The general equations relating selected observables to the looked for model are written down in Sect. 2.1 below. Various techniques for their solution might possibly be developed. The one adopted in this work involve several consecutive steps:

- derivation of an
*asymptotic disk model*together with associated local bulge parameters (see Sect. 2.2) - inward continuation of this outer model
*without constraints*upon the solution (see Sect. 2.3) - improvement of the preliminary model by introducing ad hoc constraints derived from the preceeding results (see Sect. 2.4)
- control of the solution (see Sect. 2.5).

Consider an elliptical reference contour CR having the same MajA
orientation as the galaxy to be analysed: its axis ratio is *q*_{0} and its
MajA length *a*_{0}. Through the tip of the MajA are passing:

. one bulge isophote of SuBr and axis ratio

. one disk isophote of SuBr and axis ratio

Through a point M of the contour of eccentric anomaly are also
passing one bulge isophote and one disk isophote. Their major axis
are respectively

with the definitions ; ;
and

The contributions to the SuBr in M are respectively
and . Upon the contour RC one may *measure* the
quantities

with *n*=0, 2, 4, 6....
These observables are related to the bulge and disk models through the
equations

with *n*=0, 2, 4, 6..., where the contributions of the bulge
and of the disk are expressed by replacing by
or in the integrals (3).

Note that these integrals involve limited ranges of the bulge and disk
models, i.e. for the bulge, and
for the disk. These ranges are here
termed the *working ranges* of the model.

In principle a family of reference contours could be arbitrarily selected
to relate the bulge and disk models to observed quantities, provided of
course that their *q*_{0} is intermediate to the and
at each MajA value. It is however *convenient to select the (unknown!)
bulge isophotal contours as reference*. Then the system (4) is greatly
simplified: for , and vanishes for
. Also note that the equation for is the only one
containing the bulge SuBr; all the other equations involve only the disk
model for . *Unless otherwise noted, this choice of the
contour RC is assumed in the following*.

The applications of the general equations with the choice of the bulge
contours for RC's, necessitates the knowledge of the disk SuBr and profiles *outside the range* of *a*_{0} where the complete
representation may be sought, eventually outside the field of the available
image! The quantity defined in Sect. 2.1 as "working range of a model"
may indeed be much larger than *a*_{0} if the ratio is
large. This knowledge is provided by an asymptotic exponential disk model,
defined by its SuBr, value and scale factor at the outermost
attainable *a*_{0}.

If the starting *a*_{0} is well inside an exponential disk, i.e. if the disk
can be considered exponential within the working range,
while the variations can also be
neglected, then the integrals can be rewritten as

with *n*=0, 2, 4....The quantity is the scale factor of the
disk for . Now the equations for contain only the
four unknowns , , and
while the equation for could give to complete the
model. It has been verified that the harmonics for do not bring
useful information, so that we need another equation to complete the
system. This is obtained by using an auxiliairy contour of MajA *a*_{0}^{'}
somewhat larger than *a*_{0}. Since the disk SuBr at *a*_{0}^{'} is known
from and , this second contour adds 4
equations for but no new unknown. Only one of these
equations is sufficient to make the problem fully determined.

This "local exponential model" is of great practical interest, because
many galaxies
contain extended stretches where the disk is exponential with constant
, while the bulge has reached a quasi constant .
Therefore the straightforward solution of this restricted problem at one or
several values of *a*_{0}, will give essential information about local values
of , , and their variations. If the
outer parts of the studied galaxy contain an exponential disk, the present
technique readily provides the needed asymptotic disk model.

For many galaxies however the geometrical evidence for the
two-components structure disappears at some distance from the center. In
this case attempts to derive an external exponential disk model do not
give reliable results: tests made at several nearby values of *a*_{0} lead
to systematically varying , and
solutions.

It is then necessary to select carefully the starting value of *a*_{0} and
the corresponding asymptotic model. One may be guided by Michard and
Marchal's classification of the envelopes of E-S0 galaxies, derived from
their Carter-like analysis. For the *spH* case, it is assumed that *
the disk vanishes* in an envelope which may be formally considered as
an extension of the bulge. For the *exD* case, the bulge becomes very faint
and the envelope is *an extension of the disk*. for the intermediate
*thD* case, the thick disk envelope may be considered either as an extension
of the disk with a nearly vanishing bulge, or as an extension of the bulge
with a vanishing thin disk. Depending upon the above cases, specific
constraints have to be introduced in the determination of the asymptotic
disk model to make sure it is compatible with the parameters for the
one-component envelope.

