THE MILKY WAY'S CIRCULAR-VELOCITY CURVE BETWEEN 4 AND 14 kpc FROM APOGEE DATA

Our approach to determining the Milky Way's rotation curve from the APOGEE data is to fit a kinematical, axisymmetric model to the observed line-of-sight velocities. In its most basic form, such a kinematical model consists of (1) a model for the rotation curve, (2) a model for the distribution of peculiar velocities with respect to circular motion as a function of Galactocentric radius, and (3) a set of parameters describing the transformation of positions and velocities from the heliocentric to the Galactocentric frame. These latter parameters are the distance R₀ from the Sun to the Galactic center, and the Sun's velocity with respect to the (dynamical) center of the Galaxy. We will ignore the small projection effects that arise at non-zero Galactic latitude and the vertical dependence of V_c, since these are all at the sub-km s⁻¹ level for |b| < 15 and distances ≲ 10 kpc (e.g., Bovy & Tremaine 2012). Therefore, we are only concerned with motions in the plane of the Galactic disk, and we ignore vertical motions.

The kinematical model provides the probability distribution of line-of-sight velocities as a function of position (Galactocentric radius R and azimuth ϕ, or distance d and Galactic longitude l), after marginalizing over the component of the velocity tangential to the line of sight. This requires knowing the distance, which is only weakly constrained for giants (see Appendix A). Therefore, we additionally marginalize the kinematical model over the distance to each star, to obtain the probability of the observed line-of-sight velocity of a star, given its position (l, b), photometry (J₀, H₀, K_{s, 0}), the iron abundance [Fe/H] (taken as representative of the overall metallicity of the star), the kinematical model (represented by the circular-velocity curve V_c(R) and the distribution of peculiar velocities, distribution function (DF)), and the coordinate-transformation parameters R₀ and V^gal_☉ = (V_{R, ☉}, V_{ϕ, ☉}) (where we label this parameter as "gal" to emphasize that this is the full velocity with respect to the center of the Galaxy, not just the Sun's motion with respect to the local circular velocity):

$${\fontsize{9}{11}\selectfont{\begin{eqnarray} && p(V_{\mathrm{los}}|l,b, (J-K_s)_0,H_0,[\mathrm{Fe/H}],\; V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{DF},\mathrm{iso}) \nonumber\\ && \quad= \sum _{\mathrm{dwarf/giant}} P(\mathrm{dwarf/giant})\nonumber\\ && \qquad \times \int dd\, p(V_{\mathrm{los}}| d,l,b,V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{DF}) \nonumber\\ && \qquad \times p(d|l,b,(J-K_s)_0,H_0,[\mathrm{Fe/H}], \; \mathrm{DF},\mathrm{iso},\mathrm{dwarf/giant}). \nonumber\\ \end{eqnarray}}} \tag{ 1 }$$

The first factor within the integral is produced by marginalizing the kinematical model over the velocity component tangential to the line of sight at a given position. The second factor is given by the photometric-distance PDF, discussed in Appendix A; we include "iso" in the prior information to emphasize that we use an isochrone and IMF model to obtain the photometric-distance PDFs. We also marginalize over whether a star is a giant or a dwarf star, which changes the photometric-distance PDF; the dwarf contamination probability P(dwarf) is a single free parameter in our model. Equation (1) is the likelihood for a single star, and the total likelihood for the sample is calculated by multiplying together the individual likelihoods for all the 3365 data points. This total likelihood is what we optimize to fit models to the data.

We can also calculate the distribution of line-of-sight velocities for individual fields, which we will use later to show comparisons between the best-fit model and the data. For the distribution of V_los of a field centered at (l_field, b_field), we obtain

$$\begin{eqnarray} && p(V_{\mathrm{los}} |l_{\mathrm{field}},b_{\mathrm{field}},V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{DF},\mathrm{iso}) \nonumber\\ && \quad= \int dl\,db \int d(J-K_s)_0\,dH_0\, d[\mathrm{Fe/H}] \nonumber\\ && \qquad \times p((J-K_s)_0,H_0,[\mathrm{Fe/H}],l,b|l_{\mathrm{field}},b_{\mathrm{field}},\mathrm{DF},\mathrm{iso}) \nonumber\\ && \qquad \times \int dd\,p(V_{\mathrm{los}}|l,b,(J-K_s)_0,H_0,[\mathrm{Fe/H}],\nonumber\\ && \,\,\qquad V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{DF},\mathrm{iso})\,, \end{eqnarray} \tag{ 2 }$$

where p(J − K_s, H, [Fe/H], l, b|l_field, b_field, DF, iso) is taken directly from the field's data (that is, we sum over the data (l,b,(J − K_s)₀,H₀,[Fe/H])).

We now describe in detail the full model that is fit to the data. Our fiducial model for the distribution of velocities in the Galactocentric rest frame is that it is a single biaxial Gaussian, with a mean radial velocity of zero because of the assumption of axisymmetry, and a mean rotational velocity given by the local circular velocity V_c(R), adjusted for the local asymmetric drift V_a(R; σ_R, h_R, h_σ), which is a function of the velocity dispersion, σ_R, and the radial and radial-velocity-dispersion scale lengths of the population (h_R and h_σ, respectively, see below). The only free parameters of the distribution of peculiar velocities are therefore the radial and rotational velocity dispersions, as the mixed radial–azimuthal moment vanishes, again because of the assumption of axisymmetry. The mixed moment, or equivalently the vertex deviation, in the solar neighborhood does not vanish for the warm disk population (Dehnen & Binney 1998), but this seems likely to result from the presence of moving groups, and not from the smooth underlying distribution having a strong non-zero vertex deviation (Bovy et al. 2009b).

Assuming that the distribution of velocities in the Galactocentric cylindrical frame is Gaussian allows us to analytically marginalize the velocity distribution over the velocity component tangential to the line of sight. The resulting one-dimensional distribution is itself Gaussian, with mean (V_c(R) − V_a(R; σ, h_R, h_σ))sin (ϕ + l) and variance σ²_R(1 + sin ²(ϕ + l) (X² − 1)), where X² is the ratio of the rotational and radial-velocity variances X² ≡ σ²_ϕ/σ_R². The Galactocentric line-of-sight velocity for a star is calculated from its heliocentric line-of-sight velocity

$$\begin{equation} V_{\mathrm{los}}^{\mathrm{gal}} = V_{\mathrm{los}}^{\mathrm{helio}} - V_{R,\odot }\,\cos l + \Omega _{\odot } \,R_0\,\sin l\,, \end{equation} \tag{ 3 }$$

where we have written the Sun's velocity in terms of its angular velocity Ω_☉. This angular velocity is expected to be close to the observed proper motion of Sgr A* in the plane of the Galaxy (30.24 km s⁻¹ kpc⁻¹; Reid & Brunthaler 2004), but we leave it as a free parameter, in addition to V_{R, ☉}.

