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2. Principles and implementation of the method

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:

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

2.1. General equations

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 tex2html_wrap_inline1196 of the MajA are passing:

. one bulge isophote of SuBr tex2html_wrap_inline1198 and axis ratio tex2html_wrap_inline1200

. one disk isophote of SuBr tex2html_wrap_inline1202 and axis ratio tex2html_wrap_inline1204

Through a point M of the contour of eccentric anomaly tex2html_wrap_inline1206 are also passing one bulge isophote and one disk isophote. Their major axis are respectively

with the definitions tex2html_wrap_inline1208; tex2html_wrap_inline1210; tex2html_wrap_inline1212 and

The contributions to the SuBr in M are respectively tex2html_wrap_inline1214 and tex2html_wrap_inline1216. 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 tex2html_wrap_inline1222 and of the disk tex2html_wrap_inline1224 are expressed by replacing tex2html_wrap_inline1226 by tex2html_wrap_inline1228 or tex2html_wrap_inline1230 in the integrals (3).

Note that these integrals involve limited ranges of the bulge and disk models, i.e. tex2html_wrap_inline1232 for the bulge, and tex2html_wrap_inline1234 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 tex2html_wrap_inline1238 and tex2html_wrap_inline1240 at each MajA value. It is however convenient to select the (unknown!) bulge isophotal contours as reference. Then the system (4) is greatly simplified: tex2html_wrap_inline1242 for tex2html_wrap_inline1244, and vanishes for tex2html_wrap_inline1246. Also note that the equation for tex2html_wrap_inline1244 is the only one containing the bulge SuBr; all the other equations involve only the disk model for tex2html_wrap_inline1250. Unless otherwise noted, this choice of the contour RC is assumed in the following.

2.2. The asymptotic exponential disk

The applications of the general equations with the choice of the bulge contours for RC's, necessitates the knowledge of the disk SuBr and tex2html_wrap_inline1240 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 tex2html_wrap_inline1258 is large. This knowledge is provided by an asymptotic exponential disk model, defined by its SuBr, tex2html_wrap_inline1240 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 tex2html_wrap_inline1238 variations can also be neglected, then the integrals tex2html_wrap_inline1224 can be rewritten as
with n=0, 2, 4....The quantity tex2html_wrap_inline1272 is the scale factor of the disk for tex2html_wrap_inline1250. Now the equations for tex2html_wrap_inline1276 contain only the four unknowns tex2html_wrap_inline1202, tex2html_wrap_inline1238, tex2html_wrap_inline1240 and tex2html_wrap_inline1272 while the equation for tex2html_wrap_inline1244 could give tex2html_wrap_inline1198 to complete the model. It has been verified that the harmonics for tex2html_wrap_inline1290 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 tex2html_wrap_inline1202 and tex2html_wrap_inline1272, this second contour adds 4 equations for tex2html_wrap_inline1302 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 tex2html_wrap_inline1240, while the bulge has reached a quasi constant tex2html_wrap_inline1238. Therefore the straightforward solution of this restricted problem at one or several values of a0, will give essential information about local values of tex2html_wrap_inline1238, tex2html_wrap_inline1240, tex2html_wrap_inline1272 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 tex2html_wrap_inline1238, tex2html_wrap_inline1240 and tex2html_wrap_inline1272 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.

2.3. A general technique for the inward continuation of the model

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 tex2html_wrap_inline1338, tex2html_wrap_inline1340, tex2html_wrap_inline1202 and tex2html_wrap_inline1198: these 4 unknowns are to be derived from the 4 equations (4) with tex2html_wrap_inline1346. The integrals tex2html_wrap_inline1224 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 tex2html_wrap_inline1354 is small, the tex2html_wrap_inline1224 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 tex2html_wrap_inline1354 large enough, and interpolate back the solution to a MajA value a0' closer to a1. If however tex2html_wrap_inline1354 is large, the needed interpolation of disk parameters do not favour accuracy in the evaluation of the integrals tex2html_wrap_inline1224.

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 tex2html_wrap_inline1238 and tex2html_wrap_inline1240, and thus guide the choice of constrained, and hopefully better, solutions. It has been found useful, in order to get better estimates of the tex2html_wrap_inline1238 and tex2html_wrap_inline1240 profiles, to repeat the operation described here with two or more choices of the asymptotic model, starting at different a0 values.

2.4. Inward continuation of the model using ad hoc

To improve the solutions obtained at the preceding stage Sect. 2.3, constraints are introduced upon the run of tex2html_wrap_inline1240, or tex2html_wrap_inline1238, 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.
  1. constant tex2html_wrap_inline1240 as input, the remaining unknowns being tex2html_wrap_inline1238, tex2html_wrap_inline1202 and tex2html_wrap_inline1198. This corresponds to the assumption described in Sect. 1.1 as the Oblique Thin Disk Model. The constant tex2html_wrap_inline1240 is however selected as the mean of the values obtained by the general technique Sect. 2.3, not by trial and error.
  2. tex2html_wrap_inline1240 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.
  3. when one has to deal with a disk dominated or bulge dominated object, it is preferred to have more stringent constraints. Both tex2html_wrap_inline1240 and tex2html_wrap_inline1238 are then taken from an input table, again derived from the previous step, and the remaining unknowns tex2html_wrap_inline1202 and tex2html_wrap_inline1198 are taken from an ad hoc combination of the Eq. (4).
  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.
  5. a special case also considered is the assumption of constant input tex2html_wrap_inline1238 and tex2html_wrap_inline1240, for galaxies of moderate inclinations, with small deviations from ellipses of the observed isophotes.

2.5. Control of the solution

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 tex2html_wrap_inline1158 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 tex2html_wrap_inline1240 and/or tex2html_wrap_inline1238 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 tex2html_wrap_inline1158 within the significant area.

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