Trigonometric Parallaxes of High-mass Star-forming Regions: Our View of the Milky Way

Our view of the Milky Way from its interior does not easily reveal its properties. The Sun is near the mid-plane of the Galaxy, resulting in multiple structures at different distances being superposed on the sky. Directly mapping the spiral structure of the Milky Way has proven to be a challenging enterprise, since distances are very large and dust extinction blocks most of the Galactic plane at optical wavelengths. Thus Gaia, even with a parallax accuracy of ±0.02 mas, will not be able to freely map the Galactic plane. However, Very Long Baseline Interferometry (VLBI) at radio wavelengths is unaffected by extinction and can detect molecular masers associated with massive young stars that best trace spiral structure in galaxies. Current parallax accuracy for VLBI allows distance measurements across most of the Milky Way. The Bar and Spiral Structure Legacy (BeSSeL) Survey¹⁴ and the Japanese VLBI Exploration of Radio Astrometry (VERA) project¹⁵ have now measured approximately 200 parallaxes for masers with accuracies typically about ±0.02 mas. Indeed, recently Sanna et al. (2017) measured the parallax of a maser of 0.049 ± 0.006 mas, placing its young massive star at a distance of 20 kpc, or about 12 kpc farther than the Galactic center.

VLBI parallaxes offer a unique opportunity to determine where we are in the Milky Way in three dimensions, as well as to reveal its spiral structure and kinematics. Massive young stars located within 7 kpc of the Galactic center are distributed in a plane with a small perpendicular dispersion (≈20 pc), and thus can be used to define the Galactic plane and locate the Sun relative to it. This allows us to robustly determine the orientation of the plane and the distance of the Sun perpendicular to it (${Z}_{\odot }$). In addition, proper motions, when coupled with parallax distances, give linear motions on the sky and, when combined with line-of-sight velocities, provide three-dimensional space motions. These can be fitted to simple models of Galactic rotation to yield our distance from the Galactic center (R₀), as well as the three-dimensional motion of the Sun (${U}_{\odot },{{\rm{\Theta }}}_{0}+{V}_{\odot },{W}_{\odot }$) in its orbit about the Galaxy (where the peculiar motion components are defined as ${U}_{\odot }$ toward the Galactic center, ${V}_{\odot }$ toward 90° longitude, and ${W}_{\odot }$ toward the north Galactic pole, and ${{\rm{\Theta }}}_{0}$ is the circular rotation of the Galaxy at the Sun).

Reid et al. (2009b, 2014) and Honma et al. (2012) have summarized VLBI parallaxes available at the time. Now, about twice as many parallaxes exist than are presented in those papers, and we update our understanding of Galactic structure and kinematics. In Section 2 we collect published parallaxes, as well as those in preparation or submitted for publication from the BeSSeL Survey group through 2018. Using this large data set, we improve upon our view of the spiral structure of the Milky Way in Section 3, and we fit the space motions in order to estimate fundamental Galactic and solar parameters in Section 4. With these parameters, we then calculate non-circular (peculiar) motions in Section 5. Using stars interior to the solar orbit where the Galactic plane is very flat, we evaluate the orientation of the IAU-defined plane and estimate the Sun's location perpendicular to the plane in Section 6. Based on motions of massive young stars perpendicular to the plane, we place limits on the precession of the Galactic plane in Section 7. We discuss some implications of our results in Section 8, and look forward to future advances in Section 9.

VLBI arrays have now been used to measure parallaxes and proper motions for about 200 maser sources associated with young massive stars, and these are listed in Table 1. They include results from the National Radio Astronomy Observatory's Very Long Baseline Array (VLBA), the Japanese VERA project, the European VLBI Network, and the Australian Long Baseline Array. For sources that have multiple parallax measurements (indicated with multiple references), either based on different masing molecules, transitions, and/or measured with different VLBI arrays, we present averaged results. Some judgment was used in how to weight the values. Typically we used variance weighting, but for results in tension we evaluated the robustness of each result and adjusted weights accordingly. Note that, in these cases, source coordinates correspond to a measurement of one of the masing molecules and transitions and are not averages; one should consult the primary references when using coordinates.

Table 1.  Parallaxes and Proper Motions of High-mass Star-forming Regions

Source	Alias	R.A.	Decl.	Parallax	${\mu }_{x}$	${\mu }_{y}$	${v}_{\mathrm{LSR}}$	Spiral	References
		(hh:mm:ss)	(dd:mm:ss)	(mas)	(mas y−1)	(mas y−1)	(km s−1)	Arm
G305.20+00.01		13:11:16.8912	−62:45:55.008	0.250 ± 0.050	−6.90 ± 0.33	−0.52 ± 0.33	−38 ± 5	CtN	61
G339.88−01.25		16:52:04.6776	−46:08:34.404	0.480 ± 0.080	−1.60 ± 0.52	−1.90 ± 0.52	−34 ± 3	CtN	37
G348.70−01.04		17:20:04.0360	−38:58:30.920	0.296 ± 0.026	−0.73 ± 0.31	−2.83 ± 0.59	−9 ± 5	CtN	1
G351.44+00.65	NGC6334	17:20:54.6010	−35:45:08.620	0.752 ± 0.069	0.31 ± 0.58	−2.17 ± 0.90	−8 ± 3	CrN	2,76
G359.13+00.03		17:43:25.6109	−29:39:17.551	0.165 ± 0.031	−3.98 ± 0.36	−6.50 ± 0.79	−1 ± 10	GC	52
G359.61−00.24		17:45:39.0697	−29:23:30.265	0.375 ± 0.021	1.00 ± 0.40	−1.50 ± 0.50	21 ± 5	CtN	59,52
G359.93−00.14		17:46:01.9183	−29:03:58.674	0.181 ± 0.029	−1.33 ± 0.44	−1.80 ± 0.78	−10 ± 10	GC	52
G000.31−00.20		17:47:09.1092	−28:46:16.278	0.342 ± 0.042	0.21 ± 0.39	−1.76 ± 0.64	18 ± 3	ScN	53
G000.37+00.03		17:46:21.4012	−28:35:39.821	0.125 ± 0.047	−0.77 ± 0.28	−2.67 ± 0.40	37 ± 10	3kF	7
G000.67−00.03	SgrB2	17:47:20.0000	−28:22:40.000	0.129 ± 0.012	−0.78 ± 0.40	−4.26 ± 0.40	62 ± 5	GC	3

