1. INTRODUCTION - uni-bonn.desuyu/Papers/AndromedaModels.pdf · barlike triaxial ellipsoida l bulge. Beyo nd r # 20 kpc , meas-urements wer e obtaine d for spira larm segme nts on

DISK-BULGE-HALO MODELS FOR THE ANDROMEDA GALAXY

LawrenceM.Widrow, KathrynM. Perrett,1 and Sherry H. Suyu2

Department of Physics, Queen’s University, Kingston, ONK7L 3N6, Canada; [email protected], [email protected] 2002 November 6; accepted 2003 January 7

ABSTRACT

We present a suite of semianalytic disk-bulge-halo models for the Andromeda galaxy (M31) that satisfythree fundamental conditions: (1) internal self-consistency, (2) consistency with observational data, and (3)stability of the disk against the formation of a central bar. The models are chosen from a set first constructedby Kuijken & Dubinski. We develop an algorithm to search the parameter space for this set in order to bestmatch observations of the M31 rotation curve, inner velocity dispersion profile, and surface brightnessprofile. Models are obtained for a large range of bulge and disk masses; we find that the disk mass must bed8! 1010 M" and that the preferred value for the bulge mass is 2:5! 1010 M". N-body simulations are car-ried out to test the stability of our models against the formation of a bar within the disk. We also calculate thebaryon fraction and halo concentration parameter for a subset of our models and show that the results areconsistent with the predictions from cosmological theories of structure formation. In addition, we describehow gravitational microlensing surveys and dynamical studies of globular clusters and satellites can furtherconstrain the models.Subject headings: cosmology: miscellaneous — galaxies: individual (M31) — galaxies: spiral —

galaxies: structure — gravitational lensing — methods:N-body simulations

1. INTRODUCTION

Owing to its size, proximity (distance#770 kpc), and longhistory of observation, the Andromeda galaxy (M31) o!ersa unique opportunity to study in detail the components of alarge spiral (Sb) galaxy. The aim of this paper is to presentnew disk-bulge-halo models for M31 that are (1) internallyself-consistent, (2) consistent with published observationswithin 30 kpc, and (3) stable against the rapid growth ofbarlike modes in the disk. Our models are drawn from ageneral class of three-component models constructed byKuijken & Dubinski (1995, hereafter KD), chosen to fitavailable observations of the M31 rotation curve, surfacebrightness, and bulge velocity profiles. The models span aparameter space defined by the disk and bulge masses, flat-tening parameter, and tidal radius of the galactic halo.

One advantage of the KD models is that they provide thefull phase-space distribution function (DF) for each of thecomponents. The DFs are simple functions of three inte-grals of motion: the energy, the angular momentum, and anapproximate integral that describes the vertical motions ofparticles in the disk. This feature insures that the models are(very nearly) in dynamical equilibrium. The KD models areeach specified by 15 parameters. Of these, 10 have a directe!ect on the rotation curve, velocity dispersion, and surfacebrightness profile of the model galaxy. A search of thisparameter space yields a set of models that fit the observa-tional data to within quoted uncertainties.

The data considered in this paper constrain the inner#30kpc of M31 but say little about the mass distribution atlarger radii. For this, one must turn to dynamical tracerssuch as dwarf satellites (Mateo 1998; Evans et al. 2003) andglobular clusters (Perrett et al. 2002). Recently, Evans &Wilkinson (2000) derived an estimate for the total mass of

M31 based on observations of satellites and outer halo glob-ular clusters. Their analysis assumed simple analytic formsfor the gravitational potential of M31 and the DFs of thetracer populations. The models and methods presented inour paper may provide the basis for future investigationsalong similar lines and, in particular, a unified and self-consistent treatment of the full data set associated withM31.

Two gravitational microlensing surveys toward M31 arecurrently underway (Crotts & Tomaney 1996; Crotts et al.2001; Kerins et al. 2001; Calchi Novati et al. 2002). Similarsurveys toward the Magellanic Clouds have so far yieldedinconclusive results. Roughly 20 microlensing events havebeen observed toward the LMC and SMC (Alcock et al.2000; Lasserre et al. 2000), but one cannot say whether thelenses responsible for these events are indeed MACHOs orsimply stars within the Magellanic Clouds or Milky Waydisk. The M31 microlensing experiments should be able toresolve this question. M31 is highly inclined, and thereforelines of sight toward the far side of its disk pass throughmore of its halo than do lines of sight toward the near side.If the halo of M31 is composed (at least in part) ofMACHOs, there will be more microlensing events occurringtoward the far side of the galaxy (Crotts 1992). Previouse!orts to compute theoretical optical depth and event ratemaps for M31 assumed ad hoc models for the disk, bulge,and halo. Our models represent a significant improve-ment over these models since they are both internally self-consistent and consistent with published data on M31. Wewill describe how one can compute optical depth and eventrate maps for the models constructed in this study.

This paper takes the following form. In x 2 we review pub-lished observations of the rotation curve, velocity dispersionprofile, and surface brightness profile ofM31. Section 3 pro-vides a summary of the main features of the KD models. Inx 4 we describe a method to search parameter space formodels that best fit the observations. The results of thissearch are presented in x 5. Promising models are found thatspan a wide range in disk mass and in halo tidal radius and

1 Current address: Department of Astronomy and Astrophysics, Uni-versity of Toronto, 60 St. George Street, Toronto, ONM55 348, Canada.

2 Current address: Department of Physics, California Institute of Tech-nology, Pasadena, CA 91125.

The Astrophysical Journal, 588:311–325, 2003May 1# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

311

shape. We compute various properties of these models suchas the disk and bulge mass-to-light ratios and the baryonmass fraction. For a subset of the models, we calculate themass distribution and line-of-sight velocity dispersionprofiles. We also fit the density profile to the NFW profile(Navarro, Frenk, & White 1996) and thus obtain an esti-mate for the halo concentration parameter. In x 6 we des-cribe a technique to construct theoretical event-rate mapsand present one example. Our conclusions and a discussionof possible extensions of this work are presented in x 7.

2. OBSERVATIONS

Our analysis utilizes published measurements of the gal-axy’s rotation curve, average surface brightness profile, andbulge velocity profiles. There is a considerable amount ofM31 observational data of dissimilar quality available fromthe literature. In this section, we briefly describe the dataselected for use in fitting the KDmodels.

2.1. Rotation Curve

Rotation curves for the M31 disk have been obtainedfrom optical and radio observations spanning variousranges in galactocentric radius (e.g., Rubin & Ford 1970,1971; Gottesman & Davies 1970; Deharveng & Pellet 1975;Kent 1989; Braun 1991). An optimal combination of suchdata sets requires a good understanding of any associatedcalibration errors and uncertainties. In this study, the com-posite rotation curve for the galaxywas obtained by combin-ing velocity data from the studies of Kent (1989) and Braun(1991). Kent (1989) obtained velocities with estimated statis-tical errors of #6 km s$1 for 30 H ii emission regions alongthe major axis of the galaxy, with galactocentric radii in therange of 6–25 kpc. The Braun (1991) measurements ofneutral hydrogen within the gaseous disk of M31 yieldedvelocitymeasurements out to a radius of r # 30 kpc. Braun’sdata within 2 kpc of the M31 center were neglected herebecause of possible distortions arising from the presence of abarlike triaxial ellipsoidal bulge. Beyond r # 20 kpc, meas-urements were obtained for spiral arm segments on only oneside of the galaxy; hence data in this region were also notincorporated into our fitting. Details of the rotation curveinterpolation can be found in Braun (1991).

