4 System stability

In this section, we test the stability of the t system with respect to morphological type, size of atmospheric seeing disc, degree of smoothing and limiting isophote. These tests are based on CCD surface photometry of four galaxies of different morphological types. Details of the original observations and of the early stages of the reduction procedures are listed and/or referenced below for each galaxy individually.

Spiral (Sbc) at: $(\alpha,\delta)_{1950.0}=$22$^{\rm h}$01$^{\rm m}$06$^{\rm s}$, -20$^{\circ}$07$^\prime$24$^{\prime\prime}$. One 200-s B-band exposure of this galaxy was used; the frame having been taken in 1989 using the RCA CCD chip at the prime focus of the 2.5-m Isaac Newton Telescope. The FWHM of the seeing disc at the time of observation was 1.$^{\prime\prime}$A 300$\times$300 0.$^{\prime\prime}$-pixel subsection of the field was used after being cleaned of stars and other galaxies. For details of the reduction and calibration procedures adopted, see Metcalfe et al. (1995).

Lenticular: NGC 7180. One 600-s B-band exposure of this galaxy was used; the frame having been taken in 1989 with the TI CCD chip on the 0.9-m telescope of the Cerro Tololo Inter-American Observatory. The FWHM of the seeing disc was about 1.$^{\prime\prime}$The original 396$\times$396 array of $0\hbox{$.\!\!^{\prime\prime}$}494$ pixels was binned up to a 197$\times$197 array of 0.$^{\prime\prime}$pixels (by omitting the peripheral pixels in the original array). A 120$\times$120 subsection of the resulting frame was then used after it had been cleaned of stars and other galaxies. For details of the reduction and calibration procedures adopted, see Metcalfe et al. (1995).

Classical elliptical: NGC 6411. One 200-s B-band exposure of this galaxy was used; the frame having been taken using a Loral CCD at the prime focus of the 4.2-m William Herschel Telescope on La Palma. The observation was made at UT 22:08 on 1997 September 4, when the FWHM of the seeing disc was 1.$^{\prime\prime}$Stellar images on the frame were removed using Starlink's GAIA package and the photometric zero point was based on observations of Landolt (1983) standard stars. The original 2048$\times$2048 array of 0.$^{\prime\prime}$pixels was binned up to a 512$\times$512 array of 1.$^{\prime\prime}$pixels. A 360$\times$360 pixel subsection of the binned-up array was then used.

Dwarf elliptical: NGC 147. This galaxy is a member of the Local Group and lies within the vicinity of M 31. Nine 600-s and four 900-s exposures of this galaxy were made of it using the CCD camera on the 60/90-cm F/3 Schmidt Telescope of Beijing Astronomical Observatory's (hereunder BAO) Xing Long Station. The observations were made between UT 14:03 and 17:08 on 1996 October 18, when the FWHM of the seeing disc was 2.$^{\prime\prime}$The CCD chip used was a Ford device which had an array size of 2048$\times$2048 pixels, and a corresponding field of view of 54$^\prime$37$^{\prime\prime}$$\times$54$^\prime$37$^{\prime\prime}$. In the absence of a broad-band filter, an i-band Beijing-Arizona-Taipei-Connecticut (hereunder BATC) survey filter was employed. The BATC filter system is described at length by Fan (1995) and briefly by Fan et al. (1996) who refer to the i-band filter as Filter No. 9. This filter's transmission curve peaks at a wavelength of 6600 Å and has a FWHM of 480 Å. After bias subtraction, flat fielding and the removal of spurious images caused by cosmic-ray events; all thirteen CCD frames were stacked, thereby yielding a single frame whose effective integration time was 9000 s. The procedure adopted for these reductions was the same as adopted by Fan et al. (1996). As Hodge (1976) found no evidence for any global colour gradient in NGC 147, we were able to transform the i-band images directly to the B system by calibration with the $r_{\rm TG}$-band surface-brightness profile of Kent (1987) and the transformation, $B=r_{\rm TG}+1.21$ from Young & Currie (1994). After the calibration process, the stacked frame was binned up to one with 15.$^{\prime\prime}$pixels.

Although NGC 6411 and NGC 147 were not observed by Metcalfe et al. (1995), the same software as used by those authors was applied to the reduced but unsmoothed frames of these galaxies in order to generate Kron-system total magnitudes for them. The values obtained were B_K=12.88 and 10.36 respectively.

