Modelling the Milky Way through adiabatic compression of cold dark matter halo

0.31 MB·English

by  V. F. Cardone

Astronomy & Astrophysics manuscript no. mwmodels˙aa˙2nd February 2, 2008
(DOI: will be inserted by hand later)
Modelling the Milky Way through adiabatic compression of cold
dark matter halo
V. F. Cardone1 and M. Sereno2,3
1 Dipartimento di Fisica “E.R. Caianiello”, Universit`a di Salerno, and INFN, Sez. di Napoli, Gruppo Coll. di
5 Salerno, Via S.Allende, 84081 - Baronissi (Salerno), Italy
0 2 Dipartimento di Scienze Fisiche, Universit`a di Napoli, and INFN, Sez. di Napoli, Complesso Universitario di
0 Monte S. Angelo, Via Cinthia, 80126 Napoli, Italy
2 3 IstitutoNazionalediAstrofisica-OsservatorioAstronomicodiCapodimonte,SalitaMoiariello16,80131Napoli,
Italy
n
a
J Received / Accepted

6
2 Abstract. Weusetheadiabatic compression theory tobuild a physically well-motivated Milky Way mass model
inagreementwiththeobservationaldata.ThevisiblemassoftheGalaxyisdistributedinaspheroidalbulgeand

1 amulti-componentsdiscparametrized bythreegalactic parameters, theSundistancetothegalactic centre,R0,
v
the total bulge mass, Mbulge, and the local disc surface density, Σ⊙. To model the dark matter component, we
7
6 adiabatically compress a Navarro, Frenk and White (NFW) halo (with concentration c and total mass Mvir) for
fixed values of the spin parameter, λ, the fraction of the mass in baryons, m , and the thin disc contribution to
5 b
total angular momentum, j . An iterative selection procedure is used to explore in very detail the wide space of
1 d
0 parameters onlyselecting thosecombinationsof {R0,Mbulge,Σ⊙,λ,mb,jd,c,Mvir}thatgiverisetoaMilkyWay
5 model in agreement with theobservational constraints. This analysis leads usto concludethat only models with
0 R0 = 8.5kpc, 0.8×1010 M⊙ < Mbulge < 1.6×1010 M⊙ and 49 M⊙ pc−2 ≤ Σ⊙ ≤ 56 M⊙ pc−2 can be reconciled
/ with the set of observational constraints. As regard the parameters entering the adiabatic compression, we find
h
0.03 ≤ λ ≤ 0.10 and 0.04 ≤ m ≤ 0.10, while the final estimates of the parameters describing the initial halo
p b
- profile turn out to be 5∼< c∼< 12 and 7×1011 M⊙ ∼< Mvir ∼< 17×1011 M⊙ (all at 95.7% CL).
o
r
Key words.Galaxy: kinematics and dynamics – Galaxy: structure– galaxies: formation – dark matter
t
s
a
:
v 1. Introduction still remains. Dark matter dominated objects, such as
i
X dwarf and low surface brightness galaxies, show rotation
The determination of the mass distribution of the Milky
r curves inconsistent with the central density cusp pre-
a Way is a classical task of astronomy (Schmidt 1956,
dicted by CDM cosmology (McGaugh & de Block 1998,
Caldwell & Ostriker 1981, Dehnen & Binney 1998). The
Alam et al. 2002).
usual framework for the origin of structures in
the Universe is provided by the Cold Dark Matter The understanding of galaxy formation also re-
(CDM) paradigm. This standard cosmological the- mains unsolved in the standard hierarchical model.
ory has proved very successful on large scale, in DissipationlessCDMhaloesareassumedto formbottom-
explaining both the abundance and clustering of up via gravitational amplification of initial density fluc-
galaxies (Peacock et al. 2001, Verde et al. 2002) and the tuations. Gas carried with such haloes cools and con-
power spectrum of the cosmic microwave background tracts within them to form luminous, independent self-
anisotropies (de Bernardis et al. 2000), but is experienc- gravitating units, which can form stars, at the halo
ing a number of difficulties on the scales of galax- centres (Fall & Efstathiou 1980, Blumenthal et al. 1986,
ies and dwarf galaxies. CDM paradigm predicts an Mo et al. 1998). The halo profile affects both gas cooling
over-abundance of satellites around the Milky Way and infall since it determines the structural properties of
and M31 by an order of magnitude (Klypin et al. 1999, the resultant heavy discs and dense nuclear bulges. Even
Moore et al. 1999). While the presence of a photoion- if the growth of dark haloes is not much affected by the
izing background can solve this “sub-structure” prob- baryoniccomponents,thehalogravitationallyrespondsto
lem (Somerville 2002), difficulty about density profile the dissipative baryonic infall and the present day CDM
haloes can have density profiles quite different from the
Send offprint requests to: [email protected] original CDM prediction.
2 V. F. Cardone and M. Sereno: Modeling theMilky Way
In this context, we want to address the density profile 2.1. The bulge
problem by fitting CDM models to the observed prop-
The morphology of the Galactic bulge (defined as the
erties of the Milky Way. The Milky Way seems to be a
spheroidwithinthegalacticcoordinates l <20oand b <
typical system on a mass scale of 1012M⊙, mostly con- 10o region around the Galactic centre) i|s|much harde|r|to
tributed by exotic particles, such as weakly interacting
ascertain than that of the bulges in many external galax-
massive particles or axions. Direct searches for dark com-
ies, because of obscurationby interstellar dust due to our
pact objects, such as MACHOs, in the Milky Way halo
position in the Galactic plane. In the recent years, how-
have been performed by the MACHO and EROS collab-
ever, striking images of the Galactic bulge have been ob-
orations through microlensing surveys. According to the
tained in the near infrared (at wavelengths of 1.25, 2.2,
MACHOgroup(Alcock et al. 2000b),themostlikelyhalo
3.5 and 4.9 µm) by the DIRBE experiment on board the
fraction in form of compact objects with a mass in the
COBE satellite (Arendt et al. 1994, Weiland et al. 1994)
range 0.1 1 M⊙ is of about 20%; the EROS collabora-
− allowingto study with goodaccuracyits structure.These
tion (Lasserre et al. 2000) has set a 95% confidence limit
images suggest that the stellar distribution in the bulge
that objects less than 1 M⊙ contribute less than 40% of
is bar-shaped, i.e. that the bulge is not rotationally
the dark halo. These upper limits on the fraction of com-
symmetric. However, a bar-like structure can not be
pactobjectsareevidencesfavouringthatGalaxyhalocan
used since the standard adiabatic compression theory
be described by the standard cosmologicalapproach.
