(1) |

(2) |

(3) |

The structural parameters have been determined
by fitting the photometric data to the model, convolved with a circular
Gaussian seeing disk of appropriate FWHM, using a minimization.
The fitted parameters are:
the two surface brightnesses and (or
and when given in magnitudes),
the two scale lengths and , the bulge ellipticity
, and the system inclination *i*.
Because of the difficulty inherent in estimating a luminosity distribution
of unknown form (bulge) which is partially embedded in another (disk),
we did not attempt to fit the bulge index *n* explicitly (cf.
Andredakis et al. 1995).
Instead, for each galaxy the bulge was modeled
with four different values of the exponent *n*:
1, 2, 3, and 4 ( for a de Vaucouleurs bulge).
We then assessed the ``quality'' of the results for each set of 2-D fits,
and determined the value of *n* that gave the best all-round fit, thus
assigning, in effect, a shape index to the bulge.
Three quality indicators were considered in the assessment:
the successful convergence of the process,
the value of , and the mean absolute residual.

When both *J* and *K* data were available,
tests showed that bulge and disk decompositions and
colors of the components were more stable
when the two bands were fitted simultaneously.
We also found no evidence for systematic scale length
changes with wavelength of either component (cf., Evans 1994).
Hence, when two bands were available, we performed
a single fit keeping scale lengths, inclination, and bulge ellipticity
the same for both bands, and letting the surface brightnesses of both
components vary independently, but with constant color indexes
and .

The decomposition technique was extensively tested on
a set of synthetic maps of axisymmetric bulge+disk distributions,
covering a range of values for *i*, , and *B*/*D* ratio similar to ours.
We adopted an exponential thin disk, and generalized exponential laws for
the bulge.
An appropriate amount of noise was added to
simulate the typical signal-to-noise levels we achieve, and finally
the models were convolved with a range of PSF's to
simulate the effects of seeing.
The 2-D decomposition method was always able to recover the true parameter set,
within the estimated errors, independently of the starting point in
parameter space.

We also performed a 1-D fit to the radial
profile, obtained by averaging along elliptical annuli;
center and position angle of the annuli were held fixed, while the
ellipticity was allowed to vary with semi-major axis.
In the 1-D case, the convolution of the model with the PSF requires
*i* and to be assigned a priori:
as customary in 1-D techniques, was fixed at 0 and *i*
was determined from the outer *J*-band isophotes.
The best-fit bulge index *n* was determined in the same way as for
the 2-D models.
1- and 2-D techniques are compared in Sect. 4.1.

The lower left panels in Fig. 1 show the results of the
2-D decompositions in terms of cuts along the semi-major axis (*K* band).
Table 2 gives the best-fit parameters, including
the shape index *n*:
the first line reports the values from the 2-D fit,
the second reports the 1-D ones, and
the third gives the values from the non-parametric decompositions
(Sect. 3.2).
Bulge and disk
surface brightnesses are corrected to face-on values assuming optical
transparency with *C*=1:

(4) |

Line 2: 1-D parametric decompositions. Line 3: K-band 2-D non-parametric decompositions. |

Kent (1986) was the first to introduce a non-parametric (*np*) method
to separate bulge and disk;
recently this has been extended to the profile analysis of 2-D
images by Andredakis et al. (1995).
We propose here a new, completely 2-D technique.
Assuming the apparent ellipticities of the two components,
and , are known,
let us consider the average surface brightness
of an annulus of ellipticity, say, .For a sufficiently narrow annulus, *A*, the disk contribution
will be , where
*a* is the semi-major axis.
The total average brightness will be:

(5) |

(6) |

(7) |

When defining the two sets of annuli, a proper sampling of the semi-major axis is of crucial importance. A reliable estimate of the integrals in Eq. (7) requires that each annulus of either set intersects with a sufficient number of annuli of the other set. If we constrain this number to be the same for every ellipse, the sampling is linear in , i.e. the annuli become wider with increasing radius.