  

Figure 4: The simulated effect of poor to very poor seeing conditions on the surface-brightness profiles of bright galaxy images: (a, b, c and d) a spiral, (e, f, g and h) a lenticular, (i, j, k and l) a classical elliptical and (m, n, o and p) a dwarf-elliptical galaxy. This effect is analogous to the effect of ordinary seeing conditions on more distant galaxies of the same type and physical size. The image resolution function adopted was that of Moffat (1969) and the FWHM of the synthetic seeing discs are shown in arcsec. The curves represent model Sérsic profiles fitted to all plotted isophotes

  

Table 1: Best-fitting Sérsic model parameters for the synthetic seeing-distorted surface-brightness profiles depicted in Fig. 1, and for the same profiles but with brighter limiting isophotes

• $^{\rm a}$ $\nu$ represents degrees of freedom (number of isophotes minus two).
• $^{\rm b}$ These values were (after system transformation when relevant) adopted in Table 2.

For each galaxy, four synthetic low-resolution images were generated. This involved the convolution of each original or stacked image with Moffat (1969) functions of ($\sqrt{16-d^2}$)$^{\prime\prime}$, 8$^{\prime\prime}$, 16$^{\prime\prime}$ and 32$^{\prime\prime}$ FWHM in the case of the classical galaxies or with the same functions of 1$^\prime$, 2$^\prime$, 4$^\prime$and 8$^\prime$ FWHM, in the case of NGC 147; where d was the FWHM of seeing disc at the time of the original observation(s). In order to minimize edge effects, each digital image was embedded in a very much larger array of pixels (in which each pixel in the surrounding grid was set to zero) before any convolution was performed.

The Moffat function was chosen on this occasion in order to simulate both the effect of poor seeing on nearby objects and the effect of average seeing on distant objects. Note that in adopting a Moffat function here, we are actually applying a very much more stringent test on the stability of the t system than we would have done had we adopted the Gaussian function that we recommend for the purpose of smoothing. This is because the Moffat function is a much more complicated function than the Gaussian one, being similar to the Gaussian at small radial distances but falling off much more slowly at larger radial distances.

Godwin's (1976) image-segmentation software, as outlined by Carter & Godwin (1979), was used in order to fit elliptical isophotes of 0.25 mag arcsec^-2 separation (each defined by a mean radius r, an ellipticity and a position angle) to all of the synthetic low-resolution images. The isophotes were weighted according to the simple algorithm: $\sigma_{\mu}=0.05$ for $\mu\leq20.0$ or $\sigma_{\mu}=0.02(\mu-20.0)$ for $\mu\gt 20.0$. The resulting synthetic surface-brightness profiles are plotted in Fig. 1 together with the best fitting Sérsic model profiles; whilst the corresponding model parameters are tabulated in Table 1, which also lists those model parameters obtained when different limiting isophotes were applied.

As can be seen from Fig. 1 and Table 1, Sérsic's model yields very consistent results for all of the synthetic images except for the two highest resolution images of the classical elliptical (which, at 12th magnitude is in fact a very bright object) and the highest resolution image of the dwarf elliptical. In these three cases one-component profile models appear not to be completely adequate. However, once the resolution of a galaxy image has been degraded sufficiently, even if purely by seeing effects, the t-system total magnitudes obtained do appear to be stable typically to a couple of percent or so, irrespective of morphological type, the size of the seeing disc, or the limiting isophote-provided that the limiting isophote is not so bright that there are too few isophotes to fit.

Note that under normal circumstances, when the seeing disc is not almost as large as the galaxy image itself and the resolution of the image can be deliberately degraded by convolution with a Gaussian function (or even a simple Hanning function), the level of stability with respect to image resolution must be even greater than this. This is because the synthetic surface-brightness profile obtained by convolving any galaxy image with a Gaussian function of large FWHM, must be more Gaussian than the original profile and therefore more likely to be well described by Sérsic's law (as the Gaussian function, unlike the Moffat function for example, can be perfectly described by Sérsic's law).

We also tested the system for stability with respect to different weighting schemes for the isophotes, and found that whilst altering the weightings had very significant effects on the $\chi^{2}$ values obtained, and reasonably significant effects on which best-fitting parameters were adopted, the effects on the total magnitude values obtained were only at the one or two per cent level-for realistic weighting schemes at least.