Note. Columns 1 and 2 give the Galactic source name/coordinates and an alias, when appropriate. R.A. and decl. (J2000) are listed in columns 3 and 4. Columns 5 through 7 give the parallax and proper motion in the eastward (μ_x = ${\mu }_{\alpha }$ cosδ) and northward directions (${\mu }_{y}={\mu }_{\delta }$). Column 8 lists the local standard of rest velocity. Column 9 indicates the spiral arm segment in which it resides, based mostly on association with structure seen in ℓ − V plots of CO and H i emission. Starting near the Galactic center: GC = Galactic center region; Con = connecting arm; 3kN/F = 3 kpc arm (near/far); Nor = Norma (a.k.a 4 kpc) arm; ScN/F = Scutum–Centaurus arm (near/far); SgN/F = Sagittarius arm (near/far); Loc = Local arm; Per = Perseus arm; Out = Outer arm; OSC = Outer–Scutum–Centaurus arm. In addition we list two spurs: LoS = Local arm spur; AqS = Aquarius spur. Sources indicated with "???" could not be confidently assigned to an arm. Some parameter values listed here were preliminary ones and may be slightly different from final values appearing in published papers. Motion components and their uncertainties are meant to reflect that of the central star that excites the masers, and may be larger than formal measurement uncertainties quoted in some papers. Parallax uncertainties for sources with multiple (N) maser spots have been adjusted upward by $\sqrt{N}$, if not done so in the original publications. References are as follows: (1) BeSSeL Survey unpublished, (2) Wu et al. (2014), (3) Reid et al. (2009c), (4) Sato et al. (2014), (5) Sanna et al. (2009), (6) Zhang et al. (2014), (7) Sanna et al. (2014), (8) Immer et al. (2013), (9) Sato et al. (2010a), (10) Xu et al. (2011), (11) Brunthaler et al. (2009), (12) Bartkiewicz et al. (2008), (13) Kurayama et al. (2011), (14) Zhang et al. (2009), (15) Zhang et al. (2013), (16) Xu et al. (2009), (17) Sato et al. (2010b), (18) Oh et al. (2010), (19) Rygl et al. (2010), (20) Nagayama et al. (2011), (21) Xu et al. (2013), (22) Sanna et al. (2012), (23) Ando et al. (2011), (24) Moscadelli et al. (2011), (25) Rygl et al. (2012), (26) Zhang et al. (2012b), (27) Hachisuka et al. (2015), (28) Choi et al. (2014), (29) Hirota et al. (2008), (30) Moscadelli et al. (2009), (31) Moellenbrock et al. (2009), (32) Sato et al. (2008), (33) Xu et al. (2006), (34) Hachisuka et al. (2006), (35) Asaki et al. (2010), (36) Hachisuka et al. (2009), (37) Krishnan et al. (2015), (38) Honma et al. (2007), (39) Niinuma et al. (2011), (40) Reid et al. (2009a), (41) Shiozaki et al. (2011), (42) Honma et al. (2007), (43) Sandstrom et al. (2007), (44) Menten et al. (2007), (45) Kim et al. (2008), (46) Choi et al. (2008), (47) Zhang et al. (2012a), (48) Sparks et al. (2008), (49) Burns et al. (2014a), (50) Burns et al. (2017), (51) Zhang et al. (2019), (52) K. Immer et al. (2019, in preparation), (53) J. Li et al. (2019, in preparation), (54) K. L. J. Rygl et al. (2019, in preparation), (55) Wu et al. (2019), (56) B. Hu et al. (2019, in preparation), (57) Kounkel et al. (2017), (58) Sakai et al. (2019), (59) A. Sanna et al. (2019, in preparation), (60) L. Moscadelli et al. (2019, in preparation), (61) Krishnan et al. (2017), (62) Yamauchi et al. (2016), (63) Sanna et al. (2017), (64) Xu et al. (2018), (65) Quiroga-Nuñez et al. (2019), (66) Asaki et al. (2014), (67) Chibueze et al. (2016), (68) A. Bartkiewicz et al. (2019, in preparation), (69) Nagayama et al. (2014), (70) Xu et al. (2016), (71) Nagayama et al. (2015), (72) Dzib et al. (2016), (73) Sakai et al. (2015), (74) Burns et al. (2014b), (75) Imai et al. (2012), (76) Chibueze et al. (2014), (77) Kusuno et al. (2013), (78) Sakai et al. (2012).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The proper motion components, μ_x and μ_y, and local standard of rest (LSR) velocities, ${v}_{\mathrm{LSR}}$, given here are meant to apply to the central star which excites the masers. However, the proper motions are usually estimated from a small number of maser spot motions, and using them to infer the motion of the central star can involve significant uncertainty. For example, we gave preference to methanol over water maser motions, since the former generally have much smaller motions (≈5 km s⁻¹) with respect to their exciting star, compared to water masers that occur in outflows with expansion speeds of tens of kilometers per second (e.g., Sanna et al. 2010a, 2010b; Moscadelli et al. 2011). Some papers reporting proper motions give only formal measurement uncertainties, and for these we estimated an additional error term associated with the uncertainty in transferring the maser motions to that of the central star. Typically this error term was ±5 km s⁻¹ for methanol masers and ±10 km s⁻¹ for water masers, and these were added in quadrature with the measurement uncertainties. For LSR velocities, one often has extra information, for example from CO emission from associated molecular clouds. We generally treated methanol maser and CO LSR velocities as equally robust estimates of the central star's velocity, and preferred over water maser measurements.

The locations of the maser stars listed in Table 1 projected onto the Galactic plane are shown in Figure 1 superposed on a schematic plot of the Milky Way as viewed from the north Galactic pole where rotation is clockwise. An expanded view of the portion of the Milky Way for which we currently have most parallax measurements is shown in Figure 2. Distance uncertainties are indicated by the size of the dots, with sources having smaller uncertainties emphasized with larger dots. Note that for a given fractional parallax uncertainty, distance uncertainty increases linearly with distance. Thus, dot size should not be considered as, for example, a mass or luminosity indicator. The masers are color coded by spiral arms, which have been assigned in part by traces of quasi-continuous structure seen in CO and H i Galactic longitude–velocity plots, as well as Galactic latitude information. In Figure 3 we overplot the parallax sources on a longitude–velocity plot of H i emission. For the majority of cases, these arm assignments are unambiguous. For cases with uncertain arm identification, we used all information available, including the parallax and kinematic distances (from both radial and proper motions), using an updated version of our parallax-based distance estimator (see Appendix A).

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We fitted log-periodic spirals to the locations of the masers, expanding upon the approach described in Reid et al. (2014). We now allow for a "kink" in an arm, with different pitch angles on either side of the kink. The basic form of the spiral model is given by

$$\begin{eqnarray*}&&\mathrm{ln}(R/{R}_{\mathrm{kink}})=-(\beta -{\beta }_{\mathrm{kink}})\tan \psi ,\end{eqnarray*}$$

where R is the Galactocentric radius at a Galactocentric azimuth β in radians (defined as 0 toward the Sun and increasing in the direction of Galactic rotation) for an arm with a radius R_kink at a "kink" azimuth β_kink. An abrupt change in pitch angle, $\psi$, at β_kink allows for a spiral arm to be described by segments, as found in large-scale simulations by D'Onghia et al. (2013), who suggest that spiral arms are formed from multiple segments that join together. Kinks are also observed in spiral galaxies, for example, by Honig & Reid (2015), who found that arm segments have characteristic lengths of 5–8 kpc, often separated by kinks or gaps. Since we currently have parallax measurements that trace arms over typically ≲12 kpc in length, allowing for only one kink is a reasonable simplification. When assigning kink locations we often relied on apparent gaps in the spiral arms.

