* [51] C. A. Stuart, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rational Mech. Anal., 113 (1991), 65–96.
  * [52] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125 (1987), 153–169.
  * [53] J. Sylvester, An anisotropic inverse boundary value problem, Commun. Pure Appl. Math. 43 (1990), no 2, 201–232.
  * [54] J. Sylvester, G. Uhlmann, Inverse problems in anisotropic media Contemp. Math. 122(1991) 105–117.
  * [55] L. Oksanen, M. Salo, P. Stefanov, G. Uhlmann, Inverse Problems For Real Principal Type Operators, arXiv:2001.07599v2.
  * [56] Z. Sun, On a quasilinear inverse boundary value problem. Math. Z., 221(1996), no. 2, 293–305.
  * [57] Z. Sun, An inverse boundary-value problem for semilinear elliptic equations. Electronic Journal of Differential Equations (EJDE)[electronic only], 37(2010), 1–5.
  * [58] Z. Sun, G. Uhlmann, Inverse problems in quasilinear anisotropic media. American Journal of Mathematics, 119 (1997), no. 4, 771–797.
  * [59] Z. Sun, G. Uhlmann, Anisotropic inverse problems in two dimensions Inverse Problems, Inverse Problems 19 (2003), no. 5, 1001-1010.
  * [60] G. Uhlmann, Electrical impedance tomography and Calderón’s problem, Inverse Problems 25 (2009), 123011.
  * [61] G. Uhlmann, Inverse problems: seeing the unseen. Bull. Math. Sci. 4 (2014), no. 2, 209–279.
  * [62] Y. Wang, T. Zhou, Inverse problems for quadratic derivative nonlinear wave equations, Comm. Partial Differential Equations, 44 (2019), 1140–1158.


# The uncertainty in Galactic parameters

Paul J. McMillan, and James J. Binney
Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP,
UK E-mail<EMAIL_ADDRESS>
###### Abstract

We reanalyse the measurements of parallax, proper motion, and line-of-sight
velocity for 18 masers in high mass star-forming regions presented by Reid et
al. (2009). We use a likelihood analysis to investigate the distance of the
Sun from the Galactic centre, $R_{0}$, the rotational speed of the local
standard of rest, $v_{0}$, and the peculiar velocity of the Sun,
$\mathbf{v_{\odot}}$, for various models of the rotation curve, and models
which allow for a typical peculiar motion of the high mass star-forming
regions.

We find that these data are best fit by models with non-standard values for
$\mathbf{v_{\odot}}$ or a net peculiar motion of the high mass star-forming
regions. We argue that a correction to $\mathbf{v_{\odot}}$ is much more
likely, and that these data support the conclusion of Binney (2009) that
$V_{\odot}$ should be revised upwards from $5.2\,\mathrm{km\,s}^{-1}$ to
$11\,\mathrm{km\,s}^{-1}$. We find that the values of $R_{0}$ and $v_{0}$ that
we determine are heavily dependent on the model we use for the rotation curve,
with model-dependent estimates of $R_{0}$ ranging from $6.7\pm
0.5\,\mathrm{kpc}$ to $8.9\pm 0.9\,\mathrm{kpc}$, and those of $v_{0}$ ranging
from $200\pm 20\,\mathrm{km\,s}^{-1}$ to $279\pm 33\,\mathrm{km\,s}^{-1}$. We
argue that these data cannot be thought of as implying any particular values
of $R_{0}$ or $v_{0}$. However, we find that $v_{0}/R_{0}$ is better
constrained, lying in the range
$29.9-31.6\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$ for all models but one.

###### keywords:

Galaxy: fundamental parameters – methods: statistical – Galaxy: kinematics and
dynamics

##  1 Introduction

The fundamental parameters that define the Solar position and velocity within
the Galaxy remain uncertain to a remarkable degree. The major remaining
uncertainty is that in the distance from the Sun to the Galactic centre,
$R_{0}$. The most recent results from studies of stellar orbits in the
Galactic centre (Ghez et al., 2008; Gillessen et al., 2009) give values of
$R_{0}=(8.4\pm 0.4)\,\mathrm{kpc}$ and $R_{0}=(8.33\pm 0.35)\,\mathrm{kpc}$
respectively. These can be compared to the earlier estimate in the review by
Reid (1993) of $R_{0}=(8.0\pm 0.5)\,\mathrm{kpc}$.

The total velocity of the Sun about the Galactic centre is the sum of the
velocity of the local standard of rest (LSR), $v_{0}$, and the peculiar motion
of the Sun with respect to the LSR in the same direction, $V_{\odot}$. This
total velocity can be determined using the apparent proper motion of Sgr A*,
$\mu_{A*}$, since it is expected to be moving with a peculiar velocity less
than $\sim 1\,\mathrm{km\,s}^{-1}$ at the Galactic Centre. This constraint on
the peculiar motion of Sgr A* is justified by the observation that the
velocity of Sgr A* perpendicular to the plane is consistent with zero, with
uncertainties $\sim 1\,\mathrm{km\,s}^{-1}$ (Reid & Brunthaler, 2004), and
because this motion is thought to be due to stochastic forces from discrete
interactions with individual stars, so the velocity in the plane should be
similar to that perpendicular to it (Chatterjee, Hernquist, & Loeb, 2002).
Reid & Brunthaler (2004) found $\mu_{A*}=(6.379\pm 0.024)\,\mathrm{mas\
yr}^{-1}$, which corresponds to $(v_{0}+V_{\odot})/R_{0}=(30.2\pm
0.2)\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$, or a velocity about the
Galactic centre (using the Ghez et al. result for $R_{0}$) of
$v_{0}+V_{\odot}=(252\pm 11)\,\mathrm{km\,s}^{-1}$, where by far the dominant
uncertainty comes from the value of $R_{0}$. Analysis of the GD-1 stellar
stream (Koposov, Rix, & Hogg, 2009) has recently been used to suggest a
significantly lower value, $v_{0}=221^{+16}_{-20}\,\mathrm{km\,s}^{-1}$.

