---
abstract: 'Over 100 trigonometric parallaxes and proper motions for masers associated with young, high-mass stars have been measured with the Bar and Spiral Structure Legacy Survey, a Very Long Basline Array key science project, the European VLBI Network, and the Japanese VERA project. These measurements provide strong evidence for the existence of spiral arms in the Milky Way, accurately locating many arm segments and yielding spiral pitch angles ranging from about $7^\circ$ to $20^\circ$. The widths of spiral arms increase with distance from the Galactic center. Fitting axially symmetric models of the Milky Way with the 3-dimensional position and velocity information and conservative priors for the solar and average source peculiar motions, we estimate the distance to the Galactic center, $\Ro$, to be $8.34\pm0.16$ kpc, a circular rotation speed at the Sun, $\To$, to be $240\pm8$ , and a rotation curve that is nearly flat ( a slope of $-0.2\pm0.4$ ) between Galactocentric radii of $\approx5$ and 16 kpc. Assuming a “universal” spiral galaxy form for the rotation curve, we estimate the thin disk scale length to be $2.44\pm0.16$ kpc. With this large data set, the parameters  and  are no longer highly correlated and are relatively insensitive to different forms of the rotation curve. If one adopts a theoretically motivated prior that high-mass star forming regions are in nearly circular Galactic orbits, we estimate a global solar motion component in the direction of Galactic rotation, $\V=14.6\pm5.0$ . While  and $\V$ are significantly correlated, the sum of these parameters is well constrained, $\To+\V = 255.2\pm5.1$ , as is the angular speed of the Sun in its orbit about the Galactic center, $(\To+\V)/\Ro = 30.57\pm0.43$ . These parameters improve the accuracy of estimates of the accelerations of the Sun and the Hulse-Taylor binary pulsar in their Galactic orbits, significantly reducing the uncertainty in tests of gravitational radiation predicted by general relativity.'
author:
- 'M. J. Reid, K. M. Menten, A. Brunthaler, X. W. Zheng, T. M. Dame, Y. Xu, Y. Wu, B. Zhang, A. Sanna, M. Sato, K. Hachisuka, Y. K. Choi, K. Immer, L. Moscadelli, K. L. J. Rygl, & A. Bartkiewicz'
title: |
    TRIGONOMETRIC PARALLAXES OF HIGH MASS STAR FORMING REGIONS:\
    THE STRUCTURE AND KINEMATICS OF THE MILKY WAY 
---

Introduction
============

Two major projects to map the spiral structure of the Milky Way are providing parallaxes and proper motions for water and methanol masers associated with high-mass star forming regions (HMSFRs) across large portions of the Milky Way. The Bar and Spiral Structure Legacy (BeSSeL) Survey [^1] and the Japanese VLBI Exploration of Radio Astrometry (VERA) [^2] have yielded over 100 parallax measurements with accuracies typically about $\pm20$ , and some as good as $\pm5$ . This accuracy exceeds the target of the European astrometric satellite mission Gaia, launched in December 2013 and scheduled for final results in 2021-2022 [@Eyer:13]. While Gaia aims to measure $\sim10^9$ stars, far more than practical by Very Long Baseline Interferometry (VLBI), Gaia will be limited by extinction at optical wavelengths and will not be able to freely probe the Galactic plane. In contrast, VLBI at radio wavelengths is not affected by dust extinction and can yield parallaxes for massive young stars that best trace spiral structure in other galaxies, and current parallax accuracy allows measurements for stars across most of the Milky Way.

Given parallax and proper motion measurements (coupled with source coordinates and line-of-sight velocities from Doppler shifts of spectral lines), one has complete phase-space information. This provides direct and powerful constraints on the fundamental parameters of the Galaxy, including the distance to the Galactic center, , and the circular orbital speed at the Sun, . Preliminary models of the structure and dynamics of the Galaxy based on VLBI parallax and proper motions of star forming regions have been published. @Reid:09b fitted results from 16 HMSFRs and determined $\Ro=8.4\pm0.6$ kpc and $\To=254\pm16$ , assuming the solar motion in the direction of Galactic rotation, $\V$, is 5   [@Dehnen:98]. More recently @Honma:12 analyzed results from a larger sample of 52 sources, including both low-mass star forming regions and HMSFRs, and concluded that $\Ro=8.05\pm0.45$ kpc and $\To=238\pm14$ , assuming $\V=12$  [@Schoenrich:10]. Several groups have re-modeled maser parallax and proper motion data [@Bovy:09; @McMillan:10; @Bobylev:10] using different approaches and focusing on effects of parameter correlations and prior assumptions, most notably the values adopted for the solar motion (see §\[sect:priors\] and §\[sect:solar\_motion\]).

With the much larger number and wider distribution of parallaxes and proper motions of HMSFRs now available, we can provide more robust estimates of the fundamental Galactic parameters. In Section \[sect:parallaxes\], we present the combined parallax data sets from the BeSSeL and VERA groups and comment on aspects of spiral structure in Section \[sect:spiral\_structure\]. We model the combined data set to obtain better estimates of  and  in Section \[sect:modeling\], including discussion of priors, different forms of rotation curves, and parameter correlations. Finally, in Section \[sect:discussion\], we discuss the solar motion, best values for  and , and some astrophysical implications.

Parallaxes and Proper Motions {#sect:parallaxes}
=============================

Table \[table:parallaxes\] lists the parallaxes and proper motions of 103 regions of high-mass star formation measured with VLBI techniques, using the National Radio Astronomy Observatory’s Very Long Baseline Array (VLBA), the Japanese VERA project, and the European VLBI Network (EVN). We have include three red supergiants (NML Cyg, S Per, VY CMa) as indicative of HMSFRs, since they are high mass stars that have short lifetimes ($<10^7$ yr) and therefore cannot have migrated far from their birth locations. The locations of these star forming regions in the Galaxy are shown in Figure \[fig:parallaxes\], superposed on a schematic diagram of the Milky Way. Distance errors are indicated with error bars ($1\sigma$), but for many sources the error bars are smaller than the symbols.