Both sets of rotation velocity measurements and theirrespective error bars are shown in the upper panel ofFigure 1. Kernel smoothing was used to form the compositedisk rotation curve, which is shown with its rms uncertain-ties in the lower panel of Figure 1.

In determining the rotation curve of the galaxy, Braunassumes circular gas motions within the disk. It should benoted that the presence of certain dynamical anomalies maylead to systematic errors in the rotation curve determina-tion. Such anomalies will be discussed at the end of x 3.

2.2. Surface Brightness Profile

Although there have been many optical photometricstudies of M31 (e.g., Table 4.1 of Walterbos & Kennicutt1987), the task of combining such data sets is not straight-forward. Di!erences in filter bandpasses and resolutionbetween the various studies yield systematic errors that aregenerally di"cult to characterize. Discrepancies in theadopted galaxy inclination and isophote orientation alsocontribute to di!erences in the light profiles obtained by dif-

ferent authors. For these reasons, we opted to avoid com-bining di!erent data sets and instead adopted the surfacebrightness profile from Walterbos & Kennicutt (1987).These authors produced a global light profile for M31 outto r # 28 kpc by averaging the distribution of galaxy lightover elliptical rings, assuming an inclination of 77%. Theinner parts of the galaxy are dominated by light from thebulge component, which itself has a significantly higherposition angle than that of the disk: P:A: # 50% versus 38%

(Hodge &Kennicutt 1982).There is a distinct variation in the position angle of ellipti-

cal isophotes fitted to the surface brightness of M31 as onelooks out in galactocentric radius (Hodge & Kennicutt1982). Furthermore, there is a significant warp in the diskbeyond r # 22 kpc (Walterbos & Kennicutt 1987). Thesestructural features cannot be reproduced in the KD modelsand will thus contribute to fitting errors calculated for thesurface brightness profiles.

2.3. Bulge Velocity Profiles

The dynamics of the bulge can be used to deduce the massdistribution in that component of the galaxy. We utilize thestellar rotation and velocity dispersion results of McElroy(1983) along the bulge major axis (P:A: & 45%) andminor axis (P:A: & 135%) in the fits to the KD modelsdescribed later in x 3. The bulge data were smoothed usingthe same kernel-averaging technique mentioned above; therotation and dispersion results are shown in Figures 2 and3, respectively.

McElroy (1983) noted several asymmetries in the rotationcurves along various position angles of the bulge for jRj <100, although the cause of these asymmetries was unclear.The KD models are unable to reproduce such asymmetrieswithin the bulge; we will return to this point later in the nextsection.

Fig. 1.—Disk rotation curve. The rotation measurements of Kent (1989)[K89]) and Braun (1991 [B91]) are shown in the upper panel, and thesmoothed rotation profile used in the model fitting is provided in the lowerpanel.

312 WIDROW, PERRETT, & SUYU Vol. 588

3. SELF-CONSISTENT DISK-BULGE-HALO MODELS

Kuijken & Dubinski (1995) constructed a set of semi-analytic models for the phase-space DFs of disk galaxies.Their models have three components: a thin disk, a centrallyconcentrated bulge, and an extended halo. In this section wesummarize the essential features of these models.

The halo DF is taken to be a lowered Evans model(Kuijken & Dubinski 1994). Evans models (Evans 1993) areexact, two-integral distribution functions for the flattenedlogarithmic potential, ! / log x2 ' y2 ' z2=q2( ), where q isthe flattening parameter. As with the isothermal sphere,Evans models are infinite in extent. In analogy with the low-ered isothermal sphere or King model (King 1966), loweredEvans models introduce a truncation in energy such that thesystem that results has finite mass. The halo DF,

fhalo E;Lz( ) &AL2

z ' B( ) exp $E=!20

! "'C

# $

! exp $E=!20

! "$ 1

# $if E < 0 ;

0 otherwise ;

8><

>:

(1)

has five free parameters: A, B, C, !0, and !0, the centralpotential for the system. (!0 is implicit in the definition ofenergy E.) Following KD, the first three parameters arereplaced by q, a characteristic halo radius Ra, and a core

Fig. 2.—Bulge rotation velocity as a function of radius. The upper panelshows the bulge rotation measurements of McElroy (1983) for the north-east (NE) and southwest (SW) sides of the minor axis along an axis with aposition angle of 45%. The smoothed bulge rotation curve data and errorsused in the model fitting are shown in the lower panel.

Fig. 3.—Velocity dispersion measurements of McElroy (1983) along the major axis (upper left-hand panel) and minor axis (upper right-hand panel ) of thebulge. The corresponding smoothed profiles used in the model fitting are shown in the lower panels. The four outer points in theminor-axis data were neglectedbecause of their large uncertainties.

No. 1, 2003 ANDROMEDA DISK-BULGE-HALO MODELS 313

radius Rc. In practice, the latter is described in terms of acore smoothing parameter Rc=RK( )2, which specifies theratio of the core radius to the derived King radius for thehalo. TheDF is independent of the sign ofLz andmay there-fore be written as the sum of two components, one with pos-itive Lz and the other with negative Lz. Rotation can beincorporated into the model by varying the relative‘‘ weight ’’ of these two parts as specified through thestreaming fraction Sh (=

12 for no rotation).

The bulge DF is given by the lowered isothermal sphereor King model (King 1966; Binney & Tremaine 1987) andtakes the form

fbulge E( ) &"b 2#!2

b

! "$3=2exp !0 $!c( )=!2

b

# $

! $ exp E $!c( )=!2b

# $$ 1

% &if E < !c ;

0 otherwise ;

8><

>:

(2)

where "b and !b govern the central density and velocity dis-persion of the bulge. The cuto! potential, !c, controls theextent of the bulge. As with the halo, there is an additionalparameter, the bulge streaming fraction Sb, which controlsthe rotation of the bulge.

The disk DF depends on three integrals of motion: E, Lz,and an approximate third integral corresponding to theenergy associated with vertical oscillations of stars in thedisk. The disk DF can be chosen to yield a density field withthe desired characteristics. In the KD models, the densityfalls o! approximately as an exponential in the radial direc-tion and as sech2 in the vertical direction. Five parametersare used to characterize the disk: its mass Md, the radialscale length Rd, the scale height hd, the disk truncationradius Ro, and the parameter $Ro, which governs the sharp-ness of the truncation.

In a given potential, the DF for each component impliesa unique density field. For self-consistency, the density-potential pair must satisfy the Poisson equation. This isaccomplished by an iterative scheme as described byKuijken & Dubinski (1995). The essential point is that theproperties of the density fields of the bulge and halo areimplicit rather than explicit functions of the input parame-ters. In particular, the masses of the bulge and halo (Mb andMh, respectively) and the halo tidal radius (Rt) are deter-mined a posteriori. Likewise, the shapes of the bulge andhalo are determined via the Poisson-solving algorithm. Inparticular, the bulge is flattened by the contribution to thepotential from the disk.

