## 3. Application to the de Grijs sampleThe 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 From the total magnitudes in de Grijs (1998, Table 6) we obtain the integrated Using the radial scale length as
measured in the 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
Then using Eq. (15), we estimate
the vertical velocity dispersion at one photometric scale length in
the We found, that through Eq. (13) Bottema's relation (1) provides a
value for of about 5.7. So we make a
choice for
The rotation velocity version of Bottema's empirical relations
(Eq. (1)) can provide further support for the choice of where 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 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 Having adopted a value for 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 H 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) 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 H 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
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 © European Southern Observatory (ESO) 1999 Online publication: November 23, 1999 |