The distribution of dark and luminous matter inferred from extended rotation curves

1.2 MB·English

by  Roelof Bottema

Mon.Not.R.Astron.Soc.000,1–28(2015) Printed27January2015 (MNLATEXstylefilev2.2)
The distribution of dark and luminous matter inferred
from extended rotation curves.
5 Roelof Bottema1 and Jos´e Luis G. Pestan˜a2
1
1KapteynAstronomical Institute, PO Box 800, NL-9700 AV Groningen, The Netherlands, [email protected]
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2Departamento de F´ısica, Universidad de Ja´en, Campus Las Lagunillas, 23071 Ja´en, Spain, [email protected]
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Accepted: date1. Received: inoriginalform:
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ABSTRACT

A better understanding of the formation of mass structures in the universe can be
]
A obtainedby determiningthe amountanddistributionofdarkandluminous matterin
spiralgalaxies.Toinvestigatesuchmattersasampleof12galaxies,mostwithaccurate
G
distances, has been composed of which the luminosities are distributed regularly over
. a range spanning 21 orders of magnitude. Of the observed high quality and extended
h 2
p rotation curves of these galaxies decompositions have been made, for four different
- schemes,eachwithtwofreeparameters.Fora“maximumdiscfit”the rotationcurves
o
canbe wellmatched,yetalargerangeofmass-to-lightratiosforthe individualgalax-
r
t ies is required. For the alternative gravitationaltheory of MOND the rotation curves
s canbeexplainedifthefundamentalparameterassociatedwithMONDisallowedasa
a
[ free parameter. Fixing that parameter leads to a disagreementbetween the predicted
andobservedrotationcurvesforafewgalaxies.WhencosmologicallymotivatedNFW

1 darkmatter halosare assumed,the rotationcurvesfor the leastmassivegalaxiescan,
v
by no means, be reproduced; cores are definitively preferred over cusps. Finally, de-
4
compositionshavebeenmade for a pseudoisothermalhalo combinedwith a universal
2
M/L ratio.For the latter, the light of each galactic disc and bulge has been corrected
4
for extinction and has been scaled by the effect of stellar population. This scheme
6
0 can successfully explain the observed rotations and leads to sub maximum disc mass
. contributions. Properties of the resulting dark matter halos are described and a ratio
1 betweendarkandbaryonicmassof∼9fortheleast,andof∼5,forthemostluminous
0
galaxies has been determined, at the outermost measured rotation.
5
1 Key words: Galaxies:general– galaxies:halos – galaxies:kinematics and dynamics
:
v – galaxies: spiral – cosmology:dark matter.
i
X
r
a
1 INTRODUCTION nentandthatoftheouterdarkcomponentconspiretomake
acombinedcurveataconstantlevel(the“conspiracy”,van
Albada&Sancisi1986).Lateronitbecameevidentthatless
It is now a firmly established result that there is a large
luminous galaxies have rotations which continue to rise in
discrepancy between thevisible, luminous mass in a galaxy
the outer parts while the most luminous, and certainly the
and the dynamical mass. Observed rotation curves, espe-
galaxies with a more concentrated light distribution, have
cially those derived from the neutral hydrogen line, remain
a rotation which rises quickly near the centre and declines
mainlyflatintheouterregions.However,thelightdistribu-
slightlyintheouterregions.Inthatwaytheconspiracydoes
tion, which generally decreases exponentially predicts that
not exist any more (Casertano & van Gorkom 1991; Persic
these rotation curves should be declining. This discrepancy
et al. 1996).
canbeexplainedbyinvokingdarkmatter(DM)surrounding
thevisiblecomponentasanextendedmoreorlessspherical Naively one could reason that by subtracting the ro-
masscomponent(Sancisi&vanAlbada1987;Knapp&Ko- tational contribution of the luminous component from the
rmendy1987;Trimble1987).Thefirstdetailedobservations totalobservedrotation,therotational signatureofthedark
ofextendedHirotationcurves(Bosma1978;Begeman1987, haloremains.Theprincipleisright,butinpracticealthough
1989) were made of intermediate sized galaxies. These sys- theradial functionality of theluminous rotation can be de-
temsgenerallyshowarotationwhichremainsflatoveravery rived quite accurately, the absolute contribution is difficult
large range of radii. Sucha situation can only begenerated to quantify. The unknown luminous scale factor might be
if the rotational contribution of the inner luminous compo- determined by making a least squares fit of the total com-
(cid:13)c 2015RAS
2 R. Bottema and J.L.G. Pestan˜a
bined rotation to the observations. In most cases such a fit dial density functionality is used equal to that of a pseudo
convergestoamaximalcontributionfortheluminousmass, isothermal sphere (Carignan & Freeman 1988, Begeman
butacloseinspectionofthefitprocedureshowsthattheso- 1989). Objects of this sort are indeed produced by gravita-
lution is highly degenerate: nearly equally good fits can be tionalinstabilitiesofauniversalfluidhavingdensityfluctua-
achieved when exchanging dark for luminous matter (van tionsofaspecifiedplausibleform(Silk1987).Butrecentde-
Albada & Sancisi 1986; Dutton et al. 2005). Consequently, tailedcalculations andsimulationsoftheformation ofmass
whendoingsuchfits,constraintshavetobesettoeitherkind structures in the universe seem to require dark matter of a
ofmatter.Oneofsuchconstraintsisthemaximumdischy- collisionless and cold kind. Such CDM (Cold Dark Matter)
pothesis which states that the amount of luminous matter is able to explain, with considerable success, the observed
should be maximized. There are no principle reasons why filamentary and honey-comb structures of clusters and su-
this constraint should apply, but it has the useful feature perclusters (Davis et al. 1985; Bond et al. 1996), appearing
that the procedure minimizes the amount of dark matter withtherightsizesandattherighttimes.Collisionlessmat-
which is needed. Thus making the dark matter problem as ter entities get shaped by continued interactions, merging,
small as possible. Furthermore is sets a firm upper limit to andviolentrelaxationofsmalleraggregatesintolargerstruc-
any determined mass-to-light (M/L) ratio of a galactic disc tures.Inevitablethatforms darkhalos withdensityprofiles
or bulge. close to that of an NFW (Navarro, Frenk, & White 1997)
In principle the amount of mass in a disc (or bulge) shape.Suchaprofilehasaninnerfunctionalityproportional
can be determined by measuring the velocity dispersion of to R−1 (cusp) while a pseudo isothermal halo has a central
the stellar content. Such observations of stellar discs and core with constant density. In any way, it seems natural to
the subsequent analyses are not straightforward and conse- assume that dark halos of present day galaxies should have
quentlyreliablemeasurementhavesofaronlybeenobtained a shape similar to that of an NFW profile.