formally assumes that all the mass distributions have
The very detailed data, obtained from many indepen-
spherical symmetry. On the other hand, it is possible
dent techniques, which characterizethe Milky Way, make
to achieve a relatively good agreement with the near in-
it a unique testing for theories of galactic structure and
frared photometric data also using spheroidal models for
baryonicinfall.Actually,thisconsiderationhasmotivated
the bulge (Kent et al. 1991, Dwek et al. 1995). Thus, we
Klypin et al. (2002) to perform an analysis similar to the
willdescribethebulgeasaspheroidaldensitydistribution
onewepresentinthispaper.Althoughconceptuallyanal-
(Dehnen & Binney 1998):
ogous to that of Klypin et al. (2002), the procedure we
wpliollreusteheiswmhoulcehpmaroarmeedteetrasilpeadcea.nAdsaallorwessutlto, dweeepwlyillebxe- ρb =ρ0 m γ 1+ m γ−β e−m2/rt2 , (1)
(cid:18)r (cid:19) (cid:18) r (cid:19)
able not only to investigate the viability of the standard 0 0
CDM paradigm when applied to the Galaxy, but also to
where
constrain a set of parameters that are, on the contrary,
held fixed in the work of Klypin et al. (2002). m2 (R2+z2/q2)1/2 , (2)
≡
The paper is organized as follows. In Sect. 2 we de-
with R the galactocentric radius and z the height above
scribetheingredientsneededforouranalysisandinSect.3
the equatorialplane.Thusthe densityofthe bulgeis pro-
we introduce briefly the adiabatic compression theory.
portional to r−γ for r << r , to r−β for r << r << r
Section4isdevotedtothedescriptionoftheobservational 0 0 t
and softly truncated at r = r . Fitting of the model to
constraintswehavefortheMilkyWay,whileinSect.5we t
the observed infrared photometric COBE/DIRBE data
explain how we investigate the parameter space to select
yields the values of four of the five bulge parameters
only models in agreement with the data. The impact of
the different selection criteria on the parameter space is (Dehnen & Binney 1998):
discussed in Sect. 6 and the results of our analysis are
β =γ =1.8 , q =0.6 , r =1 kpc , r =1.9 kpc .
0 t
reportedinSect.7.Section8isdevotedtosomefinalcon-
siderations. The density normalization ρ is not determined from
0
thefittingrelation,butitiseasilyrelatedtothetotalmass
of the bulge, M . Integrating Eq.(1), one gets
2. Model ingredients bulge
Traditional models of spiral galaxies usually include at Mb(m)=0.518Mbulgem1.2 1F1[0.6,1.6, 0.27m2] (3)
−
leastthree dynamicalcomponents:a spheroidalbulge,an
where F is the hypergeometric function and M is
1 1 bulge
exponentialdiscandadarkhalo.Thelackofobservational
related to the central density ρ0 (in M⊙/pc3) as:
dataprecludesfromusingmoredetailedmulti-component
mass models since it is not possible to break the degener-
M =1.60851 4πqρ . (4)
bulge 0
acy among the many parameters involved. However, due ×
toourprivilegedposition,cinematicandphotometricdata Inthe adiabaticcompressionformalism,allthe galaxy
ontheMilkyWayarenumerousenoughtorequiretheuse components are assumed to be spherical. To this aim, it
of a multi-component model in order to have a unified is needed to describe the bulge mass distribution with a
picture. To this aim, we model the mass distribution in modified “spherical” version of Eq.(3). A simple and rea-
the Milky Way introducing five components (the bulge, sonable way to solve this problem is to substitute Eq.(3)
the thin and thick discs, the interstellar medium disc and with the following one:
the dark halo) that we describe in detail in the following
subsections. Msph(r) M f(r) , (5)
b ≃ bulge
V. F. Cardone and M. Sereno: Modeling theMilky Way 3
f(r)=1 e−1.867r(1+1.543r+0.1898r2+ we use the original flattened density distribution given in
−
Eq.(8).1
0.6349r3 0.6109r4+0.1491r5 0.01126r6) , (6) Although the simple formula in Eq.(8) may fit well
− −
the large-scale structure of the stellar discs, it is not able
where now r is the usual spherical radius. The spherical
to reproduce the smaller scale density fluctuations, which
mass distribution in Eq.(5) has been defined so that the
are prevalent in the ISM component. Dame et al. (1987)
mean density inside the spherical radius r is the same as
haveshownthatthere isverylittle interstellarmatter be-
the one within the elliptical radius m, M (m), i.e.:
b tween the nuclear disc at 200pc and the molecular ring
at R = 4 5kpc. These local fluctuations strongly affect
Msph(r) M (m) −
b = b . the estimation of the Oort constants. Hence, we do not
4/3 π r3 4/3 π q m3 assume an analytical expression for the ISM disc density
profile, but we use a third order polynomial interpolation
UsingEq.(5)insteadofEq.(3)introducesasystematicer-
of the data in Table D1 of Olling & Merrifield (2001)also
ror when describing the very inner regions of the Galaxy.
including a 23.8% helium contribution by mass. The 3-D
Actually, this error is expected to be not a serious one
mass distribution of the ISM disc is then evaluated as in
since the bulge contribution to the total mass budget of
Eq.(9). The ISM disc rotation curve has been evaluated
the Galaxyis indeedsmall.Moreover,aswewillseelater,
following the method described in Kochanek (2002).
most of the observational constraints we will use probe a
Tofullycharacterizethediscmodel,wehavetofixthe
region of the Galaxy that is so far from the inner bulge
geometricalparameters(R ,z )andavalueforthecentral
d d
dominated region that its dynamical effect could be de-
(or the local) surface density. We fix the geometry of our
scribed even by modelling this component as a pointlike
stellar discs giving their scale-length and scale-height as
mass. However, to further reduce this effect, we will use
follows (Dehnen & Binney 1998):
Eq.(5) for the bulge mass distribution, but we consider
the exact rotation curve (Binney & Tremaine 1987): thin disc:R =κR kpc , z =180 pc ;
d 0 d
v2(R)=4πGq R ρb(m2)m2dm . (7) thick disc:Rd =κR0 kpc , zd =1000 pc .
c Z0 R2 (1 q2)m2 where R is the distance of the Sun to the galac-
− − 0
p tic centre and κ = (0.30 0.05) a scaling constant
To fully characterize the bulge we only need its to- ±
(Dehnen & Binney 1998). The exact value of R is quite
0
tal mass, M . Dwek et al. (1995) found M =
bulge bulge uncertain, with most of the estimates ranging from
(1.3±0.5)×1010 M⊙. 7.0 to 8.5kpc (Kerr & Lynden-Bell 1986, Reid 1993,
Olling & Merrifield 2000, Olling & Merrifield 2001).