In order to extend arm fits into the fourth Galactic quadrant (awaiting parallax measurements from southern hemisphere VLBI arrays), we constrain the fitted arms to pass near observed enhancements of CO and H i emission at spiral arm tangencies. Priors for these tangencies are listed in the notes for Table 2.

As before, we fitted a straight line to [x, y] = [$\beta ,\mathrm{ln}(R/{R}_{\mathrm{kink}})]$ using a Markov chain Monte Carlo (MCMC) approach, in order to estimate the parameters by minimizing the distance perpendicular to the best-fit line. Our model included an adjustable parameter giving the (Gaussian 1σ) intrinsic width of a spiral arm, $w(R)=w({R}_{\mathrm{kink}})\,+({dw}/{dR})(R-{R}_{\mathrm{kink}})$, where (dw/dR) = 42 pc kpc⁻¹ was adopted from Reid et al. (2014). The data were variance weighted by adding in quadrature the effects of parallax–distance uncertainty and the component of the arm width perpendicular to the arm. The arm-width parameter w(R_kink) was adjusted along with the other model parameters and resulted in a reduced ${\chi }_{\nu }^{2}$ near unity.

We first weighted the data by assuming uncertainties that had a probability density function (PDF) with Lorentzian-like wings, which makes the fits insensitive to outliers (see the "conservative formulation" of Sivia & Skilling 2006), in order to identify and then remove sources with >3σ residuals (see the discussions of individual arms for source names). Then we re-fitted assuming a Gaussian PDF, and the best-fitting parameter values are listed in Table 2. Based on this approach, we find evidence for significant kinks in the Norma, Sagittarius, and Outer arms, as discussed in Section 3.2. As an example of the improved quality of fits by adding a kink, we did three fittings for the Sagittarius–Carina arm: (1) solving for a constant pitch angle which gave χ² = 34.1 for 32 degrees of freedom (dof); (2) solving for different pitch angles about a kink fixed at an azimuth of 52° which gave χ² = 30.9 for 31 dof; and (3) also solving for the kink azimuth, which gave a value of 24° with a χ² = 25.5 for 30 dof.

The best-fitting arm widths in the Galactic plane are plotted versus radius in Figure 4, extending the radial range and updating the results of Reid et al. (2014). We find that spiral arms widen with Galactocentric radius as $w(R)=336\,+36(R(\mathrm{kpc})-8.15)$ pc.

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3.1. Distributions Perpendicular to the Plane

The distribution perpendicular to the Galactic plane of the young high-mass stars with maser emission is shown in the top panel of Figure 5. Plotted are true Z-heights, adjusted for the Sun's vantage point at 5.5 pc above the plane, but with no correction for the warping of the plane. The increasing scatter in Z beyond ≈6–7 kpc from the Galactic center is a combination of intrinsic scatter and the effects of warping. The lower panel of Figure 5 is a zoomed view of the inner 8 kpc. It reveals a very flat distribution with no sign of corrugations with amplitudes in excess of ≈10 pc. This is in contrast to H i observations toward tangencies in the inner Galaxy by Malhotra (1995), which suggest corrugations with semi-amplitudes of ≈30 pc.

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In order to estimate the widths of arms perpendicular to the Galactic plane, one needs to deal with the effects of warping. We did this by smoothing the Z-heights along each arm with a window of ±20° of Galactic azimuth centered on each source. Provided there were at least five sources within the window, we then subtracted the variance-weighted mean-z. In order to minimize the effects of outliers, we iterated this process, removing 6σ, 5σ, 4σ, and finally 3σ outliers. Finally, with the effects of warping removed, the data from all arms were placed into 0.5 kpc bins in radius and rms values calculated. These rms values are displayed in Figure 4. These Z-widths can be reasonably described by ${\sigma }_{z}(R)=20+36(R(\mathrm{kpc})-7.0)$ pc, for R > 7.0 kpc, and constant at 20 pc inside that radius.

3.2. Updated Spiral Arm Model

3.3. Other Arm Segments

In order to estimate the distance to the Galactic center and rotation curve parameters, we used the Bayesian MCMC approach described in Reid et al. (2014). We treat as data the three-dimensional components of velocity and model these as arising from an axisymmetric Galactic rotation, allowance for an average streaming (non-circular) motion in the plane of the Galaxy. Previously, we adopted the "universal" form for the rotation curve (URC) of Persic et al. (1996), since it fitted the data as well as or better than other rotation curves, and it well models the rotation of a large number of external spiral galaxies. In our previous paper, we used a three-parameter formulation for the URC: $a1=V({R}_{\mathrm{opt}})$, $a2={R}_{\mathrm{opt}}/{R}_{0}$, and $a3=1.5{(L/{L}^{* })}^{0.2}$, where R_opt and $V({R}_{\mathrm{opt}})$ are the radius enclosing 83% of the optical light and the circular velocity at that radius for a galaxy with luminosity L relative to an ${L}^{* }$ galaxy with ${M}_{b}=-20.6$ mag. However, Persic et al., in their note added in proof, present a simplified two-parameter formulation with parameter a1 removed via scaling relations between optical radius, velocity, and luminosity. Since the two- and three-parameter formulations produce similar rotation curves over radial ranges of 0.5–2.0 R_opt, we now adopt the two-parameter version, with the rotation curve defined by only parameters a2 and a3. (See Appendix B for a FORTRAN subroutine that returns a circular rotation speed for a given Galactocentric radius.)

As in Reid et al. (2014), given the current uncertainty in the value for the circular component (${V}_{\odot }$) of solar motion and the magnitude of the average peculiar motions of masers associated with massive young stars, we present fits with four sets of priors:

• (A)  
  adopting a loose prior for the ${V}_{\odot }$ component of solar motion, ${U}_{\odot }=11.1\pm 1.2$, ${V}_{\odot }=15\pm 10$, ${W}_{\odot }=7.2\,\pm 1.1$ km s⁻¹, and for the average peculiar motion for the masing stars of $\overline{{U}_{s}}=3\pm 10$ and $\overline{{V}_{s}}=-3\pm 10$ km s⁻¹;
• (B)  
  using no priors for the average peculiar motion of the stars, but a tighter prior for ${V}_{\odot }=12.2\pm 2.1$ km s⁻¹ from Schoenrich et al. (2010);
• (C)  
  using no priors for the solar motion, but tighter priors on the average peculiar motion of the stars of $\overline{{U}_{s}}=3\pm 5$ and $\overline{{V}_{s}}=-3\pm 5$ km s⁻¹;
• (D)  
  using essentially no priors for either the solar motion or average peculiar motion of the stars, but bounding the ${V}_{\odot }$ and $\overline{{V}_{s}}$ parameters with equal probability within ±20 km s⁻¹ of the set-A values and zero probability outside that range.

