Mass models of the Milky Way
Abstract
We present a simple method for fitting parametrized mass models of the Milky Way to observational constraints. We take a Bayesian approach which allows us to take into account input from photometric and kinematic data, and expectations from theoretical modelling. This provides us with a bestfitting model, which is a suitable starting point for dynamical modelling. We also determine a probability density function on the properties of the model, which demonstrates that the mass distribution of the Galaxy remains very uncertain. For our choices of parametrization and constraints, we find disc scale lengths of and for the thin and thick discs respectively; a Solar radius of and a circular speed at the Sun of ; a total stellar mass of ; a virial mass of and a local dark matter density of . We find some correlations between the bestfitting parameters of our models (for example, between the disk scale lengths and the Solar radius), which we discuss. The chosen disc scaleheights are shown to have little effect on the key properties of the model.
keywords:
Galaxy: fundamental parameters – methods: statistical – Galaxy: kinematics and dynamics1 Introduction
A great deal is still unknown about the distribution of mass in the various components of the Milky Way. The major discoveries in Galactic astronomy over the past decade have almost all been related to components which comprise a small fraction of the total mass of the Milky Way, most of which either are or were dwarf galaxies (for example, the many objects observed in the “Field of Streams”, Belokurov et al., 2006). The structure of the dominant components – the disc(s) and cold dark matter (CDM) halo – remains rather uncertain.
An important element of understanding and constraining the structure of the major components of the Galaxy is creating Galaxy models which can be compared to observational data. It is important to draw a distinction between three types of Galaxy models: mass, kinematic and dynamical models. Mass models are the simplest of these, and only attempt to describe the density distribution of the various Galaxy components, and thus the Galactic potential (e.g. Klypin, Zhao, & Somerville, 2002; Dehnen & Binney, 1998, henceforth DB98). Kinematic models, such as those produced by galaxia (Sharma et al., 2011), specify the density and velocity distributions of the luminous components of the Galaxy, but do not consider the question of whether these are consistent with a steady state in any Galactic potential. Dynamical models (e.g. Widrow, Pym, & Dubinski, 2008) describe systems which are in a steady state in a given potential, because their distribution functions depend only on the integrals of motion. The Besançon Galaxy model (Robin et al., 2003) is primarily a kinematic model with a dynamical element used to determine the vertical structure of the disc.
It is clear that moving beyond simple kinematic models to full dynamical ones is an essential step in fully understanding our Galaxy. The majority of the mass of the Galaxy is expected to lie in the CDM halo, which is only observable through its gravitational effect on luminous components of the Galaxy, so purely kinematic models cannot provide any insight into its structure.
The first step towards a dynamical model is to produce a mass model that is consistent with available constraints. An influential mass model was that of Schmidt (1956), and, as observational data and understanding of galaxy structure improved, updated versions have been produced by other authors, notably Caldwell & Ostriker (1981) and DB98. Our intention in this study is to follow these authors in producing a mass model that is consistent with uptodate observational data and theoretical understanding, and to provide a simple framework for producing these models into which future data can be placed as they become available.
The major difficulty in producing a model of this kind is drawing together data from numerous different studies of the various components which make up the Milky Way in a way that is as consistent as possible. Such studies often make different underlying assumptions, and sometimes come to seemingly mutually contradictory conclusions. In principle the correct approach is to return to the raw data that each study was based upon and to synthesise them into one coherent picture. Even if this is possible in practice, it is certainly an immense undertaking, and one we do not attempt here. Instead we follow the approach of previous authors by accepting various constraints on the parameters of our models as stated by other studies, without returning to the raw data. In addition we use well understood kinematic data sets, and – for the CDM halo, about which observational data is limited – we use our best current theoretical understanding. Our approach is best thought of as Bayesian with direct constraints on the model parameters (from photometric data or theoretical insight) being our Bayesian “prior”, and kinematic data used to find the likelihood.
In this paper we present a simple method for determining both a bestfitting parametrized mass model of the Galaxy, and the full probability density function of the parameters of the model. This paper is associated with a series of papers in which we construct dynamical models to fit observational data (Binney, 2010; Binney & McMillan, 2011, McMillan et al. in preparation). Eventually these dynamical models will themselves be used to constrain the Galactic potential.
Like DB98 we restrict ourselves to axisymmetric models. It is clear that the Galaxy is not actually axisymmetric, especially the inner Galaxy (see Section 2.1), but the disc is close to axisymmetric (Jurić et al., 2008), and (as noted by DB98) axisymmetric models successfully account for observations in the 21cm line of hydrogen at Galactic longitudes , corresponding to .
To find the gravitational potential associated with a given mass model we use the publicly available code galpot, which is described by DB98 section 2.3.
In Section 2 we describe the different components that make up the bulk of the mass of the Galaxy, and how our model represents them. We also explain our Bayesian priors on the parameters that describe our model. In Section 3 we give the kinematic data we use to constrain our model. All the constraints applied to the model are summarised in Table 1. In Section 4, we describe the method used to fit the model, and in Section 5 we give our bestfitting model and the rest of our results. In Section 6 we compare our results to other data sets.
Unless otherwise stated, any measurement or derived quantity we use to constrain our model is assumed to have Gaussian uncertainties.
2 Components of the Milky Way
2.1 The bulge
The Galactic bulge has been shown in many studies to be a nearprolate, triaxial rotating bar with its long axis in the plane of the Galaxy (e.g. Binney, Gerhard, & Spergel, 1997). However, our interest in this study is in producing an axisymmetric model, so we are forced to make crude approximations in our modelling of this component. We must therefore accept that it is unwise to compare our model to measurements taken from the inner few of the Galaxy, as it can be expected to do a poor job of reproducing them.
Insight into the structure of the bulge can be gained from photometric studies, however one must be careful when doing so as there can be a major contribution to the stellar density in the inner few from the disc component. The model used for the disc in these studies will therefore have a significant effect on the properties determined for the bulge. This goes some way towards explaining why the mass of the bulge as determined by Picaud & Robin (2004), , using a photometric model with a “hole” in the centre of the disc, is so much larger than that determined by studies using kinematic data (e.g. DB98; Widrow, Pym, & Dubinski, 2008; Bissantz & Gerhard, 2002) which do not include a disc “hole” in their models.
Our density profile is based on the parametric model which Bissantz & Gerhard (2002) fit to dereddened Lband COBE/DIRBE data (Spergel, Malhotra, & Blitz, 1996), and the masstolight ratio that Bissantz & Gerhard determine from a comparison between gas dynamics in models and those observed in the inner Galaxy. This model is found on the assumption that the disc component has no central hole. This also assumes that the masstolight ratio is spatially constant, which allows us to convert a photometric model directly into a mass model for this component.
The Bissantz & Gerhard model is not axisymmetric, so we make an axisymmetrised approximation which has the density profile
(1) 
where, in cylindrical coordinates,
(2) 
with , , , and axis ratio . The Bissantz & Gerhard masstolight ratio has a quoted uncertainty , but given the extent to which we have altered this model it is prudent to recognize the need to introduce further uncertainty in our model fitting. We therefore assume that the bulge mass , with uncertainty . For this density profile, this corresponds to scale density .
2.2 The disc
The Milky Way’s disc is usually considered to have two major components: a thin disc and a thick disc (e.g. Gilmore & Reid, 1983). These are generally modelled as exponential in the sense that
(3) 
with scale height , scale length and central surface density . The total mass of a disc like this is .
As mentioned in Section 2.1, some studies of the inner Galaxy prefer models with a central “hole” in the disc (e.g. Picaud & Robin, 2004). We do not consider such models here. The kinematic data we consider (Section 3) is all related to parts of the Galaxy which lie outside any central disc hole, so, in this study, a model with a central hole should simply redistribute mass in the inner few from the disc to the bulge (note that our prior on the bulge would have to be replaced, as it is taken from a study which used a model of the disc with no central hole).
The Jurić et al. (2008) analysis of data from the Sloan Digital Sky Survey (SDSS: Abazajian et al., 2009) showed that the approximation to exponential profiles is a sensible one for the Milky Way, and produced estimates based on photometry for the scale lengths, scale heights and relative densities of the two discs.
The scaleheights of the discs are not at all well constrained by the kinematic data we use in this study, so initially we accept without question the Jurić et al. (2008) bestfitting values, and . In Section 5.1 we explore the (relatively small) effects of changing the assumed disc scaleheights.
The Jurić et al. scalelengths for the thin and thick discs are and respectively, with a quoted uncertainty of in each case. The local density normalisation is quoted as , with uncertainty . We approximate these uncertainties as Gaussian and uncorrelated, and take these as prior probability distributions on , and . Again, we assume a constant masstolight ratio which allows us to convert these photometric constraints directly into constraints on the mass density.
2.3 The darkmatter halo
For obvious reasons, we cannot use photometric data to constrain the shape of the dark matter distribution, so instead we look to cosmological simulations for insight.
In cosmological simulations that only include dark matter, halo density profiles are well fit by a universal profile, known as the NFW profile (Navarro, Frenk, & White, 1996)
(4) 
where , with the scale radius. It is clear that the baryons within CDM haloes will have an impact on the halo profile, but the nature of this impact remains uncertain because it depends on the complex physics of baryons. Cosmological simulations suggest that the condensation of baryons to the centre of galactic haloes will cause the density profile in the inner parts of the halo to be steeper than , as it is in the NFW case (Duffy et al., 2010). Observations of low surface brightness galaxies, however, appear to indicate that they lie in darkmatter haloes with constant density cores. This difference between observation and simulations (with or without baryonic physics) is known as the “cuspcore problem” (for an overview see de Blok, 2010).
Observations of low surface brightness galaxies and dwarf galaxies can used to provide more reliable information about the inner dark matter profile than galaxies like the Milky Way because they are dark matter dominated right to their centre, so the mass contribution of the baryons can be modelled with little uncertainty compared to the total mass. The Milky Way is not dark matter dominated and it is very hard to determine slope of the dark matter density profile in the inner Galaxy because the baryonic density is dominant in the inner Galaxy, and still uncertain.
Haloes in darkmatteronly cosmological simulations tend to be significantly prolate, but with a great deal of variation in axis ratios (e.g. Allgood et al., 2006). It is recognised that, again, baryonic physics will play an important role – condensation of baryons to the centres of haloes is expected to make them rounder than darkmatteronly simulations would suggest (Debattista et al., 2008). The shape of the Milky Way’s halo is still very much the subject of debate, with different efforts to fit models of the Sagittarius dwarf’s orbit favouring conflicting halo shapes (see e.g. Law, Majewski, & Johnston, 2009).
In this study we do not attempt to address the question of the shape of the Milky Way halo, or the impact of baryonic physics on the CDM density profile. We take the simple model of a spherically symmetric NFW halo (eq. 4). As better understanding of the effect of baryonic physics on CDM haloes or constraints from observations become available, these simple assumptions will have to be revisited.
