Mon Not R Astron So c Printed February
The Outer Regions of the Galactic Bulge II Analysis
Ro drigo A Ibata and Gerard F Gilmore
Institute of Astronomy Madingley Road Cambridge CB HA
Department of Geophysics and Astronomy University of British Columbia Vancouver Canada present address
ABSTRACT
Velocity chemical abundance and spatial distribution data for some K and M
giants in the Galactic Bulge which are presented in an accompanying pap er Ibata
Gilmore are analysed We provide three levels of analysis an almost mo del
indep endent determination of basic symmetry prop erties of the galactic bulge a more
detailed analysis based on conservative assumptions which investigates the basic kine
matic and chemical abundance distributions in terms of a few parameters and a more
detailed investigation of p ossible functional forms for the density and kinematic dis
tributions that are consistent with the data using maximum likelihoo d ts
The bulge has a well determined linear rotation curve over the range p c
R p c with amplitude km s kp c Several velocity disp ersion mo dels
are found to t the data We test the oblate isotropic bulge mo del of Kent
by integrating the kinematics predicted by that mo del into our Galaxy mo del and
comparing with our data We nd reasonable agreement except in the most distant
of our elds from the Galactic centre We do not nd a signi cant requirement for
asymmetry or a bar from our kinematics
Our data allow a quantitative comparison b etween the Milky Way bulge and
the spiral galaxy bulges studied by Kormendy Illingworth over a similar
Galacto centric distance range the Galactic bulge app ears to b e characteristic of the
extragalactic bulge p opulation We determine the metallicity distribution of K giants
in that subset of our elds where the photometric calibration is adequate Our metal
licity index measures the Mgb feature and is calibrated against lo cal eld standards
Thus our abundance scale assumes a similar relationship b etween Mg Fe and Fe H
in the bulge to that which is seen in the solar neighbourho o d Given that assumption
the mean metallicities in these elds is Fe H and do es not vary signi cantly
over the regions investigated This mean is very close to the mean abundance of
K giants observed recently in Baade s Window McWilliam Rich That is
there is no detectable abundance gradient in the Galactic bulge over the galacto centric
range p c R p c The overall bulge abundance distribution is in go o d
GC
agreement with a mo del that combines standard solar neighbourho o d determinations
of the abundance distribution functions for the halo and thick disk together with a
closed b ox simple mo del of chemical evolution with yield Z to describ e the
bulge comp onent
We derive the distribution function of sp eci c angular momentum for the bulge
from our data and compare it with determinations for the halo the thick disk and
the thin disk from Wyse Gilmore We con rm that the bulge and the halo
have angular momentum distributions which are indistinguishable as do the thick
disk and the thin disk The bulge halo distribution is however very di erent from the
thick disk thin disk distribution This is p erhaps the strongest available clue to the
evolutionary relationships b etween di erent Galactic structural comp onents
Key words The Galaxy Galactic bulge stellar kinematics stellar abundances
galaxy evolution galactic structure
THE SPATIAL KINEMATIC AND rich M giants Frogel et al to p c for K band
ABUNDANCE STRUCTURE OF THE BULGE surface brightness Kent et al The most recent de
termination from the DIRBE exp eriment on COBE pro
vides a scale height of some p c Weiland etal The Galactic bulge is a very centrally condensed Galactic
For comparison the scale height of the Galactic old disk structural comp onent Its half light radius along the mi
is p c while Baade s Window the innermost region nor axis is estimated variously b etween p c for metal
2 R Ibata and G Gilmore
dance gradient in the central bulge stellar p opulation in at where the bulge may b e well studied with optical data is
least some tracers e g infrared photometry of M giants is some p c from the Plane Power law r ts to inte
consistent with an abundance change of dex from grated surface brightness and to tracer ob ject number den
to Frogel et al sity b oth require Frogel et al Kent
Once the disk et al with inside jbj
Plan of this Paper
contribution is deducted the integrated K band luminos
ity Kent et al the IRAS p oint source surface density
Our aim in this pap er is to present an analysis of our kine
Harmon Gilmore Weinberg and the DIRBE
matic abundance and photometric data at three comple
data display a attening of c a This attening is
mentary levels First in a mo del indep endent way to
also seen in the inner RR Lyrae system Wesselink
identify general structural prop erties of the Galaxy which
though the relationship if any b etween these very old and
we may b e con dent are reliably determined Second in
rather metal p o or stars and the dominant p opulation in the
a mo del dep endent way with a small number of conser
central bulge which is rather metal rich remains obscure
vative assumptions to provide a larger amount of detail
Several lines of evidence show that the bulge is at least
while retaining an appreciation of the imp ortance of the
partly non axisymmetric Binney et al demonstrated
various assumptions required Third with increased mo del
that the existence of a bar extending out to kp c from
dep endence to investigate the full amount of information
the Galactic centre explains naturally the observed kine
p otentially contained in the data This provides a wealth of
matics of HI CO and CS gas in the region j j fur
information though at the exp ense of risking one s compre
thermore Blitz Sp ergel showed that the m
hension in a sea of parameters
ballo on data of Matsumoto et al within jbj is
In x we brie y summarise the data which is presented
brighter at p ositive than at negative as would b e ex
in detail in an accompanying pap er Ibata Gilmore
p ected from a barred stellar distributio n The DIRBE data
x outlines the stellar luminosity classi cation and describ es
also marginally favour such an asymmetry which was sus
the implicati ons of the reliabili ty of this classi catio n for
p ected in the IRAS data Harmon Gilmore From
the astrophysical conclusions of this pap er x presents the
an analysis of IRAS p oint sources selected from those sources
most robust and least mo del dep endent deductions from the
with jbj Weinberg nds evidence for a large
data concluding that the bulge is kinematicall y symmetric
scale stellar bar in the Galactic disk which he suggests ex
in spite of the confusing presence of the Sagittarius dwarf
tends out to kp c however this need not b e related to
galaxy in our data x discusses our photometric star count
the other non axisymmetric phenomena mentioned ab ove
data together with other surface brightness measures to
Kinematic studies of candidate bulge tracers Miras
deduce the most appropriate bulge and old disk spatial den
planetary nebulae K giants M giants and carb on stars
sity distributio ns Surprisingl y our data suggest a low inner
show that the bulge has a central disp ersion of km s
disk scale height x provides a kinematic analysis utilis
which decreases outwards see e g the minor axis data com
ing Kent s isotropic oblate rotator dynamical mo del and the
piled by Tyson In a given region there is a tendency
density distribution of x The agreement of our data with
for lower abundance tracers to have a higher velocity disp er
the predictions of this mo del at least in the inner few kp c
sion as predicted by dissipatio nal collapse mo dels such as
of the Galaxy is rather go o d In x we extend the kinemat
the well known ELS mo del Eggen et al However
ical analysis to a more wide ranging investigation of p ossi
available information on the mean rotation v of the bulge
ble kinematic mo dels rotation curves disp ersion pro les
as a function of cylindrical co ordinates R z in the Galaxy
and velocity ellipsoi d orientations and shap es We employ
is confusing b ecause di erent tracers or similar tracers with
a maximum likelihoo d metho d to compare mo dels with the
di erent abundance or disp ersion display markedly di er
data and analyse the data b oth eld by eld and en masse
ent v a compilation of rotation data showing this e ect is
A robust conclusion is that the bulge has a linear rotation
displayed in Figure of Menzies
curve with well determined amplitude over the whole range
The shap e of the bulge velocity ellipsoid has b een con
of our data x takes up this robust determination and
strained only in Baade s Window very near the centre
derives the corresp onding distributi on function of sp eci c
Spaenhauer et al obtained prop er motions for stars
angular momentum for the bulge We compare this to simi
in that region their ndings are consistent with the nuclear
lar distribution functions for other Galactic comp onents the
bulge having nearly isotropic velocity disp ersion
halo the thick disk and the thin disk to show how the bulge
A recent high resolution sp ectral survey of bulge K
is closely related to the halo but not to the disks In x we
giants in Baade s Window McWillia m Rich showed
combine the abundance and kinematic data and use these
that the bulge has mean metallicity Fe H That
to isolate the abundance distribution of the bulge We then
work contradicts earlier ndings e g Rich Frogel
compare this to the closed b ox simple mo del of chemical
Whitford Geisler Friel Tyson which
evolution generating a rather go o d t with plausible param
