On the kinematic signature of a central Galactic bar in observed star samples

15 Pages·0.33 MB·English

by  P. Vauterin

On the kinematic signature of a central Galactic bar in observed star
samples
P. Vauterin
Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussel, Belgium
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9 H. Dejonghe
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Universiteit Gent, Sterrenkundig Observatorium, Krijgslaan 281(S9), B–9000 Gent, Belgium
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2 ABSTRACT

1 A quasi self–consistent model for a barred structure in the central regions of our
v
Galaxy is used to calculate the signature of such a triaxial structure on the kinematical
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4 properties of star samples. We argue that, due to the presence of a velocity dispersion,
2 such effects are much harder to detect in the stellar component than in the gas. It
1
might be almost impossible to detect stellar kinematical evidence for a bar using only
0
8 l-v diagrams, if there is no a priori knowledge of the potential. Therefore, we propose
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some test parameters that can easily be applied to observed star samples, and that also
/
h incorporate distances or proper motions. We discus the diagnostic power of these tests
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as a function of the sample size and the bar strength. We conclude that about 1000
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o stars would be necessary to diagnose triaxiality with some statistical confidence.
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: Subject headings: Galaxy: kinematics and dynamics, Galaxy: structure, Galaxy: center,
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Galaxy: stellar content, methods: statistical
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1. Introduction 2. Amodelforthestellarcomponentofabar
in the centre of the Galaxy
Duringthelastyears,therehasbeenmountingevi-
dencethatthecentralpartofourMilkyWaycontains We applya modelwhichwe constructedinanear-
atriaxial,barlikestructure. This theoryissupported lier paper (Vauterin & Dejonghe, 1997; hereafter pa-
by the observation of the “parallellogram” structure per I). It is based on the nonlinear extension of an
in the kinematics of the gas (see e.g. Binney et al, unstablelinearmodeoccurringinaninitiallyaxisym-
1991), and by the asymmetry in both the integrated metric,two–dimensionalmodel. Thegalaxyconsistof
light(e.g. Blitz&Spergel,1991)andthestarsamples two components:
(e.g. Whitelock & Catchpole, 1992).
• Anunperturbed,axisymmetricandtime–independent
However, as pointed out already by de Zeeuw (de
part with a binding potential V and a distri-
Zeeuw, 1994), it is striking that there is currently al- 0
bution function f . The parameters (scale and
most no evidence for such a bar in the stellar kine- 0
rotation curve) are adjusted in order to fit the
matics. Theonlyobservationswhichseemtopointin
centralregionoftheGalaxy. Therotationcurve
that direction are the vertex deviations in the sam-
of this model is displayed in fig. 1, and addi-
ples of K-giants (Zhao, Spergel & Rich, 1994), but
tional details about this model can be found in
these data suffer from large statistical uncertainties.
paper I.
Put differently, triaxial and axisymmetric models fit
almost equally well kinematical data (Blum et al.,
• Abarlikeperturbation,whichisnon–axisymmetric
1995). Of course, this is an unsatisfactory situation,
and time–dependent. The binding potential is
becausethestellarcomponentisthemostmassiveone ′
denoted by V , and the distribution function is
and thus should play a dominant role in the creation ′
f . The time and angular dependency of this
and evolution of such a bar.
part are of the following form:
Therefore,wewilladdresstwoquestionsinthispa-
per: (1) which are the dominant effects caused by a ℜ ei(mθ−ωt) , (1)
triaxialstructureinthekinematicsofstellarsamples, h i
and (2) what is the magnitude of these effects. In where m is the symmetry number, ℜ(ω)/m the
this way, we will try to determine where the triaxial- rotation speed and ℑ(ω) the growth rate.
ity shows up, as well as the required minimal sample
sizeandobservationalaccuracyinorderto be ableto The motion of the stars in the perturbation is a
detect it. solution of the Boltzmann equation:
Inordertoachievethisgoal,weuseaself–consistent ′
∂f
′ ′ ′ ′ ′
model for the stellar component of a barred galaxy ∂t −[f0,V ]=[f0,V ]+[f ,V ]. (2)
which we developed in an earlier paper (Vauterin &
Dejonghe, 1997). The various scale parameters are In addition, for a self–consistent perturbation, the
adjusted in such a way that the model is a fair fit to Poisson equation applies:
the centralregionof ourMilky Way. Since the distri-
′ ′
∇V =4πGρ. (3)
butionfunctionisknownwithhighaccuracy,itiseasy
to draw synthetic stellar samples from this model.