The velocity of the Sun with respect to the local standard of rest,
$\mathbf{v_{\odot}}$, is often assumed to be well known and constrained to
within $\sim 0.5\,\mathrm{km\,s}^{-1}$ because of the analysis of the dynamics
of nearby stars conducted by Dehnen & Binney (1998, henceforth DB98), and
again more recently, using identical techniques, by Aumer & Binney (2009).
DB98 found

$\begin{array}[]{ll}\mathbf{v_{\odot}}&\equiv(U_{\odot},V_{\odot},W_{\odot})\\\
&=(10.00\pm 0.36,\,5.25\pm 0.62,\,7.17\pm 0.38)\,\mathrm{km\,s}^{-1},\\\
\end{array}$ (1)

which has been widely accepted and used. However Binney (2009, henceforth B09)
suggests that the value for $V_{\odot}$ determined in these papers may be an
underestimate by $\sim 6\,\mathrm{km\,s}^{-1}$. This is because the analysis
by DB98 uses Stromberg’s equation, which is derived under the assumption that
the Galactic potential is axisymmetric, and extrapolates to zero velocity
dispersion. In practice, the Galactic potential is not axisymmetric, and the
smaller the velocity dispersion of a population, the more it is affected by
departures from axisymmetry. B09 uses a more global approach, ensuring that
more emphasis is placed on stellar populations with high velocity dispersions,
which one would expect to be less affected by the non-axisymmetry of the
Galactic potential.

Reid et al. (2009) brought together observations of masers seen in high mass
star-forming regions (hmsfrs) in the Milky Way. They used simple statistical
tools in an effort to determine the values of $R_{0}$ and $v_{0}$, using the
DB98 value of $\mathbf{v_{\odot}}$, initially under the assumption that the
hmsfrs were moving on circular orbits in a flat rotation curve, for which they
offer best-fitting parameters $R_{0}=(8.24\pm 0.55)\,\mathrm{kpc}$ and
$v_{0}=(265\pm 26)\,\mathrm{km\,s}^{-1}$, but with a high $\chi^{2}$ value.

In addition they considered a model in which the hmsfrs were moving with a
characteristic velocity with respect to their local circular velocity. They
found that they achieved a significantly improved fit to their data using this
model with a peculiar velocity $\sim 15\,\mathrm{km\,s}^{-1}$ in the opposite
direction to rotation. This fit yielded $R_{0}=(8.40\pm 0.36)\,\mathrm{kpc}$
and $v_{0}=(254\pm 16)\,\mathrm{km\,s}^{-1}$. They briefly considered a model
in which the value of $\mathbf{v_{\odot}}$ was allowed to vary, finding an
acceptable fit with $\mathbf{v_{\odot}}=(9,20,10)\,\mathrm{km\,s}^{-1}$.
However, believing the DB98 result to be “well determined” they did not pursue
the matter further.

In this paper we re-examine the data described in Reid et al. (2009), and
conduct a likelihood analysis for various models of the velocity distribution
of the maser sources. This enables us to exploit these data more thoroughly.
We also consider the implications of the results from B09 on the
interpretation of these data. In Section 2 we explain what the data consist
of, describe our models and our statistical technique; in Section 3 we give
the raw results and examine their significance. We discuss the implications of
these results in Section 4.

##  2 Methods

###  2.1 The data

These data, as given in Table 1 of Reid et al. (2009), consist of measurements
for 18 masers of: Galactic coordinates ($l_{i}$, $b_{i}$), which we can assume
to be exact; parallaxes $\pi_{i}$; proper motions $\mu_{x,i}$ and $\mu_{y,i}$;
line-of-sight velocities $v_{\mathrm{LSR},i}$. For each quantity Reid et al.
give an error and we assume that this error together with the measured value
of the quantity defines a Gaussian probability distribution for the true value
of the quantity. For the proper motions $\mu_{x}$ and $\mu_{y}$ we shall use
the more conventional notation $\mu_{\alpha}$ and $\mu_{\delta}$. The line-of-
sight velocity $v_{\rm LSR}$ is relative to an obsolete estimate of the LSR.
Fortunately the underlying heliocentric line-of-sight velocity $v_{r}$ can be
recovered from $v_{\rm LSR}$ without impact on the associated uncertainty
(Reid et al., 2009, Appendix).

###  2.2 Our models

The motion of both the Sun and the masers is dominated by circular motion
around the Galactic centre with velocity

$-v_{c}(R)\mathbf{e_{\phi}},$ (2)

where $v_{c}(R)>0$ and the minus sign reflects the fact that the Galaxy
rotates clockwise in our coordinate system. We explore three forms for
$v_{c}(R)$

  * •

A flat rotation curve, $v_{c}(R)=v_{0}$, with $v_{0}$ being a (positive) free
parameter.

  * •

A power-law rotation curve $v_{c}(R)=v_{0}(R/R_{0})^{\alpha}$, with $v_{0}$,
$R_{0}$ and $\alpha$ being free parameters.

  * •

A rotation curve corresponding to that given by the Galactic potential Model I
in §2.7 of Binney & Tremaine (2008, henceforth GDII), linearly scaled to
variable values of $R_{0}$ and $v_{0}$.

In each case, we take the parameters of $v_{c}(R)$ to have uniform prior
probability distributions. This choice ensures that we determine what _these_
data tell us, without prejudice.

Figure 1: Diagram showing a pole-on view of the Galaxy illustrating the
various peculiar velocities we consider (e.g. equation 4). This therefore only
shows the in plane components. The Sun (represented by a solid circle) is
placed at $(0,8)\,\mathrm{kpc}$, and components of its velocity
$\mathbf{v_{\odot}}$ are indicated. Three points (empty squares) are plotted
to represent masers, and the directions of the components of any systematic
offset from circular velocity in each case ($\mathbf{v}_{\mathrm{SFR}}$) are
shown, as is the peculiar velocity due to any bias in the observed radial
velocity, $v_{\mathrm{m}}$. This diagram is purely illustrative and does not
show any real data.