It is common to describe CDM haloes in terms of their virial mass – which is the mass contained within the virial radius – and the concentration , where is the radius at which the logarithmic slope of the density profile (for an NFW profile, ). The virial radius is defined as the radius of a sphere centred on the halo centre which has an average density of times the critical density, but the definition of varies between authors. The definitions that are relevant to this paper are (when using this definition we will use the notation , and for the virial mass, radius, and the concentration), and a value of which varies with redshift, derived from spherical tophat collapse models (e.g. Gunn & Gott, 1972), with today (for which we use the notation , , ).
Guo et al. (2010) compared the distribution of halo virial masses found in the Millennium and MillenniumII simulations (Springel et al., 2005; BoylanKolchin et al., 2009) to the distribution of galaxy stellar masses found by Li & White (2009) using Sloan Digital Sky Survey data (Abazajian et al., 2009). Guo et al. (2010) found that the ratio of total stellar mass to halo mass was well fit by
(5) 
with an intrinsic scatter of order in , where , , , and .
To gain insight into the likely value of the concentration parameter we consider the study of BoylanKolchin et al. (2010). This examined haloes taken from the MillenniumII simulations at redshift zero, in the mass range – a mass range that the Milky Way’s halo is likely to lie in – and determined that the probability distribution of the concentration was well fit by a Gaussian distribution in , with
(6) 
Cosmological simulations predict that typical halo concentration varies with halo mass, but typically only weakly (e.g. , Neto et al., 2007) when compared to the intrinsic scatter in at a given virial mass, so we accept the relationship given by BoylanKolchin et al. (2010) as stated, independent of . Baryonic physics is likely to have an effect on the concentration, much as it does on the inner density profile (Duffy et al., 2010), but we neglect that in this study.
The Galaxy’s darkmatter halo is far more massive than its stellar halo, which has a mass (Bell et al., 2008). We therefore treat the stellar halo as a negligible fraction of the Galactic halo, and do not consider it further.
3 Kinematic data
3.1 The Solar position and velocity
If we do not know the position and velocity of the Sun it is extremely difficult to interpret any other kinematic observations of the Milky Way. Unfortunately there is still significant uncertainty on the key parameters, namely the distance from the Sun to the Galactic Centre , the rotational speed of the local standard of rest (LSR) , and the velocity of the Sun with respect to the LSR (see e.g. McMillan & Binney, 2010). In this work we take from the study of stellar orbits around the supermassive black hole at the Galactic Centre, Sgr A*, by Gillessen et al. (2009)
(7) 
though we could equally have taken the value from Ghez et al. (2008), found from similar data.
A series of recent papers (Binney, 2010; McMillan & Binney, 2010; Schönrich, Binney, & Dehnen, 2010) have reexamined evidence regarding the value of . We use the value determined by Schönrich, Binney, & Dehnen:
(8) 
where is the velocity towards the Galactic Centre, is the velocity in the direction of Galactic rotation, is the velocity perpendicular to the Galactic plane. We quote the systematic uncertainties here, because these dominate the total uncertainty (especially in ).
Many studies that can be used to constrain the value of are ones which actually constrain the ratio – for example using the Oort constants and , which can be determined locally (e.g. Feast & Whitelock, 1997) where . In this study, we use the proper motion of Sgr A* in the plane of the Galaxy, as determined by Reid & Brunthaler (2004)
(9) 
Since Sgr A* is expected to be fixed at the Galactic Centre to within , this proper motion is thought to be almost entirely due to the motion of the Sun around the Galactic Centre . This measurement is sufficiently accurate (and the assumed velocity of Sgr A* sufficiently small) that the greatest uncertainty in the value of required for our models actually comes from the uncertainly in the value of !
This measured proper motion is inconsistent with the value of found by Feast & Whitelock (1997) – who give the most commonly used values for and . Therefore we do not attempt to use the Oort constants as constraints.
3.2 Terminal velocity curves
For circular orbits in an axisymmetric potential, the peak velocity of the interstellar medium (ISM) along any lineofsight at Galactic coordinates for corresponds to the gas at the tangent point, at Galactocentric radius . This is known as the terminal velocity, and is given by
(10) 
Malhotra (1994, 1995) produced data which gave this peak velocity for a range of lines of sight in the Galaxy from a number of surveys (Weaver & Williams, 1974; Kerr et al., 1986; Bania & Lockman, 1984; Knapp et al., 1985).
To constrain our model, we have to take account of the effects that nonaxisymmetric structure in the Galaxy and noncircular motion of the ISM will have on these data. To do so we follow DB98 in allowing for an uncertainty of in at any given , and restricting ourselves to data at , which is unlikely to be significantly affected by the bar.
3.3 Maser observations
In recent years it has become possible to use Galactic maser sources as targets for astrometric measurements that are sufficiently accurate for parallaxes with uncertainties to be determined for them (e.g. Reid et al., 2009). McMillan & Binney (2010) showed that these observations were consistent with models that placed the masers on near circular orbits with similar to the value implied by the proper motion of Sgr A* (Section 3.1). However by itself (or, equivalently, ) was shown to depend quite strongly on the shape of the rotation curve. We constrain our mass model using these data by applying a version of the likelihood analysis used by McMillan & Binney (2010).