The asymmetric drift in the plane of the Galaxy of a population of stars is given by (e.g., Binney & Tremaine 2008)

$$\begin{equation} \frac{V_c(R)V_a(R)}{\sigma ^2_R(R)} = \frac{1}{2}\left[X^2-1-\frac{\partial \ln \left(\nu _*\sigma ^2_R\right)}{\partial \ln R}\right]\,, \end{equation} \tag{ 4 }$$

assuming that the radial–vertical velocity moment σ_RZ = 0 in the Galactic plane, for symmetry reasons. For an exponential-disk tracer population (stellar tracer density $\nu _*\propto e^{-R/h_R}$) with an exponentially declining radial-velocity dispersion ($\sigma _R(R) \propto e^{-R/h_\sigma }$), this formula becomes

$$\begin{equation} \frac{V_c(R)V_a(R)}{\sigma ^2_R(R)} = \frac{1}{2}\left[X^2-1+R\left(\frac{1}{h_R}+\frac{2}{h_\sigma }\right)\right]\,. \end{equation} \tag{ 5 }$$

The main unknown in this relation is X² and its dependence on R. In the fit below, we assume a constant X² with R for calculating the model's Gaussian line-of-sight velocity dispersion (see above), but to model the asymmetric drift more realistically, we use an X²(R) coming from an axisymmetric equilibrium model for the DF f(E, L) in a disk with a constant circular velocity with radius. For this DF, we use a Dehnen DF (Dehnen 1999) given by

$$\begin{equation} f_{{\rm Dehnen}}(E,L) \propto \frac{\nu _*(R_e)}{\sigma _R^2(R_e)} \, \exp \left[ \frac{\Omega (R_e)\left[L-L_c(E)\right]}{\sigma _R^2(R_e)}\right]\,, \end{equation} \tag{ 6 }$$

where R_e, L_c, and Ω(R_e) are the radius, angular momentum, and angular velocity, respectively, of the circular orbit with energy E. Using the procedure given in Section 3.2 of Dehnen (1999), we choose the ν_*(R) and σ_R(R) functions such that they reproduce a disk with exponential surface density and velocity-dispersion profiles to an accuracy of a fraction of a percent at all radii.

The asymmetric drift for the Dehnen DF for a population with σ_R(R₀) = 0.2 V_c(R₀) ≈ 44 km s⁻¹ is shown in Figure 4 for h_R = 3 kpc, h_σ = 8 kpc. The same is shown for a population with σ_R(R₀) = 0.1 V_c(R₀) ≈ 22 km s⁻¹, and the difference, normalized for the difference in σ_R between them, is small. For different values of h_R and h_σ, we correct this function using Equation (5). As different values for h_R and h_σ lead to a different X²(R), this approach is slightly incorrect, but tests show that the difference in V_a is ≲ 5% for reasonable values of h_R and h_σ. As the asymmetric drift is typically ≲ 20 km s⁻¹ for our sample, this leads to changes ≲ 1 km s⁻¹, which we can ignore. Similarly, differences in X² for our sample from this model are typically ≲ 0.2, also leading to changes ≲ 1 km s⁻¹, such that our assumption of a constant X²(R) does not strongly bias the value of the inferred circular velocity. In the exponential-disk model, the asymmetric drift for our sample is ≲ 25 km s⁻¹ at all radii, even in the innermost disk (Freeman 1987).

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The middle panel of Figure 4—now for σ_R(R₀) = 0.14 V_c, the best-fit value below—shows the difference in asymmetric drift for different values of h_R and h_σ; the main difference is at small R. This figure shows that the dependence of the asymmetric-drift correction on h_R and h_σ at R₀ is mostly ≲ 0.02 V_c—as it is unlikely that the scale length is as small as 2 kpc for this sample of stars (Bovy et al. 2012c), or that the radial-velocity-dispersion scale length is much shorter than 8 kpc—which is the same as the statistical uncertainty on the circular velocity that we infer below. Therefore, the systematic uncertainty due to the scale length of the population of stars is ≲ 4 km s⁻¹ at R₀, and likely ≲ 2 km s⁻¹. We could directly measure the scale length of our sample, but as this requires a detailed understanding of the dust distribution in the plane of the Galaxy, and as it does not contribute greatly to the error budget of our measurement, we do not attempt it here.

Our best-fit model below has a radial-velocity dispersion that is approximately flat over the range in R considered here. In this case, the asymmetric-drift correction increases with R, as is clear from Equation (5), and the difference between this dispersion profile and that considered above is shown in the middle panel of Figure 4. The bottom panel of Figure 4 shows the actual asymmetric-drift correction applied in our best-fit model below. It is clear that this correction is small (≲ 6 km s⁻¹ at R < 10 kpc); hence it does not drive our inferred value for V_c below. The bottom panel of Figure 4 also illustrates that, if the dispersion scale length were smaller than that in our best-fit model below, then the circular-velocity curve would drop more steeply than what we infer below.

In addition to the kinematical model discussed so far, we include an outlier model that consists of a Gaussian with a width of σ = 100 km s⁻¹, centered on a mean value that is a free parameter in the model. This outlier model exists primarily to deal with data-processing outliers, stars with a color and magnitude that places them beyond the disk, and close binaries with large velocity amplitudes (as we do not currently have a sufficient number of APOGEE epochs for each star to confidently remove these from the sample). In all of the fits below, the outlier fraction is ≲ 1%, and thus it does not influence our fits.

The dwarf contribution to the likelihood in Equation (1) could in principle be treated in the same way as the giant contribution, by integrating over the p(d|l, b, (J − K_s)₀, H₀, [Fe/H], DF, iso, dwarf) obtained from a similar isochrone and IMF model as for the giants. However, this is problematic, because the main sequence is much narrower than the giant branch, such that this integral is dominated by a narrow range of distances. As all of the dwarfs in our sample are faint main-sequence stars, they are expected to all be within a few 100 pc from the Sun, and thus be relatively unaffected by the radial and azimuthal gradients in velocity distribution that we model for the giant stars. Our approach is therefore to replace the integration over distance with a single evaluation at zero distance from the Sun for the dwarf part of the likelihood. The dwarf stars are otherwise modeled using the same distribution of peculiar velocities as the giant stars, i.e., we assume that they are drawn from a similar population. The dwarf contamination in our best-fit model below is 7.5%, and is similar for other fits considered below. This is close to the value expected from stellar-population-synthesis models (see Section 2).

The procedure described above makes non-trivial approximations to an ideal axisymmetric fit to the data. The most important of these is that, although we include the asymmetric drift as part of our model, we do not include the deviations from a Gaussian rotational-velocity distribution that are expected for a warm population of stars (with the same physical origin as the asymmetric drift). In addition to this assumption, we make small approximations to the full asymmetric-drift expression in Equation (4). These simplifications are expected to have small effects on our analysis: the skew of the rotational-velocity distribution is only fully visible near the tangent point where the line-of-sight velocity is equal to the rotational velocity, but our sample covers a large range of distances for each line of sight, and 85% of our sample lies at 90° < l < 270°, where there is no tangent point. As such, most stars are drawn from a distribution that is close to Gaussian.

To investigate these simplifications, we conduct a series of mock-data tests. In these tests, we sample line-of-sight velocities from the best-fit flat-rotation-curve model below, except that we use the full Dehnen DF of Equation (6)—uncorrected, such that it does not quite reproduce the assumed exponential disk—rather than the Gaussian approximation used in the fit (we ignore the best-fit X² as the Dehnen DF has X² built-in). We then fit these mock data using the approximate methodology.