[llrrrrrrll]{} G348.70$-$01.04 & &17:20:04.04 &$-$38:58:30.9 & 0.296$\pm$ 0.026 & $-$0.73$\pm$ 0.19 & $-$2.83$\pm$ 0.54 & $-$7$\pm$ 6 &... &1\
G351.44$+$00.65 &NGC 6334 &17:20:54.60 &$-$35:45:08.6 & 0.744$\pm$ 0.074 & 0.40$\pm$ 0.51 & $-$2.24$\pm$ 0.64 & $-$8$\pm$ 3 &Sgr &2\
G000.67$-$00.03 &Sgr B2 &17:47:20.00 &$-$28:22:40.0 & 0.129$\pm$ 0.012 & $-$0.78$\pm$ 0.40 & $-$4.26$\pm$ 0.40 & 62$\pm$ 5 &... &3\
G005.88$-$00.39 & &18:00:30.31 &$-$24:04:04.5 & 0.334$\pm$ 0.020 & 0.18$\pm$ 0.34 & $-$2.26$\pm$ 0.34 & 9$\pm$ 3 &Sct &4\
G009.62$+$00.19 & &18:06:14.66 &$-$20:31:31.7 & 0.194$\pm$ 0.023 & $-$0.58$\pm$ 0.13 & $-$2.49$\pm$ 0.29 & 2$\pm$ 3 &4$-$k &5\
G010.47$+$00.02 & &18:08:38.23 &$-$19:51:50.3 & 0.117$\pm$ 0.008 & $-$3.86$\pm$ 0.19 & $-$6.40$\pm$ 0.14 & 69$\pm$ 5 &Con &7\
G010.62$-$00.38 &W 31 &18:10:28.55 &$-$19:55:48.6 & 0.202$\pm$ 0.019 & $-$0.37$\pm$ 0.50 & $-$0.60$\pm$ 0.25 & $-$3$\pm$ 5 &3$-$k &7\
G011.49$-$01.48 & &18:16:22.13 &$-$19:41:27.2 & 0.800$\pm$ 0.033 & 1.42$\pm$ 0.52 & $-$0.60$\pm$ 0.65 & 11$\pm$ 3 &Sgr &2\
G011.91$-$00.61 & &18:13:58.12 &$-$18:54:20.3 & 0.297$\pm$ 0.031 & 0.66$\pm$ 0.28 & $-$1.36$\pm$ 0.41 & 37$\pm$ 5 &Sct &4\
G012.02$-$00.03 & &18:12:01.84 &$-$18:31:55.8 & 0.106$\pm$ 0.008 & $-$4.11$\pm$ 0.07 & $-$7.76$\pm$ 0.27 & 108$\pm$ 5 &3$-$k &7\
G012.68$-$00.18 & &18:13:54.75 &$-$18:01:46.6 & 0.416$\pm$ 0.028 & $-$1.00$\pm$ 0.95 & $-$2.85$\pm$ 0.95 & 58$\pm$ 10 &Sct &8\
G012.80$-$00.20 & &18:14:14.23 &$-$17:55:40.5 & 0.343$\pm$ 0.037 & $-$0.60$\pm$ 0.70 & $-$0.99$\pm$ 0.70 & 34$\pm$ 5 &Sct &8\
G012.88$+$00.48 &IRAS 18089$-$1732&18:11:51.42 &$-$17:31:29.0 & 0.400$\pm$ 0.040 & 0.15$\pm$ 0.25 & $-$2.30$\pm$ 0.39 & 31$\pm$ 7 &Sct &8,10\
G012.90$-$00.24 & &18:14:34.42 &$-$17:51:51.9 & 0.408$\pm$ 0.025 & 0.19$\pm$ 0.80 & $-$2.52$\pm$ 0.80 & 36$\pm$ 10 &Sct &8\
G012.90$-$00.26 & &18:14:39.57 &$-$17:52:00.4 & 0.396$\pm$ 0.032 & $-$0.36$\pm$ 0.80 & $-$2.22$\pm$ 0.80 & 39$\pm$ 10 &Sct &8\
G013.87$+$00.28 & &18:14:35.83 &$-$16:45:35.9 & 0.254$\pm$ 0.024 & $-$0.25$\pm$ 2.00 & $-$2.49$\pm$ 2.00 & 48$\pm$ 10 &Sct &4\
G014.33$-$00.64 & &18:18:54.67 &$-$16:47:50.3 & 0.893$\pm$ 0.101 & 0.95$\pm$ 1.50 & $-$2.40$\pm$ 1.30 & 22$\pm$ 5 &Sgr &9\
G014.63$-$00.57 & &18:19:15.54 &$-$16:29:45.8 & 0.546$\pm$ 0.022 & 0.22$\pm$ 1.20 & $-$2.07$\pm$ 1.20 & 19$\pm$ 5 &Sgr &2\
G015.03$-$00.67 &M 17 &18:20:24.81 &$-$16:11:35.3 & 0.505$\pm$ 0.033 & 0.68$\pm$ 0.32 & $-$1.42$\pm$ 0.33 & 22$\pm$ 3 &Sgr &10\
G016.58$-$00.05 & &18:21:09.08 &$-$14:31:48.8 & 0.279$\pm$ 0.023 & $-$2.52$\pm$ 0.37 & $-$2.33$\pm$ 0.35 & 60$\pm$ 5 &Sct &4\
G023.00$-$00.41 & &18:34:40.20 &$-$09:00:37.0 & 0.218$\pm$ 0.017 & $-$1.72$\pm$ 0.14 & $-$4.12$\pm$ 0.33 & 80$\pm$ 3 &4$-$k &11\
G023.44$-$00.18 & &18:34:39.19 &$-$08:31:25.4 & 0.170$\pm$ 0.032 & $-$1.93$\pm$ 0.15 & $-$4.11$\pm$ 0.13 & 97$\pm$ 3 &4$-$k &11\
G023.65$-$00.12 & &18:34:51.59 &$-$08:18:21.4 & 0.313$\pm$ 0.039 & $-$1.32$\pm$ 0.20 & $-$2.96$\pm$ 0.20 & 83$\pm$ 3 &... &12\
G023.70$-$00.19 & &18:35:12.36 &$-$08:17:39.5 & 0.161$\pm$ 0.024 & $-$3.17$\pm$ 0.12 & $-$6.38$\pm$ 0.16 & 73$\pm$ 5 &4$-$k &7\
G025.70$+$00.04 & &18:38:03.14 &$-$06:24:15.5 & 0.098$\pm$ 0.029 & $-$2.89$\pm$ 0.07 & $-$6.20$\pm$ 0.36 & 93$\pm$ 5 &Sct &4\
G027.36$-$00.16 & &18:41:51.06 &$-$05:01:43.4 & 0.125$\pm$ 0.042 & $-$1.81$\pm$ 0.11 & $-$4.11$\pm$ 0.27 & 92$\pm$ 3 &Sct &10\
G028.86$+$00.06 & &18:43:46.22 &$-$03:35:29.6 & 0.135$\pm$ 0.018 & $-$4.80$\pm$ 0.30 & $-$5.90$\pm$ 0.30 & 100$\pm$ 10 &Sct &4\
G029.86$-$00.04 & &18:45:59.57 &$-$02:45:06.7 & 0.161$\pm$ 0.020 & $-$2.32$\pm$ 0.11 & $-$5.29$\pm$ 0.16 & 100$\pm$ 3 &Sct &6\
G029.95$-$00.01 &W 43S &18:46:03.74 &$-$02:39:22.3 & 0.190$\pm$ 0.019 & $-$2.30$\pm$ 0.13 & $-$5.34$\pm$ 0.13 & 98$\pm$ 3 &Sct &6\
G031.28$+$00.06 & &18:48:12.39 &$-$01:26:30.7 & 0.234$\pm$ 0.039 & $-$2.09$\pm$ 0.16 & $-$4.37$\pm$ 0.21 & 109$\pm$ 3 &Sct &6\
G031.58$+$00.07 &W 43Main &18:48:41.68 &$-$01:09:59.0 & 0.204$\pm$ 0.030 & $-$1.88$\pm$ 0.40 & $-$4.84$\pm$ 0.40 & 96$\pm$ 5 &Sct &6\
G032.04$+$00.05 & &18:49:36.58 &$-$00:45:46.9 & 0.193$\pm$ 0.008 & $-$2.21$\pm$ 0.40 & $-$4.80$\pm$ 0.40 & 97$\pm$ 5 &Sct &4\
G033.64$-$00.22 & &18:53:32.56 &$+$00:31:39.1 & 0.153$\pm$ 0.017 & $-$3.18$\pm$ 0.10 & $-$6.10$\pm$ 0.10 & 60$\pm$ 3 &... &1\
G034.39$+$00.22 & &18:53:18.77 &$+$01:24:08.8 & 0.643$\pm$ 0.049 & $-$0.90$\pm$ 1.00 & $-$2.75$\pm$ 2.00 & 57$\pm$ 5 &Sgr &13\
G035.02$+$00.34 & &18:54:00.67 &$+$02:01:19.2 & 0.430$\pm$ 0.040 & $-$0.92$\pm$ 0.90 & $-$3.61$\pm$ 0.90 & 52$\pm$ 5 &Sgr &2\
