The Tully-Fisher relation for S0 galaxies

Abstract

We present a study of the local B and K-band Tully-Fisher Relation (TFR) between absolute magnitude and maximum circular speed in S0 galaxies. To make this study, we have combined kinematic data, including a new high-quality spectral data set from the Fornax Cluster, with homogeneous photometry from the RC3 and 2MASS catalogues, to construct the largest sample of S0 galaxies ever used in a study of the TFR. Independent of environment, S0 galaxies are found to lie systematically below the TFR for nearby spirals in both optical and infrared bands. This offset can be crudely interpreted as arising from the luminosity evolution of spiral galaxies that have faded since ceasing star formation. However, we also find a large scatter in the TFR. We show that most of this scatter is intrinsic, not due to the observational uncertainties. The presence of such a large scatter means that the population of S0 galaxies cannot have formed exclusively by the above simple fading mechanism after all transforming at a single epoch. To better understand the complexity of the transformation mechanism, we have searched for correlations between the offset from the TFR and other properties of the galaxies such as their structural properties, central velocity dispersions and ages (as estimated from line indices). For the Fornax Cluster data, the offset from the TFR relates with the estimated age of the stars in the individual galaxies, in the sense and of the magnitude expected if S0 galaxies had passively faded since being converted from spirals. This correlation implies that a significant part of the scatter in the TFR arises from the different times at which galaxies began their transformation. Comment: 17 pages, 11 figures, 3 tables, accepted for publication in MNRAS

Full text

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arXiv:astro-ph/0609076v1 4 Sep 2006
Mon. Not. R. Astron. Soc. 000, 1–18 (2002) Printed 4th February 2008(MN LATEX style file v2.2)
The Tully–Fisher relation for S0 galaxies
A.G. Bedregal1⋆, A. Arag´ on-Salamanca1and M.R. Merrifield1
1School of Physics and Astronomy, Centre of Astronomy and Particle Theory,
University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Accepted ***. Received ***; in original form ***
ABSTRACT
We present a study of the local B- and Ks-band Tully–Fisher Relation (TFR) between
absolute magnitude and maximum circular speed in S0 galaxies. To make this study,
we have combined kinematic data, including a new high-quality spectral data set
from the Fornax Cluster, with homogeneous photometry from the RC3 and 2MASS
catalogues, to construct the largest sample of S0 galaxies ever used in a study of the
TFR. Independent of environment, S0 galaxies are found to lie systematically below
the TFR for nearby spirals in both optical and infrared bands. This offset can be
crudely interpreted as arising from the luminosity evolution of spiral galaxies that
have faded since ceasing star formation.
However, we also find a large scatter in the TFR. We show that most of this
scatter is intrinsic, not due to the observational uncertainties. The presence of such a
large scatter means that the population of S0 galaxies cannot have formed exclusively
by the above simple fading mechanism after all transforming at a single epoch. To
better understand the complexity of the transformation mechanism, we have searched
for correlations between the offset from the TFR and other properties of the galaxies
such as their structural properties, central velocity dispersions and ages (as estimated
from line indices). For the Fornax Cluster data, the offset from the TFR correlates
with the estimated age of the stars in the individual galaxies, in the sense and of
the magnitude expected if S0 galaxies had passively faded since being converted from
spirals. This correlation implies that a significant part of the scatter in the TFR arises
from the different times at which galaxies began their transformation.
Key words: galaxies: elliptical and lenticular – galaxies: fundamental parameters
1 INTRODUCTION
The Tully–Fisher relation (TFR; Tully & Fisher 1977) is
one of the most important physically-motivated correlations
found in spiral galaxies. The correspondence between lumi-
nosity and maximum rotational velocity (Vmax) is usually
interpreted as a product of the close relation between the
stellar and total masses of galaxies or, in other words, as
the presence of a relatively constant mass-to-light ratio in
the local spiral galaxy population (e.g. Gavazzi 1993, Zwaan
et al. 1995). Such a general property in spirals puts strong
constraints on galaxy formation scenarios (e.g. Mao et al.