By design, the KD models are axisymmetric, and thereinlies the main limitation to the development of a truly realis-tic model for M31. Observational studies indicate that M31cannot be described adequately by an axisymmetric model.Dynamical anomalies for M31 include factors such as thepresence of a triaxial bulge (Stark 1977; Gerhard 1986), per-turbations caused by its companions M32 and NGC 205(Byrd 1978; Sato & Sawa 1986), the e!ects of significant var-iations in the inclination of the galactic plane as a functionof radius (Hodge & Kennicutt 1982), areas of infall motiontoward the galaxy center (Cram, Roberts & Whitehurst1980), and local anomalies attributable to fine structure andshocks within the spiral arms of the galaxy (Braun 1991).Small local perturbations in spectra obtained through dustlanes in the central regions of the galaxy may also be afactor. These dust patches would have the e!ect of slightly

increasing the local radial velocity measurements, therebyinducing local errors in the rotation curve of the bulge(McElroy 1983). Furthermore, the bulge dispersion may bea!ected by residual rotation caused by disk contaminationalong its minor axis.

4. FITTING KD MODELS TO THE OBSERVATIONS

The KD models are specified by 15 parameters: four forthe bulge, five for the disk, and six for the halo. Our goalis to determine the parameter set that yields a model thatbest fits the observations. This is accomplished by mini-mizing the composite %2 statistic that is calculated bycomparing the model rotation curve, surface brightnessprofile, and bulge velocity profiles with the data sets asdescribed in x 2.

4.1. Minimization Strategy

Unfortunately, the data considered in this paper are notsu"cient to determine a global minimum in the full 15-dimensional parameter space.We therefore adopt a strategyin which a best-fit model is found for targeted values of thedisk and bulge masses, flattening parameter, and in somecases, tidal radius. In addition, we do not attempt to mini-mize over the disk thickness, disk truncation radius, or halostreaming fraction (i.e., the parameters Ro, $Ro, hd, or Sh).Our analysis proceeds as follows:

1. The surface brightness profile at radii where the diskdominates is very nearly exponential. In the R band, thescale radius of the disk is 5.4 kpc. We therefore fix Rd to thisvalue at the outset.2. Since the data used in our analysis have little to say

about the parametersRo, $Ro, and hd, we set them to be typi-cal values forM31 of 40 kpc, 1 kpc, and 300 pc, respectively.In addition, we assume that the halo does not rotate, i.e.,Sh & 0:5.3. We consider models with disk and bulge masses in the

following ranges:

3! 1010 M" * Md * 16! 1010 M" ;

1! 1010 M" * Mb * 4! 1010 M" :

Here Md is an input parameter, but Mb is a complicatedfunction of "b, !b, !c and, to a lesser extent, the otherparameters. Therefore, to force the minimization routineto select a model with the desired Mb, we minimize a‘‘ pseudo-%2 statistic ’’ that includes the term +Mb desired( )$Mb model( ),2=!2

Mb. The user-specified parameter !Mb con-

trols the accuracy with which Mb is fitted to the desiredvalue. Of course, the physically relevant quantity is the %2

statistic associated with the fits to the observational data,and this is the one quoted herein.4. As with Mb, the tidal radius of the halo is an implicit

function of the other input parameters. Therefore, for thosecases where we wish to specify the tidal radius, an additionalterm Rt desired( ) $ Rt model( )+ ,2=!2

Rtis included as part of

the pseudo-%2 statistic in order to drive Rt to the desiredvalue. Again, the user specifies the value of !Rt .5. The flattening parameter q governs the shape of the

halo potential and is specified explicitly. The shape of themass distribution of the halo is determined implicitlythrough the Poisson-solving algorithm.


A given model is constructed by minimizing the pseudo-%2

statistic for the fit. This minimization procedure is describedin the next section.

4.2. MultidimensionalMinimization Technique

Minimization of the pseudo-%2 statistic over the multi-dimensional KD parameter space is performed by employingthe downhill simplex algorithm (see, e.g., Press et al. 1986).AnN-dimensional simplex is a geometrical object consistingof N ' 1 points or vertices and all of the line segments thatconnect them. Thus, a simplex encloses a finite volume in anN-dimensional space. For the case at hand, the space isdefined by the 10 free parameters as described above. An ini-tial guess at the values of these parameters fixes one vertexof the initial simplex. The remaining vertices of the initialsimplex are constructed by stepping in each direction ofparameter space by some appropriate distance, which istypically set to 10% of the parameter value.

The downhill simplex algorithm proceeds through a seriesof iterations as follows. The pseudo-%2 statistic is calculatedat each vertex of the simplex. The algorithm then reflects thevertex with the highest value of pseudo-%2 through theopposite face of the simplex to search for a lower functionvalue. If a lower value is found at this new location, the algo-rithm proceeds by testing the point twice as far along thisline. The vertex in the original simplex with the highestpseudo-%2 is then replaced by the reflected position with thelower function value. If a lower value is not found at thereflected position, the simplex is contracted about its vertexwith the lowest function value. In this manner, the simplexsteps through parameter space and gradually contractsaround the point with the minimum pseudo-%2, therebyhoning in on the best fit to the observed data and thetargeted values for the component masses and/or tidalradius.

The downhill simplex method has a number of advan-tages over minimization procedures that are based on gra-dients of the function (e.g., the method of steepest descent;see Press et al. 1986 and references therein). Gradientmethods appear to be more susceptible to the presence oflocal minima in the complicated %2 surface. In addition, thesimplex method is computationally e"cient, requiring rela-tively few function evaluations. Between 10 and 100 itera-tions are needed in order to locate a position in parameterspace with the minimum value of pseudo-%2. An iterationrequires between 1 and 10 function evaluations, each ofwhich takes a minute or so of CPU time. Therefore, a modelcan typically be generated within 1–2 CPU-hours using a1 GHz desktop computer.

5. THE MODELS

We begin with a survey of models in the Md-Mb plane,focusing on the quality of the fits to the observational data,the stability of the disk, and the mass-to-light ratios of thedisk and bulge components. We next consider constraintson the mass distribution beyond 30 kpc from dynamicalstudies of satellite galaxies, globular clusters, and planetarynebulae. We also check whether our preferred models areconsistent with predictions from the baryonic Tully-Fisherrelation and cosmological constraints such as the baryonfraction. Toward this end we construct a series of modelswith tidal radii between 80 and 160 kpc and also explore theimplications of replacing the lowered Evans halo of a KDgalaxy, which has a sharp cuto! in density at the tidalradius, with an NFW halo. Finally, we investigate modelswith flattened halos such that q < 1.

5.1. The Disk and BulgeMassModels

We have constructed and analyzed over 20 M31 modelsin the Mb-Md plane with q & 1 and Rt unconstrained.Table 1 summarizes the results for a select subset of thesemodels. In addition to the disk and bulge masses, Table 1provides the composite %2 statistic for the fits to the obser-vational data (col. [4]), the R-band mass-to-light ratios forthe disk and bulge (cols. [5] and [6]), the mass interior to asphere 30 kpc in radius (col. [7]), the mass of the halo(col. [8]), and the tidal radius (col. [9]).