2.2. The disc Given the importance of this parameter in the modeling
oftheGalaxy,wewillexploremodelswithdifferentvalues
Contrary to the bulge, the structure of the Milky Way
of R .
0
disc is quite easy to investigate given the large amount
Weusethefollowingestimateforthelocalsurfaceden-
of available data. Disc is made up of three compo-
sity (Kuijken & Gilmore 1989):
nents, namely the thin and the thick stellar discs and
the interstellar medium (ISM) disc. We model the two Σ⊙ =(48 8) M⊙ pc−2 .
±
stellar discs with the usual double exponential profile
(Freeman 1970, Dehnen & Binney 1998): Σ⊙ accountsforthecontributionofthediscs(bothstellar
and ISM), not taking into account the halo contribution.
Σ R z The total mass of each sub-disc is fixed specifying the
d
ρ (R,z)= exp | | . (8)
d 2z (cid:18)−R − z (cid:19) fractional contribution of each one to Σ⊙. Using the rela-
d d d
tions given in Dehnen & Binney (1998), we fix:
The total mass of each stellar disc is M = 2πΣ R2. For
d d d Σ (R )=14 Σ (R ) ,
the adiabatic compression, we will adopt a 3-D mass dis- thin 0 thick 0
tribution such that: while, for the ISM disc, the adopted value is
+∞ r (Olling & Merrifield 2001):
M (r)=2π ρ (R,z)RdRdz . (9)
d Z−∞ Z0 d ΣISM(R0)=14.5 M⊙ pc−2 .
InEq.(9),M (r)hassphericalsymmetrywhichisnotfor- It is noteworthy that there are also other estimates
d
mallycorrectsincethediscisahighlyflattenedstructure. of Σ⊙ significantly lower than the one used here
Actually, this approximationintroduces a negligible error (Olling & Merrifield 2001, Gerhard 2002).
as it is witnessed by the good agreement found between 1 To this aim we use a C++ code kindly provided us by
the predictions of the adiabatic compression theory and W. Dehnen which implements a modified multipole technique
the numerical simulations (Jesseit et al. 2002). However, developed by Kuijken & Dubinski (1994, see also Dehnen &
to evaluate the disc rotation curve and the vertical force Binney 1998).
4 V. F. Cardone and M. Sereno: Modeling theMilky Way
2.3. The halo on the angular momentum of the galactic components
(baryons and CDM). The effects of the baryonic infall
Observationaldatamaybefittedbyawiderangeofmod-
are treated here following the approach of the adiabatic
els, even unphysical ones. This is why there are a lot of
compression (Blumenthal et al. 1986, Flores et al. 1993,
different dark halo models which are claimed to describe
Mo et al. 1998). If the disc is assembled slowly, we can
wellthedensityprofileofthiscomponent.Toobtainphys-
assumethatthehalorespondsadiabaticallytothemodifi-
ically interesting models, it is thus important to impose
cationsofthegravitationalpotentialanditremainsspher-
constraints based on a physical theory of halo formation
icalwhilecontracting.Theangularmomentumofthedark
and to select models which are both compatible with the
matter particles is then conserved and a particle which is
data and also physically well motivated. From this point
initially at a mean radius r ends up at a mean radius r
ofview,numericalsimulationsofgalaxyformationinhier- i
where:
archicalCDMscenariosareveryhelpfulsincetheypredict
the initial shape of the dark matter distribution. In this
M (r) r =M(r ) r (15)
f i i
paper, we assume a NFW profile (Navarro et al. 1997) as
initialdarkmatterhalo.ThemainpropertiesoftheNFW being M(r ) the initial total mass distribution within r
i i
model are: and M (r) the total finalmass within r. M (r) is the sum
f f
ρs ofthedarkmatterinsidetheinitialradiusri andthemass
ρ(r)≡ x (1+x)2 , x=r/rs (10) contributed by the baryonic components. We thus have:
M (r)=(1 m )M(r )+M (r) (16)
M(r)=4πρsrs3 f(x)=Mvirf(x)/f(c) , (11) f − b i bar
x where Mbar(r) is the sum of the contributions from the
f(x) ln(1+x) , (12)
baryonic components and m is the fraction of the total
≡ − 1+x b
mass that ends up in the visible components. Note that
c rvir/rs , (13) we are implicitly assuming that the baryons had initially
≡
thesamemassdistributionastheCDMparticlesandthat
4πδ
M = thΩ ρ r3 , (14) those which do not form the luminous units still remain
vir 3 M crit vir distributed as the CDM.
