TRIGONOMETRIC PARALLAXES OF HIGH MASS STAR FORMING REGIONS: THE STRUCTURE AND KINEMATICS OF THE MILKY WAY

We adjusted the Galactic parameters so as to best match the data to the spatial-kinematic model using a Bayesian fitting approach. The posteriori PDFs of the parameters were estimated with McMC trials that were accepted or rejected by the Metropolis–Hastings algorithm. While a simple axi-symmetric model for the Galaxy may be a reasonable approximation for the majority of sources, a significant minority of outliers are expected for a variety of well known reasons. For example, the gravitational potential of the Galactic bar (or bars), which extend 3–4 kpc from the Galactic center (Liszt & Burton 1980; Blitz & Spergel 1991; Hammersley et al. 2000; Benjamin et al. 2005) is expected to induce large non-circular motions for sources in its vicinity. Indeed, some of these sources show large peculiar motions, although based on a nearly flat rotation curve extrapolated inward from measurements outside this region (Sanna et al. 2014). Therefore, we removed the eight sources within 4 kpc of the Galactic center (i.e., excluding G000.67−00.03, G009.62+00.19, G010.47+00.02, G010.62−00.38, G012.02−00.03, G023.43−00.18, G023.70−00.19, G027.36−00.16) before model fitting.

In the Galaxy's spiral arms, super-bubbles created by multiple supernovae can accelerate molecular clouds to ≈20 km s⁻¹ (Sato et al. 2008). It is probably not possible, prior to fitting, to determine which sources have been thus affected and are likely kinematically anomalous. Therefore, we initially used an "outlier-tolerant" Bayesian fitting scheme described by Sivia & Skilling (2006) as a "conservative formulation," which minimizes the effects of deviant points on estimates of the fitted parameters. For this approach, one maximizes

$$\begin{equation*} \sum _{i=1}^N \sum _{j=1}^3 \; {\ln \, \left((1-e^{-R_{i,j}^2/2})/R_{i,j}^2 \right)}, \end{equation*}$$

where the weighted residual R_{i, j} = (v_{i, j} − m_{i, j})/w_{i, j} (i.e., the data (v) minus model (m) divided by the uncertainty (w) of the i^th of N sources and j^th velocity component). For large residuals, this formulation assigns a 1/R² probability, compared to a Gaussian probability of $e^{-R^2/2}$ which vanishes rapidly. Thus, for example, a 5σ outlier has a reasonable (4%) probability with the outlier-tolerant approach, compared to ≈10⁻⁶ probability for Gaussian errors in the least-squares method, and will not be given excessive weight when adjusting parameters. Once the outliers were identified and removed, we assumed Gaussian data uncertainties and fitted data by maximizing

$$\begin{equation*} \sum _{i=1}^N \sum _{j=1}^3 {-R_{i,j}^2/2}, \end{equation*}$$

essentially least-squares fitting.

Our choice of weights (w) for the data in the model fitting process was discussed in detail in Reid et al. (2009b). We include both measurement uncertainty and the effects of random (Virial) motions of a massive young star (with maser emission) with respect to the average motion of the much larger and more massive HMSFR when weighting the differences between observed and modeled components of motion. Specifically the proper motion and Doppler velocity weights were given by $w(\mu) = \sqrt{{{{\sigma ^2_\mu + \sigma ^2_{{\rm Vir}}/d^2_s}}}}$ and $w({v_{{\rm Helio}}}) = \sqrt{{{{\sigma ^2_v + \sigma ^2_{{\rm Vir}}}}}}$, where $\sigma ^2_{{\rm Vir}}$ is the expected (one-dimensional) Virial dispersion for stars in an HMSFR. We adopted σ_Vir = 5 km s⁻¹, appropriate for HMSFRs with ∼10⁴ M_☉within a radius of ∼1 pc, and did not adjust this value. As will be seen in Section 4.1, the vast majority of the velocity data can be fit with a $\chi ^2_\nu$ near unity with these weights. Note that we were fairly conservative when assigning motion uncertainties for individual stars based on the maser data (see Section 2), and this may result in a slightly low σ_Vir value in order to achieve unity $\chi ^2_\nu$ fits.

In order to model the observations, one needs prior constraints on the non-circular motion of our measurement "platform" (i.e., the solar motion parameterized by U_☉, V_☉, W_☉) and/or the average peculiar motion of the sources being measured (parameterized by $\overline{U_s}$, $\overline{V_s}$, $\overline{W_s}$). Allowing for a non-zero average source peculiar motion can be thought of as a first approximation of the kinematic effects of spiral structure. In Reid et al. (2009b), we assumed the solar motion determined by Dehnen & Binney (1998) based on Hipparcos measurements and concluded that HMSFRs lagged circular orbital speeds by 15 km s⁻¹ (i.e., ${\overline{V_s}}=-15$ km s⁻¹). The observed orbital lag (${\overline{V_s}}<0$) is insensitive to the value adopted for Θ₀, but it is strongly correlated with the adopted solar motion component, V_☉ (Reid et al. 2009b; Honma et al. 2012). Recently, the value of the solar motion component in the direction of Galactic rotation (V_☉) has become controversial. Motivated in part by the large $\overline{V_s}$ lag in Reid et al. (2009b), Schoenrich et al. (2010) re-evaluated the standard "asymmetric-drift" approach used by Dehnen & Binney (1998) and concluded that it was biased by coupled metallicity/orbital-eccentricity effects. They suggested new solar motion values; specifically they argued for a substantial increase for V_☉ from 5 to 12 km s⁻¹. This change would decrease the average orbital lag of HMSFRs ($\overline{V_s}$) by ≈7 km s⁻¹ to a more theoretically appealing value near 8 km s⁻¹.