When the perturbation is sufficiently small, one
We discuss the effect of the bar on the l-v dia-
can linearize the equations. In this case, the last,
gram, and further construct various test parameters
quadratic term of the Boltzmann equation is omit-
for triaxiality, based on the kinematical properties of ′ ′
ted. Solutions (V ,f ) that satisfy both the Poisson
individual stars in stellar samples. These tests in-
equation and the linearized Boltzmann equation are
volve the radial velocities, distances and/or proper
called linear modes. There exist various methods in
motions. Using Monte Carlo simulations, we calcu-
the literature to solve this problem (e.g. Kalnajs,
late the distribution of these parameters for different
1977,Hunter,1992,Vauterin&Dejonghe,1996). The
sample sizes, in order to determine minimal sample
analyses show that, for each value of m, an infinite
sizes.
series of linear modes exists, with different rotation
speeds and growth rates. The mode with the high-
est growth rate is the dominant instability. In most
2
cases, it turns out to be an m = 2 barlike structure fore, the l-v diagramalone is usually not sufficient to
(as is the case for the present model). prove triaxiality, and one needs to incorporate other
Byconstruction,linearmodesareonlyvalidforin- information as well. This is in contrast to the situa-
finitesimallysmallamplitudesoftheperturbation. In tion for the gas, where the bar introduces orbits that
order to construct finite amplitude perturbations, we are “forbidden” by an axisymmetric potential. Un-
performed a nonlinear extension. The total potential fortunately, there exist no such forbidden regions for
is taken from the linear mode, V =V +ǫV′, but stars, due to the presence of a velocity dispersion.
TOT 0
the corresponding distribution function is obtained In addition, the features induced by triaxiality are
bysolvingthefull,nonlinearBoltzmannequationnu- muchless pronouncedinthe stellarl–v diagramthan
merically. The details about these calculations are in the gaseous counterpart. Again, this is mainly
given in paper I. due to the important velocity dispersion of the stars,
The total distribution function of such a barred which tends to smooth out the various properties of
model can be accurately calculated for any point in the different orbital families (see also paper I). The
phase space. In addition, it is possible to calculate situationforthe gasismuchsimpler,since,ina good
derived quantities, such as mass densities, streaming approximation, the behaviour of the system at each
velocities, etc., by numerical integration over a grid point is completely determined by the single, closed
in the velocity coordinates orbit passing through that point.
In this way, we constructed a physically meaning-
4. Tests including additional information
ful barred model, which is self-consistent to a large
degree. The bar has a semi–major axis of about 2
As mentioned in the previous section, the galac-
kpc, a semi–minor axis of about 1 kpc, and has a 15
tic length l and the radial velocity v of the stars in
r
degrees tilt angle with respect to the direction of the
a sample are not sufficient to make a clear distinc-
sun. In addition, we have put the sun at a distance
tion between a triaxial and an axisymmetric system.
of 8 kpc from the center of the Galaxy. These values
Therefore, we will investigate two additional types of
seem to be more or less compatible with most of the
information: distances and proper motions. It is well
observations (see e.g. Zhao, 1994).
knownthoughthatsuchinformation,ifpresentatall,
is subject to rather large errors. A test should there-
3. Stellar l–v diagrams of the bar
forenotrelytoocriticallyontheexactvaluesofthese
quantities, but should be rather robust to errors.