These mock-data tests are described in detail in Appendix B. The results from these tests show that the approximations made in the analysis do not produce any significant bias in the fitted parameters. Fits with non-flat rotation curves to the mock-data samples indicate that we can reliably determine the shape of the rotation curve between 4 kpc < R < 14 kpc. The uncertainties in the fitted parameters for the mock data are approximately the same as those for the real data (see below), which indicates that the uncertainties in the Galactic parameters derived below for the real data are correct.

We briefly discuss here how our data are sensitive to the parameters of the model. The mean heliocentric line-of-sight velocity at a position (R, ϕ) = (d, l) is given by (see Equation (3))

$$\begin{equation} \bar{V}_{\mathrm{los}} = \bar{V}_\phi \,\sin \left(\phi +l\right)+V_{R,\odot }\,\cos l-V_{\phi ,\odot }\,\sin l\,, \end{equation} \tag{ 7 }$$

where $\bar{V}_\phi = V_c - V_a$. We have treated the position and velocity of the Sun as free parameters, without imposing any prior on R₀, and without assuming that the solar rotational velocity is given by the local circular velocity corrected for the (previously measured) velocity of the Sun with respect to the local standard of rest (LSR). We can determine the full space motion of the Sun from our sample, because the longitudinal dependence of the correction for the solar motion of the line-of-sight velocity of a star at (R, ϕ) ≡ (d, l) is sinusoidal (∝V_{ϕ, ☉} sin l), while the dependence on the rotational velocity at the position of the star varies differently for the d ≈ 1 to 10 kpc sample of APOGEE stars (∝(V_c − V_a) sin (ϕ + l)). Mock-data tests in Appendix B show that our sample spans a sufficiently wide range in Galactic longitude and distance to disentangle these two effects, and measure the solar Galactocentric motion. Because we know the Sun's rotational frequency from assuming that the radio source Sgr A* at the Galactic center is at rest with respect to the disk, we have an independent check on our inferred value for the solar motion (but we do not use the measured proper motion of Sgr A* as a prior on the solar motion).

We can then turn a measurement of $\bar{V}_{\phi }(R)$ into a measurement of V_c(R) by correcting for the asymmetric drift, V_a(R), using Equation (5). This transformation requires a measurement, foremost, of the radial-velocity dispersion, σ_R(R). We can measure σ_R almost directly using the l = 180° line of sight. By modeling σ_R(R) as an exponential and assuming a constant σ²_ϕ/σ_R², we can measure σ_R(R) from the line-of-sight velocity dispersion at each l. The only subtlety here is that, in the first quadrant (R < R₀, l < 90°), we typically sample positions where V_los is close to V_ϕ, while in the second and third quadrants (R > R₀, 90° < l < 270°), V_los is mainly composed of V_R. We can then use the measurement of σ_R(R) to correct for V_a and derive V_c.

The fact that our data sample is composed of the intermediate-age to old disk population is crucial for our ability to measure the full shape of the rotation curve. Traditional tangent-point analyses of the H i emission at l < 90°, or the measurement of V_c(R) at 90° < l < 270° using the thickness of the H i layer, are insensitive to solid-body (uniform angular speed) contributions to the rotation curve, because these do not give rise to line-of-sight velocities for circular orbits. Formally, the H i velocities are invariant under the transformation V_c(R) → V_c(R) + Ω R. However, this is not the case for a warm tracer population, as, for example, the mean rotational velocity transforms as

$$\begin{equation} \bar{V}_{\mathrm{los}} \rightarrow \bar{V}_{\mathrm{los}} + V_a(\Omega _0)\left(1-\frac{1}{1+\Omega /\Omega _0(R)}\right)\,\sin \left(\phi +l\right)\,, \end{equation} \tag{ 8 }$$

where Ω₀(R) = V_c(R)/R. Thus, the addition of Ω R to V_c(R) changes $\bar{V}_{\mathrm{los}}$ by ∼V_a Ω/Ω₀, and Ω/Ω₀ ∼ 10% gives rise to a few km s⁻¹ changes in $\bar{V}_{\mathrm{los}}$, which we can detect with our data. Equation (8) clearly shows that a warmer population, i.e., a population with a larger asymmetric drift, is more sensitive to the local slope of the rotation curve than a colder population.

In this section, we discuss the results from fitting the basic kinematical model of a single population with exponential ν_*(R) and σ_R(R) profiles to the APOGEE disk midplane data from Section 2. These are the main results of this paper. In a subsequent section, we discuss the effects of various systematics on these basic model fits, but we find in the end that they do not significantly bias the results for the Galactic parameters in this section.

The results from fitting a flat rotation curve to the APOGEE data are given in the left column of Table 2. The right column of this table gives the results when fitting a power-law model for the rotation curve V_c(R) = V_c(R₀) (R/R₀)^β. The best-fit power-law index in this case is approximately zero, such that both models for the rotation curve yield similar best-fit values for all parameters. The only significant difference between the two models for the rotation curve is the uncertainty interval for V_c(R₀), which extends to lower values of V_c in the case of the power-law fit, although the upper limit on V_c stays approximately the same. We find that V_c is tightly constrained in the flat-rotation-curve model: V_c = 218 ± 6 km s⁻¹ (68% confidence). When fitting a power law we find V_c = 218⁺⁴_{− 19} km s⁻¹, with a strong correlation between the power-law index β (or, equivalently, the local derivative dV_c/dR of the circular velocity with respect to R) and V_c. Other parameters do not exhibit any strong correlation with V_c (see discussion below).

Table 2. Results for Galactic Parameters and Tracer Properties

Parameter	Flat Rotation Curve	Power-law Vc(R) = Vc(R0) (R/R0)β
Vc(R0) (km s−1)	218 ± 6	218+4− 19
β	...	0.01+0.01− 0.10
dVc/dR(R0) (km s−1 kpc−1)	...	0.2+0.2− 2.8
A (km s−1 kpc−1)	13.5+0.2− 1.7	13.5+0.2− 1.0
B (km s−1 kpc−1)	−13.5+1.7− 0.2	−13.7+3.3− 0.1
(B2 − A2)/(2πG) (M☉ pc−3)	...	0.0002+0.0002− 0.0025
Ω0 (km s−1 kpc−1)	27.0+0.3− 3.5	27.3+0.4− 4.2
R0 (kpc)	8.1+1.2− 0.1	8.0+0.8− 0.1
VR, ☉ (km s−1)	−10.5+0.5− 0.8	−10.3+1.1− 0.1
Vϕ, ☉ (km s−1)	242+10− 3	241+5− 17
Vϕ, ☉ − Vc (km s−1)	23.9+5.1− 0.5	23.1+3.6− 0.5
$\mu _{\mathrm{Sgr\ A}^{^*}}\ (\mathrm{mas\ yr}^{-1})$	6.32+0.07− 0.70	6.36+0.09− 0.86
σR(R0) (km s−1)	31.4+0.1− 3.2	32.2+0.2− 2.6
R0/hσ	0.03+0.01− 0.27	0.06+0.01− 0.17
X2 ≡ σ2ϕ/σR2	0.70+0.30− 0.01	0.64+0.18− 0.02

