3. Application to the de Grijs sample
The sample of edge-on disk galaxies of de Grijs (1998) contains 46 systems for which the structural parameters of the disks have been determined (including a bulge/disk separation in the analysis). From this sample we take those for which rotation velocities have been derived in a uniform manner (Mathewson et al. 1992; data collected in de Grijs 1998, Table 4) as well as those for which the Galactic foreground extinction in the B -band is less than 0.25 magnitudes. For this remaining sample of 36 galaxies we perform the following calculations:
From the total magnitudes in de Grijs (1998, Table 6) we obtain the integrated B - and I -magnitudes of the disk.
Using the radial scale length as measured in the I -band we calculate the central (face-on) surface brightness of the disk from its I -band integrated luminosity. So, we do not use Eq. (10) for a constant central surface brightness.
Then we use one or both of the two Bottema relations (1) and (2) to estimate the radial velocity dispersion at one photometric scale length (by definition in the B -band). Where we can do this with both relations, the ratio between the two estimates is 1.11 0.19. This is not trivial, as the rotation velocities and disk luminosities are determined completely independently (and by different workers) and only the one using the absolute magnitude needs an assumption for the distance scale.
Then using Eq. (15), we estimate the vertical velocity dispersion at one photometric scale length in the I -band. For this we need a value for the mass-to-light ratio and we will discuss this first.
We found, that through Eq. (13) Bottema's relation (1) provides a value for of about 5.7. So we make a choice for Q rather than for . It has become customary to assume values of Q of order 2, mainly based on the numerical simulations of Sellwood & Carlberg (1984), who find their disks to settle with 1.7 at all radii. In principle we can use the observed properties of the Galaxy to fix Q from Eq. (17). We have (in the solar neighbourhood, but assume for the sake of the argument also at ) and (see Sackett 1997 for a recent review), so that indeed . We will make the general assumption that Q = 2, in agreement with the considerations above; then = 2.8.
The rotation velocity version of Bottema's empirical relations (Eq. (1)) can provide further support for the choice of Q, along the lines of the discussion in van der Kruit & Freeman (1986). In the first place we recall the condition for the prevention of swing amplification in disks (Toomre 1981), as reformulated by Sellwood (1983)
where m is the number of spiral arms. For a flat rotation curve this can be rewritten as
With Eq. (1) this becomes . Considering that the coefficient in Eq. (1) has an uncertainty of order 15%, this tells us that we have to assume Q at least of order 2 to prevent strong barlike (m =2) disturbances in the disk.
A similar argument can be made using the global stability criterion of Efstathiou et al. (1982). This criterion states that for a galaxy with a flat rotation curve and an exponential disk, global stability requires a dark halo and
Here is the total mass of the disk. This can be rewritten as
and, when evaluated at , yields with Eq. (1) , and therefore also implies that Q should be at least about 2. Efstathiou et al. have also come to this conclusion for our Galaxy, with the use of local parameters for the solar neighbourhood.
Having adopted a value for Q and through this a value for , we will have to convert it to . For this we need a colour for the disks. From the fits of de Grijs (1998) we find that the total disk magnitudes show a rather large variation in colour; for the sample used here has a mean value of 1.9, but the r.m.s. scatter is 0.8 magnitudes. In his discussion, de Grijs (1998) suspects a systematic effect of the internal dust in the disks (particularly on the B -magnitudes, which is another reason for us to use the -version of the Bottema relation (Eq. (1)) in our derivation in the previous section). Instead we turn to the discussion of de Jong (1996b), who compares his surface photometry of less inclined spirals to star formation models. From his Table 3, we infer that for single burst models with solar metallicity and ages of 12 Gyr . So we will use an of 1.4.
There is a further refinement required. In order to take into account the fact that in late-type galaxies the gas contributes significantly to the gravitational force, we have to correct for a galaxy's gas content as a function of Hubble type. In the following, we will discuss the observational data regarding the HI and the H2 separately.
