Appendix A: Photometric uncertainties

Accurate detection and characterization of faint surface brightness features in galaxies requires a thorough understanding of the uncertainties in CCD photometry at very low light levels. Both systematic and statistical uncertainties are present, and can affect the photometry over different spatial scales. We consider here seven different sources of photometric uncertainty, and combine them to create an error budget for an area of arbitrary size in the combined mosaic of the ESO 342-G017 field. The error bars presented in Figs. 7 and 8 are calculated according to this error budget.

Our analysis of photometric uncertainties must reflect the process by which the deep, masked mosaics from which we derive surface brightness profiles were generated. In what follows, all fluxes and uncertainties will be expressed as numbers of electrons e^-, and i will be used as an index to label one of the $N_{\rm f}$ individual frames that were co-added to form the mosaic at that position.

The flux $F_{\rm\sc SM} (x,y)$ at any sky position in the mosaic is defined by:

$$\begin{eqnarray*}F_{\rm\sc SM} (x,y)& =& \left( \frac{\zeta}{\sum_i^{N_{\rm f}} ... ... &\equiv& {C_{\sc {\rm SM}}} \, \sum_i^{N_{\rm f}} {e}^-_i(x,y), \end{eqnarray*}$$

where ${\cal S}_i$ is the sky value (in e^-) on frame i, that is, the median of all background (non-object, non-masked) pixels in frame i. The quantity $\zeta$ is a normalization constant defined by

$$\zeta \equiv { \frac{1}{N_{\rm f}} \sum_i^{N_{\rm f}} \left[ \frac{1}{N_{\rm sky}}\sum_j^{N_{\rm sky}} {e}^-_{ij} \right] }$$

where j runs over all sky pixels on frame i, and i runs over all frames used anywhere in the matrix. On the other hand, note that if frame i did not contribute to the flux at position (x,y), due to it being masked, then e^-_i(x,y) = 0 and ${\cal S} _i = 0$. Thus $C_{\sc {\rm SM}}$ is a property of submosaics - portions of the mosaic composed from the same individual frames - and is of the order $N_{\rm\sc SM}^{-1}$, where $N_{\rm\sc SM}$is the number of frames contributing to submosaic SM. The electron flux at position (x,y) is thus the total flux of all frames contributing to the mosaic at that position, weighted by the total sky value on each part of the submosaic in order to account for frame-to-frame differences in transparency and total exposure time.

We now consider separately individual sources of photometric uncertainty within detector regions composed of $N_{\rm p}$ pixels spread over a total area A, $N'_{\rm p}$ of which are unmasked pixels that combine to form an unmasked area A'. Throughout, we will assume that all portions of the subarea A on the mosaic were constructed from the same individual CCD frames. All expressions for uncertainties are expressed in terms of numbers of electrons.

A.1 Read noise: ${\sigma _{RN}}$

The noise associated with reading the charge collected in the CCD array is associated with every pixel of the array. For the UT1 test camera, this noise has a random distribution with an rms (root-mean-square) value of 7.2e^-. Over an unmasked area A' on the detector, the uncertainty in the flux contributed by read noise is thus

$$\sigma_{\rm RN}(A') = 7.2 \, C_{\sc {\rm SM}} \, \sqrt{N_{\rm f} N'_{\rm p}},$$

assuming that each of the $N_{\rm f}$ frames contributing to the mosiac area A' had  $N'_{\rm p}$ unmasked pixels.

A.2 Flat-fielding: ${\sigma _{FF}}$

Flat-fielding was performed by constructing supersky flats from moonless UT1 Hubble Deep Field South and EIS images in the same bands taken during a 10-day period coinciding with our ESO 342-G017 observations. Dithering helped to ensure that sky objects did not fall on the same portion of the physical detector and could thus be removed in the median process (see Sect. 3.1). Nevertheless, Table 3 demonstrates that the fractional rms scatter $\widetilde \sigma _{\rm FF}(A)$in the flat field averaged over different size scales A does not scale with $1/\sqrt{A}$, a clear sign that the flat-fielding errors are not purely statistical. For scales larger than the Gaussian FWHM of the seeing disk ($\sim$1 $\hbox{$^{\prime\prime}$ }$), subtle extended light from sky objects may not be entirely removed by the supersky flat median process, creating an increase in the flat-fielding residuals on these scales. On the largest scales, systematic errors are nearly an order of magnitude larger than those due to counting statistics in the flat-fields.