Based on the first year of data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE), Bovy et al. (2012) argue that the Sun's motion relative to a circular orbit in the Galaxy (i.e., a "rotational standard of rest") is 26 km s⁻¹ in the direction of Galactic rotation, suggesting that the entire solar neighborhood, which defines the LSR, leads a circular orbit by 14 km s⁻¹. Taking into account these developments, we considered a conservative prior of V_☉ = 15 ± 10 km s⁻¹, that encompasses the values of V_☉ from 5 to 26 km s⁻¹ within approximately the ±1σ range.

One could argue on theoretical grounds that HMSFRs should, on average, lag circular orbits by only a few km s⁻¹ (McMillan & Binney 2010). We observe masers in HMSFRs that are very young and the gas out of which their exciting stars formed could have responded to magnetic shocks when entering spiral arms, leading to departures from circular speeds by ≲ 10 km s⁻¹ (Roberts & Yuan 1970), apportioned between components counter to rotation and toward the Galactic center. In addition, radial pressure gradients can also reduce orbital speeds of gas slightly (Burkert et al. 2010), contributing to a small lag of ≈1 km s⁻¹. Allowing for such effects, we consider priors for $\overline{U_s}$ of 3 ± 10 km s⁻¹ and $\overline{V_s}$ of −3 ± 10 km s⁻¹ as reasonable and conservative.

Given the current uncertainty in (1) the value for the circular (V_☉) component of solar motion and (2) the magnitude of the average peculiar motions of HMSFRs, we tried four sets of priors when fitting the data:

• Set-A.  
  Adopting a loose prior for the V_☉ component of solar motion, U_☉ = 11.1 ± 1.2, V_☉ = 15 ± 10, W_☉ = 7.2 ± 1.1 km s⁻¹, and for the average peculiar motions for HMSFRs of ${\overline{U_s}}= 3\pm 10$ and ${\overline{V_s}}=-3\pm 10$ km s⁻¹.
• Set-B.  
  Using no priors for the average peculiar motions of HMSFRs, but tighter priors for the solar motion of U_☉ = 11.1 ± 1.2, V_☉ = 12.2  ±  2.1, W_☉ = 7.2  ±  1.1 km s⁻¹ from Schoenrich et al. (2010).
• Set-C.  
  Using no priors for the solar motion, but tighter priors on the average peculiar motions of HMSFRs of ${\overline{U_s}}= 3\pm 5$ and ${\overline{V_s}}=-3\pm 5$ km s⁻¹.
• Set-D.  
  Using essentially no priors for either the solar or average peculiar motions of HMSFRs, but bounding the V_☉ and $\overline{V_s}$ parameters with equal probability within ±20 km s⁻¹ of the Set-A initial values and zero probability outside that range.

Using the 95 sources with Galactocentric radii greater than 4 kpc.¹¹ the outlier-tolerant Bayesian fitting approach, and the Set-A priors as described above, we obtained the parameter estimates listed in Table 4 under fit A1. As expected for a sample with some outliers (see discussion in Section 4.1), we found a χ² = 562.6, greatly exceeded the 277 degrees of freedom, owing to a number of sources with large residuals.

Table 4. Bayesian Fitting Results

	A1	A5	B1	C1	D1
Parameter Estimates
R0(kpc)	8.15 ± 0.25	8.34 ± 0.16	8.33 ± 0.16	8.30 ± 0.19	8.29 ± 0.21
Θ0(km s−1)	238 ± 11	240 ± 8	243 ± 6	239 ± 8	238 ± 15
$d\Theta \over dR$(km s−1 kpc−1)	−0.1 ± 0.7	−0.2 ± 0.4	−0.2 ± 0.4	−0.1 ± 0.4	−0.1 ± 0.4
U☉(km s−1)	10.4 ± 1.8	10.7 ± 1.8	10.7 ± 1.8	9.9 ± 3.0	9.6 ± 3.9
V☉(km s−1)	15.1 ± 7.3	15.6 ± 6.8	12.2 ± 2.0	14.6 ± 5.0	16.1 ± 13.5
W☉(km s−1)	8.2 ± 1.2	8.9 ± 0.9	8.7 ± 0.9	9.3 ± 1.0	9.3 ± 1.0
$\overline{U_s}$(km s−1)	3.7 ± 2.4	2.9 ± 2.1	2.9 ± 2.1	2.2 ± 3.0	1.6 ± 3.9
$\overline{V_s}$(km s−1)	−2.4 ± 7.4	−1.6 ± 6.8	−5.0 ± 2.1	−2.4 ± 5.0	−1.2 ± 13.6
Fit statistics
χ2	562.6	224.9	225.1	224.7	224.1
Ndof	277	232	232	232	232
Nsources	95	80	80	80	80
$\quad r_{{R_0},{\Theta _0}}$	0.61	0.46	0.74	0.66	0.44

Notes. Fit A1 used the 95 sources in Table 1 for which Galactocentric radii exceeded 4 kpc, an "outlier tolerant" probability distribution function for the residuals (see Section 4.1), and Set-A priors: Gaussian solar motion priors of U_☉ = 11.1 ± 2.0, V_☉ = 15 ± 10, W_☉ = 7.2 ± 2.0 km s⁻¹ and average source peculiar motion priors of ${\overline{U_s}}=3\pm 10$ and ${\overline{V_s}}=-3\pm 10$ km s⁻¹. Fit A5 removed 15 sources found in fit A1 to have a motion component residual greater than 3σ, used a Gaussian probability distribution function for the residuals (i.e., least-squares), and the same priors as A1. Fits B1, C1 and D1 were similar to A5, except for the priors: B1 used the solar motion priors of Schoenrich et al. (2010) (U_☉ = 11.1 ± 2.0, V_☉ = 12.2 ± 2.1, W_☉ = 7.2 ± 2.0) km s⁻¹ and no priors for source peculiar motions; C1 used no solar motion priors and source peculiar motion priors of ${\overline{U_s}}=3\pm 5$ and ${\overline{V_s}}=-3\pm 5$ km s⁻¹; and D1 used flat priors for all parameters except $V^{{\rm Std}}_\odot$ and $\overline{V_s}$, which were given unity probability between ±20 km s⁻¹of the initial Set-A values and zero probability outside this range. The fit statistics listed are chi-squared (χ²), the number of degrees of freedom (N_dof), the number of sources used (N_sources), and the Pearson product-moment correlation coefficient for parameters R₀ and Θ₀ ($r_{{R_0},{\Theta _0}}$).

