Some effects of galaxy structure and dynamics on the Fundamental Plane. II. A Virgo-Fornax distance modulus

13 Pages·0.31 MB·English

by  Alister W. Graham

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Some effects of galaxy structure and dynamics on the
Fundamental Plane. II. A Virgo-Fornax distance modulus
⋆
Alister W. Graham
Mount Stromlo and Siding Spring Observatories, The Australian National University,
Private Bag, Weston Creek PO, ACT 2611, Australia.
8
9
9
ABSTRACT
1

n The influence of broken structural homology, and the implied broken dynamical
a
homology, is examined for the Fundamental Plane (FP). Requiring a symmetrical
J
treatment of the FP variables, a bisector method of linear regression was applied,
5
in 3-dimensions, to derive the best FP. A bootstrapping procedure has been used
1
to estimate the uncertainties associated with the slope of the FP. For 25 E and S0

Virgo galaxies, the ‘standard’ FP, constructed using R1/4 model parameters for the
1
v effective radii (Re,4) and the mean surface brightness within this radius (Σe,4) and
3 using centralvelocity dispersion(CVD) measurements (σ0), gavea relationdescribed
4 by Re,4 ∝ σ01.10±0.14Σ−e,40.55±0.09. Using Sersic R1/n light profile model parameters
1 andthe projected,infinite aperture,velocity dispersion(σtot,n), derivedfromapplica-
1
tion of the Jeans equation to the observed intensity profiles, gave an ‘improved’ FP
80 described by the relationRe,n ∝σt1o.3t,7n±0.16Σ−e,n0.76±0.05. This result, based on indepen-
dent data, supports the previous finding by Graham & Colless (1997a) that assump-
9
/ tions of structural and dynamical homology are partly responsible for the departure
h
of the observed FP from the plane expected by the virial theorem, which predicts
p R∝σ2Σ−1. Upon removal of the known S0 galaxies from the sample of Virgo galax-
o- ies, the above planes were observed to change to Re,4 ∝ σ01.19±0.21Σ−e,40.60±0.11 and
tr Re,n ∝σt1o.7t,2n±0.24Σ−e,n0.74±0.09. The perpendicular rms residuals about these planes are
s 0.084 and 0.050 dex, respectively.
a
The Fornaxcluster wassimilarly treated,althoughremovalofthe S0 galaxiesleft
:
v a sample of only 7 ellipticals which had published CVD measurements. Treating the
i rangeofstructuralanddynamicalprofilespresentinthis sampleproducedaFPgiven
X
r by the relation Re,n ∝ σt2o.0t,3n±0.78Σ−e,n1.07±0.30, in tantalizing agreement with the plane
expectedfromthevirialtheorem,butwithdiscouraginglylargeerrorsduetothesmall
a
sample size. Similarly to Virgo, the perpendicular rms residuals about this plane is
0.050 dex.
The FP was also constructedwith the purpose of using it as a distance indicator,
achieved by minimising the distance-dependent quantity logR against the distance-
independent quantities Σ and logσ. A Virgo-Fornaxdistance modulus was computed
using Working-Hotelling confidence bands (Feigelson & Babu 1992). The ‘standard’
FPparametersgaveavalueof0.45±0.16mag,where-asthe‘improved’FPparameters
gave a value of 0.25±0.12 mag. However, a full treatment of the uncertainties on the
FP slopes, derived through a bootstrapping procedure of the 3-dimensional FP data
set, revealed that the analytical expressions for the uncertainties on the estimated
distance moduli, given above, should be increased by a factor of ∼5.
Key words: galaxies:fundamentalparameters– galaxies:structure –galaxies:kine-
matics and dynamics. galaxies: elliptical and lenticular, cD – distance scale
1 INTRODUCTION
Ourknowledgeaboutthestructureofelliptical galaxieshas
⋆ E-mail:[email protected] progressed considerably in the last decade. No longer are
(cid:13)c 0000RAS
2 A.W.Graham
we satisfied with fitting a de Vaucouleurs (1948,1953) R1/4 namical homology appeared to be responsible for some of
intensitydistributiontothelight profilesof elliptical galax- thetiltoftheFP,i.e.itsdeparturefromtheplanepredicted
ies.Modernimagingtechniquesareprovidingever-morepre- bythe virial theorem, R∝σ2Σ−1.
cise measurements, revealing a range of light profile shapes Ultimately, this broken dynamical and structural ho-
which have been shown to be better described by the Ser- mology needs to be dealt with if one is to understand the
sic(1968)R1/n model(Capaccioli1989;Caon,Capaccioli& natureofellipticalgalaxiesandusethemformeasuringdis-
D’Onofrio1993;Grahametal.1996,andreferenceswithin). tances(Dressleretal.1987).Giventhesubstantialimprove-
Through varying the parameter n in the Sersic model, one mentsindescribingelliptical galaxies, it isapt thattheFP,
can reproduce an exponential light distribution (n=1), the and its application to distance estimates, is revisited. Us-
classic de Vaucouleurs profile (n=4), approximate a power- inganewdatabase(takenfromCaon etal.1994) obtained
lawforlargevaluesofn(approximately>15),oranyofthe by different observers with a different instrument and us-
intermediate forms which galaxies exhibit (Michard 1985; ing a different reduction procedure, the influence of broken
Schombert1986;Binggeli&Cameron1991;Caonetal.1993; structural and broken dynamical homology upon the FP is
Graham & Colless 1997a, hereafter Paper I, to mention a againinvestigated.Thedatausedinthispaperisthesurface
few). Consequently, for those galaxies which are not well brightness profiles along the major axis of each galaxy, as
described by an R1/4 profile, the effective radius and mean opposed to the integrated aperture magnitudes used in Pa-
surface brightness within this radius will be different when perI.TheinfluenceofS0galaxies upontheslopeoftheFP
obtained using the de Vaucouleurs model and the Sersic isalsoinvestigated,withcomparisonoftheFPsconstructed
model.Moreimportantly,thesedifferenceshavebeenshown with, and without, their inclusion.
tohaveasystematictrend:galaxieswithn<4havetheiref- ThispaperpresentstheFPforthegalaxy clustersFor-
fective half-light radii over-estimated by the R1/4 model, nax and Virgo, two key clusters in the extra-galactic dis-
andgalaxies betterdescribed withvaluesofn>4havetheir tance scale (Jacoby et al. 1992; McMillan, Ciardullo & Ja-
effectivehalf-lightradiiunder-estimatedbytheR1/4 model. coby 1993; Bureau, Mould & Staveley-Smith 1996). These
Similarly, the mean surface brightness is under- and over- clusters are sufficiently distant to possess significant mo-
estimated by the R1/4 model when n<4 and n>4 respec- tion due to the Hubble expansion and yet they are close
tively (Graham & Colless 1997b). enoughthattheirdistancesmaybeobtainedwithaplethora
Observed variations in the galaxy structure, as de- of independentobservational techniques.The following sec-
scribed by the R1/n model, imply a variety of dynami- tionpresentsthegalaxysampleandtheobservational/model
cal structures (Ciotti 1991; Paper I) are required to main- datawhich isused inthisstudy.TheFundamentalPlane is
tain these predominantly pressure-supported stellar sys- computed, and its method of construction, including a new
tems. This conjecture is borne out in the observations of error analysis routine, is described in Section 3. Using two
elliptical galaxy dynamics (Davies et al. 1983; Illingworth different methods of analysis, a distance modulus between
1983;Capaccioli&Longo1994)andhasimplicationsforthe Virgo and Fornax is calculated in Section 4. This result is
construction of theFundamentalPlane (FP). In particular, comparedwithotherestimatesoftheVirgo-Fornaxdistance
the use of a galaxy’s central velocity dispersion (CVD, σ0) modulus in Section 5, along with a discussion of the other
to represent the overall kinetic energy of a galaxy has been results. Section 6 provides a summary and concluding re-
shown tobeinappropriate,(Jørgensen, Franx&Kjærgaard marks.