G035.19$-$00.74 & &18:58:13.05 &$+$01:40:35.7 & 0.456$\pm$ 0.045 & $-$0.18$\pm$ 0.50 & $-$3.63$\pm$ 0.50 & 30$\pm$ 7 &Sgr &14\
G035.20$-$01.73 & &19:01:45.54 &$+$01:13:32.5 & 0.306$\pm$ 0.045 & $-$0.71$\pm$ 0.21 & $-$3.61$\pm$ 0.26 & 42$\pm$ 3 &Sgr &14\
G037.43$+$01.51 & &18:54:14.35 &$+$04:41:41.7 & 0.532$\pm$ 0.021 & $-$0.45$\pm$ 0.35 & $-$3.69$\pm$ 0.39 & 41$\pm$ 3 &Sgr &2\
G043.16$+$00.01 &W 49N &19:10:13.41 &$+$09:06:12.8 & 0.090$\pm$ 0.007 & $-$2.88$\pm$ 0.20 & $-$5.41$\pm$ 0.20 & 10$\pm$ 5 &Per &15\
G043.79$-$00.12 &OH 43.8$-$0.1 &19:11:53.99 &$+$09:35:50.3 & 0.166$\pm$ 0.005 & $-$3.02$\pm$ 0.36 & $-$6.20$\pm$ 0.36 & 44$\pm$ 10 &Sgr &2\
G043.89$-$00.78 & &19:14:26.39 &$+$09:22:36.5 & 0.121$\pm$ 0.020 & $-$2.75$\pm$ 0.30 & $-$6.43$\pm$ 0.30 & 54$\pm$ 5 &Sgr &2\
G045.07$+$00.13 & &19:13:22.04 &$+$10:50:53.3 & 0.125$\pm$ 0.005 & $-$2.98$\pm$ 0.45 & $-$6.26$\pm$ 0.45 & 59$\pm$ 5 &Sgr &2\
G045.45$+$00.05 & &19:14:21.27 &$+$11:09:15.9 & 0.119$\pm$ 0.017 & $-$2.34$\pm$ 0.38 & $-$6.00$\pm$ 0.54 & 55$\pm$ 7 &Sgr &2\
G048.60$+$00.02 & &19:20:31.18 &$+$13:55:25.2 & 0.093$\pm$ 0.005 & $-$2.89$\pm$ 0.13 & $-$5.50$\pm$ 0.13 & 18$\pm$ 5 &Per &15\
G049.19$-$00.33 & &19:22:57.77 &$+$14:16:10.0 & 0.189$\pm$ 0.007 & $-$2.99$\pm$ 0.40 & $-$5.71$\pm$ 0.40 & 67$\pm$ 5 &Sgr &2\
G049.48$-$00.36 &W 51 IRS2 &19:23:39.82 &$+$14:31:05.0 & 0.195$\pm$ 0.071 & $-$2.49$\pm$ 0.14 & $-$5.51$\pm$ 0.16 & 56$\pm$ 3 &Sgr &16\
G049.48$-$00.38 &W 51M &19:23:43.87 &$+$14:30:29.5 & 0.185$\pm$ 0.010 & $-$2.64$\pm$ 0.20 & $-$5.11$\pm$ 0.20 & 58$\pm$ 4 &Sgr &17\
G052.10$+$01.04 &IRAS 19213+1723&19:23:37.32 &$+$17:29:10.5 & 0.251$\pm$ 0.060 & $-$2.60$\pm$ 2.00 & $-$6.10$\pm$ 2.00 & 42$\pm$ 5 &Sgr &18\
G059.78$+$00.06 & &19:43:11.25 &$+$23:44:03.3 & 0.463$\pm$ 0.020 & $-$1.65$\pm$ 0.30 & $-$5.12$\pm$ 0.30 & 25$\pm$ 3 &Loc &16\
G069.54$-$00.97 &ON 1 &20:10:09.07 &$+$31:31:36.0 & 0.406$\pm$ 0.013 & $-$3.19$\pm$ 0.40 & $-$5.22$\pm$ 0.40 & 12$\pm$ 5 &Loc &19,20,21\
G074.03$-$01.71 & &20:25:07.11 &$+$34:49:57.6 & 0.629$\pm$ 0.017 & $-$3.79$\pm$ 1.30 & $-$4.88$\pm$ 1.50 & 5$\pm$ 5 &Loc &21\
G075.29$+$01.32 & &20:16:16.01 &$+$37:35:45.8 & 0.108$\pm$ 0.005 & $-$2.37$\pm$ 0.11 & $-$4.48$\pm$ 0.17 & $-$58$\pm$ 5 &Out &22\
G075.76$+$00.33 & &20:21:41.09 &$+$37:25:29.3 & 0.285$\pm$ 0.022 & $-$3.08$\pm$ 0.60 & $-$4.56$\pm$ 0.60 & $-$9$\pm$ 9 &Loc &21\
G075.78$+$00.34 &ON 2N &20:21:44.01 &$+$37:26:37.5 & 0.261$\pm$ 0.030 & $-$2.79$\pm$ 0.55 & $-$4.66$\pm$ 0.55 & 1$\pm$ 5 &Loc &23\
G076.38$-$00.61 & &20:27:25.48 &$+$37:22:48.5 & 0.770$\pm$ 0.053 & $-$3.73$\pm$ 3.00 & $-$3.84$\pm$ 3.00 & $-$2$\pm$ 5 &Loc &21\
G078.12$+$03.63 &IRAS 20126+4104&20:14:26.07 &$+$41:13:32.7 & 0.610$\pm$ 0.030 & $-$2.06$\pm$ 0.50 & 0.98$\pm$ 0.50 & $-$4$\pm$ 5 &Loc &24\
G078.88$+$00.70 &AFGL 2591 &20:29:24.82 &$+$40:11:19.6 & 0.300$\pm$ 0.024 & $-$1.20$\pm$ 0.72 & $-$4.80$\pm$ 0.66 & $-$6$\pm$ 7 &Loc &25\
G079.73$+$00.99 &IRAS 20290+4052&20:30:50.67 &$+$41:02:27.5 & 0.737$\pm$ 0.062 & $-$2.84$\pm$ 0.50 & $-$4.14$\pm$ 0.70 & $-$3$\pm$ 5 &Loc &25\
G079.87$+$01.17 & &20:30:29.14 &$+$41:15:53.6 & 0.620$\pm$ 0.027 & $-$3.23$\pm$ 1.30 & $-$5.19$\pm$ 1.30 & $-$5$\pm$ 10 &Loc &21\
G080.79$-$01.92 &NML Cyg &20:46:25.54 &$+$40:06:59.4 & 0.620$\pm$ 0.047 & $-$1.55$\pm$ 0.57 & $-$4.59$\pm$ 0.57 & $-$3$\pm$ 3 &Loc &26\
G080.86$+$00.38 &DR 20 &20:37:00.96 &$+$41:34:55.7 & 0.687$\pm$ 0.038 & $-$3.29$\pm$ 0.45 & $-$4.83$\pm$ 0.50 & $-$3$\pm$ 5 &Loc &25\
G081.75$+$00.59 &DR 21 &20:39:01.99 &$+$42:24:59.3 & 0.666$\pm$ 0.035 & $-$2.84$\pm$ 0.45 & $-$3.80$\pm$ 0.47 & $-$3$\pm$ 3 &Loc &25\
G081.87$+$00.78 &W 75N &20:38:36.43 &$+$42:37:34.8 & 0.772$\pm$ 0.042 & $-$1.97$\pm$ 0.50 & $-$4.16$\pm$ 0.51 & 7$\pm$ 3 &Loc &25\
G090.21$+$02.32 & &21:02:22.70 &$+$50:03:08.3 & 1.483$\pm$ 0.038 & $-$0.67$\pm$ 1.56 & $-$0.90$\pm$ 1.67 & $-$3$\pm$ 5 &Loc &21\
G092.67$+$03.07 & &21:09:21.73 &$+$52:22:37.1 & 0.613$\pm$ 0.020 & $-$0.69$\pm$ 0.60 & $-$2.25$\pm$ 0.60 & $-$5$\pm$ 10 &Loc &21\
G094.60$-$01.79 &AFGL 2789 &21:39:58.27 &$+$50:14:21.0 & 0.280$\pm$ 0.030 & $-$2.30$\pm$ 0.60 & $-$3.80$\pm$ 0.60 & $-$46$\pm$ 5 &Per &18,28\
G095.29$-$00.93 & &21:39:40.51 &$+$51:20:32.8 & 0.205$\pm$ 0.015 & $-$2.75$\pm$ 0.20 & $-$2.75$\pm$ 0.25 & $-$38$\pm$ 5 &Per &28\
G097.53$+$03.18 & &21:32:12.43 &$+$55:53:49.7 & 0.133$\pm$ 0.017 & $-$2.94$\pm$ 0.29 & $-$2.48$\pm$ 0.29 & $-$73$\pm$ 5 &Out &27\
G100.37$-$03.57 & &22:16:10.37 &$+$52:21:34.1 & 0.291$\pm$ 0.010 & $-$3.77$\pm$ 0.60 & $-$3.12$\pm$ 0.60 & $-$37$\pm$ 10 &Per &28\