forafewsamplesofgalaxies.Yetsuchstudiesnowgenerate Despite its success in explaining the large scale struc-
amountingevidencethatadiscissubmaximal.Frommea- tures, CDM predicts a number of details which are not
surements of stellar velocity dispersions of a sample of in- in agreement with observations. The three disagreements
clinedgalaxiesBottema(1993)determinedthat,onaverage, which standout are:theexpectedcuspswhile coresare ob-
themaximumcontributionofastellardisctothetotalrota- served, the expected large numberof dwarf satellites which
tion is 63% with an error of some 10%. These observations should orbit a larger galaxy, which are not there, and the
and findings have been confirmed by Kregel et al. (2005) expectedtriaxialityofDMhaloswhichisnotobserved.The
for edge-on disc galaxies, by Herrmann & Ciardullo (2009) first item of this list has been considered and discussed ex-
by measuring the kinematics of planetary nebulae, and by tensively in the recent literature. For example, there are a
Martinssonetal.(2013)formoreface-onsystems.Thelatter numberofobservationsoftherotationinsmallorLSBgalax-
analysisevenseemstoindicatethatthediscrotationalcon- ies,wheretheDMshoulddominate,whichclearlyshowthat
tribution is smaller than the 63% mentioned above. There the dark halos of these galaxies have cores (de Blok et al.
is additional evidence for such a sub maximum disc from a 2001a,b, 2003; Salucci et al. 2003; Simon et al. 2003; Blais-
statisticalanalysisofrotationcurveshapesinrelationtothe Ouellette et al. 2004; Gentile et al. 2004). Several mecha-
compactness of discs (Courteau & Rix 1999). nisms have been invented to alleviate or modify the CDM
An alternative assessment of the matter of disc con- predictions(seePen˜arrubiaetal.2012,andreferencesthere
tribution can be made by considering mass-to-light ratios. in). One of the most promising seems to be cusp flattening
TheInitialMassFunction(IMF)seemstobeuniversalfora by supernova explosions after an idea by Read & Gilmore
rangeofnormalgalacticconditions(Kroupa2001).Thatim- (2005) and studied extensively by Governato et al. (2010)
plies that identical stellar populations have identical mass- and by Pontzen & Governato (2012). Yet, in order to ac-
to-light ratios. By considering a broad spectrum of galaxy complish this, baryonic matter has to flow into the cen-
formation scenarios and a number of different population tral regions of primeval galaxies until over there the den-
synthesis codes, Bell & de Jong (2001) establish a tight re- sity becomes comparable to that of the dark matter. Star
lation between the optical colour of a galaxy and its M/L formationhasthentobesuppresseduntilsuchdensitiesare
ratio in every optical and near infra-red passband. This re- reachedandsubsequentlyanorderofmagnitudeofthebary-
lationiswellestablished,buttheabsolutevalueoftheM/L onicmasshastobeexpelledbySNexplosions,takingalong
ratio depends on the adopted IMF. If the low mass end of thecentralDMto theouterregions. Itremains a matterof
theIMFchangesthenalsotheM/Lchanges,typicallyfora debate if this mechanism can actually work or whether it
standardSalpeterIMF(Salpeter1955)withslope1.35,M/L is in agreement with observed star formation histories and
m−0.35 wherem isthelowmasscutoff.Thus,oneisback metallicity content of galaxies. For thesmaller systems and
∝ l l
at squareone: it is not possible to knowtheluminousmass certainlyfordwarfspheroidalstheenergyrequirementscon-
contribution a priori. Anyhow, the relative functionality of tradict such a scenario (Pen˜arrubia et al. 2012; Garrison-
Bell&deJongisveryusefulbecauseitallowsamassscaling Kimmel et al. 2013).
of the observed population, like the preliminary procedure As an alternative to dark matter the suggestion has
used by Bottema (1997). For instance, a red population is been put forward that the usual law of Newtonian gravity
relativelyoldanddeficientinlightandsoitsassociatedmass breaks down on the scale of galaxies. In particular there is
should be scaled up. On the other hand, a blue population the proposal by Milgrom (1983) that the effective law of
is young and bright and its light has to be scaled down to attraction becomes more like 1/r in the limit of low ac-
get theappropriate mass. celerations. This proposal designated as MOND: Modified
Historically, whendealingwith thedistributionofdark NewtonianDynamics,hasbeensuccessfulinexplainingcer-
matter in the context of extended rotation curves, a ra- tainaspectsofthedifferencebetweenluminousanddynam-
(cid:13)c 2015RAS,MNRAS000,1–28
Distribution of dark and luminous matter inferred from rotation curves 3
ical rotation on the scale of galaxies and groups of galaxies oftheirsamplecanwellbefitted;leavingdistanceasanad-
(Sanders1990;Sanders&McGaugh2002).MONDhasbeen ditional free parameter all galaxies can be accommodated,
generalized and moulded into a relativistic theory with ap- as noted above. In a recent studyof Randriamampandry &
propriateyetcomplicatedfieldequations(Bekenstein2004). Carignan (2014) for a sample of 15 nearby galaxies, partly
IfthistheoryprovestobecomparableorsuperiortotheDM coincidingwiththesampleofGentileetal,whilekeepingthe
paradigmitcannotbediscardedasatoymodel,anditcould (M/L)3.6µ ratioandthevalueofa0fixed,findthatonly60%
bethe expression of a more fundamentalprinciple. of the galaxies can be made in agreement with the MOND
Let’sgobacktotherealworld.MONDfittinghasbeen prescription.
applied to the rotation curves of galaxies in a number of As part of a broader investigation of the distribution
studies. This fitting can and has been done in a few differ- of dark matter in the present paper, also MOND has been
ent ways depending on the number of allowed free parame- considered.Infirstinstancedistancesarenowassumedtobe
ters. In principle MOND is governed by one single acceler- known accurately. Then MOND-like fits are made to inves-
ation parameter a0,which hastobeequalfor allstructures tigatehowMONDwith twofreeparameterscompares with
in the universe. For galaxies this leaves as a free parame- the procedure of adding DM with the same degree of free-
ter the amount of Newtonian contribution to the rotation, dom. In second instance it has been checked if the MOND
which can be parameterized by the M/L ratio of the lumi- philosophyworks: can all rotation curvesbeexplained with
nous component. The contribution of the gasmass is fixed. oneuniversala0 valueforreasonableadjustmentsofthedis-
Abandoning MOND as a fundamental theory, the acceler- tances.