Foragivenrotationcurve,v (R),theangularmomen-
where c is the concentration parameter, M the virial c
vir
tum of the thin disc is:
mass and r the virial radius2. The model is fully de-
vir
scribedby twoindependentparameters,whichweassume rvir
J =2π v (R)Σthin(R)R2dR . (17)
to be c and Mvir. A correlation between c and Mvir has d Z c d
0
beenfoundinnumericalsimulations(Navarro et al. 1997,
Bullock et al. 2001, Colin et al. 2004), but we do not use Theupperintegrationlimitcanbesettoinfinitysincethe
such a relation since it is affected by a quite large scatter disc surface density Σd(R) drops exponentially andrvir is
( 25%). much larger than the disc scale-length. We assume Jd to
∼
The NFW model is not the only model proposed be a fraction jd of the initial angular momentum of the
to fit the results of numerical simulations. Some au- halo, J (i.e. Jd = jdJ). J can be expressed in terms of a
thors (Moore et al. 1998, Ghigna et al. 2000) have pro- spin parameter λ defined as:
posed models with a central slope steeper than the NFW
one. However, the difference between these models and λ≡J|E|1/2G−1Mv−ir5/2 , (18)
the NFW one is verysmallfor radiilargerthan0.5%-1%
with E, the total energy of the NFW halo,
the virialradiusand it is further washedout by the bary-
onic infall. For these reasons,we will not consider models
GM2
different from the NFW one. E = virfc . (19)
− 2r
vir
Some simple algebra allows us to rewrite Eq.(17) as
3. Adiabatic compression
(Mo et al. 1998):
The present day dark matter halo has a different shape
with respect to the original NFW model since the gravi- R = 1 jd λr f−1/2f (λ,c,m ,j ) (20)
tationalcollapseofthebaryonicmatter,whichformsboth d √2(cid:18)md(cid:19) vir c R d d
the bulge and the disc, changes the overall gravitational
with:
potential of the system. The halo structure is thus modi-
fied by the forces of the collapsed baryonsdepending also c 1 1/(1+c)2 2(1+c)−1ln(1+c)
f = − − , (21)
2 The virial radius is defined such that the mean density c 2 [c/(1+c) ln(1+c)]2
−
withinrvir isδthtimesthemeanmatterdensityoftheuniverse
ρ¯ = ΩMρcrit. We assume a flat universe with (ΩM,ΩΛ,h) = f (λ,c,m ,j )=2 ∞e−uu2vc(Rdu)du −1 . (22)
(0.3,0.7,0.72) and δth =337 (Bryan & Norman 1998). R d d (cid:20)Z0 vvir (cid:21)
V. F. Cardone and M. Sereno: Modeling theMilky Way 5
In Eqs.(20) and (22), m is the fraction of the total mass wherez R tanb,withbthegalacticlatitudeofthefield
d 0 0
≃
competingtothethindisc,u R/R andv isthetotal target.Thisisonlyalowerestimateoftheminimumbary-
d vir
≡
circularvelocityatthevirialradiusr .Itisimportantto onicmassinsidethesolarcircleanditisquiteindependent
vir
stressthatthe rotationcurveenteringEq.(17)isthe total on the density profile of the baryonic components. Using
one, i.e. it is: thelatestmeasuredvalueoftheopticaldepthtowardsthe
Baade window from the analysis of 52 events in which a
v2(r)=v2 (r)+v2 (r) , (23)
c c,bar c,DM clump giant is lensed, τ = (2.0 0.4) 10−6 (Popowski et
± ×
where the first term is simply evaluated given the distri- al.2000),wecanthus selectonly thosesets ofparameters
bution of the baryonic components, while the latter is: (R0,Mbulge,Σ⊙) which produce a baryonic mass within
R higherthanwhatispredictedbyEq.(25).Notethatwe
v2 (r)=G[M (r) M (r)]/r . (24) 0
c,DM f − bar areimplicitly assumingthatallthe observedmicrolensing
The full set of equations allows one to determine the events are due to stellar lenses which is true only if there
final distribution of the DM particles provided that a isnocompactdarkmatter(suchasMACHOs)inthedisc.
model for the density profiles of the baryons has been Shouldthisassumptionnottobetrue,theminimumbary-
assigned and the parameters (λ,m ,m ,j ,c,M ) have onic mass shouldbe lowered.However,we stressthat this
b d d vir
been fixed. We will see later in Sect. 4 that these param- constraintis not very selective so that changing the value
eters are not all independent since it is possible to find of Mmin does not affect our results.
bar
somephysicallymotivatedrelationsamongsomeofthem. Whilst microlensing towards the bulge provides con-
It is worth stressing that there is some debate about straints onthe inner Galaxy,satellite dynamics andmod-
the validity of the adiabatic compression formalism. eling of the Magellanic Clouds motion probe the Galaxy
Jesseit et al. (2002) have found a substantial agreement mass distribution on a large scale ( 50-100kpc). Lin
≃
between the final dark matter distribution in numerically et al. (1995) have used the dynamics of the Magellanic
simulatedhaloesandthatpredictedbytheadiabaticcom- Clouds to infer the mass of the Milky Way inside 100kpc
pressionapproach.Ontheotherhand,thisresulthasbeen obtainingM(r<100kpc)=(5.5 1) 1011M⊙.Thisisin
± ×
contradicted on the basis of a set of higher resolution nu- goodagreementwiththevaluefoundbyKochanek(1996)
merical simulations recently carried out by Gnedin et al. using escape velocity and motions of the satellite galax-
(2004). According to these authors, the standard adia- ies which give(5 8) 1011 M⊙.Taking into accountthe
− ×
batic compression formalism systematically overpredicts different techniques used and a dependence of the results
thedarkmatterdensityprofileintheinner5%ofthevirial on the halo modeling, we follow Dehnen & Binney (1998)
radius. It is worth noting, however, that only one of the assuming as our constraint:
eight simulations considered by Gnedin et al. refers to a
galactic(ratherthana cluster)halo.The bottompanelof M(r <100 kpc)=(7.0 2.5) 1011 M⊙ . (26)
± ×
Fig.4intheirpapershowsthattheadiabaticcompression
A third and quite efficient constraint is given by the
formalism overpredict the dark matter density less than
Oort’s constants defined as:
10% at r/r 0.1, while the error quickly decreases
vir
∼ ∼
forlargervaluesofr/rvir.Thiserroris muchsmallerthan A= 1 vc(R) dvc(R) , (27)
the uncertainties we have on the observational quantities 2 (cid:20) R − dR (cid:21)
sothatweareconfidentthatusingthestandardadiabatic
formalism does not introduce any bias in the results. 1 vc(R) dvc(R)
B = + . (28)
−2 (cid:20) R dR (cid:21)
4. Observational constraints
Dehnen & Binney (1998) have reviewed the estimates of
We wanttoselectmassmodels ofthe Milky Waythatare theOort’sconstantspresentinliteraturefinallyproposing
physicallywellmotivatedandareinagreementwithobser- the following values:
vationaldata.We discussinthissectionthe observational
A=(14.5 1.5) km s−1kpc−1 , (29)
constraints used to test each model. ±
Only compact baryonic objects can cause microlens- B =( 12.5 2.0) km s−1kpc−1 , (30)
ing events towards the Galactic bulge. The number − ±
of microlensing events observed towards the galactic A B =(27.0 1.5) km s−1kpc−1 . (31)
− ±
bulge (Alcock et al. 2000a, Popowski et al. 2000) deter-
From the definitions of the Oort’s constants, it immedi-
mines the minimum baryonic mass in the inner Galaxy
ately turns out that (A B)R = v (R ). We can thus
that can yield the measured value of the optical depth 0 c 0
−
replacetheconstraintsonA B withaconstraintsonthe
τ. For an axisymmetric mass density which decreases
−
local circular velocity:
moving vertically away from the plane, it is possible
to demonstrate that the minimum baryonic mass is v (R )=R (27.0 1.5) km s−1 . (32)
c 0 0
(Binney & Evans 2001): × ±
ec2z τ The vertical force Kz at some height above the plane
Mbmarin = G0 (25) places a condition on the local mass distribution. Using
6 V. F. Cardone and M. Sereno: Modeling theMilky Way
K stars as a tracer population, Kujiken & Gilmore (1989, onlyaminoreffectonthiscriterium.Theloweristhevalue
1991) have deduced: of Mmin, the higher is the number of models passing this
bar
preliminary test, but all of the models added by lowering
Kz,1.1 ≡|Kz(R0,1.1kpc)|=2πG×(71±6)M⊙ pc−2 .(33) Mbmarin will be finally excluded by the selection procedure
described later.