Since we expect large non-circular motions in the vicinity of the Galactic bar, we removed the 19 sources that are within 4 kpc of the Galactic center. Next, we removed 22 sources whose fractional parallax uncertainties exceeded 20%, in order to avoid significant complications arising from highly asymmetric PDFs when inverting parallax to estimate distance (e.g., Bailer-Jones 2015). Finally, as discussed in Reid et al. (2014), we expect some outliers in the motion data, owing for example to the effects of super-bubbles that can accelerate gas which later forms the stars with the masers we observe. Therefore, we used the "conservative formulation" of Sivia & Skilling (2006), which uses a "Lorentzian-like" PDF for motion uncertainties when fitting a preliminary model of Galactic rotation to the data. The best-fitting parameters are listed in Table 3 in column A1. This fit is largely insensitive to outliers, and allows us to identify and remove them in a prescribed and objective manner. Defining an outlier as having greater than a 3σ residual in any motion component, we removed 11 sources from further consideration.¹⁶

Table 3.  Bayesian Fitting Results

	A1	A5	B	C	D
Fitted Parameters
R0 (kpc)	8.22 ± 0.22	8.15 ± 0.15	8.15 ± 0.14	8.04 ± 0.17	8.04 ± 0.19
${U}_{\odot }$ (km s−1)	10.8 ± 1.2	10.6 ± 1.2	10.7 ± 1.2	8.1 ± 2.5	8.2 ± 2.8
${V}_{\odot }$ (km s−1)	13.6 ± 6.7	10.7 ± 6.0	11.9 ± 2.1	10.7 ± 4.8	6.5 ± 8.4
${W}_{\odot }$ (km s−1)	7.6 ± 0.9	7.6 ± 0.7	7.7 ± 0.7	7.9 ± 0.8	7.9 ± 0.9
$\overline{{U}_{s}}$ (km s−1)	6.1 ± 1.9	6.0 ± 1.4	6.2 ± 1.5	3.6 ± 2.6	3.7 ± 3.0
$\overline{{V}_{s}}$ (km s−1)	−2.1 ± 6.5	−4.3 ± 5.6	−3.1 ± 2.2	−4.4 ± 4.5	−8.7 ± 7.9
a2	0.96 ± 0.08	0.96 ± 0.05	0.95 ± 0.05	0.95 ± 0.05	0.96 ± 0.05
a3	1.62 ± 0.03	1.62 ± 0.02	1.62 ± 0.02	1.61 ± 0.02	1.62 ± 0.03
Calculated Values
${{\rm{\Theta }}}_{0}$ (km s−1)	237 ± 8	236 ± 7	236 ± 5	233 ± 6	238 ± 8
(${{\rm{\Theta }}}_{0}$ + ${V}_{\odot }$) (km s−1)	249 ± 7	247 ± 4	247 ± 4	244 ± 5	244 ± 6
(${{\rm{\Theta }}}_{0}$ + ${V}_{\odot }$)/R0(km s−1 kpc−1)	30.46 ± 0.43	30.32 ± 0.27	30.32 ± 0.28	30.39 ± 0.28	30.36 ± 0.28
Fit Statistics
${\chi }^{2}$	809.7	425.0	427.3	426.2	425.7
Ndof	465	432	432	432	432
Nsources	158	147	147	147	147
${r}_{{R}_{0},{{\rm{\Theta }}}_{0}}$	0.57	0.45	0.80	0.63	0.36

Note.  Fit A1 used the 158 sources in Table 1 with Galactocentric radii >4 kpc and fractional parallax uncertainty <20%, an "outlier tolerant" probability distribution function for data uncertainty (see Section 4), and set-A priors: Gaussian solar motion priors of ${U}_{\odot }=11.1\pm 1.2$, ${V}_{\odot }=15\pm 10$, ${W}_{\odot }=7.2\pm 1.1$ km s⁻¹ and average source peculiar motion priors of $\overline{{U}_{s}}=3\pm 10$ and $\overline{{V}_{s}}=-3\pm 10$ km s⁻¹. Fit A5 removed 12 sources found in Fit A1 to have a motion component residual greater than 3σ, and used a Gaussian probability distribution function for the residuals (ie, least-squares), and the same priors as A1. Fits B, C, and D were similar to A5, except for the priors: B used the solar motion priors of Schoenrich et al. (2010) $({U}_{\odot }=11.1\pm 1.2,{V}_{\odot }=12.2\pm 2.1,{W}_{\odot }=7.2\pm 1.1)$ km s⁻¹ and no priors for source peculiar motions; C used no solar motion priors and source peculiar motion priors of $\overline{{U}_{s}}=3\pm 5$ and $\overline{{V}_{s}}=-3\pm 5$ km s⁻¹; D used flat priors for all parameters, except ${V}_{\odot }^{\mathrm{Std}}$ and $\overline{{V}_{s}}$ which were given unity probability between ±20 km s⁻¹ of the set-A prior values and zero probability outside this range. The fit statistics listed are chi-squared (${\chi }^{2}$), the number of degrees of freedom (N_dof), the number of sources used (N_sources), and the Pearson product-moment correlation coefficient for parameters R₀ and ${{\rm{\Theta }}}_{0}$ (${r}_{{R}_{0},{{\rm{\Theta }}}_{0}}$).

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With the resulting "clean" data set of 147 sources, we assumed a Gaussian PDF for the data uncertainties (equivalent to least-squares fitting) and the best-fitting parameter values are given in Table 3 in column A5. Note that we present estimates of ${{\rm{\Theta }}}_{0}$ in Table 3, calculated from the rotation curve, even though it was not a fitted parameter. We also combine parameters to generate the marginalized PDF for the full circular rotation rate of the Sun in linear, (${{\rm{\Theta }}}_{0}$+${V}_{\odot }$), and angular, (${{\rm{\Theta }}}_{0}$+${V}_{\odot }$)/R₀, units. We adopt the A5 fit as the best model. However, for completeness we follow Reid et al. (2014) and also present model fits using the different priors listed above. Table 3 summarizes the best-fitting parameter estimates for priors B, C, and D in the last three columns, which yield parameter values similar to those for fit A5. Note that Quiroga-Nuñez et al. (2017) have shown that fits of mock data sets demonstrate that parameter estimates should be unbiased.

The correlation between parameters R₀ and ${{\rm{\Theta }}}_{0}$ in fit A5 is modest: ${r}_{{R}_{0},{{\rm{\Theta }}}_{0}}=0.45$. This is similar to our previous value reported in Reid et al. (2014), as well as found in simulations by Quiroga-Nuñez et al. (2017), since it depends on the range and distribution of parallax sources, which has not significantly changed. Parallax measurements from a VLBI array in the southern hemisphere are needed to further reduce this correlation. As we have previously noted, ${{\rm{\Theta }}}_{0}$, ${V}_{\odot }$ and $\overline{{V}_{s}}$ can be highly correlated, hence the need for priors for some of these parameters. For the set-A priors, we find the following correlations: ${r}_{{{\rm{\Theta }}}_{0},{V}_{\odot }}=-0.74$, ${r}_{{{\rm{\Theta }}}_{0},\overline{{V}_{s}}}=+0.74$, and ${r}_{{V}_{\odot },\overline{{V}_{s}}}=-0.99$.