1998, van den Bosch, 2000) and cosmological models (e.g.
Giovanelli et al. 1997, Sakai et al. 2000). Also, the low scat-
ter in the TFR (only ∼ 0.35 mag in I-band, according to
Giovanelli et al. 1997, Sakai et al. 2000, Tully & Pierce 2000
and Verheijen 2001) permits us to use this tool as a good
distance estimator (e.g. Yasuda et al. 1997).
⋆E-mail:ppxapgg@nottingham.ac.uk
Attempts to ascertain whether S0 galaxies follow a sim-
ilar TFR have two main motivations. First, if there is a TFR
of S0s, it would prove useful for estimating distances in the
nearby universe, particularly in clusters where S0s are very
prevalent (Dressler 1980). Second, and more related to the
present study, a possible scenario where S0 galaxies are the
descendants of evolved spirals at higher redshifts (Dressler
1980, Dressler et al. 1997) could leave traces of this evolu-
tion in the observed TFR of S0s. Different mechanisms have
been proposed as the channels for such evolution, like small
mergers (Schweizer 1986), gas stripping in later types (Gunn
& Gott 1972), halo stripping (Bekki et al. 2002) and galaxy
harassment (Moore et al. 1998). If this kind of picture is
correct, it would be expected that S0s retain some memory
of their past as spirals, in particular through their TFR, and
perhaps even some clues as to which of the channels they
evolved down.
Only a few studies of the TFR for S0 galaxies can be
found in the literature. The first effort, made by Dressler
& Sandage (1983), found no evidence for the existence of
a TFR for S0 galaxies. However, the limited spatial extent

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2 A.G. Bedregal, A. Arag´ on-Salamanca and M.R. Merrifield
of their rotation curves, the inhomogeneous photographic
photometry employed and the large uncertainties in the dis-
tances to their sample made it almost certain that any cor-
relation between luminosity and rotational velocity would
be lost in the observational uncertainties.
Fifteen years later, Neistein et al. (1999) explored the
existence of a TFR for S0s in the I-band, using a sample
of 18 local S0s from the field. Although some evidence for
a TFR was uncovered in this study, they also found a large
scatter of 0.7 magnitudes in the relation, suggesting the pres-
ence of more heterogeneous evolutionary histories for these
galaxies when compared to spirals. Also, a systematic shift
0.5 magnitudes was found between their galaxies and the
relation for local spirals.
In two papers, Hinz et al. (2001, 2003), explored the
I- and H-band TFRs for 22 S0s in the Coma Cluster and 8
S0s in the Virgo Cluster. By using cluster data, they avoided
some of the errors that arise from the uncertainty in abso-
lute distances estimation. The analysis of I-band data from
the Coma Cluster revealed very similar results to the study
by Neistein et al. (1999), implying that the larger scatter
of the latter could not be attributed to distance errors or
the heterogeneous nature of the data. In the H-band, an
even larger scatter of 1.3 magnitudes was found, but with a
smaller offset from the corresponding spiral galaxy TFR of
only 0.2 magnitudes. Interestingly, there was no evidence for
any systematic difference between the results for the Virgo
and Coma Clusters, despite their differences in richness and
populations, implying that these factors could not be re-
sponsible for the scatter in the TFR. Given the large scatter
and small shift in the H-band TFR for S0s compared to
spirals, it was concluded that these galaxies’ properties are
not consistent with what would be expected for spiral galax-
ies whose star formation had been truncated; instead they
suggested that other mechanisms such as minor mergers are
responsible for the S0s’ TFR.