had deduced that the Baade s Window bulge was extremely
eters In x we compare our bulge with data for external
metal rich with a mean abundance Fe H The ex
galaxies which we are able to do over a similar range of
istence of an abundance gradient in the inner Galaxy is cur
galacto centric distances for the rst time x summarises
rently p o orly constrained partly b ecause there is now reason
and concludes
to b elieve that the Washington photometric system up on
which many of the relevant studies have b een based Geisler
SUMMARY OF THE DATA
Friel Tyson Harding Morrison suf
In an accompanying pap er Ibata Gilmore hence fers from calibration limitations in the relevant abundance
forth Paper I present the results of an analysis of photo range Nonetheless there is suggestive evidence for an abun
The Outer Galactic Bulge II Analysis 3
Contamination in the Metallicity metric and sp ectroscopic data for a large sample of bulge
Distributions giants The stellar data are presented in Table B of Paper
in detail Each star has a p osition a colour a velocity
The stellar sample was divided into into giants and dwarfs
a metallicity estimate a sp ectral classi cation and an es
using a Principal Comp onent Analysis technique Ibata
timate of its luminosity class dwarf or giant A total of
Irwin in preparation This classi cation pro cedure is a
stars were observed sp ectroscopicall y In summary
function of the colour of the star and a co e cient similar
i Five carb on stars were discovered in the sp ectroscopic
to its linestrength The metallicity estimates we obtained cf
survey Four of these stars lie in the elds are
Paper I are the result of an interpolatio n in an empirical
members of the Sagittarius dwarf galaxy and are discussed
correlation b etween colour and linestrength calibrated by
further in Ibata Gilmore Irwin hereafter IGI
data for many standard stars Thus we exp ect our classi
ii Forty two M giants were found in the complete sample
cation and metallicity estimates to b e strongly correlated
of which also lie in the elds Again they are
We note that the metallicity derived for a dwarf is not cal
members of the Sagittarius dwarf and are discussed further
ibrated in this pro ject so that such metallicity estimates
by IGI
do not corresp ond to meaningful abundances In particular
iii Some K dwarfs of the sample were classi
no zero p oint calibration of the dwarf abundance estimates
ed and identi ed b oth by visual insp ection of the sp ectra
has b een attempted
and by use of a new classi catio n technique based on Prin
We examine the shap e of the pseudo metallici ty dis
cipal Comp onent Analysis This technique will b e discussed
tribution of stars classi ed as dwarfs so as to see the way
further elesewhere Ibata Irwin in preparation
that the K giant metallicity distribution s would b e con
iv The stars of direct interest as members of the Galactic
taminated by any residual mis classi cati on The distribu
bulge the K giants numbered making up of
tions are shown in Figure ob jects classi ed as dwarfs are
the observed sample
strongly concentrated in the region Fe H Thus any
errors in luminosity classi cation will preferentially provide
contamination to the K giant metallicity distribution also in
DWARF CONTAMINATION
that range Recall also pap er I that the metallicity distri
Since we wish to determine parametric descriptions of the
butions are reliable only in those regions where a reliable cal
kinematics of the Galactic bulge it is imp ortant that any
ibration to the APM photometry is available that is not
stars in our sample which lie in the foreground disk or the
in the elds at b or b
background Sagittarius dwarf galaxy are reliably identi ed
cf Paper I
For present purp oses we describ e such stars as contami
nants Dwarf giant luminosity classi cation has b een car
ried out for all our stars Giant carb on stars and M giants
MODEL INDEPENDENT DIFFERENCES IN
can b e classi ed very reliably cf Paper I in large part since
THE VELOCITY DISTRIBUTIONS
they are apparently bright and so have high signal noise
Using the non parametric Kolmogorov Smirnov KS test
ratio sp ectra The fainter K dwarfs may however b e less
we compare the observed radial velocity distributions of dif
reliably classi ed Thus we check by testing for their e ect
ferent parts of the Galactic bulge A cumulative probability
on the abundance and kinematic distribution s Before this
distribution is constructed from the radial velocity data set
however we emphasise as discussed in pap er I that the
in each region such that the probabili ty of the lower veloc
number of stars in our sample classi ed as disk K dwarfs
ity b ound is zero and the probability of the upp er b ound is
is in excellent agreement with the number predicted from
unity the value of these b ounds will b e stated later The
our Galaxy mo del There is therefore no reason to exp ect
KS statistic D is found this is simply the maximum value
that any signi cant error in the luminosity classi cation has
of the absolute di erence b etween the two cumulative prob
o ccured
ability distributio ns The KS test allows one to calculate
the probabili ty P D observed that D should b e greater
Contamination in the Velocity Distributions
than the observed value if the distribution s are drawn ran
domly from the same parent p opulation If this probability The reliabili ty of K dwarf giant discriminati on signi cantly
is either very small or very near unity then the distributio ns a ects the conclusions that will b e drawn from this data set
are signi cantly di erent We therefore rst investigate whether the ob jects identi ed
The rst question to b e addressed is is there any ev as K dwarfs have an exp ected velocity signature Any K
idence that the observed bulge region stars are kinemati dwarfs in the sample will b e foreground disk stars some
cally non axisymmetric This question can b e answered di kp c from the Sun cf Paper I They will therefore have
rectly in a mo del indep endent way by comparing pairs of a clear kinematic distributio n characteristic of the old disk
lines of sight which are situated at p oints re ected symmet The observed velocity distribution s of all stars classi ed as
rically ab out the bulge minor axis The regions that we dwarfs are plotted in Figure where we have sup erimp osed
can test in this way are those at b and the exp ected distributio n according to our galaxy mo del All
b Then the ab ove question can b e stated ts are go o d to b etter than the level so we are con dent
as if the map v v is applied to the velocity distribu that the classi cation scheme do es indeed pick out dwarfs
tions at negative longitudes is P D observed such that and do es not select a signi cant number of giants The
the distribution s are signi cantly di erent relationship of disk and bulge kinematics down these lines
The answer to this dep ends on the choice of velocity of sight which determines the e ect of any p ossible residual
b ounds for the KS test Two distribution s are formed by dwarf contamination in the giant sample is discussed later
4 R Ibata and G Gilmore
Figure Comparison b etween the exp ected K dwarf velocity distributions smo oth curves and those for the stars classi ed as K
dwarfs The agreement b etween the data and the mo del prediction lends con dence to the classi cation The lines of sight are a
b b b c b d b e b and f
b
mean of a gaussian distribution of standard deviation selecting stars from the b and
b elds starting from a lower velocity b ound
1
Z
of km s and ending at an upp er velocity b ound v
p
exp t dt P D Observed
We then calculate the probabili ty P D Observed and
x
equate this probabili ty with the probability that an event
should b e found more than x standard deviations from the Thus solving for x we nd the probability P D Observed
in units of this is shown in Figure Clearly up to
v km s the radial velocity distributions in the two
regions are distingui shab le only b elow the level which
The Outer Galactic Bulge II Analysis 5
Figure The pseudo metallicity distribution of those stars classi ed as dwarfs is shown for a b b
b c b d b e b and f b These abundances are neither
calibrated nor reliable Their signi cance is to show that any residual dwarf contamination in the K giant star sample will have greatest
e ect at the high metallicity end of the K giant distribution for Fe H
Galactic bulge the two symmetric elds do not di er sig is not signi cant but b eyond v km s they are sig
ni cantly in their kinematics Figure Thus we conclude ni cantly di erent The statisticall y signi cant
that withing the precision of the present data the bulge is kinematic feature at km s in the elds is due to
kinematicall y symmetric the hitherto unknown Sagittarius dwarf galaxy evident in
The next question to b e asked is is there evidence for the elds Ibata Gilmore Irwin IGI It
a change in kinematics with vertical distance from the is the presence of the Sagittarius dwarf in our survey elds
Galactic plane in the regions investigated This may b e which leads to the need for the relatively complex consid
tested directly with the radial velocity data obtained from eration of velocity b ounds in Figure At velocities b e
the three elds parallel to the Galactic minor axis low km s where the data exclusively represent the
6 R Ibata and G Gilmore
selection functions would give rise to di erent prop ortions of
disk thick disk and halo stars in the two elds and one ob
tains a spurious comparison In section we shall mo del the
kinematic distribution s and amend this de ciency by taking
the selection function into account Since this sophistica