Formally, this estimate depends on the galactic constants
The parameters (λ,m ,m ,j ,c,M ) characterize
b d d vir
R andv (R ),butOlling&Merrifield(2001)haveshown
0 c 0 the dark halo and are needed to solve the adiabatic com-
that the result is quite robust against variations in these
pression equations. Furthermore, they give scaling rela-
parameters. Thus, we can use the previous value to con- tionsbetweenthebaryoniccomponentsandthetotalmass
strain our models.
distribution. First, we note that m can be expressed as
d
Finally, another constraint comes from the rotation function of m as:
b
curve of our Galaxy. We can reconstruct this quantity
from the measurements of the velocity field. For an ax- Mthin
m = m f m ,(35)
d b d b
isymmetric galaxy,the radialvelocityrelativeto the local M +M +M +M × ≡
bulge thin thick ISM
standardofrest, v , ofa circularorbiting object at galac-
r
withM ,M andM ,respectively,the totalmass
ticcoordinates(l,b)andgalactocentricradiusRisrelated thin thick ISM
of the thin, thick and ISM disc. It is also possible to ex-
to the circular speed by:
press the virial mass M as function of M and m
vir thin b
vr R0 simply as (Mo et al. 1998):
W(R) = v (R) v (R ) (34)
c c 0
≡ sinlcosb R −
M M
thin thin
M = = . (36)
with the following relation between R and the distance d vir m f m
d d b
to the object:
Equation (36) simply states that the final mass of the
R2 =d2cos2b+R02−2dR0cosbcosl . systemisthesameastheinitialone.Thus,wearenowre-
duced to only four parameters, namely (λ,m ,j ,c). We
b d
Severalstudiesareavailablewithmeasurementsofbothd can further reduce the number of parameters observing
and vr for objects which ought to be on a nearly circular that Eqs.(15), (16) and (20) are a system of three in-
orbits,sothattheMilkyWayrotationcurvevc(R)canbe dependent equations which can be iteratively solved to
reconstructed. We use here the data on HII regions and determine the initial radius r (r), the final mass M (r)
i f
molecularcloudsinBrand&Blitz(1993)andtheoneson and one of the four parameters, once fixed the remaining
a sample of classical Cepheids in the outer disc obtained threeones.Aswewillseelater,whiletherearesomehints
byPontetal.(1997).FollowingDehnen&Binney(1998), about the distribution of the other parameters, little is
we reject objects with either 155o l 205o or W < 0 known about the value of the concentration c. We have
≤ ≤
or d < 1kpc when vr is very likely dominated by non- thus decided to solve the set of equations with respect
circular motions. to (r ,M ,c) having fixed the parameters(λ,m ,j ). The
i f b d
equations are highly non linear and must be solved itera-
tively3, so that, to speed up the calculations,we have im-
5. Exploring the parameter space
posed a priori that c should be in the range (5,25) which
Toapplytheformalismoftheadiabaticcompressioninor- is a quite conservative estimate for spiral galaxies similar
dertohavethepresentdayhalomassprofile,weneedthe to the Milky Way (Jimenez et al. 2003).
initialhaloshapeandthetodaydensityprofileofthebary- Toexploreindetailthe spaceofparameters,webuild,
onic components. As discussed above, the baryons have for each model with given values of (R0,Mbulge,Σ⊙,κ),
been distributed in the bulge and in the three sub-discs a set of models individuated by the values of (λ,m ,j ).
b d
according to the observational data. We briefly explain how we define the grid. We fix a value
We havehoweverstillanindeterminationonthebary- for the spin parameter λ. The distribution of λ for haloes
onic components since the Sun distance to the galactic generated in numerical N-body simulations is well ap-
centre R0, the total mass of the bulge, Mbulge, the local proximated by a log-normal distribution with param-
surface density, Σ⊙, and the discs scale-lengths (fixed by eters nearly independent on the cosmological parame-
the scaling constant κ) are known with uncertainties. As ters, halo mass and redshift (Barnes & Efstathiou 1987,
a first step, we consider a grid of models with R0 ranging Lemson & Kauffmann 1999, Vitvitska et al. 2002). Using
from7.0to8.5kpcinstepsof0.5kpc,Mbulge from0.80to the parameters in Vitvitska et al. (2001), the maximum
14.080to×5160M10⊙Mp⊙c−w2itihnastsetpeps ooff 10..612M5×⊙10p1c0−M2 ⊙an,dΣ⊙κ ffrroomm ca3nMbeouestedal.to(1e9s9t8im)aptreopdoirseecdtlyancafpropmroxEimq.(a2t0e)rtehlautsioanvowidhiinchg
0.25to0.35instepof0.05.Amongthesemodelsweselect
theiterativeprocedure.However,theirapproximationhasbeen
only the ones which pass the test on the minimum bary-
obtained neglecting the bulgeand assuming a single exponen-
onic mass within R0. We stress that changing the value tial disc instead of the three sub-discs we are using. We have
of Mmin to consider the (quite unlikely) possibility that checkedthattheirformulamayleadtounderestimatestrongly
bar
somemicrolensingeventsarenotduetostellarlenseshave thevalue of c.