Results of the *np* decomposition technique are given in the
third line of Table 2,
and in the second line of Table 3.
The non-parametric parameters are defined as follows. The bulge
is the semimajor axis of the ellipse enclosing half light;
is the bulge surface brightness at ; is the
semimajor axis of the ellipse enclosing 0.264 of the disk light,
as in the exponential case. Since the central disk is not
tightly constrained by the *np* fits, is the central brightness
of an exponential disk having scale length and the same
total luminosity as the *np* disk. The *np* colors of the components are
derived by the total component luminosities in the two bands.

The *np* decomposition algorithm was tested
on the same set of synthetic maps
used for the parametric tests.
Since the method applies only to components with different apparent ellipticity,
we only tested models with
at least 10% smaller than .
Under these conditions, to within the noise level,
the method always proved to be able to recover the true
bulge and disk distributions given the correct values
of the ellipticities.

When errors on the ellipticities are introduced,
the shape of the resulting components is affected in a systematic way:
the value of affects mainly the inner part of the disk,
while determines the shape of the outer bulge.
In particular, if is underestimated the inner disk
is too steep, whereas if the bulge is too
elliptical the inner disk develops a hole which gets deeper as increases.
If is too small the outer bulge is too steep and viceversa.
Although qualitatively these are expected trends, they
are difficult to quantify in practice,
since they depend on the
shapes of the distributions, on the ellipticities,
and on the seeing.
Especially when the two components have similar
ellipticities, even small errors on and (<0.1)
perceptibly affect parameters such as the
luminosity of the components or the *B*/*D* ratio, as well as the
shape of the profiles.

Simulations show that, if no correction is applied,
seeing significantly affects *np* decompositions.
The effects are stronger in the inner parts,
the main feature being a central
hole in the disk, whose extension is proportional to the seeing
width.
In fact, the seeing makes
inner isophotes rounder, causing the component of higher ellipticity
to be depressed. The disk profile, however, can be
corrected by extrapolating the outer points to the center.

To recover the intrinsic bulge profile, we introduced a second iterative algorithm which proved to be quite effective on synthetic maps. The effect of seeing is accounted for by defining a correction coefficient , for each point of the bulge profile, as the ratio of the convolved bulge distribution and an estimate of the true one. Each decomposition is performed using to degrade the assumed true distribution. After decomposition, the coefficients are recomputed by convolving the extracted bulge profile with the PSF, and a new decomposition is performed until both the coefficients and the profiles converge. The initial estimate of is provided by the convolution of the bulge profile extracted without any correction.

To determine the apparent ellipticity of the components,
we extracted from each image a radial
profile of the ellipticity by fitting
isophotes of variable ellipticity and fixed position angle.
We then selected
the lowest value beyond one seeing
disk from the center as ,
and the average value in the outer parts as (see also Andredakis et al. 1995).
The values for the *J* and *K* bands were then averaged
to obtain a single value of and for each
galaxy.
The accuracy of these estimates is difficult to assess:
bright bulges can affect the
isophotes even at large radii, leading to
an underestimate of , while bars
and spiral arms may often lead to overestimates.
In turn, can be easily underestimated because of the
seeing.
From simulations, it seems
that a reasonable estimate for the uncertainty in is
about 0.1; it is certainly worse in ,
0.2 or more in the worst cases.

We note that comparing the observed image to the extracted
bulge+disk distribution, for example in terms of residuals,
is of little use to test the ellipticities,
since it is often possible to find excellent agreement
for a wide range of and .A more stringent constraint is provided by the *color* profiles of bulge
and disk, which turn out to be quite sensitive to the values of and
. This has ultimately allowed us to verify the ellipticities
a posteriori (in particular ) by
letting them vary within the estimated errors
and checking the plausibility of the resulting color profiles.
The final accuracy is estimated to be better than 0.05.

Line 2: 2-D non-parametric decompositions. |

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