tion in mo delling will come at the cost of introducing many
free parameters into the discussion we b elieve this simplistic
discussion valuable to complement the later complexity
BULGE MODEL STAR COUNTS
The contributions to the total kinematics from the thin disk
thick disk and halo are kept xed with the exception of the
old disk scale height as given by the parameters and rela
tions detailed in our star count mo del of the Galaxy This
mo del will b e describ ed more fully in its present form else
Figure Testing for any p ossible di erence b etween the
where Gilmore Wyse Hewett in preparation but is ba
b and b elds Two cu
sically that describ ed by Gilmore by Gilmore Wyse
mulative distributions are formed b etween km s and an
Kuijken and by Gilmore King van der Kruit
upp er b ound v from the two data sets The KS test statistic D
The imp ortant new features here are the parame
the maximum value of the absolute di erence b etween the two
ters sp eci c to the Galactic bulge which we outline in this
cumulative distributions is found The plot shows the probabil
section
ity P D Observed in units of see the text as a function
The luminosity function LF adopted for the bulge
of the upp er b ound v Note that b elow v km s the two
distributions cannot b e considered di erent The velocities shown
comp onent is taken from the disk LF by Wielen with
are Galacto centric radial velocities this term is explained in
the LF for M giants in Baade s Window by Frogel Whit
Paper I
ford app ended on at the bright end An arbitrary
normalisation is used b ecause the zero p oint of the bulge
b b and b
density law is not known
It is found that constrained by any upp er or lower velocity
The colour magnitude CM relation used to represent
b ounds the distributions are di erent at less than Fit
the bulge is that of M compiled by Chiu This disk
ting a gaussian to the data in the helio centric radial velocity
cluster is of slightly higher metallicity Fe H than
range km s v km s so as to avoid bias
the bulk of our K giant sample see section
ing the t with stars from the Sagittarius dwarf galaxy we
nd that the mean helio centric velocities of the maximum
Star Count Data and the Galaxy Mo del
likelihoo d gaussian ts in the elds at b b
and b are km s km s
We investigate the ability of the Galaxy mo del to pre
and km s resp ectively We therefore conclude
dict stellar number counts given the selection function im
that there is no evidence from the data for vertical varia
p osed on our data Stars were chosen for observation
tions in the kinematics of bulge region stars in the range
cf Paper I from the colour magnitude parameter space
kp c at Note that this do es not z
the exp ected B R distri B R R
J J
necessarily imply that there is no vertical variation in bulge
bution of these stars in the b eld is
kinematics since we have not considered the exp ected kine
shown in Figure where we have sup erimp osed the ob
matics of the disk thick disk and halo p opulations yet
served distribution In this simulation the mo del works
It has b een noted that bulges of external galaxies that
acceptably well in the red b eyond B R where our
J
have so called b oxy isophotes generally display cylindrica l
data lie but gives a p o or t in the blue The steep gradient
rotation Kormendy Illingworth Shaw et al
on the red side of Figure clearly shows that it is necessary
this phenomenon is observed in many low luminosity bulges
to calibrate the zero p oint of the B R photometry to b etter
J
m
Jarvis Kent et al nds that the m Galac
than in order to obtain star counts accurate to b et
tic bulge emission has slightly b oxy contours which has sug
ter than Recall that in the b
gested the p ossibili ty of cylindrica l rotation The recent
b and b elds no reliable
m
DIRBE data however Weiland etal show this shap e
CCD calibration is available Reddening errors of
to b e an artefact of patchy reddening Since b oth the mean
in the Burstein Heiles maps will introduce similar
velocity and the velocity distributions in our elds at ap
errors in the starcounts Since we have neither precise photo
p ear to b e indistin gui shab le over a factor of two in distance
metric data nor accurate reddening determinations in most
from the plane our data are consistent with the bulge b eing
of our elds the data presented here are not well suited for
a cylindrical rotator
the study of stellar number density
It should b e noted that the results presented in this sec
tion should b e considered only as suggestive This is b ecause
Bulge Density Laws
we have not taken the selection function into account in our
Given the assumed luminosity function and colour magni calculations stars in the di erent elds were not selected
tude relation for the bulge a density relation must b e in an identical way Thus even if the kinematics of stars in
chosen to t the starcounts However we show in IGI two elds were intrinsicall y identical the di erences in the
The Outer Galactic Bulge II Analysis 7
The Old Disk Scale Height
The disk comp onents of spiral galaxies show constant verti
cal scale height h indep endent of R see e g the chapters
z
by van der Kruit in Gilmore King van der Kruit
Assuming that the presence of the bulge has no e ect on the
disk scale height in the central regions of the Milky Way one
would exp ect z constant through the Milky Way Then
h
z can b e set by the solar neighbourho o d value for instance
h
Kuijken Gilmore t h p c to lo cal K dwarfs
z
Kent et al on the other hand nd a b est t to m
photometry with a bulge and a disk mo del in which the disk
scale height is constant z p c out to a certain radius
h
R p c but b eyond which it increases linearly out
wards so as to b e h p c at the solar neighbourho o d
z
The obvious interpretation is that the m data are dom
p c inated by the young disk which has a scale height
Figure The B R distribution of stars along the line of sight
J
and not by the old disk We now check to see if a similar
b in the magnitude interval R is
e ect app ears in our data
shown the star symbols represent the exp ected distribution
In the regions investigated the ratio of exp ected num
while dots represent the observed distribution
b er counts b etween a mo del with h p c and one with
z
h p c is this makes the di erence b etween
z
a small disk contribution and one that has in some elds
that the stars with helio centric radial velocities in the range
more than twice the observed number of stars Since we have
v km s in the elds b elong to the Sagit
three elds at we can easily compare the vertical b e
tarius dwarf galaxy a p opulation that is not accounted for
haviour of the disk density at R p c We deduce that
in the galaxy mo del in those elds the t of the mo del
p c then either the adopted Wielen lumi if z
h
to the starcounts will therefore b e p erformed on stars with
nosity function or the colour magnitude relation of M are
helio centric radial velocities v km s With this re
inappropriate approximations for the inner parts of the disk
striction we calculate the total number of stars exp ected
or there are signi cant zero p oint di erences in the adopted
M and observed N in each eld i The zero p oint of the
i i
photometry cf Paper I b etween the elds The problems
adopted bulge density relation is obtained by minimising
P
with such a large scale height are demonstrated in Figure
M N
i i
i
p c it is p ossible to t the For z
h
Kent et al mo deled m S P AC E LAB data
elds to within the starcount error cf section
Kent et al in the regions relevant to this investiga
p c we are unable to t the kinematics of the but if z
tion the b est t density relation they nd is K s
eld at b at b etter that ab out the
this gives b oxy pro jected density contours where K is a
level unless the bulge velocity disp ersion in that eld has
mo di ed Bessel function and s R z The
the unrealisti call y low value of km s when the t
b est t also contains a double exp onential disk with a scale
is acceptable at the level However these problems are
length h p c and a scale height which is a constant
R
resolved if we adopt a scale height z p c and we can
h
h p c from R to R p c but b eyond which it
z
make acceptable ts to b oth the starcounts and kinematics
increases linearly so as to b e h p c at R The prin
z
in all elds
cipal contributors to the m emission are late K and
Thus in this resp ect it app ears that the disk of the
M giants Jones et al so we convert standard K band
Milky Way is not representative of the disk p opulations of
ux into K and M giant number counts using the K band
spiral galaxies The extent to which this is a structural
luminosity function by Jones et al The combined
feature of the Galaxy indicates that our survey is still over
mo del for b oth bulge and disk works acceptably well cf
sensitive to young disk rather than old disk stars or indi
section we nd that the exp ected number counts do
cates limitation s in the calibration of our photometric data
not di er from the observed number counts by more than
remains unclear
in any of the elds This con rms that the data of Kent
et al is not dominated by a few very bright M giants
Sellwoo d Sanders t a maximum disk mo del
COMPARISON TO KENT S ISOTROPIC
to m ballo on data by Matsumoto et al and to
OBLATE ROTATOR BULGE MODEL
m data by Becklin Neugebauer The bulge
density law they nd is r r kp c which
The only kinematic mo del to date that has b een constructed
from a t to the large scale mass structure of the Galactic clearly asymptotes to r We therefore also try an r
bulge and inner Galactic disk as obtained from m emis like mo del then the b est t to K giant number density has
s s where s R z and s p c
sion with an assumption of the mass to light ratio is that
of Kent He assumes that the bulge is an axisym However we nd that as long as the old disk contribu
metric oblate spheroid of constant M L and with isotropic tion to the total K giant number counts is less than
velocity disp ersion so that the distribution function f is a z p c di erent bulge density laws that are able
h
function only of total energy E and the z comp onent of to t the number counts do not signi cantly a ect the
angular momentum L The Jeans equations then take the bulge kinematic parameters derived b elow z
8 R Ibata and G Gilmore
Figure Comparison b etween the observed velocity distributions and the Galaxy mo del where we have adopted a scale height of
z p c for the disk comp onent The lines of sight are a b b b c b
h
d b e b and f b In each panel the dashed single dotted curve represents the halo
the dotted curve represents the thick disk the dashed triple dotted curve represents the bulge and the dashed curve represents the disk
We have normalised the counts so as to t the distribution in f The ts in a and b can b e rejected at ab out the level while c
and d have to o many stars by and can b e rejected at the level
form where R z are cylindrical co ordinates is stellar num
Z
b er density is the Galactic p otential v is the mean ro
1
tational velocity of the bulge and is the bulge velocity
R z dz a
R z
z
z
disp ersion
The forces R and z are computed using a
technique describ ed in Kent from the mass distribu
R
R
b v R z R
tion of the disk and bulge as found by Kent et al
R R
The Outer Galactic Bulge II Analysis 9
fi mg elds containing n stars the statistic s is with the assumption that M L A central M black
i
found from hole is also included in the mo del though is not relevant for
n the present study An algorithm to calculate R and
m
i
X X
G v
j
z was kindly made available to us by Steve Kent pri
ln s
T
i
vate communication The mean velocity and disp ersion
j i
resulting from equation are in go o d agreement with obser
where G is the exp ected line of sight velocity distribution
R
vations for jl j p c and jbj kp c ie interior to almost
T G v dv and v are the velocity values
i j
all our data but has hitherto not b een tested in more distant
The statistic s is minimised by varying the chosen pa
regions from the Galactic centre
rameters in the mo del with an amo eba minimising algo
The velocity distributio n of the bulge p opulation at
rithm from Press et al
each p oint along the line of sight is contructed from the esti
mates of R z and v obtained from equations We use
the Galaxy mo del to integrate disk thick disk and halo con
Bo otstrapping
tributions and add Kent s mo del for the bulge comp onent
With conventional Monte Carlo simulation one estimates
this gives the distributions shown in Figure According to
the con dence limits on the parameters a tted to a data
KS tests all the ts except that in Figure e are unaccept
set by nding new parameters a which are tted to a large
i
able at ab out the level
number i of synthetic data sets The exp ectation is that the
Since the observed distribution s are more p eaked than
probability distribution of a a mirrors a a where
i i true
the ab ove mo del predicts we ask the question are these
a are the parameters one would nd from a hypothet
true
bad ts due to contamination from the lo cal dwarf p opula
ical in nitely precise exp eriment The synthetic data sets
tion Ob jects classi ed as dwarfs contaminate the sample
are created according to one s understanding of the pro cess
mostly at the high metallicity end as we showed in Figure
b eing investigated and the related measurement errors
To investigate this question we apply the ab ove kinematic
However since it is di cult to quantify the errors in the
mo del to only those stars with metalliciti es Fe H
input parameters to our starcount mo del we use instead the
The resulting ts are shown in Figure now all are accept
so called b o otstrap metho d which is a Monte Carlo simu
able at b etter than the level except for the eld at
lation where the synthetic data sets are created by drawing
b which can b e discarded at the
N p oints with replacement from the original data set of
level Thus Kent s mo del do es not work well at Galacto cen
N p oints Taking i synthetic data sets in each of the
tric distances R kp c It also app ears that either the
observed lines of sight we sample the distributio n a a
i true
mo del do es not predict su cient numbers of dwarf stars
from the i maximum likelihoo d ts a For a discussion of
i
the apparent excellent agreement b etween mo del prediction
this technique see Press et al
and classi cation of dwarfs noted in x is chance and that
there is sign cant residual dwarf contamination in the giant
sample or that there is a signi cant dep endence of velocity
The Fits
disp ersion on metallicity in the bulge p opulation these
p ossibili tie s will b e investigated more thoroughly in section We now investigate some p ossible functional forms of R z
and v in the bulge nding parameters that are consistent
b elow
with our data The most likely bulge rotation relations given
the data are dep endent on the bulge velocity disp ersion re
lations and up on the scale height z of the disk comp onent
h
FUNCTIONAL FITS TO BULGE
in the inner regions of the Galaxy We argued ab ove in sec
KINEMATICS
tion our preference for the values z p c Given
h
that this is a ma jor source of uncertainty in the ts b elow
Maximum Likelihoo d
we will present three p ossibili ties with each t N no
When examining the mo dels that can b e allowed or refuted
disk K for z p c as given by Kent et al
h
by the data we x most of the relations in the galaxy mo del
for z p c the largest scale height that app ears to b e
h
We wish to nd the most likely parameters of the remaining
consistent with our data The KS test probabili tie s for all
relations The likeliho o d L H jD of a hypothesis or mo del
the ts discussed b elow are given in table except when
H given data D is de ned to b e prop ortional to the prob
otherwise stated they are all acceptable at b etter than the
ability P D jH of the data given the hypothesis see e g
level
Edwards Maximising the likeliho o d L
Each of the observed velocity distribution s has infor
n
mation on the kinematics of the Galaxy over a large range
Y
of Galacto centric radius Thus the kinematic mo del of the
L P D jH
i
Galaxy should b e t simultaneously to the velocity distribu
i
tions in all of our elds However the maximum likeliho o d
is equivalent to minimising the quantity
ts will naturally b e biased towards nding parameters that
X
t the b etter sampled elds b est We therefore rst in
s ln P D jH
i
vestigate in the next two paragraphs b elow what can b e
i n
said ab out linear and cylindrical rotation in the bulge if
the kinematic distribution s from each eld are t indep en where D fD D D g It is customary to include
n
dently from one another The results of these ts can later the factor in the de nition of s so as to emphasize the
b e compared to those in which the velocity distribution s are analogy with thermo dynamic entropy Given the observed
10 R Ibata and G Gilmore
Figure Comparison b etween the observed velocity distributions and the exp ected distribution according to a mo del which has
standard parameters for the disk thick disk and halo p opulations but whose bulge p opulation is an oblate isotropic rotator as t by
Kent Kent s density mo del for the bulge and thin disk works reasonably well giving number counts that agree with observations
to Note that the total counts in the mo deled distributions b elow have b een normalised to the counts in the data and not
predicted The lines of sight are a b b b c b d b e
b and f b The t in e is acceptable at the level though all the rest are p o or at ab out the
level
s s where s x y z and s p c t simultaneously
cf section The KS test comparisons b etween kine
matic distributio ns from data on opp osite sides of the bulge
strongly supp ort this mo del cf section Oblate bulge
We investigate mo dels in which the bulge has an
a Is the bulge a linear rotator oblate stellar density distribution of the form R z
The Outer Galactic Bulge II Analysis 11
Figure The velocity distributions of K giant stars with metallicities Fe H are compared to a mo del similar to that shown
in Figure but without a lo cal dwarf p opulation The lines of sight are a b b b c
b d b e b and f b All ts except that in a are acceptable at b etter
than the level The t in a can b e discarded at the level This mo del therefore app ears to work well in the central parts
of the Galaxy
velocity v to b e a linear function of Galacto centric radius The velocity distributio n in each of our elds contains
R so to rst order we can attempt to t v with a func information ab out the mean azimuthal velocity and the ve
tion v R R We pro ceed to t the co e cient lo city disp ersion tensor over a large range of helio centric
b b
indep endently for each eld Then if the tted parame distance However the largest contribution to the distribu
ters are the same over all elds to within the measuring tion will come from regions near the tangent p oint to the
b
errors the bulge can indeed b e said to b e a linear rota line of sight in the particular eld under study where the
tor For these ts we select only those stars classi ed as stellar number density is highest In a small distance range
giants with metallicities Fe H this should allevi near the tangent p oint we can exp ect the mean azimuthal
12 R Ibata and G Gilmore
