## 7. Interpretation of the type effect in terms of the mass- luminosity structureWhy should depend on the galaxy type at a
constant ## 7.1. Type effect at constant logD: a simple interpretationThe maximum rotational velocity is determined by the total mass inside the radius (, , , respectively, for disc, bulge and dark halo mass): We make the assumption that at these radii beyond
where is effectively
measured, the dark mass is a constant fraction
of the total mass, i.e. proportional to the total luminous mass of the
galaxy. This assumption, which seems to be the simplest possible, has
the advantage that the proportionality constant
drops off from our final result. In addition, in Sect. 7.3 we find
that this assumption is needed to understand in the simplest manner
the slope of the inverse TF relation, i.e. when one looks at galaxies
of different sizes instead of looking at one fixed Let us assume that the measurement refers to
a radius which is ( Here the latter assumed to be constant, as well as the ratios themselves. Now from the dynamical law together with the proportionality , follows the simple result that Fig. 12 shows the observed zero-point shifts against the
predictions of Eq. (15), with put equal to
2.5. A nice agreement is seen, especially in the type range 2 - 7.
Apparently, the simple model explains the zero-point shifts for
different types in this
## 7.2. Discs with different coloursAccording to Casertano & van Albada (1990) is smaller for late types, because these are presently producing more young stars per unit mass. This interpretation is on line with the view that the systematic photometric changes along the Hubble sequence are basically due to different star formation rates (SFR) (Kennicutt et al. 1994). This might be seen as a corresponding change in the disc's total () ratio (it certainly should be seen in the stellar disc's ). Now the luminous mass is proportional to where for the disc depends on the type, on the average, and one may decide to keep the bulge mass-to-light ratio constant. In order to play with some numbers, we take the "S86" relation between the colour and for an evolving disc population, from Table 1 of Kennicutt et al. (1994) and adopt the average colours for types, obtained from our sample, as given in Table 3.
According to this model = 3.72 for the case
of very small SFR, while = 1.22 for the colour corresponding to
In addition to stars we must here take into account the gas mass. Let be the fraction of the total mass of HI and the corresponding number for the molecular gas. We take representative figures from Roberts & Haynes (1994) for and add to these 0.02 (0.0 to types 7,8) to account for the molecular component. Because is measured relative to the total mass, now the dark mass component enters the end result (factor ). The total mass within the measuring radius becomes: where is taken from Table 3 and = 3.7. The zero- point shifts in are again 0.5 . The thus predicted zero-point shifts for the different types are
given in Table 4 with the observed ones, using the dark mass
fraction = 0.75 or 0.80 . The last value gives
good agreement with the observed zero-point shifts for types 2 to 7.
From Table 4 one may see that adopting for
From the model that we have used above, one may easily calculate
for each type the ratios referring to inside
the radius . These values are given in
Table 3 for the case = 0.8 . They follow
well the behaviour of the averages in the vs.
type diagram of Roberts & Haynes (1994), though are shifted
upwards by a factor of about 1.8 (when = 70 km
s One possibility to decrease the value of , is to increase the total gas mass fraction (Eq. 17). E.g., with = 0.7, we get equally good predictions for the zero-point shifts, if we increase by a factor of 1.5 . Another set of may be obtained from the ratios of molecular to atomic gas derived by Young & Knezek (1989) from a large sample of spiral galaxies. Multiplying by (from their Table 1),one gets in Table 3. Then a good agreement with the zero-point shifts is given by =0.5-0.55 for the types 2-7 and =0.1 for the late type 8. Clearly the present method has potentials for investigating the value of in different galaxy types, though the above examples show the importance of a good knowledge of the trends in the gas contents. ## 7.3. Slope of the inverse diameter TF-relationAbove we have assumed, supported by the observations, that there is
a common slope of about 0.5 for all types. This value is expected for
a pure disc with the total mass proportional to
. Having in mind the TF-relation where
measures the maximum rotation velocity, we may
calculate the radius where that maximum is
reached. For this we need the relation giving the luminosity inside
radius If the mass follows the luminosity, then is
proportional to , an expression seen to contain
What implication the slope of 0.5 has on these components? If we
fix the bulge-to-total ratio where (see Sect. 6.1) If ## 7.4. Expected zero-point shifts for the magnitude inverse TF relationIn the magnitude inverse TF diagram, the total magnitude is kept
fixed. Then for a given The additional term makes the shifts smaller than in the diameter
inverse TF relation. Do we see this in the observations? Fig. 13 shows
the zero-point shifts for the diameter (open circles) and magnitude
(stars) iTF relations, normalized to 0.0 at
© European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |