Galactic rotation curve and spiral density wave parameters from 73 masers

Based on kinematic data on masers with known trigonometric parallaxes and measurements of the velocities of HI clouds at tangential points in the inner Galaxy, we have refined the parameters of the Allen-Santillan model Galactic potential and constructed the Galactic rotation curve in a wide range of Galactocentric distances, from 0 to 20 kpc. The circular rotation velocity of the Sun for the adopted Galactocentric distance R ₀ = 8 kpc is V ₀ = 239 ± 16 km s^?1. We have obtained the series of residual tangential, ΔV _θ , and radial, V _R , velocities for 73 masers. Based on these series, we have determined the parameters of the Galactic spiral density wave satisfying the linear Lin-Shu model using the method of periodogram analysis that we proposed previously. The tangential and radial perturbation amplitudes are f _θ = 7.0±1.2 km s^?1 and f _R = 7.8±0.7 km s^?1, respectively, the perturbation wave length is λ = 2.3±0.4 kpc, and the pitch angle of the spiral pattern in a two-armed model is i = ?5.2° ±0.7°. The phase of the Sun ζ _⊙ in the spiral density wave is ?50° ± 15° and ?160° ± 15° from the residual tangential and radial velocities, respectively.     

Parameters of the spiral structure of the galaxy from data on δ Cephei stars

We determined the locations of Galactic spiral arm segments for various age groups from the available data on the positions, ages, radial velocities, and proper motions of 440 δ Cephei variables using a previously developed technique. We obtained such parameters of the Galactic spiral structure as the arm pitch angle, , and the pattern speed, Ω_P = 21.7 ± 2.8 km s^?1 kpc^?1, which are comparable to and Ω_P = 20.4 ± 2.5 km s^?1 kpc^?1, respectively, determined previously from open star clusters. Based on the radial velocities and proper motions of the sample stars, we derived the rotation curve of the Galaxy for the range of Galactocentric distances approximately from 6 to 15 kpc. Using the pattern speed, we determined the positions of the corotation region and the inner and outer Lindblad resonances. We estimated the perturbation amplitudes of the Galactic velocity field, f _R = ?1.8 ± 2.5 km s^?1 and f _? = +4.0 ± 3.4 km s^?1.     

Kinematics of stars in the northern and southern galactic hemispheres

We use vector spherical harmonics for a kinematic analysis of the proper motions of stars from the Hipparcos, Tycho-2, and UCAC3 catalogues in the northern and southern Galactic hemispheres. We have found that the statistically reliable values of the Ogorodnikov-Milne model parameters M ₃₂ ⁺ and M ₃₂ ^? have different signs in different hemispheres. This is a consequence of the Galaxy??s rotational retardation with distance from the principal Galactic plane. Based on various samples of stars from the above catalogues, we have obtained the following estimate for the magnitude of the vertical gradient of Galactic rotation velocity in the solar neighborhoods: (20.1 ± 2.9) < |?V_??/?z| < (49.2 ± 0.8) km s^?1 kpc^?1. Another result that is revealed by our analysis of the parameters M ₁₃ ^? and M ₁₃ ⁺ in different Galactic hemispheres is that the vertical gradient of expansion velocity for the stellar system ?V _R /? _z is positive in the northern hemisphere and negative in the southern one. This suggests that the expansion velocity V _R increases with distance fromthe Galactic plane. We show that both these gradients give rise to an apparent acceleration of the solar motion along the x and y axes of the rectangular Galactic coordinate system. Our analysis of the parameters M ₂₁ ^? and M ₁₂ ⁺ shows no significant differences in both hemispheres and has allowed us to determine the Oort parameters, to estimate the Galactic rotation velocity and period in the solar neighborhood, and to calculate the ratio of the epicyclic frequency to the angular velocity of Galactic rotation in the solar neighborhood. The derived diagonal elements of the velocity field deformation tensor suggest that the orientation of the rectangular Galactic coordinate system in space must be determined by taking into account not only the geometrical factors but also the dynamical ones. All these results agree well with these quantities estimated over the entire sphere by various authors.     