or metallicities for stars in the eld at b ate contamination problems with lo cal K dwarfs see Figure
however comparing the b eld to that at and distant disk giants whose mean metallicity we can
b gives a large vertical baseline which is exp ect to b e approximately solar We also select only the
crucial in answering the cylindrical rotation question Fur elds at b b and
thermore if we accept the preliminary photometric calibra b The elds at b
tions for the b eld cf Paper I and choose and b are not analysed b ecause we wish
only those stars with Fe H we would obtain the to suppress vertical terms in the ts while the eld at
velocity distribution s shown in Figure which have to o few b has a velocity distribution that is sub
stars for reliable mo deling stantially contaminated with stars b elonging to the Sagit
We will t the velocity distributio n of all the stars of tarius dwarf galaxy this is discussed in IGI We will as
one eld at a time nding and for each eld In the sume that the R comp onent of the velocity disp ersion is
b
next section we will attempt to t and simultaneously almost constant in the neighbourho o d of the tangent p oint
b
in all our elds Again we assume that R is almost i e R constant
RR RR
constant in the neighbourho o d of the tangent p oint
a i Assuming that the bulge has isotropic velocity dis
b i Assuming that the velocity ellipsoid is isotropic i e p ersion i e we nd
RR z z
we nd b km s kp c and
RR z z b
N b km s kp c and km s
b
km s b km s kp c and
b
b km s kp c and km s
b
km s b km s kp c and
b
b km s kp c and km s
b
km s
K b km s kp c and a ii Constraining in the ratio of solar
b RR z z
km s neighbourho o d disk disp ersions gives
b km s kp c and b km s kp c and
b b
km s km s
b km s kp c and b km s kp c and
b b
km s km s
L b km s kp c and b km s kp c and
b b
km s km s
b km s kp c and
b
km s a iii Constraining in the ratio of solar
RR z z
b km s kp c and neighbourho o d halo disp ersions gives
b
km s b km s kp c and
b
Thus with the ab ove assumptions the bulge may clearly km s
b e a cylindrical rotator in N In K the mean rotational b km s kp c and
b
velocity in the b eld is lower than that in km s
the elds at b and b by b km s kp c and
b
standard deviations so cylindric al rotation cannot b e km s
ruled out However cylindrical rotation in L can b e ruled Clearly is not very sensitive to the assumed velocity
b
out at the level disp ersion mo del the detailed relation b etween and
b
will b e explored b elow Thus to go o d approximation the
b ii With in the ratio of solar neighbour bulge as sampled with our data set is a linear rotator
RR z z
ho o d disk disp ersions we nd whose rotation rate is km s kp c It is interesting
b
N b km s kp c and to note that extrap olated to the solar neighbourho o d this
b
km s p opulation would have a mean azimuthal velocity identical
b km s kp c and to that of the solar neighbourho o d thin disk though this is
b
km s p erhaps coincidental
b km s kp c and
b
km s
K b km s kp c and b Is the bulge a cylindrical rotator
b
km s Some calculations presented in section suggested that the
b km s kp c and mean rotation of the bulge do es not vary signi cantly with
b
km s height ab ove the Galactic plane We now re investigate this
b km s kp c and p oint using data from the three elds at b
b
km s b and b We would ide
L b km s kp c and ally like to choose as ab ove only those stars with say
b
km s Fe H that way the contribution of the lo cal and
b km s kp c and distant disk p opulation s to any vertical variations in the
b
km s mean velocity would b e reduced Unfortunately we have
b km s kp c and no reliable photometry and hence no reliable classi cati ons b
The Outer Galactic Bulge II Analysis 13
All three ts can b e rejected at the level km s
With the ab ove assumptions cylindrica l rotation is consis
c iii Next we try constraining in the ratio
RR z z
tent at b etter than the level with scenarios N K
of solar neighbourho o d halo disp ersions This
and L
gives
N km s kp c and km s
b RR
b iii While if is in the ratio of solar
RR z z
This t can b e rejected at the level
neighbourho o d halo disp ersions then
K km s kp c and km s
b RR
N b km s kp c and
b
This t can b e rejected just at the level
km s
L km s kp c and km s
b RR
b km s kp c and
b
Thus our data are consistent with the bulge b eing a lin
km s
ear rotator in all the mo dels considered ab ove The bulge
b km s kp c and
b
rotation rate is well determined in all mo dels The velocity
km s
disp ersion in these elds is much lower than that observed
K b km s kp c and
b
in the central regions of the bulge near Baade s window
km s
where the velocity disp ersion of most tracer ob jects is in
b km s kp c and
b
excess of km s see e g the data compiled by Tyson
km s
Tyson s compilation of minor axis observations shows
b km s kp c and
b
a clear decrease in velocity disp ersion with increasing Galac
km s
to centric distance with tailing o to km s at
RR
L b km s kp c and
b
This is consistent with the ab ove ts The veloc jbj
km s
ity disp ersion mo del which has the velocity ellipsoi d in the
b km s kp c and
b
shap e of the solar neighbourho o d disk disp ersions is not con
km s
sistent with the data at the level though it should
b km s kp c and
b
b e noted that the other ts are only marginally b etter see
km s
app endix A
With the ab ove assumptions cylindrica l rotation is consis
tent at b etter than the level with scenarios N K
and L
d isothermal cylindrical rotation
According to the ab ove ts to our data cylindrical ro
Next we demand that the bulge velocity disp ersion is
tation is consistent with our data except if the bulge velocity
not a function of p osition in the Galaxy i e R
ellipsoid is isotropic and the disk scale height is greater than
z and also that the bulge b ehaves as a cylindrical
p c Cylindrical rotation in the bulge will b e discussed
rotator whose mean rotation increases linearly with R over
in more detail b elow where we will mo del the Galaxy less
the region for which we have data i e v R z R
b
simplistical l y dropping the assumption that the observed
velocity distributio n is only due to stars near the tangent
d i Constraining such that the velocity ellipsoi d is ev
p oint
erywhere isotropic i e we nd
RR z z
N km s kp c and km s
b
In the paragraphs b elow we will t mo dels to the ve
K km s kp c and km s
b
lo city distribution s of several elds at a time see the intro
This t can b e rejected just at the level
duction to this section
L km s kp c and km s
b
d ii Constraining in the ratio of solar
RR z z
c isothermal linear rotation curve
neighbourho o d disk disp ersions gives
We demand that the bulge velocity disp ersion is not a
N km s kp c and km s
b RR
function of p osition in the Galaxy i e R z
K km s kp c and km s
b RR
and that it has linear rotation v R R where is
b b
L km s kp c and km s
b RR
constant We select data from only those elds at b
All three ts can b e rejected at the level
so as to suppress vertical dep endence in the mo del ts
d iii Next we try constraining in the
RR z z
ratio of solar neighbourho o d halo disp ersions
c i Constraining such that the velocity ellipsoid is ev
This gives erywhere isotropic i e we nd
RR z z
N km s kp c and km s
N km s kp c and km s
b RR
b
This t can b e rejected just at the level
K km s kp c and km s
b
K km s kp c and km s This t can b e rejected just at the level
b RR
This t can b e rejected just at the level
L km s kp c and km s
b
L km s kp c and km s
This t can b e rejected at the level
b RR
It is clear from the ab ove that the assumption of cylin
drical rotation do es not signi cantly a ect the tted rota c ii Constraining in the ratio of solar
RR z z
tion and velocity disp ersion values listed in sub paragraph neighbourho o d disk disp ersions gives
c ab ove Again the velocity disp ersion mo del which has N km s kp c and km s
b RR
the bulge velocity ellipsoi d in the shap e of the solar neigh K km s kp c and km s
b RR
b ourho o d disk disp ersions is not consistent with the data at L km s kp c and km s
b RR
14 R Ibata and G Gilmore
Can the outer bulge b e triaxial the level The probabili ties of the ts are summarised in
Table Thus our data are just consistent with a mo del Though many studies have found that the bulge is non
in which the bulge velocity disp ersion is either isotropic or axisymmetric cf section it should b e stressed that the
as anisotropic as the solar neighbourho o d halo and where data we have obtained can b e adequately describ ed by sim
the bulge is a rigid cylindric al rotator ple axisymmetric mo dels Perhaps when the velocities of
many more outer bulge K giants are known it may b e
necessary to resort to a less simple explanation Neverthe
e exp onential linear rotation less we investigate whether our data are compatible with
RR
non axisymmetric mo dels Next we ask the question assuming that the bulge is a
v R R can the velocity disp ersion Published non axisymmetric bulge mo dels vary signi linear rotator
b
b e t by an exp onential function cantly according to the bulge tracer they are t to and so R z
RR RR
are not well established To reduce the large numbers of free kp c exp R R The scale length R is a free param
s s
parameters we assume the the bulge has a triaxial form sim eter in the t This b ehavior of the velocity disp ersion with
ilar to that found by Binney et al which was t to the R is observed in thin disk stars see e g Gilmore Wyse
central Galactic bar the axial ratios are b a c a Kuijken So as to eliminate most of the dep endence
and the bar is viewed at an angle of to its ma jor axis of on z in the ts we consider only the four elds at
We adopt the density pro le used by Binney et al b