1993; Bender,Saglia &Gerhard 1994). Thevelocity disper-
sion derivedfrom thekineticenergyofrandom motion over
theentiregalaxysurfacehasbeenshowntobeproportional
to σ01.6 rather than σ02 (Busarello et al. 1997). A procedure 2 GALAXY DATA
is described in Paper I,based uponthetheoretical studyof
2.1 The galaxy sample
Ciotti (1991), which enables the velocity dispersion to be
measured in a consistent fashion between galaxies. Rather ThegalaxysamplefromCaonetal.(1994)hasbeenusedfor
than simply using CVD measurements, that sample differ- this investigation. This data set has been used in a bench-
entfractionsofdifferentgalaxies’velocitydispersionprofiles mark study (Caon et al. 1993) which gave insight into the
(in terms of the half-light radii), through application of the natureof elliptical galaxy light profiles byquantifyingtheir
Jeansequation,PaperIpresentedamethodtocomputethe deviations from the R1/4 model. Knowing that this galaxy
asymptotic value of the velocity dispersion within a pro- sample does not posses structural homology amongst its
jected apertureofinfiniteradius.This measure,denotedby members makes it an ideal sample to explore the necessar-
σtot, has the added significance of being equal to one-third ily implied broken dynamical homology and the combined
of the virial velocity for spherical systems (Ciotti 1994). influenceoftheseeffectsupontheFP.Fromthesub-sample
In Paper I the light profile data (aperture magnitudes of galaxies for which major-axis profileshavebeen modeled
from Bower, Lucey & Ellis 1992) from a sample of 26 (see Caon et al. 1993 for a discussion of some of the prob-
E/S0Virgogalaxieswasanalysed.DeparturesfromtheR1/4 lems in doing this), those which have no published velocity
model were found tobewell-described bytheSersicmodel, dispersion data have been excluded here. This reduced the
andtheimpliednon-homologyinthevelocitydispersionpro- sample of Virgo galaxies from 33 (Caon et al. 1993) to 25,
filewasexplored.UsingtheSersicmodelparameters,rather and thesample of Fornax galaxies from 16 (D’Onofrio, Ca-
thanR1/4 model parameters, andusing thecomputed,infi- paccioli & Caon 1994) to 10. The 10 Fornax galaxies were
nite aperture velocity dispersion rather than the CVD, the furtherreducedtoasampleof9aftertheexclusionofNGC
FPwasobservedtochangefromRe,4∝σ01.33±0.10Σe−,40.79±0.11 1399,which,withitsshallowlightprofile(n>15),couldnot
to Re,n∝σt1o.4t,4n±0.11Σe−,n0.93±0.08. Broken structural and dy- have its velocity dispersion profile accurately modeled due
(cid:13)c 0000RAS,MNRAS000,000–000
A Virgo-Fornax distance modulus 3
tothelargeuncertaintiesassociatedwiththeeffectiveradius
R .The remaining galaxies are listed in Table 1.
e
2.2 Surface brightness profiles and structural
model parameters
The B band surface brightness profiles from Caon et al.
(1994) havebeenusedinthisstudy.Theywere constructed
by these authors using the ‘global mapping’ procedure of
Capaccioli & Caon (1989) which couples CCD data for the
bright inner galaxy profile with photographic data for the
outer profile. This technique allows very deep photometry
ofgalaxies,andalleviatestheproblemofuncertainskysub-
traction errors that is associated with the use of small area
CCD’s. The method gives internal errors of less than 0.1
mag at µ =26 mag.
B
In this study the major-axis surface brightness profiles
havebeentruncatedatthesameinnerandouterlimitsthat Figure 1. The shape parameter n is plotted against the R1/n
were applied by Caon et al. (1993) and D’Onofrio et al. modeleffectiveradiusRefortheVirgoandFornaxgalaxieslisted
(1994) for the Virgo and Fornax galaxies respectively. In inTable1.Theassociated99%confidenceintervalisalsodrawn,
general, this meant fitting the entire light profile after the asdetailedinthetext.
exclusion of the inner portion affected by seeing, and the
outer portion where sky subtraction uncertainties became
persion is preferred over the CVD measurement because it
large(>0.25mag).Thisleftatypicalrangeinsurfacebright-
not only allows for the range of velocity dispersion profiles
ness from µ =20 to27 mag.
TheSerBsic(1968)R1/n lightprofilemodelwasfittedto present, but also samples different galaxies in a consistent
fashion (CVDmeasurementssampletypicallyanywherebe-
theremainingprofile,withtheintensityIgivenasafunction
tween0.01-0.2R ).However,duetothephysicalsignificance
of radius R such that e
oftheprojectedvelocitydispersionwithinanapertureofin-
R 1/n finiteradius,thisterm,whichalsosamplesdifferentgalaxies
I(R)=I exp −b(n) −1 . (1)
e " "(cid:18)Re(cid:19) ## inaconsistentfashion,ispreferredintheanalysiswhichfol-
lows.
Ie is the intensity at the radius Re, and b(n) (≈1.9992n- FollowingCaonetal.(1993)(theirFig.3),thelightpro-
0.3271, Capaccioli 1989) is defined such that Re is the ra- file shape parameter is plotted against galaxy scale-size in
dius that effectively encloses half of the total light of the Figure 1. Although our methods of parameterisation differ
profile model (Caon et al. 1993; Graham & Colless 1996). (Caon et al. 1993 have used model-independent estimates
The ‘shape parameter’ n describes the curvature of each for the half-light radius), the same trend of increasing val-
light profile. Application of a non-linear least-squares fit of ues of n with R is evident here. The present study also
e
the R1/n model to the galaxy surface brightness data, with includes Fornax galaxies as well as those from Virgo. Also
equalweightforeachdatapoint,producedsimilarvaluesfor shown in thefigurearethe99% confidenceintervals for the
theshapeparameterstothosederivedbyCaonetal.(1993) model parameters. These were computed by evaluating the
andD’Onofrioetal. (1994),confirmingthatthemodelsare chi-square values for a fine grid of points around the opti-
being fitted consistently between authors. mumsolution.Thecontoursdrawncorrespondto∆χ2=9.21
Table 1 lists the parameters from the best fitting R1/n (i.e.4σ),scaledafternormalisation ofthereducedχ2 ofthe
modelandalsothosefromthebestfittingR1/4model,which optimum solution. †
was fitted in an identical fashion. Σe is the mean surface In exploring the sensitivity of the R1/n model parame-
brightness within the effective radius Re, computed using terstotheradialrangeofthesurfacebrightnessprofile,the
thetechniquedescribedinAppendixAofPaperI.Thesub- Sersic model was fitted to that data within the inner 40′′.