The velocity of a maser can be expected to differ from the local circular
speed. We separate this difference into a random component and (in some cases)
two systematic components. Like Reid et al. (2009), we consider the
possibility that the velocity of hmsfrs has a systematic offset from the
circular velocity, $\mathbf{v}_{\mathrm{SFR}}$. We also consider the
possibility that the expansion of the shell within which a maser occurs will
displace the maser’s velocity from that of the star, and if we are biased
towards seeing masers on the near side of that shell, we will see a bias in
radial velocity. We represent this as an average peculiar motion
$v_{\mathrm{m}}\mathbf{e_{*}}$, where $\mathbf{e_{*}}$ is the unit vector from
the Sun to the maser. We assume that the random component of the maser
velocities has a Gaussian distribution of dispersion $\Delta_{v}$. In total we
take the probability distribution of the velocities of masers to be

$p(\mathbf{v})\,\mathrm{d}^{3}\mathbf{v}=\frac{\,\mathrm{d}^{3}\mathbf{v}}{(2\pi\Delta_{v}^{2})^{3/2}}\exp{\left(\frac{-|\mathbf{v}-\overline{\mathbf{v}}|^{2}}{2\Delta_{v}^{2}}\right)},$
(3)

where

$\overline{\mathbf{v}}=-v_{c}(R)\mathbf{e_{\phi}}-\mathbf{v}_{\mathrm{SFR}}+v_{\mathrm{m}}\mathbf{e_{*}}.$
(4)

Notice that positive values of $v_{\phi,\rm SFR}$ imply that the masing stars
lead Galactic rotation, and that motion of the masers towards us will be
reflected in negative values of $v_{\mathrm{m}}$. In most cases we fix both
$\mathbf{v}_{\mathrm{SFR}}=0$ and $v_{\mathrm{m}}=0$. The various velocities
are illustrated in Fig. 1.

We sometimes take the Sun’s motion with respect to the LSR,
$\mathbf{v_{\odot}}$, to be specified, and sometimes we fit it to the data.

For given values of $l_{i}$, $b_{i}$, the heliocentric distance $s_{i}$, and
$R_{0}$, we deduce the probability distributions in proper motion and line-of-
sight velocity given the above probability distribution of the velocity (eq.
3). Since our focus is on the Galaxy’s parameters, we marginalise over the
$s_{i}$. We do this under the assumption that the probability distribution of
the $s_{i}$ is that implied by the assumption of Gaussian errors in the
parallaxes.

###  2.3 Statistical analysis

####  2.3.1 Likelihood function

In order to determine the best-fitting model parameters, we maximise the
likelihood function. This is

$\mathcal{L}(\mbox{\boldmath$\theta$})\propto\prod_{i}\int\,\mathrm{d}s_{i}\,p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$}),$
(5)

where $p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$})$ is the conditional
probability of the $i$th observation, given the model with parameters

$\mbox{\boldmath$\theta$}\equiv(v_{0},R_{0},\alpha,\mathbf{v_{\odot}},\Delta_{v},s_{i},\ldots).$
(6)

For each distance $s_{i}$, the model provides a multivariate Gaussian
probability distribution in $v_{r,i}$, $\mu_{\alpha,i}$ and $\mu_{\delta,i}$.
The data also provide a multivariate Gaussian distribution for the underlying
values of these variables. We obtain
$p(\mathrm{data}\,|\,\mbox{\boldmath$\theta$})$ by integrating the product of
these two Gaussian distributions over all three variables.

The likelihood function defines the a posteriori probability distribution of
the model parameters $\theta$. We use a Metropolis algorithm (Metropolis et
al., 1953) to identify the peak in this probability distribution, and to
characterise its width around the most probable model. The Metroplis algorithm
is a Markov Chain Monte Carlo method for drawing a representative sample from
a probability distribution, such as the likelihood function. We start with
some choice for the parameters $\theta$, and calculate the associated
likelihood $\mathcal{L}(\mbox{\boldmath$\theta$})$. We then

  1. 1.

choose a trial parameter set $\mbox{\boldmath$\theta$}^{\prime}$ by moving
from $\theta$ in all directions in parameter space, by an amount chosen at
random (from a Gaussian distribution);

  2. 2.

determine $\mathcal{L}(\mbox{\boldmath$\theta$}^{\prime})$;

  3. 3.

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

  4. 4.

if
$\mathcal{L}(\mbox{\boldmath$\theta$}^{\prime})/\mathcal{L}(\mbox{\boldmath$\theta$})>r$,
accept the trial parameter set, and set
$\mbox{\boldmath$\theta$}=\mbox{\boldmath$\theta$}^{\prime}$. Otherwise do not
accept it.

  5. 5.

Return to step (i).

The first few values of $\theta$ are ignored as “burn-in”, which helps to
remove the dependence on the initial value of $\theta$. We repeat the
procedure until the chain of $\theta$ values constitutes a fair sample of the
probability distribution – we establish that the burn-in period is
sufficiently long by comparing chains that have different starting $\theta$
(e.g. Gelman & Rubin, 1992).

####  2.3.2 Bayesian evidence

In addition to comparing models that differ only in the values taken by a
given set of parameters, we have to assess the value of adding a parameter to
a model. Adding a parameter is guaranteed to increase the maximum likelihood
achievable for given data, but is that increase statistically significant or
just the consequence of an enhanced ability to fit noise? The Bayesian
methodology for making such assessments is now well established and described
by Heavens (2009), for example.

One calculates the “evidence” for the model under different priors for
whatever parameters are either fixed a priori or varied. The evidence is the
total probability of the data after integrating over all parameters:

$p({\rm data}|{\rm
Model})=\int\,\mathrm{d}^{n}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm Model})\,p(\mbox{\boldmath$\theta$}|{\rm
Model}).$ (7)

Here $p(\mbox{\boldmath$\theta$}|{\rm Model})$ is the prior on the parameters.
For a parameter $\theta_{i}$ that is varied, we take the prior to be uniform
within a range of width $\Delta\theta_{i}$ that is large enough to encompass
any plausible value of $\theta_{i}$ (so $p(\theta_{i}|{\rm
Model})\,\mathrm{d}\theta_{i}=d\theta_{i}/\Delta\theta_{i}$ over this range).
The priors on parameters that are always varied play little role because we
are interested the ratio of the evidence when a parameter $\theta_{n}$ is
fixed to when it is varied. When $\theta_{n}$ is fixed, its prior is a delta
function at the chosen value, so the ratio of the evidences when $\theta_{n}$
is fixed to when it is varied is

${p({\rm data}|{\rm Model},\theta_{n}\,\mathrm{fixed})\over p({\rm data}|{\rm
Model})}={\int\,\mathrm{d}^{n-1}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm
Model})\over\int\,\mathrm{d}^{n}\mbox{\boldmath$\theta$}\,p({\rm
data}|\mbox{\boldmath$\theta$},{\rm Model})}\Delta\theta_{n},$ (8)

where upper integral excludes $\theta_{n}$ and the lower one includes it.

By Bayes theorem the ratio of the a posteriori probabilities of the model when
$\theta_{n}$ is fixed to when it is varied is the ratio of the corresponding
evidences times the ratios of the prior on $\theta_{n}$ being fixed to that on
it being variable. We take this second factor to be unity, so the a posteriori
probabilities of the models is simply the ratio of the evidences.