In the top panels of fig. 2, we compare the stellar
l-v diagramofthe barredmodelto the one ofthe un- In fig. 3, the streaming velocities of the stars in
perturbed system. Apart from density changes, the a barred system are shown in an exaggerated and
presenceofthebaralsoclearlyhassomeeffectsonthe schematic way. The streamlines have an elliptical
kinematical properties of the l–v diagram. The most form, aligned along the major axis of the system. In
importantdifferenceisthattheaverageradialvelocity addition, the mean velocity is larger for stars that
ofthestarsishigherthantherotationcurveoftheax- crosstheminoraxisthanforthoseonthemajoraxis.
isymmetric potential. This is easily explained by the Itis clearthattests fortriaxialityshouldtakeadvan-
fact that we see the bar almost along the major axis, tage of these properties in some way.
andthatthe starscrossingtheminoraxisaremoving Thetestparameterspresentedinthefollowingsec-
faster than those crossing the major axis of the bar tions are functions of the observable parameters of
(seee.g. fig. 7ofpaperI).Thesamefigurealsoshows a star sample taken from the centre of our Galaxy.
the l–v diagram of the barred model, observed from Therefore,weneedsimulatedstarsamples,inorderto
differentpoints. Thepresenceofabaralsoclearlyin- be able to discuss the properties of these parameters.
troducessubstantialchangesinthe structureofthese Onecaneasilydrawsuchastarsamplefromthestel-
diagrams. lar distribution function using a try–and–rejecttech-
However, if the potential is not known a priori, it nique. In this approach,a random number generator
is not possible to easily conclude that these l–v di- creates star coordinates (positions ~r and velocities~v,
agrams are caused by a barred system. They could both in rectangular reference systems) that have a
verywellbecompatiblewithanaxisymmetricsystem rectangular distribution over the whole area of inter-
that has a steeper or a slower rotation curve. There- est. Foreachstar,anadditionalrandomnumberRis
3
further generated, having a rectangular distribution Because we take only ratios of stars in one give
between 0 and the maximum value reached by dis- distance class, the parameter is insensitive to photo-
tribution function (for our models, this corresponds metric effects (caused by the fact that there is a bias
to the central value). The coordinates (~r,~v) are ac- against seeing stars at the far side of the bar), and
ceptedasanewmemberofthestarsampleonlyifthe is thus mostly determined by the kinematical prop-
valueofRissmallerthatthevalueofthedistribution erties of the sample. It only requires crude distance
functionatthatpoint; itisrejectedintheothercase. information, which is a prerequisite condition since
This process is continued until the sample contains distances are usually subject to large errors.
the desired number of stars. Itisimportanttonoticethatlargenumbersofstars
are summed in each subclass, resulting in reduced
4.1. Distance information and radial veloci-
noise effects. In addition, it is a so–called ”robust”
ties
test parameter,because only numbers ofstars arein-
volvedinstead of measuredvalues. Such a parameter
As shown in fig. 3, the line of zero mean line–
of–sight (LOS) velocity in a barred system with the is relatively insensitive to outliers.
adoptedorientationisingeneralnotalignedanymore
4.2. Proper motions
with the line l = 0 (as is the case for an axisymmet-
ric system). As a consequence, there is a symmetry A second test only involves proper motions. It is
breakdown in the l-v diagram: the part of the dia- based on the fact that, in a barred system which is
gram that contains stars that are closer to the sun not aligned with the line of sight, the mean proper
than the galactic centre is not identical anymore to motions for positive and negative galactic longitudes
the part that contains the stars lying farther away. have opposite sign (see also fig. 3 and fig. 5). This
Inthissection,wewillexploitthisasymmetry,and asymmetry is quantified using a second test parame-
use it as a basis for a test parameter for the triaxial- ter, which measures the difference in proper motion
ity ofthe system. Tothis end, we subdivide the stars (PM) at positive an negative galactic longitudes.
into eight different groups, labelled (xxx), where x
can be − or + (see also fig. 4). The first sign in- T2 =hPMl>0i−hPMl<0i. (5)
dicates the distance from the sun ( − means closer
than the galactic centre and + means further away), Unfortunately, this parameter is not robust with
thesecondsignisrelatedtothe galacticlongitude(− respect to errors in the observed proper motions, so
for negative and + for positive longitudes), and the it might be less useful in practical circumstances.
last sign is determined by the radial velocity (− for
4.3. Proper motions and radial velocities
negative values and + for positive values).