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We iteratively removed the sources with the largest residuals. Using the outlier-tolerant Bayesian fitting approach (see Section 4.1) minimizes potential bias, based on assumed "correct" parameter values, when editing data. However, to further guard against any residual bias, we first removed sources with >6σ residuals, followed by re-fitting and removal of those with >4σ residuals, and finally re-fitting and removal of those with >3σ residuals (fits A2, A3 and A4, not listed here). In total, 15 sources¹² were removed.

With the resulting "clean" data set of 80 sources, we performed a least-squares fit (assuming a Gaussian PDF for the data uncertainties). We used the same loose priors (Set-A) as for model A1, namely solar motion components U_☉ = 11.1 ± 1.2, V_☉ = 15 ± 10, W_☉ = 7.2 ± 1.1 km s⁻¹ and average peculiar motions for HMSFRs of ${\overline{U_s}}= 3\pm 10$ and ${\overline{V_s}}=-3\pm 10$ km s⁻¹. This resulted in the parameter estimates listed under fit A5 in Table 4. This model produced a good χ² = 224.9 for 232 degrees of freedom and estimates of R₀ = 8.34 ± 0.16 kpc and Θ₀ = 240 ± 8 km s⁻¹. We find dΘ/dR = −0.2 ± 0.4 km s⁻¹ kpc⁻¹, indicating a very flat rotation curve for the Milky Way between radii of ≈5 and 16 kpc from the Galactic center.

Compared to the preliminary results of Reid et al. (2009b) based on 16 sources, where the Pearson product-moment correlation coefficient for R₀ and Θ₀ was high, $r_{{R_0},{\Theta _0}}=0.87$, with the larger number of sources and a better distribution across the Galaxy, these parameters are significantly less correlated, $r_{{R_0},{\Theta _0}}=0.46$. The joint and marginalized PDFs for these fundamental Galactic parameters are displayed in Figure 3.

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The circular velocity parameters are still correlated (see Section 4.10), but linear combinations of these parameters are well determined: Θ₀ + V_☉ = 255.2 ± 5.1 and ${V_\odot }-{\overline{V_s}}=17.1\pm 1.0$. Also, the angular rotation rate for the Sun's orbit about the Galactic center is constrained to ±1.4% accuracy: (Θ₀ + V_☉)/R₀ = 30.57 ± 0.43 km s⁻¹ kpc⁻¹. This value is consistent with the reflex of the apparent motion of Sgr A*, the assumed motionless supermassive black hole at the center of the Galaxy, which gives 30.26 ± 0.12 km s⁻¹ kpc⁻¹ (Reid & Brunthaler 2004).

The component of solar motion in the direction of Galactic rotation, V_☉, estimated to be 15.6 ± 6.8 km s⁻¹ is better constrained than the prior of 15 ± 10 km s⁻¹. It is consistent with the local estimate (relative to solar neighborhood stars) of 12 km s⁻¹ (Schoenrich et al. 2010) and the global estimate of Bovy et al. (2012) of 26 ± 3 km s⁻¹ (relative to stars across the Milky Way).

In order to explore the sensitivity of the modeling to our priors, we fit the clean data set using the Set-B priors: adopting the latest Hipparcos measurement of the solar motion of U_☉ = 11.1 ± 1.2, V_☉ = 12.2 ± 2.1, W_☉ = 7.2 ± 1.1 km s⁻¹ (Schoenrich et al. 2010) and no prior information on the average peculiar motion of the HMSFRs. This resulted in parameter estimates similar to those of model A5, e.g., R₀ = 8.33 ± 0.16 kpc and Θ₀ = 243 ± 6 km s⁻¹. The quality of fit, as measured by χ² = 225.1 for 232 degrees of freedom, was comparably good as for model A5. The average velocity lag of the HMSFRs relative to circular orbits, which was not constrained by priors, was ${\overline{V_s}}= -5.0\, \pm \, 2.1$ km s⁻¹. This is comparable to that found by Reid et al. (2009b), after correcting for the 7 km s⁻¹ difference in the adopted solar motion values.

Given the current uncertainty in the V_☉ component of solar motion, we fit the data with the Set-C priors, assuming no prior information for the solar motion, but using a stronger prior than for model A5 for the average peculiar motion of the HMSFRs: ${\overline{U_s}}= 3\pm 5$ and ${\overline{V_s}}=-3\pm 5$ km s⁻¹. As for model B1, we found most parameter estimates to be similar to model A5, e.g., R₀ = 8.30 ± 0.19 kpc and Θ₀ = 239 ± 8 km s⁻¹. For the solar motion, we find U_☉ = 9.9 ± 2.0, V_☉ = 14.6 ± 5.0, and W_☉ = 9.3 ± 1.0 km s⁻¹. The V_☉ value is consistent with revised Schoenrich et al. (2010; 12 km s⁻¹) solar motion, but differs by 2σ from the Bovy et al. (2012) estimate.