Our analysis indicates that there is a trough in %2 runningroughly parallel to theMd-axis and centered onMb ’ 2:5!1010 M". Both models A andD lie near the minimum of thistrough, while models A, B, and C trace out the cross sectionof the trough at Md & 7! 1010 M". Models along theminimum of the trough have low values of %2 (typically%2 ’ 0:6 1), indicating an excellent overall fit to the obser-vations. This point is illustrated in Figures 4 and 5, wherewe compare the theoretical predictions for models A and Dwith the observational data. The agreement is particularlygood for the gas rotation curve and inner velocity dispersionprofiles along the galaxy’s major and minor axes. Further-more, the exponential disk does an excellent job of fittingthe surface brightness profile beyond 5 kpc, although the fitis not as good in the transition zone between bulge- anddisk-dominated regions of the galaxy (2 kpcdrd4 kpc).Moreover, the inner rotation curve for the model appears tohave the wrong shape: the model curves are relatively flatwhile the data suggest a rotation curve that is rising slowly.These two discrepancies may, in part, reflect the fact thatour models assume an axisymmetric bulge, whereas the

TABLE 1

Models with q & 1 and Rt Unconstrained

Model(1)

Md

(1010M")(2)

Mb

(1010M")(3)

%2

(4)M=LR( )d

(5)M=LR( )b

(6)

M30

(1010M")(7)

Mh

(1010M")(8)

Rt

(kpc)(9)

A........... 7 2.5 0.70 4.4 2.7 36 32 80B ........... 7 1 2.29 4.5 1.3 38 27 38C........... 7 4 1.55 4.4 4.1 29 33 129D .......... 14 2 0.63 8.6 2.4 32 32 201K1......... 16 4 1.75 9.9 4.8 28 6.5 137K2......... 3 4 1.28 2.0 3.2 50 120 155


actual bulge of M31 is triaxial. We note that the mass distri-bution in the bulge component of the model is flattened—but still axisymmetric—as a result of its interaction with thegravitational potential of the disk. The major-to-minor axisratio is found to be #0.8, in good agreement with the valuefound by Kent (1989).

Figures 4 and 5 help to explain the existence of the troughin %2. The contributions to the outer rotation curve (re5kpc) from the halo and disk are both rather broad in radius,and therefore one can be played o! the other.

Figure 5 shows that the disk dominates the rotation curveof model D for 4 kpcdrd30 kpc. This feature, common toall high-Md models, reveals a fundamental problem withthese models, namely, a susceptibility to the bar instability.A dynamically cold, self-gravitating disk is unstable to barformation (Hohl 1971), and since a strong bar instabilitycompletely disrupts the disk, any model in which one ispresent is unacceptable.

In general, bar instabilities can be suppressed by anextended halo, a bulge that dominates the dynamics of the

Fig. 4.—Comparison of theoretical fits with observational data for model A from Table 1. The upper left-hand panel shows the net rotation curve for thegalaxy (solid line), along with the profiles for the individual components: bulge (dotted line), disk (dashed line), and halo (long-dashed line). The data points anderror bars are those of the smoothed rotation profile given in Fig. 1. The lower left-hand panel shows the surface brightness profile measurements with themodel fits of the bulge (dotted line), disk (dashed line), and total light (solid line). The upper right-hand panel provides the bulge velocity profile data andmodelfit. Themiddle and lower right-hand panels show the inner velocity dispersion data and resulting fits along the galaxy’s major andminor axes, respectively.

Fig. 5.—Comparison of theoretical fits with observational data for model D (see Table 1). The plots and line types are as described previously in Fig. 4.


inner part of the galaxy, significant vertical velocities amongthe disk stars, or a combination thereof (Ostriker & Peebles1973; Sellwood 1985). We have performed a series ofN-body experiments to test the stability of our models. Weuse the algorithm of Dehnen (2002), which has the advant-age over tree and mesh codes in that the computation timescales as the number of simulation particles, N, rather thanN ln(N). The simulations were performed with 2! 105 par-ticles for each of the three components of the model andwere run for 8 dynamical times as measured at 1 scaleradius. Model D develops a bar within a few dynamicaltimes, as expected, given that the disk dominates the gravi-tational potential within a radius of r # 30 kpc (see Fig. 5).Model A, in which the inner rotation curve is dominated bythe bulge and the outer rotation curve is dominated by thehalo (Fig. 4), appears to be stable. These results are illus-trated in Figure 6 where we show face-on and Milky Wayobserver views of the evolved disk-particle distribution formodels A and D. Further simulations suggest that thedemarcation between stable and unstable models occurs inthe neighborhood ofMd ’ 8! 1010 M". Models with a diskin this mass range show signs of weak spiral and barlikestructures (Stark 1977). The presence of spiral structurein the disk of M31, as well as a triaxial barlike bulge, sug-gests that weak instabilities operate in this galaxy. Thus,Md & 8! 1010 M" should not be interpreted as a strictupper bound on the disk mass ofM31 but rather as an inter-esting region of parameter space in which dynamical evolu-

tion may give rise to nonaxisymmetric structures similar towhat is observed in M31. What we can say for sure is thatmodels like D and K1 are violently unstable and thereforeruled out. Conversely, model K2 may be so stable thatthe mechanisms that drive spiral structure are suppressed(Sellwood 1985).

Model K1 assumes values forMd andMb from the popu-lar small-bulge model of Kent (1989). This model does a rea-sonable job of fitting the surface brightness profile as well asthe rotation curve beyond 6 kpc (Fig. 7). However, modelK1 predicts a velocity dispersion in the bulge region that istoo large. This discrepancy is common among all modelswith Mb ’ 4! 1010 M" (e.g., models C and K2). It isalready evident in Figure 3 of Kent (1989) if one focuses onthe McElroy (1983) data at 1–2 kpc. The discrepancy isworse for the self-consistent models considered here. Kentassumed a constant density halo: when compared with arealistic model halo (i.e., one in which the density is monot-onically falling with radius), the contribution to the rotationcurve from a constant density halo is relatively low at smallradii. For a fixed halo contribution to v2circ at 30 kpc, Kent’smodel underestimates the halo contribution in the region ofthe bulge as compared with more realistic models.

Model K2 assumes values for Md and Mb used in thepixel-lensing study from Kerins et al. (2001). This modelprovides an excellent fit to the surface brightness profile,matching almost perfectly the data through the transitionzone between bulge and disk dominated regions of the

Fig. 6.—Results ofN-body simulations for models A andD after several dynamical times. Upper panels provide face-on views of the disk for model A (left)andmodel D (right). Lower panels show the corresponding ‘‘ observer’s view ’’ with the disk tilted to an inclination angle 77%.


galaxy (Fig. 8). However, the model rotation curve appearsto have the wrong shape between 7 and 12 kpc and perhapsalso beyond 20 kpc. In addition, the model velocity disper-sion profile is systematically high, as discussed above.

5.2. Mass-to-Light Ratios

We now consider the disk and bulge mass-to-light ratiofor the models in Table 1 (see cols. [5] and [6]). The (M=LR)values for model K1 are in excellent agreement with theresults obtained by Kent (1989), who found M=Lr( )d & 10and M=Lr( )b & 5. This agreement serves as a consistencycheck of our method, although it is worth noting that Kentuses the r-band filter of the Thuan & Gunn system, which

di!ers slightly from the R-band filter of the standardUBVRI system used by Walterbos & Kennicutt (1987) (seeTable 2.1 of Binney &Merrifield [1998] for a comparison ofthe di!erent filter characteristics).