Notes. Maximum-likelihood best-fit results; uncertainties are given as 68% intervals of the marginalized PDF for each parameter. The first block of parameters are the basic Galactic parameters: circular velocity V_c(R₀) at the Sun's location, the derivative of the circular-velocity curve with R, and various other representations of these two basic parameters: the Oort parameters A and B, the rotational frequency Ω₀ at the Sun's location, and the combination (B² − A²)/(2πG) that is relevant in the determination of the local dark-matter density using the cylindrical Poisson equation. The second block of parameters describes the Sun's phase-space position in the Galaxy: distance R₀ to the Galactic center, the Sun's radial velocity V_{R, ☉} (away from the Galactic center), the Sun's Galactocentric rotational velocity V_{ϕ, ☉}, and the Sun's peculiar velocity with respect to local Galactic rotation V_{ϕ, ☉} − V_c. We also give the proper motion of Sgr A* derived from the previous parameters assuming that Sgr A* is at rest with respect to the disk. The final block of parameters describe the tracer population sampled by APOGEE: radial-velocity dispersion σ_R(R₀) at R₀, the radial scale length h_σ of the radial-velocity dispersion (given as R₀/h_σ), and the ratio of the tangential to the radial-velocity variance σ²_ϕ/σ_R².

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When fitting a power-law model for the rotation curve, the local slope of the rotation curve is constrained to be −3 km s⁻¹ kpc⁻¹ < dV_c/dR < 0.4 km s⁻¹ kpc⁻¹ (68% confidence), with a best-fit value near the upper end of that range, 0.2 km s⁻¹ kpc⁻¹. When fitting a linear model for the circular velocity the constraints are similar, and we do not discuss them further; the top, middle panels of Figure 5 show the joint PDF for V_c and β (for the power-law fit) and dV_c/dR (for the linear fit). In Table 2, we also give the Oort constants A and B corresponding to the V_c and dV_c/dR inferred directly for our sample (for the case of a flat rotation curve, A and B are equal in magnitude and opposite in sign, by definition). We also present the combination (B² − A²)/(2πG), which is the correction term that must be added to the local dark-matter density to account for a non-flat rotation curve, when the local dark-matter density is inferred from local vertical kinematics using the cylindrical Poisson equation (e.g., Bovy & Tremaine 2012). It is clear that we constrain this correction term to be smaller than a fraction of the local dark-matter density: −0.0025 M_☉ pc⁻³ < (B² − A²)/(2πG) < 0.0004 M_☉ pc⁻³, compared to a local dark-matter density of 0.008 ± 0.003 M_☉ pc⁻³ (Bovy & Tremaine 2012). Thus, the correction for the non-flatness of the rotation curve for our best-fit parameters is approximately zero, and at 68% confidence it lies between −0.1 GeV cm⁻³ and 0.0 GeV cm⁻³. We also give the rotational frequency Ω₀ at the Sun's location.

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Using a power-law rotation-curve model, we estimate the uncertainty in the rotation curve by evaluating the range of V_c(R) at each R for 10,000 samples from the PDF shown in Figure 5. Thus, every sample has a power-law rotation curve, but the range at each R does not have to follow a power law itself. This exercise shows whether, in the power-law model, the value of V_c(R) is well constrained or not for each R; The result is shown in the left panel of Figure 6. The same is shown in the right panel of that figure for a fit assuming a cubic-polynomial model for the rotation curve. However, in the latter model we have imposed a prior that R₀ < 9 kpc, as a large number of samples from the PDF have R₀ > 9 kpc, which we assume to be implausible from prior data (e.g., Ghez et al. 2008; Gillessen et al. 2009). We emphasize that we do not include such a prior in any other part of the analysis presented here. We see that the rotation curve is best constrained at small radii (4 kpc < R < 8 kpc), and that it is largely consistent with being close to flat over the full range of Galactocentric radii considered here.

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We also determine the full planar phase-space position of the Sun in the Galactocentric reference frame. The current APOGEE data are relatively insensitive to the value of R₀. The 68% interval for R₀ for both the flat-rotation-curve and power-law fits is about 8 kpc < R₀ < 9 kpc, which is similar to that found for the mock data in Appendix B. The best-fit value is at the lower end of this range: 8.1 kpc when assuming a flat rotation curve, and 8 kpc for the power-law rotation-curve fit. The fully marginalized PDF for R₀ in the top left panel of Figure 5 is almost entirely flat between 8.0 kpc < R₀ < 9.3 kpc.

The Galactocentric motion of the Sun is approximately V_{R, ☉} = −10.5 ± 1.0 km s⁻¹, and V_{ϕ, ☉} = 242⁺¹⁰_{− 3} km s⁻¹ (in the case of a flat rotation curve) or V_{ϕ, ☉} = 242⁺⁵_{− 17} km s⁻¹ (for a power-law fit to the rotation curve). In the latter fit, there is a strong correlation between V_c and V_{ϕ, ☉}; the difference between the two is much better constrained: V_{ϕ, ☉} − V_c = 23.1^+3.6_{− 0.5} km s⁻¹. The marginalized PDF for V_{ϕ, ☉} − V_c in Figure 5 is well described by V_{ϕ, ☉} − V_c = 26 ± 3 km s⁻¹. We discuss the consequences of this solar motion in detail in Section 5.3, but we note here that the estimate of the angular motion of the Galactic center that we obtain from combining our fits for V_{ϕ, ☉} and R₀ is consistent with the proper motion of Sgr A* as measured by Reid & Brunthaler (2004): our estimate is $\mu _{\mathrm{Sgr\ A}^*} = 6.3^{+0.1}_{-0.7}\,\mathrm{mas\ yr}^{-1}$, compared to the direct measurement of 6.379 ± 0.024 mas yr⁻¹. We discuss the apparent discrepancy between our agreement with the proper motion of Sgr A* and our low value for V_c in Section 5.3.

The final block of parameters in Table 2 describes the tracer population. The velocity dispersion that we infer for the tracer stars is close to that expected for an old disk population: σ_R(R₀) ≈ 32.0^+0.5_{− 3} km s⁻¹ for the flat-rotation-curve and power-law fits. The ratio of the tangential-to-radial-velocity dispersions squared is 0.69 < X² < 1.0, with the best-fit value at the lower end of this range. This value is higher than expected from the epicycle approximation for a flat or falling rotation curve, which is X² ⩽ 0.5. However, this expectation holds only for a cold disk, and corrections due to the temperature of the old disk population always increase X² near R₀ (Kuijken & Tremaine 1991): the Dehnen disk DFs of Equation (6) have X² that varies spatially, and reaches approximately 0.65 near R₀ (Dehnen 1999). The best-fit value for R₀/h_σ is approximately zero, with non-zero positive values ruled out by the data: the 68% interval is −0.24 < R₀/h_σ < 0.03. Thus, the radial-velocity dispersion does not drop exponentially with radius with a scale length between 2 R₀/3 and R₀; such a drop would be expected from previous measurements of the radial dispersion as a function of R (Lewis & Freeman 1989), or from the observed exponential decline of the vertical velocity dispersion (Bovy et al. 2012b) combined with the assumption of constant σ_z/σ_R. We have attempted fits with two populations of stars with different radial scale lengths (3 kpc and 5 or 6 kpc) and radial-velocity dispersions, but the same radial-dispersion scale length. The best-fit R₀/h_σ remains zero, such that it does not seem that we are seeing a mix of multiple populations that conspire to form a flat σ_R profile.