s s s where s x y z
and s p c We further assume that the closed x e i We constrain such that at a p oint
RR
orbits Contopoulos Mertzanides follow approxi The resulting ts have
z z
mately the iso density surface and that the velocity ellipsoi d N km s kp c R kp c and
b s
is aligned with it such that one diagonali sin g axis u p oints R z kp c km s
RR
out normal to the iso density surface another diagonali si ng K km s kp c R kp c and
b s
axis v p oints along the iso density surface in a plane paral R z kp c km s
RR
lel to the Galactic plane while the third diagonali si ng axis L km s kp c R kp c and
b s
w is p erp endicular to the other two We try mo dels where R z kp c km s
RR
bulge rotation is a linear function of Galacto centric radius
R indep endent of z and bulge geometry e ii We also try such that at a p oint
RR z z
is in the ratio Then
a With the ab ove assumptions we can reject at the N km s kp c R kp c and
b s
level the most likely t to a mo del that has v R R z kp c km s
b RR
and s constant K km s kp c R kp c and
uu v v w w uu b s
R z kp c km s
RR
b Assuming also that v R s constant L km s kp c R kp c and
b uu b s
s constant and s constant we nd R z kp c km s
w w w w RR
N km s kp c This t can b e rejected at the level
b
km s km s and
uu v v w w
km s e iii We constrain such that at a p oint
RR
K km s kp c is in the ratio Then
b z z
km s km s and N km s kp c R kp c and
uu v v w w b s
km s R z kp c km s
RR
L km s kp c K km s kp c R kp c and
b b s
km s km s and R z kp c km s
uu v v w w RR
km s L km s kp c R kp c and
b s
These ts require a very non isotropic velocity ellipsoid R z kp c km s
RR
This would imply a signi cant change in the shap e of the It is interesting to note that esp ecially in the isotropic
velocity ellipsoi d from our innermost eld to Baade s win velocity disp ersion situation the tted radial scale length R
s
dow where the velocity disp ersion is observed to b e isotropic is broadly similar to that of the thin disk and that assumed
Spaenhauer et al This calls the physical plausibi li ty for the thick disk Recall that simple dynamical mo dels for
of this t into question unless the nuclear bulge is unre the disk suggest that the scale length for variation of the
lated to the outer bulge stellar velocity disp ersion is just twice that for variation of
the spatial density Thus with the assumption that R
RR
c We can reject again at the level the most likely is an exp onential function it is p ossible to t the bulge with
t to a mo del that has v R and velocity disp ersion mo dels where the velocity ellipsoid is as
b uu v v w w
s s exp s s where s is a variable anisotropic as that of the solar neighbourho o d disk
s s
uu uu
d An isotropic velocity ellipsoid can b e t to our data if In summary our data set implies a well constrained lin
we free the orientation of the prolate bulge ear rotation curve in the bulge but it allows many bulge ve
N km s kp c lo city disp ersion relations the shap e of the velocity ellipsoi d
b
km s and can b e isotropic or as attened as the solar neighbourho o d
K km s kp c disk and there may or may not b e radial variations in the
b
km s and velocity disp ersion This is discussed further in section
The Outer Galactic Bulge II Analysis 15
the line of sight is m M and the reddening is L km s kp c
b
E By sup erimp osin g CM relations for four km s and
B V
globular clusters of di erent metalliciti es shifted to the tan Though is at o dds with the central bar
gent p oint we illustrate the metallicity range of stars that mo del of Binney et al it is consistent with the triaxial
have b een observed near the bulge The colour magnitude HI mo del of Blitz Sp ergel Thus with several
system B R R was converted into B V V using simplifying assumptions it is p ossible to t our data to bulge
J
the colour equations presented in Paper I The clusters are mo dels with non axisymmetric geometry
from left to right M M M Tucanae and NGC
The last example ab ove demonstrates the problem of
At the tangent p oint in this eld we therefore exp ect to
tting non axisymmetric mo dels to our data the constraints
nd few halo stars signi cantly more metal p o or than M
on the geometry of the bulge from extant studies allow to o
Fe H Djorgovski but more metal rich
great a range of mo dels
p opulations should easily b e detected if present
In a high resolution sp ectroscopic study of bulge K
giants McWilli am Rich nd that Mg and Ti in
SPECIFIC ANGULAR MOMENTUM
their stars are more abundant by dex than in the Sun
DISTRIBUTION OF THE BULGE
over the range of Fe H they observed while Ca and Si
are present in similar ratios to those of disk giants If that
The distribution of sp eci c angular momentum is a primary
deduction is correct we cannot simply infer Fe H from
attribute of a Galactic comp onent see Wyse Gilmore
Mg H as measured by the Mg index de ned in Paper I
To obtain this distributio n one clearly needs the
which was calibrated against lo cal K giants Further data
rotation and density distribution through the comp onent
would b e necessary to establish the bulge Mg Fe gradient
We established in the previous section that the rotation
When the correctness of the bulge element ratio results has
p c at b is R in the bulge from p c
b een established a correction will need to b e applied to the
very well constrained Almost indep endent of the mo del
abundances derived here
assumed for the velocity disp ersion the bulge rotates at
With the ab ove health warning we show the relation
km s kp c Kent gives a spatial density
b
ship b etween metallicity and radial velocity for ob jects clas
mo del of the bulge which is t to m emission for the
si ed as K giants in Figure Recall that reliable cal
outer bulge regions and to the data of Allen et al
ibration of the photometry is not available for the elds
and Becklin Neugebauer for the central regions
b and b so that for these
The mo del has luminosity density such that
elds Fe H is really only a ranking in order of relative
s L p c
abundance Furthermore we show in IGI that the low ve
lo city disp ersion feature with mean velocity km s
for s p c and
seen in Figure d Figure e and Figure f is the Sagit
K s L p c
tarius dwarf an ob ject that is not part of the Milky Way
at least just yet We therefore can con dently analyse the
for s p c In this t s is an oblate cylindrical co ordinate
abundance radial velocity data of Figure a Figure c
such s R z and K is a mo di ed Bessel
Figure d and Figure e and for the latter two elds we
function
select only those stars with Galacto centric radial velocity
This density distribution is integrated through the
v km s we pro ceed to do this in the remainder of
Galaxy to give the mass M out to radius R From M R one
this section
can directly obtain the distributio n of sp eci c angular mo
Recall that of ob jects in our sample classi ed as
mentum M h by substituting for the angular momentum
K dwarfs have formally calcula ted Fe H The tech
h R The distribution of sp eci c angular momentum
b
nique for determining these values was calibrated on K gi
of the bulge is displayed in Figure where we also show the
ants only so for K dwarfs the calculated metallicity val
sp eci c angular momentum distribution of the halo disk
ues will b e very di erent from the actual metallicity values
and thick disk as given by Wyse Gilmore The
In most elds the exp ected number of dwarf stars will b e
bulge under the ab ove assumptions has an even more dis
while mainly photometric errors will induce
sipated sp eci c angular momentum distribution than found
misclassi cati on Given that of ob jects classi ed
by Wyse Gilmore This is of course consistent with
as K giants have Fe H we should exp ect K
the hypothesis that the bulge is the dissipated core of the
dwarf contamination in the K giant sample for stars with
halo but is not consistent with the hypothesis that the bulge
calculated Fe H Thus the observed metallicity dis
is closely related in an evolutionary sense to the thick disk
tributions are unreliabl e for Fe H
or the thin disk
The velocity disp ersion in the four elds at
b b b
and b decreases with increasing metallicity
THE BULGE METALLICITY
We divide the stars into three bins these are de ned as
DISTRIBUTION FUNCTION
i bin Fe H that is a region containing
stars more metal p o or than the mean halo abundance We show a representative colour magnitude CM diagram
Laird et al of the elds studied Figure is the CM diagram in B R
J
o o
ii bin Fe H the region b etween the and R as found for the b eld The star
mean halo abundance and the abundance where we ex symbols represent the stars observed sp ectroscopicall y In
p ect dwarf and distant disk giant contamination to b e this eld the distance mo dulus to the tangent p oint to
16 R Ibata and G Gilmore
Table KS test probabilities for the bulge functional ts for all the mo dels describ ed in the text
Test
a i
a ii
a iii
b i N
b i K
b i L
b ii N
b ii K
b ii L
b iii N
b iii K
b iii L
c i N
c i K
c i L
c ii N
c ii K
c ii L
c iii N
c iii K
c iii L
d i N
d i K
d i L
d ii N
d ii K
d ii L
d iii N
d iii K
d iii L
e i N
e i K
e i L
e ii N
e ii K
e ii L
e iii N
e iii K
e iii L
b N
b K
b L
d N
d K
d L
The Outer Galactic Bulge II Analysis 17
Figure The sp eci c angular momentum distributions for the four ma jor Galactic comp onents The solid curve is the distribution of
the bulge comp onent as given by the rotational velocity found in this work together with a density mo del of Kent The other
curves are taken from Wyse Gilmore the dashed dotted curve represents the halo the dotted curve represents the thick disk
and the dashed curve represents the disk
