TRIGONOMETRIC PARALLAXES OF MASSIVE STAR-FORMING REGIONS. VI. GALACTIC STRUCTURE, FUNDAMENTAL PARAMETERS, AND NONCIRCULAR MOTIONS

The Milky Way is known to possess spiral structure. However, revealing the nature of this structure has proven to be elusive for decades. The Georgelin & Georgelin (1976) study of H ii regions produced what has been generally considered the "standard model" for the spiral structure of the Galaxy. However, after decades of study there is little agreement on this structure. Indeed, we do not really know the number of spiral arms (Simonson 1976; Cohen et al. 1980; Bash 1981; Vallée 1995; Drimmel 2000; Russeil 2003) or how tightly wound is their pattern. The primary reason for the difficulty is the lack of accurate distance measurements throughout the Galaxy. Photometric distances are prone to calibration problems, which become especially severe when looking through copious dust to distant objects in the plane of the Galaxy. Thus, most attempts to map the Galaxy rely on radio frequency observations and kinematic distances, which involve matching source Doppler shifts with those expected from a model of Galactic rotation. However, because of distance ambiguities in the first and fourth quadrants (where most of the spiral arms are found) and the existence of sizeable noncircular motions, kinematic distances can be highly uncertain (Burton & Bania 1974; Liszt & Burton 1981; Gómez 2006).

We are measuring trigonometric parallaxes and proper motions of sources of maser emission associated with high-mass star-forming regions (HMSFRs), using the National Radio Astronomy Observatory's¹⁰ Very Long Baseline Array (VLBA) and the Japanese VLBI Exploration of Radio Astronomy (VERA) project. The great advantage of trigonometric parallaxes and proper motions is that one determines source distances directly and geometrically, with no assumptions about luminosity, extinction, metallicity, crowding, etc. Also, from the same measurements, one determines proper motions, and if the time sampling is optimal there is little if any correlation between the parallax and proper motion estimates. Thus, the magnitude of the proper motion does not affect the parallax accuracy. Combining all of the observational data yields the full three-dimensional locations and velocity vectors of the sources.

Results for 12 GHz methanol (CH₃OH) masers toward 10 HMSFRs, carried out with the VLBA (program BR100), are reported in the first five papers in this series (Reid et al. 2009; Moscadelli et al. 2009; Xu et al. 2009; Zhang et al. 2009; Brunthaler et al. 2009), hereafter Papers I through V, respectively. Eight other sources with H₂O or SiO masers have been measured with VERA (Honma et al. 2007; Hirota et al. 2007; Choi et al. 2008; Sato et al. 2008) and with methanol, H₂O or continuum emission with the VLBA (Hachisuka et al. 2006; Menten et al. 2007; Moellenbrock et al. 2007; Bartkiewicz et al. 2008; Hachisuka et al. 2009). In this paper, we collect these parallaxes and proper motions in order to study the spiral structure of the Galaxy. Combining positions, distances, Doppler shifts, and proper motions, allows us not only to locate the HMSFRs that harbor the target maser sources in three dimensions, but also to determine their three-dimensional space motions. In Section 2, we map the locations of the HMSFRs and measure the pitch angles of some spiral arms. In Section 3, we use the full three-dimensional spatial and kinematic information to examine the noncircular (peculiar; tar file) motions of these star-forming regions. We also fit the data with a model of the Galaxy and estimate the distance from the Sun to the Galactic center (R₀) and the circular orbital speed at the Sun (Θ₀). The nature of the rotation curve and its effect on estimates of R₀ and Θ₀ is also discussed. In Section 4, we compare kinematic distances with those determined by trigonometric parallax and offer a new prescription to improve such distance estimates. In Section 5, we discuss limitations of the current definition of Galactic coordinates and suggest a new system based partly on dynamical information. Finally, we discuss the broader implications of our results in Section 6.

Table 1 summarizes the parallax and proper motions of 18 regions of high-mass star formation measured with very long baseline interferometry (VLBI) techniques. The locations of these star-forming regions in the Galaxy are shown in Figure 1, superposed on an artist's conception of the Milky Way. Distance errors are indicated with error bars (1σ), but for most sources the error bars are smaller than the dots.

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2.1. Spiral Arms

2.2. Pitch Angles

Given measurements of position, parallax, proper motion, and Doppler shift, one has complete three-dimensional location and velocity vectors relative to the Sun. One can then construct a model of the Milky Way and adjust the model parameters to best match the data. We model the Milky Way as a disk rotating with speed $\Theta (R)=\Theta _0+\frac{d\Theta}{dR}(R-R_0)$, where R₀ is the distance from the Sun to the Galactic center. We started the fitting process by assuming a flat rotation curve (i.e., $\frac{d\Theta}{dR}=0$). Later, we relaxed this assumption and solved for $d\Theta \over dR$, followed by an investigation of other forms of the Galactic rotation curve. Since all measured motions are relative to the Sun, we need to remove the peculiar (noncircular) motion of the Sun, which is parameterized by U_☉ toward the Galactic center, V_☉ in the direction of Galactic rotation, and W_☉ toward the north Galactic pole (NGP). Table 2 summarizes these and other parameters.

We adjusted the Galactic parameters so as to best match the data to the spatial-kinematic model in a least-squares sense. For each source, we treated the measured parallax (π_s), two components of proper motion (μ_x, μ_y), and the heliocentric velocity (v_Helio) as data. The observed source coordinates are known to extremely high accuracy and were treated as independent variables. The model is a smoothly rotating galaxy given by the parameters listed in Table 2. Specifically, the model parallax is calculated from a kinematic distance, based on the observed Doppler shift. The three-dimensions of motion relative to the Sun (proper motion and heliocentric Doppler shift) are calculated from the source location, taking into account the size and rotation curve of the galaxy model, and the solar and source peculiar motions. We adopt the Hipparcos determination of solar motion (Dehnen & Binney 1998) as definitive and generally did not vary these parameters. However, in one least-squares fit, we solved for these parameters for illustrative purposes in order to compare solar motion results from Hipparcos stars and our HMSFRs.

Our choice of weights for the data in the least-squares fitting process requires some comment. While the heliocentric velocity of any maser spot can be measured with very high accuracy, it may not exactly reflect the motion of the HMSFR. The internal motions of methanol masers are generally small and cause uncertainty of ≈3 km s⁻¹ (Moscadelli et al. 2002), whereas H₂O masers can be associated with fast outflow and, if not accurately modeled, can lead to larger uncertainty in the motion of the exciting star. In addition, the virial motion of an individual massive star (associated with the masers) with respect to the entire HMSFR is likely to be ≈7 km s⁻¹ per coordinate (e.g., for a region of mass of ∼3 × 10⁴M_☉ and radius of ∼1 pc). Therefore, we allow for a deviation of the measured motion from the center of mass of its associated HMSFR by adding an uncertainty of σ_Vir = 7 km s⁻¹ in quadrature with the internal motion estimates (between 3 and 5 km s⁻¹). Specifically, the variance weights for the v_LSR data, w(v_LSR), are calculated from $w(v_{\rm LSR}) = 1/(\sigma _{v_{\rm LSR}}^2 + \sigma _{\rm Vir}^2)$.

Since the parallax data is compared to a kinematic model, we considered both the parallax measurement uncertainty and a modeling uncertainty for the kinematic distance, σ_kd, owing to the total uncertainty in the heliocentric velocity of the associated HMSFR. These two components were added in quadrature when calculating the weights: w(π_s) = 1/(σ²_π + σ²_kd/d⁴_s), where d_s = 1/π_s. Similarly, the proper motion weights allowed for both measurement uncertainties and the possible deviation of the measured maser motions from the center of mass of the HMSFR. The latter term was set by the uncertainty in the heliocentric velocity divided by the distance. Thus, for either proper motion component, w(μ) = 1/(σ²_μ + σ²_Vir/d²_s).

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3.1. Galactic Three-Dimensional Motions

3.2. Fundamental Galactic Parameters

3.3. Rotation Curves

Figure 6 compares the locations of the star-forming regions determined by trigonometric parallax and by kinematic distances. The kinematic distances were computed for the IAU standards R₀ = 8.5 kpc and Θ₀ = 220 km s⁻¹ and the standard definition of LSR. For 13 of 18 regions (11 of 16 in the basic sample), the kinematic distance exceeds the true source distance; in three cases the discrepancy is over a factor of 2. The kinematic distances for (these) star-forming regions tend to over-estimate the source distances.

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As shown above, HMSFRs on average orbit the Galaxy ≈15 km s⁻¹ slower than the circular rotation speed. Taking this into account, a prescription for a "revised" kinematic distance for a HMSFR could be as follows:

• 1.  
  add back the (old) standard solar motion corrections to the LSR velocities, returning them to the heliocentric frame;
• 2.  
  apply "best values" for the solar motion to calculate a revised "LSR" velocity, v^r_LSR;
• 3.  
  subtract −15 km s⁻¹ from the velocity component in the direction of Galactic rotation;
• 4.  
  calculate a kinematic distance using values for the fundamental parameters of the Milky Way, e.g., R₀ = 8.4 kpc and Θ₀ = 254 km s⁻¹, that are consistent with astrometric measurements; and
• 5.  
  when determining the uncertainty in the kinematic distance, include a systematic contribution allowing for the possibility of a 7 km s⁻¹ uncertainty in v^r_LSR.

Table 4 shows parallax distances, standard (old) kinematic distances, and revised kinematic distances and uncertainties (using the above prescription) for all 18 HMSFRs listed in Table 1. (We provide the FORTRAN source code (tar file) used to calculate revised kinematic distances in the online material.) Note that our prescription for the uncertainty in kinematic distances performs reasonably well for our basic sample (excluding the two sources G 23.6 − 0.1 and NGC 7538 which we earlier noted as outliers). The mean difference between the parallax and kinematic distances is near zero and the differences divided by their uncertainties average to near unity. Now only half (8 of 16) of the sources in the basic sample have kinematic distances that exceed the true source distance. There are no cases for which the discrepancy is a factor of 2, and the estimated uncertainties reasonably account for differences between the parallax and kinematic distances.

While the prescription outlined above results in some improvement in kinematic distances compared to the standard approach, the improvement is not as great as one might at first expect. This occurs because the definition of the LSR uses the standard solar motion. While the standard solar motion differs only slightly from the Hipparcos solar motion for components toward the Galactic center (U_☉) and the north Galactic pole (W_☉), there is a large discrepancy for the component in the direction of Galactic rotation. The standard value is V^Std_☉ = 15.3 km s⁻¹, whereas the Hipparcos value is V^H_☉ = 5.25 km s⁻¹. The +10 km s⁻¹ "error" in V^Std_☉ partially compensates for the 15 km s⁻¹ slower Galactic orbits of HMSFRs shown in Section 3.1. (Note that a positive change in the solar motion component V_☉ results in a negative change in a source peculiar motion component $\overline{V_s}$). Even with the improved prescription for kinematic distances, one cannot really hope to discern spiral structure using kinematic distances.