Determination of galactic rotation parameters and the solar galactocentric distance R 0 from 73 masers

We have determined the Galactic rotation parameters and the solar Galactocentric distance R ₀ by simultaneously solving Bottlinger’s kinematic equations using data on masers with known line-of-sight velocities and highly accurate trigonometric parallaxes and proper motions measured by VLBI. Our sample includes 73 masers spanning the range of Galactocentric distances from 3 to 14 kpc. The solutions found are Ω₀ = 28.86 ± 0.45 km s^?1 kpc^?1, Ω′₀ = ?3.96 ± 0.09 km s^?1 kpc^?2, Ω″₀ = 0.790 ± 0.027 km s^?1 kpc^?3, and R ₀ = 8.3 ± 0.2 kpc. In this case, the linear rotation velocity at the solar distance R ₀ is V = 241 ± 7 km s^?1. Note that we have obtained the R ₀ estimate, which is of greatest interest, from masers for the first time; it is in good agreement with the most recent estimates and even surpasses them in accuracy.     

Kinematics of Tycho-2 red giant clump stars

Based on the Ogorodnikov-Milne model, we analyze the proper motions of 95 633 red giant clump (RGC) stars from the Tycho-2 Catalogue. The following Oort constants have been found: A = 15.9 ± 0.2 km s^?1 kpc^?1 and B = ?12.0±0.2 km s^?1 kpc^?1. Using 3632 RGC stars with known proper motions, radial velocities, and photometric distances, we show that, apart from the star centroid velocity components relative to the Sun, only the model parameters that describe the stellar motions in the XY plane differ significantly from zero. We have studied the contraction (a negative K effect) of the system of RGC stars as a function of their heliocentric distance and elevation above the Galactic plane. For a sample of distant (500–1000 pc) RGC stars located near the Galactic plane (|z| < 200 pc) with an average distance of d = 0.7 kpc, the contraction velocity is shown to be Kd = ?3.5 ±0.9 km s^?1; a noticeable vertex deviation, l _xy = 9 _· ^o 1 ± 0 _· ^o 5, is also observed for them. For stars located well above the Galactic plane (|z| ≥200 pc), these effects are less pronounced, Kd = ?1.7 ± 0.5 km s^?1 and l _xy = 4 _· ^o 9 ± 0 _· ^o 6. Using RGC stars, we have found a rotation around the Galactic X axis directed toward the Galactic center with an angular velocity of ?2.5 ± 0.3 km s^?1 kpc^?1, which we associate with the warp of the Galactic stellar-gaseous disk.     

Galactic kinematics from a sample of young massive stars

Based on published sources, we have created a kinematic database on 220 massive (> 10 M _⊙) young Galactic star systems located within ≤3 kpc of the Sun. Out of them, ≈100 objects are spectroscopic binary and multiple star systems whose components are massive OB stars; the remaining objects are massive Hipparcos B stars with parallax errors of no more than 10%. Based on the entire sample, we have constructed the Galactic rotation curve, determined the circular rotation velocity of the solar neighborhood around the Galactic center at R ₀ = 8kpc, V ₀ = 259±16 km s^?1, and obtained the following spiral density wave parameters: the amplitudes of the radial and azimuthal velocity perturbations f _R = ?10.8 ± 1.2 km s^?1 and f _θ = 7.9 ± 1.3 km s^?1, respectively; the pitch angle for a two-armed spiral pattern i = ?6.0° ± 0.4°, with the wavelength of the spiral density wave near the Sun being λ = 2.6 ± 0.2 kpc; and the radial phase of the Sun in χ _⊙ = ?120° ± 4°. We show that such peculiarities of the Gould Belt as the local expansion of the system, the velocity ellipsoid vertex deviation, and the significant additional rotation can be explained in terms of the density wave theory. All these effects decrease noticeably once the influence of the spiral density wave on the velocities of nearby stars has been taken into account. The influence of Gould Belt stars on the Galactic parameter estimates has also been revealed. Eliminating them from the kinematic equations has led to the following new values of the spiral density wave parameters: f _θ = 2.9 ± 2.1 km s^?1 and χ _⊙ = ?104° ± 6°.     

Galactic kinematics from data on open star clusters from the MWSC catalogue

Open star clusters from the MWSC (Milky Way Star Clusters) catalogue have been used to determine the Galactic rotation parameters. The circular rotation velocity of the solar neighborhood around the Galactic center has been found from data on more than 2000 clusters of various ages to be V ₀ = 236 ± 6 km s^?1 for the adopted Galactocentric distance of the Sun R ₀ = 8.3 ± 0.2 kpc. The derived angular velocity parameters are Ω ₀ = 28.48 ± 0.36 km s^?1 kpc^?1, Ω’₀ = ?3.50 ± 0.08 km s^?1 kpc_?2, and Ω″₀ = 0.331 ± 0.037 km s^?1 kpc^?3. The influence of the spiral density wave has been detected only in the sample of clusters younger than 50 Myr. For these clusters the amplitudes of the tangential and radial velocity perturbations are f ^θ = 5.6 ± 1.6 km s^?1 and f _R = 7.7 ± 1.4 km s^?1, respectively; the perturbation wavelengths are λ _θ = 2.6 ± 0.5 kpc (i _θ = ^?11? ± 2?) and λ _R = 2.1 ± 0.5 kpc (i _R = ?9? ± 2?) for the adopted four-armed model (m = 4). The Sun’s phase in the spiral density wave is (χ_⊙)_θ = ?62? ± 9? and (χ_⊙)_R = ?85? ± 10? from the residual tangential and radial velocities, respectively.     

Corrections for the Lutz-Kelker bias for Galactic masers

Based on published data, we have collected information about Galactic maser sources with measured distances. In particular, 44 Galactic maser sources located in star-forming regions have trigonometric parallaxes, proper motions, and radial velocities. In addition, ten more radio sources with incomplete information are known, but their parallaxes have been measured with a high accuracy. For all 54 sources, we have calculated the corrections for the well-known Lutz-Kelker bias. Based on a sample of 44 sources, we have refined the parameters of the Galactic rotation curve. Thus, at R ₀ = 8kpc, the peculiar velocity components for the Sun are (U _⊙, V _⊙, W _⊙) = (7.5, 17.6, 8.4) ± (1.2, 1.2, 1.2) km s^?1 and the angular velocity components are ω ₀ = ?28.7 ± 0.5 km s^?1 kpc^?1, ω ₀′ = +4.17 ± 0.10 km s^?1 kpc^?2, and ω₀″ = ?0.87 ± 0.06 km s^?1 kpc^?3. The corresponding Oort constants are A = 16.7 ± 0.6 km s^?1 kpc^?1 and B = ?12.0 ± 1.0 km s^?1 kpc^?1; the circular rotation velocity of the solar neighborhood around the Galactic center is V ₀ = 230 ± 16 km s^?1. We have found that the corrections for the Lutz-Kelker bias affect the determination of the angular velocity ω ₀ most strongly; their effect on the remaining parameters is statistically insignificant. Within themodel of a two-armed spiral pattern, we have determined the pattern pitch angle $i = - 6_.^ \circ 5$ and the phase of the Sun in the spiral wave χ ₀ = 150°.     

Stability of ion-acoustic waves in a pair-ion plasma with a third species of ions: application to cometary plasmas

A popular model of a cometary plasma is hydrogen (H⁺) with positively charged oxygen (O⁺) as a heavier ion component. However, the discovery of negatively charged oxygen (O^?) ions enables one to model a cometary plasma as a pair-ion plasma (of O⁺ and O^?) with hydrogen as a third ion constituent. We have, therefore, studied the stability of the ion-acoustic wave in such a pair-ion plasma with hydrogen and electrons streaming with velocities $V_{d\mathrm{H}^{+}}$ and V _de , respectively, relative to the oxygen ions. We find the calculated frequency of the ion-acoustic wave with this model to be in good agreement with the observed frequencies. The ion-acoustic wave can also be driven unstable by the streaming velocity of the hydrogen ions. The growth rate increases with increasing hydrogen density $n_{\mathrm{H}^{+}}$ , and streaming velocities $V_{d\mathrm{H}^{+}}$ and V _de . It, however, decreases with increasing oxygen ion densities $n_{\mathrm{O}^{+}}$ and $n_{\mathrm{O}^{-}}$ .     

Physical properties of X-ray gas in galaxy cluster CL0024+17

We analyzed the X-ray data obtained by the Chandra telescope for the galaxy cluster CL0024+17 (z = 0.39). The mean temperature of the cluster is estimated (kT = 4.35 _?0.44 ^+0.51 keV) and the surface brightness profile is derived. We generated the mass and density profiles for dark matter and gas using numerical simulations and the Navarro-Frenk-White dark matter density profile (Navarro et al., 1995) for a spherically symmetric cluster in which gas is in hydrostatic equilibrium with the cluster field. The total mass of the cluster is estimated to be M ₂₀₀ = 3.51 _?0.47 ^+0.38 × 10 _Sun ¹⁴ within a radius of R ₂₀₀ = 1.24 _?0.17 ^+0.12 Mpc of the cluster center. The contribution of dark matter to the total mass of the cluster is estimated as ${{M_{200_{DM} } } \mathord{\left/ {\vphantom {{M_{200_{DM} } } {M_{tot} }}} \right. \kern-0em} {M_{tot} }} = 0.89$ .     

