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Astron. Astrophys. 319, 435-449 (1997)

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7. Interpretation of the type effect in terms of the mass- luminosity structure

Why should [FORMULA] depend on the galaxy type at a constant logD? We give a simple explanation which presently satisfies us, though it clearly needs more elaboration. As was seen in Sect. 6, the size [FORMULA] is determined exclusively by the disc, excepting small galaxies, and fixing logD keeps the discs approximately identical. Then what changes, when one goes from late to early types, is the increasing contribution to light and mass by the bulge. Though in light this effect does not seem so strong (relatively small k mostly), the larger mass-to-luminosity ratio of the bulge enhances its importance. Things are complicated by the fact that the discs of different types may have different [FORMULA] ratios, especially when one considers only the stellar discs. Kennicutt et al. (1994) have suggested that the star formation is going on more strongly in the discs of the late types which (the stellar discs) should then have smaller ([FORMULA]) ratios. We give first the simple explanation which assumes that the discs (stars + gas) have similar [FORMULA] independent of type.

7.1. Type effect at constant logD: a simple interpretation

The maximum rotational velocity [FORMULA] is determined by the total mass inside the radius [FORMULA] ([FORMULA], [FORMULA], [FORMULA], respectively, for disc, bulge and dark halo mass):


We make the assumption that at these radii beyond [FORMULA] where [FORMULA] is effectively measured, the dark mass is a constant fraction [FORMULA] 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 [FORMULA] 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 logD. Hence we have, at a constant disc size:


Let us assume that the [FORMULA] measurement refers to a radius which is [FORMULA] (g is roughly constant, independent of type). This radius being quite probably well beyond the optical edge, we may write for the mass inside [FORMULA]:





the latter assumed to be constant, as well as the [FORMULA] ratios themselves. Now from the dynamical law together with the proportionality [FORMULA], follows the simple result that


Fig. 12 shows the observed zero-point shifts against the predictions of Eq. (15), with [FORMULA] 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 T -range. Such values of [FORMULA] have appeared in the discussions of the disc-bulge-halo model of galaxies (Whitmore & Kirshner 1981), though this question has not been widely discussed. Kent (1986) when decomposing the disc and the bulge and fitting mass-models on rotation curves, derives also [FORMULA] ratios for the two components. Depending on the assumptions in the solutions, different ratios were obtained. For Sb galaxies the three solutions gave average values of [FORMULA] of about 1.03, 1.85 and 1.73 (negative and one excessively large [FORMULA] omitted from the data given by Kent's Tables III, IV, and V). However, these results refer to the r -band, and need a correction to the B -band. For a very rough correction, let us assume that for the bulge [FORMULA] = 1.3, for spirals typically (disc + bulge) [FORMULA] = 0.9 (Kent 1984) , and k = 0.24. This would require for the disc alone [FORMULA] 0.7 and a correction factor to [FORMULA] (in r) of about 1.7 in order to get [FORMULA] in the B -band. Then the above values of [FORMULA] become 1.75, 3.15, and 2.94, and one may say that Kent's study is not in disagreement with the value of 2.5 that was needed above.

[FIGURE] Fig. 12. Comparison between observed zero-points and predicted ones using the simple model described by Eq. 12 (stars) and the more accurate one described by Eq. 17 (open circles). For type 8, we obtained a better agreement with [FORMULA] =0.75 instead of 0.8 .

7.2. Discs with different [FORMULA] colours

According to Casertano & van Albada (1990) [FORMULA] 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 ([FORMULA]) ratio (it certainly should be seen in the stellar disc's [FORMULA]).

Now the luminous mass is proportional to


where [FORMULA] for the disc depends on the type, on the average, and one may decide to keep the bulge mass-to-light ratio [FORMULA] constant. In order to play with some numbers, we take the "S86" relation between the [FORMULA] colour and [FORMULA] for an evolving disc population, from Table 1 of Kennicutt et al. (1994) and adopt the average [FORMULA] colours for types, obtained from our sample, as given in Table 3.


Table 3. column 1: morphological type (LEDA);
column 2: average total colour index (using LEDA);
column 3: disc colour index (this paper);
column 4: disc mass-to-light ratio (using Kennicutt et al., 1994);
columns 5,6: HI and total gas masses (Roberts, Haynes, 1994);
column 7: mass-to-luminosity ratio (this paper);
column 8: total gas mass (this paper, using Young and Knezek, 1989).

According to this model [FORMULA] = 3.72 for the case of very small SFR, while = 1.22 for the colour corresponding to Sc. However, we need typical colours for the discs, and these may be derived using the above total colours, the colour for the bulge (= 0.90), and the bulge-to-total light ratio k. The result of this calculation is also shown above as well as the corresponding [FORMULA] from Kennicutt et al. (1994).

In addition to stars we must here take into account the gas mass. Let [FORMULA] be the fraction of the total mass of HI and [FORMULA] the corresponding number for the molecular gas. We take representative figures from Roberts & Haynes (1994) for [FORMULA] and add to these 0.02 (0.0 to types 7,8) to account for the molecular component. Because [FORMULA] is measured relative to the total mass, now the dark mass component enters the end result (factor [FORMULA]). The total mass within the measuring radius [FORMULA] becomes:


where [FORMULA] is taken from Table 3 and [FORMULA] = 3.7. The zero- point shifts in [FORMULA] are again 0.5 [FORMULA].