The velocity disp ersions in Figure a are found to b e
km s km s and km s in the rst second
and third bins resp ectively while in Figure c the velocity
disp ersions are km s km s and km s in
the rst second and third bins resp ectively We do not
calculate similar quantities in Figure d and Figure e
b ecause of the contamination in those elds
We mo del the metallicity distributi on of the halo
as a gaussian of intrinsic width dex centred at
Fe H Laird et al and we assume that the
shap e of the thick disk metallicity distribution towards the
South Galactic Pole Gilmore Wyse Jones is the
same as that in the elds studied here That is we assume
that there are no radial abundance gradients in either the
halo or the thick disk So we p ose the question is it p ossible
to t the observed metallicity distribution s with a comp osite
mo del that consists of the sum of the ab ove halo and thick
disk mo dels together with a simple mo del of chemical evo
Figure B R R colour magnitude diagram of stars mea
J
lution to represent the bulge
o o
sured in a region of sky in the b eld
The star symbols represent the stars observed sp ectroscopically
We rst need to know the relative numbers of halo and
The four colour magnitude relations sup erimp osed are from left
thick disk stars in our sample in table we show the num
to right M M M Tucanae from Sandage and
b er of halo and thick disk stars predicted by the galaxy
M from Chiu all shifted to the distance of the tangent
mo del in the four elds investigated
p oint in this line of sight
The simple or closed b ox mo del of Galactic chemical
come signi cant
evolution is discussed in standard texts and is discussed by
Rich in the context of the Milky Way bulge The iii bin Fe H where we exp ect signi cant
probability density function dN d Fe H it predicts is given dwarf and distant disk giant contamination
18 R Ibata and G Gilmore
Figure The relationship b etween velocity and metallicity for stars classi ed as giants The elds are a b b
b c b d b e b and f b
However it is useful to have a simple parameterisation of the by
data and this sp eci c limitation in application is relatively
dN z ln z
Fe H Fe H
exp
minor compared to the rather extreme nature of some of the
d Fe H y y
physical assumptions underlying the simple mo del
To simulate measuring errors in the ab ove comp osite where y is the yield and z is the solar metal abundance In
mo del we convolve the empirical thick disk distributio n of an evolved p opulation which has turned its gas into stars
Gilmore Wyse Jones with a gaussian of z y the mean abundance equals the yield i e
the di erence of the measuring errors b etween their and Since we are not con dent of the observed distributions
our Fe H measurements while the theoretical bulge and in the region Fe H we will truncate b oth the mo del
halo distribution s are convolved with our metallicity error and the data at Fe H We stress that this cuto
distribution cf Paper I The only free parameter is the will have the e ect of altering the yield of the resulting ts
The Outer Galactic Bulge II Analysis 19
Table The predicted number of halo and thick disk stars with
Fe H according to the Galaxy mo del In the
and elds we have also only selected stars with Galac
to centric radial velocity v km s
Comp onent
Total
Halo
Thick disk
thick disk stars as given in table then t the comp osite yield of the simple mo del which we nd by maximising
mo del to the data in all bins The resulting yield in units the likelihoo d of the t The results are shown in Figure
of z is found to b e y y all these ts are acceptable at b etter than the level The
y and y in the eld at yield y in units of z is found to b e
b b b and in Figure a Figure b Figure
and b resp ectively the distribution s are c and Figure d resp ectively The reader can use the
acceptable at b etter than the level Comparison with agreement or otherwise b etween these derived yields as an
the results ab ove shows that imp osing the abundance cut indication of the ability of the mo del to represent the data
marginally reduces the derived yield as exp ected adequately
The kinematic information can now b e used to check for
In summary the yield found in each eld is dep endent
self consistency in this mo del As usual we will not use the
on the imp osed metallicity cut and slightly dep endent on the
elds at b and b b ecause
assumed normalisatio n of thick disk and halo stars Never
of the contamination from the dwarf galaxy Note that ac
theless it app ears that the bulge comp onent can b e ade
cording to the adopted thick disk metallicity distribution
quately well describ ed for Fe H by the abundance
less than of stars in bin are thick disk stars while
distribution predicted by the simple or closed b ox mo del
thin disk giants comprise less than of the stars in bin
of chemical evolution From tting this simple mo del we
and contribute negligibl y to bin We then t the kinemat
obtain the yield of a simple mo del t to the K giant p opula
ics of bulge stars in bins and separately using our Galaxy
tion in the neighbourho o d of the tangent p oint in each eld
mo del standard kinematic and space density distributions
this is consistent with no metallicity gradient through the
are assumed for the halo and thick disk with normalisations
region in which we have data i e kp c R kp c
in bins and as detailed ab ove while the bulge comp o
Recall also that recent developments McWilliam Rich
nent is mo deled as an oblate rotator with isotopic velocity
have shown that K giants in Baade s window typi
disp ersion throughout Descriptions of these distributions
cally have abundance Fe H this would imply
may b e found in Gilmore King van der Kruit and
a zero metallicity gradient throughout almost the whole of
in Gilmore Wyse Kuijken The bulge is assumed
the bulge
to rotate rigidly at a rate km s kp c as derived
in x which leaves the bulge disp ersion as the only free
parameter in the t
A maximum likeliho o d t to the data in the
A Connection with the Thick Disk
b eld gives in bin and
in bin while a t to the data in the b
Since the thick disk metallicity distribution in Figure
eld gives in bin and in bin We
lo oks similar to that of the bulge we will investigate whether
therefore conclude that either there are signi cantly more
we can t the observed metallicity distribution s with a
halo stars in the eld at b than the stan
mo del similar to that discussed ab ove but whose bulge com
dard halo mo del predicts so as to increase the ratio of halo
p onent has the same metallicity distribution as the thick
to bulge stars in bin or there is a signi cant dep endence
disk
of velocity disp ersion on metallicity in the bulge
We nd a go o d b etter than t by adopting the Alternatively if we demand that there is no change in
Galaxy mo del halo star prediction of halo stars in the bulge disp ersion with metallicity in the b
b eld this is displayed in Figure a eld then there must b e approximately halo stars in the
If the mo del predictions for the number of halo stars in the sample a factor of more than the standard mo del and
other elds are correct then we can reject at the level the the bulge disp ersion b ecomes km s consistent with
hypothesis that the bulge has the same abundance distribu km s If this assumption is true we nd that the
tion as the thick disk in those elds To obtain agreement comp osite metallicity mo del describ ed ab ove which has a
at b etter than the level we require halo stars in the simple mo del with yield y representing the
b eld halo stars in the b bulge comp onent is acceptable at b etter than the level
eld and halo stars in the b eld Recall the t is shown in Figure
that such stars are exp ected These ts are shown in Fig How much are the ab ove ts a ected by our decision
ure to ignore bin data We take the numbers of halo and
20 R Ibata and G Gilmore
Figure The metallicity distributions in a the eld at b in b at b in c at
b and in d at b These are compared to the mo del describ ed in the text in which the bulge
comp onent is represented by a simple closed b ox chemical evolution mo del with a yield tted indep endently to each eld In each
panel the dashed line represents the halo comp onent the dashed dotted line represents the thick disk the dotted line represents the
bulge and the full line is the sum of all three comp onents
attened by anisotropic velocity disp ersions COMPARISON TO EXTERNAL GALAXIES
An interesting feature of the data of KI is the lack of
spatial overlap b etween their data for external bulges with Kormendy Illingworth hereafter KI compared
extant data for the Milky Way The only regions of the the imp ortance of rotational supp ort in spiral galaxy bulges
Milky Way not dominated by the disk where the kinemat to that in elliptic al galaxies The ratio of the energy in or
ics of K giants were well known are the Baade s Window dered motion to that in random motion was estimated from
bulge and the sub dwarf halo in the solar neighbourho o d the ratio of the maximum rotational velocity to the cen
Comparable regions in the KI edge on spiral galaxies were tral velocity disp ersion V V is approximately
m m
either hidden by the galactic disks or were to o faint to b e equal to the mass weighted averaged quantities V
detected KI quote the limiting surface brightness in their in isotropic oblate spheroid mo dels Binney It is the
data as B mag arcsec while Morrison estimates mean stellar rotation that provides supp ort and attening
that the surface brightness of the halo as viewed edge on in these mo dels Similarl y Davies et al a showed that
from outside the Galaxy at the solar radius is V the kinematics of elliptica l galaxies of low absolute luminos