Estimation of the galactic spiral pattern speed from Cepheids

To study the peculiarities of the Galactic spiral density wave, we have analyzed the space velocities of Galactic Cepheids with propermotions from the Hipparcos catalog and line-of-sight velocities from various sources. First, based on the entire sample of 185 stars and taking R ₀ = 8 kpc, we have found the components of the peculiar solar velocity (u _⊙, v _⊙) = (7.6, 11.6) ± (0.8, 1.1) km s^?1, the angular velocity of Galactic rotation Ω₀ = 27.5 ± 0.5 km s^?1 kpc^?1 and its derivatives Ω′₀ = ?4.12 ± 0.10 km s^?1 kpc^?2 and Ω″₀ = 0.85 ± 0.07 km s^?1 kpc^?3, the amplitudes of the velocity perturbations in the spiral density wave f _R = ?6.8 ± 0.7 and f _θ = 3.3 ± 0.5 km s^?1, the pitch angle of a two-armed spiral pattern (m = 2) i = ?4.6° ± 0.1° (which corresponds to a wavelength λ = 2.0 ± 0.1 kpc), and the phase of the Sun in the spiral density wave χ_⊙ = ?193° ± 5°. The phase χ_⊙ has been found to change noticeably with the mean age of the sample. Having analyzed these phase shifts, we have determined the mean value of the angular velocity difference Ω _p ? Ω, which depends significantly on the calibrations used to estimate the individual ages of Cepheids. When estimating the ages of Cepheids based on Efremov’s calibration, we have found |Ω _p ? Ω₀| = 10 ± 1_stat ± 3_syst km s^?1 kpc^?1. The ratio of the radial component of the gravitational force produced by the spiral arms to the total gravitational force of the Galaxy has been estimated to be f _r0 = 0.04 ± 0.01.     

Galactic Rotation Based on OB Stars from the Gaia DR2 Catalogue

We have studied a sample containing ~6000 OB stars with proper motions and trigonometric parallaxes from the Gaia DR2 catalogue. The following parameters of the angular velocity of Galactic rotation have been found: Ω₀ = 29.70 ± 0.11 km s^-1 kpc^-1, Ω'₀ = -4.035 ± 0.031 km s^-1 kpc^-2, and Ω ₀ = 0.620 ± 0.014 km s^-1 kpc^-3. The circular rotation velocity of the solar neighborhood around the Galactic center is V₀ = 238 ± 5 km s^-1 for the adopted Galactocentric distance of the Sun R₀ = 8.0 ± 0.15 kpc. The amplitudes of the tangential and radial velocity perturbations produced by the spiral density wave are f_θ = 4.4 ± 1.4 kms^-1 and f_R = 5.1 ± 1.2 kms^-1, respectively; the perturbation wavelengths are λ_θ = 1.9 ± 0.5 kpc and λ_R = 2.1 ± 0.5 kpc for the adopted four-armed spiral pattern. The Sun's phase in the spiral density wave is χ_⊙ = -178° ± 12°.

     