ation parameter a0 can be considered as a free parameter Presently the approach of BBS has been taken over:
too. This kind of fitting will be referred to as MOND-like quality is preferred over quantity. A sample of 12 galax-
and in first instance it can be used to check if indeed an ies has been compiled of which 7 are in common with the
equalvaluefora0exists.Inpracticeandcertainlybeforeac- sample of BBS. The requirement of a rotation curve ex-
curately measuredCepheid distancesbecame available, dis- tending beyond the edge of the optical disc is essential. If
tancecanbeusedasathirdfreefittingparameter,towithin not,thencertainlyforthemoremassivegalaxies, anyfitto,
certainlimits,ofcourse.SoinitsbroadestsenseMONDfit- or determination of the DM properties is usually possible,
tinghasthreefreeparametersandassuchcanexplainevery butmeaningless.Fourdifferentschemeshavebeenemployed
available rotation curvewith ease. in decomposing the observed rotation curves. Each scheme
AfirstdetailedanalysisoftheMONDmethodhasbeen uses a fitting method with two free parameters and are in
donebyBegeman,Broeils&Sanders(1991,hereafterBBS). thatsensecomparable.Anextensiveanalysisismadeofthe
Theyselectedasampleofonly10galaxieswithhighquality quality and probability of each scheme; some work better,
andextendedrotationcurves.Mostofthesecurvescouldbe others are even ruled out. This all in an effort to find the
fitted with the MOND prescription using a universal value most likely distribution of dark matter and to restrict the
of a0 equal to 1.21 10−8 cm s−2 with only slight adjust- range of possible distributions.
ments to the Hubble distances. With one exception, being Thispaperisorganizedasfollows.Insection2thesam-
NGC 2841 which needed a MOND preferred distance twice ple is discussed and peculiarities of each galaxy have been
as large as its Hubble distance. Well, one exception should specified.Section3givesadescriptionoftheextinctionand
confirm the rule. However, during the course of theHubble population corrections in order to get a reliable amount of
key distance project a few Cepheid distances became avail- representativelightforeachsystem.Insection4therotation
able for galaxies in the sample of BBS. These distances are curvedecompositionprocedureisdescribedinageneralway.
accurate to approximately 10% and appeared to differ in That procedure has been applied to the sample in section
somecasesconsiderablyfromtheHubbledistancesusedbe- 5 for a maximum disc fit with pseudo isothermal halo. In
fore. Notably NGC 3198 was further away and NGC 2841 section 6 MOND-likefitting and subsequentanalysis of the
appeared to be closer than the preferred MOND distance. MOND philosophy is presented. In order to relate the cos-
As analysed and discussed by Bottema et al. (2002, here- mologically motivatedNFWhalostopresentdaygalaxiesit
after BPRS), it is really difficult to reconcile the distances isnecessarytoconsidertheprocessofadiabaticcontraction.
ofboththesegalaxies withauniversala0 parameter,unless Thatprocessisdescribedinsection7andthenincorporated
one or more Cepheid distances are considerably wrong. intothefittingofNFWhalospresentedinsection8.Incase
A few more studies have been done investigating the ofthemorephysicallyandobservationallymotivatedscheme
applicability of MOND in general and its consistency with of an equaluniversalM/L ratio thefittingresults aregiven
observed rotation curves specifically. Different studies use insection 9,where also anextensivediscussion is presented
different galaxy samples and a different number of free fit- concerning the uncertainties and the Tully-Fisher relation
tingparameters and are onlygenerally comparable. Forex- (Tully & Fisher 1977). Finally in section 10 a more general
ample de Blok & McGaugh (1998) consider a sample of 15 discussionandpointbypointconclusionshavebeenputto-
LowSurfaceBrightness(LSB)galaxies, whichshouldreside gether.Throughout,aHubbleconstantof75kms−1Mpc−1
largely in theweak acceleration or MOND regime and con- has been adopted.
clude that the MOND fit is excellent in 75% of the cases.
Swaters et al. (2010) reach essentially the same conclusion
foralargersampleofLSBgalaxies.Gentileetal.(2011)con-
sider12nearbygalaxies forwhich rotation curvesareavail- 2 THE SAMPLE OF GALAXIES
able from the THINGS sample. For a MOND-like fit these
A sample of 12 galaxies has been composed based on the
authorsfindanaverage valuefor a0 essentially equaltothe following criteria:
standard number of BBS. Fixing that value three quarters
(cid:13)c 2015RAS,MNRAS000,1–28
4 R. Bottema and J.L.G. Pestan˜a
(i) Rotationcurvesderivedfromatwodimensionalradial mon.Especially,inordertodeterminetheDMdistribution,
velocity field. theseouterdatapointsoftheRCareofutmostimportance
(ii) The velocity field should be regular and non- andtherefore,usingtheTHINGSrotationcurvesinsteadof
distorted, without large scale asymmetries. thepresent ones would not improveon theresults.
(iii) Itshouldextendbeyondtheopticaledge( 5scale- The rotation curves appeared to be nearly identical at
∼
lengths) of the disc. the positions where both are determined, except for DDO
(iv) Inclinationslargerthan50◦,inordertoderive,inde- 154 where for the inner 3 kpc THINGS gives a larger rota-
pendently, the inclinations and rotation velocities from the tion by a few km s−1 compared with the rotation curve of
velocity field. Carignan &Purton(1998) which hadbeen used.Likely the
(v) Inclinations less then 80◦, to avoid composite veloci- latter suffers from beam smearing caused by the very lim-
ties along theline of sight. ited spatial resolution. Therefore it appeared useful to take
(vi) Photometryisavailableinabandredderorequalto overtheTHINGS rotation curvefor this specific case.
R. Over the past few years 3.6µ infrared photometry has
becomeavailableforanumberofgalaxiesfromSINGS(Ken-
and based on thefollowing preferences: nicutt et al. 2003), which has been used by de Blok et al.