G105.41$+$09.87 & &21:43:06.48 &$+$66:06:55.3 & 1.129$\pm$ 0.063 & $-$0.21$\pm$ 1.20 & $-$5.49$\pm$ 1.20 & $-$10$\pm$ 5 &Loc &21\
G107.29$+$05.63 &IRAS 22198+6336&22:21:26.73 &$+$63:51:37.9 & 1.288$\pm$ 0.107 & $-$2.47$\pm$ 1.40 & 0.26$\pm$ 1.40 & $-$11$\pm$ 5 &Loc &29\
G108.18$+$05.51 &L 1206 &22:28:51.41 &$+$64:13:41.3 & 1.289$\pm$ 0.153 & 0.27$\pm$ 0.50 & $-$1.40$\pm$ 1.95 & $-$11$\pm$ 3 &Loc &19\
G108.20$+$00.58 & &22:49:31.48 &$+$59:55:42.0 & 0.229$\pm$ 0.028 & $-$2.25$\pm$ 0.50 & $-$1.00$\pm$ 0.50 & $-$49$\pm$ 5 &Per &28\
G108.47$-$02.81 & &23:02:32.08 &$+$56:57:51.4 & 0.309$\pm$ 0.010 & $-$2.45$\pm$ 1.00 & $-$3.00$\pm$ 0.70 & $-$54$\pm$ 5 &Per &28\
G108.59$+$00.49 & &22:52:38.30 &$+$60:00:52.0 & 0.398$\pm$ 0.031 & $-$5.55$\pm$ 0.40 & $-$3.38$\pm$ 0.40 & $-$52$\pm$ 5 &Per &28\
G109.87$+$02.11 &Cep A &22:56:18.10 &$+$62:01:49.5 & 1.430$\pm$ 0.080 & 0.50$\pm$ 1.50 & $-$3.70$\pm$ 1.00 & $-$7$\pm$ 5 &Loc &30\
G111.23$-$01.23 & &23:17:20.79 &$+$59:28:47.0 & 0.288$\pm$ 0.044 & $-$4.28$\pm$ 0.60 & $-$2.33$\pm$ 0.60 & $-$53$\pm$ 10 &Per &28\
G111.25$-$00.76 & &23:16:10.36 &$+$59:55:28.5 & 0.294$\pm$ 0.016 & $-$2.45$\pm$ 0.60 & $-$2.10$\pm$ 0.60 & $-$43$\pm$ 5 &Per &28\
G111.54$+$00.77 &NGC 7538 &23:13:45.36 &$+$61:28:10.6 & 0.378$\pm$ 0.017 & $-$2.45$\pm$ 0.24 & $-$2.44$\pm$ 0.25 & $-$57$\pm$ 5 &Per &30\
G121.29$+$00.65 &L 1287 &00:36:47.35 &$+$63:29:02.2 & 1.077$\pm$ 0.039 & $-$0.86$\pm$ 0.76 & $-$2.29$\pm$ 0.82 & $-$23$\pm$ 5 &Loc &19\
G122.01$-$07.08 &IRAS 00420+5530&00:44:58.40 &$+$55:46:47.6 & 0.460$\pm$ 0.020 & $-$3.70$\pm$ 0.50 & $-$1.25$\pm$ 0.50 & $-$50$\pm$ 5 &Per &31\
G123.06$-$06.30 &NGC 281 &00:52:24.70 &$+$56:33:50.5 & 0.355$\pm$ 0.030 & $-$2.79$\pm$ 0.62 & $-$2.14$\pm$ 0.70 & $-$30$\pm$ 5 &Per &32\
G123.06$-$06.30 &NGC 281W &00:52:24.20 &$+$56:33:43.2 & 0.421$\pm$ 0.022 & $-$2.69$\pm$ 0.31 & $-$1.77$\pm$ 0.29 & $-$29$\pm$ 3 &Per &19\
G133.94$+$01.06 &W 3OH &02:27:03.82 &$+$61:52:25.2 & 0.512$\pm$ 0.010 & $-$1.20$\pm$ 0.32 & $-$0.15$\pm$ 0.32 & $-$47$\pm$ 3 &Per &33,34\
G134.62$-$02.19 &S Per &02:22:51.71 &$+$58:35:11.4 & 0.413$\pm$ 0.017 & $-$0.49$\pm$ 0.35 & $-$1.19$\pm$ 0.33 & $-$39$\pm$ 5 &Per &35\
G135.27$+$02.79 &WB 89$-$437 &02:43:28.57 &$+$62:57:08.4 & 0.167$\pm$ 0.011 & $-$1.22$\pm$ 0.30 & 0.46$\pm$ 0.36 & $-$72$\pm$ 3 &Out &36\
G160.14$+$03.15 & &05:01:40.24 &$+$47:07:19.0 & 0.244$\pm$ 0.006 & 0.87$\pm$ 0.35 & $-$1.32$\pm$ 0.29 & $-$18$\pm$ 5 &... &1\
G168.06$+$00.82 &IRAS 05137+3919&05:17:13.74 &$+$39:22:19.9 & 0.130$\pm$ 0.040 & 0.50$\pm$ 0.24 & $-$0.85$\pm$ 0.17 & $-$27$\pm$ 5 &Out &37,38\
G176.51$+$00.20 & &05:37:52.14 &$+$32:00:03.9 & 1.038$\pm$ 0.021 & 1.84$\pm$ 1.00 & $-$5.86$\pm$ 1.00 & $-$17$\pm$ 5 &Loc &21\
G182.67$-$03.26 & &05:39:28.42 &$+$24:56:32.1 & 0.149$\pm$ 0.011 & 0.16$\pm$ 0.32 & $-$0.17$\pm$ 0.32 & $-$7$\pm$ 10 &Out &37\
G183.72$-$03.66 & &05:40:24.23 &$+$23:50:54.7 & 0.570$\pm$ 0.013 & 0.13$\pm$ 1.20 & $-$1.40$\pm$ 1.20 & 3$\pm$ 5 &Per &28\
G188.79$+$01.03 &IRAS 06061+2151&06:09:06.97 &$+$21:50:41.4 & 0.496$\pm$ 0.103 & $-$0.10$\pm$ 0.50 & $-$3.91$\pm$ 0.50 & $-$5$\pm$ 5 &Per &39\
G188.94$+$00.88 &S 252 &06:08:53.35 &$+$21:38:28.7 & 0.476$\pm$ 0.006 & 0.02$\pm$ 0.30 & $-$2.02$\pm$ 0.30 & 8$\pm$ 5 &Per &18,40\
G192.16$-$03.81 & &05:58:13.53 &$+$16:31:58.9 & 0.660$\pm$ 0.040 & 0.70$\pm$ 0.78 & $-$1.80$\pm$ 0.86 & 5$\pm$ 5 &Per &41\
G192.60$-$00.04 &S 255 &06:12:54.02 &$+$17:59:23.3 & 0.628$\pm$ 0.027 & $-$0.14$\pm$ 0.67 & $-$0.84$\pm$ 1.80 & 6$\pm$ 5 &Per &19\
G196.45$-$01.67 &S 269 &06:14:37.08 &$+$13:49:36.7 & 0.189$\pm$ 0.012 & $-$0.42$\pm$ 0.20 & $-$0.12$\pm$ 0.20 & 19$\pm$ 5 &Out &42\
G209.00$-$19.38 &Orion Nebula &05:35:15.80 &$-$05:23:14.1 & 2.410$\pm$ 0.030 & 3.30$\pm$ 1.50 & 0.10$\pm$ 1.50 & 3$\pm$ 5 &Loc &43,44,45\
G211.59$+$01.05 & &06:52:45.32 &$+$01:40:23.1 & 0.228$\pm$ 0.007 & $-$0.93$\pm$ 0.24 & 0.71$\pm$ 0.26 & 45$\pm$ 5 &... &1\
G229.57$+$00.15 & &07:23:01.84 &$-$14:41:32.8 & 0.221$\pm$ 0.014 & $-$1.34$\pm$ 0.70 & 0.81$\pm$ 0.70 & 47$\pm$ 10 &Per &28\
G232.62$+$00.99 & &07:32:09.78 &$-$16:58:12.8 & 0.596$\pm$ 0.035 & $-$2.17$\pm$ 0.38 & 2.09$\pm$ 0.60 & 21$\pm$ 3 &Loc &40\
G236.81$+$01.98 & &07:44:28.24 &$-$20:08:30.2 & 0.298$\pm$ 0.018 & $-$3.10$\pm$ 0.63 & 2.12$\pm$ 0.63 & 43$\pm$ 7 &Per &28\
G239.35$-$05.06 &VY CMa &07:22:58.33 &$-$25:46:03.1 & 0.855$\pm$ 0.057 & $-$2.80$\pm$ 0.58 & 2.60$\pm$ 0.58 & 20$\pm$ 3 &Loc &46,47\
G240.31$+$00.07 & &07:44:51.92 &$-$24:07:41.5 & 0.212$\pm$ 0.021 & $-$2.36$\pm$ 0.23 & 2.45$\pm$ 0.30 & 67$\pm$ 5 &Per &28\
\[table:parallaxes\]