Kinematics of the Galaxy from OB Stars with Data from the Gaia DR2 Catalogue

We have selected and analyzed a sample of OB stars with known line-of-sight velocities determined through ground-based observations and with trigonometric parallaxes and propermotions from the Gaia DR2 catalogue. Some of the stars in our sample have distance estimates made from calcium lines. A direct comparison with the trigonometric distance scale has shown that the calcium distance scale should be reduced by 13%. The following parameters of the Galactic rotation curve have been determined from 495 OB stars with relative parallax errors less than 30%: (U, V,W)_⊙ = (8.16, 11.19, 8.55)± (0.48, 0.56, 0.48) km s^?1, Ω₀ = 28.92 ± 0.39 km s^?1 kpc^?1, Ω'₀ = ?4.087 ± 0.083 km s^?1 kpc^?2, and Ω″ ₀ = 0.703 ± 0.067 km s?1 kpc^?3, where the circular velocity of the local standard of rest is V0 = 231 ± 5 km s^?1 (for the adopted R₀ = 8.0 ± 0.15 kpc). The parameters of the Galactic spiral density wave have been found from the series of radial, V_R, residual tangential, ΔV_circ, and vertical, W, velocities of OB stars by applying a periodogram analysis. The amplitudes of the radial, tangential, and vertical velocity perturbations are f_R = 7.1± 0.3 km s^?1, f_θ = 6.5 ± 0.4 km s^?1, and f_W = 4.8± 0.8 km s^?1, respectively; the perturbation wavelengths are λ_R = 3.3 ± 0.1 kpc, λ_θ = 2.3 ± 0.2 kpc, and λ_W = 2.6 ± 0.5 kpc; and the Sun’s radial phase in the spiral density wave is (χ_⊙)_R = ?135? ± 5?, (χ_⊙)θ = ?123? ± 8?, and (χ_⊙)_W = ?132? ± 21? for the adopted four-armed spiral pattern.     

Low density structures in the local universe. I. Diffuse agglomerates of galaxies

This paper is the first of a series considering the properties of distribution of nearby galaxies in the low density regions. Among 7596 galaxies with radial velocities V _LG < 3500 km/s, absolute magnitudes M _K < ?18?4, and Galactic latitudes |b| > 15° there are 3168 field galaxies (i.e. 42%) that do not belong to pairs, groups or clusters in the Local universe. Applying to this sample the percolation method with a radius of r ₀ = 2.8Mpc, we found 226 diffuse agglomerates with n ?? 4 number of members. The structures of eight most populated objects among them (n ?? 25) are discussed. These non-virialized agglomerates are characterized by amedian dispersion of radial velocities of about 170 km/s, the linear size of around 6 Mpc, integral K-band luminosity of 3 × 10¹¹ L _??, and a formal virial-mass-to-luminosity ratio of about 700M _??/L _??. The mean density contrast for the considered agglomerates is only $\left\langle {\Delta n/\bar n} \right\rangle $ ?? 5, and their crossing time is about 30?C40 Gyr.     

Estimation of the solar galactocentric distance and galactic rotation velocity from near-solar-circle objects

We have tested the method of determining the solar Galactocentric distance R ₀ and Galactic rotation velocity V ₀ modified by Sofue et al. using near-solar-circle objects. The motion of objects relative to the local standard of rest has been properly taken into account. We show that when such young objects as star-forming regions or Cepheids are analyzed, allowance for the perturbations produced by the Galactic spiral density wave improves the statistical significance of the estimates. The estimate of R ₀ = 7.25 ± 0.32 kpc has been obtained from 19 star-forming regions. The following estimates have been obtained from a sample of 14 Cepheids (with pulsation periods P > 5 ^d ): R ₀ = 7.66 ± 0.36 kpc and V ₀ = 267 ± 17 km s^?1. We consider the influence of the adopted Oort constant A and the character of stellar proper motions (Hipparcos or UCAC4). The following estimates have been obtained from a sample of 18 Cepheids with stellar proper motions from the UCAC4 catalog: R ₀ = 7.64 ± 0.32 kpc and V ₀ = 217 ± 11 km s^?1.     

Probing the mass and anisotropy of the Milky Way gaseous halo: sight-lines toward Mrk 421 and PKS 2155-304