The thus predicted zero-point shifts for the different types are given in Table 4 with the observed ones, using the dark mass fraction [FORMULA] = 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 T =8 the total colour 0.39 (" [FORMULA] ") or, especially, taking [FORMULA] equal to 0.75, while keeping it equal to 0.8 for the other types, gives a better agreement for this very late galaxy type (see Fig. 12). In view of the uncertainties in the various steps leading to these predictions, we do not put very much importance on such deviations nor on the exact value of [FORMULA]. However, it may be asked whether such a large [FORMULA] is believable.


Table 4. Predicted and observed zero-point shifts.
column 1: morphological type (LEDA);
columns 2,3: predicted zero-point shifts using the second model;
column 4: predicted zero-point shifts using the first simple model;
column 5: observed zero-point shifts

From the model that we have used above, one may easily calculate for each type the [FORMULA] ratios referring to inside the radius [FORMULA]. These values are given in Table 3 for the case [FORMULA] = 0.8 . They follow well the behaviour of the averages in the [FORMULA] vs. type diagram of Roberts & Haynes (1994), though are shifted upwards by a factor of about 1.8 (when [FORMULA] = 70 km s-1 Mpc-1). As those authors calculated the mass using [FORMULA] as the size 2R in the formula [FORMULA], that factor would in principle give the value of [FORMULA], i.e. where [FORMULA] is typically measured.

One possibility to decrease the value of [FORMULA], is to increase the total gas mass fraction [FORMULA] (Eq. 17). E.g., with [FORMULA] = 0.7, we get equally good predictions for the zero-point shifts, if we increase [FORMULA] by a factor of 1.5 . Another set of [FORMULA] may be obtained from the ratios of molecular to atomic gas derived by Young & Knezek (1989) from a large sample of spiral galaxies. Multiplying [FORMULA] by [FORMULA] (from their Table 1),one gets [FORMULA] in Table 3. Then a good agreement with the zero-point shifts is given by [FORMULA] =0.5-0.55 for the types 2-7 and [FORMULA] =0.1 for the late type 8. Clearly the present method has potentials for investigating the value of [FORMULA] 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-relation

Above 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 [FORMULA]. Having in mind the TF-relation where [FORMULA] measures the maximum rotation velocity, we may calculate the radius [FORMULA] where that maximum is reached. For this we need the relation giving the luminosity inside radius r:


If the mass follows the luminosity, then [FORMULA] is proportional to [FORMULA], an expression seen to contain r everywhere as [FORMULA]. Differentiation hence gives the maximum rotation at [FORMULA] where g = 1.8 . Then it is seen that [FORMULA] is proportional to [FORMULA] and to [FORMULA]. This means that for a pure disc the slope of the inverse diameter TF- relation is expected to be 0.5 . This result is, of course, not new, but is given here in order to emphasize the remarkable thing that the observations give this same slope for a range of galaxies, containing in addition to the disc, a bulge and dark matter.

What implication the slope of 0.5 has on these components? If we fix the bulge-to-total ratio k, consider galaxies of different scale lengths [FORMULA], assume that [FORMULA] is measured around [FORMULA] (g is assumed roughly constant regarding to the general shape of spiral rotation curves), and require that the dark mass inside [FORMULA] is [FORMULA] which may depend in some special way on [FORMULA], then one may write for the rotational velocity at [FORMULA]:


where [FORMULA] (see Sect. 6.1)

If logV is strictly proportional to [FORMULA], then it is clear that [FORMULA] must be proportional to [FORMULA], and hence the dark mass inside [FORMULA] must be proportional to [FORMULA] and to [FORMULA]. In this way, considering galaxies of a fixed type but different sizes, we arrive at the same assumption that above was found useful while interpreting the zero-point shifts for different types with fixed size.

7.4. Expected zero-point shifts for the magnitude inverse TF relation

In the magnitude inverse TF diagram, the total magnitude is kept fixed. Then for a given k, the constant luminosity [FORMULA] is related to the disc luminosity as [FORMULA]. Above it was shown that at a constant size, i.e. at a constant [FORMULA] the zero-point shift is given by Eq. (15). Evidently, at a constant [FORMULA], [FORMULA] changes with k as [FORMULA], hence [FORMULA] as [FORMULA] (instead of being constant as in the diameter iTF). Then, an additional term is needed to Eq. (15), resulting in the total shift (in the simple model of Sect. 6.2) :


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 T = 8 where the correction term is practically zero. The shifts in the magnitude relation are smaller, as expected. The broken line shows what happens when the correction term is substracted from the magnitude relation shifts: the agreement with the observed zero-point shifts in the diameter relation is good.

[FIGURE] Fig. 13. Observed zero-point shifts for the inverse Tully-Fisher relation in magnitude (stars) and in diameter (open circles). The broken line shows the expected values for the diameter relation when the zero-point shifts in magnitude relation are known (see Eq. 15 and Eq. 17).
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998