By contrast, Mathieu, Merrifield & Kuijken (2002)
found in their detailed dynamical modeling of six disk-
dominated field S0s that these galaxies obey a tight I-band
TFR with a scatter of only 0.3 magnitudes, but offset from
the spiral galaxy TFR by a massive 1.8 magnitudes. The
authors therefore concluded that these objects were consis-
tent with being generated by passively fading spirals that
had simply stopped producing stars. This result does not
appear to be consistent with the previous studies, although
it should be borne in mind that the galaxies in this study
were selected to be disk-dominated, so they morphologically
resemble spiral galaxies more than those in other work. In
addition, their field locations means that they are less likely
to have had their evolution complicated by mergers. It is
therefore possible that these S0s really are just passively-
fading spirals where those in clusters have led more compli-
cated lives.
As can be seen, there is no general consensus as to ei-
ther the scatter or the shift in the TFR for S0 galaxies when
compared to spirals, and so no agreement as to their inter-
pretation. We therefore revisit the TFR of S0 galaxies in
this paper, adding our new spectral dataset from the For-
nax Cluster to the existing results from literature. In addi-
tion to the new homogeneous spectral data set, we can also
take advantage of the 2MASS photometry for these galaxies
(Jarrett et al. 2003), to obtain consistent infrared magni-
tudes and structural parameters for all the galaxies. With
this more systematic study of S0s, it is to be hoped that we
can avoid any past problems that may have arisen due to
the heterogeneous nature of the available data, to reveal the
underlying physics that dictates the TFR in S0 galaxies.
The remainder of the paper is laid out as follows. In
Section 2, we describe the different samples of S0s used in
this study, and the data that we have adopted. Section 3
presents the main results and then discusses their implica-
tions. Finally in Section 4 our conclusions are summarised.
2 THE DATA
2.1Kinematics
To build the TFR of local S0 galaxies, we have collated the
data on their kinematics from four previous studies. These
works are: Neistein et al. (1999), hereafter N99; Hinz et
al. (2001, 2003), hereafter H01 and H03, respectively; and
Mathieu, Merrifield & Kuijken (2002), hereafter M02. From
the sample of N99, we exclude the galaxy NGC4649 as it
has a low degree of rotational support and it presents evi-
dence of interaction with a neighbouring system. From H01,
the Sab spiral galaxy IC4088 was also excluded. To these
data, we have added the observations that we have recently
obtained using the VLT of S0 galaxies in the Fornax Clus-
ter (Bedregal, Arag´ on-Salamanca & Merrifield 2006, here-
after Paper I); of the galaxies observed in this cluster, seven
are rotationally supported systems, so are suitable for this
study. This combined data set provides us with 60 S0 galax-
ies with measured kinematics, the largest sample yet used in
a study of the S0 TFR. The collated kinematic data values
for the maximum rotation speed, Vmax (for all the sample),
and the central velocity dispersion, σ0 (for 51 galaxies of the
sample), are listed in Table A1.
2.2 Photometry
In the present study, we have adopted Ks-band photometry
from the Two Micron All Sky Survey (2MASS, Jarrett et
al. 2003) and B-band photometry from the Third Reference
Catalogue of Bright Galaxies (RC3, de Vaucouleurs et al.
1991). Of the complete sample of 60 galaxies, photometry
in Ks-band is available for all objects, while 54 galaxies have
photometry in B-band.
In order to convert these data to absolute magnitudes,
we need distances to all objects in the sample. In the clus-
ters, we adopted distance moduli of 31.35 for members of the
Fornax Cluster (Madore et al. 1999), a redshift of 0.0036
for the Virgo Cluster members (Ebeling et al. 1998), and
a redshift of 0.0227 for Coma Cluster galaxies (Smith et
al. 2004). For the field galaxies from N99, the distance
moduli of Tonry et al. (2001) were used. Finally, for the
M02 sample, redshifts from the NASA/IPAC Extragalac-
tic Database were used where available. For two of their
galaxies (NGC1611 and NGC2612), an estimate of the dis-
tance was calculated by the authors, using the systemic
velocity derived from their spectra. A Hubble constant of
70kms−1Mpc−1was adopted in converting redshifts into
distances. Galactic extinction corrections were calculated us-
ing the Schlegel, Finkeiner & Davis (1998) reddening curve

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The S0 Tully–Fisher relation3
description, Aλ
for λ = B and Ks, respectively. In the B-band, we applied
the k-correction from Poggianti et al. (1997); no correction
was applied for the Ks-band, as it is negligible. No internal
extinction correction was applied to the apparent magni-
tudes; there is no definitive study on the internal extinction
of S0 galaxies, but the apparent lack of dust in these sys-
tems suggests that such a correction would be very small.