Both the proper motion, $\mu_x$ and $\mu_y$, and Local Standard of Rest (LSR) velocity, , values and their uncertainties are meant to apply to the central star (or stars) that excite the masers. (Note that “LSR velocities” are [*defined*]{} based on the Standard Solar Motion values of 20  toward $18^h$ Right Ascension and $30^\circ$ Declination in 1900 coordinates, which translate to Galactic cartesian components of $\Uo=10$, $\Vo=15$ and $\Wo=7$ .) For the  values we adopted methanol maser values, when available, or CO emission values from associated giant molecular clouds. Since some of the references reporting parallax and proper motion present only measurement uncertainty, for these we estimated an additional error term associated with the uncertainty in transferring the maser motions to that of the central star. These were added in quadrature with the measurement uncertainties. For methanol masers, which typically have modest motions of $\lax10$   with respect to the central star, we estimated the additional error term to be $\pm5$  for  and a corresponding value for the proper motion components at the measured distance. While some water masers have expansion motions comparable to methanol masers, others display much faster outflow motions. High velocity outflows are usually associated with water masers that have spectra rich in features, spread over many tens of . We, therefore, evaluated the richness and spread of the water spectra (with respect to the systemic velocity as indicated by CO emission) and assigned the additional error term for $\mu_x$ and $\mu_y$ values between 5 and 20 .

Spiral Structure {#sect:spiral_structure}
================

Spiral arms in the Milky Way have long been recognized as presenting coherent arcs and loops in Galactic longitude–velocity ($\ell-V$) plots of atomic and molecular emissions. However, transforming velocity to distance (kinematic distances) has been problematic, owing to near-far distance ambiguities in the first and fourth Galactic quadrants and significant distance errors owing to large peculiar motions for some arm material (see,  @Xu:06 [@Reid:09b]). While one cannot accurately place spiral arms on a plan view of the Milky Way from $\ell-V$ plots, one can in most cases unambiguously assign HMSFRs to spiral arms by association with CO and  emission features. We have done this for the vast majority of the HMSFRs for which parallax and proper motions have been measured [@Hachi:13; @Choi:13; @Zhang:13; @Xu:13; @Wu:13; @Sato:13; @Sanna:13], as indicated in Table \[table:parallaxes\] and Figure \[fig:parallaxes\]. This avoids using the measured distances (parallaxes) and subjective judgment based on spatial location for arm assignments.

There are two avenues for checking that the arm assignments are reliable. Firstly, and most straightforwardly, looking at a plan view of the Milky Way (see Fig. \[fig:parallaxes\]) on which star forming regions with parallax distances are located, one can see that the pattern of sources for any given arm traces a continuous arc that resembles a spiral arm in external galaxies. Also, there are clear inter-arm regions with few, if any, HMSFRs between the Outer, Perseus, Local, Sagittarius, and Scutum arms. However, as one looks to the inner Galaxy, the current parallax data are not adequate to clearly separate arms, presuming significant separations even exist.

Secondly, once sources are assigned to arms based on $\ell-V$ information, one can then attempt to fit their radial and azimuthal locations to log-periodic spiral forms using measured distances. In the papers cited above, we fitted spiral patterns to arm segments, adopting a log-periodic spiral defined by $$\ln{(R/R_{ref})} = -(\beta - \beta_{ref}) \tan{\pa}~~,$$ where $R$ is the Galactocentric radius at a Galactocentric azimuth $\beta$ (defined as 0 toward the Sun and increasing with Galactic longitude) for an arm with a radius $R_{ref}$ at reference azimuth $\beta_{ref}$ and pitch angle $\pa$. We fitted a straight line to ($x,y$)=($\beta,\ln{(R/R_{ref})}$) using a Bayesian Markov chain Monte Carlo (McMC) procedure to estimate the parameters $R_{ref}$ and $\pa$. (The reference azimuth, $\beta_{ref}$, was arbitrarily set near the midpoint of the azimuth values for the sources in an arm). We minimized the “distance” perpendicular to the fitted straight line by rotating ($x,y$) through the angle $\pa$ to ($x_r,y_r$), $$x_r = x~\cos{\pa} + y~\sin{\pa};~~~y_r = y~\cos{\pa} - x~\sin{\pa}~~,$$ such that the best-fitting line lay in the $x_r$ axis.

Uncertainties in the source parallax “map” into both coordinates and were estimated numerically by randomly drawing trial parallax values (consistent with the measured values and uncertainties) and calculating the root-mean-squares for trial $\ln{(R/R_{ref})}$ and $\beta$ values. The locations of the HMSFRs deviated from fitted spirals by more than could be explained by parallax uncertainties. This is expected for spiral arms with intrinsic widths of several hundred parsecs. In order to allow for (and estimate) the scatter in location expected from the width of the spiral arm, before calculating trial $\ln{(R/R_{ref})}$ values, we added random scatter to the trial $R$ values via $R \leftarrow R + g a_w \cos{\pa}$, where $g$ is a random number drawn from a Gaussian distribution with zero mean and unity standard deviation and $a_w$ is an arm-width parameter, adjusted to give a post-fit $\chi^2_\nu$ near unity. The uncertainties in ($\beta,\ln{(R/R_{ref})}$) were then rotated by angle $\pa$ to match the data.