scriptsnand4onthemodelparametersareusedtodenote This outer truncation is the same as the outermost data
themodelfromwhichtheparticularparametershavecome; pointsusedinPaperI,enablingafurthercomparisonofthe
n signifying R1/n model parameters, and 4 signifying R1/4 present data with that used in Paper I. The inner trunca-
model parameters. tion used byCaon et al. (1993) and D’Onofrio et al. (1994)
Ciotti (1991), and also Paper I, explored the relation- was again employed, except for NGC 4473 and NGC 4636
ship between theprojected and spatial (deprojected) quan- for which only the inner 3′′ was excluded. In Figure 2, the
tities.Bydefinition,thesurfacebrightnesstermshadaone- resultingmodelparametersareshownplottedagainst those
to-onemapping,whiletherewasanapproximatelyconstant obtained using the entire surface brightness profile (Table
offset (logR∼0.13, Ciotti 1991) between the effective radii.
Therefore either of these quantities could be used for the
construction of theFP.The deprojected velocity dispersion †
It is noted that these confidence intervals only encompass the
at the effective radius was found not to have such a simple uncertainty of the parameters derived from the analytical fit to
relation with its projected counterpart, the central veloc- the data. They do not incorporate measurement errors or zero
ity dispersion measurement. This deprojected velocity dis- pointoffsetswhichmaybepresentinthedata.
(cid:13)c 0000RAS,MNRAS000,000–000
4 A.W.Graham
have negative extinctions, with a mean Virgo cluster value
of A =0.04±0.07 (standard deviation) for the sample. The
B
negative values are due to the variable ratio of H gas-to-
I
dust in the Galaxy (Burstein & Heiles 1978), and have no
physical meaning. Such values arise in the method used by
Burstein&Heilesforregionswherethereiszeroornegligible
reddening,andtheyshouldbesetequalto0.00.Corrections
for Galactic extinction (Burstein & Heiles 1984; Burstein
et al. 1987) were not applied by Caon et al. (1994). This
practice has been maintained here.
Ataredshiftofaround0.003,thek-correctionandred-
shiftdimmingwillbeequallysmallforbothclusters,at0.015
mag and 0.013 respectively. Consequently these terms have
not been incorporated into theanalysis.
2.3 Dynamical model parameters
CVDsare typically used as a measure of the kineticenergy
within elliptical and S0 galaxies. There are many problems
associatedwithdoingthis.Firstly,rotationalenergies,which
are likely to be significant for S0 galaxies, are ignored. Sec-
ondly,fixedaperturesizes(onthesky)inwhichtheCVDis
measured correspond to different physical sizes for galaxies
inclustersatdifferentdistances.Thiseffectalonemaylikely
introducesystematicerrorsintoclusterdistancedetermina-
tions which have incorporated fixed aperture sizes to mea-
sure the CVD. Thirdly, given that galaxies have a range of
sizes and structural profiles, they will also have a range of
dynamicalprofileswhichsupportthevariousstructuresthat
areobserved.Therefore,evenwithinclusters,fixedaperture
sizeswillsampledifferentfractionsofindividualgalaxies,as
measured in terms of theireffective radii.
In an attempt to find a better measure of the kinetic
energy within each galaxy, a method of computing the ve-
locity dispersion in a consistent manner for all galaxies was
presented in Paper I. Deprojecting the galaxy light profile,
Figure2.TheSersicR1/n modelparametersfromthefittothe the spatial luminosity density (Ciotti 1991) was computed.
entirelightprofile(subscriptb)areplottedagainstthoseparam-
Assuming a constant M/L ratio within each galaxy, such
eters obtained from fit to the light profile within 40′′ (subscript
knowledge of the internal density profile coupled with the
a). The circles represent Virgo E’s (18), the triangles are Virgo
application of the Jeans hydrodynamical equation enables
S0’s (7), the squares are Fornax E’s (7) and the stars represent
onetocomputetheinternalvelocitydispersionprofile.This
FornaxS0’s(2).
can then be projected outward to give the projected radial
velocity dispersion profile, which in turn can be integrated
1).It isnoted thatthelight profiles from thesmaller radial overincreasingaperturesizestogivetheprojectedaperture
extension(<40′′)maynotberepresentativeofthegalaxyas velocity dispersion profile. This is then calibrated using the
awhole,andthereforemaynotrepresenttheover-allglobal measured CVD.
galaxy shape. This is more likely to be true for the larger Thecomputedspatial(deprojected)velocitydispersion
galaxies,especiallythosehavingR >40′′.Itisalsotruethat atthespatialhalf-light radiuswasusedinPaperItorepre-
e
theSersic indexnis lesssensitive at larger values(Graham sent the kinetic energy of the galaxies. Unlike use of CVD,
et al. 1996). Thus, Figure 2 offers an alternative estimate this approach consistently samples the velocity dispersion
for the uncertainty of the R1/n model parameters. Some of profile in the same way for all galaxies. As an alternative,
the more discrepant points have been labeled in the plot. thecomputedvelocitydispersionwithinanapertureofinfi-
It is noted that they are predominantly the S0 galaxies. If niteradiuswas also used.This quantityhas been shown by
oneistothinkofthesegalaxiesaspossessinganR1/4(n=4) Ciotti (1994) to be equal to one-third of the virial velocity
centralbulgeand an exponential(n=1) outerdisk,or some dispersion,independentofanyorbitalanisotropiesthatmay
approximately equivalent Sersic form, then this could ac- be present. This measure for the kinetic energy is adopted
count for thelarger index obtained using thesmaller radial here, not only for the consistent fashion in which galaxies
rangefortheS0galaxiesNGC1380,4339,and4550.Infact, are sampled but also due to the significance of the infinite
such measurements offer themselves as possible diagnostics aperture velocity dispersion σ . These values are listed in
tot
for distinguishing S0 galaxies from ellipticals. Table 1, where the subscripts 4 and n refer to use of the
FortheFornaxgalaxies,thetabulatedextinctionsrange R1/4 andR1/n modelsrespectively.AlsolistedaretheCVD
from AB=-0.05 to -0.09. Some of the Virgo galaxies also measurements, σ0, from McElroy (1995).
(cid:13)c 0000RAS,MNRAS000,000–000
A Virgo-Fornax distance modulus 5
Table 1. Galaxy model parameters. Thegalaxy sampleis aselection from Caonet
al. (1994). The parameters are from the best fitting R1/4 and R1/n models, with
the exception ofσ0 whichis the tabulated central velocity dispersionfromMcElroy
(1995).Re,4andRe,naretheeffectiveradiioftheR1/4andtheR1/n modelsrespec-
tively, with Σe,4 and Σe,n the mean surface brightness within Re. σtot,4 and σtot,n
aretheinfiniteaperturevelocitydispersionsdescribedinthetext.