(2008) to characterize the luminous radial density distribu-
(i) Rotation curves extending as far as possible beyond
tion. When comparing these radial luminosity profiles with
theoptical edge.
the ones of Kent (1986, 1987, hereafter K86 and K87) and
(ii) A well determined distance, five galaxies have mea-
Wevers et al (1986), in the Red bands one is struck by the
sured Cepheid distances.
similarity. This should not be obvious; radial absorption,
(iii) Avelocity field andrelated rotation curvewhich are
population, and metallicity gradients might generate a dif-
sampled bya sufficiently large numberof independentdata
ference between the red and infrared profiles. An explana-
points.
tionislefttoothers.Anyway,usingthe3.6µprofilesinstead
(iv) Optical emission line radial velocities in the inner
of the present red profiles would not make a difference for
regions when at those positions beam smearing of the Hi
the subsequent analysis or results. On the other hand, us-
data might bepresent.
ing the infrared luminosities would pose a serious problem
(v) Measured stellar velocity dispersions of the disc to
for a population and a necessary metallicity correction for
havea handleon thelocal mass density.
galaxies as a whole. Stellar population analyses in the in-
(vi) To generate a sample which is evenly spread over a
frared are exceptionally difficult and moreover theemission
large mass range.
maybecontaminatedbyacontributionofPAHs(polycyclic
Without any doubt there will be more than the 12 galax- aromatic hydrocarbons).Forphotometryin theRed,popu-
ies in the sample which satisfy the requirements, but we lation corrections are well described and the light is barely
did not havetheintention to becomplete. The criteria and dependenton metallicity.
preferences above, can only be met when galaxies are ob- For all the galaxies a short description is now given
served in the neutral hydrogen line with an interferometer. of the adopted distances, photometry, and rotation curves.
A rotation curve is usually derived by a least squares fit to Uncertainties,peculiaritiesandreferencesarequoted.InTa-
the velocity field of a collection of rings with radially vari- ble 1 a number of important parameter values are summa-
ableinclinations,linesofnodes,andcircularvelocities.This rized.
procedure is described in detail by Begeman (1987, 1989).
When Hi datahavebeen replaced by optical emission lines
in the inner regions, the latter have on occasion been ra-
dially resampled to match the Hi sampling. That ensures
an equal radial weighting when doing the RC fit, which is
a choice. Having more data points at a specific region may
2.1 NGC 2841
slightly change therelative contributions of the disc, bulge,
and dark halo, but will not change any of the main results Adistance of 14.1 1.5 Mpchas been determinedby HST
±
(see Blais-Ouellette et al. 1999). measurements of Cepheids (Macri et al. 2001). Photometry
Theparametersandpropertiesofthesamplehavebeen is available by K87 in the Thuan & Gunn (1976) r-band
collected in the period 2003 to 2007 after which the anal- giving both, the photometric profile and total magnitude.
ysis as presented in the remainder of this paper has been Because there is a substantial difference in observed ellip-
carried out. Because of personal reasons the work on this ticity between the bulge and disc a decomposition of the
paper had to be abandoned for a while. During that time light of these components according to Kent’s procedure is
results of the THINGS (The Hi Nearby Galaxy Survey; de well determined. The Hi rotation curve has initially been
Blok et al, 2008) had become available. Of the 12 galaxies measuredbyBosma(1981) andislaterrefinedbyBegeman
in the present sample 6 are in common with the THINGS (1987).Thegasdistributionissymmetric,butthekinemat-
sample: NGCs 2403, 2841, 2903, 3198, 7331 and DDO 154. ics displays asizable amount of warping, making thedeter-
When comparing the Hi rotation curves one can conclude minedrotationintheouterregionsslightlylesscertain.The
thatTHINGShasabetterradialsamplingbecauseobserva- rotation curve remains more or less at a constant level all
tions have been done at a higher spatial resolution. But in thewayinwardstothecentre.TheTHINGSrotation curve
general the observations do not have the signal-to-noise of reachesoutto36kpcanddisplaysasizableuncertaintybe-
the rotation curves already considered. The THINGS data tween 25 and 36 kpc, while the present one goes out to 64
thenhaveashorterradialextentforallthegalaxiesincom- kpc,both are fully consistent.
(cid:13)c 2015RAS,MNRAS000,1–28
Distribution of dark and luminous matter inferred from rotation curves 5
2.2 NGC 3992 with only a minor bulge. For Kent’s bulge-disc decomposi-
tionmethodthebulgehasaluminosityofonly2%ofthatof
NGC3992isoneofthemostprominentmembersoftheUrsa
the entire galaxy. Therefore, presently, the whole galaxy is
Major clusterofgalaxies. Tully&Pierce(2000) determined
considered as a disc structure,which effectively means that
adistanceof18.6Mpctothiscluster,whichwillbeadopted
the mass-to-light ratio of the bulge is equal to that of the
as the distance for NGC 3992. There is photometry in the disc. An accurate Hi rotation curve has been determined
BRIK′ bandsbyTully etal. (1996) ofwhich theI profileis
by Broeils (1992a), which has unfortunately not been pub-
used to calculate the disc rotation and the R-band for the
lished in the refereed literature. The velocity field is very
total light. The signature of the bulge is clearly present in
symmetric and regular and the gas extends far beyond the
theobserved radial light profileand consequently thebulge optical edge. Because of its large distance the Hi structure
to disc decomposition is straightforward. The Hi rotation
is not as well resolved as that of the more nearby galaxies
curvehas been measured in detail by Bottema & Verheijen
and consequently the rotation curve is more sparsely sam-
(2002), who also give a rotation curve decomposition. This
pled.Moreover,theobservedkinematicsintheinnerregions
galaxyhasabarandthisbarregionisdevoidofgassothat
is affected bybeam smearing and cannot be used.
no rotation is available for theinnerregions.
Absorption line spectroscopy is obtained along the
whole major axis of NGC 2998, with a (1σ) velocity res-
olution of 25 km s−1 (Bottema & Kregel 2014). From that,
2.3 NGC 7331 stellar radial velocities and stellar velocity dispersions have
been derived extending to approximately two and a halve
The HST Cepheid distance to this galaxy is 14.72 0.60
± optical scalelengths. In addition the emission lines of Hβ
Mpc (Hughes et al. 1998; Freedman et al. 2001). Photom-
and[OIII]5007˚Aareobserved,whichhavebeencombined
etry is given by K87 in the r-band, but unfortunately the
to generate the rotation in the inner 40′′, there where the
ellipticity of the central bulge and outer disc appear to be
Hidataarenotuseful.Theemissionlinekinematicsissym-
equal and Kent’s decomposition procedure cannot be used.
metric, but slightly irregular, in the sense of showing some
Adetailedstellarandemissionlinekinematicalstudyofthe
corrugationbylessthan10kms−1.Thestellarvelocitydis-
galaxy has been made by Bottema (1999). In that paper
persion decreases radially as expected for an exponential
a decomposition into separate bulge and disc is presented,
disc. Depending on the exact parameterization of the disc,
partlybasedontheobservedstellarabsorptionlineprofiles.
a maximal contribution of the rotation of the stellar com-
The bulge appears to be quitedominant and nearly spheri-
cal while the disc has an inclination consistent with theHi ponenttothatofthetotalrotationiscalculated.Thisvalue
kinematicalvalueof75◦.Unfortunatelyanerrorhasturned ranges between 0.64 and 0.72 with an error of 10%.