The integration over the space of parameters that is required to calculate the
evidence is exceedingly costly if done by brute force. We approximate it by
assuming that in the vicinity of its peak, $p({\rm
data}|\mbox{\boldmath$\theta$},{\rm model})$ can be approximated by a Gaussian
$p\propto\exp(-\mbox{\boldmath$\theta$}^{T}\cdot
K\cdot\mbox{\boldmath$\theta$})$, where $K$ is a matrix whose eigenvectors and
eigenvalues can be estimated from the output of the Metropolis algorithm. With
this approximation, the integral over parameters becomes analytic.

The Metropolis algorithm yields a set of points in parameter space which
sample $p({\rm data}|{\rm Model})$. Principal component analysis of this
sample yields the eigenvectors and eigenvalues of $K$.

##  3 Results

Table 1: Log likelihoods for best-fitting models with flat rotation curves ($\alpha=0$), power law rotation curves ($\alpha\neq 0$) and rotation curves taken from GDII. The first 9 likelihoods are for models in which the sources are assumed to have no typical velocity – for the first 3 the value of $\mathbf{v_{\odot}}$ is that found in DB98; the next 3 are for models in which $\mathbf{v_{\odot}}$ is assumed to be that suggested by B09; and the next 3 are for models in which $\mathbf{v_{\odot}}$ is allowed to vary freely; the final 3 are for cases in which $\mathbf{v_{\odot}}$ is taken to be the DB98 value, and the mean source peculiar motion with respect to their local standard of rest, $\mathbf{v}_{\mathrm{SFR}}$, is allowed to vary freely. | $v_{0}$ | $R_{0}$ | $\alpha$ | $U_{\odot}$ | $V_{\odot}$ | $W_{\odot}$ | $\Delta_{v}$ | $v_{0}/R_{0}$ | $\log(\mathcal{L})$
---|---|---|---|---|---|---|---|---|---
| $(\mathrm{km\,s}^{-1})$ | $(\mathrm{kpc})$ |  | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1})$ | $(\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1})$ |
DB98 | $200\pm 20$ | $6.7\pm 0.5$ | $0$ | $10.0$ | $5.2$ | $7.2$ | $10.0\pm 1.3$ | $30.1\pm 1.7$ | $26.8$
$209\pm 26$ | $6.9\pm 0.7$ | $0.10\pm 0.16$ | $10.0$ | $5.2$ | $7.2$ | $10.0\pm 1.3$ | $30.4\pm 1.7$ | $27.1$
$218\pm 20$ | $7.1\pm 0.5$ | GDII | $10.0$ | $5.2$ | $7.2$ | $9.3\pm 1.2$ | $30.8\pm 1.6$ | $30.7$
B09 | $215\pm 19$ | $7.0\pm 0.5$ | $0$ | $10.0$ | $11.0$ | $7.2$ | $8.1\pm 1.1$ | $30.5\pm 1.5$ | $35.7$
$228\pm 24$ | $7.4\pm 0.6$ | $0.15\pm 0.13$ | $10.0$ | $11.0$ | $7.2$ | $8.0\pm 1.1$ | $30.8\pm 1.5$ | $36.5$
$235\pm 19$ | $7.6\pm 0.5$ | GDII | $10.0$ | $11.0$ | $7.2$ | $7.6\pm 1.0$ | $31.1\pm 1.5$ | $38.9$
$\mathbf{v_{\odot}}$ free | $232\pm 24$ | $7.7\pm 0.6$ | $0$ | $8.1\pm 2.9$ | $18.6\pm 2.4$ | $9.7\pm 2.0$ | $7.1\pm 1.0$ | $30.0\pm 1.8$ | $42.5$
$258\pm 32$ | $8.6\pm 0.9$ | $0.27\pm 0.13$ | $8.1\pm 2.8$ | $19.5\pm 2.5$ | $10.1\pm 2.0$ | $7.1\pm 1.0$ | $29.9\pm 1.7$ | $45.4$
$246\pm 24$ | $8.1\pm 0.6$ | GDII | $8.3\pm 2.8$ | $16.5\pm 2.4$ | $9.9\pm 2.0$ | $7.0\pm 1.0$ | $30.3\pm 1.8$ | $43.5$
| $v_{0}$ | $R_{0}$ | $\alpha$ | $v_{R,\mathrm{SFR}}$ | $v_{\phi,\mathrm{SFR}}$ | $v_{z,\mathrm{SFR}}$ | $\Delta_{v}$ | $v_{0}/R_{0}$ | $\log(\mathcal{L})$
$\mathbf{v}_{\mathrm{SFR}}$ free | $241\pm 24$ | $7.7\pm 0.6$ | $0$ | $3.3\pm 2.8$ | $-12.9\pm 2.4$ | $2.5\pm 2.0$ | $7.0\pm 1.0$ | $31.1\pm 1.7$ | $42.9$
$279\pm 33$ | $8.9\pm 0.9$ | $0.25\pm 0.12$ | $2.0\pm 2.8$ | $-14.8\pm 2.5$ | $3.0\pm 2.0$ | $6.6\pm 1.0$ | $31.5\pm 1.6$ | $45.5$
$259\pm 23$ | $8.2\pm 0.6$ | GDII | $2.5\pm 2.8$ | $-11.0\pm 2.4$ | $2.8\pm 2.0$ | $7.1\pm 1.0$ | $31.6\pm 1.7$ | $43.8$

Figure 2: Plot showing contours of the Likelihood function (marginalised over
$\Delta_{v}$) for a model with the GDII rotation curve, and the DB98
$\mathbf{v_{\odot}}$. There is clearly a strong correlation between the values
found for $v_{0}$ and $R_{0}$. Contours are drawn at likelihood differing from
the maximum likelihood $\Delta\mathcal{L}=\,0.25$, $0.5$, $1$, $2$.