In order to facilitate the use of the results presented here with other Galactic parameter estimates, we performed a fit with essentially no informative priors. We did this by taking the A5 (Set-A) initial parameter values and assuming flat priors for all parameters except for V_☉ and $\overline{V_s}$. For these parameters we assumed equal probability for values within ±20 km s⁻¹ of the initial values and zero probability outside this range in order to exclude unreasonable parameter values. The parameters that remain well determined include R₀ = 8.29 ± 0.21 kpc, Θ₀ = 238 ± 15 km s⁻¹, dΘ/dR = −0.1 ± 0.4 km s⁻¹ kpc⁻¹, U_☉ = 9.6 ± 3.9 km s⁻¹, W_☉ = 9.3 ± 1.0 km s⁻¹, and ${\overline{U_s}}= 1.6 \,{\pm}\, 3.9$ km s⁻¹. The correlated velocity terms, V_☉ and $\overline{V_s}$ displayed nearly flat posteriori PDFs over their allowed ranges. However, linear combinations involving these parameters are very well constrained, Θ₀ + V_☉ = 253.8 ± 6.4 km s⁻¹ and ${V_\odot }-{\overline{V_s}}= 17.2\pm 1.2$ km s⁻¹, as well as the angular rotation rate of the Sun about the Galactic center, (Θ₀ + V_☉)/R₀ = 30.64 ± 0.41 km s⁻¹ kpc⁻¹.

Next, we investigated the sensitivity of the fundamental Galactic parameters, R₀ and Θ₀, to alternative rotation curves. When fitting, we replaced the simple linear form, Θ(R) = Θ₀ + (dΘ/dR)(R − R₀), with the empirically determined functions of Θ(R) of Clemens (1985), the power-law parameterization of Brand & Blitz (1993), a polynomial, and the "universal" rotation curve of Persic et al. (1996). We adopted the Set-A priors in order to facilitate comparisons with the A5 fit. Table 5 presents the fitting results for these rotation curves.

Table 5. Rotation Curve Results

	C-10	C-8.5	BB	Poly	Univ
Parameter Estimates
R0(kpc)	8.36 ± 0.16	8.12 ± 0.14	8.34 ± 0.16	8.34 ± 0.17	8.31 ± 0.16
Θ0(km s−1)	237 ± 8	221 ± 8	240 ± 9	241 ± 9	241 ± 8
U☉(km s−1)	10.1 ± 1.8	10.5 ± 1.8	10.5 ± 1.8	10.7 ± 1.7	10.5 ± 1.7
V☉(km s−1)	19.4 ± 6.8	25.0 ± 6.8	15.5 ± 6.8	14.7 ± 6.8	14.4 ± 6.8
W☉(km s−1)	8.9 ± 1.0	8.9 ± 1.0	8.8 ± 1.0	8.8 ± 0.9	8.9 ± 0.9
$\overline{U_s}$(km s−1)	2.4 ± 2.1	2.6 ± 2.0	2.8 ± 2.0	2.8 ± 2.0	2.6 ± 2.1
$\overline{V_s}$(km s−1)	+3.4 ± 6.8	+8.5 ± 6.8	−1.5 ± 6.8	−1.4 ± 6.8	−1.4 ± 6.8
a1(km s−1)	...	...	240 ± 9	241 ± 9	241 ± 8
a2	...	...	0.00 ± 0.02	0.5 ± 3.7	0.90 ± 0.06
a3	...	...	...	−15.1 ± 8.4	1.46 ± 0.16
Fit statistics
χ2	229.7	248.1	225.2	221.9	214.5
Ndof	233	233	231	230	230
Nsources	80	80	80	80	80
$\quad r_{{R_0},{\Theta _0}}$	0.46	0.36	0.48	0.47	0.47

Notes. Rotation curves C-10 and C-8.5 are from Clemens (1985) for old and revised IAU recommended values of (R₀ = 10 kpc, Θ₀ = 250 km s⁻¹) and (R₀ = 8.5 kpc, Θ₀ = 220 km s⁻¹), respectively. These curves have been scaled by the fitted values for these fundamental parameters. Note the higher χ² value for the C-8.5 model compared to the others in the table. The "BB" rotation curve from Brand & Blitz (1993) is a power law in radius: $\Theta (R) = {a_1}(R/{R_0})^{a_2}$. The "Poly" model is second-order polynomial in radius: Θ(R) = a₁ + a₂ρ + a₃ρ², where ρ = (R/R₀) − 1. The "Univ" curve is a universal rotation curve (Persic et al. 1996), where a₁ is the rotation speed at the optical (83% light) radius, and the other parameters are dimensionless and provide the shape. For the latter three models, Θ₀ is not an independently adjustable parameter; instead it is calculated from a₁, a₂, and a₃.

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Clemens (1985) supplied two curves with different shapes: one assuming the old IAU constants (C-10) of R₀ = 10 kpc and Θ₀ = 250 km s⁻¹ and the other assuming the revised constants (C-8.5) of R₀ = 8.5 kpc and Θ₀ = 220 km s⁻¹ currently in widespread use. The C-10 model has rotational speeds that rise faster with radius than the C-8.5 model. For either model, we fitted for different values of R₀ (which we used to scale model radii) and Θ₀ (which we used to scale rotation speeds).

Brand & Blitz (1993) parameterize their rotation curve (BB) as a power law in Galactocentric radius, R, with potentially three adjustable parameters: $\Theta (R) = {a_1}(R/{R_0})^{a_2}+ {a_3}$. For a flat rotation curve (a₂ = 0), parameters a₁ and a₃ become degenerate. Since the Galaxy's rotation curve is nearly flat over the range of radii we sample (see, e.g., model A5 above), we held a₃ at zero, solving only for a₁ and a₂. Indeed, we find the power law exponent, a₂ = −0.01 ± 0.01, essentially flat. For this formulation, a₁ = Θ₀, and in Table 5 we copy a₁ to Θ₀ to facilitate comparison with other models.