There is excellent agreement between the two independent VLBI measurements of Θ₀/R₀: the measurement based on parallaxes and proper motions of HMSFRs (this paper) and based on the proper motion of Sgr A* (Reid & Brunthaler 2004). This gives us confidence that (1) we can well model the Galaxy with parallax and proper motions of HMSFRs and (2) Sgr A* is indeed a supermassive black hole at the dynamical center of the Milky Way. These findings offer an independent definition of the Galactic plane and Galactic coordinates. Currently, the IAU definition of the Galactic plane is based primarily on the thin distribution of neutral hydrogen 21 cm emission (Blaauw et al. 1960). The Sun is defined to be precisely in the plane and the origin of longitude was set by the centroid of the radio emission of the large, complex source Sgr A. The Sun is now known to be ≈20 pc north of the plane (see Reed 2006 and references therein) and the supermassive black hole, Sgr A*, is offset by a few arcmin from the IAU defined center.

In the future, one could consider redefining Galactic coordinates based, in part, on the proper motion of Sgr A*, which, after correction for the well-determined solar motion component perpendicular to the Galactic plane, gives the orbital plane of the LSR. The zero of longitude would be best defined by the position of Sgr A*. This would place our supermassive black hole at the origin of Galactic coordinates, and one could rotate the reference frame to remove the Sun from its special location precisely in the Galactic plane.

VLBI now routinely yields parallax measurements with accuracies of ∼10 μas, corresponding to 10% uncertainty at a distance of 10 kpc, and proper motions that are usually accurate to ∼10 μas y⁻¹ or better than ∼1 km s⁻¹ at similar distances. Target sources include molecular masers associated with star formation and red giant stars, as well as nonthermal continuum emission associated with young T Tau stars and cool dwarfs. Combining the first results of parallaxes for HMSFRs from the VLBA and the Japanese VERA project has allowed us to begin to investigate the spiral structure and kinematics of the Galaxy.

We have accurately located three of the spiral arms of the Milky Way and directly measured a pitch angle of 16° for a portion of the Perseus spiral arm. This pitch angle is similar to those of spiral arms in other galaxies of type Sb to Sc (Kennicutt 1981). Two armed spirals can account for most of the known large H ii regions only if the arms wrap twice around the Galaxy; this requires pitch angles of ≈8°. With pitch angles greater than ≈12°, the Galaxy needs to have four arms in order to account for the approximate locations of H ii regions (Georgelin & Georgelin 1976; Taylor & Cordes 1993). There has been considerable discussion in the literature concerning the number of spiral arms in the Galaxy (Simonson 1976; Bash 1981; Vallée 1995; Drimmel 2000; Drimmel & Spergel 2001; Benjamin et al. 2005; Nakanishi & Sofue 2006; Steiman-Cameron et al. 2008), with Spitzer GLIMPSE survey results suggesting that only two arms can be traced in the redder, older population of stars (Benjamin 2008). Perhaps, the VLBI and infrared survey results can be reconciled if the Milky Way exhibits a hybrid structure, consisting of two dominant spiral arms, populated by both young and old stars and with pitch angles near 16°, and two weaker arms traced only by young stars.

Our finding that HMSFRs on average orbit the Galaxy ≈15 km s⁻¹ slower than expected for circular orbits has implications for star formation and spiral density wave theory. The plot of the apparent solar motion in the direction of Galactic rotation (V_☉) versus stellar dispersion (Figure 4) can be interpreted as a time sequence, with stellar age increasing with dispersion. The 15 km s⁻¹ slower orbital speed of HMSFRs displays as a positive departure of the apparent solar motion with respect to the asymmetric drift (the fitted trend to the Hipparcos data shown in Figure 4), since the Sun appears to orbit faster when measured against such stars. One explanation for this finding is that HMSFRs are born in elliptical Galactic orbits, near apocenter, with orbital eccentricity of about 0.06. As young stars continue to orbit the Galaxy, their orbits become more circularized, as evidenced by the lesser departure of the youngest Hipparcos bin (mostly late B-type stars) from the asymmetric drift line compared to the HMSFRs. The gradual transfer of angular momentum from gas to stars in the Galaxy proposed by Chakrabarti (2008) may explain this. At a stellar dispersion of ≈300 (km s⁻¹)², which corresponds to A2- to A5-type stars with colors B − V = 0.1 and characteristic main-sequence lifetimes of ∼1 Gy, the stars join the asymmetric drift. As stars continue to age, their orbits are progressively "randomized" and they (again) become part of a slower orbiting population, which appears as a larger apparent V_☉.