Remark: The here described derivation of an external exponential disk model is one of a family of "restricted problems" involving the local representation of the model by a small set of unknowns. It would perhaps be fesible to obtain a complete general model by merging a number of such local approximate models.

When the outer part of a disk model is known up to some *a*_{1} value,
initially as an asymptotic disk model, it will be continued inwards by
selecting an *a*_{0} contour, with *a*_{0} < *a*_{1}, with the unknowns , , and : these 4
unknowns are to be derived from the 4 equations (4) with . The
integrals can be evaluated from the known part of the disk
model completed by an ad hoc interpolation for the interval between *a*_{1}
and *a*_{0}. If is small, the integrals will be
quite unsensitive to the change of the disk parameters in this range. To
have a significant solution at *a*_{0} one should take large
enough, and interpolate back the solution to a MajA value *a*_{0}^{'}
closer to *a*_{1}. If however is large, the needed
interpolation of disk parameters do not favour accuracy in the evaluation
of the integrals .

According to these remarks, our software for continuing inwards the
asymptotic model *in the general case*, that is without introducing
constraints upon one or more of the variables, may give imprecise results,
*specially if one of the two components is much fainter than the other*.
It is however a necessary step in the present technique, because it will
indicate the eventual variations of and , and thus
guide the choice of constrained, and hopefully better, solutions. It has
been found useful, in order to get better estimates of the and
profiles, to repeat the operation described here with two or
more choices of the asymptotic model, starting at different *a*_{0} values.

constraints

- constant as input, the remaining unknowns being , and . This corresponds to the
assumption described in Sect. 1.1 as the
*Oblique Thin Disk Model*. The constant is however selected as the mean of the values obtained by the*general technique*Sect. 2.3, not by trial and error. - given in an input table, with the same remaining unknowns. This input table is again written from the results of stage Sect. 2.3. One of these treatments is adequate when the two components are not of too widely different SuBr.
- when one has to deal with a disk dominated or bulge dominated object, it is preferred to have more stringent constraints. Both and are then taken from an input table, again derived from the previous step, and the remaining unknowns and are taken from an ad hoc combination of the Eq. (4).
- another algorithm has been successfully tried, where the calculated SuBr of the models are bound to equal the measured ones at the tips of the two major and minor axis of the reference contours.
- a special case also considered is the assumption of constant input and , for galaxies of moderate inclinations, with small deviations from ellipses of the observed isophotes.

After the above operations have been performed, a complete model for both
bulge and disk has been obtained, extending from the asymptotic disk model
inclusive, up to an innermost MajA so selected that the region *much
affected by seeing or unsufficient sampling* is excluded. The output file
containing the SuBr and *q* profiles of the bulge and disk is then used
to synthesize the 2D image of both components, add these and compare the
results to the input galaxian image. This is done by calculating a map
of the of local SuBr. The solution is deemed satisfactory if
these residuals are less than 10% everywhere, while they are usually much
smaller near the MajA and MinA.
If this is not the case, one may try to repeat the last phase of the
procedure as described in Sect. 2.4, modifying the adopted
and/or profiles and recalculating the corresponding solution.

One should also consider the possibility that the object under study cannot be well represented by the sum of a bulge and disk component, both with elliptical isophotes. This might well be the case for edge-on galaxies where the transverse and longitudinal scale lengths of the disk are not necessarily strongly coupled. One should also keep in mind the frequent occurence of boxy bulges. Finally two-components disks, as found by Seifert and Scorza for several nearly edge-on galaxies, will not necessarily be well represented by concentric ellipses. If the assumption of elliptical isophotes for the disk seems to fail, one could accept the bulge model and obtain an unconstrained disk model by substracting the assumed bulge from the galaxian image. Unfortunately there is no way to control the quality of such a solution.

Remarks: The synthesis of images required for the 2D test of a solution as described above is made within the contour of an ellipse enclosing the part of the image where the model is assumed valid. For aesthetical reasons, the disk and bulge parameters are extrapolated to the galaxian center in the innermost 1-3 arcsec. A routine is available to measure the elementary statistical parameters of the residuals within the significant area.

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