The sum of the squares of the residuals divided by their uncertainties in the $y_r$ direction were minimized. Since preliminary estimates of $\pa$ affect these quantities, we iterated the fitting to convergence. Final parameter values were estimated from marginalized posteriori probability density distribution functions (PDFs) for each parameter based on McMC trials that were accepted or rejected following the Metropolis-Hastings algorithm; the values reported in Table \[table:pitchangles\] assume $\Ro=8.34$ kpc (see §\[sect:modeling\]). Based on the fitted parameter values, we plot the trace of the centers and $1\sigma$ widths of each arm on Fig. \[fig:parallaxes\].

The intrinsic widths of the spiral arms, estimated from the $a_w$ parameters, show an interesting pattern in Fig. \[fig:armwidths\]. The estimated arm widths increase nearly linearly with Galactocentric radius at a rate of 42 pc kpc$^{-1}$ between radii of 5 to 13 kpc. Spiral pitch angles vary between $7^\circ$ and $20^\circ$ as listed in Table \[table:pitchangles\]. The significant range of pitch angles among arms suggests that no single value applies to all arms and, possibly, cannot be applied to the full length of an arm as it winds around the Galaxy [@Savchenko:13]. However, these pitch angles are characteristic of spiral galaxies of Sb to Sc class [@Kennicutt:81], further supporting the identification of $\ell-V$ tracks as spiral arms for the Milky Way.

[lrlrrr]{} Scutum &17 &27.6 ($+3\rightarrow101$) &$5.0\pm0.1$ & $0.17\pm0.02$ &$19.8\pm2.6$\
Sagittarius &18 &25.6 ($-2\rightarrow68$) &$6.6\pm0.1$ & $0.26\pm0.02$ &$6.9\pm1.6$\
Local &25 &8.9 ($-8\rightarrow27$) &$8.4\pm0.1$ & $0.33\pm0.01$ &$12.8\pm2.7$\
Perseus &24 &14.2 ($-21\rightarrow88$) &$9.9\pm0.1$ & $0.38\pm0.01$ &$9.4\pm1.4$\
Outer &6 &18.6 ($-6\rightarrow56$) &$13.0\pm0.3$ & $0.63\pm0.18$ &$13.8\pm3.3$\
\[table:pitchangles\]

The HMSFRs with measured parallaxes are clearly tracing the major spiral arms of the Milky Way (see Fig. \[fig:parallaxes\]), and details of the locations and properties of the individual arms can be found in the primary references [@Hachi:13; @Choi:13; @Zhang:13; @Xu:13; @Wu:13; @Sato:13; @Sanna:13]. Interestingly, some surprising results are already evident. We are finding that the Perseus arm, thought to be one of the major spiral arms of the Milky Way, has little massive star formation over a 6 kpc-long arc between Galactic longitudes of $50^\circ$ and $80^\circ$ [@Choi:13; @Zhang:13]. On the other hand, the Local (Orion) arm, often called a “spur” and considered a minor structure [@Blaauw:85], has comparable massive star formation to its adjacent Sagittarius and Perseus arms [@Xu:13].

Modeling the Galaxy {#sect:modeling}
===================

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. As in @Reid:09b, we model the Milky Way as a disk rotating with speed $\Theta(R)=\To+\Tdot~(R-\Ro)$, where  is the distance from the Sun to the Galactic center and  is the circular rotation speed at this distance. We then evaluate the effects of different forms for the rotation curve. Since all measured motions are relative to the Sun, we need to model the peculiar (non-circular) motion of the Sun, parameterized by  toward the Galactic center,  in the direction of Galactic rotation, and  towards the north Galactic pole (NGP). Table \[table:model\] summarizes these and other parameters.

[ll]{} & Distance of Sun from GC\
& Rotation Speed of Galaxy at\
& Derivative of $\Theta$ with $R$: $\Theta(R)=\To+\Tdot~(R-\Ro)$\
\
& Solar motion toward GC\
& Solar motion in direction of Galactic rotation\
& Solar motion toward NGP\
\
& Average source peculiar motion toward GC\
& Average source peculiar motion in direction of Galactic rotation\
& Average source peculiar motion toward NGP\
\[table:model\]

For each source, we treated the 3-dimensional velocity components (two components of proper motion, $\mu_x$ and $\mu_y$, and the heliocentric Doppler velocity, , as data to be compared to a model. The source coordinates ($\ell,b$) and parallax distance ($1/\pars$) were treated as independent variables. This approach is slightly different than in @Reid:09b, where the parallaxes were also treated as data in the least-squares fitting. While that approach adds some extra information ( for sources near the Galactic tangent points, distance is very sensitive to Doppler velocity, but not vice versa), it brings correlated data into the fitting, which will lead to slightly underestimated parameter uncertainties. We tested the inclusion versus exclusion of parallax with simulated data sets and found little difference and no bias between the methods. However, in order to avoid the need to adjust formal parameter uncertainties, as well as subtle issues associated with resolving the near/far distance ambiguities for sources in the first and fourth Galactic quadrants, we used the more conservative “velocity-only” fitting as done, for example, by others [@Bovy:09; @McMillan:10; @Bobylev:10; @Honma:12].

Bayesian fitting {#sect:Bayesian}
----------------

We adjusted the Galactic parameters so as to best match the data to the spatial-kinematic model using a Bayesian fitting approach. The posteriori PDFs of the parameters were estimated with Markov chain Monte Carlo (McMC) trials that were accepted or rejected by the Metropolis–Hastings algorithm. While a simple axi-symmetric model for the Galaxy may be a reasonable approximation for the majority of sources, a significant minority of outliers are expected for a variety of well known reasons. For example, the gravitational potential of the Galactic bar (or bars), which extend 3 to 4 kpc from the Galactic center [@Liszt:80; @Blitz:91; @Hammersley:00; @Benjamin:05] is expected to induce large non-circular motions for sources in its vicinity. Indeed, some of these sources show large peculiar motions, although based on a nearly flat rotation curve extrapolated inward from measurements outside this region [@Sanna:13]. Therefore, we removed the eight sources within 4 kpc of the Galactic center (excluding G000.67$-$00.03, G009.62$+$00.19, G010.47$+$00.02, G010.62$-$00.38, G012.02$-$00.03, G023.43$-$00.18, G023.70$-$00.19, G027.36$-$00.16) before model fitting.