As an alternative to a power law rotation curve, we fitted a second-order polynomial (Poly) in ρ = (R/R₀) − 1: Θ(R) = a₁ + a₂ρ + a₃ρ². The model fit parameters for this form of a rotation curve are similar to those from models C-10, BB and Univ.

The universal (Univ) rotation curve of Persic et al. (1996) includes terms for an exponential disk and a halo. It can have three adjustable parameters: a₁, the circular rotation speed at the radius enclosing 83% of the optical light (R_opt); a₂ = R_opt/R₀; and a₃, a core-radius parameter for the halo contribution, nominally 1.5 for an L* galaxy. With flat priors for the three rotation curve parameters, the posteriori PDF for a₂ was bimodal, with the dominant peak at a₂ = 0.9 and a second peak with 50% of the primary's amplitude at a₂ = 0.1. Since the secondary peak seems unlikely, we refit the data using a prior for a₂ of 1.2 ± 0.5. We then obtained similar parameter values as other models (see Table 5), with the three adjustable rotation curve parameters of a₁ = 241 ± 8 km s⁻¹, a₂ = 0.90 ± 0.06, and a₃ = 1.46 ± 0.16.

All but one of the rotation curve models lead to similar values for the fundamental Galactic parameters R₀ and Θ₀ as our A5 fit. Only the Clemens "R₀ = 8.5 kpc; Θ₀ = 220 km s⁻¹" (C-8.5) rotation curve results in a marginally significant change in estimates of R₀ and Θ₀. However, this fit has a significantly poorer quality (χ² = 248.1 for 233 degrees of freedom) than, for example, the A5 fit (χ² = 224.9 for 232 degrees of freedom), and we do not consider this model further. We conclude that the fundamental Galactic parameters R₀ and Θ₀ are reasonably insensitive to a wide variety of rotation curve shapes.

With full three-dimensional location and velocity information, we can transform our heliocentric velocities to a Galactocentric reference frame and calculate the tangential (circular) speed for each HMSFR. Figure 4 plots these speeds for all sources in Table 1. Most published rotation curves for the Milky Way have come from only one component of velocity (radial), often using kinematic distances and assuming a value for Θ₀. As such, the data in Figure 4 represent a considerable advance. See also the analysis of this data set by Xin & Zheng (2013).

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It is important to remember that the transformation from heliocentric to Galactocentric frames requires accurate values of R₀, U_☉, and, most importantly, Θ₀ + V_☉, since the motion of the Sun has (by definition) been subtracted in the heliocentric frame. For most sources, increasing or decreasing the assumed value of Θ₀ + V_☉ would, correspondingly, move each data point up or down by about the same amount. Thus, the level of this, and essentially all published, rotation curves is determined mostly by Θ₀ + V_☉. Our results are the first to use fully three-dimensional data to strongly constrain all three parameters: R₀, U_☉ and Θ₀ + V.

The dashed line in Figure 4 represent the linear rotation curve from the A5 fit, based only on sources with R > 4 kpc. Sources used in the fit are plotted with filled symbols and the sources not used with open symbols. The dashed line indicates the expected rotation for sources in circular Galactic orbit (i.e., ${\overline{U_s}}={\overline{V_s}}=0$). There are now sufficient data to clearly indicate that the rotation curve drops at Galactocentric radii ≲ 4 kpc. However, given the likelihood for a significant non-axisymmetric gravitational potential within ≈4 kpc of the center, more measurements are needed before extending a rotation curve to this region as azimuthal terms may be needed.

Figure 5 shows the peculiar (non-circular) motions of all sources in Table 1 with motion uncertainties less than 20 km s⁻¹. Similar results were described in the primary papers presenting the parallaxes and proper motions for each arm (Sato et al. 2014; Wu et al. 2014; Xu et al. 2013; Choi et al. 2014; Zhang et al. 2013; Hachisuka et al. 2014). For uniformity, here the motions were calculated using the A5 fit parameters (see Table 4), but with zero correction for the average source peculiar motions. Typical peculiar motions are ≈10 km s⁻¹, but some sources have much larger values. For example, many sources in the Perseus arm in the Galactic longitude range ≈100° to ≈135° display peculiar motions ≳ 20 km s⁻¹. Many sources within ≈4 kpc of the Galactic center display even larger peculiar motions, probably indicating that the rotation curve used here is inadequate to describe their Galactic orbits, especially in the presence of the Galactic bar(s).

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The Pearson product-moment correlation coefficients, r, for all parameters from fit A5 are listed in Table 6. In the preliminary analysis of 16 HMSFRs with parallaxes and proper motions of Reid et al. (2009b), the estimates of R₀ and Θ₀ were strongly correlated (${r_{{R_0},{\Theta _0}}} =0.87$). However, with the much larger set of HMSFRs that covers a larger portion of the Galaxy, the correlation between R₀ and Θ₀ estimates is now moderate: ${r_{{R_0},{\Theta _0}}} =0.465$ for our reference A5 fit. However, there remains a significant anti-correlation between Θ₀ and V_☉ ($r_{{\Theta _0},{V_\odot }}=-0.809$), as well as a strong correlation between V_☉ and $\overline{V_s}$ ($r_{{V_\odot },{\overline{V_s}}}=0.990$). As suggested by the fitted parameter values in Table 4, our data strongly constrain the following combinations of these correlated parameters: Θ₀ + V_☉ = 255.2  ±  5.1 km s⁻¹ and ${V_\odot }- {\overline{V_s}}=17.1\,{\pm}\, 1.0$ km s⁻¹. Also, the combination of parameters that yield the angular orbital speed of the Sun about the Galactic center, (Θ₀ + V_☉)/R₀ = 30.57  ±  0.43 km s⁻¹ kpc⁻¹, is more tightly constrained than the individual parameters. Figure 6 shows the marginalized PDFs for these combinations of parameters.