Figure 3:  Residual velocities left after the best-fitting model velocities
are subtracted, for models with GDII rotation curves and $\mathbf{v_{\odot}}$
taking the DB98 value (top-left); the B09 value (top-right); or taken to be a
free parameter (bottom-left). We also plot the residual velocities left after
subtracting the best fitting model velocities for a model with the B09
$\mathbf{v_{\odot}}$ value, and $v_{\mathrm{m}}$ taken to be a free parameter
(bottom-right). _Open squares_ correspond to the objects for which the
likelihood improves most significantly between the best-fitting DB98 model and
the best-fitting free Solar velocity one ($\Delta(\log\mathcal{L}_{i})>1$),
and _crosses_ to all other sources. The Sun is represented by a solid circle.
The _solid lined ellipses_ around each point are $1\,\sigma$ measurement error
ellipses, where the velocity error perpendicular to the line of sight in each
case is due to a combination of the uncertainty in the proper motion and in
the parallax. The uncertainty in the parallax also means that the residual
velocities shown here are not an ideal illustration of the difference between
the model and the data, because the peculiar motions can only be given for a
chosen position (in this case the position corresponding to the quoted
parallax). However this does provide a useful guide, especially for sources
relatively close to the Sun, which have small position uncertainties. The
_dashed ellipse_ in the bottom-right plot is drawn around the three objects
for which the likelihood improves most significantly when $v_{\mathrm{m}}$ is
taken to be a free parameter ($\Delta(\log\mathcal{L}_{i})>1$ for B09
$\mathbf{v_{\odot}}$)

We first investigate models in which the masers are at rest with respect to
their host stars (i.e., $v_{\rm m}=0$).

Table 1 gives the peak likelihoods of our models, as well as the best-fitting
parameters and the corresponding uncertainties. The best-fitting value of
$v_{0}$ varies in the range $(200-279)\,\mathrm{km\,s}^{-1}$, and that of
$R_{0}$ between $6.7$ and $8.9\,\mathrm{kpc}$. As Fig. 2 illustrates, $v_{0}$
and $R_{0}$ are strongly correlated with the result that $v_{0}/R_{0}$ is
confined to the relatively narrow range
$29.9-31.6\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$.

When $\mathbf{v_{\odot}}$ is a free parameter, the best-fitting values of
$U_{\odot}$ and $W_{\odot}$ are close to the DB98 values, and essentially
independent of the form of the rotation curve, whereas $V_{\odot}$ varies in
the range $16.5-19.5\,\mathrm{km\,s}^{-1}$. Similarly, when
$\mathbf{v_{\odot}}$ is fixed at the DB98 value and
$\mathbf{v}_{\mathrm{SFR}}$ is a free parameter, only $v_{\phi,\rm SFR}$ takes
a value that is far removed from that which one would naively expect – it
moves in the range $-11.0$ to $-14.8\,\mathrm{km\,s}^{-1}$. Thus the data
suggest either that the Sun is circulating significantly faster than the
circular speed, or that the hmsfr have an appreciable rotational lag.

The fits are almost perfectly degenerate between $\mathbf{v_{\odot}}$ and
$\mathbf{v}_{\mathrm{SFR}}$: they constrain only the difference between these
velocities. However, we shall argue in Section 4 that significantly non-zero
values of $\mathbf{v}_{\mathrm{SFR}}$ are physically implausible, so here we
focus on what can be inferred about $\mathbf{v_{\odot}}$ given
$\mathbf{v}_{\mathrm{SFR}}=0$. In view of the degeneracy between
$\mathbf{v_{\odot}}$ and $\mathbf{v}_{\mathrm{SFR}}$ we do not report results
obtained when both $\mathbf{v_{\odot}}$ and $\mathbf{v}_{\mathrm{SFR}}$ are
varied.

When $\mathbf{v}_{\mathrm{SFR}}=0$, the peak likelihood is higher when
$\mathbf{v_{\odot}}$ is fixed to the B09 value than when it is fixed to the
DB98 value. To assess the significance of this increase in likelihood, we
calculate the ratio of the evidences for the two models as described in
Section 2.3.2

$\frac{p(\mathrm{DB98}|\mathrm{data})}{p(\mathrm{B09}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}2\times
10^{-4}&\;\mbox{for }\alpha=0\\\ 1\times 10^{-4}&\;\;\alpha\mbox{ variable}\\\
4\times 10^{-4}&\;\;\mbox{GDII}.\\\ \end{array}\right.$ (9)

Thus regardless of the adopted rotation curve, the data strongly favour upward
revision of $V_{\odot}$ from $5.2\,\mathrm{km\,s}^{-1}$ to
$11\,\mathrm{km\,s}^{-1}$.

When $\mathbf{v_{\odot}}$ is a free parameter, the peak likelihood of the B09
model is surpassed at yet larger values of $V_{\odot}$. To determine whether
the increase in likelihood that occurs when $\mathbf{v_{\odot}}$ is set free
from the B09 value, we again calculate the relevant ratio of the evidences.
Since the two models now differ in whether $\mathbf{v_{\odot}}$ is fixed or
free, the priors on the components of $\mathbf{v_{\odot}}$ now become relevant
(cf eq. 8). We have adopted $\Delta U_{\odot}=\Delta V_{\odot}=\Delta
W_{\odot}=100\,\mathrm{km\,s}^{-1}$. With these values we have

$\frac{p(\mathrm{B09}|\mathrm{data})}{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}5&\;\mbox{for
}\alpha=0\\\ 0.6&\;\;\alpha\mbox{ variable}\\\ 40&\;\;\mbox{GDII}.\\\
\end{array}\right.$ (10)

Therefore under these assumptions the increase in likelihood attained on
setting $\mathbf{v_{\odot}}$ free from the B09 value is not statistically
significant. The ratio of evidences for the case when $\mathbf{v_{\odot}}$ is
set to the DB98 value or is set free is

$\frac{p(\mathrm{DB98}|\mathrm{data})}{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}\simeq\left\\{\begin{array}[]{ll}8\times
10^{-4}&\;\mbox{for }\alpha=0\\\ 7\times 10^{-5}&\;\;\alpha\mbox{ variable}\\\
0.01&\;\;\mbox{GDII}.\\\ \end{array}\right.$ (11)

Therefore, even with a generous choice of $\Delta U_{\odot}$, etc., the data
reject the possibility that the Sun has the DB98 value of
$\mathbf{v_{\odot}}$. Reducing the widths $\Delta U_{\odot}$, etc., of the
priors on $\mathbf{v_{\odot}}$ would strengthen the case against the DB98
value of $\mathbf{v_{\odot}}$.