Galaxy Morph. σ0 n logRe,n Σe,n σtot,n logRe,4 Σe,4 σtot,4
Ident. Type kms−1 [kpc] [mag] kms−1 [kpc] [mag] kms−1
NGC1339 E4 160 1.80 0.20 20.78 138 0.04 19.74 134
NGC1351 E5 143 3.38 0.46 21.52 112 0.46 21.46 110
NGC1374 E0 186 3.28 0.34 20.93 149 0.32 20.77 147
NGC1375 S0 53 1.36 0.29 21.27 53 0.40 21.59 41
NGC1379 E0 119 2.07 0.32 20.96 100 0.26 20.57 95
NGC1380 S0/a 225 1.63 0.59 20.61 191 0.72 21.01 166
NGC1404 E2 250 4.42 0.35 19.82 193 0.34 19.79 196
NGC1419 E0 133 6.41 -0.06 20.35 110 0.02 20.89 112
NGC1427 E4 155 3.42 0.50 21.36 121 0.49 21.27 118
NGC4168 E2 186 6.16 0.78 22.82 125 0.64 22.35 139
NGC4261 E2 326 8.01 0.99 22.82 196 0.74 21.91 240
NGC4339 S0(0) 114 2.90 0.36 21.48 92 0.38 21.54 89
NGC4365 E3 261 6.45 1.13 22.83 164 0.96 22.14 190
NGC4374 E1 296 8.19 1.09 22.33 173 0.91 21.69 216
NGC4387 E5 112 1.92 0.06 20.35 99 0.00 19.89 95
NGC4431 dS0,N 68 1.57 0.30 22.29 58 0.60 23.50 51
NGC4434 E0 115 3.85 0.07 20.63 96 0.07 20.63 96
NGC4458 E1 106 2.56 0.21 21.30 89 0.24 21.29 85
NGC4459 S0(2) 178 4.84 0.60 21.20 130 0.57 21.14 134
NGC4464 E3 121 2.41 -0.22 19.48 111 -0.38 18.44 112
NGC4472 E2 303 5.82 1.36 22.74 193 1.21 22.15 221
NGC4473 E5 193 3.12 0.58 20.88 151 0.53 20.51 146
NGC4476 S0(5) 81 3.01 0.15 20.77 68 0.06 20.19 68
NGC4478 E2 143 1.87 0.03 19.70 127 -0.10 18.92 124
NGC4486 E0 333 5.35 1.15 22.29 221 1.01 21.78 242
NGC4550 S0(7) 83 1.80 0.30 20.25 71 0.01 18.38 70
NGC4551 E2 113 1.81 0.09 20.56 99 -0.03 19.90 96
NGC4552 S0(0)a 269 13.86 1.30 23.73 117 0.76 21.60 198
NGC4564 E6 160 1.51 0.36 20.59 137 0.07 18.63 133
NGC4621 E4 230 5.22 0.85 21.61 159 0.70 21.05 170
NGC4623 E7 89 1.18 0.38 21.41 89 0.30 20.71 70
NGC4636 E1 207 4.40 1.33 23.43 147 1.30 23.29 152
NGC4649 S0(2) 339 5.00 1.01 21.70 232 0.95 21.48 247
NGC4660 E3 185 2.66 0.08 19.39 159 -0.03 18.78 158
a Alsoknown as M89, this galaxy iscatalogued as type E inde Vaucouleurs et al.
(1991).
3 CONSTRUCTION OF THE FUNDAMENTAL thebisectormethodoflinearregressionisthepreferredtech-
PLANE nique (Isobe et al. 1990; Feigelson & Babu 1992). This ap-
proach requires that threeFPs becomputed,each of which
ThequantitieswhichareusedinthisinvestigationoftheFP isobtainedbyminimisingtheresidualsofadifferentparam-
aretheeffectiveradius,Re,andthemeansurfacebrightness eter.Themean slope angles of these3planes arethenused
withinthisradius,Σe,derivedfromthemodelfitstothesur- to represent thefinal FP.
face brightness profiles. These values are listed in Table 1. Theuncertaintiesassociated withtheFPslopearealso
This approach differs from D’Onofrio, Longo & Capaccioli important.Feigelson &Babu(1992) notonlyshowthatthe
(1996), who used model-independentvalues for these quan- standard formula (e.g. Bevington 1969) for calculating the
tities in their construction of the FP. They also only used uncertaintiesonthederivedslopeandinterceptisingeneral
CVDs for their velocity dispersion measure. mathematically incorrect, but they also show that the de-
In this section, the Fundamental Plane is constructed rived asymptotic, analytic expressions for the variance will
with the goal of comparison with theoretical predictions. under-estimate the true variance when the sample size is
Consequently,the3parameterswhichareusedtodefinethe small (approximately ≤50). They therefore recommend the
FP (σ,R,Σ) are treated as independent of each other. This use of a bootstrapping procedure to gauge the variance on
requires a symmetrical treatment of these parameters, and thefittedslopeanditsintercept.Thisapproachwasfollowed
(cid:13)c 0000RAS,MNRAS000,000–000
6 A.W.Graham
Table 2. Fundamental Plane R∝σAΣB for a sample of 25 E & Table 4.FundamentalPlaneR∝σAΣB forasampleof7EFor-
S0Virgogalaxiesconstructedusingthebisectormethodoflinear nax galaxies constructed using the bisector method of linear re-
regression in 3 dimensions. The errors are from a bootstrapping gression in 3 dimensions. The errors are from a bootstrapping
procedure.TheapproachreferredtoasR1/4[σ0]hasusedtheef- procedure.
fectiveradiusfromthe R1/4 model, themeansurface-brightness
within this radius, and the projected CVD, σ0, in the construc- model R∝σAΣB ⊥rms
tion of the FP. R1/n[σtot,n] uses the infinite aperture velocity A B residual
dispersion term, σtot,n, and the effective radius and associated
surfacebrightnesstermfromtheR1/n model. R1/4,σ0 1.69±0.83 -0.85±0.30 0.064
R1/n,σ0 1.79±0.79 -0.98±0.28 0.049
model R∝σAΣB ⊥rms R1/4,σtot,4 1.85±0.86 -1.00±0.35 0.070
A B residual R1/n,σtot,n 2.03±0.78 -1.07±0.30 0.050
R1/4,σ0 1.10±0.14 -0.55±0.09 0.099
R1/n,σ0 1.12±0.17 -0.64±0.07 0.084 Table 5. Fractional Variance from the Principal Component
R1/4,σtot,4 1.17±0.15 -0.60±0.08 0.100 Analysisonthesampleof25VirgoE&S0galaxies.