The likelihood analysis for these data requires integrating over heliocentric radius the probability density of the observations (parallax, proper motion and radial velocity) given a model for the expected velocity distribution. The maser sources are associated with high mass star formation regions, and are expected to be on nearcircular orbits. We model the velocity distribution as circular rotation (with a velocity which depends on the potential), plus a random component that has a Gaussian probability distribution, uniform in direction, with width . The interested reader can find details in McMillan & Binney (2010). However the analysis in this study differs from that of McMillan & Binney (2010) in that (a) the rotation curve, including , is defined by the mass model; (b) the value of is assumed to be that in eq. (8); (c) we include data from Rygl et al. (2010) and Sato et al. (2010), which were not available when the McMillan & Binney (2010) study was performed; (d) in the interests of simplicity we fix , close to the bestfitting values found when was allowed to vary in the previous study, and (e) we assume the masers have no systematic velocity offset from the rotation curve – one of the conclusions of McMillan & Binney (2010) was that no such offset is required to explain the data.
3.4 Vertical force
Kuijken & Gilmore (1991) used observations of K stars to find the vertical force at above the plane at the Solar radius, . We adopt the value they found
(11) 
as a constraint.
3.5 Observational constraints on the structure at large radii
Property constrained  Constraint  Section described  Source 
Bulge profile  see equation (1)  2.1  Bissantz & Gerhard (2002) 
2.1  Bissantz & Gerhard (2002)  
Disc profile  Double exponential  2.2   
2.2  Jurić et al. (2008)  
2.2  Jurić et al. (2008)  
2.2  Jurić et al. (2008)  
2.2  Jurić et al. (2008)  
2.2  Jurić et al. (2008)  
Halo profile  NFW profile  2.3  Navarro et al. (1996) 
see equation (5)  2.3  Li & White (2009)  
2.3  BoylanKolchin et al. (2010)  
3.1  Gillessen et al. (2009)  
3.1  Reid & Brunthaler (2004)  
3.4  Kuijken & Gilmore (1991)  
, see eq. (12)  3.5  Wilkinson & Evans (1999)  
Kinematic data  Section described  Source  
Terminal velocities  3.2  Malhotra (1994, 1995)  
Maser observations  3.3  Reid et al. (2009); Rygl et al. (2010);  
Sato et al. (2010) 
It is extremely difficult to gain useful constraints on the structure of the Milky Way at large radii from observational data. Any population of dynamical tracers suffers from small number statistics and/or poor observational constraints (especially on proper motion) and because of uncertainty over the associated distribution function, especially the velocity anisotropy, and whether or not all of the population is in fact bound to the Milky Way. This leads to total halo mass estimates which have fractional uncertainties of order unity, e.g. (Wilkinson & Evans, 1999), to (Battaglia et al., 2006).
It is possible to take cosmologically motivated simulations and use them to provide some insight into the expected distribution function of a tracer population. Xue et al. (2008) did this for a sample of blue horizontal branch (BHB) stars and found that the mass at radii less than , under this assumption, was . However this value is entirely dependent on the assumption that these cosmological simulations (and their prescriptions for starformation, feedback etc.) can be relied upon to predict accurately the distribution function of BHB stars.
Another possible source of information is the motion of the Magellanic clouds, but this can only be used straightforwardly if they can be assumed to be bound to the Milky Way – Besla et al. (2007) have shown that this may not be a valid assumption.
Other authors (notably Smith et al., 2007) have attempted to determine the local escape velocity from the Milky Way, and use this to constrain the mass distribution at large radii. However this is determined under the assumption that the population of stars close to the escape velocity can be described by a steadystate distribution function that is depends on energy alone. Stars that have velocities that lie close to the escape velocity are on orbits with very long periods, which means that there is no reason to expect them to be in a steady state, especially since it is likely that a large fraction of halo stars are associated with streams. The assumed distribution function is isotropic (because it only depends on energy), and this assumption is unlikely to be valid – halo stars in cosmological simulations typically have a distribution function which is radially anisotropic.
The “Timing Argument” has been used to constrain the mass of the local group or that of the Milky Way itself. In its simplest form, this assumes that the currently observed radial velocity of M31 towards the Milky Way must be due to the gravitational interaction of the two galaxies overcoming the overall cosmic expansion, and that the corresponding orbital motion must have happened within the age of the universe, and shows that this implies a lower limit on the mass of the two galaxies. Li & White (2008) extended this argument by searching cosmological simulations for comparable galaxy pairs that were moving towards one another in the simulations. They also looked for galaxy pairs comparable to the Milky Way and Leo I, which are moving away from one another, and used the Timing Argument under the assumption that Leo I is moving towards apocentre for a second time. This yielded a probability distribution on with lower and upper quartiles and a confidence lower limit of . This approach rather relies upon Leo I being bound to the Milky Way, which it may not be, and the authors note that, compared to the Milky Way/M31 system, the “complex dynamical situation offers greater scope for uncertainty”.
While Wilkinson & Evans (1999) quoted a very uncertain figure for the total halo mass, they also found that the mass within was a more robustly determined quantity: . The quoted uncertainty is very asymmetrically distributed about the most likely value, and we choose to see this result as providing an upper bound on the mass inside , which we take into account in our analysis through the probability distribution
(12) 
where , , and is a normalisation constant. The only effect of using this constraint is to penalise models with . For models with this would be an inappropriate choice but, as we show in Section 5, that is not the case for this study.