As expected, the disk and bulgeM/L values for the othermodels in Table 1 scale roughly with the mass of the corre-sponding component. In particular, the M/L values formodel A are approximately a factor of 2 smaller than thosefor model K1. The values for model A compare well withthe predictions from stellar synthesis studies. Bell & de Jong(2001) presented a correlation between the stellar mass-to-light ratios and the optical colors of integrated stellar popu-lations for spiral galaxies. Based on the relationships they

Fig. 7.—Comparison of theoretical fits with observational data for model K1 (Kent 1989; see Table 1). The plots and line types are as described previouslyin Fig. 4.

Fig. 8.—Comparison of theoretical fits with observational data for model K2 (Kerins et al. 2001; see Table 1). The plots and line types are as describedpreviously in Fig. 4.


obtained under di!erent assumptions of initial mass func-tion combined with typical M31 disk and bulge colors fromTable 1 of Walterbos & Kennicutt (1988), one expectsM=LR # 2 5 within the di!erent regions ofM31.

Despite this general agreement between the model mass-to-light ratios and the predictions from stellar populationstudies, the question arises as to why, for all of the models ofTable 1 except K2, M=LR( )d is significantly greater thanM=LR( )b. In general, one expects the mass-to-light ratio ofthe bulge to be comparable to, if not greater than, that ofthe disk (although see Heraudeau & Simien 1997). We pro-pose two explanations for this apparent di"culty. First, thequotedM/LR values presented in this paper do not incorpo-rate corrections for foreground extinction. Since the e!ectsof obscuration by dust are likely to be more severe in M31’sdisk than in its bulge (Walterbos & Kennicutt 1988), thiscorrection would drive down theM=L of the disk relative tothat of the bulge and bring the two values more into line(Kent 1989). Moreover, for typical Sb spirals like M31, gascontributes between 2% and 20% of the mass of the disk (seeFig. 8.20 of Binney &Merrifield 1998, and therefore boosts,by the same fraction, theM=L value relative to the value forthe stellar population.We therefore conclude that the valuesfor M=L( )d and M=L( )b are quite reasonable.

It is nevertheless instructive to consider a second possibil-ity, namely, that the M=L values obtained for model Areflect a genuine problem that is perhaps connected with thepoor fit to the surface brightness profile in the disk-bulgetransition region (Fig. 4). Deviations of the structure ofM31’s bulge from a simple oblate spheroid may also haveresulted in an underestimate of its e!ective radius in the fit-ting procedure, causing the bulge to get somewhat short-changed in mass. One might then argue that model K2provides a superior fit to M31. However, the gains in thesurface brightness profile and in the mass-to-light ratiosmade by employing model K2 come at the cost of the qual-ity of the rotation curve fit. Model K2 also has the unusualfeature that the bulge mass is greater than the disk mass,contrary to what is commonly found for bright Sb galaxies.

5.3. Mass Distribution

The total mass interior to a sphere 30 kpc in radius, M30,is provided in column (7) of Table 1. These values wereobtained by generating anN-body realization of each modeland tabulating the mass interior to the prescribed sphere.The mass M30 is constrained almost entirely by the outerrotation curve: a circular rotation speed at 30 kpc ofvr ’ 215 km s$1 (an average of the outer four points in Fig.1) corresponds to a mass estimate v2r r=G & 32! 1010 M".This is in good agreement with the values obtained for mostof our models. The most prominent outlier is model K2,where the model rotation curve rises to vr ’ 265 km s$1

(v2r r=G ’ 50! 1010 M"), in conflict with the observations.

Columns (8) and (9) of Table 1 provide the halo mass andtidal radius, respectively. Since the data considered in thispaper probe only the inner 30 kpc of M31, it is not surpris-ing that Mh and Rt vary considerably. Dynamical tracerssuch as globular clusters and dwarf galaxy satellites havebeen used to constrain the mass distribution of M31 beyondr & 30 kpc. To better understand how this type of datamight help constrain the models, we construct a sequence ofmodels with 80 kpcdRtd160 kpc, q & 1, and Md and Mbfixed to the values from model A. The results for model A(Rt ’ 80 kpc) andmodel E (Rt ’ 160 kpc) are given in Table2 for comparison. In addition to %2, Rt, and Mh, we recordthe baryon fraction, fB (col. [8]). We also provide four quan-tities derived by substituting an NFW halo for the loweredEvans model. These quantities, to be discussed in x 5.4, arethe halo concentration parameter c (col. [5]), the virialradius R200 and mass M200 (cols. [6] and [7]), and an alter-nate estimate for the baryon fraction based on the NFWhalo, fB;NFW (col. [9]).

The value of %2 is found to rise rapidly with increasing Rt.The main source of the discrepancy is with the rotationcurve fits, as illustrated in Figure 9. Models with Rt > 160kpc lead to even larger discrepancies between the predictedand observed rotation curves and thus provide completelyunacceptable fits to the data.

Dynamical studies of globular clusters as well as plane-tary nebulae are especially useful for obtaining mass

Fig. 9.—Comparison of the rotation curves for models A and E (seeTable 2).

TABLE 2

Models withMd & 7! 1010 M",Mb & 2:5! 1010 M", and q & 1

Model(1)

Rt

(kpc)(2)

%2

(3)

Mh

(1010M")(4)

c(5)

R200

(kpc)(6)

M200

(1010M")(7)

fB(8)

fB;NFW

(9)

A........... 80 0.70 32 11.5 224 130 0.23 0.07E ........... 160 1.30 67 7.9 193 81 0.13 0.10


estimates at intermediate radii. Federici et al. (1993) ana-lyzed spectroscopic observations for several dozen globularclusters in M31 between 10 and 30 kpc and derived a rangefor M30 of (50 80) ! 1010 M". Using the projected massestimator of Bahcall & Tremaine (1981) and Heisler,Tremaine, & Bahcall (1985), Perrett et al. (2002) obtained amass estimate of #40! 1010 M" based on data from 319globular clusters out to r ’ 27 kpc and under the assump-tion of isotropic orbits. As with most mass estimates thatare based on dynamical tracers, the uncertainty is duelargely to our ignorance of the true orbit distribution for theglobular cluster system.

Evans & Wilkinson (2000) considered a sample of M31globular cluster candidates and planetary nebulae at largegalactocentric radii and employed a more sophisticatedmethod for estimating the mass of the galaxy. They assumedsimple analytic forms for the globular cluster DF and forthe halo-density-profile/gravitational-potential pair. Twoparameters characterize their model potential, and thesewere determined by performing a maximum likelihoodanalysis over the data. Evans & Wilkinson (2000) obtaineda mass estimate at r & 40 kpc of 47! 1010 M" for the globu-lar cluster data, while for the planetary nebulae data theyfoundM30 ’ 28! 1010 M".

At present, virtually all of the information available forthe outer halo of M31 comes from dynamical studies ofits satellite galaxies. Courteau & van den Bergh (1999)used radial velocity data for Local Groupmembers to calcu-late a dynamical mass of the Andromeda subgroup of(133- 18) ! 1010 M". Using data from high-resolutionechelle spectroscopy from the Keck Telescope, Cote et al.(2000) applied the projected mass estimator to derive anM31 mass of #79! 1010 M" under the assumption of iso-tropic satellite orbits. In the cases of circular and radialorbits, the estimated enclosed masses change to #37! 1010

and #215! 1010 M", respectively. Based on the dynamicalmodeling of a similar set of data, Evans et al. (2000)obtained an M31 mass in the range of (70 100) ! 1010 M".For comparison, Evans & Wilkinson (2000) derived anestimate for the total mass of M31 within #550 kpc of123'180

$60 ! 1010 M". This result was obtained by applyingthe maximum likelihood DF method described above to thecombined globular cluster, planetary nebula, and satellitedata set.

