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Astron. Astrophys. 351, 869-882 (1999)

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Appendix A: the optical growth curve

Let [FORMULA] be defined by

[EQUATION]

where [FORMULA] is such as [FORMULA].

According to Prugniel & Héraudeau (1998), [FORMULA] is defined by [FORMULA], where [FORMULA], if [FORMULA] or [FORMULA] - corresponding to a Sérsic (1968) luminosity profile [FORMULA] -, and by

[EQUATION]

if [FORMULA], i.e., interpolated between the de Vaucouleurs profile and the exponential profile valid respectively for giant ellipticals and pure-disk galaxies.

Appendix B: computation of NIR magnitudes and uncertainties

Let us first define [FORMULA], [FORMULA], where [FORMULA] is the weight attributed to the [FORMULA] point, [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

Two methods have been used, depending essentially on the number of observations and their accuracy. In the first one, the total NIR magnitude [FORMULA] is computed assuming [FORMULA] by minimizing [FORMULA] with respect to [FORMULA]. We obtain

[EQUATION]

[EQUATION]

and the corresponding effective magnitude is

[EQUATION]

[EQUATION]

We assume here that, for want of constraint on s, the uncertainty [FORMULA] on s is equal to the intrinsic scatter [FORMULA] for this type.

In the second method, s is considered as a free parameter and [FORMULA] is computed by minimizing

[EQUATION]

with respect to [FORMULA] and s. We then get

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

Note that we have made the questionable assumption that the errors in the aperture magnitudes of a given galaxy are independent.

We are however not primarily interested in the best fitting growth curve, which would be determined assuming [FORMULA], but rather in the best estimate of the asymptotic magnitude. When there is only one point, the uncertainty on [FORMULA] is [FORMULA], i.e., the uncertainty on [FORMULA] decreases with the size of the aperture. It is therefore reasonable to adopt [FORMULA] for all the points to give more weight to the large apertures.

To deal with outliers, we apply an iterative procedure. We initially assume [FORMULA] for the "slope" s and [FORMULA] for all the points. At each step, we fit the growth curve to the data, compute [FORMULA] from above equations, and estimate new values of the uncertainties by [FORMULA]. This procedure is a compromise between our a priori uncertainty [FORMULA] and the a posteriori estimate [FORMULA], where [FORMULA] or 2 is the number of fitted parameters (and also the index of the method!). It has the advantage of reducing the weight of outliers and takes the scatter around the growth curve automatically into account. Practically, it improves the fit and gives a reasonable estimate of the uncertainties. The convergence is usually achieved in a few iterations.

The second method provides a better fit than the first one but is less secure, because the "slope" s of the growth curve is free. It is used only when following conditions are fulfilled simultaneously:

  1. [FORMULA], which allows to check the validity of the fit,

  2. [FORMULA]

  3. and

    [EQUATION]

    i.e., the fit provided by (2) is better enough than (1) to justify the deviation of s relatively to [FORMULA].
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© European Southern Observatory (ESO) 1999

Online publication: November 16, 1999
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