Given the difficulties described here, we have decided not to constrain our model with the results of any other study. We do compare our bestfitting model to them (Section 6)
4 Fitting the models
We use Bayesian statistics to find the probability density function (pdf) for our model parameters given the kinematic data described in Section 3, and the prior probabilities given in Section 2 (and the value of , given in Section 3). We refer to the parameters collectively as , and the data as . Bayes theorem tells us that this pdf is then
(13) 
where the total likelihood is the product of the likelihoods associated with each kinematic dataset or constraint described in Section 3, and is the probability of a parameter set given the prior probability distributions described in Section 2 (and that on ). The Bayesian Evidence, , is often a very important quantity, but in this study it is an unimportant normalisation constant, so we ignore it.
Given the constraints we have applied to the components described in Section 2, we are left with 8 model parameters that we allow to vary: The scalelengths and density normalisations of the thin and thick discs (); the density normalisation – and thus mass – of the bulge (); the scalelength and density normalisation of the CDM halo (); and the solar radius (). While each of these parameters is free to vary, each one is explicitly or implicitly associated with at least one prior probability distribution.
To explore the pdf we use the Metropolis algorithm (Metropolis et al., 1953), which is a Markov Chain Monte Carlo method for drawing a representative sample from a probability distribution. This allows us both to find the peak of the pdf (to reasonable accuracy) and to characterise its shape. We start with some choice for the parameters , and calculate . We then

choose a trial parameter set by moving from in all directions in parameter space, by an amount chosen at random from a “proposal density” (see below);

determine ;

choose a random variable from a uniform distribution in the range [0,1];

if , accept the trial parameter set, and set . Otherwise set .

Return to step (i), replacing with .
The first few values of are ignored as “burnin”, which helps to remove the dependence on the initial value of .
The pdf is quite narrow in some directions in the parameter space, but these directions are not simply parallel to the coordinate axes. Therefore it is efficient to use a proposal density which is aligned (to a reasonable approximation) with the pdf – this is allowed by the Metropolis algorithm as the only constraints on are that it is symmetrical with respect to swapping and , and makes it possible to reach any point in phase space. We therefore use the Metropolis algorithm in two phases – first with a proposal density which is simply made up of Gaussian distributions in each parameter individually, with associated step size chosen by hand. The resultant chain (of a relatively short length, and after a burnin) is then used to construct a covariance matrix that is then used to define a new . This new proposal density is, again, a multivariate Gaussian but with the principal axes now aligned to the eigenvectors of the covariance matrix, and with the step size in each direction related to the respective eigenvalues. The chain produced with this second proposal density is then used in all calculations.
5 Results
Best  816.6  2.90  209.5  3.31  95.6  0.00846  20.2  8.29  
Convenient  753.0  3  182.0  3.5  94.1  0.0125  17  8.5  
Mean  
Std. Dev.  123  0.22  110  0.56  6.9  0.006  4.3  0.16  
Best  239.1  77.7  63.9  0.087  0.0104  0.122  1.38  
Convenient  244.5  8.84  65.1  1340  75.4  60.3  0.083  0.0111  0.121  1.12 
Mean  76.5  62.0  0.085  0.0106  0.120  1.29  
Std. Dev.  4.8  0.65  6.3  240  5.3  7.6  0.010  0.0010  0.012  0.30 
1  

1  
0.483  1  
0.468  1  
0.167  0.037  1  
0.328  0.151  1  
0.588  0.221  0.003  1  
0.602  0.157  0.373  1 
In Figures 1 to 3 we plot the probability density functions of various quantities associated with our models, marginalised over all parameters, as determined by the Metropolis algorithm. Where appropriate we also plot the prior probability distribution directly associated with each. The value associated with our bestfitting model is indicated on each plot with a dashed vertical line. In Table 2 we give the parameters of our bestfitting model, and those of what we call the “convenient” model (see below), along with the mean and standard deviation for each parameter, marginalised over other parameters in the pdf.
The marginalised distributions do not give a sense of the correlations between parameters, so in Table 3 we show the correlation matrix of the various parameters . For the ,th component, , this takes the value
(14) 
where is the covariance. A value of 1 corresponds to a perfect correlation, corresponds to a perfect anticorrelation, and 0 corresponds to no correlation. The correlation matrix is manifestly symmetric, so in Table 3 we only show half of it.
The strongest correlations or anticorrelations are between parameter pairs that describe a single component – this explains, for example, why the spread in is so much smaller than the spread in or , and why the standard deviation of can be as large as while that in is (and that in is even smaller, see below). Other fairly strong correlations are noticeable between and virtually every other parameter, and between and .
Figure 1 shows the pdfs associated with the disc scalelengths, and . Both have peaks around , and the bestfitting values also lie quite near . That the kinematic data (primarily the terminal velocity curve) drives the two scalelengths towards a similar value is unsurprising – the two discs have effects on the dynamics of the Galaxy that scale very similarly. The posterior pdf on is much narrower than the prior, which is not true to the same extent for ; this is not surprising given that the thin disc is more massive than the thick disc so the constraints based on kinematic data are more likely to dominate the photometric constraints in the thin disc pdf.
DB98 considered mass models with four different scalelengths between and , and halo density profiles which could take on a wide range of powerlaw slopes in their inner and outer parts. They found that the models with disc scalelengths of or smaller required haloes which, unrealistically, had density profiles which increased with radius () in their inner parts, whereas the models with scalelengths or greater did not. Therefore, given that we require a halo profile that increases as in its centre, we should not be surprised that this favours models with disc scalelengths larger than .