In Figure 10 we plot the total mass distribution as a func-tion of radius for the models listed in Table 2. The dynami-cal mass estimates described above are included forcomparison. Based on these results, it would appear thatboth models A and E provide adequate descriptions of theouter halo, with model E coming closest to the estimates.

In a forthcoming publication, we will study in detail theconstraints on our models from observations of dynamicaltracers. Typically, for each member of the tracer popula-tion, one knows its angular position and line-of-sight veloc-ity. As an illustration of how one might use such data,Figure 11 shows the line-of-sight velocity dispersion as afunction of projected radius for models A and E. The veloc-ity dispersion along a particular line of sight as a function ofthe projected position vector s is given by

!2LOS s( ) & 1

" s( )

Z 1

l0

dl" l; s( )hv2l l; s( )i ; (3)

where l0 corresponds to the position of the observer and

" s( ) &R1l0

dl" l; s( ) is the surface density along the line ofsight. The density " l; s( ) and velocity dispersion hv2l l; s( )i arecalculated from the DF in the usual way:

" l; s( ) &Z

d3vf l; s( ) ; (4)

Fig. 10.—Mass distribution for models of Table 2. Plotted is the totalmass interior to the radius r for model A (solid line), model E (dotted line),and modified model A with an NFW halo (dashed line). Also shown arepublished mass estimates from studies of dynamical tracers: globular clus-ter data from Perrett et al. (2002, open triangle), Evans & Wilkinson (2000;open circle), and Federici et al. (1993; open square); satellite data from Coteet al. (2000; filled triangle), Evans et al. (2000; filled pentagon), Evans &Wilkinson (2000; filled circle), and Courteau & van den Bergh (1999; filledsquare); planetary nebula data from Evans &Wilkinson (2000, cross).

Fig. 11.—Line-of-sight velocity dispersion as a function of projectedradius s. The solid line refers to model A; the dotted line refers to model E.


and

" l; s( )hv2l l; s( )i &Z

d3vv2l f l; s( ) : (5)

Here vl is the component of the velocity along the line ofsight.

KD provide a simple algorithm that allows one to gener-ate an N-body representation for any of their models. AnN-body representation can be used to perform a MonteCarlo evaluation of the integrals in equations (3), (4), and(5). The DF is written as a sum over the particles:

f x; v( ) & mi

X

i

$3 x$ xi( )$3 v$ vi( ) ; (6)

where mi, xi, and vi are the mass, position, and velocity ofthe ith particle. The surface density is given by

" s( ) & AX

i2Vmi ; (7)

where V is a volume corresponding to a thin tube centeredon the line of sight and A is the cross-sectional area of thetubes. Likewise,

!2LOS s( ) & 1

" s( )AX

i2Vmiv

2s;i : (8)

The curves in Figure 11 represent an average over positionangle: di!erences between the line-of-sight dispersionprofiles along the major and minor axes were found to beinsignificant. Our results may be compared with those fromthe analytic halo-only model of Evans et al. (2000; see theirFig. 3). More to the point, the model prediction for the line-of-sight velocity dispersion profile together with data fordynamical tracers can be incorporated into an improvedversion of our model-finding algorithm.

As a final consistency check, we turn to the Tully-Fisherrelation (Tully & Fisher 1977), which, in its original form,describes a tight correlation between the total luminosity ofa spiral galaxy and vflat, the circular rotation speed in the flatpart of the rotation curve. For our purposes, a variantknown as the baryonic Tully-Fisher relation (see McGaughet al. 2000), which described a correlation between vflat andthe total baryonic mass in gas and stars, Mbaryons, is moreuseful. The baryonic Tully-Fisher relation takes the form

Mbaryons & Avbflat ; (9)

where A and b are constants. McGaugh et al. (2000) findthat b is statistically indistinguishable from 4 (however, seeBell & de Jong 2001) and that if it is fixed to this value, thenormalization is A ’ 20 h$2 M" km$4 s4, where h is theHubble constant in units of 100 km s$1 Mpc$1. Note thatthere are large uncertainties inA (see, e.g., McGaugh [2001],where the acceptable range for A is given as 34–85 in unitsofM" km$4 s4. For M31, the circular rotation speed in theflat part of the rotation curve is vflat ’ 230 km s$1. Assum-ing h & 0:75 and A ’ 35 M" km$4 s4, this value of vflatimplies a baryonic mass of 9:8! 1010 M" in good agree-ment with the total baryon mass (taken to be Md 'Mb inmodel A). Note that somewhat higher values of Md can betolerated without introducing too strong a bar instability.Thus, one would be able to find a consistent model even if ahigher value ofA is assumed.

5.4. Connection with Cosmology

Theories of structure formation within the context of thestandard # CDM cosmology (cold dark matter plus a cos-mological constant) are now sophisticated enough to makepredictions for the density profiles and masses of dark mat-ter halos. Indeed, one may use such predictions as the basisof halo models for specific galaxies such as M31. This wasthe approach of Klypin, Zhao, & Somerville (2002) whoconstructed models ofM31 based on a theory of disk forma-tion that assumes adiabatic compression of the baryonswithin a cold dark matter halo. It is worth noting that theirfavored models have disk and bulge masses similar to thoseof our model A.

In this section, we take a di!erent approach and calculatecosmological diagnostics for various KD models and com-pare these to predictions from structure formation theories.The results will help further constrain the models and pro-vide the basis for further study.

We first consider the baryon fraction. In general, weexpect the baryon fraction of a spiral galaxy such as M31 tobe comparable to the baryon fraction of the universe. Underthe assumption that dark matter makes a negligible contri-bution to the mass of the disk and bulge, we havefB * Md 'Mb( )= Md 'Mb 'Mh( ), where fB is the baryonfraction of the galaxy as a whole. The inequality incorpo-rates the possibility that some baryons may reside in thehalo. Values for fB are given in column (8) of Table 2 andcan be compared with estimates from cosmology and astro-physics. Based on big bang nucleosynthesis constraints,Burles, Nollett, & Turner (2001) derived an estimate of$Bh2 & 0:020- 0:002, where $B is the density of baryons inunits of the critical density. Similar results have been ob-tained from analyses of microwave background anisotropymeasurements. For example, de Bernardis et al. (2002)found $Bh2 & 0:022'0:004

$0:003 for data from the BOOMER-ANG experiment. With values of h & 0:7- 0:07 and$matter & 0:31- 0:13 (the latter is also from de Bernardis etal. 2002), one finds fB & 0:13- 0:06, the baryon fraction ofthe universe. This result is consistent with estimates of thegas fraction in X-ray clusters (Armaud & Evrard 1999).

Figures 9 and 10 and the results for the baryon fraction inmodels A and E point to a potential problem with usinglowered Evans models for the halo of M31. The baryonfraction for model A ( fB & 0:23) is too high. Moreover, thetotal mass of M31 in model A falls in the lower range ofacceptable values based on the dynamics of satellites andglobular clusters. Model E does better on both counts, butits rotation curve provides a poor fit to the data.