Using the Galactic parameters and solar motion values from fit A5, we can transform to a reference frame that rotates as a function of radius within the Galaxy. The resulting non-circular or peculiar motions in the plane of the Galaxy are shown in Figure 6. We only plot sources whose motion uncertainties are $\lt 20$ km s⁻¹. While the vast majority of sources have moderate peculiar motions of ≲10 km s⁻¹, we can identify two regions in the Galaxy with significantly larger peculiar motions. The first anomalous region is in the Perseus arm at Galactic longitudes between about $105^\circ$ and $135^\circ$. This anomaly has been well documented in the literature (e.g., Humphreys 1978; Xu et al. 2006; Sakai et al. 2019). The second anomalous region is within a Galactocentric radius of $\approx 5\,\mathrm{kpc}$. Large peculiar motions are seen near the end of the long bar for sources in the 3 kpc, Norma, and Scutum arms.

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The average peculiar motion of our sources for all fits in Table 3 indicates small positive values for $\overline{{U}_{s}}$ (toward the Galactic center) and small negative values for $\overline{{V}_{s}}$ (in the direction of Galactic rotation), possibly resulting from streaming motions of massive young stars toward the Galactic center and counter to Galactic rotation. Adopting fit A5, which has loose priors for the solar motion component in the direction of Galactic rotation (${V}_{\odot }^{\mathrm{Std}}=15\pm 10$ km s⁻¹) and for $\overline{{U}_{s}}$ and $\overline{{V}_{s}}$ (3 ± 10 and −3 ± 10 km s⁻¹, respectively), we find $\overline{{U}_{s}}=6.0\pm 1.4$ and $\overline{{V}_{s}}\,=-4.3\pm 5.6$ km s⁻¹. The $\overline{{U}_{s}}$ value appears significant and qualitatively consistent with theoretically expected values for the formation of stars from gas that was shocked when entering a spiral arm of low pitch angle (e.g., Roberts 1969).

Given a source's distance and Galactic coordinates, we can calculate its three-dimensional location in the Galaxy. Since one expects massive young stars to be distributed very closely to the plane, this offers an opportunity to refine the parameters of the IAU-defined Galactic plane, as recently investigated by Anderson et al. (2019). Here we fit for the Sun's location perpendicular to the plane and a two-dimensional tilt of the true plane with respect to the IAU-defined plane. Then, with improved parameters, we estimate the scale height of our sources.

We define Cartesian Galactocentric coordinates $(X,Y,Z)$, where X is distance perpendicular to the Sun–Galactic center line (positive in the first and second quadrants), Y is distance from the Galactic center toward the Sun, and Z is distance perpendicular to the plane (positive toward the north Galactic pole). In order to avoid complications of Galactic warping, we follow Gum et al. (1960) and Blaauw et al. (1960) and restrict the range of sources fitted to within a Galactocentric radius of 7.0 kpc, giving a sample of 116 sources. Anticipating a scale height for massive young stars near 20 pc (see below), we also remove possible "outlying" sources more than 60 pc from the plane. Some outliers would be expected if gas compressed and accelerated by superbubbles forms high-mass stars. With these restrictions, we retained 96 massive young stars and plot their three-dimensional locations in the Galaxy relative to the IAU-defined plane in Figure 7.

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These stars are very tightly distributed in a plane. However, the distribution about the IAU-defined plane is slightly asymmetric, with two-thirds (65) of the sources lying below that plane. The mean offset is −7.3 pc with a standard error of the mean of ±2.1 pc. In general, this can be explained by a combination of the Sun being offset from the true Galactic plane and/or a tilt of the true plane from the IAU plane. Hence, we fitted an adjusted plane to these data. We treat the observed (${Z}_{\mathrm{IAU}}$) values as data and model them as follows:

$$\begin{eqnarray*}&&{Z}_{\mathrm{IAU}}=-{Z}_{\odot }+{d}_{p}\,\cos l\,\tan {\psi }_{Y}+{d}_{p}\,\sin l\,\tan {\psi }_{X},\end{eqnarray*}$$

where ${Z}_{\odot }$ is the Z-offset of the Sun, d_p is the distance of a source from the Sun projected in the plane, l is Galactic longitude, and ${\psi }_{X}$ and ${\psi }_{Y}$ are tilt angles of the true plane with respect to the IAU-defined plane (referred to as "roll" and "tilt," respectively, by Anderson et al. 2019). The ${Z}_{\mathrm{IAU}}$ data were fitted using an MCMC technique, accepting or rejecting trials with the Metropolis–Hastings algorithm. Data uncertainties were based on parallax uncertainties and assumed to be Gaussian. In order to minimize the effects of distance biases, owing to asymmetric distance PDFs when converting parallaxes to distances (e.g., Bailer-Jones 2015), we used only the 80 sources with fractional parallax uncertainties less than 20%.

Figure 8 displays the MCMC trials and marginalized PDFs for the three parameters. The centers of the 68% confidence ranges for the marginalized PDFs give the following parameter estimates: ${Z}_{\odot }=5.5\pm 5.8$ pc, ${\psi }_{X}=-0\buildrel{\circ}\over{.} 08\pm 0\buildrel{\circ}\over{.} 10$, and ${\psi }_{Y}=0\buildrel{\circ}\over{.} 00\pm 0\buildrel{\circ}\over{.} 10$. Thus, we find very small changes in the orientation ("tilt" parameters) of the true Galactic plane relative to the IAU-defined plane. Anderson et al. (2019) find similarly small values of ${\psi }_{X}$ (between $-0\buildrel{\circ}\over{.} 04$ and +015) and ${\psi }_{Y}$ (between $-0\buildrel{\circ}\over{.} 11$ and +008) for different sub-samples. Therefore, for simplicity, in the following we assume the orientation of the plane follows the IAU definition, but we correct for the Sun's Z-height above the plane.

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Placing the Sun 5.5 pc above the IAU-defined plane, we calculate Z-offsets relative to the true (adjusted) Galactic plane, which we designate Z_true. Figure 9 presents a binned histogram of Z_true values for our full sample of 120 massive young stars within 7.0 kpc of the Galactic center and 200 pc of the plane, along with the best-fitting exponential distribution. The exponential distribution fits the data very well and gives a scale height of 19 ± 2 pc.