Figure 2 shows the posterior pdfs for the position of the Sun and the circular speed at the Sun in our models. The posterior pdf on is much narrower than the prior, with a standard deviation of , compared to an uncertainty on the prior of . The posterior pdf on is of a similar width to that which would be seen if our only information on its value was the proper motion of Sgr A*, but it is slightly displaced to higher values of . This displacement is most likely due to the maser data (Section 3.3), which McMillan & Binney (2010) showed are best fit by models in which is slightly larger than the value implied by equation (9). The corresponding value of , , has a significantly smaller uncertainty than simply combining the Gillessen et al. (2009) value of and the Reid & Brunthaler (2004) value of gives, primarily because of the tighter constraints on .
The posterior pdf on bulge masses (Figure 3, top left) is close to our prior from Bissantz & Gerhard (2002). The bottom right panel of Figure 3 shows that the stellar mass of the models is tightly peaked at values towards the high end of those we would expect given the virial mass, and our constraint on this relationship given in equation (5). This can equivalently be thought of as corresponding to low halo masses, given the stellar masses. Compared to the prior pdfs taken from theoretical considerations (Section 2.3), the haloes have low masses and high concentrations ( for the bestfitting model, and with mean and standard deviation and respectively). The halo is likely to be subdominant at in the sense that the gravitational force on the Sun is mostly from the stellar component, though this is not the case for all models (ratio of gravitational forces ). The mass inside of our bestfitting model is , with mean and standard deviation . This is close to the mass limit we take from Wilkinson & Evans (1999), indicating that the latter is an important upper limit on our models.
The local density normalisation is for our bestfitting model. The mean and standard deviation are , which is exactly what one would expect from our prior alone. This is due to the fact that our kinematic data does not provide us with any great ability to discriminate between mass in either of the two discs. Therefore, the value of almost entirely defines (with the disc scaleheights) our constraints on the relative contributions of the thick and thin disc.
Catena & Ullio (2010) used a rather similar method to the one described in this paper with the single intention of determining the local dark matter density, finding a value . We have made a number of different assumptions, and taken into account different constraints, but many of the key assumptions (axisymmetry, exponential disc and a halo profile motivated by CDM simulations) are the same, and we find a similar local dark matter density . It is well worth noting that the uncertainty we quote here is purely the statistical uncertainty associated with this set of parametrized models, and the constraints we have chosen to apply. In particular, we have made the approximations that the discs are well described by exponential profiles at all radii, and that the halo density profile is well described by a spherically symmetric NFW profile. These assumptions and others, including ones which are not not directly related to the dark matter profile, will have a significant effect on the value found for the local dark matter density. We expect that the systematic uncertainties on this value are much larger than the statistical uncertainty we state here.
It is clear from Figures 1 to 3 that there is a wide range of models that fit our constraints almost as well as our bestfitting model. It is often convenient to use models in which certain key parameters are chosen to take simple values (for example it is common to take or ), and we provide such a model in Table 2. To take account of the correlations between parameters we chose to build this “convenient” model one step at a time. We first hold constant at a suitable value determined from Figure 2, and find the pdf associated with these models, which we use to choose a suitable thin disc scale length . We then repeat this process to find a suitable value for , and then again for . Following this procedure we take , , and .
In Figure 4 we compare the terminal velocity curve associated with our bestfitting model, and with the “convenient” model, to the observational data (Section 3.2). The difference between the curves of the two models is very small, and both provide a good fit to these data. In Figure 5 we plot the rotation curves of the bestfitting and convenient models, and the decomposition into the components due to the bulge, discs, and halo. The two models are more clearly distinguished in this plot, primarily because of the larger value of chosen for the convenient model, which results in a higher value for , because of the strong correlation between the two.
5.1 Effect of changing the disc scaleheights
In all of the models discussed thus far (and shown in Figures 1–5 and Tables 2 & 3) the scaleheights of the discs have been held constant at and for the thin and thick discs respectively. These values were chosen because they are the bestfitting scaleheights given by Jurić et al. (2008), and were held constant because we do not expect the kinematic data described in Section 3 to strongly constrain the vertical density profile. It is, however, important that we check this assumption by considering different scaleheights for the disc. Therefore we now consider models which have a thin disc scaleheight of or and a thick disc scaleheight of or , in all nine possible combinations.
One change to our prior must be considered because of the effect of changing the scaleheight of the disc. The local density normalisation, , is defined at the Sun’s position, so if the scaleheights change, the value of changes without the surface densities of the two discs changing. It has been noted (by, for example, Reylé & Robin, 2001) that there is an anticorrelation between the values of and found by studies using star counts, which can indeed be seen in figures 21 & 24 of Jurić et al. (2008). We therefore approximate that simply changes with , taking the values for respectively, with an uncertainty in of in all cases (as previously). These values are taken by eye from figure 21 of Jurić et al. (2008), with a correction for the recognised biases in the data. We ignore any possible correlation between and – an anticorrelation of this type is seen in figures 21 & 24 of Jurić et al. (2008), but could easily be related to the relationship between and , as the values they derive for and are also correlated.
We give the mean and standard deviation for the parameters (and derived quantities) of all these models in the appendix. Here we discuss the most significant findings.
Most importantly, the changes in scale heights have very little impact on the overall structure of the Galaxy models. The bulge mass and virial mass of the Galaxy are virtually unchanged, and the total stellar mass changes by at most , which is much less than the standard deviation of the value. The disc scale lengths are also barely changed (changes of at most in and in ). The surface density at the Sun is also largely unchanged, with changes of in and the local disc surface density changing by .