R1/n,σtot,n 1.37±0.16 -0.76±0.05 0.077
model Var1 Var2 Var3
Table 3. Fundamental Plane R∝σAΣB for a sample of 18 E R1/4,σ0 76.03% 21.99% 1.98%
Virgo galaxies constructed using the bisector method of linear R1/n,σ0 82.79% 15.80% 1.41%
regression in 3 dimensions. The errors are from a bootstrapping R1/4,σtot,4 72.30% 25.57% 2.13%
procedure. R1/n,σtot,n 75.18% 23.44% 1.38%
model R∝σAΣB ⊥rms
A B residual the data sets are defined by a 2-dimensional plane within
the 3-space of observables. Table 5 displays the fractional
R1/4,σ0 1.19±0.21 -0.60±0.11 0.084 variance of the data along the major eigenvectors of the
R1/n,σ0 1.22±0.25 -0.66±0.12 0.071 FP parameter space. This reveals that greater than 97% of
R1/4,σtot,4 1.28±0.24 -0.65±0.11 0.085 the variance in the data indeed lies in a plane, confirming
R1/n,σtot,n 1.72±0.24 -0.74±0.09 0.050 the appropriateness of constructing a plane to describe the
properties of elliptical galaxies.
inthisstudy.However,thisnumericalresamplingtechnique
was modified for application to the small sample of Fornax
4 THE VIRGO-FORNAX
elliptical galaxies in order to exclude 10% of the 400 boot-
DISTANCE-MODULUS
strappingsimulations with anegative,ratherthan positive,
slope. These simulations were excluded from the computa- Onlytheellipticalgalaxypopulationisusedinthefollowing
tion of the final slope variance due to their obviously wild determinationoftheVirgo-Fornaxdistance-modulusdueto
departurefromtheexpectedvalue.SuchFP’sareduetothe the contaminating influence of the S0 galaxies on the FP.
smallsamplesizeof7pointsinFornax.Withthesomewhat Contaminating in the sense that S0 galaxies may possess a
larger sample of Virgo galaxies (25) this problem did not significant degree of rotational energy which is ignored in
arise in any of the2000 simulations used. thepresent analysis.
The total sample of Virgo galaxies contains 18 E’s The problem of computing the offset between two par-
and 7 S0’s. It was therefore possible to compute the allel data sets, with an emphasis on cosmic distance scale
FP using both the entire sample, and using only el- calibrations, is well described in Isobe et al. (1990) and
liptical galaxies. In addition, given the four sets of Feigelson & Babu (1992). If the objective in construct-
FPparameters(logσ0,logRe,4,Σe,4),(logσ0,logRe,n,Σe,n), ing the FP is for the purpose of computing distances be-
(logσtot,4,logRe,4,Σe,4) and(logσtot,n,logRe,n,Σe,n),a to- tween galaxy clusters, then the FP parameters should be
tal of eight FP’s have been computed for the Virgo clus- treated differently than they were treated in Section 3
ter. The results are shown in Table 2 and Table 3, where (where the FP was constructed for comparison with a the-
the rms residual is that perpendicular to the fitted plane. oretical prediction). In this case, one wishes to predict the
The uncertainties on the estimated slopes are those from value of a distance-dependent quantity,namely logR, from
the bootstrapping procedure, and are therefore larger than the distance-independent quantities logσ and Σ. Conse-
the standard analytically determined uncertainties derived quently,the plane that is fitted to the data should be com-
fortheFP.Usingthesampleof7availableFornaxelliptical putedthroughanordinaryleast-squaresminimisationofthe
galaxies an additional four planes were computed using the distance-dependent quantity, rather than using the bisec-
abovementionedfoursetsofparameters.Table4showsthe tor method of linear regression which treats all variables
results. equally (Section 3). This approach will minimise the un-
APrincipalComponentAnalysis(PCA)wasperformed certainty on the estimate of the relative distance between
on each of the four 3 dimensional data sets from the com- clusters. Such an approach was taken with the two ex-
binedE&S0Virgogalaxysample.ThecodefromMurtagh tremeVirgoandFornaxdatasets,(logσ0,logRe,4,Σe,4)and
&Heck(1987)wasimplementedtoshowthedegreetowhich (logσtot,n,logRe,n,Σe,n). (These are extreme in the sense of
(cid:13)c 0000RAS,MNRAS000,000–000
A Virgo-Fornax distance modulus 7
theFornax cluster data does indeed definea plane which is
Table 6. Fundamental Plane R∝σAΣB using only elliptical
consistentwithbeingparalleltotheVirgoclusterdata.This
galaxies. Constructed using ordinary linear regression, in 3-
dimensions, that minimised the logR residuals. The errors are is true when R1/4 model parameters and CVD’s are used,
fromabootstrapping procedure. andalsowhenR1/n modelparametersandinfiniteaperture
velocity dispersions are used to construct the FP.
model R∝σAΣB (logR)rms In the following method of analysis, a linear regression
A B residual is performed on one of the cluster data sets, and the re-
sulting regression line from this calibration is then applied
Virgo:R1/4,σ0 1.11±0.27 -0.59±0.13 0.134 to the second (parallel) cluster. One technique for measur-
Fornax:R1/4,σ0 1.14±0.84 -0.59±0.29 0.120 ing the intercept offset between parallel data sets was pi-
Virgo:R1/n,σtot,n 1.62±0.27 -0.74±0.12 0.100 oneered by Working & Hotelling (1929) and further devel-
Fornax:R1/n,σtot,n 1.47±0.82 -0.81±0.34 0.110 oped by Feigelson & Babu (1992) to permit the x-values
(i.e. distance-independent values) to be random variables
rather than fixed quantities. With this method, the cali-
Table 8. Distance moduli given by 5(∆logR), where ∆logR brated regression line for Virgo was applied individually to
is the computed intercept offset between the cluster FPs, as de- the data points in Fornax and a weighted mean offset at-
scribed in the text. These have been obtained using a general- tained; the method is discussed in Section 3.2 of Feigelson
isation of the Working-Hotelling confidence bands (Feigelson &
& Babu (1992) and their AppendixA2. This procedure en-
Babu1992),usingbothVirgoandFornaxasthecalibrationsam-
compasses the fact that the confidence intervals about the
ple.Theuncertaintiesarefromtheasymptoticformulafitsgiven
y-value one is trying to predict, given its x-value, will be
inTable7.
larger for those data points at the ends of the distribution.
Calibrator R1/4[σ0] R1/n[σtot,n] Thisisbecausetheuncertaintyintheslopeofthecalibration
line/planegivesgreaterfreedomofmovementtothecalibra-
sample mag.offset mag.offset
tion line at the ends of the distribution than in the center.