The resulting absolute magnitudes in Ks and B-bands are
listed in Tables A1 and A2, respectively.
In addition to the absolute magnitudes, we derived
structural parameters from the spatially-extended photom-
etry available for these galaxies. The data used were the
publicly-available “postage stamp” images in the Ks-band
from 2MASS. Bulge-plus-disk models were fitted directly
to these images using GIM2D (Simard et al. 2002), with
a S´ ersic law adopted for the bulge distribution, and an ex-
ponential for the disk. In this way, we derived values for the
bulge effective radius, Re, its S´ ersic index, n, the disk scale-
length, Rd, the half-light radius, Rhalf, the bulge-to-total
fraction, B/T, and the galaxy inclination i. In a few cases,
the derived bulge scale length was found to be smaller than
the seeing (∼ 2.5 arcsec). In those cases, the structural pa-
rameters are not well constrained by the observations, so the
values were excluded. The structural parameters derived for
the remaining 48 galaxies are listed in Table A1.
b= RλE(B − V ), where Rλ=4.32 and 0.37
2.3 Line indices, ages and metallicities
We have full spectral data for 7 Fornax Cluster galaxies from
which we can extract Lick indices. A detailed description of
the line indices calculation is given in our forthcoming study
of the stellar populations of these galaxies (Bedregal et al.
2007, hereafter Paper III). Briefly, we convolved the spec-
tral bins with appropriate gaussians (including galactic and
instrumental dispersions) in order to achieve the 3˚ A resolu-
tion of Bruzual & Charlot (2003) (hereafter BC03) simple
stellar populations models. The indices Hβ, Mgb, Feλ5270
and Feλ5335 were measured within Re/8 (“Central” val-
ues), between 0.75 and 1.25 Re (“1Re” values) and between
1.5 and 2.5 Re (“2Re” values).The resulting values are pre-
sented in Tables A2 and A3.
Central line indices for a handful of objects in our sam-
ple can be found in the literature (Fisher et al. 1996, Ter-
levich et al. 2002, Denicolo et al. 2004) mainly corresponding
to field S0s from the N99 subsample. Unfortunately, these
datasets mainly include the brightest objects of the sample,
so it is difficult to make meaningful comparisons with the
fainter galaxies from Fornax. Also, differences between the
two samples could arise because of the superior quality of
Fornax data, effect which is difficult to quantify. In conse-
quence, we will focus our analysis of the line indices, ages
and metallicities on the data from the Fornax Cluster only.
From Fornax galaxies’ indices, one can derive measures
of the luminosity-weighted ages and metallicities. To this
end, we have used the simple single-age stellar population
models of BC03 to translate the measured line indices into
estimates of age and metallicity. As a check on the un-
certainties inherent in this process, we have repeated the
calculations using the Mgb index and the combined in-
dices ?Fe? (Gorgas, Efstathiou & Arag´ on-Salamanca 1990)
and [MgFe]′(Gonz´ alez 1993; Thomas, Maraston & Bender
Figure 1. B-band Tully–Fisher relation of S0 galaxies. Solid line
represents the TFR of spiral galaxies from Tully & Pierce (2000);
dashed line represents the spiral TFR by Sakai et al. (2000); dot-
ted line is the best fit to the S0 data-points using the slope from
Tully & Pierce (2000). The error bars in the right down corner
correspond to the median value for each subsample, while for the
Fornax Cluster data we plot the errors for each data-point.