The number of stars Nxxx in each subclass is used It also follows from fig. 3 that, in a barred system
in order to construct a test parameter: that is not aligned with the direction of the sun, the
line of zero LOS velocity is not equal to the l = 0
line (as is the case for axisymmetric systems). This
T = 1 N−−+ / N−−− + N++− / N+++ ×100. phenomenon causes differences in the partial l-v dia-
1
2(cid:18)N−+− / N−++ N+−+ / N+−−(cid:19) gramscontainingstarswithrespectivelyonlypositive
(4)
and negative proper motions. In order to quantify
The denominators of both fractions contain the this difference, two subset are drawn from the star
fraction of counterrotating stars in the minor axis sample: (1) stars havingpositive proper motions and
quadrants,atthe nearside(firstfraction),andatthe lying in the upper 25% of the proper motion distri-
farside(secondfraction). Ontheotherhand,thenu- bution, and (2) stars having negative proper motions
meratorscontainthefractionofcounterrotatingstars and lying in the lowest 25%. We rejected the mid-
inthe majoraxisquadrantsinthe correspondingdis- dle 50%, because this is the “gray area”, where large
tances classes. Therefore, this parameter essentially measurementerrorscauseuncertaintiesonthesignof
expresses the fact that the quadrants containing the the proper motion. Fig. 6 displays the l-v diagrams
major axis of the bar (i.e. N−+x and N+−x) contain of both subsets, and shows that the first subset has,
more counterrotating stars than the other quadrants on average, larger radial velocities than the second.
(N++x and N−−x).
4
This difference is measuredby the third test parame- 4.5.1. Verification of a hypothesis using observed
ter,definedasthe differencebetweenthemeanradial data
velocities of both subsets:
The combination of part [2] and [3] of the his-
togram (part [2] is the overlap between both distri-
T =hv i−hv i. (6)
3 r, pm>75% r, pm<25% butions) givesthe probabilitydistributionfor the pa-
Again, this test parameter turns out to be fairly rameters in the case of an axisymmetric model. If
insensitive to photometric effects. It is also robust one has calculated a test parameter using actual ob-
with respect to errors on the measurement of proper servations of our Milky Way, the corresponding dis-
motions, since it depends only on their sign rather tribution can be used to check whether the value is
than on their exact values. The fact that it is not consistent with an axisymmetric model or not, given
robust with respect to the radial velocities is not a a specified level of confidence.
disadvantage, because these values can be measured On the other hand, part [1] and [2] correspond to
with rather high accuracy. the probability distribution for the parameters of the
As one can infer from fig. 3, the symmetry break- barred model. Again, when actual observations are
down of the radial motions and the proper motions present, this curve can be used to check the consis-
turn out to work in the same direction for the value tency with this model.
of T . Obviously, this effect has a positive influence
3
4.5.2. Determinationoftheresolvingpowerofatest
on the discriminating power of this parameter.
