Astron. Astrophys. 327, 966-982 (1997)

4. Discussion

4.1. Main observational results

As shown in the previous section, our main observational results are the following:

• The mean levels for the sharpness of the -band luminosity peaks indicate that the vertical luminosity distributions are more peaked than expected for the intermediate sech(z) distribution, but rounder than exponential.
• In the majority of our sample of edge-on disk galaxies, we find that the sharpness of the peak of the vertical profiles, characterized by the exponent 2/n of the generalized family of fitting functions ( 6) (van der Kruit 1988), varies little with position along the major axis. This result is independent of galaxy type.
• Due to the unpredictable contamination by in-plane dust, -band observations are preferred to I -band measurements, although statistically we cannot distinguish between the results obtained in either of the two passbands. In general, the I -band determinations of the best-fitting 2/n values are much noisier than those obtained in the near-infrared.
• The behaviour of the maximum I -band optical depth as a function of galaxy type suggests an increasingly important dust contribution from the lenticular and early spiral galaxies towards later types, although for the latest galaxy types the dust content seems to diminish relative to the intermediate types.
• For those galaxies with the faintest absolute magnitudes, we find a lack of "rounder" profiles compared to those galaxies with brighter absolute magnitudes.

4.2. Projection effects

In interpreting the observed surface brightness distributions as surface density representations, we should be cautious: different stellar populations may have both different mass-to-light ratios and different velocity dispersions. Moreover, the observed galaxy surface brightness distributions are line-of-sight integrations of the light contributions and therefore include contributions from very different locations in the disk. However, as noted by, e.g., Dove & Thronson (1993), these effects are normally considered to be small.

Kylafis & Bahcall (1987) showed that the main effects of scattering are the reduction of the extinction in the dust lane due to forward scattering and an apparent thickening of the central layer of high luminosity. This would not lead to a more sharply peaked profile if the underlying light (or density) distribution were isothermal. Moreover, scattering is not expected to be important in the near-infrared for edge-on galaxies (Kuchinski & Terndrup 1996).

Since the differences between the models discussed here are small, we have investigated the effects of projection. Projection effects lead to an apparent thickening of the high-luminosity midplane of the galaxy. Deviations from an inclination of 90 cause a slight increase of the scale height, as can be seen in Fig. 12a. The effect is comparable for the exponential and the isothermal models. We are confident that none of our sample galaxies have inclinations lower than .

The effect on the 2/n values of deviations from a inclination is shown in Fig. 12b for an exponential distribution at the galaxy center, at 1 and at 2 radial scale lengths, respectively. For exponential distributions the effects of projection are most noticeable, compared to more flattened vertical luminosity distributions. In fact, the effects of projection can be so large, that all our galaxies can have intrinsically exponential vertical surface brightness distributions.

Fig. 12. Importance of projection effects on the results obtained. An intrinsic axis ratio of was assumed (Guthrie, 1992). Projection effects are shown for a vertical surface brightness profile at the galaxy center (short-dashed line), at 1 radial scale length (dotted line) and at 2 radial scale lengths (long-dashed line). a  Projection effects cause an increasing scale height with increasing deviations from a inclination; b  The influence of projection effects on the 2/n parameter.

4.3. Disk heating

In this paper we show that the universal vertical structure that is observed away from the galaxy planes is maintained even when going all the way down to very low z distances. Models that aim to explain this observational fact should also be able to account for intrinsically exponential vertical surface brightness or density distributions.

As Dove & Thronson (1993) warn, if the vertical distribution of stars cannot be fit well by the isothermal sheet approximation, which model is based on physical principles, then rather than invoking an arbitrary alternative function, a more physical function should be that of a nonisothermal stellar distribution. A nonisothermal distribution of stars can be represented by a linear combination of isothermal components (Oort 1932).

It is possible that once a coeval population of stars has been formed, they do not (or only over a long period of time) interact with other components and are therefore quasi-independent isothermal components (Dove & Thronson 1993). Such a model would be physically realistic, since if stars have formed at different times with different velocity dispersions and have not yet reached an equilibrium state, or if the stars are dynamically heated during their evolution, the resulting distribution of stars would not be well approximated by a single isothermal model. From observational evidence we know that stellar subpopulations of different ages have different velocity dispersions. Star formation is generally believed to be a continuous process. Therefore, a more accurate model for a galaxy disk is a superposition of a very large number of components, or, as Kuijken (1991) proposed, an integral representation.