Parallaxes measurements alone generally cannot yield R₀ (except for a parallax of Sgr A*). However, since galaxies rotate in a fairly smooth fashion, a kinematic model can be directly compared with distance and relative motion measurements in order to estimate R₀ and Θ₀. In this paper, we have demonstrated that parallax and proper motion measurements for HMSFRs across large portions of the Galaxy can separate estimates of R₀ and Θ₀, although because of the somewhat restricted coverage of the Galaxy currently available, we have a significant correlation between these parameters. Our best estimate of R₀ is 8.4 ±  0.36 ±  0.5 kpc, where the second uncertainty is systematic and comes from our lack of detailed knowledge of the rotation curve of the Galaxy. This estimate is consistent with the "best" R₀ of 8.0 ±  0.5 kpc from a combination of many methods reviewed by Reid (1993). Also, recent direct estimates of R₀ from radial velocities and elliptical paths of stars that orbit Sgr A* have converged on values of 8.4 ±  0.4 kpc (Ghez et al. 2008) and 8.33 ±  0.35 (Gillessen et al. 2009). (These estimates assume that Sgr A* is nearly motionless at the Galactic center. Relaxing this assumption decreases the estimates to about 8.0 kpc.) Of course, many other less direct estimates of R₀ can be found in the literature and span a much greater range.

The characteristic rotation speed of the Galaxy (Θ₀) is a crucial parameter not only for Galactic dynamics and kinematic distance determinations, but also for estimating the total mass in dark matter and the history and fate of the Local Group of galaxies (Loeb et al. 2005; Shattow & Loeb 2008). Estimates of the rotation speed of the Galaxy from the recent literature span a very large range between 184 km s⁻¹ (Olling & Merrifield 1998) and 272 km s⁻¹ (Méndez et al. 1999). Most estimates of Θ₀ are based on analyses of the shear and vorticity of large samples of stars in the (extended) solar neighborhood and thus really measure Oort's A and B parameters. Quoted values of Θ₀ then come by assuming a value for R₀ and using the relation Θ₀ = R₀(A − B). Our result that Θ₀ = 254 ±  16 km s⁻¹ was obtained by fitting for both R₀ and Θ₀ using full three-dimensional locations and motions of sources well beyond the extended solar neighborhood and, thus, does not assume a value for R₀. However, as discussed in Section 3.2, with the present distribution of sources there is considerable correlation between R₀ and Θ₀ parameters, which is reflected in the ±16 km s⁻¹ formal uncertainty for Θ₀.

Our estimate of the ratio Θ₀/R₀ of 30.3 ±  0.9 km s⁻¹ kpc⁻¹ is determined more accurately than either parameter individually and is nearly independent of the value of R₀ over the range of likely values between about 8.0 and 8.5 kpc. This value differs considerably from that determined from the IAU values of Θ₀/R₀ = 220 km s⁻¹/8.5 kpc = 25.9 km s⁻¹ kpc⁻¹ and differs marginally from the Feast & Whitelock (1997) analysis of Hipparcos Cepheids of 27.19 ±  0.87 km s⁻¹ kpc⁻¹. Recent studies, using samples of OB-type stars within 3 kpc of the Sun (excluding Gould's Belt stars and based on Hipparcos data augmented with photometrically determined distances), arrive at A − B between 30 km s⁻¹ kpc⁻¹ (Uemura et al. 2000) and 32 km s⁻¹ kpc⁻¹ (Miyamoto & Zhu 1998; Elias et al. 2006), with uncertainties of about ±1.5 km s⁻¹.