parameter
4.4. Monte Carlo simulations of the test pa-
One can use the information in fig. 7 to estimate
rameters
the diagnostic power of a particular test for a given
WeusedaMonte-Carlosimulationtechniquetonu- sample size. Let assume that, in reality, the centre
merically calculate the statistical distributions of the of our Milky Way is barred and more or less consis-
testparametersfora givengalaxymodel. A largeset tent with our model. Suppose further that one wants
of randomstar samples, consistent with the distribu- to be able to reject the null hypothesis (“an axisym-
tion function of the model, is constructed, and the metric system”) with a specified confidence level α
values of the various test parameters are calculated (e.g. 95%). To this end, anyobservedtest parameter
for each individual sample. The parameter values of has to lie outside the 1−α region of the probability
all samples are further binned into a histogram in curveoftheaxisymmetricmodel[2]+[3]. Sinceweas-
order to calculate a numerical estimate of the distri- sumed that the galaxy is compatible with the barred
butions of these parameters. Obviously, one should model,weknowapriorithatthisparameterobeysthe
incorporate a sufficient number of synthetical sam- distribution [1]+[2]. Using this information, one can
ples in order to obtain a histogram with enough res- calculate the probability that the parameter indeed
olution. We use histograms with 10 intervals, and a lies outside this region.
simulationsetcontaining100samplesforeachmodel. In fig. 8, the probabilities for rejection of the null
We have checkedthat this number is sufficient to ob- hypothesis are shown for several confidence levels, as
tain a reasonable degree of accuracy by checking the afunctionofthestrengthofthebar(avalueof1cor-
consistency of the results with estimations based on responds to the model described in section 2). These
larger simulation sets. results are calculated using Gaussian fits to the dis-
tributions of the test parameters. The estimates for
4.5. Probability distribution functions of the barstrengthsotherthan1arebasedonalinearinter-
test parameters polation of the parameter distributions between the
axisymmetricandthebarredmodel. Thisapproxima-
We calculated the distribution function of the pa-
tionisjustifiedbythefactthatourmodelistoahigh
rameters T , T and T for two different galaxy
1 2 3
degreelinear,andthatthetestparametersofbothex-
models: an axisymmetric one and a barred system
trememodelshavemoreorlessthesamedistribution.
with parametersadjusted to those of our Milky Way.
Of course, one should realize that these numbers are
These distribution functions werecalculatedfor sam-
based on a limited number Monte-Carlo simulations,
plescontaining700and1400stars. Theresultingdis-
and these values should therefore be considered only
tributions are shown in fig. 7.
5
as rough estimates. inthiscase,oneneedsalargesamplesize(oftheorder
Thethirdtestparameterclearlyturnsouttobethe of 1000 stars) in order to have a good chance to be
most discriminating one. Presumably, this is a con- able to rule out an axisymmetric model with a high
sequence of the fact that the deviation of the proper degree of confidence.
motions and the radial velocities happen to work in
REFERENCES
the samedirection. Forsamplescontaining700stars,
it is the only test that offers a reasonable chance to
Binney, J.J., Gerhard, O.E., Stark, A.A., Bally, J.,
ruleoutanaxisymmetricmodelatahighlevelofcon-
Uchida, K.I., 1991, MNRAS, 252, 210
fidence. The first test turns out to be less sensitive,
but still offers a reasonable discriminating power for Blitz, L., Spergel, D.N., 1991, ApJ, 379, 631
samples containing (at least) 1400 stars. The sec-
Blum, R. D., Carr, J. S., Sellgren, K., Terndrup, D.
ondtest, whichonlyinvolvespropermotions,hasthe
M., 1995,ApJ, 449, 623
poorest score.
These probability numbers indicate the a priori Hunter C., 1992, in Hunter J.H., Wilson R.E., eds,
chancethataparticulartypeofobservationswilllead Astrophysical disks, New York Academy of Sci-
to positive results. In this way, they are very useful ences
for the estimation of the required sample size.