V
mag arcsec The various contributions to the light pro le ity are also well describ ed by such mo dels while intrinsically
observed by KI and the consequences thereof were dis bright ellipti cal galaxies b ear most resemblance to mo dels
cussed in some detail by Shaw Gilmore The essen which are supp orted by random motion ie pressure and
The Outer Galactic Bulge II Analysis 21
for the non cylindri cal rotation case we obtain V
m
KI approximate central disp ersion to an average
out to r where r is the e ective radius of a de Vau
e e
couleurs r law t No K giant data are available
over the same Galacto centric distance range However cen
tral Miras r p c from the Galactic centre have velocity
disp ersion km s Feast et al This value is
compatible with the velocity disp ersion of km s of
Carb on stars at r p c found by Tyson Rich
Thus V if the bulge is a cylindrical rotator
m
or V if it resembles the other mo del ab ove
m
Therefore if the bulge is similar to the oblate rotating
mo dels with isotropic velocity disp ersion Binney to
which KI compare their data see KI their gure
then we nd that the elliptici ty of the Milky Way bulge
should b e This value of bulge elliptici ty is very
similar to the value found by Kent et al
Figure Comparison of the data towards b
with the comp osite mo del describ ed in the text We have as
sumed that there are halo stars in the sample rather than
CONCLUSIONS
the exp ected The dashed line represents the halo comp onent
the dashed dotted line represents the thick disk the dotted line
Our analysis of the data discussed in Paper I has shown that
represents the bulge and the full line is the sum of all three com
the outer regions of the Galactic bulge b etween kp c
p onents
kp c can b e adequately de jz j kp c and R
scrib ed by axisymmetric mo dels with linear rotation
tial p oint is that no data were available to compare available
km s kp c The data also constrain the Galactic disk
to have vertical scale height z km s in the central sp ectroscopic studies of external spiral bulges with the bulge
h
of the Milky way We now have the necessary kinematic data
kp c Since the vertical scale height of the Galactic
to compare the Milky Way bulge to the spiral galaxy bulges
disk in the solar neighbourho o d is z p c Kuijken
h
Gilmore the Galactic disk app ears to b e di erent observed by KI over a similar range of R h and z h
R z
where h and h are characteristic bulge scale lengths in the
from pure disk galaxies where the disk scale height is inde
R z
R z directions We also investigate whether the Milky Way
p endent of galacto centric distance Alternatively our selec
tion function for K giants continues to select preferentially bulge is typical of spiral galaxy bulges Since the bulge spa
tial density pro le is now reasonably well known e g Kent
in favour of the young disk in preference to the old disk
Freeman we also investigate whether the ab ove
The data allow several velocity disp ersion R mo dels If
R is isothermal then the bulge velocity ellipsoid can only mentioned assumption used by KI that V V
m m
holds true in our Galaxy
b e ab out as anisotropic as that of the solar neighbourho o d
halo However if R decreases exp onentially with R the In most of the galaxies they studied KI were not
bulge velocity ellipsoid can b e as anisotropic as that of the able to measure V and directly due to problems with
m
solar neighbourho o d disk More data in several more elds disk contamination but used instead interpolatio ns to es
will b e required to con rm or reject cylindrical rotation and timate those parameters We repro duce their technique on
to constrain b etter the radial dep endence of the shap e size the Milky Way bulge
and orientation of the velocity ellipsoid The simplest al KI calculated V from those bulge regions where
m
lowed mo del which is consistent with this data set do es not V r reaches a well de ned plateau or maximum at radii
require non axisymmetric geometry dominated by bulge light This is the case in our l
We nd acceptable agreement with the axisymmetric b eld where the mean line of sight ve
bulge mo del devised by Kent which uses a Galactic lo city has increased by less than km s over a range of
mass mo del t to m luminosity with the assumption kp c from l b Since the mean rota
M L to predict self consistently the kinematic struc tion in most galaxy bulges decreases with height ab ove the
ture of the bulge Thus the p otential of the inner Galaxy is galactic plane KI extrap olated the V values down to
m
currently reasonably well established V z The extrap olation was p erformed by using the
m
We show that the shap e of the observed abundance dis functional form of V z as obtained from two dimensional
tribution for Fe H where we b elieve are data are velocity maps of elliptical galaxies by Davies and Illingworth
reliable can b e adequately represented by a mo del that con b V z from these elliptical galaxies was scaled by the
sists of the sum of the abundance distributio ns for the halo length z the value of the e ective radius of a Vaucouleurs
e
and thick disk observed in the solar neighbourho o d together r law t to the minor axis pro le We use z p c for
e
with an abundance distribution for the bulge as predicted the Milky Way bulge from Frogel et al Where the
by the simple mo del of chemical evolution The maximum external galaxy bulge displayed cylindrical rotation KI
likelihoo d tted yield in each eld do es not vary signi cantly used V z V However whether or not the
m m
over the Galactic regions for which we have data and is con Milky Way bulge is a cylindrical rotator is not as yet estab
sistent with a zero metallicity gradient in the bulge K giant lished cf section so we will treat b oth cases separately
p opulation from the Baade s Window region to R kp c Cylindrical rotation would then give V km s at
m
Thus the bulge is a well mixed p opulation b while applying the ab ove V z correction
22 R Ibata and G Gilmore
Figure We test the hypothesis that the bulge has a similar metallicity distribution to the empirical thick disk distribution at the
Solar Galacto centric distance determined by Gilmore Wyse Jones The metallicity distributions corresp ond to a the eld at
b in b to b in c b and in d b The number of halo stars
required to make these ts acceptable at b etter than the level are and stars in a b c and d resp ectively while
are exp ected in each eld The dashed line represents the halo comp onent the dotted line represents the bulge and the full line is
the sum of the two comp onents
that the bulge is the dissipated core of the halo The cor The bulge region investigated in this volume covers ap
resp onding distributio ns of sp eci c angular momentum for proximately the same Galacto centric distance range mea
the disks are very di erent Thus our results are not eas sured in bulge scale lengths as did the kinematic study
ily consistent with any mo del in which the bulge is closely of external spiral galaxy bulges presented by Kormendy
related in an evolutionary sense to the Galactic disks Illingworth Our data allow the rst detailed kine
Finally we note that our study is of what we have matic comparison of the Milky Way bulge to the p opulation
termed the Outer Bulge Although this includes most of the of external spiral galaxy bulges We nd that the Galactic
volume of the central Galactic bulge in the sense that this bulge is representative of the p opulation of external spiral
term is applied to external galaxies most recent studies of galaxy bulges
the central Galaxy have b een more concerned with the very With our measured linear rotation curve and adopting
inner parts of the Bulge This part of the Galaxy is nicely the mass mo del derived from the observed near IR surface
seen in recent IRAS and COBE maps but is nearly com brightness of the bulge by Kent we derive the sp eci c
pletely obscured from short wavelength star by star study angular momentum distribution of the bulge This distribu
This very central and very compact structure is the central tion is very similar to though slightly more dissipated than
highest density part of all Galactic structural comp onents that of the halo and so is consistent with the hypothesis
The Outer Galactic Bulge II Analysis 23
It is p ossible that something else also lives there We are Kuijken K Gilmore G MNRAS
Laird J Carney B Rup en M Latham D AJ
con dent that the order of magnitude improvements in spa
Matsumoto T Hayakawa S Koizumi H Murakami H
tial resolution and sensitivity which will b e provided by ISO
in Riegler G Blandford R eds AIP Conf no The
at wavelengths appropriate to study the dominant central
Galactic Centre American Institute of Physics New York
stellar p opulation will provide substantial improvements in
McWilliam A Rich R ApJS
our knowledge of this very central Galactic core In the in
Menzies J in Jarvis B J Terndrup D M eds Pro c
terim we refrain from sp eculation as to the relationship if
ESO CTIO Workshop on Bulges of Galaxies
any b etween that region and the outer bulge which has b een
Morrison H AJ
the sub ject of this investigation
Press W Flannery B Teukolsky S Vetterling W
Numerical Recip es Cambridge
Rich R M ApJ
Rich R AJ
ACKNOWLEDGEMENT
Sandage A ApJ
We thank Steve Kent for providing us with the to olkit to
Sellwoo d J Sanders R MNRAS
facilitate comparison of his mass mo del with our kinematic
Shaw M Gilmore G MNRAS
Shaw M Wilkinson A Carter D A A
data
Spaenhauer A Jones B Whitford A AJ
Tyson N in Warner B ed ASP Conf No Variable
Stars Galaxies San Francisco
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