Our value for Θ₀/R₀ is in excellent agreement with that determined directly from the apparent proper motion of Sgr A* (the supermassive black hole at the Galactic center) of 6.379 ±  0.024 mas yr⁻¹ (Reid & Brunthaler 2004). One expects a supermassive black hole to be stationary at the dynamical center of the Galaxy to better than ∼1 km s⁻¹ (Chatterjee et al. 2002; Dorband et al. 2003; Reid & Brunthaler 2004). Hence, Sgr A*'s apparent motion should be dominated by the effects of the Galactic orbit of the Sun. After correcting for the solar motion of 5.25 km s⁻¹ in the direction of Galactic rotation (Dehnen & Binney 1998), Sgr A*'s apparent motion yields a global estimate of Θ₀/R₀ = 29.45 ±  0.15 km s⁻¹ kpc⁻¹. Thus, there is excellent agreement between this and our global and direct method for measuring Θ₀/R₀.

Coupling the Sgr A* motion result of Θ₀/R₀ = 29.45 ±  0.15 km s⁻¹ kpc⁻¹ with estimates of R₀ from stellar orbits in the Galactic center (Ghez et al. 2008; Gillessen et al. 2009) of 8.4 ±  0.4 kpc yields Θ₀ = 247 ±  12 km s⁻¹. This result is also a direct and global measurement of Θ₀ and is independent of our result from parallaxes and proper motions of star-forming regions. Combining the Galactic center and star-forming region estimates gives Θ₀ = 250 ±  10 km s⁻¹.

It seems clear that Θ₀ is near the upper end of the range of estimates in the literature. We note that both the Galactic center stellar orbit and the star-forming region parallax results assume the Hipparcos solar motion of 5.25 km s⁻¹ in the direction of Galactic rotation. Only if the interpretation of the asymmetric drift is incorrect or if the entire solar neighborhood orbits the Galactic center ∼30 km s⁻¹ slower than the Galaxy spins could Θ₀ be equal to the IAU recommended value of 220 km s⁻¹.

We have determined the rotation speed of the Milky Way at the radius of the Sun to be ≈250 km s⁻¹ and the rotation curve to be nearly flat or slightly rising with distance from the Galactic center. These values are nearly identical to those of the Andromeda galaxy (M31) as shown in Figure 7. The rotation curve of Andromeda, determined from H  i emission by Carignan et al. (2006) based on interferometric observations of Unwin (1983), indicates a speed of 251 km s⁻¹ at a radius of 8 kpc, a slightly rising curve out to about 15 kpc, and a slow dropoff to about 225 km s⁻¹ beyond 20 kpc. The most straightforward interpretation of the similarities of the rotation curves for the Milky Way and Andromeda is that these two galaxies are nearly equal in size and mass.

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Finally, we note that Reid & Brunthaler (2004) placed a strong upper limit of −0.4 ±  0.9 km s⁻¹ for the component of peculiar motion of Sgr A* perpendicular to the plane of the Galaxy. However, the determination of the component in the direction of Galactic rotation was considerably less accurate: 18 ±  7 km s⁻¹, as one must remove the uncertain effects of the solar orbit. Reid and Brunthaler did this by removing 27.19 ±  0.87 km s⁻¹ kpc⁻¹, based on Hipparcos measurements of Oort's constants (A − B) by Feast & Whitelock (1997), from the observed motion of Sgr A* in the Galactic plane of 29.45 km s⁻¹ kpc⁻¹. This method assumes that Θ₀/R₀ = A − B and that estimates of the shear and vorticity of nearby stars from Hipparcos data indicate the large-scale differential rotation of the Galaxy and are not subject to local irregularities in the solar neighborhood. Since we now have a direct, global estimate of Θ₀/R₀ = 30.3 ±  0.9 km s⁻¹ kpc⁻¹, we find the peculiar motion of Sgr A* in the direction of Galactic rotation to be −7.2 ±  8.5 km s⁻¹, with little sensitivity to R₀ (adopted to be 8.5 kpc here). This adds additional strong evidence that Sgr A* is a supermassive black hole, which is nearly stationary at the dynamical center of the Galaxy.