For 25 of de Grijs' sample galaxies HI observations are available, so that we can estimate the gas-to-total disk mass. For this we apply a correction of a factor 4/3 to the HI in order to take account of helium and use de Grijs' (1998) I -band photometry and our adopted ratio of 2.8 (see below) to estimate the total disk mass. As a function of Hubble type we then find
We find no dependence on rotation velocity:
So, the HI mass is about half the stellar mass in disks of Sb's and about similar to that in Sc's and Sd's. But there is no dependence on rotation velocity. But this is not what we need; we should use surface densities rather than disk masses. Now, the HI is usually more extended than the stars and has a shallower radial profile. So the ratios in the tables above are definite upper limits. In order to take into account the effect that in late-type systems the gas contributes significantly to the gravitational force we have "added" for types Scd and Sd a similar amount of gas as in stars and half of that for Sc's.
The distribution of H2 in spiral galaxies is a more complex matter; it is often centrally peaked, although some Sb galaxies exhibit central holes (for a recent review, see Kenney 1997). The molecular fraction of the gas appears to be lower in low-mass and late-type galaxies, assuming that the conversion factor from CO to molecular hydrogen is universal 3. Since our sample galaxies are generally low-mass, later-type systems, we believe that the corrections for molecular gas are small, and therefore contribute little to the correction for the presence of gas.
We added (a) the galaxies from van der Kruit & Searle (1982) to the sample, (b) our Galaxy using the Lewis & Freeman (1989) velocity dispersion and the structural parameters in van der Kruit (1990), and (c) the observational results for NGC488 from Gerssen et al. (1997). We leave the few early type (S0 and Sa) galaxies out of the discussion, because the component seperation in the surface brightness distributions is troublesome and some of our assumptions (in particular the self-gravitating nature of the disks) are probably seriously wrong.
In order to be able to trace the origin of our results, we first show in Fig. 1 the radial and vertical scale lengths of the sample as a function of the rotation velocity. Both increase with , which would be expected intuitively. The main result is presented in Figs. 2 and 3. From Fig. 2 we see that the vertical velocity dispersions, that have been derived from hydrostatic equilibrium, increase with the rotation speed (the radial velocity dispersions do the same automatically as a result of the use of the Bottema relations). For the slowest rotation speeds the predicted vertical velocity dispersion is on the order of 10-20 km s-1, which is close to that observed in the neutral hydrogen in face-on galaxies (van der Kruit & Shostak 1984).
The distribution of the axis ratio of the velocity ellipsoid with morphological type is as follows:
Not much of a trend is seen here. It is in order to comment here briefly on the effects of our corrections for the gas to obtain vertical velocity dispersions. From our discussion above we conclude that there would be no systematic effect introduced as a function of rotation velocity. Furthermore, taking away our correction altogether reduces the values for the average axis ratio in the table just given to about 0.55 for Scd and Sd galaxies. Even in this unrealistic case of not allowing for the presence of the gas, we believe the trend to be hardly significant in view of the uncertainties. Since some correction for the gas mass as a function of morphological type must be made, we cannot claim that we find any evidence for a change in the velocity anisotropy with Hubble type.
Fig. 3 shows the axis ratio of the velocity ellipsoid of all the galaxies versus their rotation velocity. In view of the fact that the two dispersions that go into this ratio are determined from different observational data ( from integral properties such as total luminosity and amplitude of the rotation curve; from photometric scale parameters and surface brightness) and that we have made rather simplifying assumptions, the scatter is remarkably small. No systematic trends are visible (and would probably not be significant!) in the data. The points closest to unity in the dispersion ratio generally have low rotation velocities and inferred velocity dispersions. One of these ( = 95 km s-1, = 0.75) is NGC5023. Bottema et al. (1986) have shown that the stars and the gas in this galaxy are effectively coexistent; the radial and vertical distributions are very similar. This would imply that the velocity dispersions of the gas and the stars are the same. The vertical velocity dispersion found here is about 20 km s-1, which is significantly higher than that observed in larger spirals. It would be of interest to measure the HI velocity dispersion in this galaxy. Since the HI would be expected to have an isotropic velocity distribution from collisions between clouds, the vertical dispersion should be equal to that in the line of sight in edge-on galaxies.
© European Southern Observatory (ESO) 1999
Online publication: November 23, 1999