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We now investigate the possibility that the Galaxy precesses, possibly owing to torques from its triaxial halo and/or from Local Group galaxies. A rigid-body precession of the plane could be inferred from the Z-component of velocity of our massive young stars, W_s, following

$$\begin{eqnarray}&&{W}_{s}={W}_{1}^{x}\,(X/{R}_{0})+{W}_{1}^{y}\,(Y/{R}_{0})-{W}_{\odot },\end{eqnarray} \tag{ 1 }$$

where ${W}_{1}^{x}$ and ${W}_{1}^{y}$ are the speeds of vertical motion at radius R₀ and location in the plane given by X and Y. We used the estimate of Schoenrich et al. (2010) of the vertical motion of the Sun relative to the solar neighborhood of ${W}_{\odot }=7.2\,\pm 0.6$ km s⁻¹ as a strong prior. After removing W_s measurements with uncertainties $\gt 10$ km s⁻¹, we fitted the data using a Bayesian MCMC procedure similar to that described in Section 4, first using the "conservative formulation" of Sivia & Skilling (2006) to identify and remove 20 outliers and then refitting with Gaussian data uncertainties. Using sources at all Galactocentric radii, we find ${W}_{1}^{x}=2.0\pm 1.6$ km s⁻¹, ${W}_{1}^{y}\,=-0.1\pm 0.8$ km s⁻¹, and ${W}_{\odot }=7.9\pm 0.5$ km s⁻¹. Dividing the data into inner and outer regions, we find for $R\lt 7$ kpc, ${W}_{1}^{x}=1.2\pm 2.1$ km s⁻¹, ${W}_{1}^{y}=-2.5\pm 1.4$ km s⁻¹, and ${W}_{\odot }\,=7.4\pm 0.6$ km s⁻¹, and for $R\gt 7\,\mathrm{kpc}$, ${W}_{1}^{x}=3.1\pm 2.2$ km s⁻¹, ${W}_{1}^{y}=+0.7\pm 1.0$ km s⁻¹, and ${W}_{\odot }=7.6\pm 0.6$ km s⁻¹. The estimates of ${W}_{1}^{y}$ for the inner and outer Galaxy differ by 3.2 ± 1.7 km s⁻¹ with nearly 2σ significance, hinting at the possibility of a radial dependence.

The Galaxy is known to exhibit significant warping, starting between about 7 and 8 kpc radius and reaching an amplitude of about 300 pc at a radius of 12 kpc (Gum et al. 1960). Since the warping is likely dynamic in origin, one would expect vertical motions leading the warp to vary with radius. Were the warping at 12 kpc radius to develop over a timescale of $\sim {10}^{8}$ yr (roughly one-third of an orbital period at that radius), that would correspond to a characteristic vertical speed of ∼3 km s⁻¹. In order to investigate this possibility, we added second-order terms to Equation (1) yielding

$$\begin{eqnarray}\begin{array}{rcl}{W}_{s} & = & {W}_{1}^{x}\,(X/{R}_{0})+{W}_{2}^{x}\,{\left(X/{R}_{0}\right)}^{2}+{W}_{1}^{y}\,(Y/{R}_{0})\\ & & +\,{W}_{2}^{y}\,{\left(Y/{R}_{0}\right)}^{2}-{W}_{\odot }.\end{array}\end{eqnarray} \tag{ 2 }$$

Re-fitting with no radial restriction, we find ${W}_{1}^{x}=-1.7\,\pm 3.7$ km s⁻¹, ${W}_{2}^{x}=5.4\pm 4.8$ km s⁻¹, ${W}_{1}^{y}=-0.9\pm 1.2$ km s⁻¹, ${W}_{2}^{y}=1.4\pm 1.3$ km s⁻¹, and ${W}_{\odot }=7.8\pm 0.5$ km s⁻¹. Neither the first- nor second-order parameters differ significantly from zero, and for the y-terms reasonable upper limits for their magnitudes (and hence for vertical motions at $R={R}_{0}$) are $\approx 4$ km s⁻¹.

8.1. An Updated View of the Milky Way

Is the Milky Way a two- or four-arm spiral? Based on the model described in Section 3.2, the answer is a four-arm spiral as traced by massive young stars. The four arms do not include sub-structures such as the 3 kpc arms, since they are likely a phenomenon associated with the bar, and the Local arm, which appears to be an isolated segment. Were the Milky Way a two-arm spiral, its arms would have to wrap twice around the center in order to accommodate the parallax data. This would require pitch angles to average near 5°. However, based on the data in Table 2 and weighting the pitch angles by their segment lengths, we find an average pitch angle of 10° for the major arms (Norma–Outer, Sct–Cen–OSC, Sgr–Car, and Perseus).

The form of spiral arm segments obtained in Section 3.2 can be used to update the input model for the parallax-based distance estimator of Reid et al. (2016) (see Appendix A for details). Coupling our revised model of spiral arm locations with line-of-sight velocities that trace arms in CO or H i emission in longitude–velocity plots, we arrive at a revised spatial-kinematic model for arms. Incorporating the revised arm model into our parallax-based distance estimator, we can now take longitude–latitude–velocity values from surveys of spiral-arm tracers (e.g.,  H ii regions, molecular clouds, star-forming masers) and estimate distances more accurately than from standard kinematic distances. Taking as input the catalogs of water masers from Valdettaro et al. (2001), methanol masers from Pestalozzi et al. (2005) and Green et al. (2017), H ii regions from Anderson et al. (2012), and red MSX sources from Urquhart et al. (2014), we now construct an improved visualization of the spiral structure of the Galaxy, shown in Figure 10.

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While the locations of spiral arms can be obtained from a sample of maser parallaxes that is far from complete, the input catalogs for our visualization of the Milky Way have better defined samples and are more complete. For example, the red MSX catalog of Urquhart et al. (2014) is estimated to be complete for star-forming regions with bolometric luminosities $\gt 2\times {10}^{4}$ ${L}_{\odot }$ to a distance of 18 kpc. While the map in Figure 10 should be much more complete than that in Figure 1, we suspect that it under-represents distant sources, owing to their intrinsic weakness as well as effects of confusion. Therefore, as a first attempt to correct for incompleteness, for sources more distant than 3 kpc we randomly "sprinkle" a number of points given by ${(D(\mathrm{kpc})/3)}^{2}$ (capped at 10 points) from a Gaussian distribution whose width increases with Galactocentric radius as found in Section 3.

The visualization of the pattern of sources associated with spiral arms in Figure 10 shows a dearth of sources toward the Galactic center within a cone of half angle $\approx 12^\circ$. This is likely a result of survey bias against identifying distant, weak sources in the presence of numerous strong nearby sources. Also, most sources in the general direction of the Galactic center have LSR velocities close to zero, which can limit the information used to discriminate among arm assignments.