We recently found that the halo of the Milky Way contains a large reservoir of warm-hot gas that accounts for large fraction of the missing baryons from the Galaxy. The average physical properties of this circumgalactic medium (CGM) are determined by combining average absorption and emission measurements along several extragalactic sightlines. However, there is a wide distribution of both, the halo emission measure and the O?vii column density, suggesting that the Galactic warm-hot gaseous halo is anisotropic. We present Suzaku observations of fields close to two sightlines along which we have precise O?vii absorption measurements with Chandra. The column densities along these two sightlines are similar within errors, but we find that the emission measures are different: 0.0025±0.0006 cm^?6?pc near the Mrk 421 direction and 0.0042±0.0008 cm^?6?pc close to the PKS 2155-304 sightline. Therefore the densities and pathlengths in the two directions must be different, providing a suggestive evidence that the warm-hot gas in the CGM of the Milky Way is not distributed uniformly. However, the formal errors on derived parameters are too large to make such a claim. In the Mrk 421 direction we derive the density of \(1.6^{+2.6}_{-0.8} \times 10^{-4}~\mbox{cm}^{-3}\) and pathlength of \(334^{+685}_{-274}~\mbox{kpc}\) . In the PKS 2155-304 direction we measure the gas density of \(3.6^{+4.5}_{-1.8} \times10^{-4}~\mbox{cm}^{-3}\) and path-length of \(109^{+200}_{-82}~\mbox{kpc}\) . Thus the density and pathlength along these sightlines are consistent with each other within errors. The average density and pathlength of the two sightlines are similar to the global averages, so the halo mass is still huge, over 10 billion solar masses. With more such studies, we will be able to better characterize the CGM anisotropy and measure its mass more accurately. We can then compare the observational results with theoretical models and investigate if/how the CGM structure is related to the larger scale environment of the Milky Way. We also show that the Galactic disk makes insignificant contribution to the observed O?vii absorption; a similar conclusion was also reached by Henley and Shelton (2013) about the emission measure. We further argue that any density inhomogeneity in the warm-hot gas, be it from clumping, from the disk, or from a non-constant density gradient, would strengthen our result in that the Galactic halo path-length and the mass would become larger than what we estimate here. As such, our results are conservative and robust.     

Kinematic parameters of young subsystems and the galactic rotation curve

We analyze the space velocities of blue supergiants, long-period Cepheids, and young open star clusters (OSCs), as well as the H I and H II radial-velocity fields by the maximum-likelihood method. The distance scales of the objects are matched both by comparing the first derivatives of the angular velocity Ω′ determined separately from radial velocities and proper motions and by the statistical-parallax method. The former method yields a short distance scale (for R₀=7.5 kpc, the assumed distances should be increased by 4%), whereas the latter method yields a long distance scale (for R₀=8.5 kpc, the assumed distances should be increased by 16%). We cannot choose between these two methods. Similarly, the distance scale of blue supergiants should be shortened by 9% and lengthened by 3%, respectively. The H II distance scale is matched with the distance scale of Cepheids and OSCs by comparing the derivatives Ω′ determined for H II from radial velocities and for Cepheids and OSCs from space velocities. As a result, the distances to H II regions should be increased by 5% in the short distance scale. We constructed the Galactic rotation curve in the Galactocentric distance range 2–14 kpc from the radial velocities of all objects with allowance for the difference between the residual-velocity distributions. The axial ratio of the Cepheid+OSC velocity ellipsoid is well described by the Lindblad relation, while σ_u≈σ_v for gas. The following rotation-curve parameters were obtained: Ω₀=(27.5±1.4) km s^?1 kpc^?1 and A=(17.1±0.5) km s^?1 kpc^?1 for the short distance scale (R₀=7.5 kpc); and Ω₀=(26.6±1.4) km s^?1 kpc^?1 and A=(15.4±0.5) km s^?1 kpc^?1 for the long distance scale (R₀=8.5 kpc). We propose a new method for determining the angular velocity Ω₀ from stellar radial velocities alone by using the Lindblad relation. Good agreement between the inferred Ω₀ and our calculations based on space velocities suggests that the Lindblad relation holds throughout the entire sample volume. Our analysis of the heliocentric velocities for samples of young objects reveals noticeable streaming motions (with a velocity lag of ～7 km s^?1 relative to the LSR), whereas a direct computation of the perturbation amplitudes in terms of the linear density-wave theory yields a small amplitude for the tangential perturbations.     

Kinematics of the galaxy from Cepheids with proper motions from the Gaia DR1 catalogue

A sample of classical Cepheids with known distances and line-of-sight velocities has been supplemented with proper motions from the Gaia DR1 catalogue. Based on the velocities of 260 stars, we have found the components of the peculiar solar velocity vector (U, V, W)_⊙ = (7.90, 11.73, 7.39) ± (0.65, 0.77, 0.62) km s^?1 and the following parameters of the Galactic rotation curve: Ω₀ = 28.84 ± 0.33 km s^?1 kpc^?1, Ω′₀ = ?4.05 ± 0.10 km s^?1 kpc^?2, and Ω″₀ = 0.805 ± 0.067 km s^?1 kpc^?3 for the adopted solar Galactocentric distance R ₀ = 8 kpc; the linear rotation velocity of the local standard of rest is V ₀ = 231 ± 6 km s^?1.