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:
Consider an elliptical reference contour CR having the same MajA
orientation as the galaxy to be analysed: its axis ratio is q0 and its
MajA length a0. 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 q0 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 a0 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 a0 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 a0.
If the starting a0 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 a0'
somewhat larger than a0. Since the disk SuBr at a0' 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 a0, 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 a0 lead
to systematically varying , and
solutions.
It is then necessary to select carefully the starting value of a0 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 a1 value,
initially as an asymptotic disk model, it will be continued inwards by
selecting an a0 contour, with a0 < a1, 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 a1
and a0. 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 a0 one should take large
enough, and interpolate back the solution to a MajA value a0'
closer to a1. 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 a0 values.
, or
, or both. These are derived from an examination and
interpolation of the preliminary solutions. A number of programs have been
written to continue inward an asymptotic model, with various constraints
upon the axis ratio of one, or both, of the two components.
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.
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).
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.