The choice of $\Delta U_{\odot}\sim 100\,\mathrm{km\,s}^{-1}$ is reasonably
generous, and it is sensible to ask what value of $\Delta U_{\odot}$, etc.,
would bring ${p(\mathrm{B09}|\mathrm{data})}/{p(\mathbf{v_{\odot}}\mathrm{\
free}|\mathrm{data})}$ down to unity. In the case where the GDII rotation
curve is used, it would require a reduction to a value $\Delta U_{\odot}\sim
30\,\mathrm{km\,s}^{-1}$ – this is smaller than is reasonable given that the
value for $V_{\odot}$ found when it is allowed to vary is already $\sim
15\,\mathrm{km\,s}^{-1}$ greater than the canonical DB98 value.

It is worth noting that the case for setting $\mathbf{v_{\odot}}$ free from
either the DB98 value of the B09 value is weakest when the best-motivated
rotation curve is adopted – the GDII curve.

$v_{0}(\,\mathrm{km\,s}^{-1})$ | $R_{0}(\,\mathrm{kpc})$ | $U_{\odot}$ | $V_{\odot}$ | $W_{\odot}$ | $v_{\mathrm{m}}$ | $\Delta_{v}$ | $v_{0}/R_{0}(\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1})$ | $\log(\mathcal{L})$
---|---|---|---|---|---|---|---|---
$233\pm 20$ | $7.3\pm 0.5$ | $10.0$ | $5.2$ | $7.2$ | $-8.1\pm 2.7$ | $8.0\pm 1.1$ | $31.6\pm 1.5$ | $36.2$
$247\pm 19$ | $7.8\pm 0.4$ | $10.0$ | $11.0$ | $7.2$ | $-6.2\pm 2.4$ | $6.8\pm 0.9$ | $31.4\pm 1.4$ | $43.1$
$259\pm 25$ | $8.2\pm 0.5$ | $10.7\pm 3.3$ | $15.0\pm 2.7$ | $10.0\pm 2.0$ | $-4.8\pm 2.9$ | $6.5\pm 1.0$ | $31.4\pm 1.9$ | $46.0$
$v_{0}(\,\mathrm{km\,s}^{-1})$ | $R_{0}(\,\mathrm{kpc})$ | $v_{R,\mathrm{SFR}}$ | $v_{\phi,\mathrm{SFR}}$ | $v_{z,\mathrm{SFR}}$ | $v_{\mathrm{m}}$ | $\Delta_{v}$ | $v_{0}/R_{0}(\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1})$ | $\log(\mathcal{L})$
$271\pm 25$ | $8.2\pm 0.6$ | $2.0\pm 3.1$ | $-9.7\pm 2.8$ | $-2.5\pm 2.2$ | $-5.8\pm 3.3$ | $6.4\pm 1.1$ | $32.9\pm 2.0$ | $46.0$
Table 2:  Similar to Table 1. This shows the best-fitting parameters (and
corresponding log likelihoods) for models with a GDII rotation curve, and with
the value $v_{\mathrm{m}}$ allowed to vary.

The lowest values of both $v_{0}$ and $R_{0}$ are found for the models with
the DB98 value of $\mathbf{v_{\odot}}$, and $\mathbf{v}_{\mathrm{SFR}}$ set to
zero. Both $v_{0}$ and $R_{0}$ take larger values for models with the B09
value of $\mathbf{v_{\odot}}$, and larger still for models with either
$\mathbf{v_{\odot}}$ or $\mathbf{v}_{\mathrm{SFR}}$ allowed to vary. The
lowest values of $v_{0}$ and $R_{0}$ for the models in which the peculiar
velocities are allowed to vary are $232\,\mathrm{km\,s}^{-1}$ and
$7.7\,\mathrm{kpc}$ respectively.

Fig. 3 suggests a reason why the value of $\mathbf{v_{\odot}}$ or
$\mathbf{v}_{\mathrm{SFR}}$ in the model has such a large impact on the best
fitting values of $v_{0}$ and $R_{0}$. It shows the residual velocities of the
objects after the expected velocity is subtracted (ignoring the uncertainty in
parallax). The objects that provide the strongest indication that some change
in peculiar motion is required (indicated by open squares) are all relatively
close to the Sun. Consequently, changes in $R_{0}$ and $v_{0}$ have a
relatively small “lever arm” with which to bring the model closer to the data,
with rather large changes required in order to have any significant impact on
the expected values of the observables for those objects. The introduction of
a new peculiar velocity (either of the sources or the Sun) allows the model to
compensate for the significant discrepancy between the velocity of these
objects and those expected from the rotation curve, without producing a major
effect on the best-fitting values of $R_{0}$ and $v_{0}$.

###  3.1 Maser line-of-sight velocities

Finally we consider models in which we allow for a systematic difference
between the measured line-of-sight velocity of a maser and the line-of-sight
velocity of the exciting star – such a difference would arise if masing
occurred on the near side of an expanding shell around the star. We have done
this for all the rotation curves described in this paper, but the results are
sufficiently similar to one another that we only present (in Table 2) the
results for the GDII rotation curve.

We again compare the different models by calculating the ratios of their
evidences, using the prior for $v_{\mathrm{m}}$ uniform in a range of width
$\Delta v_{\mathrm{m}}=20\,\mathrm{km\,s}^{-1}$. We find

$\frac{p(v_{\mathrm{m}}=0|\mathrm{data})}{p(v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq\;\left\\{\begin{array}[]{ll}0.01&\;\mbox{for
DB98}\\\ 0.06&\;\;\mbox{B09}\\\ 0.3&\;\;\mathbf{v_{\odot}}\mathrm{\ free}\\\
0.3&\;\;\mathbf{v}_{\mathrm{SFR}}\mathrm{\ free}\\\ \end{array}\right.$ (12)

so in each case the data support adding the extra free parameter.

If we accept this extra parameter in our models, we have to reconsider the
evidence for a change in $\mathbf{v_{\odot}}$ (or
$\mathbf{v}_{\mathrm{SFR}}$). The relevant ratios of evidences are:

$\frac{p(\mathrm{DB98},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathrm{B09},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 0.001,$ (13)
$\frac{p(\mathrm{DB98},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathbf{v_{\odot}},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 0.3,$ (14)
$\frac{p(\mathrm{B09},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}{p(\mathbf{v_{\odot}},v_{\mathrm{m}}\mathrm{\
free}|\mathrm{data})}\simeq 220.$ (15)

So the B09 value of $\mathbf{v_{\odot}}$ is still favoured. Moreover, when
$\mathbf{v_{\odot}}$ is taken to be free, the likelihood still peaks at an
even larger value of $V$ than that of B09.