These di"culties no doubt arise from the fact that thelowered Evans models employed in the current implementa-tion of the KD algorithm incorporate a sharp cuto! in den-sity at the tidal radius. A natural alternative is to use amodel halo whose density profile falls o! more graduallywith radius. Such a substitution is consistent with ourcurrent understanding of structure formation. In the hier-archical clustering scenario, each halo is a subsystem of alarger halo and in this sense, the lowered Evans models areunphysical.

N-body simulations based on the cold dark matter modelof structure formation suggest that the density profiles ofdark matter halos have a simple universal shape. This resultwas first noticed by (Navarro et al. 1996, hereafter NFW),who found that the spherically averaged density profiles of


halos in their simulations, which spanned 4 orders of magni-tude in mass, could be fitted by a function of the form

"NFW(r) & "s

r=rs 1' r=rs( )2: (10)

Here rs and "s are free parameters that characterize theradius and density in the transition region between the r$1

central cusp and the r$3 outer halo. While there has beensome debate over the slope of the density profile at smallradii, there is widespread agreement that the NFW profilecaptures the general features of dark matter halos in cosmo-logical simulations.

It is common practice to define the virial radius as R200,the radius within which the mean density is 200 times thebackground density. The concentration parameter, c, isdefined as the ratio of the virial radius to rs: c . R200=rs.Navarro et al. (1996; see also Bullock et al. 2001) have foundthat there is a tight correlation between M200 (the massinterior toR200) and c, the implication being that the densityprofiles of simulated halos are characterized by a singleparameter.

In order to make contact with the results from cosmologi-cal simulations, we determine the NFW profile that mostclosely matches the halo profile of a particular KD model.To be precise, we vary rs and "s in equation (10) so as to min-imize the rms deviation between the halo contribution to therotation curve for the KD model and the rotation curvederived from NFW (eq. [10]). Since our models weredesigned to fit rotation curve and velocity dispersion datafor 1 kpc < r < 30 kpc, only this range in r is used to deter-mine the ‘‘ best-fit ’’ NFWmodel.

We have carried out this exercise for models A and E. Thevalues obtained for c, R200, and M200 are given in columns(5)–(7) of Table 2. The density, mass distribution, and rota-tion curve profiles are shown in Figure 12. We see that forthe range in radius probed by the data considered in thispaper, the rotation curve of an NFWmodel can be matchedclosely to the halo contribution to the rotation curve for ourmodels. Moreover, the values obtained for c are consistentwith what is predicted by the cosmological simulations(Navarro et al. 1996; Bullock et al. 2001).

Not surprisingly, the halo profiles for the KD models dif-fer significantly from the NFW profile at small and largeradii. The lowered Evans models have a constant densitycore, whereas the NFWmodels have an r$1 cusp. The halosin the self-consistent models develop a weak cusp, but theslope is closer to 0 than to $1. However, the discrepancy inmass at small radii between the two models is actually quitesmall: at r & 1 kpc, the cumulative mass for the NFWmodelexceeds that of the KDmodel by an amountd108M".

The discrepancy at large radii is more interesting. Thelowered Evans models have a sharp cuto! in density,whereas the NFW profile falls o! as r$3. The discrepancy atlarge radii is especially apparent in model A, where R200 isnearly a factor of 3 larger than Rt and M200 is a factor of 4larger thanMh.

Figure 12 suggests a resolution to the baryon fractionproblem discussed above, namely, to replace the halo inmodel A with anNFWprofile selected to match the rotationcurve within 30 kpc. The excellent overall fit to the data isretained, and the baryon fraction is brought down to a valuecomfortably within the range predicted by cosmology. Atthis stage, a cautionary remark is in order. The KD models,

by design, describe dynamically self-consistent systems, andone cannot simply substitute a di!erent halo with an NFWprofile without spoiling the self-consistency. Several authorshave derived DFs for NFW models (e.g., Zhao 1997;Widrow 2000; Lokas &Mamon 2000) that, with some mod-ification, could be incorporated into future implementationsof the KD algorithm.

5.5. Halo Shape

Our final set of models explores variations in the shape ofthe dark halo. Results for two of these models are providedin Table 2. For model Q1, we set q & 0:85 and leave Rtunconstrained, whereas for model Q2 the target value of Rtis set to be 60 kpc. In Table 3 we present %2 (col. [3]), Rt(col. [4]), and Mh (col. [5]). We also provide qh, an e!ectiveflattening parameter for the mass distribution (col. [6]). Thevalue of qh is calculated by taking the ratio of the massmoments along the short and long axes. It di!ers from q fortwo reasons. First, the halo ‘‘ responds ’’ to the disk throughthe Poisson-solving algorithm of KD. Thus, even for q & 1,the mass distribution is flattened slightly and qh < 1. Sec-ond, qh reflects the mass distribution, which tends to be

Fig. 12.—Comparison of the halo used in model A with a closelymatchedNFWhalo. The top panel is the density profile, the middle panel isthe mass distribution, and the bottom panel is the halo contribution to therotation curve. Solid lines refer to model A; dashed lines refer to the NFWhalo.

TABLE 3

Models withMd & 7! 1010 M",Mb & 2:5! 1010 M",and Rt Unconstrained

Model q %2Rt

(kpc)Mh

(1010M") qh

A........... 1.00 0.70 80 32 0.93Q1......... 0.85 0.63 37 17 0.59Q2......... 0.85 1.08 58 23 0.61


more aspherical than the gravitational potential. Therefore,qh < q for models Q1 and Q2.

Flattened models appear to favor smaller tidal radii.Model Q1, for example, provides an excellent fit to the databut assumes a tidal radius that is very small. This model isprobably inconsistent with observations of satellites beyond30 kpc. Model Q2 has a somewhat larger tidal radius, but,as with model E, the fit to the rotation curve is significantlydegraded (Fig. 13).

6. GRAVITATIONAL MICROLENSING

Gravitational microlensing is an attractive means bywhich to detect MACHOs in the halo of our galaxy and thehalos of our nearest neighbors. Results from the 5.7 yrLMC data set of the MACHO collaboration suggest that asignificant new component of the Galaxy—namely, onecomposed ofMACHOs—has been discovered (Alcock et al.2000). The MACHO collaboration detected 13–17 micro-lensing events, a number too large to be accounted for byknown populations. Within the context of a specific halomodel, a maximum likelihood analysis yields an estimatefor the MACHO halo fraction of 20%, with the most likelyMACHOmass to be between 0.15 and 0.9M". This result ispuzzling since it requires rather extreme assumptions aboutstar formation and galaxy formation. Moreover, the resultis in conflict with the upper limits on the halo mass fractionfound by the EROS collaboration (Lasserre et al. 2000).One possibility is that the lenses responsible for theobserved microlensing events are in the Magellanic cloudsor in the Galactic disk rather than the Galactic halo.

Microlensing surveys toward M31 have the potential toresolve this question. The main advantage of looking toM31 is that one can probe a variety of lines of sight acrossthe M31 disk and bulge and through its halo. In particular,a massive spherical halo of MACHOs will yield more eventstoward the far side of the disk than toward the near side.

This front-back asymmetry is an unambiguous signature ofaMACHO halo (Crotts 1992).