Even with the best-fit flat radial-dispersion profile, the disk is stable over most of the range in R considered here. The Toomre Q parameter—Q = σ_R κ/(3.36GΣ) (Toomre 1964)—for a flat rotation curve is

$$\begin{eqnarray} Q &=& 1.72\,\left(\frac{\sigma _R}{32\,\mathrm{km\,s}^{-1}}\right)\, \left(\frac{V_c}{220\,\mathrm{km\,s}^{-1}}\right)\, \nonumber\\ && \times \left(\frac{R}{8\,\mathrm{kpc}}\right)^{-1}\, \left(\frac{\Sigma }{50\,M_\odot \,\mathrm{pc}^{-2}}\right)^{-1}\,. \end{eqnarray} \tag{ 9 }$$

This expression has Q > 1 down to 4.9 kpc and Q = 0.91 at R = 4 kpc for a constant σ_R(R) and a surface density Σ∝e^{−R/(3 kpc)}. Although the disk is marginally unstable in our best-fit model, this conclusion depends strongly on the assumed radial scale length: for h_R = 3.25 kpc, Q > 1 everywhere at R > 4 kpc. The flatness of the inferred σ_R profile also depends on the assumed constancy of X². Actual equilibrium axisymmetric disks, such as those having a Dehnen DF (Equation (6)), have a radially dependent X², with X² at R = 4 kpc typically smaller than at R = 8–16 kpc (Dehnen 1999, Figure 4). At R = 4 kpc, which for the present data sample is only reached for the l = 30° line of sight, the line-of-sight velocity is entirely composed of the tangential-velocity component, such that any decrease in X² leads to an increase in σ_R to sustain the same σ_ϕ. Therefore, the true σ_R(R) profile presumably is falling with R, and the entire disk at R > 4 kpc should be stable in our model.

Full PDFs for all of the parameters of the basic models discussed in this section are given in Figure 5. It is clear that with the exception of V_c and the derivative of the rotation curve—β in the power-law model and dV_c/dR in the linear model—there are no strong degeneracies among the parameters. Also included in this figure are the results from fitting all but one of the 14 APOGEE field for each field: these leave-one-out results show that no single field drives the analysis for any of the parameters.

We compare the best-fit flat-rotation-curve model with the data in Figures 7 and 8. Figure 7 shows the raw heliocentric velocities versus Galactic longitude for all of the stars in the sample, as well as the mean for each field, and the prediction for the mean from the best-fit model. This "model mean" is computed by constructing the predicted line-of-sight velocity distribution using Equation (2) and then calculating its mean. The full predicted line-of-sight velocity distribution for each field is shown in Figure 8 as the smooth curve, which is to be compared with the histogrammed data points for each field. It is clear from Figure 8 that the fits capture the overall smooth features of the distribution of line-of-sight velocities in each field, including any asymmetry in the distribution. However, it is also obvious that some of the model predictions are shifted from the observed histograms. This effect also appears in the comparison between the data mean and model mean for each field in the lower panel of Figure 7. There appears to be a distinct pattern in these residuals. Thus, the best-fit axisymmetric model fails to account for ≈10 km s⁻¹ smooth deviations in the velocity field. These deviations are likely due to non-axisymmetric streaming motions, which are expected at this level of accuracy; we discuss this further in Section 5.3. With the exception of the l = 60° field, the data seem to follow the smooth velocity distribution quite well, albeit somewhat shifted on average.

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The basic models presented in the previous section are potentially subject to systematic uncertainties, which we discuss in this section. These uncertainties are mainly related to the distances of the stars in our sample, as these are obtained from somewhat uncertain models of the color–magnitude distribution of giant stars in the near-infrared J, H, and K_s bands. We discuss here the influence of our assumed value for the scale length of the stars, of potential systematic offsets in the preliminary APOGEE metallicities used to inform the photometric distances, of systematic distance uncertainties, and of the impact of misestimated interstellar extinction and absorption. We also discuss the effect of binary contamination, of using a different set of stellar isochrones for distance estimation, and of modeling the data using many underlying populations of stars, with velocity dispersions smoothly varying with age.

In our fits, we have simply assumed a value of 3 kpc for the radial scale length, h_R, of the sample of stars used in the analysis. This assumption is based on the fact that, locally, the dominant solar-metallicity disk population has this scale length (Bovy et al. 2012c). The parameter h_R enters our analysis in two ways, the first being in the asymmetric-drift correction (Equation (5)), and the second being in the prior on the distance distribution (Equation (A1)). The former of these is the most important for determining V_c(R). Changing h_R to 2 kpc—which is highly unlikely, given that such a short scale length is not observed for any of the metal-rich, "thin disk" populations in the solar neighborhood (Bovy et al. 2012c)—increases the best-fit value to V_c = 223 km s⁻¹ for a model with a flat rotation curve, and to V_c = 220 km s⁻¹ for a power-law rotation curve. Increasing h_R to 4 kpc decreases the best-fit value to V_c(R₀) = 217 km s⁻¹. Similar changes happen for the mock data sets described in Appendix B. Therefore, we conclude that the assumed radial scale length has a negligible influence on the inferred V_c.

To determine the influence of systematic offsets in the metallicities used in the photometric distances, we have performed fits allowing for a single metallicity offset, Δ[Fe/H], for all of the stars in the sample, for a model with a flat rotation curve. We find that the best-fit Δ[Fe/H] = −0.15 ± 0.02 dex, but the best-fit value of V_c is unchanged, as is its uncertainty (see Figure 5 for the Δ[Fe/H]–V_c PDF). Although an offset of −0.15 dex might seem large, this offset should not be thought of as a measurement of [Fe/H] for the stars in our sample, because the photometric-distance PDFs, such as that in Figure 11, are relatively insensitive to large changes in [Fe/H], especially at the high-metallicity end. We have also performed a fit with a different Δ[Fe/H] for inner (l ⩽ 90°) and outer-disk fields. Such a fit finds Δ[Fe/H] = −0.12 ± 0.03 dex for the outer-disk fields, and Δ[Fe/H] = 0.70 ± 0.06 dex for the inner-disk fields, with a best-fit V_c = 220 km s⁻¹; see Figure 5 for the full PDF of the Δ[Fe/H] and V_c. As discussed previously, the photometric-distance PDFs are largely insensitive to changes in [Fe/H] at high [Fe/H]; the Padova isochrones also have an upper limit of [Fe/H]_max = 0.45 dex, such that increasing a star's [Fe/H] beyond this has no effect, because we then use [Fe/H]_max. Therefore, even a change of Δ[Fe/H] = 0.7 dex has a negligible influence on the inner, high-metallicity, disk fields. In none of these fits is our inferred range for R₀ affected.