V. F. Cardone and M. Sereno: Modeling theMilky Way 7
of the distribution is at λ = 0.035 while there is a 90% onbothv (R)andtheGalacticconstants,whichenterthe
c
probability that λ is in the range (0.02,0.10). We thus estimate of v (R) through Eq.(34), are not normal. This
c
let λ change in this range in steps of 0.01. Next, we meansthatahighvalueofχ2foragivenmodelmightbea
have to fix a value for m . This parameter is poorly consequence of an intrinsically wrong model or of the not
b
constrained since we only know that it cannot be larger normaloriginoftheerrors.Furthermore,thedataofPont
thantheuniversalbaryonfractionΩ /Ω .Thislatterhas etal.(1997)ontheradialvelocitiesoftheouterdiscclassi-
b M
been inferred by observations involving completely differ- calCepheids aregivenwithoutany uncertainties,so that,
entphysicalprocesses(Turner 2002).Thepowerspectrum for these data points, σ is only determined by the propa-
i
ofmatterinhomogeneitiesfromobservationsoflarge-scale gationoftheerrorsonthegalacticconstantsanditisthus
structure is sensitive to Ω /Ω ; the Two Degree Field underestimated.Sowehavedecidedtostillretaintheχ2-
b M
GalaxyRedshiftSurveyhasreportedavalueof0.15 0.07 testaselectioncriterium,butwealsoconsiderthemedian
±
(Percival et al. 2001).Measurementsoftheangularpower statistics. As shown in Gott et al.(2001; see also Avelino
spectrum of the CMBR also provide a very significantes- et al. 2002, Chen & Ratra 2003, Sereno 2003), median
timate. The combined analysis in Jaffe et al. (2001) of statistics provide a powerful alternative to χ2 likelihood
several data set gives Ω /Ω = 0.186+0.010. We thus let methods. Fewer assumptions about the data are needed.
b M −0.008
m change from 0.01 to 0.20 in steps of 0.01. Finally, A proper median statistics assume that (i) experimental
b
we have to fix j . This parameter also is not constrained results are statistically independent; (ii) there are no sys-
d
either theoretically or by numerical simulations. On one tematic effects. Statistical errors are not required to be
hand, it is reasonable to assume j =m and indeed this either knownorgaussianlydistributed.Since ouranalysis
d d
seemstobenecessarytofitspiralgalaxiesrotationcurves is based on not normal errors, performing a test without
(Mo et al. 1998). On the other hand, numerical simula- using the errors themselves turns out to be a very con-
tions have found j /m significantly less than unity. We servative approach.Furthermore,median statistics is also
d d
have varied this parameter from 0.005 to m in steps of less vulnerable to the presence of bad data and outliers.
d
0.01. To compute the likelihood of a particular set of pa-
The grid we build in this way is quite detailed. For rameters, we count how many data points are above or
each given (R0,Mbulge,Σ⊙,κ) the total number of sets below each model prediction and compute the binomial
(λ,m ,j )is 800 1200(dependingonthevalueofm ), likelihoods. Given a binomial distribution, if we perform
b d b
∼ −
so that the parameterspace is indeed checkedintensively. N measurements, the probability of obtaining k of them
To select among this large number of models we use, above the median is given by
foreachmodelparameterization,amulti-stepprocedure.
2−NN!
First, we solve iteratively the set of Eqs.(15), (16) and P(k)= . (38)
k!(N k)!
(20) so that we have the full halo mass distribution. We −
arethusable toestimate Mf(r =100kpc),the totalmass We count how many of the 115 experimental points are
inside r = 100kpc. A model passes to the next step only above the expected velocity rotation curve, and retain a
if this value is consistent with the estimate in Eq.(26). modelifthenumberofoverestimatesisbetween43and72.
In the second step, we compare the local circular velocity GiventhedistributioninEq.(38),theprobabilitythatthe
vc(R0) to the constraint in Eq.(32) rejecting the model median of 115 sorted entries falls in this range is 99.73%.
if there is no agreement. Next, the Oort’s constant are Itisworthtonotethatourfinalresultsdonotdepend
evaluatedandthe modelisretainedifthe valuesofAand on the sequence of the tests.
B areinagreementwith the values givenbyEqs.(29)and
(30).Then,wecomputeK andacceptthemodelifthe
z,1.1 6. Analysis of the selection criteria
resultingvalueisinagreementwiththe estimatereported
in Eq.(33). For the surviving models, we estimate the χ2 Before presenting the results of our analysis, it is inter-
defined as: esting to investigate at what stage in our selection pro-
cedure certain types of models are excluded, i.e. we want
χ2 = 1 vc2,data(Ri)−vc2,model(Ri) (37) to study the impact of each criterium on the parameter
N σ2 space. To this aim, for a given set of galactic parameters
X i
(R0,Mbulge,Σ⊙,κ),wefirstselectallthe modelswithcin
where σ is the error on the i-th measurements and the
i the range (5,25) and then apply to this set of models the
sum runs over the N data points. Note that we compute
selection criteria introduced in the previous section sepa-
the χ2 asthe laststepofourselectionprocedurewhenall
rately. The main results of this analysis are presented in
the model parameters are fixed so that we do not change
Figs.1–5 and discussed below as regard the two param-
theirvaluesinordertominimizetheχ2.Wedeemamodel
eters entering the adiabatic compression equations, i.e. c
as acceptable if χ2 < 1.33. For our data set of N = 115
and m .
b
entries, this corresponds to a 99.7% confidence level.
Actually,asstressedbyOlling&Merrifield(2001),se- – Constraint on M(r < 100 kpc). As Fig.1 shows, the
lectingamongdifferentmodelsonthebasisoftheχ2value application of this criterium tends to flatten the his-
is not a statistically correct procedure because the errors togram of the c values, being however slightly more
8 V. F. Cardone and M. Sereno: Modeling theMilky Way
14 10 14 10
12 12
ber180 ber 68 ber180 ber 68
Num 46 Num 24 Num 46 Num 24
2 2
0 0 0 0
7.51012.51c517.52022.525 0.04 0.06m0.08 0.1 7.51012.51c517.52022.525 0.04 0.06m0.08 0.1
b b
Fig.1. Histogram of the number distribution of c and Fig.3. Same as Fig.1 for the test on the Oort constants.
of mb the for the models with (R0,Mbulge,Σ⊙,κ) =
(8.5,1.0,54,0.3) and c between 5 and 25. R0 is in kpc, 14 10
12
rMefbeurlsgetoint1h0e1m0 Mod⊙el,sΣp⊙asisninMg⊙th/epct2e.sTthonesMha(dre<dh1i0s0tokgpracm). Number14680 Number 2468
2
14 10 0 0
ber11802 ber 68 7.51012.51c517.52022.525 0.04 0.06m0b.08 0.1
Num 46 Num 24 Fig.4. Same as Fig.1 for the test on Kz,1.1.