2003) as the metallicity-sensitive index. Solar abundance ra-
tios were assumed for the models. The resulting ages, metal-
licities and their uncertainties (which include the effects of
covariance between the two parameters) are also shown in
Tables A2 and A3.
3 RESULTS AND DISCUSSION
The basic result of this analysis is presented in the Tully–
Fisher plots of rotation velocity versus absolute magni-
tude shown in Figures 1 and 2 (for the B- and Ks-band
respectively). In these plots, the solid line represents the
TFR of spirals in local clusters found by Tully & Pierce
(2000, hereafter TP00), shifted by −0.207 magnitudes in or-
der to be consistent with the adopted Hubble constant of
H0 = 70kms−1Mpc−1. The long-dashed line in the B-band
represents the TFR of cluster spirals by Sakai et al. (2000,
hereafter Sak00). The difference between these lines give an
indication of the remaining systematic uncertainty in the
spiral galaxy TFR with which we seek to compare the S0s.
3.1Shift between the spiral and S0 TFRs
The first point that is immediately clear from Figures 1 and
2 is that, as found by previous authors, whichever spiral
galaxy TFR we adopt, the S0s lie systematically below it.
It is also interesting to note that this result holds true for
the S0 data from all environments, from the poorest field
objects to fairly rich clusters, so it is clearly a very general
phenomenon. We therefore now seek to quantify and model
the possible origins of such an offset.

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4 A.G. Bedregal, A. Arag´ on-Salamanca and M.R. Merrifield
Figure 2. Ks-band Tully–Fisher relation of S0 galaxies. Solid
line represents the TFR of spiral galaxies from Tully & Pierce
(2000); dotted line is the best fit to the S0 data-points using the
slope from Tully & Pierce (2000).The error bars in the right down
corner correspond to the median value for each subsample, while
for the Fornax Cluster data we plot the errors for each data-point.
3.1.1 Observational results
One problem in trying to quantify the offset in the TFR is
that the incompleteness in magnitude of the data presented
in Figures 1 and 2 will bias a conventional fit. We therefore
adopt the approach of Willick (1994), which involves fitting
the inverse function,
log(Vmax) = a + bMλ, where λ = B or Ks, (1)
to minimise this source of bias. The slope b is fixed to match
the slope for the spiral galaxy TFR, and a is varied to find
the least-squares fit between this function and the data, with
each point i weighted by
wi =
1
σ2
i
, (2)
where
σ2
i= σ2
log(Vmax),i+ b2σ2
Mλ,i+ σ2
int, (3)
to account for the uncertainty in the measured maximum
velocity, σlog(Vmax),i, and that in the absolute magnitude,
σMλ,i. The quantity σint is set to quantify the intrinsic scat-
ter in the relation such that the reduced χ2of the fit comes
out at unity; this procedure is discussed in more detail in
Section 3.2.