Similarly, we have performed fits allowing for a single distance-modulus offset, Δμ, to be applied to all of the stars in our sample, or for a separate offset for inner- and outer-disk fields. In the former case, we find Δμ = −0.27 ± 0.06, corresponding to a 12% ± 3% distance offset, without any influence on the inferred value of V_c. In the inner/outer offset fit, we find 23% smaller distances at l ⩽ 90°, and 20% larger distances in the outer disk, again with a negligible influence on the best-fit V_c (V_c = 221 km s⁻¹), but with a second minimum around V_c = 190 km s⁻¹, and a wider V_c PDF that is, however, at V_c < 235 km s⁻¹ at 99% confidence. A more physical effect is to allow for an offset in the extinction, ΔA_H, again using a single offset for the whole sample, or splitting the sample into inner- and outer-disk fields. A single extinction offset is well constrained to be small: ΔA_H = −0.01 ± 0.01. Similarly, the best-fit extinction offset in the outer disk is ΔA_H = 0 ± 0.02, while in the inner disk we find some evidence for misestimated extinction, with ΔA_H = −0.2 ± 0.03. In both of these cases, the best-fit V_c increases by 1 km s⁻¹ or less, and the uncertainties remain approximately the same (see Figure 5).

As discussed in Section 2, we do not limit the sample to stars for which multiple velocity epochs are available that indicate that they are not part of a multiple system. Such a cut significantly reduces the longitudinal coverage of our sample. When we do apply this cut, we find that the best fits for V_c and all other parameters are unchanged from the results for our basic flat-rotation-curve model in Section 4.1, albeit with larger uncertainties. We find in particular that V_c = 217 ± 7 km s⁻¹, but the uncertainties in the position and velocity of the Sun are much increased.

In Appendix A, we describe how we employ Padova isochrones to estimate photometric distances to the stars in our sample. When instead we use isochrones from the BaSTI library (Pietrinferni et al. 2004), using the filter transformation from Carpenter (2001) to transform K_s to the K-band filter used by BaSTI, we find that the results of our basic models (both with a flat and a power-law rotation curve) are essentially unchanged (with changes in V_c of about 1 km s⁻¹).

To determine whether our results change when we fit the data as a mix of multiple stellar populations, with velocity dispersions varying from that of a young population to that of the oldest disk population, we have performed a fit where the likelihood for V_los in Equation (1) is generalized to

$${\fontsize{9.6}{11.6}{\begin{eqnarray} && p(V_{\mathrm{los}}|l,b, (J-K_s)_0,H_0,[\mathrm{Fe/H}], \; V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{iso}) \nonumber\\ && \quad = \sum _i P(i) \, p(V_{\mathrm{los}}|l,b, (J-K_s)_0,H_0,[\mathrm{Fe/H}],\nonumber\\ && \qquad V_c(R),R_0,V_{R,\odot },V_{\phi ,\odot },\mathrm{DF}_i,\mathrm{iso}), \end{eqnarray}}} \tag{ 10 }$$

where DF_i is the single-Gaussian population model used before, with dispersion at age τ_i

$$\begin{equation} \sigma _{R,i} = \sigma _{R,0}\,\left(\frac{\tau _i + 0.1\,\mathrm{Gyr}}{10.1\,\mathrm{Gyr}}\right)^{0.38}\, \end{equation} \tag{ 11 }$$

and using an exponentially declining star formation history, such that $P(i) \propto e^{\tau _i/(8\,\mathrm{Gyr})}$. This model follows the fit of Aumer & Binney (2009) in the solar neighborhood. We use a mix of 20 equally age-spaced populations between 1 and 10 Gyr, and find V_c = 216 ± 7 km s⁻¹, showing that this alternative model does not change our results; the full V_c–R₀ PDF is shown in Figure 5. This model has σ_{R, 0} = 38 ± 2 km s⁻¹, as expected for the old, "thin disk" population.

In the fits in Section 4, we have assumed axisymmetric models for the velocity field in the Galactic disk. However, the Milky Way is known to have some non-axisymmetric structures, such as the central bar (Blitz & Spergel 1991; Binney et al. 1991), spiral arms (e.g., Drimmel & Spergel 2001), and a potentially triaxial halo (Law et al. 2009) that may influence the measurement performed in this paper. Because we use a warm disk population as dynamical tracers, we can expect the influence of any non-axisymmetry to be smaller than for the colder gas tracers used in other studies (e.g., Levine et al. 2008), because warm tracers respond less strongly to non-axisymmetric perturbations to the Milky Way potential than cold tracers (Lin et al. 1969).

We have calculated the velocity field for a warm tracer population in various non-axisymmetric models (see J. Bovy 2013, in preparation, for full details). We follow the procedure of Dehnen (2000) and Bovy (2010) to calculate the velocity distribution at a given position for a given non-axisymmetric model for the disk, by performing backward orbit integrations until before the onset of the non-axisymmetric perturbation in an axisymmetric disk. At that time, the DF of the pre-existing axisymmetric disk is evaluated. By the conservation of phase-space density, this probability is equal to the current probability of the phase-space starting point of the orbit integration. We assume a flat rotation curve for the axisymmetric part of the potential, and we model the initial, axisymmetric DF as a Dehnen DF (Equation (6)) with a velocity dispersion of 0.2 V_c, a radial scale length of R₀/3 and a dispersion scale length of R₀. These parameters are close to the best-fit parameters for the APOGEE tracer population.

We calculate the mean velocity field for the bar model of Dehnen (2000), and use it to construct the mean, non-axisymmetric V_los field. This field is then applied as part of the model, and the result for the mean line-of-sight velocity in each field is shown in Figure 9—there is of course much more discretionary power in the full DFs. In the same manner, we calculate the mean V_los field for a two-armed logarithmic spiral with a fractional potential amplitude of 1% of the background, axisymmetric potential, a pitch angle of −15°, an angular frequency of 0.65 Ω₀ (placing the Sun near the 4:1 inner Lindblad resonance), and an angle between the Sun–Galactic-center line and the line connecting the peak of the spiral pattern at the solar radius of 20°. We do the same for models with a flat elliptical (stationary, m = 2) distortion to the potential (see Kuijken & Tremaine 1994), with a fractional amplitude of 5% and position angles of 0° and −45° with respect to the Sun–Galactic-center line. All of these perturbations were adiabatically grown. We see in Figure 9 that the influence of the bar and spiral structure is negligible in the mean V_los velocity field. An elliptical, m = 2, perturbation could distort the mean velocity field in a way that would affect our analysis. Naively, the model in the bottom panel of Figure 9 significantly reduces the residuals between the best-fit $\bar{V}_{\mathrm{los}}$ and the data $\bar{V}_{\mathrm{los}}$.

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It is clear that the data used in this paper could be used to constrain non-axisymmetric perturbations to the axisymmetric potential assumed here. We defer a full treatment of this to a subsequent paper.