2
0 0
7.51012.51c517.52022.525 0.04 0.06m0b.08 0.1 rupted. Furthermore, as we have observed comparing
Fig.3 with Figs.1–2, applying only the constraint on
Fig.2. Same as Fig.1 for the test on vc(R0). A,B gives results that are consistent with those ob-
tained using the two constraints on M(r < 100 kpc)
andv (R )sothatanysystematicerrorintheestimate
c 0
effective in cutting out models with values of c in
of the Oort constants is washed out in our multi-step
the tails of the distribution. On the contrary, this cut
procedure.
tends to suppress the high end of the m histogram,
b – ConstraintonK .Thistestworksexcludingmodels
z,1.1
onlyretainingthosemodelswithsmallerm .Thislat-
b withvaluesofcandm inthetailsofthedistribution.
b
ter result may be qualitatively explained considering
This constraint is very effective when selecting among
Eq.(36) which shows that the higher is m , the lower
b different galactic parameters leading to reject models
is Mvir and thus M(r <100 kpc). with Σ⊙ <49 M⊙/pc2. Actually, the lower is Σ⊙, the
– Constraint on v (R ). Fig.2 shows the impact of the
c 0 higheristhepercentageofmodelsexcludedbythetest
test on the local circular velocity. High values of c
on M(r <100 kpc) or by that on the Oort constants.
are excluded by this constraint.As regardthe m his-
b Thisisaclearevidenceagainstmodelswithlowvalues
togram,thisselectioncriteriumturnsouttobealmost
ofthelocalsurfacedensityinagreementwithwhatthe
orthogonaltothe previousonesincenowlowvaluesof
results of the application of the constraint on K
z,1.1
m are clearly disfavoured.
b claim.
– Constraint on the Oort constants. This test turns out – Constraint on χ2. The application of this constraint
to be a sort of compromise between the two previous
allowstoflattenthehistogramofthecvalueslowering
ones. On one hand, it is quite effective in excluding
the peak in Fig.5 for low values. However, the high
models with very low values of c (look at the lowest
end of the histogram is erased thus suggesting that
bin), as it is (with much less efficiency) for the con-
modelswithveryhighvaluesofcarenotabletofitthe
straintonM(r <100kpc). Onthe otherhand,it cuts
Milky Way rotation curve. This is not an unexpected
away the low end of the m histogram in the same
b result since both N-body simulations and fitting of
way as the selection criterium based on the value of
the adiabatically compressed NFW model to external
v (R ) do. This is a quite important result since it
c 0 galaxies show that small values of c are best suited to
shows that the eventual exclusion of this constraint
describegalacticdarkhaloes(Jimenez et al. 2003).As
does not alter the final results of the multi-step selec- concernm ,the constraintontheχ2 worksasthaton
b
tion procedure. The presence of local fluctuations in
the local circular velocity selecting models with high
the ISM disc density strongly affects the derivatives
values of this parameter. This is a reasonable result
of the gravitational potential thus leading to possible
sincebothconstraintsarerelatedtothesamephysical
errors in the evaluation of the Oort constants for a
quantity,i.e.the rotationcurve.The medianstatistics
givensetofgalacticparameters.One couldthus argue works the same way as the χ2-test, but it is more
that it should be better to not use the Oort constants
stringent.
as a selection criterium, but only their difference and
hence the localcircularvelocity.However,the use ofa As a final remark, we want to stress that the obser-
third order polynomial interpolation of the measured vational data we have reviewedin Sect.3 are “consensus”
ISMdiscdensityalleviatesthisproblemsothatweare valueswith“consensus”errorssothattheselattercannot
confidentthattheestimatedvaluesofA,Barenotcor- be treated as “statistical” uncertainties. This is the rea-
V. F. Cardone and M. Sereno: Modeling theMilky Way 9
14 10 Table 1. Median values, 68% and 95% regions for the
12
Number14680 Number 2468 misoedxeplrepsasreadmineteurnsitosfosfam10p1l1eMA.⊙T. he total halo mass Mvir
2 Parameter Median 68% range 95% range
0 0
7.51012.51c517.52022.525 0.04 0.06m0.08 0.1 c 6.89 5.49 – 9.45 5.04 – 11.88
b
Mvir 8.68 7.53 – 10.20 6.87 – 16.60
Fig.5. Same as Fig.1 for the test on χ2 value. mb 0.08 0.07 – 0.09 0.04 – 0.10
λ 0.06 0.04 – 0.09 0.03 – 0.10
λ′ 0.029 0.026 – 0.035 0.018 – 0.042
son why we have decided to adopt a multi-step selection
procedure insteadof the usualχ2 minimization technique
basedonthe definitionofaχ2 enteringalltheconstraints Table 2. Median values, 68% and 95% regions for the
at the same time. Our filtering approach and the discus- model parameters of sample B. The total halo mass M
vir
sion presented in this section allow one to avoid all the is expressed in units of 1011 M⊙
problems connected with the statistics of not normal er-
rors and makes it possible to understand how the results Parameter Median 68% range 95% range
could vary changing one of the constraints. Actually, the c 6.48 5.49 – 7.26 5.08 – 9.75
procedure we have implemented is quite robust since the Mvir 8.44 7.58 – 9.39 7.03 – 11.16
different constraints select different regions of the param- mb 0.08 0.07 – 0.09 0.06 – 0.09
λ 0.06 0.04 – 0.09 0.03 – 0.10
eterspacethusallowingtonarrowtherangesforboththe
λ′ 0.029 0.027 – 0.032 0.023 – 0.035
adiabatic compression parameters and the galactic con-
stants.
17.5
7. Results
15
Theselectionprocedurewehaveemployedisquiteefficient
12.5
allowingustorejectthemostofthemodels.Sincewehave
r
used two alternative test as last constraint (the χ2 value e 10
b
or the median statistics), we define two samples. Sample m
u 7.5
A contains those models passing all the selection criteria N
andhavingχ2 <1.33,while Sample B is made outbythe 5
modelspassingthetestonthemedianstatistics.Thefinal 2.5
number of models is 116 in the Sample A and 34 in the
0
Sample B from an initial set of 106. Actually, it turns
out that the Sample B is a sub∼set of Sample A, i.e. all 6 8 c10 12 14
the models in Sample B belong to Sample A too. This is
Fig.6. Histogram of the number distribution of the con-
expected since the median statistics is a more restrictive
test than the χ2 analysis. centration parameter c for the models in Sample A and
Sample B (shaded histogram).