Virgo: –0.46±0.23 –0.32±0.22 This method of analysis gave an intercept offset, along the
Fornax: +0.45±0.16 +0.25±0.12 logRaxisbetweentheFP’sdefinedbytheVirgoandFornax
data sets, of ∆logR=-0.092±0.045 dex and -0.064±0.043
dex for the (R1/4,σ0) parameters and the (R1/n,σtot,n) pa-
the ‘standard’ versus ‘improved’ treatment of galaxy struc- rameters, respectively.
tureanddynamics.)Thebestfittingplanes,underthismin- In a similar fashion, the Fornax cluster has been used
imisation scheme,aregiveninTable6,asarethermsresid- as the calibration sample, where the fitted regression line
uals of thedata projected along thelogR axis. for Fornax is applied to the Virgo cluster. This was done
In order to use the methods in Feigelson & Babu with the Working-Hotelling confidence bands. Using the
(1992), these 3-dimensional data sets have been reduced to ‘standard’ FP parameters (σ0,Re,4,Σe,4), an intercept off-
2-dimensions by combining the distance-independent vari- set of +0.091±0.031 dex was computed. Using the ‘im-
ables logσ and Σ. These were combined in such a way as proved’FPparameters(σtot,n,Re,n,Σe,n),aninterceptoffset
to preserve the previously obtained optimal linear regres- of+0.051±0.025 dexwas computed.Theseinterceptoffsets
sionsolution.Thismeantcomputingthe‘mixing’valueb=– translateintoadistancemodulusof+0.45±0.16mag(using
B/(2.5A), where A and B are the FP exponents given in R1/4 model parameters and σ0) and +0.25±0.12 mag (us-
Table 6, giving the quantity logσ+b<µ>, where <µ> is ingR1/nmodelparametersandσ ).Theformerestimate
tot,n
themean surface brightnessinmag units.‡ A furtherlinear placing Fornax some (23±8)% further away than theVirgo
regression calculation was performed, minimising the logR cluster, compared to thelatter estimate of (12±6)%. These
residuals at the expense of the (logσ + b<µ>) residuals. solutions are fully consistent with those obtained using the
The code SLOPES, discussed in Isobe et al. (1990), was Virgo cluster as the calibration sample (see Table 8).
usedtodothis,andtheresultsarepresentedinTable7.As Analternativeapproachforcomputingtheoffsetofpar-
expected,duetothechoiceofthemixingvalue,theasymp- alleldatasetswasachievedfollowingthemethodsinSection
totic formula gave the same slope as derived with the pre- 3.1ofFeigelson&Babu(1992)andtheirAppendixA1.The
vious treatment of the FP variables in 3-dimensions. The FP data set from the Virgo cluster of galaxies was used to
slopes obtained with bootstrap and jackknife sampling are calibrate the regression line which was then applied to the
seentovaryslightlyandhavedifferentassociateduncertain- FPdatasetoftheFornaxclusterasawhole(andvice-versa).
ties,asexpectedforsmallsamplesizes.Theerrorsfrom the This was done in order to derive the intercept offset, along
bootstrapping procedure for the Fornax data set have been the logR axis, between the FP’s defined by the Virgo and
restricted as discussed previously. Fornaxclusters.Thesolutionsobtainedweresimilar,within
The two techniques described in Feigelson & Babu 0.001 dex, to those calculated using the Working-Hotelling
(1992) for estimating the offset of parallel data sets have confidencebands,althoughtheassociateduncertaintieswere
bothbeen implementedhere.Thesemethodsaresubject to generallyequalorlarger.Thisservedasausefulconsistency
the condition that the Virgo and Fornax elliptical galaxy check on the determination of the cluster-cluster distance
population define FPs which are parallel to each other. modulus.
GiventheuncertaintiesonthederivedFPslopes (Table7), With the Working-Hotelling confidence bands, the un-
certainty on theintercept offset, and hence thedistance es-
timate,isreducedbythesquare-rootofthesamplesizeone
‡ Σisthemeansurfacebrightness inlinearunits. is applying the calibrated solution to. Consequently,apply-
(cid:13)c 0000RAS,MNRAS000,000–000
8 A.W.Graham
Table7.Ordinaryleast-squaresregressionanalysisforlogRe versus(logσ+bΣe),using
theprogramSLOPES(Feigelson&Babu1992).Ineachcase,thevalueofb=−B/(2.5A)
isderivedfromthebestfittingplaneshowninTable6.
method AsymptoticFormula Bootstrap Jackknife
Slope(A) Slope(A) Slope(A)
Virgo,R1/4[σ0]:b=0.21 1.11±0.08 1.10±0.08 1.11±0.09
Fornax,R1/4[σ0]:b=0.21 1.14±0.27 1.21±0.45 1.14±0.38
Virgo,R1/n[σtot,n]:b=0.18 1.62±0.09 1.62±0.09 1.62±0.10
Fornax,R1/n[σtot,n]:b=0.22 1.47±0.38 1.40±0.79 1.41±0.75
ing the Fornax regression line to Virgo, with a sample size 5 DISCUSSION
of 18, results in a more accurate solution than achieved by
In the interest of statistical completeness, galaxies with a
applyingtheVirgo regression linetoFornax,with a sample
CVD of less than 100 km s−1 were not explicitly removed
size of only 7. However, this effect does have to compete
fromtheanalysis.DuetothegreaterscatterintheFPbelow
with the fact that the uncertainty on the regression line is
thisvalue,paststudieshaveenforcedthisarbitrarycut-offin
larger for a smaller calibration sample. As the generalised
theirgalaxysample.However,itseemsplausibletoattribute
Working-Hotellingconfidencebandspropagatetheerrorsin
thehigherlevelofscattertoaninadequatetreatmentofthe
the calibrated line, this effect cannot be avoided, and the
kineticenergyforthesegalaxies. Asthisisanissuewhichis
sizeoftheseerrorsinfluencetheassociatederrorsforthein-
addressedinthisstudy,itmadesensetoleavethesegalaxies
tercept offset. Thelack of symmetry for theerror estimates
in the analysis.
whenusingVirgoorFornaxasthecalibratorisduetofollow-
Contributions from rotational energy to the total ki-
ing: The error estimates on the slope of the regression line,
neticenergy of agalaxy can in principle influencetheslope
which were propagated in the above analysis, were derived
of the FP. Djorgovski & Santiago (1993) noted that gains
fromtheanalyticalexpressionfortheseterms.However,the
from including a rotational energy measurement would be
errorsontheslopeandinterceptwillbeunder-estimatedby
offsetbytheintroductionofadditionalmeasurementerrors.
the analytical expressions when the number of data points
Indeed, a marginal effect has been reported many times
is small. This is seen in the FP solutions shown in Table 7,
(Simien & Prugniel 1992; Busarello, Longo & Feoli 1992;
where the jackknife and bootstrapping methods of analysis
Prugniel&Simien1994).Anasymmetrictrendbetweenthe
provide a better estimate of the true size of these errors.
galaxy FP residuals and their maximum rotational velocity
Consequently, the error estimates on the previously given
hasalso been found byD’Onofrio et al. (1996). More quan-
distance moduli should be multiplied by approximately the
titative estimates from Prugniel & Simien (1996a,b), using
ratio of thebootstrapping error (or jackknife error if this is
asampleof400galaxies,claimthatrotationcanaccountfor
larger) to theasymptotic error (Feigelson & Babu 1992).