X.W.Z., B.Z., and Y.X. were supported by the Chinese National Science Foundation, through grants NSF 10673024, 10733030, 10703010 and 10621303, and by the NBPRC (973 Program) under grant 2007CB815403.

Since we have measured the position, distance, LSR velocity, and proper motion of each source, we know its full three-dimensional location in the Galaxy and full space motion relative to the Sun. Given a model of Galactic rotation, we can then calculate the noncircular (peculiar) velocity of each source. While this calculation is conceptually simple, in practice, there are some subtleties and sign convention issues that can lead to errors, and so here we present the necessary formulae (and FORTRAN source code (tar file) in the online material).

The required Galactic and solar motion parameters are given in Table 5, and those associated with the source are defined in Table 6. A schematic depiction of these parameters is given in Figure 8. We assume that the Sun is in the Galactic plane and calculate a source's peculiar motion (i.e., with respect to a circular Galactic orbit) as follows.

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We convert v_LSR to a heliocentric frame, v_Helio, by adding back the component of the standard solar motion in the line-of-sight direction that had been removed from the observed Doppler shift to calculate v_LSR. Note that one needs to use the (old) standard solar motion, which defines the LSR frame, and not the best values available today. Generally, observatories have adopted a value of 20 km s⁻¹ toward α(1900) = 18^h, δ(1900) = +30^d for the standard solar motion. Precessing these coordinates to the epoch of observation (≈2006) and converting to Galactic Cartesian coordinates yields the (U^Std_☉, V^Std_☉, W^Std_☉) values listed in Table 5. Then,

We rotate the motion vector from the equatorial heliocentric frame (μ_α, μ_δ, v_Helio) to a Galactic heliocentric frame (μ_l, μ_b, v_Helio). This is a rotation about a radial axis and is defined by the IAU in B1950 coordinates (Blaauw et al. 1960). For coordinates in J2000, Reid & Brunthaler (2004) give the right ascension and declination of the NGP as α_P = 12^h51^m262817 and δ_P = 27°07'42013, respectively, and the zero of longitude is the great semicircle originating at the NGP at the position angle θ = 122932. Galactic latitude can be obtained from

A useful angle ϕ can be determined (between 0° and 360°) from

and

and then Galactic longitude follows from

Proper motion in Galactic coordinates (μ_l, μ_b) can be easily calculated from the motion in equatorial coordinates (μ_α, μ_δ) by differencing (ℓ, b) values for coordinates determined, say, one year apart. This usually requires 64 bit precision in the calculations. Note that μ_l will naturally be defined positive in the direction of increasing Galactic longitude, which is counter to Galactic rotation.

Convert the proper motions to linear speeds (by multiplying by distance) via

where μ_lcos b is the actual motion tangent to the sky in the direction of Galactic longitude.

We now convert from spherical to Cartesian Galactic coordinates at the location of the Sun.

Next, add the full orbital motion of the Sun, using the best values of the solar motion and the circular rotation of the Galaxy at the position of the Sun (U^H_☉, V^H_☉ + Θ₀, W^H_☉),

The Galactocentric distance to the source projected onto the Galactic plane is given by

where D_p = Dcos b. The angle β between the Sun and the source as viewed from the Galactic center can be determined in all cases (i.e., from 0° to 360°) from

Rotate the vector (U₂, V₂, W₂) through the angle β in the plane of the Galaxy and remove circular Galactic rotation at the location of the source to yield (U_s, V_s, W_s),

The vector (U_s, V_s, W_s) gives the noncircular (peculiar) motion of the source in a Cartesian Galactocentric frame, where U_s is radially inward toward the Galactic center (as viewed by the source), V_s is in the local direction of Galactic rotation and W_s is toward the north Galactic pole.