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Table 6. Parameter Correlation Coefficients

	R0	Θ0	dΘ/dR	U☉	V☉	W☉	${\overline{U_s}}$	${\overline{V_s}}$
R0	1.000	0.465	0.103	0.452	0.023	−0.003	0.517	−0.002
Θ0	0.465	1.000	0.136	0.243	−0.796	−0.009	0.171	−0.809
dΘ/dR	0.103	0.136	1.000	−0.124	−0.009	0.025	−0.094	−0.018
U☉	0.452	0.243	−0.124	1.000	−0.014	−0.017	0.839	0.025
V☉	0.023	−0.796	−0.009	−0.014	1.000	0.011	−0.006	0.990
W☉	−0.003	−0.009	0.025	−0.017	0.011	1.000	−0.002	0.010
${\overline{U_s}}$	0.517	0.171	−0.094	0.839	−0.006	−0.002	1.000	0.028
${\overline{V_s}}$	−0.002	−0.809	−0.018	0.025	0.990	0.010	0.028	1.000

Notes. Pearson product-moment correlation coefficients for the A5 fit calculated from 10⁶ McMC trial parameter values thinned by a factor of 10. Parameter definitions are given in the text and the notes in Table 3.

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Other groups have analyzed parallax and proper motion data sets from the BeSSeL Survey and the VERA project, focusing on different assumptions and results. Bovy et al. (2009) confirmed the counter rotation of HMSFRs (assuming V_☉ = 5 km s⁻¹) noted by Reid et al. (2009b) and argued for a comparable value for Θ₀ (246  ±  30 km s⁻¹), but with considerably lower significance. Alternatively, McMillan & Binney (2010) found that the V_☉-component of solar motion of 5 km s⁻¹, provided by Dehnen & Binney (1998), should be raised to ≈12 km s⁻¹, thereby reducing the estimated counter-rotation of HMSFRs. Bobylev & Bajkova (2010), using 28 parallaxes available at that time and a Fourier analysis technique, estimated Θ₀ = 248  ±  14 km s⁻¹ and V_☉ = 11.0  ±  1.7 km s⁻¹, assuming R₀ ≡ 8.0 kpc. Finally, Honma et al. (2012), using 52 parallaxes, including some low-mass star forming regions, estimated R₀ = 8.05  ±  0.45 kpc and Θ₀ = 238  ±  14 km s⁻¹ for V_☉ ≡ 12 km s⁻¹.

The Bovy et al. (2009) re-analysis of our preliminary data employed a different approach than that of Reid et al. (2009b). Bovy et al. treat the elements of the velocity dispersion tensor of the HMSFRs as free parameters. These parameters give the expected deviations (variances and covariances) of the velocity data from a smooth, axi-symmetric model of Galactic rotation and are used to adjust the weights applied to the different velocity components when fitting the data. However, while Bovy et al. found a significant trace for the tensor, the velocity dispersion parameters were only marginally constrained; formally none of the diagonal components had >2.8σ formal significance. Also, their values for the radial and tangential components were nearly identical, suggesting that little is gained by making these free parameters versus adopting a single physically motivated value (σ_Vir) as we have done. Note that our value for σ_Vir is comparable to the dispersion parameter (Δ_v) values found by McMillan & Binney (2010), which range from about 6 to 10 km s⁻¹, but is considerably smaller than those of Bovy et al. (2009) of ≈20 km s⁻¹. The reason for this difference is unclear, but might reflect different treatments of outlying data and/or increased parameter correlations associated with the six extra parameters used in solving for the tensor elements.

If one adopts the theoretically motivated prior that HMSFRs have small peculiar motions (Set-C with no prior on the solar motion), then model fit C1 indicates V_☉ = 14.6  ±  5.0 km s⁻¹. This is a global measure of the peculiar motion of the Sun and, as such, is relative to a "rotational standard of rest" as opposed to an LSR, defined relative to stellar motions in the solar neighborhood. If solar neighborhood stars (extrapolated to a zero-dispersion sample) are, on average, stationary with respect to a circular orbit, then these two solar motion systems will be the same. Our estimate of V_☉ is consistent with the 12 km s⁻¹ value of Schoenrich et al. (2010), measured with respect to solar neighborhood stars, but there is some tension between our global estimate of V_☉ and that of Bovy et al. (2012) of 26  ±  3 km s⁻¹, as these two estimates differ by about 2σ. However, if one drops the prior that HMSFRs have small peculiar motions, then our result loses significance.

The large counter-rotation of HMSFRs, originally suggested by Reid et al. (2009b), was based on the initial Hipparcos result of Dehnen & Binney (1998) that V_☉ = 5 km s⁻¹. As the outcome of the Schoenrich et al. (2010) re-analysis of Hipparcos data, which gives V_☉ = 12 km s⁻¹, supersedes that lower V_☉ value, it now appears that any average counter-rotation of HMSFRs is ≲ 5 km s⁻¹. Given that we strongly constrain ${V_\odot }-{\overline{V_s}}=17.1\,{\pm}\, 1.0$ km s⁻¹, were one to independently constrain V_☉ with ±2 km s⁻¹ accuracy, the issue of HMSFR counter rotation could be clarified.