25±15% of the departure of the FP from the virial expec-
Inaddition,itshouldbenotedthattheFPslopeuncer-
tation. While this figure may be true for FP’s constructed
tainties in Table 7 were estimated when the mixing value,
usingearly-typegalaxies (EandS0),which includesalmost
b, was held fixed and assumed to have zero error. It should
all FP’s constructed to date, this figure can be reduced by
be remembered that it is the offset between two parallel
the exclusion of S0 galaxies, known to have rotating disks.
planes, not two parallel lines, that is being measured. In
This was hinted at in the study by Busarello et al. (1997),
order to fully estimate the uncertainties on the derived dis-
who used only bona fide ellipticals to obtain a contribution
tancemodulus,onemustallow forerrorsintheoptimalFP
of 15±28% to the tilt to the FP due to galaxy rotational
slopes for all axes. In so doing, one should return to the
energies.
bootstrapping errors obtained with the 3-dimensional data
Inthispaper,theFPwasconstructedusingaculledset
set(Table6).Workingin3-dimensions,theerrorsassociated
ofgalaxies,removingtheS0sandkeepingtheEs.Contribu-
with the FP exponent A, are obtained by also letting the
tions from possible rotational support tothe kineticenergy
exponent B (and hence b) vary, as permitted by the boot-
term are expected to be minimal in this study of the FP.
strappinganalysisofthedata.Therefore,theerrorobtained
Four of the nine S0 galaxies which were excluded did ac-
in the 3-dimensional analysis represents the true error esti-
tually have CVD’s less than 100 km s−1, leaving only one
mateoftheexponentA.Allowing forthis,theerrorstothe
galaxyintheremainingsampleof25ellipticalswithaCVD
solution for the Virgo-Fornax distance modulus are again
less than 100 km s−1. It may indeed be that the increased
multiplied by the ratio of the 3-D bootstrapping errors to
scatter observed for the FP at the low velocity dispersion
the 2-D asymptotic errors. This procedure does not effect
endisduetothepresenceofS0galaxiesthatharbourrotat-
theoptimalsolution, onlytheassociated uncertainties.The
resulting solutions place Fornax (23±44)% (R1/4,σ0), and ingdiskswithsignificantrotationalenergiesthataresimply
(12±36)% (R1/n,σtot,n) furtheraway than Virgo. not measured through CVD measurements. For the Virgo
cluster, exclusion of theseobjects resulted in a reduction of
some 15-35% in the scatter about the FP, as evidenced by
therms residuals listed in Table 2 and Table 3.
(cid:13)c 0000RAS,MNRAS000,000–000
A Virgo-Fornax distance modulus 9
For the Virgo cluster, exclusion of the S0’s caused weight, and does not fit the central portion of the profile.
the final FP solution (incorporating the observed range of In addition, the use of aperture magnitude data samples
galaxyprofilesandimplieddynamics)tochangefromRe,n ∝ light from all position angles, and in so doing represents a
σt1o.3t,7n±0.16Σe−,n0.76±0.05 to Re,n ∝ σt1o.7t,2n±0.24Σe−,n0.74±0.09. This meanlightdistributionforthegalaxyasawhole.Thepres-
suggeststhattheinclusionoftheS0populationnotonlyin- ence of ellipticity gradients will result in the major- and
creasesthescatterabouttheFP(seeTables2and3)butis minor-axis light profiles being described by different shape
alsopartlyresponsibleforthedepartureoftheFPfrom the parameters so that n 6=n . This is clearly evident in
maj min
relation R ∝ σ2Σ−1, predicted by the virial theorem. This therangeofshapeparameters,n,forthemajor-andminor-
isagaineasilyunderstoodasduetotheneglectofrotational axis,showninthestudyoftheVirgoellipticalsbyCaonetal.
energies in the construction of the observed FP. It is there- (1993). Consequently, the mean light distribution obtained
foresuggestedthatinadditiontotreatingbrokenstructural through theuseof circular aperturemagnitudes should not
anddynamicalhomology,theconstructionoftheFPalways be expected to be identical to the light distribution along
adopt theprocedureofonly usingbona fideelliptical galax- themajor-axis. Finally, thedataof Bower et al. (1992) was
ies, unless oneallows for rotational energies. taken in the V-band, and that of Caon et al. (1993) in the
It is found that by addressing the range of structural B-band. Possible colour gradients, although expected to be
anddynamicalprofilesevidentamongsttheellipticalgalaxy small, may also contribute to differingresults.
population,thedepartureoftheFPfromtheplanepredicted
Despite the above factors, there is still fair agreement
by the virial theorem is reduced, confirming the results in
betweentheSersicmodelparametersobtainedusingthedif-
PaperI.Inaddition,thescatterofthedatapointsaboutthe
ferentdatasets.Galaxyscale-sizeandlightprofileshapefor
FPisreducedbysome20%forthesampleofFornaxellipti-
those galaxies in common between this Paper and Paper I
calsandbythesameamountforthesampleofVirgoEand
aredisplayedinFigure 3.Forcomparison between thedata
S0 galaxies. For the sample of Virgo ellipticals, the mean
sets, the Sersic model was fitted to the same radial extent
rms scatter about the FP is reduced by 40% after treating (<40′′) for each data set. This outer radius cut-off comes
therangeofgalaxystructuresanddynamics.Thisisevident
from the aperture magnitude profile limit of Bower et al.
in Tables2,3and 4,which givethebest fittingFPrelation
(1992), and has been applied, for this comparison, to the
when onedoesand does nottreat brokenstructuralhomol-
major-axis profiles of Caon et al. (1994). The correlation
ogy, and when one does and does not consistently measure
seen between the shape parameters has a linear correlation
thekineticenergyofthegalaxies(i.e.usesσ orCVD’s,re-
tot coefficient of 0.55, which increases to 0.82 upon removal of
spectively).Thefirstrowofthese3tablesgivestheFPwhich
the 3 more discrepant points that are labeled in Figure 3.
wasconstructedassumingthatallgalaxies aredescribedby
The correlation coefficient between the effective half-light
an R1/4 profile and uses CVD’s to represent the kineticen-
radiiisstrongerat0.93,suggestingthattherangeofgalaxy
ergyofeachgalaxy;Thisproceduregivesthegreatestdepar-
sizes, and structures, can be largely reproduced from both
tures of the FP from the virial plane. The final row in each
types of observational data. The galaxies which appear to
TablegivestheFPthatwasconstructedallowingforarange
havethemorediscrepantshapeparametersaretheflattened
of structural profiles amongst the galaxies, as described by
S0galaxy NGC4550, NGC4486, andNGC4564 whichhas
the R1/n model, and measures the kinetic energy of each
a value of n=4.64 (Paper I) lying between n =1.51 and
maj
galaxy in a consistent manner by using the computed ve-
n =5.08 (Caon et al. 1993) and is therefore not a dis-
min
locity dispersion within an aperture of infinite radius; This
crepant point after all.
procedure gives the closest agreement between the FP and
the virial plane. In fact, the formal solution for the FP of Despite the above differences, it is interesting to com-
tbheeinFovrenrayxcclloussetearg(rReeem,ne∝ntσwt2o.i0tt,3hn±t0h.7e8Σpe−r,en1d.0i7c±ti0o.3n0)ofistfhoeunvidritaol pseatrse.tFhiersrtelysu,latsinsugmFiPn’gssotbrtuacitnuerdaluhsionmgotlhoegyindanepdenudsienngtRda1t/a4
theorem,butdoeshavelargeuncertaintiesassociatedwithit model parameters and CVD measurements for the Virgo
becauseofthesmallnumber(7)ofFornaxellipticalgalaxies cluster of galaxies, the FP was found in Paper I to be de-
in the sample. Overall, it is concluded that addressing the scribed by Re,4 ∝ σ01.33±0.10Σe−,40.79±0.11. This is to be com-
range of structural profiles amongst the elliptical galaxies paredwithRe,4 ∝σ01.10±0.14Σe−,40.55±0.09 foundinthisstudy.