Setting b equal to the inverse of the spiral TFR slope
determined by TP00, we can determine the zero-point pa-
rameter a and hence the offset in magnitudes from the TP00
TFR relations in the B- and Ks-band. The resulting best-fit
lines are shown dotted in Figures 1 and 2. The offsets from
the TP00 relations are
∆MB,TP00 = −1.7 ± 0.4 (4)
and
∆MKs,TP00 = −1.2 ± 0.4, (5)
where the quoted error includes the uncertainty in zero point
of both the S0 and spiral TFRs. To test the robustness of
this result against the uncertainty in the spiral TFR, we
repeated the analysis using the Sak00 B-band relation to fix
the slope and measured the offset from their relation. This
analysis resulted in an offset of
∆MB,Sak00= −1.3 ± 0.1, (6)
within the errors of the previous analysis but somewhat
smaller. Sak00 did not publish a Ks-band TFR, but the
parallel nature of the Sak00 and TP00 TFRs in Figure 1
suggest that most of the difference arose from a zero-point
shift due to a different distance-scale calibration. We might
therefore extrapolate from the Sak00 B-band results on the
assumption that ∆MB,TP00 − ∆MKs,TP00 ≈ ∆MB,Sak00−
∆MKs,Sak00, to infer a corresponding Ks-band offset of
∆MKs,Sak00= −0.8 ± 0.4. (7)
These new estimates for the offset between spiral and
S0 TFRs tend to lie toward the upper end of earlier esti-
mates, mainly because of the more recent refinements in the
calibration to the spiral galaxy TFR that we have included
in this analysis. However, even neglecting these systematic
changes, the values obtained here lie within the range of off-
sets in the TFR suggested by previous studies, implying that
this quantity can be fairly reliably determined, particularly
with the larger and more homogeneous data set presented
here.
3.1.2 Interpretation
The most natural interpretation of the offset between the S0
and spiral TFRs is that it represents a simple fading as a
spiral galaxy’s star formation is truncated when it mutates
into an S0. One complication in attempting to quantify this
scenario is that one needs to know what luminosity the spi-
ral galaxy started at in this evolution. However, observations
out to redshifts beyond unity seem to show that there has
been essentially no evolution in either the slope or zero-point
of the spiral galaxy TFR in optical or infrared wavebands
over this period (Vogt et al. 1997, Conselice et al. 2005, Bam-
ford et al. 2006; although see Rix et al. 1997 and B¨ ohm et
al. 2004 for alternative views). This lack of evolution means
that the starting point for fading spirals is the same spiral
galaxy TFR that we see today, and the offset between the
nearby galaxy spiral and S0 TFRs does provide an accurate
measure of the degree to which the S0 galaxies must have
faded if this picture is correct.
To see if this scenario is plausible and to quantify the
timescales involved, we have used the BC03 synthesis mod-
els to calculate the fading of a stellar population that started
with a constant star formation rate of 1M⊙yr−1for 5 Gyrs,
then stopped. The stellar population was assumed to have
solar metallicity and the initial mass function of Chabrier
(2003). By matching the decrease in luminosity to the ob-
served shifts between spiral and S0 TFRs, we can determine
how long ago the truncation in star formation must have
occurred to be consistent with our observations. Using the

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The S0 Tully–Fisher relation5
TP00 offsets in the B- and Ks-bands given in equations 4
and 5, we thus obtain times since truncation of
τtrunc
B,TP00= 1.6+1.2
−0.7Gyrs(8)
and
τtrunc
Ks,TP00= 6.2+∞
−3.1Gyrs, (9)
respectively. These values are somewhat inconsistent with
each other, but this may just reflect the calibration of the
spiral TFR in TP00; if we instead use the values that we
infer from the Sak00 calibration (equations 6 and 7), we
obtain more consistent timescales since truncation of
τtrunc
B,Sak00= 0.66+0.63
−0.30Gyrs (10)
and
τtrunc
Ks,Sak00= 1.1+1.6
−1.0Gyrs. (11)
The shorter timescale simply reflects the smaller offset in
the TFR that the Sak00 calibration implies. However, this
timescale is if anything worryingly short: it would be quite a
coincidence if we are living at an epoch so close to the point
at which all these galaxies transformed from spirals into S0s.
One possible resolution is that the star formation his-
tory in transforming a spiral into an S0 might be somewhat
more complex. Indeed, Poggianti et al. (1999) have sug-
gested that cluster galaxies with k+a/a+k spectra observed
at intermediate redshifts are the best candidates for S0 pro-
genitors because of their spectro-photometric characteris-
tics and the predominance of disk-dominated morphologies.
Since such spectra are usually identified with post-starburst
galaxies, it would seem that spiral galaxies may undergo
a “last gasp” burst of star formation when they start their
transitions into S0s. To investigate this possibility, we added
a burst of star formation, amounting to 10% of the total
stellar mass, to the above truncated star formation model.