The fractional flat-fielding uncertainties $\widetilde \sigma _{\rm FF}(A')$ from Table 3 must be multiplied by the total unmasked flux in area A' on frame i, and then combined to yield the flat-fielding uncertainty within an area A' on the mosaic. Since the science frames were dithered by more than 10 $^{\prime\prime}$ in each direction, larger than any area considered here, any (x,y) position on the mosaic is constructed with images that were flat-fielded at different positions on the physical detector. Thus the flat-fielding uncertainties of individual frames can be treated as being independent and added in quadrature. For the mosaic we thus have

$$\begin{eqnarray*}\sigma_{\rm FF}(A') &=& C_{\sc {\rm SM}} \, \sqrt{ \sum_i^{N_{\... ...^{N_{\rm f}} \left( \sum_j^{N'_{\rm p}} e^-_{ij} \right) ^2 }~, \end{eqnarray*}$$

where j runs over all unmasked pixels in the area A' on frame i and $\widetilde \sigma _{\rm FF}(A')$ can be pulled outside the sum because the masking is performed on the mosaic and thus is identical for all individual frames i.

A.3 Photon noise: ${\sigma _{PN}}$

The photon noise is essentially uncorrelated over areas larger than the FWHM of the PSF, so that it is given by the square root of the number of electrons within that area. For a given frame, we thus compute the uncertainty due to photon noise within areas  $A_{\rm PSF}$ comparable to the PSF, and then add these in quadrature. The uncertainties for individual frames are independent, and can be added in quadrature to yield the total uncertainty due to photon noise within an unmasked area A' on the mosaic. Since the uncertainties are proportional to the square root of the number of electrons but are then added in quadrature, for any area $A' > A_{\rm PSF}$, the resulting photon noise is

$$\begin{eqnarray*}\sigma_{\rm PN}(A') &=& C_{\sc {\rm SM}} \, \sqrt{ \sum_i^{N_{\... ... \, \sqrt{ \sum_i^{N_{\rm f}} \sum_j^{N'_{\rm p}} {e}^-_{ij} } \end{eqnarray*}$$

where $N'_{\rm p}/N_{\rm PSF}$ is the number of PSF-sized areas within A'. Note that this is nothing more that the square root of the total number of electrons recorded by all unmasked pixels from all frames contributing to the area A' of the mosaic. In all of our work, we will only quote surface photometry for $A' \geq A_{\rm PSF}$ so that all points in our surface brightness profiles are independent, and so that the formulation above can be used to calculate photon noise.

A.4 Sky subtraction: ${\sigma _{SS}}$

Sky subtraction introduces the same systematic uncertainty to every position in the mosaic. The determination of the sky values and their uncertainties $\delta {\cal S}$ ( ${\cal S}_{R} \pm \delta {\cal S}_{R} = {\rm 16651.5 \pm 0.4~e^{-}~pix^{-1}}$, ${\cal S}_{V} \pm \delta {\cal S}_{V} = {\rm 2950.2 \pm 0.2~e^{-}~pix^{-1}}$) were discussed in Sects. 3.4 and 4.1. Since the sky values are determined from the mosaic itself, the normalization factor $C_{\sc {\rm SM}}$ is already contained in these values. We have then simply

$$\sigma_{\rm SS}(A') = N'_{\rm p} \, \delta{\cal S} .$$

A.5 Absolute calibration: sigma_CAL

The absolute calibration, or transformation of our surface brightness photometry to a standard system, is not of primary importance to many of our scientific results since relative measurements from one portion of the mosaic to other portions are more relevant. Nevertheless, as explained in Sect. 3.5, all absolute measurements have a fractional uncertainty of $\widetilde \sigma _{\rm CAL}(A') \approx 0.05$ due to errors in the absolute calibration. Thus, for absolute quantities we must also consider

$$\sigma_{\rm CAL}(A') = \widetilde \sigma _{\rm CAL}(A') \, F_{\rm\sc \, SM}(A').$$

Again, because the calibration is performed on the mosaic itself, the normalization factor $C_{\sc {\rm SM}}$ is already contained in these values.

A.6 Mosaicing: ${\sigma _{M}}$

If the image of ESO 342-G017 had been formed without mosaicing, then the total normalization constant $C_{\sc {\rm SM}}$ in the first equation of the appendix would not be present and thus would introduce no uncertainty in the final photometry. (The uncertainty in $C_{\sc {\rm SM}}$ is dominated by the uncertainty in $(\sum^{N_{\rm f}}_i {\cal S} _i ) ^{-1}$; we ignore here the very much smaller uncertainty in $\zeta$.) With mosaicing, relative errors related to the uncertainty in the quantity $(\sum^{N_{\rm f}}_i {\cal S} _i ) ^{-1}$ may be introduced between different parts of the mosiac. We consider, therefore, the mosaicing uncertainty of photometry in submosaic SM relative to the fiducial submosaic SM$_{\rm O}$ (taken to be central submosaic containing flux from all ESO 342-G017 frames), to be

$$\sigma_{\rm M}(A') \equiv \sqrt{ (\delta C_{\sc {\rm SM}})^2 ... ..._O}})^2 } \, \sum_i^{N_{\rm f}} \sum_j^{N'_{\rm p}} {e}^-_{ij}$$ $$= \frac{ \sqrt{ C^4_{\sc {\rm SM}} \sum_{\sc {\rm SM}} (\del... ...}}{\zeta} \, \sum_i^{N_{\rm f}} \sum_j^{N'_{\rm p}} {e}^-_{ij}$$

where the $(\delta {\cal S} _i)^2$ are the uncertainties in the median sky values of individual frames summed over those frames contributing to the indicated submosaic.

A.7 Surface brightness fluctuations: ${\sigma _{L}}$

In outlining a method for determining extragalactic distances, Tonry & Schneider (1988) derive an expression for the intrinsic variations in an elliptical galaxy or spiral galaxy bulge. This fluctuation in surface brightness is due to the counting statistics of a finite number of unresolved stars contributing flux to each pixel of a CCD image.

The fluctuations in a single pixel is (Tonry & Schneider 1988):

$$\sigma_{\rm L}^2 = \overline{g}~t~(\frac{10\,{\rm pc}}{d})^2~10^{-0.4(\overline{M} - m_1)},$$

where $\overline{g}$ is the mean number of counts due to the galaxy alone ( $\sum_i^{N_{\rm f}}{e}^-_{i}-\sum_i^{N_{\rm f}}{\cal S}_{i}$), t is the single exposure integration time (seconds), d is the distance to the source in parsecs, and m₁ is the magnitude corresponding to 1 count per pixel per second in the final image. This equation, however, does not take into account the effects of seeing, which strongly reduce the apparent brightness fluctuations. This reduction is significantly greater than 1/$\sqrt{n}$, where n is the total number of pixels, for spatial bins larger than the seeing PSF (Morrison et al. 1994). Because of the large distance to ESO 342-G017, surface brightness fluctuations make a very small contribution to our total error budget. Therefore, rather than simulating the effects of seeing at various binning scales (as done by Morrison et al. 1994), we compute this error as an upper limit, and simply scale it to our bin sizes using only a 1/$\sqrt{n}$ factor.

For our data, we take t=600 s, d=102Mpc, m₁(R)=21.6 mag/sqarcsec and m₁(V)=21.3 mag/sqarcsec, and $\overline{M}(R)=-0.7$ and $\overline{M}(V)=-0.3$ (Tonry et al. 1990).

A.8 The total error budget

We are now in a position to combine these different sources of uncertainty to arrive at an error budget for our surface brightness photometry of ESO 342-G017. The read noise, flat-fielding, photon noise, mosaicing and intrinsic surface brightness fluctuation uncertainties are all independent and statistical, and so can be added in quadrature, so that

$$\begin{eqnarray*}\sigma_{\rm STAT}(A') =\sqrt{ \sigma^2_{\rm RN}(A') + \sigma^2_... ..._{\rm PN}(A') + \sigma^2_{\rm M}(A') + \sigma^2_{\rm L}(A')}\,. \end{eqnarray*}$$

This is the statistical error that will cause random scatter in our surface brightness profiles. The sky subtraction and absolute calibration uncertainties are systematic in that all measurments in the mosaic will be affected in the same way by errors in these derived quantities. Sky subtraction errors will systematically change the slope of the surface brightness profiles; calibration errors will shift the profiles by a constant amplitude. All profiles that we display in Sect. 5 have statistical error bars computed as described in this appendix. The effect of sky subtraction and calibration uncertainties must be considered separately.