up in Table 3 of Bottema (1999). For the favourite decom-
position labelled “lpd” the total amount of disc and bulge 2.5 NGC 2903
light have been interchanged. The total light of disc and
bulge should be 11.4 109LI and 12.7 109LI respectively The Hubbledistance to this galaxy is 6.3 Mpc, while a dis-
and the b/d ratio is 1.1 in⊙stead of 0.9. The⊙radial profile tance estimate based on the brightest stars (Drozdovsky &
Karachentsev 2000) is 8.9 1.9 Mpc. Presently we take
in the I band is given by Prada et al. (1996) which is used
±
as distance the average of the two: 7.6 Mpc. Photometry is
presently, although we have some doubt regarding the ab-
measuredbyK87givingtheprofileandtotallight.Therota-
solute calibration of this I-band photometry. Total R-band
lightisderivedfromK87.TheHirotationisdeterminedby tioncurveiscomposedofHαemissionlinedatabyMarcelin
et al. (1983) for the inner 100′′ and Hi data by Begeman
Begeman (1987), extending out to a radius of 37 kpc, and
he supplements the rotation in the inner region with data (1987) beyond, out to 29 kpc. The Hα radial velocity mea-
of Rubin et al. (1965). However, as demonstrated by Bot- surements have been transformed into a rotation curve us-
ingthesameinclinationof62◦asfortheHikinematics.The
tema (1999) the gas kinematics in the inner regions of this
galaxy is very unusual and certainly not representative for THINGS rotation curvegoes out to25 kpc.
thegravitationalpotential.Therefore,atthosepositionsthe
rotation as inferred from the stellar kinematics is adopted.
2.6 NGC 3198
The THINGS rotation curve only extends from a radius of
4 kpc to 24 kpc. The measured stellar velocity dispersion The distance of 13.8 0.5 Mpc is derived from the HST
±
of the disc suggests that the contribution of the disc to the Cepheid observations of this galaxy (Kelson et al. 1999;
total rotation is small. This measurement is somewhat un- Freedman et al. 2001), and is considerably larger than the
certain, however, because of the presence of a considerable Hubble distance of 9.4 Mpc. Again, photometry by K87 is
amount of bulge light. usedofwhichthevaliditytorepresentthemassdistribution
is confirmed by K′ band photometry by BPRS. Because
the Hi rotation curve (Begeman 1989) reaches very far out
NGC3198hasbecometheclassiccaseofaspiralgalaxyev-
2.4 NGC 2998
idencing a large mass discrepancy in its outer regions (van
NGC 2998 is the most distant galaxy of the sample. The Albadaetal.1985).Numerousre-observationsoftheHiand
radial velocity corrected for Virgo-centric flow gives a Hub- re-determinationsoftherotation curvehavebeenmade.At
ble distance of 67.4 Mpc. Considering the deviations from theflatpartnoneofthesewarrantachangetowhathadal-
the Hubble flow, this distance is nearly as accurate as the readybeendeterminedbyBegeman(1989).Additionallyall
Cepheid distances for the galaxies more close by. Photom- observationsendattheradiusofnearly45kpcprobablybe-
etry is presented in K86; the galaxy is close to exponential causeat larger galactocentric distances theHigasbecomes
(cid:13)c 2015RAS,MNRAS000,1–28
6 R. Bottema and J.L.G. Pestan˜a
ionized. The rotation curve of THINGS ends at a radius of et al. (1995) give a distance of 3.5 0.7 Mpc based on
±
38kpc.Howeverthatcurveseemstoshowasomewhatlower the TF relation. The average of the three amounts to 3.1
rotationintheinnerrisingpart,forR.5kpc.Whetherthis Mpc. Broeils (1992b) presents photometry in the B-band
isrealormightbecausedbyabarfeature,needstobecon- and Swaters & Balcells (2002) in the R-band. The profile
firmed.Inthisinnerregion ofNGC3198 mattersappearto of the latter authors declines slightly steeper and was used
be quite complicated and a thorough analysis could easily torepresentthemassdistributionofthedisc.TotalR-band
fill a complete paper and is certainly beyond the scope of luminosityalsofrom Swaters.Forradiibeyond120′′ theHi
this study. Presently Begeman’s curve has been taken over rotation curve of Broeils (1992b) is adopted. Because this
entirely with its relatively large errors in the inner region, galaxyisratheredge-on(i 80to82◦)theavailableHimay
∼
nicely representing thecurrent uncertainty. be compromised by beam smearing and integration effects.
ForitssizeandmassNGC3198 hasindeedarelatively ThereforelongslitHαdataofdeBlok&Bosma(2002,which
large amount of gas present (see Table 1). Stellar velocity havekindlybeenmadeavailablebytheseauthors)havebeen
dispersionsofthediscofthisgalaxyhavebeenmeasuredby convertedtorotationalvelocities.Forsuchahighinclination
Bottema (1988), which for an assumed average disc thick- galaxy slit data are not ideal. Yet this galaxy is large on
nessimplyasubmaximumdisccontributiontotherotation the sky and one Hα RC data point is the average of a lot
curve. of individual observations, showing quite some scatter, as
expected. This average and associated error are judged to
be sufficiently accurate. Between 120 and 200′′ the Hα and
2.7 NGC 2403 Hirotationsareequal,forradiilessthan120′′,theemission
lines indicate a slightly larger rotation by approximately a
This galaxy has a distance based on Cepheid variables
few to 10 km s−1. Therefore, at those positions the Hα is
(Freedman & Madore 1988; Freedman et al. 2001). Pho-
tometrybyK87andHirotation curvebyBegeman (1987). used for therotation.
Likefor NGC3198 thisgalaxy has also been re-observedin
the Hi on numerous occasions, which has never led to any
alteration of the rotation curveused here.