The only significant change is in the values of and , which show a transfer of mass from the thick disc to the thin disc as increases. The mean value of rises from for models with to for those with , while the value of declines from to over the same range. This then affects the fraction of the total disc mass found in each disc, and the value of .
In every case the posterior pdf of is almost precisely the same as the prior. This indicates that the prior on is, in all cases, the dominant factor determining the how the Galaxy model’s total disc mass is divided between the two discs.
We can therefore see that holding the disc scaleheights at fixed values to produce the statistics in Table 2 does not significantly alter the quoted standard deviations, except for parameters related to the division of the disc material into the thick and thin discs. Here the true uncertainies are significantly larger than the quoted values because the uncertainty in scaleheights must be taken into account.
6 Comparison to other studies
The thin disc scale length of our bestfitting model is at the larger end of the Jurić et al. (2008) range, which we used as a prior. It is therefore also larger than the values in the range to found by some other recent studies (Ojha et al., 1999; Chen et al., 1999; Siegel et al., 2002), and the value used for the old (age ) disc in the Besançon Galaxy model (, Robin et al., 2003). On the other hand it lies close to the scale length found by *KeDaFa91, , and towards the lower end of the range found by LópezCorredoira et al. (2002) . The stellar mass of our bestfitting model is (with mean and standard deviation . This compares to the value found as a “back of the envelope” estimate by Flynn et al. (2006). The local disc surface density ( bestfitting; mean standard deviation) is somewhat larger than suggested by studies which count identified matter in the Solar neighbourhood, such as that of Kuijken & Gilmore (1989) who found , or Flynn et al. (2006) who found .
Our bestfitting bulge mass is close to the centre of the prior distribution we take from Bissantz & Gerhard (2002). This is significantly larger than most of the values found by DB98, and comparable to the bulge masses found by Widrow et al. (2008). This is somewhat smaller than the masses determined by many purely photometric studies (e.g. Dwek et al., 1995; Picaud & Robin, 2004; Launhardt et al., 2002), but, as noted in Section 2.1, there is an important dependence on assumptions made about the disc. We note that the stellar mass within the inner in our bestfitting model, , is very close to the bulge mass found by Picaud & Robin (2004) assuming a central hole in the disc: .
The baryonic TullyFisher relation describes the relationship between total baryonic mass and the circular speed at some radius, observed in external galaxies. McGaugh et al. (2010) found for disc galaxies, with a scatter that is entirely consistent with observational uncertainty, where was the actual observed circular speed. Our models have , suggesting that the Milky Way has a significantly higher rotational speed (or, equivalently, lower baryonic mass) than the TullyFisher relation predicts. A similar offset from the baryonic TullyFisher relationship was found for the Milky Way by Flynn et al. (2006).
In Section 3.5, we noted the difficulty of finding rigorous constraints on Galactic structure at large radii, and described some of problems with commonly cited constraints. It is still instructive to compare our models to the results of these previous studies. The escape speed at the Sun of our bestfitting model is , with mean and standard deviation over all models of and respectively. This compares to the quoted confidence interval to of Smith et al. (2007). The mass inside is (bestfitting), with mean and standard deviation . This is rather larger than the Xue et al. (2008) value of . Xue et al. used cosmological simulations to predict the velocity anisotropy of the BHB population they were studying – this leads to predictions of rather high radial anisotropy, which drives the mass estimate towards lower values than more isotropic or tangentially anisotropic velocity distributions would suggest. DB98 adopted the constraint on , the mass inside , , based on thenavailable data. Our bestfitting model () and the ensemble of models as a whole (mean standard deviation: ) fit well within this range. The virial mass of our bestfitting model (mean standard deviation: ) lies well within the ranges quoted by Wilkinson & Evans (1999) or Battaglia et al. (2006), and slightly below the lower quartile of the pdf given by Li & White (2008), but well above their confidence limit.
7 Conclusions
We have presented a simple Bayesian method for applying photometrically and kinematically derived constraints, and theoretical understanding of galaxy structure, to parametrized axisymmetric models of the Galaxy in order to investigate the distribution of mass in its various components. We have applied this method to models with an axisymmetric bulge, exponential discs and an NFW halo.
The method we have described is sufficiently general that it could be applied to any sensible parametrized axisymmetric mass model, with a wide range of kinematic, photometric, theoretical or other constraints. The specific constraints we apply are summarised in Table 1. We have shown that these constraints still allow a wide range of Galaxy models, and have found a bestfitting model, as well as a model that is bestfitting after key parameters have been fixed at convenient values. These models will provide a suitable starting point for producing fully dynamical Galaxy models.
We have also shown that the main features of our models are unchanged when we consider different disc scaleheights, except to the extent that they alter our prior probability distribution on the relative contributions of the thin and thick discs, which alters their relative contributions in the models. It should therefore be noted that the (already weak) constraints we have on the relative contributions of the two discs are even weaker when the uncertainty in disc scaleheights is taken into account. This constraint on the ratio of thick to thin disc contributions is almost entirely that from our Jurić et al. (2008) prior, and a different choice of prior could result in a significantly different ratio.
The kinematic data we consider here does not help us to constrain the vertical density profile of the Galactic discs, but it is clear that kinematic data can be used in combination with star counts to provide greater insight into the Galactic potential above the plane (e.g. Burnett, 2010), which could then be used to improve these models.