We note that the sources providing the strongest evidence that a non-zero
value of $v_{\mathrm{m}}$ is needed are those ringed in the the bottom right
plot of Fig. 3, all associated with the Perseus spiral arm (Reid et al.,
2009). If we exclude these sources, the best fitting value of $v_{\mathrm{m}}$
is $\sim-3\,\mathrm{km\,s}^{-1}$ for any assumption about $\mathbf{v_{\odot}}$
we consider. This is approximately within the uncertainty on $v_{\mathrm{m}}$,
and if we considered only this subset of the data in equation (12), the
evidence would not support adding the extra free parameter $v_{\mathrm{m}}$
(though it should be noted that it is still a radial velocity _towards_ the
Sun). It is, therefore, possible that what we have modelled as an offset in
the radial velocities of all observations is actually primarily due to a large
peculiar velocity of the objects in the Perseus arm, directed approximately in
the direction of the Sun.

##  4 Discussion

These results, and in particular the large variation in $v_{0}$ and $R_{0}$
depending on the other model parameters, makes it impossible to constrain
tightly either $v_{0}$ or $R_{0}$ from these data – the smallest best-fitting
values of these parameters are, respectively, 40 per cent and 33 per cent
smaller than the largest best-fitting values. The choice of rotation curve has
a significant impact, and we do not investigate all possible rotation curves.
It is worth noting that our two most favoured models in Table 1 (according to
Bayesian evidence, so somewhat dependent on our choice of priors) have best-
fitting values of $v_{0}$ that differ by $30\,\mathrm{km\,s}^{-1}$ and of
$R_{0}$ that differ by $1\,\mathrm{kpc}$ from one another (the models are
those with B09 $\mathbf{v_{\odot}}$ $+$ GD rotation curve and free
$\mathbf{v_{\odot}}$ $+$ power-law rotation curve, respectively).

However, for all but one of our models the best fitting $v_{0}/R_{0}$ lie in
the narrow range $(29.8-31.5)\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$, with
typical uncertainties $\sim 1.5\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$. This
corresponds to best fitting $(v_{0}+V_{\odot})/R_{0}$ in the range
$(30.9-32.5)\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$ (again, depending on
the model), with similar uncertainties. These values are slightly larger than
the value of $(v_{0}+V_{\odot})/R_{0}=30.2\pm
0.2\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}$ found from the proper motion of
Sgr A* (Reid & Brunthaler, 2004), but consistent to within (less than) twice
the uncertainties on our values.

If we allow for a bias in the maser radial velocities with respect to the Sun
by allowing the parameter $v_{\mathrm{m}}$ to take a non-zero value in
equation (4), this does improve the fit. However we have seen that this is
primarily due to a group of maser sources in the Perseus spiral arm, so the
cause may be a large peculiar motion in that part of the Galaxy. Even if the
bias in measured radial velocity is real, it does not affect the result: (i)
that in different models these data are fit best by very different values of
$v_{0}$ and $R_{0}$; (ii) that the DB98 $\mathbf{v_{\odot}}$ is rejected by
these data, and (iii) that the B09 $\mathbf{v_{\odot}}$ is favoured over
making $\mathbf{v_{\odot}}$ a free parameter.

Reid et al. (2009) focused on the idea that these data show that the hmsfrs
are orbiting the Galaxy with a velocity that lags the circular speed by
$-v_{\phi,\mathrm{SFR}}\sim 15\,\mathrm{km\,s}^{-1}$. Our analysis of similar
models shows that the data are fit better by a slightly smaller offset of
$(11-14.8)\,\mathrm{km\,s}^{-1}$ to the circular speed. We have also shown
that an equivalent fit to the data can be obtained by instead increasing to
$\sim 17\,\mathrm{km\,s}^{-1}$ the amount $V_{\odot}$ by which the Sun is
assumed to circulate faster than the circular speed. In fact, the sources that
provide the strongest statistical support for
$v_{\phi,\mathrm{SFR}}\sim-15\,\mathrm{km\,s}^{-1}$ are found near the Sun
(Fig. 3), so a change in $\mathbf{v_{\odot}}$ has a very similar effect to a
change in $\mathbf{v}_{\mathrm{SFR}}$. More maser data at different Galactic
azimuths could break this degeneracy, as well as reducing the correlation
between the values of $v_{0}$ and $R_{0}$ seen in Fig. 2.

If the proposal of Reid et al. (2009) that
$v_{\phi,\mathrm{SFR}}\simeq-15\,\mathrm{km\,s}^{-1}$ were correct, the hmsfrs
would all have to be close to apocentre. Since the lifetime of an hmsfr is
short compared to a typical epicycle period and spiral structure must play a
significant role in their formation, it is not inherently implausible that the
maser stars are all close to apocentre. However, two arguments make it
unlikely that the maser stars are on orbits as eccentric as is implied by a
$15\,\mathrm{km\,s}^{-1}$ offset from circular motion at apocentre.

First, non-axisymmetric structure in the Galaxy’s potential modulates the
tangential velocity of gas by only $\sim 7\,\mathrm{km\,s}^{-1}$ (e.g. Binney
& Merrifield, 1998, §9.2.3), so the hmsfrs would be moving with a peculiar
velocity significantly higher than the gas from which they formed. Second, by
GDII (equation 3.100), a quarter of an epicycle period later the $U$
velocities of these stars would be
$(2\Omega/\kappa)15\,\mathrm{km\,s}^{-1}\simeq 22\,\mathrm{km\,s}^{-1}$, where
$\Omega$ and $\kappa$ are the circular and radial frequencies at the star’s
location. Hence the radial velocity dispersion of a population of such stars
would be at least $\sim 22/\surd 2\simeq 15\,\mathrm{km\,s}^{-1}$. The
velocity dispersion of stars observed locally increases with age on account of
heating by irregularities in the Galaxy’s gravitational field (e.g. GDII
§10.4.1) and the velocity dispersion of the bluer stars in the Hipparcos
catalogue is $\sim 10\,\mathrm{km\,s}^{-1}$ (Aumer & Binney, 2009) rather than
$15\,\mathrm{km\,s}^{-1}$. Thus the conjecture of Reid et al. (2009) not only
requires the masing stars to be confined to apocentre but also requires them
to have more eccentric orbits than the generality of young stars. Recognising
this problem, Reid et al. suggested that the orbits of these stars became more
circular early in their lives. However, any random scattering process spreads
stars more widely in phase space and therefore increase the mean eccentricity
of the masing stars. Only a dissipative process could increase the phase-space
density of these stars by moving them to more circular orbits, and no such
process is known.