2.11 NGC 3109
2.8 NGC 6503
For this galaxy a Cepheid distance is available of 1.36
A distance of 5.2 1.1 Mpc is assumed based on mea- ±
± 0.10 Mpc by Musella et al. (1997). These authors use the
surements of the luminosities of the brightest blue stars
samedistancetotheLMCastheHSTCepheiddistancekey
(Karachentsev & Sharina 1997), being in agreement with
project and is in that sense comparable. Yet there may be
theHubbledistanceof4MpcandTully-Fisherdistanceof6
small systematic differences between the two methods. Re-
Mpc(Rubinetal.1985).Theluminosityprofileisacompos-
cently,Soszyn´skietal.(2006) determinedadistanceof1.30
iteofphotographicKodakIIIa-FmeasurementsofWeverset
0.04 Mpc by extending the Cepheid observations to the
al.(1986)intheouterregionsandR-bandCCDphotometry ±
infra-red. This confirms the value of Musella et al. and the
by Bottema (1989) in the inner regions. All is converted to
R-bandmagnitudes.TheHirotationcurveisfromBegeman difference is so small that we maintained the value of 1.36
Mpc.UncalibratedI-bandphotometryhasbeenobtainedby
(1987). As for NGC 3198, stellar velocity dispersion obser-
Jobin&Carignan(1990)ofwhichtheradialprofilewasused
vations(Bottema1989)implythatthemasscontributionof
to calculate the rotation of the disc. A total R-band lumi-
thedisc is sub maximal.
nositywasderivedfromtheESOLVcatalogue(Lauberts&
Valentijn1989) byextrapolatingtheR-bandaperturevalue
2.9 NGC 5585 at R26 equalto the B-band functionality.
The construction of the rotation curve is somewhat
In this case the Hubble distance of 6.2 Mpc is the most
complicated. From the I-band photometry, assuming q0 =
reliablevalueavailable.PhotometryintheB,V,andR-bands 0.11 one has an inclination of 75◦, which seems well deter-
hasbeenobtainedbyCˆot´eetal.(1991)ofwhichtheR-band mined. Hi observations of Jobin & Carignan suggest an in-
profile is used to calculate the rotation curve of the disc. clinationof70◦ butisratheruncertain.Thereforetheerrors
This profile has been integrated to give the total R-band on the Hi rotation have been increased to include this un-
luminosity. An Hi rotation curve is determined by Cˆot´e et certainty in the inclination. For the inner regions (< 350′′)
al. from aregular though slightly warped velocity field.For
therotationaldatahavebeensupplementedwithHαFabry-
radiiwithin120′′therotationofBlais-Ouelletteetal.(1999)
Perot observations by Blais-Ouellette et al. (2001). In that
based on Hα Fabry-Perot observations is used. That inner paper an inclination is used of 88◦, which is clearly in con-
rotation curveshowsaspecificbumpyfeaturewhichcan be
tradictionwithotherdeterminations.Thereforetherotation
matchedwiththemoderatecentralcuspinthephotometry.
ofBlais-Ouelletteetal.hasbeenconvertedtoaninclination
75◦. Moreover, thequoted errors appeared to be unrealisti-
cally small and have for all Hα rotational data points been
2.10 NGC 1560
increasedto5kms−1.Thatshouldthenalsoincludeanyun-
The distance to this galaxy is taken as theaverage of three certainties associated with the asymmetric drift correction.
values. At first the Hubble distance of 3.25 Mpc. Secondly This might all seem a bit tricky,butfor a galaxy with such
adistance of 2.5 0.1 Mpc(Lee & Madore 1993) based on a large inclination small changes in the inclination never
±
bright stars and Tully-Fisher relation, and thirdly Krismer substantially affect therotation.
(cid:13)c 2015RAS,MNRAS000,1–28
Distribution of dark and luminous matter inferred from rotation curves 7
2.12 DDO 154
Adescriptionofdistances,photometry,androtation canbe
foundinthepaperofCarignan &Beaulieu (1989). Thedis-
tanceof3.8Mpcisbasedonacombinationofbrighteststar
considerations andassociation with theCanesVenaticorum
Icloud.PhotometryisavailableintheB,V,andRbands.An
additional measurement in the R-band by Swaters & Bal-
cells (2002) is consistent, both concerning the profile and
total luminosity.
TheHirotationcurveofCarignan &Beaulieu exhibits
a decline beyond a radius of 300′′. In a following study
∼
by Carignan & Purton (1998, hereafter CP98) additional
observations reveal a more extended low level Hi struc-
ture. The outer Hi distribution is lopsided but the veloc-
ity field appears rather regular. Again the derived rotation
curve starts to decline abruptly and continues so out to a
radiusbeyond540′′.Thederivationoftherotationcurveby
theseauthorsseemssolid.Asmentionedabove,therotation
curve of THINGS qualifies to replace the one of CP98 be-
causetheirinnerdatapointshaveprobablybeenaffectedby
beam smearing. Moreover theerrors which havebeen given
are unrealistically small. Figure1.Mass-to-lightratiointheR-bandasafunctionofB-V
Then we encountered a problem. In the paper of De colour (Bell & de Jong 2001), lowered by 15.5% to give a value
Bloketal.(2008)twodifferentrotationcurvesforDDO154 of1.0atB-V=0.6.Thisrelationisusedtoscaletheamountof
are presented, one slightly rising in the outer parts in their lightinagalaxytoarepresentativevaluefortheamountofmass.
figures 15 and 46, and one nearly constant or decreasing
slightlyintheirfigure81.Thedifferentrotationcurveshave
stellar mass. Uncorrected parameters have been given the
been derived for a differently adopted inclination function-
subscript“obs”,thosecorrectedonlyforextinction“ec”and
ality between 300′′ and 400′′, both of which are consistent
thosecorrected for extinction pluspopulation aregiven the
with the THINGS observations. For this galaxy at those
subscript “epc”.
positions the tilted rings used to fit the velocity field are
only partially and irregularly filled with data on hence an
independent inclination determination is not possible. The 3.1 Galactic extinction
observations of CP98 extend further out, until a radius of
approximately 540′′; their velocity field appears to be reg- ExtinctionfromthelocalGalaxy,designatedbyAb istaken
ular over there with an inclination nearly constant at 60◦, fromSchlegeletal.(1998).Itsvaluesforthesampleofgalax-
iescanbefound,withotherforthcomingcorrections, inTa-
which coincides with the choice of De Blok et al. in their
ble2.
figure 81. Consequently that rotation curve of THINGS is
to be preferred and as such is completely compatible with
therotationcurveofCP98.Inthisstudythatrotationcurve 3.2 Internal extinction to face-on
has been taken over, supplemented with the data points at
450′′ and540′′ ofCP98withanerrorincreasedtoarealistic Acorrectionforthisextinction(Ai−0)isgivenbyTullyetal.
value of 5 km s−1 for an adopted inclination uncertainty of (1998) and depends on the absolute luminosity of a galaxy
10◦. corrected for Galactic and internal extinction (MRb,i−0) and
The radial distribution of the gas has been taken from on the observed aspect ratio a/b as
Carignan & Beaulieu (1989) for the inner regions. It has a
Ai−0=γ log , (1)
been extended to larger radii by a smooth matching to it R R b
with thedistribution given in Fig. 4 of CP98. (cid:16) (cid:17)
with
γ = 0.24(16.06+Mb,i−0)
R = −0 if Mb,i−0 > 1R6.06. (2)
R −
3 TOTAL LIGHT IN R, EXTINCTION AND
This correction has been applied separately to thedisc and
POPULATION CORRECTIONS
thebulge (see Table 2).