There have been many previous determinations of the local circular velocity, and we discuss here how our new measurement compares to these. Determinations of the local circular velocity can be roughly divided into two groups: measurements of the Sun's velocity with respect to an object or population assumed to be at rest with respect to the Galactic center (e.g., Sgr A* or a population of halo objects), or direct measurements of the local radial force (e.g., by determining the Oort constants or the orbit of a stream of stars). The former directly measure the Sun's Galactocentric velocity, V_{ϕ, ☉}, but must assume a value for the Sun's motion with respect to the circular orbit at R₀ to arrive at V_c(R₀). The most direct measurement of this kind is the combination of the precisely measured proper motion of Sgr A* (Reid & Brunthaler 2004) with the distance to the Galactic center determined from the Keplerian orbits of S stars in the innermost parsec (Ghez et al. 2008; Gillessen et al. 2009). These measurements give R₀ ≈ 8 kpc, which combined with the proper motion of Sgr A* of 30.24 km s⁻¹ kpc⁻¹ yields a total solar velocity of 242 km s⁻¹. As noted before, this result is entirely consistent with our inferred value of the angular motion of the Galactic center, and with our V_{ϕ, ☉} = 242⁺¹⁰_{− 3} km s⁻¹. The discrepancy between the value of V_c = 229 km s⁻¹ in Ghez et al. (2008; or higher measurements also based on the proper motion of Sgr A*) and our V_c = 218 km s⁻¹ is therefore entirely due to our different value for the solar velocity with respect to V_c, and intrinsically these measurements are consistent. Similarly, the recent measurements of V_{ϕ, ☉} = 276 ± 23 km s⁻¹ and V_{ϕ, ☉} = 244 ± 14 km s⁻¹ using the kinematics of the Sgr stream (Carlin et al. 2012) are consistent with our measurement of the solar velocity.

Measurements based on the Oort constants (e.g., Feast & Whitelock 1997), or the dynamics of a cold stellar stream (Koposov et al. 2010), also indicate that V_c ∼ 220 km s⁻¹. The measurement of the Oort constants from Hipparcos proper motions by Feast & Whitelock (1997) directly measures the angular frequency of the local circular orbit, independent (in principle) from the solar motion: Ω₀ = V_c/R₀ = 27.2 ± 0.9 km s⁻¹ kpc⁻¹. This value is consistent with our determination (see Table 2) Ω₀ = 27.0^+0.3_{− 3.5} km s⁻¹ kpc⁻¹. The determination of V_c = 221 ± 18 km s⁻¹ of Koposov et al. (2010) from fitting an orbit to the cold GD-1 stellar stream at R ∼ 10 kpc is clearly consistent with our measurement, although note that they assumed the V_{ϕ, ☉} − V_c = 5.25 km s⁻¹ value for the solar motion from Dehnen & Binney (1998).

The recent measurement of V_c = 254 ± 16 km s⁻¹, from the kinematics of masers in the Galactic disk (Reid et al. 2009), was shown to have an overly optimistic precision by Bovy et al. (2009a), who concluded from a more general model for the DF of the masers that these data only imply V_c = 246 ± 30 km s⁻¹. The measurement of V_c from the maser kinematics also assumes that V_{ϕ, ☉} is given by V_c plus the locally determined solar motion of ∼10 km s⁻¹. Apart from indicating a high value for V_c, albeit with a large uncertainty, the masers were also found to be lagging with respect to circular motion by about 15 km s⁻¹, a large offset for a young and relatively cold tracer population. This offset cannot be explained by the asymmetric drift and it is physically implausible since it requires the masers to have more eccentric orbits than most of the young stars (McMillan & Binney 2010). Requiring the masers to be on circular orbits assuming a flat rotation curve leads to a solar motion of V_{ϕ, ☉} − V_c = 18.6 ± 2.4 km s⁻¹ and V_c = 232 ± 24 (McMillan & Binney 2010), in good agreement with our measurements. Our measurement, however, relies on the theoretically well-motivated asymmetric-drift correction that is a direct consequence of the collisionless Boltzmann equation.

Therefore, we conclude that our measurement of V_c is consistent with previously reported values. Compared to other determinations, our measurement of V_c is one of the most highly precise, and unlike many other measurements, it does not require assumptions about the relation between the solar motion and V_c.

Our data strongly rule out that the Milky Way's circular velocity at the solar radius is >235 km s⁻¹. Marginalizing over all of the systematics discussed in Section 4.2, V_c < 235 km s⁻¹ at >99% confidence. Fixing V_c to 250 km s⁻¹ with R₀ = 8 kpc or 8.4 kpc leads to a best fit with much larger χ²: Δχ² = 47 (∼5σ) and Δχ² = 34 (∼3.7σ), respectively.

Our measurement that the Milky Way's rotation curve is essentially flat over 4 kpc < R < 14 kpc agrees with measurements based on the kinematics of H i emission (e.g., Gunn et al. 1979; Merrifield 1992) and with the local measurement of the Oort constants (Feast & Whitelock 1997).

We found in Section 4, in both the flat- and power-law-rotation-curve models, that the Sun's velocity with respect to the center of the Galaxy—a distinct parameter from V_c in our fit—is larger than V_c(R₀) by ∼24 km s⁻¹. Defining the rotational standard of rest (RSR; Shuter 1982) as the circular velocity in the axisymmetric approximation to the full potential, this solar motion with respect to the RSR is much larger than that measured by applying Strömberg's asymmetric-drift relation to local samples of stars. The asymmetric-drift relation is used to estimate the velocity of the zero-dispersion orbit by extrapolating the relation given in Equation (5) from warmer samples of stars to σ_R = 0; it is this zero-dispersion orbit that is typically what is meant by the LSR (Fich & Tremaine 1991). Such analyses of the Hipparcos data yield a solar motion with respect to the LSR that is somewhere between 5 and 13 km s⁻¹ (Dehnen & Binney 1998; Hogg et al. 2005; Schönrich et al. 2010); 13 km s⁻¹ is also the offset of the average rotational velocity of nearby stars (e.g., Allende Prieto et al. 2004). The discrepancy between our determination of the solar velocity with respect to the RSR, and that with respect to the LSR, is clearly shown in Figure 10, where the PDF for V_{ϕ, ☉} − V_c, V_{R, ☉} is shown for the flat-rotation-curve model, together with the locally measured motion with respect to the LSR of Schönrich et al. (2010). The analogous figure for the power-law rotation-curve fit has an essentially identical appearance. It is clear that the locally measured solar motion V_{ϕ, ☉} is not equal to our globally measured solar motion to very high significance. Or, equivalently, the motion of the RSR is not the same as that of the LSR: the LSR seems to rotate ∼12 km s⁻¹ faster than the RSR.

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This discrepancy may result from a breakdown in the assumptions of the locally measured solar motion using Strömberg's asymmetric-drift relation. As convincingly shown by Schönrich et al. (2010), the neglect of the radial metallicity gradient in the analysis of Dehnen & Binney (1998) leads to a correction of ∼7 km s⁻¹, but further improvements in the chemical-evolution model of Schönrich et al. (2010) might lead to further corrections. The local velocity distribution has also long been known to contain various streams or "moving groups" (e.g., Kapteyn 1905; Schwarzschild 1907; Eggen 1986; Dehnen 1998; Bovy et al. 2009b) that are likely of a dynamical origin (Bensby et al. 2007; Famaey et al. 2008; Bovy & Hogg 2010; Sellwood 2010) and contain a significant fraction of the nearby stars (Bovy et al. 2009b). These moving groups make an accurate determination of the solar motion challenging. As such, the locally determined correction for the Sun's motion with respect to the LSR may well be incorrect by ∼5 km s⁻¹ or more. A solar motion of ∼24 km s⁻¹ rather than ∼12 km s⁻¹ is not much more unlikely given the DF for a ∼5 Gyr population: using a Dehnen DF (Equation (6)) with a radial dispersion of 30 km s⁻¹ (Equation (11)) and assuming our best-fit model gives P(V_{ϕ, ☉} − V_c > 12 km s⁻¹) ≈ 0.2 and P(V_{ϕ, ☉} − V_c > 24 km s⁻¹) ≈ 0.1.