If the B09 value of $V_{\odot}$ is correct, the lag to circular motion
required to optimise the fit to the data is so small ($\sim
5\,\mathrm{km\,s}^{-1}$) that the above objections to the proposal that hmsfr
systematically lag rotation become moot. However, the formalism of Bayesian
inference says that when the B09 value of $V_{\odot}$ is accepted, there is no
convincing evidence for a systematic lag of the hmsfr.

Throughout this paper we have proceeded under the assumption that the velocity
of the LSR is the same thing as the circular velocity at $R_{0}$. It is worth
noting that this is not necessarily the case. The LSR is defined to be the
velocity of a closed orbit as it passes the current Solar position, which will
only be a circular orbit if the Galactic potential is axisymmetric. Therefore
the velocity of the LSR may be offset from the circular velocity curve.
However (much like the possibility of a systematic lag of the hmsfr) we should
note that if we accept the B09 value of $V_{\odot}$, there is no convincing
evidence for this offset.

We conclude that these data provide a compelling case for revising $V_{\odot}$
upward. Our Bayesian analysis (equation 10) supports the revision upwards of
$V_{\odot}$ to $11\,\mathrm{km\,s}^{-1}$ suggested by B09 on the basis of
modelling predominantly old disc stars with relatively large random
velocities. It should, however, be recognised that an even better fit to
_these_ data comes from somewhat larger upward revisions to $V_{\odot}\sim
16-20\,\mathrm{km\,s}^{-1}$.

##  5 Conclusions

We have reanalysed observations of hmsfrs reported in Reid et al. (2009) using
a maximum-likelihood approach to exploit fully the data, in an effort to
determine the values of $R_{0}$, the Galactocentric radius of the Sun, and
$v_{0}$, the circular velocity of the LSR. We have found that the best-fitting
values, considered separately, are strongly dependent on the Galaxy model used
to interpret the data, but that the ratio $v_{0}/R_{0}$ is consistently found
to lie in the range $29.8-31.5\,\mathrm{km\,s}^{-1}\,\,\mathrm{kpc}^{-1}$.

We have also used these data to explore the value of the Sun’s peculiar
velocity $\mathbf{v_{\odot}}$, in light of the recent argument of B09 that the
canonical DB98 value is incorrect. We find that these data support the
conclusion that $V_{\odot}$ is significantly higher than the DB98 value. By a
small but significant margin, the data prefer models with the B09 revision of
$V_{\odot}$ to $\sim 11\,\mathrm{km\,s}^{-1}$ over models in which
$\mathbf{v_{\odot}}$ is left as a free parameter (equation 10). The best-
fitting models have $U_{\odot}$ and $W_{\odot}$ near to the DB98 values, and
the even larger value of $V_{\odot}\sim 16-20\,\mathrm{km\,s}^{-1}$.

Reid et al. (2009) suggested that hmsfrs significantly lag circular rotation.
We have investigated the possibility that the hmsfrs have a typical peculiar
velocity $\mathbf{v}_{\mathrm{SFR}}$ and find that the models only constrain
the velocity difference $\mathbf{v}_{\mathrm{SFR}}-\mathbf{v_{\odot}}$. We
have argued that models in which the hmsfrs have large peculiar velocities in
the opposite direction to Galactic rotation are neither needed nor plausible.

This work, in conjunction with B09, casts severe doubt on the accuracy of the
widely used DB98 value for $V_{\odot}$. This must be a concern for anyone
interested in the dynamics within the Milky Way because velocities are
inevitably measured with respect to the Sun. Both observations of more masers
and the development of a detailed dynamical model of the Galaxy’s spiral
structure would contribute to establishing more securely what the true value
of $V_{\odot}$ is.

## Acknowledgments

We thank John Magorrian and the other members of the Oxford dynamics group for
valuable discussions. PJM is supported by a grant from the Science and
Technology Facilities Council.

## Note added after acceptance

After this paper was accepted for publication, Rygl et al. (2009) presented
similar observations of maser sources. Incorporating these observations in our
analysis does not materially affect our conclusions.

## References

  * Aumer & Binney (2009) Aumer M., Binney J. J., 2009, MNRAS, 397, 1286
  * Binney & Merrifield (1998) Binney J., Merrifield M., 1998, Galactic astronomy. Princeton University Press
  * Binney & Tremaine (2008) Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press
  * Binney (2009) Binney J. J., 2009, MNRAS, accepted, arXiv:0910.1512 (B09)
  * Chatterjee et al. (2002) Chatterjee P., Hernquist L., Loeb A., 2002, ApJ, 572, 371
  * Dehnen & Binney (1998) Dehnen W., Binney J. J., 1998, MNRAS, 298, 387
  * Gelman & Rubin (1992) Gelman A., Rubin D. B., 1992, Statistical Science, 7, 457
  * Ghez et al. (2008) Ghez A. M., Salim S., Weinberg N. N., Lu J. R., Do T., Dunn J. K., Matthews K., Morris M. R., Yelda S., Becklin E. E., Kremenek T., Milosavljevic M., Naiman J., 2008, ApJ, 689, 1044
  * Gillessen et al. (2009) Gillessen S., Eisenhauer F., Trippe S., Alexander T., Genzel R., Martins F., Ott T., 2009, ApJ, 692, 1075
  * Heavens (2009) Heavens A., 2009, arXiv: 0906.0664
  * Koposov et al. (2009) Koposov S. E., Rix H.-W., Hogg D. W., 2009, ApJ, submitted, (arXiv: 0907.1085)
  * Metropolis et al. (1953) Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., Teller E., 1953, Journal of Chemical Physics, 21, 1087–1092
  * Reid (1993) Reid M. J., 1993, AnnRA&A, 31, 345
  * Reid & Brunthaler (2004) Reid M. J., Brunthaler A., 2004, ApJ, 616, 872
  * Reid et al. (2009) Reid M. J., Menten K. M., Zheng X. W., Brunthaler A., Moscadelli L., Xu Y., Zhang B., Sato M., Honma M., Hirota T., Hachisuka K., Choi Y. K., Moellenbrock G. A., Bartkiewicz A., 2009, ApJ, 700, 137
  * Rygl et al. (2009) Rygl K. L. J., Brunthaler A., Reid M. J., Menten K. M., van Langevelde H. J., Xu Y., 2009, A&A, accepted (arXiv:0910.0150)