The observations presented in this paper indicate that the physical process responsible for the vertical luminosity profiles is probably both global and universal in nature, since we do not find any significant radial variations of the cuspiness of the profiles, nor large variations as a function of galaxy type.

This puts interesting constraints on the dominant vertical heating mechanism in galaxy disks.

Although the dominant perturbations in a disk are the spiral waves, they have a close relationship with the giant molecular clouds (GMCs). Julian & Toomre (1966) and Julian (1967) suggested that the GMCs must acquire large wakes of material, thus increasing the effective mass of the combined spiral and GMC perturbation. These wakes can be very strong, but will also spread over a large area of the disk, which makes it more difficult to assess the importance of such wakes in enhancing the scattering efficiency. Moreover, at present there is no satisfactory explanation as to how disk heating, the rate of which must vary greatly with radius from the observed distribution of GMCs in our own and other galaxies, can naturally lead to a global and universal vertical density distribution and a constant scale height with galactocentric distance (i.e., is independent of R) (e.g., Jenkins 1992).

4.4. Isothermal versus exponential distributions

Jenkins (1992) finds, that his model disk heating process, i.e., combined spiral and GMC perturbations enhanced by disk accretion, together with constant star formation, always leads to a closely isothermal stellar population. Wielen (1977) reached a similar conclusion based on observational data.

Burkert & Yoshii (1996) show, based on realistic hydrodynamical calculations of disk evolution processes, that - if one starts from a non-equilibrium gaseous state - the final vertical stellar density profile depends strongly on the initial distribution of the protodisk gas, as opposed to the GMC heating process described in the previous section.

On the other hand, if they assume that the gas settles into isothermal equilibrium prior to star formation and gas cooling, then always an exponential density profile is formed, although the vertical scale height increases as a function of decreasing surface brightness. In fact, in de Grijs & Peletier (1997) we presented the results of a detailed study of the vertical scale height distributions in the present sample, for which we found an increasing scale height with galactocentric distance, in particular for the earlier-type galaxies. An interesting result from the calculations of Burkert & Yoshii (1996) is that when the ratio of the star formation time scale to the cooling time scale lies in the range between 0.3 and 3, the vertical stellar density distribution becomes exponential, independent of the free parameters in their modeling and also independent of the initial (isothermal) disk temperature and the initial surface density.

Therefore, the process of crucial importance is that the SFR is adjusted sooner or later to balance with the local cooling rate (Burkert & Yoshii 1996).

Just et al. (1996) find that if the SFR decreases with time, exponential luminosity profiles also grow naturally. They state that since the mass-to-light ratio of a stellar population increases with age, an exponential luminosity profile corresponds to a density profile that is slightly flattened to the galaxy plane. However, an isothermal density distribution is too thick to explain exponential light profiles. For a constant SFR an obvious luminosity excess near the plane would show up in the optical bands, when assuming a heating mechanism of the type observed in the solar neighbourhood.

The fact that we observe a more strongly peaked vertical light distribution than a sech(z) model in all our sample galaxies, independent of galaxy type, indicates that the process at work here is a process intrinsic to the disks themselves, rather than a type-dependent mechanism. The variations in the cuspiness of our profiles along the galaxies' major axes are probably due to some local mechanism, e.g. the contamination by residual dust. Although we observe a similar behaviour in S0s as in later-type galaxies, this does not necessarily mean that they all possess young populations, although the young population contributes also in the near-infrared. The dominant stellar population in is the old population, with ages of several Gyrs, which is likely present in both early and late-type galaxies. Therefore our observational result of a universal vertical density profile in the near-infrared is not incompatible with no type dependence.

Finally, the fact that we observe a lack of "rounder" profiles in the smaller galaxies compared with the larger ones, may indicate that we are hindered by an underlying dust component, which is concentrated towards the galaxy planes, and more extended in the larger galaxies than in the smaller ones, at least to an outside observer. It may be that in the smaller galaxies this dust component affects relatively fewer data points than in the larger ones, thus causing a bias towards more sharply peaked vertical profiles.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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