However, if Strömberg's relation leads to a solar motion with respect to the LSR that is different from the globally determined solar motion with respect to the RSR, the most straightforward interpretation is that LSR's orbit is not circular, but deviates from circularity by about 10 km s⁻¹ because of large-scale, non-axisymmetric streaming motions. The deviation cannot be much larger than this value, as the radial component of the solar motion as determined from local samples agrees with our global measurement. As our measurement of the circular velocity is mainly at azimuths ϕ ≲ 45°, this 10 km s⁻¹ deviation must happen over this angular scale.

There is mounting evidence for the existence of streaming motions in the Galactic disk of this magnitude. There has been a long-standing discrepancy of ∼7 km s⁻¹ between the rotation curves determined from first (l < 90°) and fourth (l > 270°) quadrant tangent-point measurements (e.g., Kerr 1962; Gunn et al. 1979; Levine et al. 2008), which could indicate streaming motions. Similarly, the analysis of the extreme line-of-sight velocity of H i emission toward l ∼ 90° leads to differences in V_c of approximately 30 km s⁻¹, albeit with large uncertainties (Knapp et al. 1979; Jackson & Kerr 1981; Jackson 1985). The fact that the maximum line-of-sight velocity in both CO and 21 cm data reaches zero at l ∼ 70°, rather than at l = 90°, has been used to argue that the LSR moves ahead of V_c with a speed of ∼7.5 km s⁻¹ (Shuter 1982; Clemens 1985), similar to the offset we find. More recently, an analysis of line-of-sight velocities from the RAVE has found a gradient, $d\bar{V}_R / dR$, in the mean radial velocity, $\bar{V}_R$, of ∼3 km s⁻¹ kpc⁻¹, and large streaming motions in both $\bar{V}_R$ and the mean tangential velocity, $\bar{V}_\phi$, within a few kiloparsecs from the Sun (Siebert et al. 2011).

As a simple exploration of this possibility, we have computed the closed orbit at R₀ in a model for the Milky Way disk where this streaming motion is due to ellipticity of the disk, following Kuijken & Tremaine (1994). We use a flat rotation curve, with a cos (2 ϕ) perturbation of constant ellipticity having an amplitude of 14 km s⁻¹. Such a perturbation could arise if, for example, the halo is triaxial (e.g., Law et al. 2009). The closed orbit is shown to scale in the inset in Figure 10; it has an eccentricity of 0.06.

Regardless of the origin of the discrepancy between our globally measured solar motion and the locally measured value, our larger preferred value of V_{ϕ, ☉} − V_c means that the Sun is likely closer to the pericenter of its orbit around the Galactic center than previously believed. Revising V_{ϕ, ☉} − V_c upward by 12 km s⁻¹ increases the eccentricity of the Sun's orbit to ∼0.1 from ∼0.06 and increases its mean Galactocentric radius to ∼8.9 kpc from ∼8.5 kpc. Such a large mean radius increases the tension between the Sun's high metallicity compared to local stars (Wielen et al. 1996), although the peak of the local metallicity distribution may be closer to solar metallicity than previously believed (e.g., Casagrande et al. 2011).

Finally, we address what our results imply for dark-matter direct-detection experiments, and for correcting the motion of Galactic and extragalactic objects for the motion of the Sun. Both of these applications essentially require knowledge of the total Galactocentric solar velocity, V_{ϕ, ☉}, to transform velocities into the Galactocentric (dark-matter-halo) rest frame.²⁵ Although this motion is often decomposed as V_c plus the Sun's motion with respect to V_c, our results in this paper show that this is unnecessarily dangerous, as it requires the strong assumption that the LSR is on a circular orbit, such that V_{ϕ, ☉} = V_c + V_{ϕ, ☉, LSR}, where the final term is the solar velocity with respect to the LSR. Our results in Section 4 strongly rule out that this assumption holds for the currently accepted solar motion of ∼12 km s⁻¹. However, this approach is entirely unnecessary for correcting velocities to the Galactocentric rest frame, as the proper motion of Sgr A* combined with the "clean" estimate of R₀ from Galactic-center dynamics shows that V_{ϕ, ☉} ≈ 242 km s⁻¹, which agrees with our measured value of V_{ϕ, ☉}. Adopting a standard V_{ϕ, ☉} = 242 km s⁻¹ for correcting velocities for the Sun's motion, and decoupling this correction from the question of the true value of V_c, would therefore be helpful.

We can use our measurement of the inner rotation curve of the Milky Way to estimate the Milky Way's total dark-halo mass. We combine the measurements in this paper with those of the Milky Way's outer rotation curve (R ≳ 20 kpc) of Xue et al. (2008; specifically, taking the measurements using simulation II in Table 3 of that paper). Based on Figure 6 and Table 2, we add our measurement as a flat rotation curve with V_c = 218 km s⁻¹, evaluated at three radii with uncertainties of 3 km s⁻¹, 6 km s⁻¹, and 10 km s⁻¹ at R = 4 kpc, R = 8 kpc, and R = 12 kpc, respectively (removing the measurements at R = 7.5 kpc and R = 12.5 kpc from Xue et al. 2008). We also use our recent measurements of the local density of dark matter, ρ_DM = 0.008 ± 0.003 M_☉ pc⁻³ (Bovy & Tremaine 2012), and of the disk scale length, h_R = 3.25 ± 0.25 kpc (Bovy et al. 2012b), revising the measurement from the latter paper downward to correct for the influence of the dark halo on that measurement.

We model the Milky Way's potential as the combination of a bulge component with a Hernquist profile with a scale length of 600 pc, a disk component with a scale height of 300 pc (Bovy et al. 2012a) and a scale length that is a free parameter, and a Navarro–Frenk–White (NFW) halo (Navarro et al. 1997), with a scale radius that is a free parameter (i.e., without constraint on the concentration). The relative contributions of these three components are free parameters. We then fit the data given in the previous paragraph, and we find that M_halo = 8⁺⁸_{− 2} × 10¹¹ M_☉. The total mass of the Milky Way is ∼8.5 × 10¹¹ M_☉.

This "low" estimate for the mass of the Milky Way likely makes it less massive than M31, which has an estimated dark-halo mass of 14 × 10¹¹ M_☉ (Watkins et al. 2010). This measurement leads to a combined mass for the Local Group of ∼2.4 × 10¹² M_☉, which is consistent with the Local-Group-timing-argument within the large cosmic scatter (van der Marel et al. 2012). Our measurement of V_c = 218 ± 6 km s⁻¹ combined with its estimated I-band magnitude of −22.3 mag makes the Milky Way underluminous with respect to the Tully–Fisher relation of external spiral galaxies by about 1σ (see Flynn et al. 2006).