To obtain an amount of light which is representative for
theamountofluminousmatterinagalaxycorrectionshave
3.3 Intrinsic extinction of a face-on galaxy
to be made to the observed total light, in this case in the
Kron-Cousins R-band. There are two main corrections to This matter is uncertain because it has never been inves-
bemade;acorrection forabsorption tobecalled extinction tigated in a systematic way. Therefore we can only give a
correction, and a correction for population. The latter is reasonable estimate. Tully & Fouqu´e (1985) and Verheijen
needed because a younger population generates more light (2001) use an amount of intrinsic extinction (Ai=0) of 0.21
R
compared to an older population for the same amount of mag. for an average galaxy in their samples. Comparing
(cid:13)c 2015RAS,MNRAS000,1–28
8 R. Bottema and J.L.G. Pestan˜a
thesesampleswiththegalaxiesinthispapersuchanaverage (B V) ofthedisccanbederived.TheB-Vofthebulge
obs
−
galaxyhasapproximatelytheluminosityofNGC3198.Itis has been assumed to be equal to the B-V colours of the
further assumed that the intrinsic extinction is smaller for smallestaperturesgivenbyLongo&deVaucouleurs(1983).
less luminous galaxies, analogous to the extinction correc- Subsequentlytheseparate(B V) colourshavebeencon-
obs
tion to face-on. Adopting Ai=0 = zero for Mb,i−0 > 16.0 vertedto(B V)0 according−totheRC3.Theresultsofthe
we use for Mb,i−0 < 16.0: R − populationco−rrectTionprocedurearegiveninTable3forthe
R −
sampleofgalaxies. Notethatthethreebulgesarerelatively
AiR=0=−0.042MRb,i−0−0.672, (3) red,thepopulationisthereforeoldandlighthastobescaled
for discs and Ai=0=0 for bulges. up considerably.
R Foralltheforthcomingfitstwokindofmass-to-lightra-
tios will be presented;as observed,(M/L) meaning with
obs
3.4 Population correction nocorrection atalland(M/L)epc withboth,theextinction
and population correction applied. In Sect. 9 a decomposi-
Bell & de Jong (2001) calculate mass-to-light ratios for tion is presented using an equal (M/L)epc for all luminous
galaxiesasafunctionoftheircolours.Theexistenceofsuch galaxy components.
a colour versus M/L ratio relation can be understood be-
cause a younger, lower M/L ratio population is relatively
bluer compared to an older, larger M/L ratio population.
4 ROTATION CURVE FITTING
Bell & de Jong demonstrate that such a relation is largely
independent of galaxy evolution scenarios and on the em- For every galaxy a model rotation curve is computed as
ployed population synthesis code. In the non near infra-red thesquaredsum oftheindividualcontributions of thedisc,
passbands it barely depends on metallicity of the popula- bulge, gas, and dark halo. The contribution of the gas is
tion. For the M/L ratio in the R-bandand B-V colour Bell fixed by observations of the neutral hydrogen gas, of which
& de Jong give a relation of the amount has been multiplied by a factor 1.4 to account
10log(M/L )= 0.66+1.222(B V) (4) forthepresenceofHeliumandasmallamountofionizedhy-
R
− − drogen. The disc and bulge light distribution are measured
As for all population synthesis analyses the M/L ratio is by the photometry of the galaxy. Scaling by the M/L ratio
only known up to a certain factor. This factor depends on then gives the mass contribution. For the dark halo gener-
theassumedlowmassendoftheIMFandmightbeaslarge allyananalyticaldensityfunctionalityisassumeddescribed
as two. We can therefore not use Eq. (4) to derive M/L by one or two parameters. Thus the composite model ro-
ratios, but we shall use it to correct the amount of R-band tation curve has as free parameters the M/L ratio of the
light for the excess amount of light of a young population, luminouscomponentsandthedescriptivesofthedarkhalo.
ordeficiencyoflightofanoldpopulation.AfiducialB-Vof This composite curve is then fitted in a least squares sense
0.6 ischosen for whichsuch apopulation correction iszero. totheobservedrotationcurve,aprocedureoftenreferredto
Then, using Eq. (4) with M/L lowered by 15.5%, in Fig. 1 as decomposition of the rotation curve.
thescalefactorisgivenwithwhichtheamountoflighthasto Inthatwaythebestfitisdesignatedasthesituationof
bemultiplied to obtain therepresentative amount of mass. aminimumχ2value.Thisprocedureisscientificinthesense
In section 9 a successful fit to all rotation curves can that it is reproduceable and therefore is preferred over, for
beachieved by usingan equalmass-to-light ratio in R,cor- example, an estimated fit by eye. Yet one has to be careful
rected for extinction and population, having a value of 1.0. when applying this method. A least squares fitting proce-
This means that the scale factor presented in Fig. 1 is then dureassumesthatthefittingfunctionisknownaprioriand
exactlyequaltothe(M/L) ratioasafunctionofB-Vcolour that thedata pointsscatter in a Gaussian way around that
R
oneneedsforgalaxiesingeneral.Asaconsequencetherela- function. For rotation curves that is not valid. Firstly a ro-
tionascalculated byBell&deJong,onthebasisofhitting tationalfunctionalityforthehaloisadoptedwhichneednot
the default lower mass cutoff in the population synthesis becorrect.Secondlytheproceduretodeterminetherotation
codesof0.1M isthen15.5%toolarge.If,instead,alower is an approximation in the sense that azimuthal symmetry
mass cutoff at⊙0.15 M would have been taken, their re- is assumed with no in or outflow. For example spiral arms
lation would be spot o⊙n the relation given in Fig. 1, and can produce small irregularities, which may lead to small
thuson what the successful rotation fit scheme of section 9 systematic deviations from the actual rotation law. In ad-
implies. dition one has to be careful when ”feeding” the fitting pro-
B-V colours of galaxies can be found in the RC3 (de cedure. At positions with a high degree of sampling the fit
Vaucouleursetal.1991),butinordertobeusableinEq.(4) is forced to a higher weight. Therefore, presently, rotation
should be corrected for absorption. For that we used the curves have, on occasion, been resampled to a nearly uni-
recipegivenintheRC3,toobtaintheparameterdesignated form radial distributiontogiveequalweight overtheentire
as (B V)0. Since rotation curve decompositions will be radial extent. Because of these matters the resulting mini-
made −withTa separate disc and bulge, it appeared neces- mum χ2 value is only a limited indicator of the quality of
sarytoapplytheextinctionandpopulationcorrectionssep- thefitanditcannotbeconcludedfromthatvalueonlythat
aratelytothesecomponentstoo.Unfortunatelythe(B V)0 a certain dark halo functionality is betteror worse.
− T
valuesarenotlisted separately,butcan beretrieved.When The radial luminosity profiles of bulge and disc are
the total observed (B V) is given (RC3), the ratio of adopted to indicate the radial mass distribution. This as-
obs
−
disc to bulge light has been determined (Table 1), and the sumes that the M/L ratio is constant in the galaxy which
observed (B V) of the bulge is known, the observed can be justified by the observed small radial colour gradi-
obs
−
(cid:13)c 2015RAS,MNRAS000,1–28
Distribution of dark and luminous matter inferred from rotation curves 9
Figure 2.Maximumdiscfits totheobserved rotation curves (dots) of the12galaxies inthe sample.Here,andinforthcomingsimilar
plots(Figs4,7,and9),thelinesarecodedasfollows:fulldrawnisthefit,dotted,short–dashed,long–dashedanddash–dotlinesrepresent
the rotation of the gas, the disc, the bulge, and the pseudo isothermal dark halo, respectively. The observations can be matched quite
accurately, exceptforNGC5585andDDO154whichshowasmalldiscrepancy.
(cid:13)c 2015RAS,MNRAS000,1–28
10 R. Bottema and J.L.G. Pestan˜a
ents,ingeneral(deJong1996),atleastfornormalquiescent
systems. Nevertheless,in ordertominimize dust andpopu-
lationeffectsandsotobeclosetotherealmassdistribution,
a profile in a passband as red as possible is preferred. Cur-
rentlyfor all galaxies of thesampletheprofileisin a(R)ed
or near (I)nfrared passband (see Table 1).
For the discs the observed radial mass distribution is
combined with an adopted sech2(z/z0) vertical mass dis-
tribution with z0 being 0.2 times the value of the radial
scalelength (van der Kruit & Searle 1981, 1982). The disc
rotationcurveisthencalculatedfollowingCasertano(1983).
For the bulge which is assumed to have a spherical distri-
bution the rotational velocity can be calculated using the
equations on page 1310 of K86. Both for the disc and the
bulgetherotationcurvescanthenbescaledupwiththeun-
known M/L ratio. For the gas the radial mass distribution
isobserveddirectly.Tocalculatetherotationathinvertical
distributionisassumed.Finallythedarkhalo.Inthispaper
a few density functionalities have been investigated usually
parameterized bytwo values. A detailed description will be
given where appropriate. The calculation of the rotations
curves and the fitting procedures have been performed us-
ingtheroutinesrotmodandrotmasintheGipsypackage.
In practice it appears that the fitting procedure is de- Figure 3. The M/L ratios, corrected for absorption and pop-
ulation effects, following from the maximum disc fits. Different
generate;onegenerallycannotdeterminetheM/Lratioand
galaxies are represented along the x-axis by their observed rota-
darkhaloparameterssimultaneously.Consequentlyitisnec-
tion at the flat, outer part of the rotation curve. There appears
essarytoimposecertainconstraints.Intheremainderofthis
tobeatrendinthesensethatthemoremassivegalaxieshavea
paper,inprinciple,constraintsaremadesuchthatthereare
smallerM/Lratio.
always two free parameters left (or three with a bulge); in
that sense the fittingschemes are thencomparable.
tobemaximizedfirstfollowed bythedisc.Forsystemswith
abulgetherotationgenerallyremainsflat,goinginwardsto
5 MAXIMUM DISC AND BULGE FITS smallradii.Thatimpliesthatifthebulgewerenotmaximal,
a dark halo with very short core radius would be required.
Ingeneralthemaximumdiscconstraintisassociatedwitha
The maximum disc/bulge adjustment procedurecan gener-
darkhalofunctionalityofapseudoisothermalsphere(Carig-
ally be done with approximately a 5% error in the bulge
nan&Freeman1988).Forconsistencythatfunctionalitywill
and disc rotation, which generates a 10% error in the M/L
also be used presently. Its density distribution ρ takes the
h ratios.
form
For the 12 galaxies maximum disc fits are presented in
R2 −1 Fig. 2. In most cases the fits to the observations are ex-
ρh=ρ0 1+ R2 , (5) cellent; even certain small scale features in the photometry
core
(cid:20) (cid:21)
which are expressed in the model rotation curve appear to
with an associated rotation vp.iso of be reflected in the observations. Nevertheless for two cases,
NGC5585andDDO154,thereisasmalldiscrepancyatthe
vp.iso =vmhaxr1− RRcorearctan(cid:16)RRcore(cid:17), (6) owuhtielermtohsetmraoddiei.lIctusreveemssprtehdaitctthaecoobnsteirnvueeddrroitsae.tioWnhdertohpesr
where the halo rotation becomes asymptotically flat at a we are witnessing a steep end to the dark halo for these
maximumvalueofvmhax which isrelated tothecentralden- two galaxies, or whether the observed drop is an artifact
sity ρ0 and core radius Rcore as remains to be investigated further. In Table 4 the fitting
parameters are presented.Errors of themass-to-light ratios
vmhax = 4πGρ0Rc2ore. (7) are estimated at 20%. That is a combination of the errors
Notethaptforsmallradiiρh constantandvp.iso R,while of the amount of light, caused by the various uncertainties
for radii much larger than≈the core radius ρ ∝R−2 and of distance, extinction, and colour corrections effects, and
h
∝
vp.iso constant. Two free parameters describe the halo: the error generated by the fitting method (Sect. 10.3). The
≈
Rcore and vmhax. determined core radius is generally comparable in size to
As mentioned is the previous section, a maximum disc theradiusofthelast measured rotation point.That isadi-
fitassumesamaximumcontributiontotherotationcurveby rect consequenceof themaximization procedure of the disc
theluminouscomponents.Forasystemwithonlyadiscthis whichleadstoaminimalamountofdarkhalomatterinthe
meansinpracticethatonescalesupthediscrotationinthe innerregions.
innerregions as much as possible tobe still consistent with The determined M/L ratios corrected for absorption
theobservations.Incaseswherethereisabulgetoo,because and population effects are given as a function of the ob-
ofitsstrongmassconcentration,thebulgecontributionhas served rotation at the flat, outer part of the rotation curve
(cid:13)c 2015RAS,MNRAS000,1–28