While our estimate of V_☉ has a large uncertainty (owing to correlations with Θ₀ and $\overline{V_s}$), we find U_☉ and W_☉ are well constrained. In fit D1, in which no informative prior was used for the components of motion either toward the Galactic center or perpendicular to the Galactic plane, we find that U_☉ = 9.6  ±  3.9 and W_☉ = 9.3  ±  1.0 km s⁻¹, respectively. Our estimate of the Sun's motion toward the Galactic center is in agreement with most other estimates, e.g., 11.1  ±  1.2 km s⁻¹ by Schoenrich et al. (2010) and 10  ±  1 km s⁻¹ by Bovy et al. (2012); see also the compilation of estimates by Coskunoglu et al. (2011).

The solar motion component perpendicular to the Galactic plane, W_☉, is generally considered to be straight forwardly determined and recent estimates typically range between 7.2 km s⁻¹ (Schoenrich et al. 2010; relative to local stars within ∼0.2 kpc) and 7.6 km s⁻¹ (Feast & Whitelock 1997; relative to stars within ∼3 kpc), with uncertainties of about ±0.5 km s⁻¹. We find a slightly larger value of W_☉ = 9.3  ±  1.0 km s⁻¹ (for model D1 which used no informative priors for the solar motion), which may be significant; the difference between the locally and our globally measured value (i.e., relative to stars across the Galaxy) is 2.1  ±  1.1 km s⁻¹. Note that one might expect a small difference between measurements with respect to a local and a global distribution of stars were the disk of the Galaxy to precess owing to Local Group torques. Simulations of galaxy interactions in a group suggest that a disk galaxy can complete one precession cycle over a Hubble time. Were the Milky Way to do this, one would expect a vertical precessional motion at a Galactocentric radius of the solar neighborhood of order R₀H₀ ∼ 0.6 km s⁻¹. It is possible that the differences in the local and global estimates of W_☉ can, in part, be explained in this manner.

Among the various forms of rotation curves that we fit to the data, the universal curve advocated by Persic et al. (1996) to apply to most spiral galaxies yielded the best fit (see discussion in Section 4.8, Table 5 and Figure 4). This rotation curve matches the flat to slightly declining run of velocity with Galactocentric radius from R ≈ 5 → 16 kpc, as well as reasonably tracing the decline in orbital velocity for R ≲ 5 kpc. However, many of the sources near the Galactic bar(s) cannot be well modeled with any axi-symmetric rotation curve.

The best fit value for our a₂ parameter (R_opt/R₀), coupled with our estimate of R₀ = 8.34  ±  0.16, locates R_opt at 7.5  ±  0.52 kpc. The a₂ parameter is sensitive to the slope of the rotation curve (near R_opt) and the radius at which it turns down toward the Galactic center. For example, setting a₂ = 0.7 steepens the rotation curve at large radii and moves the turn down radius to ≈3.5 kpc, while setting a₂ = 1.1 flattens the rotation curve and increases the turn down radius to ≈6.5 kpc. Given that the (thin) disk scale length, R_D = R_opt/3.2 (Persic et al. 1996), we estimate R_D = 2.44  ±  0.16 kpc. Estimates of R_D in the literature range from ≈1 → 6 kpc (Kent et al. 1991; Chang et al. 2011; McMillan 2011), with most consistent with a value between 2 → 3 kpc. Our estimate is also consistent with that of Porcel et al. (1998) who modeled the positions and magnitudes of 700,000 stars in the Two Micron Galactic Survey database and found R_D = 2.3  ±  0.3 kpc and, more recently, Bovy & Rix (2013), who modeled the dynamics of ≈16, 000 stars from the SEGUE survey and concluded that R_D = 2.14  ±  0.14 kpc.

Models A5, B1 and C1, which used different combinations of solar motion and/or average source peculiar motion priors, have comparable χ² values and all parameter estimates are statistically consistent. Because the priors for Model A5 are the least restrictive in keeping with current knowledge, we adopt those parameters as representative. Specifically, we find R₀ = 8.34  ±  0.16 kpc, Θ₀ = 240  ±  8 km s⁻¹ and dΘ/dR = −0.2  ±  0.4 km s⁻¹ kpc⁻¹. As noted in Sections 4.10 and 4.8, with the much larger data set now available, estimates of R₀ and Θ₀ are no longer strongly correlated and appear fairly insensitive to the assumed nature of the rotation curve. These parameter estimates are consistent with, but significantly better than, the preliminary values of R₀ = 8.4  ±  0.6 kpc, Θ₀ = 254  ±  16 km s⁻¹ and a nearly flat rotation curve reported in Reid et al. (2009b), based on parallaxes and proper motions of 16 HMSFRs and assuming V_☉ = 5 km s⁻¹, and R₀ = 8.05  ±  0.45 kpc and Θ₀ = 238  ±  14 km s⁻¹ from Honma et al. (2012), based on a sample of 52 sources and assuming V_☉ = 12 km s⁻¹.  

While there are numerous estimates of the distance to the Galactic center in the literature (e.g., Reid 1993), here we only compare those based on direct distance measurements. A parallax for the water masers in Sgr B2, a star forming region projected less than 0.1 kpc from the Galactic center, indicates R₀ = 7.9  ±  0.8 kpc (Reid et al. 2009c), consistent with, but considerably less accurate than, our current result. More competitive estimates of R₀ come from the orbits of "S-stars" about the supermassive blackhole Sgr A*. Combining the nearly two decades of data from the ESO NTT/VLT (Gillessen et al. 2009b) and Keck (Ghez et al. 2008) telescopes that trace more than one full orbit for the star S2 (aka S0-2), Gillessen et al. (2009a) conclude that R₀ = 8.28  ±  0.33 kpc. Recently the Keck group, extending their time sequence of observations by only a few years, announced a value of R₀ = 7.7  ±  0.4 kpc (Morris et al. 2012), in mild tension both with the Gillessen et al. (2009a) analysis and our parallax-based result. However, in the latest publication of the Keck group, Do et al. (2013) combined modeling of the distribution and space velocities of stars within the central 0.5 pc of the Galactic center with the stellar orbital result for star S0-2 (Ghez et al. 2008) and conclude that ${R_0}=8.46^{+0.42}_{-0.38}$ kpc, removing any tension with our estimate and that of the ESO group. We conclude that our estimate of R₀ = 8.34  ±  0.16 kpc is consistent with that from the Galactic center stellar orbits and is likely the most accurate to date.