8.2. Fundamental Galactic Parameters

The distance to the Galactic center, R₀, is a fundamental parameter affecting a wide variety of astrophysical questions (e.g., Reid 1993). As such there are a large number of estimates for R₀, and we restrict our discussion to direct methods. A trigonometric parallax measurement of water masers from Sgr B2, a massive star-forming region within $\approx 100$ pc of the Galactic center, indicates ${R}_{0}=7.9\pm 0.8\,\mathrm{kpc}$ (Reid et al. 2009c). The most recent analyses of infrared observations tracing the orbits of stars around the supermassive black hole, Sgr A*, provide geometric estimates of R₀ of 7.946 ± 0.059 kpc (Do et al. 2019) and 8.178 ± 0.026 kpc (Gravity Collaboration et al. 2019), where the improved accuracy of the latter measurement comes from infrared interferometric observations. Our result of ${R}_{0}\,=8.15\pm 0.15\,\mathrm{kpc}$ is both independent of, and consistent with, these estimates.

While there are moderate correlations among R₀, ${{\rm{\Theta }}}_{0}$, and ${V}_{\odot }$, our parallax data strongly constrain the linear and angular speeds of the Sun in its Galactic orbit. Adopting the A5-fit results, we find $({{\rm{\Theta }}}_{0}+{V}_{\odot })=247\pm 4$ km s⁻¹ and $({{\rm{\Theta }}}_{0}+{V}_{\odot })/{R}_{0}=30.32\pm 0.27$ km s⁻¹ kpc⁻¹. This can be compared to an independent and direct estimate from the apparent proper motion of the supermassive black hole, Sgr A*, assuming it is stationary at the center of the Milky Way. Reid & Brunthaler (2004) measure the apparent motion of Sgr A* in Galactic longitude to be −6.379 ± 0.019 mas yr⁻¹. This implies $({{\rm{\Theta }}}_{0}+{V}_{\odot })/{R}_{0}=30.24\pm 0.12$ km s⁻¹ kpc⁻¹. Thus, our estimate of the angular orbital speed of the Sun is in excellent agreement with the apparent proper motion of Sgr A*.

Our best-fit model rotation curve for the Galaxy (fit A5) is shown in Figure 11 as the dotted–dashed line. This curve has the URC form and is specified by only two parameters ($a2=0.96$ and $a3=1.62$). The slight bias of the curve above the data is a result of the tendency for massive young stars to lag Galactic rotation with $\overline{{V}_{s}}=-4.3$ km s⁻¹. This rotation curve peaks at 237 km s⁻¹ at a radius of 6.8 kpc and falls to 227 km s⁻¹ at a radius of 14.1 kpc, corresponding to a slope of 1.4 km s⁻¹ kpc⁻¹ or, alternatively, with a power-law index of −0.059 over that radial range.

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Plotted with black dots in the lower panel of Figure 11 are variance-weighted averages of the data, after correcting for the lag, within a window of 1 kpc full-width in steps of 0.25 kpc (see Table 4). This model-independent rotation curve shows only a very slight departure from the URC model. The data could support a small decrease of ≈5 km s⁻¹ between radii of 7 to 9 kpc, followed by a flattening out to 10 kpc. However, there is no evidence for a ≈20 km s⁻¹ dip in the rotation curve near R = 9 kpc with a width of ≈2 kpc, as has been suggested in the literature (e.g., Sofue et al. 2009). Qualitatively, our rotation curve is similar to that presented in the review by Bland-Hawthorn & Gerhard (2016) and more recently in the paper by Eilers et al. (2019) (allowing for their estimate of ${{\rm{\Theta }}}_{0}$ of 229 km s⁻¹, which is 7 km s⁻¹ below our value).

Table 4.  Rotation Curve

R (kpc)	${{\rm{\Theta }}}_{\mathrm{URC}}$	${{\rm{\Theta }}}_{\mathrm{masers}}$
0.250	50.73	...
0.500	77.36	...
0.750	98.72	...
1.000	116.96	...
1.250	132.93	...
1.500	147.06	...
1.750	159.61	...
2.000	170.76	...
2.250	180.64	...
2.500	189.38	...
2.750	197.07	...
3.000	203.80	...
3.250	209.67	...
3.500	214.73	...
3.750	219.09	...
4.000	222.79	224.06 ± 2.72
4.250	225.93	227.82 ± 1.81
4.500	228.54	228.10 ± 1.63
4.750	230.70	230.36 ± 1.62
5.000	232.46	233.51 ± 1.40
5.250	233.87	234.60 ± 1.52
5.500	234.97	236.25 ± 1.42
5.750	235.81	239.46 ± 1.19
6.000	236.41	239.88 ± 1.22
6.250	236.82	240.30 ± 1.14
6.500	237.06	240.38 ± 1.07
6.750	237.16	239.32 ± 1.17
7.000	237.14	238.53 ± 1.24
7.250	237.02	238.01 ± 1.41
7.500	236.82	234.57 ± 1.41
7.750	236.55	232.82 ± 1.08
8.000	236.23	232.83 ± 1.00
8.250	235.87	232.67 ± 1.00
8.500	235.48	233.04 ± 1.05
8.750	235.06	231.99 ± 1.33
9.000	234.63	230.78 ± 1.66
9.250	234.19	231.41 ± 1.24
9.500	233.75	231.52 ± 1.15
9.750	233.30	232.08 ± 1.21
10.000	232.86	232.46 ± 1.22
10.250	232.42	233.21 ± 1.69
10.500	231.99	232.89 ± 3.15
10.750	231.57	...
11.000	231.16	...
11.250	230.76	...
11.500	230.38	232.01 ± 6.11
11.750	230.01	225.85 ± 3.38
12.000	229.65	225.85 ± 3.38
12.250	229.31	224.66 ± 3.76
12.500	228.98	...
12.750	228.67	...
13.000	228.37	...
13.250	228.08	...
13.500	227.81	...
13.750	227.54	...
14.000	227.30	...
14.250	227.06	...
14.500	226.83	...
14.750	226.62	...
15.000	226.41	...
15.250	226.22	...
15.500	226.04	...
15.750	225.86	...
16.000	225.70	...
16.250	225.54	...
16.500	225.39	...
16.750	225.25	...
17.000	225.11	...
17.250	224.98	...
17.500	224.86	...
17.750	224.74	...
18.000	224.63	...
18.250	224.53	...
18.500	224.43	...
18.750	224.33	...
19.000	224.24	...
19.250	224.16	...
19.500	224.07	...
19.750	224.00	...
20.000	223.92	...

Note. R is Galactocentric radius, ${{\rm{\Theta }}}_{\mathrm{URC}}$ is the universal rotation curve from fit A5, and ${{\rm{\Theta }}}_{\mathrm{masers}}$ are variance-weighted, circular velocities for the masers within a radial window of 1 kpc; these values have also been adjusted upward by 4.3 km s⁻¹ to account for the average lag found in fit A5.