Applying the need for consistancy in our model allows us to find tighter constraints on individual parameters than those we begin with. This is particularly noticable in the constraints our posterior pdf place on the thin disc scalelength (), the Solar radius (), and the circular velocity at the Solar radius (). We find that the Galaxy’s darkmatter halo concentration, , is larger than the average value predicted by simulations. The Galaxy’s halo is less massive than the expected value from cosmological simulations, given the Galaxy’s stellar mass (or, equivalently, the stellar mass is higher than would be expected). In contrast, the stellar mass of the Milky Way is lower than the baryonic TullyFisher relation would suggest given its circular velocity – the discrepency between the two expectations for the stellar mass is related to the high concentration of the halo. In addition to the uncertainty on individual parameters, we are able to find the correlations between different parameters demanded by our constraints.
The results described in this paper are all dependent on the choice of parametrized model and applied constraints. In particular we have not attempted to take account of the effect on the CDM halo of baryonic processes, or to consider a CDM halo which is not spherically symmetric. The systematic uncertainties on the quantities we describe are almost undoubtedly larger than the quoted statistical uncertainties.
Acknowledgments
I am very grateful to James Binney for insightful comments, and a careful reading of a draft of this paper. I thank other members of the Oxford dynamics group, both past and present, for valuable discussions. This work is supported by a grant from the Science and Technology Facilities Council.
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Appendix A Models with different disc scale heights
For completeness we now provide a table of the mean and standard deviation of the parameters of our models in cases where we consider disc scaleheights which differ from our default values. These results are discussed in Section 5.1.
Mean  0.25  0.75  670  3.03  280  3.20  95.4  0.013  17.4  8.28  
Std. Dev.      123  0.24  125  0.56  7.0  0.006  3.9  0.16  
Mean  0.25  0.90  695  3.00  278  3.24  95.3  0.012  17.8  8.29  
Std. Dev.      119  0.22  120  0.53  6.9  0.005  3.9  0.16  
Mean  0.25  1.05  713  3.01  292  3.22  95.8  0.011  18.8  8.28  
Std. Dev.      124  0.23  137  0.57  6.8  0.005  5.1  0.16  
Mean  0.30  0.75  748  2.97  232  3.31  95.3  0.011  18.3  8.28  
Std. Dev.      118  0.21  113  0.60  6.8  0.005  4.2  0.16  
Mean  0.30  0.90  
Std. Dev.      123  0.22  110  0.56  6.9  0.006  4.3  0.16  
Mean  0.30  1.05  757  2.99  236  3.28  95.6  0.012  18.1  8.29  
Std. Dev.      116  0.21  103  0.55  6.8  0.005  4.0  0.16  
Mean  0.35  0.75  777  2.97  194  3.33  95.5  0.013  17.9  8.28  
Std. Dev.      127  0.21  81  0.54  6.9  0.007  4.6  0.16  
Mean  0.35  0.90  784  2.98  208  3.30  95.6  0.012  18.2  8.29  
Std. Dev.      131  0.21  87  0.55  6.9  0.006  4.4  0.16  
Mean  0.35  1.05  790  3.00  214  3.28  95.6  0.012  18.5  8.29  
Std. Dev.      122  0.21  98  0.57  6.8  0.006  5.0  0.16  
Mean  0.25  0.75  239.0  8.96  62.7  1240  76.4  60.3  0.097  0.0107  0.140  1.24 
Std. Dev      4.7  0.65  6.2  220  5.3  7.5  0.012  0.0010  0.012  0.28 
Mean  0.25  0.90  239.3  8.94  64.0  1250  76.1  61.7  0.096  0.0106  0.120  1.27 
Std. Dev      4.6  0.64  6.2  220  5.2  7.5  0.012  0.0010  0.012  0.28 
Mean  0.25  1.05  238.9  8.99  65.7  1280  76.6  63.7  0.099  0.0104  0.101  1.36 
Std. Dev      5.0  0.64  6.4  260  5.2  7.7  0.012  0.0011  0.012  0.33 
Mean  0.30  0.75  238.7  8.94  63.7  1280  77.1  61.4  0.086  0.0106  0.140  1.30 
Std. Dev      4.8  0.64  5.9  230  5.1  7.1  0.010  0.0010  0.012  0.29 
Mean  0.30  0.90  76.5  62.0  0.085  0.0106  0.120  1.29  
Std. Dev.      4.8  0.65  6.3  240  5.3  7.6  0.010  0.0010  0.012  0.30 
Mean  0.30  1.05  239.3  8.97  65.0  1260  76.1  62.7  0.085  0.0105  0.100  1.31 
Std. Dev      4.8  0.64  6.5  220  5.2  7.7  0.010  0.0010  0.012  0.30 
Mean  0.35  0.75  239.0  8.96  63.5  1260  76.5  61.1  0.077  0.0106  0.140  1.28 
Std. Dev      4.9  0.65  6.3  240  5.3  7.6  0.010  0.0011  0.012  0.31 
Mean  0.35  0.90  239.2  8.97  64.9  1260  76.5  62.5  0.076  0.0105  0.120  1.32 
Std. Dev      4.9  0.65  6.6  240  5.4  7.8  0.010  0.0011  0.012  0.31 
Mean  0.35  1.05  239.3  8.97  65.7  1270  76.4  63.5  0.077  0.0104  0.100  1.35 
Std. Dev      4.8  0.64  6.8  260  5.4  8.1  0.010  0.0011  0.012  0.34 