Astron. Astrophys. 319, 435-449 (1997) 6. Photometric diameter D_{25} in the presence of disc + bulgeIn trying to understand the type effect, it is best first to turn one's attention to the photometric diameter , because this diameter may change when on the disc component one adds the bulge. Although this cannot explain the type effect, it is important to know when the diameter no more reflects only the underlying disc, but is also partially determined by the bulge. 6.1. Relations for the exponential surface brightness lawWe first collect several useful results from the exponential surface brightness law of de Vaucouleurs (1959): where is the surface brightness at radius r and is the scale factor. For a disc the parameter is equal to 1 and for a bulge it is equal to 4. E.g., putting equal to the value corresponding to 25 mag (arcsec)^{-2}, one obtains for either of these components the isophotal linear size . In magnitude units one may write: Here is the effective radius, with one half of the luminosity coming from inside . µ is in units of (arcsec)^{-2} and r in pc. For such profiles the total luminosities are given by : Using the above relations and transforming the surface brightnesses into mag arcsec^{-2}, one gets the following useful relations, connecting the isophotal size and the total luminosity to each other: Here the other units are 1 pc and 1 . An important observation, and also essential for much of the present study, is the small range of for real discs (= 21.67 0.3 Freeman 1970; Bosma & Freeman 1993). From this one obtains . 6.2. Shift of with increasing bulge fractionWe shall first have a general look at the importance of the bulge in galaxies of different types and with different sizes at the isophotal distance of from the centre. To this end the photometric data given by Kent (1985), Simien & de Vaucouleurs (1986), and Kodaira et al. (1986) were inspected. Fig. 8 gives the difference in surface brightnesses of the bulge and disc components for different types as calculated at the radius , according to the data in the mentioned catalogues. Though there are be some differences in the different collections of data, it seems clear that for 1 the disc component dominates at . In the case of Sa, the bulge may become important. Below we show that even there it is important only for quite small sizes.
It is important to know how within a fixed type, the contributions from the bulge and the disc at depend on this size, as especially this may influence the type effect in the TF-relation. Here one needs more data than provided by the above references and we try to use our own sample in the following manner: For each galaxy we can calculate its absolute magnitude using the kinematic distance and its apparent B -magnitude corrected for the inclination effect using the recipes in Bottinelli et al. (1995). The absolute magnitude may then be divided into luminosities and , using the average fraction , given by Simien & de Vaucouleurs (1986): k = 0.4, 0.32, 0.24, 0.16, 0.09, 0.05, 0.02, 0.01 for T =1,..,8, respectively. The photometric profile (scale length ) for the disc may be easily obtained putting = 21.5, corresponding to the strict proportionality . Unfortunately, for the bulge one cannot assume a similar regularity. However, it is natural to think that there is still some simple relation between the effective radius and the luminosity , and inspection of Kent's (1985) data (Fig. 9) led us to write
For the derivation of this relation from Kent's r -data, we have used the colour =1.3 for the bulge. We also confirmed (Fig. 10) the value of = 21.5 from Kent's data, using now the colour = 0.7 for the disc alone (Sect. 7.1). The relation (8) differs from the steeper one derived by Kormendy (1977), but we note that he used early type spirals within a narrow absolute magnitude range for the bulges.
From , relation (8), and Eq. (5) we derive and for the bulge. Having now in hand the disc and bulge profiles and (in a statistical sense) for the given galaxy, we find the radius r where the sum of the profiles add to the surface brightness of 25 mag (arcsec)^{-2} by solving the equation or At this , the surface brightness difference is shown in Fig. 11 for different types and diameters. The different panels suggest that the bulge contribution is important only for quite small galaxies even for types 1 and 2, and almost always the radius is determined by the disc.
The above results make it unlikely that the bulge + disc composition could significantly contribute to any type dependence simply through the shift in caused by the bulge. However, some signs of this effect might be seen in the diameter TF relation for early types and small sizes. Together with the small dispersion in , this also implies that fixing the linear isophotal size logD, gives us about identical discs. We have confirmed those conclusions by calculations based on the analytical expressions for the profiles. A bulge is added on a disc and it is looked how , originally measured on the pure disc, changes. This problem again requires solving Eq. (9) similarly as above. Our calculations show that the shift caused by the bulge, even for early types (large k) and small disc sizes, is quite small. For example, when adding an Sa -bulge (defined by k = 0.4) to a small disc ( =1.2) or to a large one ( =1.8), one obtains respectively an increase of 2.65% or 1.2% for the measured diameter. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |