To derive accurate distances with the SBF technique, it is necessary that the pixel-to-pixel fluctuations are dominated by the Poisson statistics of the stars rather than by the photon shot-noise statistics or other detector artefacts (Tonry & Schneider 1988). To fulfil this constraint, one needs to acquire high S / N images (see Section 2.2). Furthermore, other fluctuations that originate from unwanted contributions (Pr) such as GCs, foreground and/or background galaxies, need to be removed as well (Blakeslee & Tonry 1995; Jensen et al. 1999). In particular, for distance moduli (m − M) > 32, the fluctuations from external sources are at least of the same order of magnitude as (or even higher than) the SBF fluctuations themselves and therefore need to be carefully removed before estimating distances (Jensen et al. 1998). This is even valid for the NIR bands, although one might expect that the relative contributions to the fluctuations from GCs to be smaller, as the stellar SBFs are much redder than (those of) GCs. Since the background is higher in the NIR and the sampling and sensitivity to faint point sources is decreased, this effect cancels out for the total fluctuation power (contribution of background galaxies is reduced by a factor of ∼ 10).
There are two main requirements for using the SBF technique as a distance
indicator: (i) The bright end of the LF among galaxies of
different type (ellipticals, S0s and spiral bulges) is universal, or
(ii) The variations in
the LF from stellar system to system can be detected and corrected for, so
that 
remains calibrated
at high precision, hence a standard
candle (see also Section 2 and equation 2).
Several studies have proven that absolute fluctuation magnitudes in both the
I and the K-band of elliptical galaxies are correlated
with the internal (V − I)0 galaxy colour
(TAL90;
Luppino & Tonry 1993;
Ajhar et al. 1997;
Tonry et al. 1997;
Jensen et al. 1998;
Mei et al. 2001;
Liu et al. 2000,
2002).
As a consequence the empirical calibration can be considered as universal
and therefore as a standard candle. The linear relationship between absolute
fluctuation magnitude and galaxy colour has been calibrated to Cepheid
distance measurements in the Virgo Cluster and the Local Group (see
Section 3.1
above). The intrinsic scatter in the Cepheid zero-point calibration is
∼ 0.05-0.09 mag, whereas the stellar population models
(Worthey 1993)
suggest an uncertainty in the
I
− (V − I)0 relation of < 0.11 mag.
The exact scatter in the theory depends on the composition and the
variations among the stellar populations.
Jensen et al. (1999)
compared their K-band data to the theoretical predictions by
Worthey (1993)
and concluded that their fluctuation magnitudes
K are almost
constant in the colour range
1.10 ≤ (V − I)0 ≤
1.30. This implies that the K-band SBF itself is a
good standard candle with a median absolute magnitude of
K =
−5.57 ± 0.19. The average scatter in
K is only
0.06 mag, but the models with solar metallicity suggest brighter
absolute magnitudes of
Ks
= −5.84 ± 0.04, but a slope (3.6 ± 0.8)
which is in agreement with the optical calibration
(Liu et al. 2002).
From this comparison it is clear that more high-quality observational data
are needed and a more careful description of the stellar populations has
to be obtained from a theoretical point of view. A more detailed
discussion on this issue with the presentation of new stellar population
models is given in Section 5.
4.1. Caveats for Distance Determinations
As we have shown, SBF magnitudes represent a universal standard candle, and hence a precise distance indicator. However, there are several possible problems and uncertainties that could affect accurate distance measurements.
4.1.1. Target Selection
The preferred targets for SBF measurements are dynamically hot stellar
systems, either GCs or elliptical galaxies. There have been attempts to
measure distances to disky S0 galaxies (NGC 1375:
Lorenz et al. 1993)
or edge-on spiral bulges (NGC 4565:
Simard & Pritchet 1994;
NGC 3115:
T01).
However, apart from differences in the analysis techniques, Cepheid
stars are only found in the dusty disks of spiral galaxies. Stellar
systems with multiple components in the light profile (bulge, disk, bar,
rings) are difficult to model for the SBF method, as the stars of the
disk need to be correlated to the place where the SBF signal was
measured. Hence, the SBF technique is mostly limited
to the dust-free stellar systems of elliptical galaxies.
An alternative approach is to go to space. Thanks to the superb
resolution of HST compared to ground-based telescopes,
different components and dust
features can be detected, modelled, and subtracted much better and the SBF
technique extended to S0 galaxies and early-type spiral bulges
(Jensen et al. 2003).
This approach appears to be promising, as Cepheids are
most common and have a higher abundance in the outer disks of spiral
galaxies, which are usually difficult to analyse with the SBF technique.
A precise stellar template is crucial for measuring the exact SBF amplitude.
All modifications of the power spectrum of the stellar template translate
directly to the power spectrum of the fluctuations. For example, a
mismatch of 3% in the normalisation of the stellar power spectrum
results in an error of 0.03 mag in the fluctuation magnitude
At the present time, the real limiting factor of (N)IR detectors is the
small field-of-view relative to their optical counterparts. As a
consequence, fewer bright field stars are available for calibration
purposes that are uncontaminated by galaxy or companion light.
Gound-based SBF measurements have a crucial limiting factor: The detection,
modelling and careful removal of faint background sources and globular
clusters. The total fluctuation amplitude (P0) must be
dominated by the SBF signal, hence the residual fluctuation of
undetected faint GCs and background galaxies
(Pr) needs to be smaller than the SBF fluctuations.
This is illustrated in Figure 6.
The superior resolution of space-based observations offers the great
advantage of much more accurate detection and sensitivity to unwanted
sources. Further, in the IR the sky background of NICMOS HST
observations is more than a factor of 100 lower compared to IR
observations from ground-based telescopes.
Figure 6. Masked (galaxy-subtracted)
residual image of NGC 3379 (adapted from the investigation of
Fritz 2002).
The unsaturated bright star right from the centre of the galaxy was
adopted as a PSF template. Sources in this image consist of GCs,
background galaxies and SBFs. From the normalized luminosity fluctuations,
a characteristic flux-weighted mean flux of
4.1.4. The Good, the Bad, and the Ugly
Considering the challenge involved in deriving a reliable estimate of the
fluctuation magnitude
For optical colours the effects of age and metallicity are largely
degenerate. Therefore, going to the IR and looking for a NIR colour
calibration would be greatly beneficial as it would allow one to
quantify differences in the stellar populations that alter the
light-averaged age and metallicity of a stellar system. A few observational
(Jensen et al. 2003)
and theoretical approaches
(Blakeslee et al. 2001a;
Lee et al. 2010)
in this direction have been conducted. However, more observations and
improved model descriptions would finally provide an accurate handle on
the age/metallicity degeneracy.
Moreover, SBF magnitudes are required to be corrected for the dust
extinction caused by our own Milky Way galaxy. Usually, the 100
µm DIRBE/IRAS dust emission maps by
Schlegel, Finkbeiner &
Davis (1998)
are adopted. In the future, improved galactic extinction values could be
derived from multi-wavelength studies involving mid-IR or
Spitzer/IRAC combined with blue ultraviolet (UV) GALEX
observations.
The uncertainty in the zero-point of the Cepheid calibration is less
than 0.1 mag (see Section 4 above). However,
the revised Cepheid calibration
(Freedman et al. 2001)
depends on the metallicity of the underlying stellar
population and needs to be accounted for
(Sakai et al. 2004;
Macri et al. 2006;
Scowcroft et al. 2009).
This additional metallicity correction is of the order
of −0.06 mag to the distance moduli and therefore adds only a small
uncertainty of 0.08 − 0.10 mag when measuring distances based on SBFs
(Jensen et al. 2003;
Blakeslee et al. 2010;
see also Section 3.1).
. Usually, having a good
stellar template as PSF is not a major problem, as there are stars
located close to or in the surrounding field of the target galaxy. In
the NIR, however, this issue is more prominent as the stars must be
detectable and separated from the much higher sky background. For
HST there is the possibility of constructing synthetic PSF
templates to account for the PSF variations and distortion effects on
the pixel-sensitive WFPC2, ACS or NICMOS chips (e.g.,
Fritz et al. 2009a,
b).
In general, for ground-based observations the mismatch between real and
synthetic PSFs is small (see
Fritz 2002),
but when real observed PSF are available these are to be preferred.
= 3.37
ADU/star/300s was measured, which corresponds to a weighted mean
fluctuation magnitude o
= 28.61 ± 0.04 mag
(Fritz 2002).,
the measurement of the galaxy colour
should represent an easy task. Instead, however, the linear correlation of
I ∝
4.5(V − I) makes an accurate colour determination
essential. Internal colour gradients in a galaxy introduce complexity and
therefore cause difficulties when deriving the colour of a stellar system.
Both and (V
− I) need to be measured over the same region in
the galaxy, which is tricky if the galaxy has multiple components of bulge
plus disk and is
established in the centre, whereas (V − I)
is obtained from the outer regions of the disk.