contribution

Since the pioneer work of Freeman (1970), the analysis of galaxies in terms of their bulge and disk components has been a fashionable exercise. Reviews of these topics have been given by Capaccioli & Caon (1990), and by Simien (1991).

Most of the effort has been devoted to bulge/disk decompositions based upon the fitting of ad hoc analytical formulae to the 1D or 2D surface brightness (SuBr) distribution. Recent publications of this kind include those by de Jong & van der Kruit (1994), Andredakis & Sanders (1994), Byun & Freeman (1995). Some authors however, have tried other approaches with less stringent a priori constraints. It was then necessary to consider objects projected at such inclinations to the line of sight, that the forms of the isophotes convey information about the two-components structure.

As an aid to derive bulge+disk models from detailed isophotal analysis, a
popular approximation has been the *Oblique Thin Disk*, adopted by
Simien & Michard (1990), and by Scorza & Bender
(1990) and (1995). A thin disk with circular symmetry will project
as a series of concentric ellipses of constant axis ratio, independently of
complexities in the radial SuBr distribution of the disk, such as ring or
lenses. This constraint will greatly help to derive the bulge and disk
models. The *Oblique Thin Disk* approximation will possibly not remain
valid if the inclination becomes so large that the intrinsic thickness of
the disk, i.e. its vertical light distribution, contributes to the geometry
of the isophotal contours.

Another approximation, here termed the *Edge-on Very Thin Disk* case,
has been introduced by Capaccioli et al. (1987), and again
used by Caon et al. (1990). This approach has been technically
improved by Seifert & Scorza (1996). It is then assumed that
the disk is so thin, and the inclination so close to 90 degrees, that the
disk does not contribute to the SuBr near the minor axis of the isophotes
but only near their major axis. Then the bulge SuBr and ellipticity
distributions may be recovered by fitting ellipses to limited isophotal
arcs surrounding the minor axis. A bulge model is thus obtained, and the
disk will be simply found by substracting this bulge model from the image.
This method might give satisfactory results if the underlying hypothesis
are valid, but unfortunately this validity cannot be tested. On the other
hand, the existence of thick disks seems to be well established. According
to the "quantitative morphology" of Michard & Marchal (1993, 1994a,
b) and Michard (1994), thick disks not unfrequently
dominate the outer envelopes of S0's galaxies. This makes the *Edge-on
Very Thin Disk* hypothesis quite uncertain for disk dominated lenticulars.

In the present contribution, the bulge and disk models of disky E and S0
galaxies are derived from image analysis within the assumption that both
components have concentric and coaxial elliptical isophotes, with
arbitrary ellipticity profiles and SuBr distributions. The *Oblique
Thin Disk* approximation, or the one adopted by Kent (1986)
(i.e. constant bulge and disk ellipticities) are special cases of our
assumptions. Since our techniques provide complete bulge and disk models,
the results can be checked by synthesising these models, adding the two
corresponding images, and comparing the sum to the input galaxian image.
The contribution of bulge and disk to the SuBr at the minor axis of an
isophote can also be calculated, and the *Edge-on Very Thin Disk*
hypothesis thus tested.

It should be emphasized however, that real galaxies do not necessarily comply with the assumptions adopted here. Even neglecting such complications as dust lanes or isophotal twists, and considering only images with two orthogonal axis of symmetry, these will not necessarily be well represented by the sum of two families of concentric and coaxials ellipses. This approximation is most likely to break in the case of edge-on, disk dominated objects.

In Sect. 2.1 are given the equations relating observable quantities
to the corresponding ones, as calculated from the two-components model.
These quantities are the mean value and low order even harmonics of the
SuBr distribution along suitable reference ellipses. The important
simplification occuring when these ellipses are the bulge isophotes is
pointed out. In Sect. 2.2 is discussed the derivation of "external
exponential models": the solution of this "restricted" problem is very
useful for the derivation of an asymptotic disk model, which is a necessary
ingredient of the complete solution. In the following sections are outlined
our techniques for deriving and checking the complete solution: the inwards
*unconstrained* continuation of the asymptotic model in Sect. 2.3;
the improvement of the solution by introducing ad hoc constraints upon
the ellipticities in Sect. 2.4; and finally the
control of the solution from the 2D relative residuals .

In Sect. 3 the results of the analysis are given for 9 test galaxies covering a large range of possible inlinations and D/B ratios. The derived models are shown graphically and commented upon in individual notes.

Finally Sect. 4 offers some comparisons with the results of
other authors. A rather good agreement is found between our models and
those obtained by Scorza & Bender (1995) with the *
Inclined Thin Disk* approximation. On the other hand the *Edge-on Very
Thin Disk* approximation, as used by Seifert & Scorza
(1996), give disk profiles in strong disagreement with our
solutions, except for galaxies with an intrinsically faint disk.

It may be useful to collect here the notations and abbreviations currently used below.

. SuBr surface brightness.

. MajA major axis; MinA minor axis.

. *a*, *c* major and minor axis of a Reference Ellipse, i.e. the one
used in Carter's representation of isophotal contours.

. *e*_{i}, *f*_{i} coefficients of cosine and sine terms in Carter's
harmonic representation of deviations from the Reference Ellipse. Note that
this ellipse is defined by its MajA *a* and the conditions
*e*_{1}=*f*_{1}=*e*_{2}=*f*_{2}=0.

. *q* axis ratio.

. ellipticity.

. *diE*, *boE*, *unE* subclassification of ellipticals as disky,
boxy or undeterminate; *p* added for peculiar envelopes.

. *spH*, *thD*, *exD* classification of envelopes, as
spheroidal haloes, thick disks, extended disks respectively.

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