Kalnajs A. J., 1977,ApJ, 212, 637
5. Conclusions
Vauterin P., Dejonghe H., 1996,A & A, 313, 465
Althoughmanyobservationspointinthe direction
Vauterin P., Dejonghe H., 1997,MNRAS, 286, 812
of a triaxial structure in the central region of our
Galaxy, there has been so far very little evidence for Whitelock, P.A., Catchpole, R.M., 1992, in The cen-
this inthe kinematicalpropertiesofthe stars. Insec- ter,BulgeandDiskofthe MilkyWay,ed.Blitz,L.
tion 3, we have shown that, to a large degree, this Kluwer, Dordrecht, p. 103
can be explained by the presence of a velocity dis-
de Zeeuw T., 1993,in Galactic Bulges,eds.Dejonghe
persion in the motion of the stars. Such a dispersion
H., Habing H.J. Kluwer, Dordrecht, p. 191
diminishes the effects of the bar on the distribution
function, by “smearing them out” over a large por-
Zhao, H., 1994,PhD Thesis, Columbia University
tion of phase space. As a consequence, it turns out
to be very hard to prove triaxiality from an l-v di- Zhao H., Spergel D. M., Rich R. M., 1994, AJ, 106,
agram alone, even if a very large number of stars is 2154
involved, unless one has some additional information
concerning the potential, obtained in an other way.
In the remainder of the article, we have shown
that if distances and/or proper motions of the stars
are known,it is possible to construct test parameters
which discriminate between axisymmetric and triax-
ial models. Because distances and proper motions
are usually subject to large errors, the tests were de-
signed to be “robust” with respect to these values.
Moreover, the use of only first order moments has
a distinct advantage over second order moments (on
which e.g. vertex deviation is based), because higher
order moments are less well constrained by the data.
Monte-Carlosimulationswerefurtherusedinorderto
estimatetheprobabilitydistributionfunctionofthese
testparametersfordifferentmodels. Itturnsoutthat
the most powerful test incorporatesa combination of
radial velocities and proper motions. However, even This 2-column preprint was prepared with the AAS LATEX
macrosv4.0.
6
Fig. 1.— Rotation curve, mean velocity and disper-
sions of the axisymmetric model.
Fig. 2.— l–v diagrams of the barred model (top
left) and the unperturbed system (top right). The
axisymmetricrotationcurveisindicatedbythewhite
line. Themassdistributionandtheorientationofthe
observerofthe systemis showninthe smallpictures.
The bottom row shows the same bar with different
orientations.
Fig. 3.— Schematic view of the streaming velocities
in a barred stellar system. Note the counterrotating
streamlinesthatarepresentinthequadrantscontain-
ing the majoraxis. This is incontrastto axisymmet-
ric systems, where counterrotating stars are present
only because of velocity dispersion.
Fig. 4.— The l-v diagrams of a simulated star sam-
ple, drawn from the barred model, for stars further
away than the galactic centre (top panel), and stars
closertothesun(bottompanel). Thegraycurvesrep-
resent the axisymmetric rotation curve, and in each
quadrantofthe graphs,the correspondingsubsample
index is marked (see text). The star sample contains
700 stars.
Fig. 5.— Proper motions of a simulated star sample
(drawn from the barred model) as a function of the
galactic length. The gray line indicates the averaged
values.
Fig. 6.— Thel-vdiagramsofasimulatedstarsample
for the upper 75% subset (top panel) and the lower
25% (bottom panel). The sample contains 700 stars,
and is drawn from the barred model. The gray lines
indicate the axisymmetric rotation curve.
Fig. 7.— Distribution functions of the test parame-
ters for different models.
Fig. 8.— Probabilities for rejection of the axisym-
metric hypothesis for the three test parameters, at
different levels of confidence, and for various bar
strengths (expressed as a fraction of the strength of
the bar in our model).
7
Rotation curve
200.0(cid:10)
Mean velocity
)
s
m/
k
(
100.0(cid:10) y
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Radial disp. ci
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Azimuthal disp.
0.0
0.0 1.0 2.0 3.0
Radius (kpc)
8
4 4
0 0
-4 -4
-4 0 4 kpc -4 0 4 kpc
250 250
m/s) m/s)
velocity (k0 velocity (k0
-250 -250
5 0 -5 5 0 -5
l (degrees) l (degrees)
4 4
0 0
-4 -4
-4 0 4 kpc -4 0 4 kpc
250 250
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-250 -250
5 0 -5 5 0 -5
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9
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10