A number of authors have computed theoretical eventrate maps (Gyuk & Crotts 2000; Kerins et al. 2001; Baltz,Gyuk & Crotts 2002) assuming an ad hoc model for thedisk, bulge, and halo of M31. Only cursory attempts weremade to insure that the models are dynamically self-consis-tent. In this regard, our models represent an improvementover the models considered in the aforementioned papers.

Following Gyuk & Crotts (2000), we consider the quan-tity d&=dA, the number of concurrent events per area on thesky. This quantity is roughly the product of the opticaldepth (number of concurrent events per source star) and thesurface density of sources and is more closely alignedwith what the experiments measure than the optical depth.As with the line-of-sight velocity dispersion, the N-bodyrepresentation of the KD models allows one to calculatetheoretical optical depth and event rate maps quickly ande"ciently (see, e.g., Widrow & Dubinski 1998). The quan-tity d&=dA is evaluated by performing a double integral oversource and lens distributions:

d&

dA&

Z 1

0dLnsource(L)

Z L

0dl"lens(l)Mlens

4GMlens

c2L$ l

L; (11)

where nsource is the number density of sources a distance Lfrom the observer, "lens is the mass density in lenses a dis-tance l from the observer, and Mlens is the mass of the lens.Given an N-body representation of a galaxy, this integralmay be evaluated by performing the double sum:

d&

dA&

X

i2source

X

j2lens

4GMlens

c2Li $ ljLi

: (12)

The map of the number of concurrent events per arcmin2

for model A is shown in Figure 14.The di!erential event rate (number of events per unit time

as a function of duration and position across the M31 disk)may be calculated in a similar manner. A semianalytic calcu-lation, on the other hand, involves multidimensional inte-grals, which may be prohibitively complicated dependingon the functional form of the DFs.

7. SUMMARY AND CONCLUSIONS

The models for M31 presented in this paper are dynami-cally self-consistent, consistent with published observations,and stable against the rapid growth of barlike modes in thedisk. To the best of our knowledge, no other model of a diskgalaxy satisfies these three criteria to the extent consideredin this paper. Although KD constructed models for theMilky Way that are dynamically self-consistent, stable, andhave roughly the correct rotation curve, they did notattempt to fit their model to other types of data such as thesurface brightness profile.

Our models span a wide range in halo size and shape. Themost successful models assume a bulge mass that is nearly afactor of 2 smaller than the oft-quoted value from Kent(1989). Our galaxy model with Mb & 2:5! 1010 M" andMd & 7! 1010 M" provides a good overall fit to observatio-nal data, yields mass-to-light ratios that are quite accept-able, and appears to be stable against bar formation. Thisdisruptive bar instability becomes more apparent inthe models with more massive disks, such as the model

Fig. 13.—Comparison of the rotation curves for models A, Q1, and Q2(see Table 3).


proposed by Kent (1989). At the other extreme lies themodel of Kerins et al. (2001), havingMb & 4! 1010 M" andMd & 3! 1010 M". Although their model is extremely sta-ble against bar formation and reproduces the surface bright-ness profile nicely within the disk-bulge transition zone, theM31 rotation curve and inner velocity dispersion profile itprovides do not fit the data at all well.

The favored model in our study is found to yield a poormatch of the estimated baryon fraction with cosmologicalpredictions, and it falls somewhat short of the total galaxymass at large radii as determined from observations ofdynamical tracers. Forcing a larger tidal radius (Rt # 160kpc) improves these at the cost of the rotation curve fit. Weattribute this failure to the form of the lowered Evans haloDF utilized in the KD algorithm and propose the use of ahalo DF that falls o! more gradually with radius. A preli-minary analysis suggests that if the halo of model A isreplaced by an appropriate NFW profile, the cosmologicalconstraints are satisfied while the quality of the fit to theobservational data is maintained.

Several improvements in the models are possible. Forexample, one can incorporate additional components of thegalaxy such as a central black hole, thick disk, and stellarhalo. The most serious drawback of the models is that theyare axisymmetric and therefore cannot capture important

aspects of M31 such as spiral structure and triaxiality of thebulge. Our models can serve as a starting point for investiga-tions of these phenomena. In particular, models withMd ’ 7! 1010 M" will be studied using numerical simula-tions to see if we can produce spiral structure and a barlikebulge similar to what is observed inM31.

The methods described in this paper are completely gen-eral. When additional data become available, the algorithmcan be rerun to determine a new suite of best-fit models.One extension that is soon to be implemented is the inclu-sion of a distribution of test-particles that are designed torepresent a population of dynamical tracers such as globu-lar clusters or satellite galaxies. In principle, the additionalinformation from observations of these populations canconstrain the extent and shape of the halo.

We are grateful to J. Dubinski andK. Kuijken for provid-ing us with their code and for invaluable assistance inrunning it. We also thank E. Baltz, A. Crotts, G. Gyuk,J. Irwin, S. Kent, and D. Sti! for useful conversations andto J. Dubinski for comments and suggestions based on anearly draft of this manuscript. We also thank the anony-mous referee for useful suggestions. This work was sup-ported, in part, by the Natural Science and EngineeringResearch Council of Canada.

TABLE 4

Input KD Parameters for Models in Tables 1, 2, and 3

Model !0 !0 q (rc=rK )2 Ra mdisk "b !c !b Sb

A........... $28.87 4.20 1.00 0.43 4.77 30.11 6.68 $16.26 2.24 0.80B ........... $24.52 4.27 1.00 0.41 4.73 30.11 7.07 $18.55 2.24 0.85C........... $32.29 4.34 1.00 0.40 5.14 30.11 6.52 $14.79 2.41 0.81D .......... $29.77 3.37 1.00 0.42 4.50 60.22 6.87 $17.96 2.05 0.77E ........... $28.98 3.68 1.00 0.40 6.03 30.11 7.35 $16.68 2.26 0.81K1......... $34.86 3.58 1.00 0.43 4.30 68.82 6.38 $16.03 2.44 0.75K2......... $35.21 3.62 1.00 0.15 5.26 12.90 4.94 $17.90 1.62 0.82Q1......... $26.57 4.05 0.85 0.42 3.63 30.11 9.70 $13.81 2.57 0.80Q2......... $26.23 3.79 0.85 0.44 4.27 30.11 10.18 $13.74 2.55 0.80

−40 −30 −20 −10 0 10 20 30 40−20

−10

0

10

20

θ (arcmin)

φ(arcmin)

0.02 0.02

0.1

0.1

0.1

0.1

0.1

0.5

Fig. 14.—Contours of concurrent microlensing events per arcmin2 for model A, assuming an all MACHO halo. Contours are, from the outside in, 0.02,0.05, 0.1, 0.2, 0.5, 1.0, and 2.0.


APPENDIX A

KD MODEL PARAMETERS

The input parameters for the KD algorithm are provided in Table 4 for the models presented in this paper. The parametersare as discussed in the text. The unit of length is 1 kpc, and the unit of velocity is 100 km s$1. We setGnewt & 1, as is the conven-tion inN-body simulations. This implies of unit of mass of 2:325! 10$9 M".

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1. INTRODUCTION - uni-bonn.desuyu/Papers/AndromedaModels.pdf · barlike triaxial ellipsoida l bulge. Beyo nd r # 20 kpc , meas-urements wer e obtaine d for spira larm segme nts on