Over the last four decades there have been many estimates of Θ₀ ranging from ∼170 → 270 km s⁻¹ (Kerr & Lynden-Bell 1986; Olling & Merrifield 1998). Focusing the discussion to the more direct measurements, two recent studies favor a lower and one a higher value of Θ₀ than our estimate of Θ₀ = 240  ±  8 km s⁻¹. Koposov et al. (2010) model the orbit of the GD-1 stream from a tidally disrupted stellar cluster in the Milky Way halo and estimate Θ₀ + V_☉ = 221  ±  18, where the Dehnen & Binney (1998) solar motion component of V_☉ = 5 km s⁻¹ was adopted. Recently, Bovy et al. (2012) modeled line-of-sight velocities of 3365 stars from APOGEE and find Θ₀ = 218  ±  6 km s⁻¹, but with a large value for the solar motion component in the direction of Galactic rotation, V_☉ = 26  ±  3 km s⁻¹. Their full tangential speed ${\Theta _0}+{V_\odot }=242^{+10}_{-3}$ is consistent with our value of 252.2  ±  4.8 km s⁻¹, suggesting the discrepancy between the Bovy et al. and our results are probably caused by differences in the solar motion. However, another recent study by Carlin et al. (2012), modeling the Sagittarius tidal stream, yields Θ₀ estimates from 232 → 264 km s⁻¹.

Our data also strongly constrain the angular rotation of the Sun about the Galactic center, (Θ₀ + V_☉)/R₀ = 30.57  ±  0.43 km s⁻¹ kpc⁻¹. This value can be compared with an independent and direct estimate based on the proper motion of Sgr A*, interpreted as the reflex motion from the Sun's Galactic orbit, of 30.24  ±  0.12 km s⁻¹ kpc⁻¹ (Reid & Brunthaler 2004). For R₀ = 8.34  ±  0.16 kpc, the proper motion of Sgr A* translates to Θ₀ + V_☉ = 252.2  ±  4.8 km s⁻¹, in good agreement with the parallax results. We conclude that Θ₀ exceeds the IAU recommended value of 220 km s⁻¹ with >95% probability provided that V_☉ ≲ 23 km s⁻¹. Clearly, independent global measures of V_☉ are critical to establish Θ₀ and ${\overline{V_s}}$ with high accuracy.

Changing the value of Θ₀ would have widespread impact in astrophysics. For example, increasing Θ₀ by 20 km s⁻¹ with respect to the IAU recommended value of 220 km s⁻¹ reduces kinematic distances by about 10%, leading to a decrease of 20% in estimated young star luminosities, a corresponding decrease in estimated cloud masses, and a change in young stellar object ages. Estimates of the total mass of the dark matter halo of the Milky Way scale as $V_{{\rm max}}^2 R_{{\rm Vir}}$. Since the maximum in the rotation curve (V_max) and the Virial radius (R_Vir) scale linearly with Θ₀, the mass of the halo scales as $\Theta _0^3$, leading to a 30% increase in the estimate of the Milky Way's (dark-matter dominated) mass. This, in turn, affects the expected dark-matter annihilation signal (Finkbeiner et al. 2009), increases the "missing satellite" problem (Wang et al. 2012), and increases the likelihood that the Magellanic Clouds are bound to the Milky Way (Shattow & Loeb 2009).

An interesting example of the effects of Galactic parameters on fundamental physics comes from the Hulse–Taylor binary pulsar. The dominant uncertainty in measuring the gravitational radiation damping of the binary's orbit comes from the need to correct for the effects of the Galactic accelerations of the Sun and the binary (Damour & Taylor 1991; Weisberg et al. 2010). These accelerations contribute ≈1% to the apparent orbital period decay. In 1993 when the Nobel Prize was awarded in part for this work, the IAU recommended values were R₀ = 8.5  ±  1.1 kpc and Θ₀ = 220  ±  20 km s⁻¹ (Kerr & Lynden-Bell 1986). Using these Galactic parameters, the formalism of Damour & Taylor, improved pulsar timing data of Weisberg, Nice, & Taylor, and a pulsar distance of 9.9 kpc, the binary's orbital period decays at a rate of 0.9994  ±  0.0023 times that prediction from general relativity (GR). Using the improved Galactic parameters from the A5 fit (R₀ = 8.34  ±  0.16 kpc and Θ₀ = 240  ±  8 km s⁻¹), gives a GR test value of 0.9976  ±  0.0008. This provides a three-fold improvement in accuracy. Both of these examples assumed a distance to the binary pulsar of 9.9 kpc (Weisberg et al. 2008). Given the improvement in the Galactic parameter values, the dominant uncertainty in the GR test now is the uncertain pulsar distance. A pulsar distance of 7.2 kpc would bring the GR test value to 1.0000 and a trigonometric parallax accurate to ±8%, which is possible with in-beam calibration with the VLBA, would bring the contribution of distance uncertainty down to that of the current Galactic parameter uncertainty. Alternatively, if one assumes GR is correct, the current improvement in Galactic parameters suggests that the Hulse–Taylor binary pulsar's distance is 7.2  ±  0.5 kpc.