Repeating the comparison between the luminosity of this
model as predicted by the BC03 population synthesis code
and the observed offsets in the TFR resulted in estimates
for the time since the burst and truncation of
τburst
B,TP00= 2.3+1.3
−0.8Gyrs, (12)
τburst
Ks,TP00= 7.8+∞
−3.7Gyrs (13)
using the TP00 calibration, and
τburst
B,Sak00= 1.1+0.7
−0.4Gyrs,(14)
τburst
Ks,Sak00= 1.7+1.9
−1.1Gyrs (15)
using the Sak00 values. As might be expected, the inclusion
of a starburst increases the age of the stellar population
compared to those found in the truncation model. These
values are therefore a little more comfortable in terms of
the timescales involved, and, at least using the Sak00 cali-
bration, produce fairly consistent results between different
wavebands.
However, there is a limit to how far it is worth pur-
suing these simple models, as it is extremely unlikely that
all S0 galaxies will have undergone the same star formation
history. Indeed, the large scatter apparent in the points in
Figures 1 and 2 indicates a relatively heterogeneous history
for these systems: the average evolution may be as described
above, but each galaxy has its own story to tell. We there-
fore now look in more detail at the scatter in the S0 TFR,
in an attempt to quantify it and explore its origins.
3.2The Scatter in TFR of S0 galaxies
3.2.1Observational results
As outlined in the previous section, we estimate the intrinsic
scatter in the TFR, σint, during the fitting process by vary-
ing its value in the weights of equation 2 until the reduced
χ2of the fit,
χ2
r=
1
n − 2
?
i
?log(Vmax,i) − a − bMλ,i
σi
?2
, (16)
was equal to unity. The inverse slope of the TFR, b, was set
to the value appropriate to either the TP00 or the Sak00
spiral TFR (the values are so similar that it made no sub-
stantial difference to the measured scatter), while the zero-
point a was allowed to vary. The presence of variables in
both numerator and denominator of equation 16 mean that
the fit is no longer linear, but we found that it could be ro-
bustly performed by a simple iterative procedure in which
at the jth iteration the estimate for σint was updated such
that
σ2
int,j= σ2
int,j−1× χ2α
r .(17)
By setting the convergence parameter α to 2/3, we found
that the iterative solution was well behaved and converged
in no more than 15 iterations. For comparison, we also cal-
culated a weighted total scatter using the formula
σ2
tot=
?
iwi(log(Vmax,i) − a − bMλ,i)2
?
For the B-band TFR of S0s (using TP00 slope), we thus
derived
iwi
. (18)
σtot,B
σint,B
=
=
0.88 ± 0.06 mag,
0.78 ± 0.06 mag,
(19)
while for Ks-band data (using TP00 slope) we found
σtot,Ks
σint,Ks
=
=
0.98 ± 0.06 mag,
0.87 ± 0.06 mag.
(20)
These values imply that ≈ 90% of the observed scatter in
Figures 1 and 2 cannot be explained by the known observa-
tional uncertainties in Mλand log(Vmax). These results seem
to be bracketed by previous estimates: H03 report a scatter
in the H-band TFR of 1.18 magnitudes in the Coma Cluster
and 1.33 in the Virgo Cluster, whereas previous I-band esti-
mates of scatter have been lower at 0.68 magnitudes (N99)
and 0.82 magnitudes (H03). However, these previous stud-
ies are not directly comparable to the current estimates,
because of their different wavebands and because they did
not use the more robust inverse-fitting approach adopted
here. In addition, it is not entirely clear whether the pre-
vious estimates have been corrected for measurement error,
particularly in the uncertain measurement of log(Vmax).
3.2.2 Interpretation
Perhaps the simplest possible explanation for the large value
σint is that the errors in the observed quantities plotted in