The main results are shown in Figs.(6-9) and summa-
rized in Tables 1 and 2. Note that the results obtained
from the two samples are in perfect agreementso that we
stressingthatthe valuesofc wegetarelowerthanex-
will not discuss them separately. We stress that the final
pected, but are not unrealistic. Jimenez et al. (2003)
samplesofmodelsmaynotbetreatedusingtheusualsta-
havefittedtherotationcurvesof400spiralgalaxiesby
tistical methods since our selection procedure is based on
modelling them with an exponential disc and a dark
constraints on different observable that are “consensus”
halo obtained by adiabatically compressing the NFW
valueswith “consensus”errors.Thatiswhy wedo notre-
profile.Their Fig. 1 shows the distribution of the con-
portasbestestimateoftheparameterstheirmeanvalues,
centration c vs the total mass M . There are indeed
but the medians which is a more conservative approach. vir
a lot of galaxies with values of c in the same range as
Becauseofthis, the quoted68%(95%)rangemustnotbe
the one found here.
considered as the 1 σ (2 σ) confidence limit, but it is
− − 2. In Fig. 7, we plot the distribution of the values
simply the range which the 68% (95%) of the values are
of the total mass M for the final set of mod-
within. With this caveat in mind, we discuss below the vir
els. The median value for Sample A (Sample B) is
distribution of the various model parameters.
8.68(8.44) 1011 M⊙,andthe95%confidenceinterval
×
1. Figure 6 shows the distribution of the values of the rangesfrom5.04(5.08)to6.74(9.75) 1011M⊙.These
×
concentration parameter c. Small values are clearly valuesarestillinagreementwiththemostrecentmea-
favoured so that the median value for Sample A surement of the Milky Way total mass estimated to
(SampleB)comesouttobec=6.89(6.44).Itisworth be 1.9+−31..67×1012 M⊙ (Wilkinson & Evans 1999). It is
10 V. F. Cardone and M. Sereno: Modeling theMilky Way
25 40
20
30
r r
e 15 e
b b
m m 20
u u
N 10 N
10
5
0 0
8·1011 1·1012 1.M2·10121.4·10121.6·1012 0.04 0.05 0.06 0m.07 0.08 0.09 0.1
vir b
Fig.7. Same as Fig. 6 for the total halo mass M . Fig.8. Same as Fig. 6 for m .
vir b
25 35
nWoitlekwinosrotnhyi,shhoiwgehveerrt,htahnatutshuealvableuienggivthene mbyosEtvoafnsth&e mber112050 mber12235050
previousestimateslowerthan1012M⊙(see,e.g.,Fig.3 Nu 5 Nu150
in Zaritsky 1999). Sample A also contains few models 0 0
with very high values of M . The value of M we 0.8 1M1.21.41.61.8 49505152Σ53545556
vir vir bulge 0
have obtained may thus be considered quite reason-
able. It is also possible to see that a clear correlation Fig.9. Histograms of the number distribution of models
existsbetweencandMvir,withthelowervaluesofthe according to the values of Mbulge (1010 M⊙, left panel)
concentration parameter c giving rise to the highest and Σ⊙ (M⊙ pc−2, right panel). Shaded regions refer to
values of the total mass. This correlation may reflect Sample B.
the existence ofsome covarianceamongthe modelpa-
rameters in the analysis procedure.
95%range for Sample A does not reject any model. It
3. The distribution of m is shown in Fig. 8. The
b is worth noting, however, that values of M in the
bulge
median value is 0.08 for both Sample A and range0.8 1.4 1010 M⊙ seemstobefavoured.Thisis
B, while the 95% range turns out to be 0.07- − ×
confirmedbyanalysingSampleBthatexcludesmodels
0.09. According to some authors (Mo et al. 1998, with Mbulge > 1.4 1010 M⊙. A similar analysis leads
Jimenez et al. 2003,Klypin et al. 2002),m shouldbe ×
b to the following constraints on the disc local surface
written as ε Ω /Ω with ε indicating the efficiency
× b M density:
of the transfer of baryons from the initial halo to the
bulge and disc. If this were correct, our results should 49 M⊙ pc−2 Σ⊙ 56 M⊙ pc−2
≤ ≤
meanε 37 48%forΩ /Ω =0.186.The efficiency
b M
is not on≃e an−d this rises the problem ofunderstanding with 54 M⊙ pc−2 as median value. If we subtract the
where the missing baryons are. contributionoftheISMdiscsurfacedensity,wegetfor
4. The median value of the spin parameter λ is 0.06 for the stellar disc Σ⋆ 40 M⊙ pc−2 in agreement with
both Sample A and B and the 95% range is 0.04 the consensus value≃(Σ⋆ =35 10 M⊙ pc−2) proposed
0.09. Models with low spin are erasedby the selectio−n byOlling&Merrifield(2001).±Only6outof116models
procedure.Wedonotreportanyresultforjd sincethis in Sample A have R0 =8.0kpc, while in all the other
parameter is found to be degenerate with λ and md, casesit is R0 =8.5kpc so thatwe may safely consider
i.e., for fixed values of (mb,R0,Mbulge,Σ⊙,κ), models R0 =8.5kpc as our final estimate of this galactic con-
having the same value of λ′ = λj /m give rise to stant. This conclusion is strengthened by noting that
d d
the same present day halo mass profile. This result is all the models in Sample B have R0 = 8.5kpc. This
notunexpectedsinceintheonlyequationcontainingλ valueissomewhathigherthanexpectedsincemostre-
and j , Eq.(20), these two parameters appear only in cent estimates predict values in the range 7.0 - 8.0
d
λ′. The median value for this latter is λ′ =0.029 with kpc (Reid 1993, Olling & Merrifield 2000). However,
a 95% range going from 0.018 to 0.042 for Sample A theestimateofR0 issomewhatmodeldependentsince
and from 0.023 to 0.035 for Sample B. its value is linked to the halo flattening. For instance,
5. Figure 9 shows the histogram of the number of mod- Olling & Merrifield (2001) have found that it is pos-
>
els according to the values of Mbulge and Σ⊙. We can sible to build galaxy models with R0 7.0kpc if
∼
draw some interesting limits. First, we observe that the halo is close to spherical which indeed is our
we are not able to significantly constraint the bulge case. Finally, there are no models with κ = 0.30
6
total mass. Indeed, the median value of the distribu- so that we conclude that the discs scale-length is
tion turns out to be Mbulge = 1.0 1010 M⊙, but the Rd =0.30 R0 =2.55kpc.Thisis lowerthanthe fidu-
× ×