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In the context of the URC formalism, $a2={R}_{\mathrm{opt}}/{R}_{0}$, implying ${R}_{\mathrm{opt}}=7.82\,\mathrm{kpc}$ and $a3=1.5{(L/{L}^{* })}^{0.2}$, yielding $L=1.47{L}^{* }$. Persic et al. (1996) give a halo mass of ${M}_{\mathrm{halo}}=1.6\,\times {10}^{12}{(L/{L}^{* })}^{0.56}$ ${M}_{\odot }$, which returns ${M}_{\mathrm{halo}}\,=2.0\times {10}^{12}$ ${M}_{\odot }$, and a disk scale length ${R}_{\mathrm{disk}}\,={R}_{\mathrm{opt}}/3.2=2.44\,\mathrm{kpc}$.

Reid et al. (2014) examined the impact of revised Galactic parameters on the Hulse–Taylor binary pulsar's orbital decay, owing to gravitational radiation as predicted by general relativity (GR). This test of GR is sensitive to (${V}^{2}/R$) accelerations from the Galactic orbits of the Sun and the pulsar. Repeating that analysis with the Galactic parameters of fit A5, we find that the binary's orbital decay rate is 0.99658 ± 0.00035 (nearly a 10σ deviation) of that expected from GR for the originally assumed pulsar distance of 9.9 kpc. However, we note that, using our estimates of the values and uncertainties in R₀ and ${{\rm{\Theta }}}_{0}$, a distance of 6.54 ± 0.24 kpc would remove the discrepancy from the GR prediction, providing a strong prediction that can be tested with a direct pulsar parallax measurement.

8.3. Displacement of the Sun from the Galactic Plane

The original five-year phase of the BeSSeL Survey used about 3500 hr on the VLBA. Results in this paper are nearly complete through the first four years of the project. The final year's observations, which preferentially targeted distant sources and those especially important to better define spiral arms, are currently being analyzed.

While Figure 10 presents the most complete picture of the spiral structure of the Milky Way to date, in regions lacking parallax data, distances to arm segments are necessarily less accurate. This model can be improved by adding parallaxes for sources in the fourth quadrant. In the near future, an extension of the BeSSeL Survey to the southern hemisphere is planned. Using the University of Tasmania's AuScope array of four antennas spanning the Australian continent (Yarragadee in Western Australia, Katherine in the Northern Territory, Ceduna in South Australia, and Hobart in Tasmania), agumented by an antenna in New Zealand (Warkworth), we anticipate starting observations in 2020. After several years of observation, we hope to obtain parallaxes for 50 to 100 6.7-GHz masers in the Galactic longitude range 240°–360°, where currently only a few have been measured.

In the long term, we would like to better explore the Milky Way well beyond its center, at distances of 10–20 kpc, in order to test and improve our current model of spiral structure. While some progress can be made with existing VLBI arrays, adding large numbers of parallaxes for these very distant regions may require the next generation of radio facilities, such as the SKA and ngVLA, with high sensitivity and long baselines.

This work was partially funded by the ERC Advanced Investigator Grant GLOSTAR (247078). It was also sponsored by the MOST grant No. 2017YFA0402701 and NSFC Grants 11933011, 11873019 and 11673066. B.Z. and Y.W.W. are supported by the 100 Talents Project of the Chinese Academy of Sciences (CAS). A.B. acknowledges support by the National Science Centre Poland through grant 2016/21/B/ST9/01455. K.L.J.R. acknowledges funding by the Advanced European Network of E-infrastructures for Astronomy with the SKA (AENEAS) project, supported by the European Commission Framework Programme Horizon 2020 Research and Innovation action under grant agreement No. 731016.

Facilities: VLBA - Very Long Baseline Array, VERA - , EVN - , Australian LBA. -

Version 2 of our parallax-based distance estimator (Reid et al. 2016), as well as the FORTRAN code and spiral arm data, can be found at http://bessel.vlbi-astrometry.org/. It incorporates the improved model for the locations of spiral arms in the Milky Way described in Section 3.2. The previous model only included the locations of arm segments in the first, second, and third Galactic quadrants. We now provisionally extend the arm models to the fourth quadrant by extrapolating arms interior to the solar circle from the first quadrant guided by the directions of arm tangencies in the fourth quadrant. We also assume that arms do not cross each other on the far side of the Galactic center, and we make use of the parallax measurement of G007.47+00.05 (Sanna et al. 2017), which ties the Scutum–Centaurus arm beyond the Galactic center with the "Outer–Scutum–Centaurus" arm. It then follows that the Norma arm connects with the Outer arm beyond the Galactic center, and that the Perseus arm is interior to the Norma arm, most likely originating near the far end of the Galactic bar. In addition, in Version 2 we include the following improvements.

• 1.  
  Source coordinates can now be entered either as J2000 (R.A., decl.) or Galactic ($l,b$). R.A. and decl. must be in hhmmss.s and ddmmss.s format, and the program assumes values smaller than 000360.0 and between ±000090.0 are Galactic coordinates in degrees, since such small values for both R.A. and decl. are far from the Galactic plane.
• 2.  
  In Version 1, the spiral arm and Galactic latitude PDFs were plotted separately. Since the arm-assignment probability terms, $\mathrm{Prob}(\mathrm{arm}| ,l,b,v,I)$, are distance independent and depend on Galactic latitude, not Z-height, they alone favor nearer arms over more distant arms. While this bias was corrected for by considering the distance-dependent Galactic latitude PDF (which uses Z-heights), for simplicity we now combine the sprial arm and Galactic latitude PDFs into a single PDF.
• 3.  
  The uncertainty in ${v}_{\mathrm{LSR}}$ can now be entered and included in the probability densities.
• 4.  
  Proper motions in both Galactic longitude and latitude and their uncertainties, if measured, can now be entered and used to estimate distances. Given a rotation curve and proper motion components, one can calculate a distance PDF from the longitude motion in an analogous fashion to using Doppler velocities. This alternative type of "kinematic distance" (Yamauchi et al. 2016; Sanna et al. 2017) has no distance ambiguity and is most effective toward the Galactic center and anticenter, the directions in which conventional kinematic distances based on radial velocities are least effective. Also, for a given velocity dispersion perpendicular to the Galactic plane, the latitude motion should decrease with distance, and one can also obtain a distance PDF for this component of motion.
• 5.  
  Since sources near the Galactic bar can exhibit large peculiar motions, we now inflate motion uncertainties for these sources in order to down-weight the impact of their kinematic distance PDFs. Starting inward from R = 6 kpc, we add a peculiar motion uncertainty in quadrature with measurement motion uncertainty. This peculiar motion uncertainty is zero at R = 6 kpc and increases linearly to 25 km s⁻¹ at R = 4 kpc; inside of R = 4 kpc we hold it constant at 25 km s⁻¹.
• 6.  
  In addition to the Galactic bar region, large peculiar motions are seen for sources in the Perseus arm in the second quadrant within a region bounded by $1.0\lt X\,\lt 3.2\,\mathrm{kpc}$ and $8.8\lt Y\lt 10.0\,\mathrm{kpc}$. For sources within this reigon, we add 20 km s⁻¹ in quadrature with measured uncertainties of each motion component.
• 7.  
  For batch processing, the FORTRAN program now reads a source-information file that contains the information for each source on a single ascii-text line.