and better treating the galaxy dynamics reduces both the The exponents on the velocity dispersion term are consis-
departure of the FP from the virial plane and the scatter tent within the 1σ joint errors, and the exponents on the
about theFP. surface brightness term are consistent within the 2σ joint
In Paper I theSersic light profilemodel was fitted toa errors. Allowing for broken structural homology and con-
sample of 26 E/S0 Virgo galaxies using the aperture mag- sistently measuring the velocity dispersion within an aper-
nitude profile data from Bower et al. (1992). Of these, 18 ture of infinite radius, it was found in Paper I that the FP
galaxies are in common with the sample of 25 used here. for Virgo is described by Re,n ∝σt1o.4t,4n±0.11Σe−,n0.93±0.08. This
Comparisonbetweenthegalaxymodelparametersobtained should be compared with Re,n ∝σt1o.3t,7n±0.16Σe−,40.76±0.05. The
in each study is not straight-forward, however. The galaxy relative consistency between the exponents is the same as
profile data used here is in the form of surface brightness before. It is noted that the uncertainties on the FP expo-
measurements along the major-axis, where as Paper I uses nents are based upon the statistics of the linear regression
circular aperturemagnitudes. Consequently,themodelsfit- to the data points; they do not incorporate possible mea-
tedinPaperIaremoreheavilyweightedbytheinnerprofile, surement errors in the data points themselves. It is also
andbynecessity incorporatethecentralpartsofthegalaxy noted that the sample of Virgo galaxies used in Paper I
light. By contrast, the treatment of the surface brightness contains 7 galaxies that this study does not include, and
profiles here gives all data points along the profile equal this study also contains 7 galaxies not present in the sam-
(cid:13)c 0000RAS,MNRAS000,000–000
10 A.W.Graham
fitted to the undisturbed portion the galaxy light profile.
This portion is free from any peculiarities of the innerlight
profiles(notedbyCaonetal.1993),isnotheavilydistorted
by the presence of a disk or dust lanes, and is uncontami-
nated from possible tidal interactions which may have per-
turbed the original galaxy profile, particularly in the outer
parts. The parameters used by D’Onofrio et al. (1996) are
better estimates of the half-light radii, in the true sense of
the term, in that they are the radii which enclose half of
thetotal light from thegalaxy,although thetotal light was
derived by extrapolating the observational data to µ =32
B
withR1/4 profiles(Caonetal.1994),Thepreferreddynam-
ical term used here differs from that used by D’Onofrio et
al.(1996).WhiletheyusedCVDvalues,thispaperhasalso
used, and in fact prefers, the infinite aperture velocity dis-
persion derived from the application of the Jeans equation
(Ciotti 1991; Paper 1). D’Onofrio et al. (1996) gave their
Virgo FP solution as Re∝σ01.26±0.09Σe−0.70±0.03. This is in
good agreement with the FP solution derived here, which
addressed the range of structural profiles but used CVD
measurements. For the sample of Virgo E and S0 galaxies
therelation Re,n∝σ01.12±0.17Σe−,n0.64±0.07 was obtained here.
D’Onofrio et al. (1996) computed a Virgo-Fornax dis-
tance modulus of 0.50 mag (using the FP) and 0.45±0.15
mag(usingtheD -σ relation). Thisresultisincloseagree-
n
mentwiththedistancemodulusderivedinthisstudybased
upon galaxy parameters from the R1/4 profile and using
CVD’s, where a value of 0.45±0.16 mag (asymptotic er-
ror) was computed. Using R1/n model parameters and in-
finite aperture velocity dispersion measurements, however,
this study found the Virgo-Fornax distance modulus to be
0.25±0.12 mag(asymptoticerror).GiventhatD’Onofrioet
al.(1996)allowedfordifferentgalaxystructuresthroughus-
ingmodel-independentestimatesofthehalf-lightradiusand
associated meansurface brightness,butusedCVD’s,thefi-
nal estimate of 0.25 mag appears to be different because of
thetreatment of the dynamical term rather than thetreat-
Figure 3. The Sersic R1/n model parameters for the effective
ment of the structural quantities. Indeed, D’Onofrio et al.
radius, Re, and shape parameter, n, are displayed for 18 galax- (1997)introducedtheirownaperturecorrectiontotheveloc-
ies from Paper I which are in common with with the sample in
itydispersionterm,findingareducedVirgo-Fornaxdistance
this study. The subscript (1) denotes that the parameters were
obtainedfromPaperI,andthesubscript(2)referstothevalues modulusof 0.30±0.05 mag (formal error).
obtainedinthisstudy. Comparison with other methods of distance determi-
nation yields good agreement with the distance modulus of
+0.25±0.12 mag obtained in Section 4. The investigation
ple of Paper I. Using only those 18 galaxies in common, byMcMillan etal.(1993), usingtheplanetarynebulalumi-
theresultingFP’swerefoundtobeinacceptableagreement nosity function, found a Virgo-Fornax distance modulus of
with each other, with the FP described by the data in Pa- +0.24±0.10 mag. Another accurate estimate has been de-
perIgivenbytherelationRe,n ∝σt1o.2t,9n±0.10Σe−,n0.89±0.05,and rived using the method of surface brightness fluctuations,
that obtained using this studies data given by the relation where a value of 0.20±0.08 mag has been obtained (Tonry
Re,n ∝ σt1o.2t,6n±0.12Σe−,n0.81±0.04. The main point is that both et al. 1997). Using a sample of E and S0 galaxies, Dressler
studies show that by treating the range of luminosity and et al. (1987) applied the Dn-σ relation to compute a value
dynamicalstructuresinherentamongsttheearlytypegalax- of +0.14±0.18 mag. An alternate approach using type Ia
iesthedepartureoftheFPfromthevirialplaneisreduced. supernovae (Hamuy et al. 1991) found a yet lower value of
D’Onofrio et al. (1996) also constructed a FP for the 0.09±0.14mag,althoughstillconsistentwithintheerrorsof
Virgo and Fornax clusters using the data of Caon et al. thevaluederived here.
(1994). The approach taken in this study differs in sev- However the result presented here is in poorer agree-
eral ways to that of D’Onofrio et al. (1996). This work mentwithstudiesusingtheglobularclusterluminosityfunc-
has used model-determined effective radii and the model- tion(-0.5±0.2mag,Bridges,Hanes&Harris1991)andwith
determined mean surface brightness within these radii, studies which have used the Tully Fisher (TF) relation for
where as D’Onofrio et al. (1996) used model-independent spiral galaxies. The H-band TF relation used by Aaronson
values for these quantities (Caon et al. 1994). The model- et al. (1989) placed the Fornax cluster closer than Virgo,
determinedquantitiesarethosederivedfromtheR1/nmodel givingadistancemodulusof-0.25±0.23magforFornaxrel-
(cid:13)c 0000RAS,MNRAS000,000–000