispersion, which through the observed shape of the rotation curve can be turned into the radial velocity dispersion. Although a time consuming programme, we believe that it is worth doing for two reasons: (1) It will set both versions of the Bottema relation on a firmer footing. (2) The uncertainties in the analysis above can likely be significantly diminished by direct measurement of the radial velocity dispersion rather than having to infer it from the rotation velocity or the disk absolute magnitude.
© European Southern Observatory (ESO) 1999
Online publication: November 23, 1999