In the Galaxy’s spiral arms, super-bubbles created by multiple supernovae can accelerate molecular clouds to $\approx20$  [@Sato:08]. It is probably not possible, prior to fitting, to determine which sources have been thus affected and are likely kinematically anomalous. Therefore, we initially used an “outlier-tolerant” Bayesian fitting scheme described by @Sivia:06 as a “conservative formulation,” which minimizes the effects of deviant points on estimates of the fitted parameters. For this approach, one maximizes $$\sum_{i=1}^N~\sum_{j=1}^3~{\ln\bigl(~(1-e^{-R_{i,j}^2/2})/R_{i,j}^2~\bigr)}~~,$$ where the weighted residual $R_{i,j} = (v_{i,j}~-~m_{i,j})/w_{i,j}$ ( the data ($v$) minus model ($m$) divided by the uncertainty ($w$) of the $i^{th}$ of $N$ sources and $j^{th}$ velocity component). For large residuals, this formulation assigns a $1/R^2$ probability, compared to a Gaussian probability of $e^{-R^2/2}$ which vanishes rapidly. Thus, for example, a $5\sigma$ outlier has a reasonable (4%) probability with the outlier-tolerant approach, compared to $\approx10^{-6}$ probability for Gaussian errors in the least-squares method, and will not be given excessive weight when adjusting parameters. Once the outliers were identified and removed, we assumed Gaussian data uncertainties and fitted data by maximizing $$\sum_{i=1}^N~\sum_{j=1}^3~{-R_{i,j}^2/2}~~,$$ essentially least-squares fitting.

Our choice of weights ($w$) for the data in the model fitting process was discussed in detail in @Reid:09b. We include both measurement uncertainty and the effects of random (Virial) motions of a massive young star (with maser emission) with respect to the average motion of the much larger and more massive HMSFR when weighting the differences between observed and modeled components of motion. Specifically the proper motion and Doppler velocity weights were given by $w(\mu) = \sqrt{\sigma^2_\mu + \sigma^2_{Vir}/d^2_s}$ and $w(\vhelio) = \sqrt{\sigma^2_v + \sigma^2_{Vir}}$, where $\sigma^2_{Vir}$ is the expected (1-dimensional) Virial dispersion for stars in a high mass star forming region (HMSFR). We adopted $\sigma_{Vir}=5$ , appropriate for HMSFRs with $\sim10^4$ within a radius of $\sim1$ pc, and did not adjust this value. As will be seen in §\[sect:Bayesian\], the vast majority of the velocity data can be fit with a $\chi^2_\nu$ near unity with these weights. Note that we were fairly conservative when assigning motion uncertainties for individual stars based on the maser data (see §\[sect:parallaxes\]), and this may result in a slightly low $\sigma_{Vir}$ value in order to achieve unity $\chi^2_\nu$ fits.

Priors {#sect:priors}
------

In order to model the observations, one needs prior constraints on the non-circular motion of our measurement “platform” ( the solar motion parameterized by , , ) and/or the average peculiar motion of the sources being measured (parameterized by , , ). Allowing for a non-zero average source peculiar motion can be thought of as a first approximation of the kinematic effects of spiral structure. In @Reid:09b, we assumed the solar motion determined by @Dehnen:98 based on Hipparcos measurements and concluded that HMSFRs lagged circular orbital speeds by 15  ($\Vsbar=-15$ ). The observed orbital lag ($\Vsbar<0$) is insensitive to the value adopted for , but it is strongly correlated with the adopted solar motion component, $\V$ [@Reid:09b; @Honma:12]. Recently, the value of the solar motion component in the direction of Galactic rotation () has become controversial. Motivated in part by the large  lag in @Reid:09b, @Schoenrich:10 re-evaluated the standard “asymmetric-drift” approach used by @Dehnen:98 and concluded that it was biased by coupled metallicity/orbital-eccentricity effects. They suggested new solar motion values; specifically they argued for a substantial increase for  from 5 to 12 . This change would decrease the average orbital lag of HMSFRs () by $\approx7$  to a more theoretically appealing value near 8 .

Based on the first year of data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE), @Bovy:12 argue that the Sun’s motion relative to a circular orbit in the Galaxy (ie, a “rotational standard of rest”) is 26  in the direction of Galactic rotation, suggesting that the entire Solar Neighborhood, which defines the local standard of rest (LSR), leads a circular orbit by 14 . Taking into account these developments, we considered a conservative prior of $\V = 15\pm10$ , that encompasses the values of  from 5 to 26  within approximately the $\pm1\sigma$ range.

One could argue on theoretical grounds that HMSFRs should, on average, lag circular orbits by only a few  [@McMillan:10]. We observe masers in HMSFRs that are very young and the gas out of which their exciting stars formed could have responded to magnetic shocks when entering spiral arms, leading to departures from circular speeds by $\lax10$  [@Roberts:70], apportioned between components counter to rotation and toward the Galactic Center. In addition, radial pressure gradients can also reduce orbital speeds of gas slightly [@Burkert:10], contributing to a small lag of $\approx1$ . Allowing for such effects, we consider priors for  of $3\pm10$  and  of $-3\pm10$   as reasonable and conservative.

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

-   Adopting a loose prior for the  component of solar motion, $\U = 11.1\pm1.2$, $\V = 15\pm10$, $\W = 7.2\pm1.1$ , and for the average peculiar motions for HMSFRs of $\Usbar = 3\pm10$ and $\Vsbar =-3\pm10$ .

-   Using no priors for the average peculiar motions of HMSFRs, but tighter priors for the solar motion of $\U = 11.1\pm1.2$, $\V = 12.2\pm2.1$, $\W = 7.2\pm1.1$  from @Schoenrich:10.

-   Using no priors for the solar motion, but tighter priors on the average peculiar motions of HMSFRs of $\Usbar = 3\pm5$ and $\Vsbar =-3\pm5$ .

-   Using essentially no priors for either the solar or average peculiar motions of HMSFRs, but bounding the  and  parameters with equal probability within $\pm20$  of the Set-A initial values and zero probability outside that range.

Models A1–A4
------------

Using the 95 sources with Galactocentric radii greater than 4 kpc[^3], the outlier-tolerant Bayesian fitting approach, and the Set-A priors as described above, we obtained the parameter estimates listed in Table \[table:fits\] under fit A1. As expected for a sample with some outliers (see discussion in §\[sect:Bayesian\]), we found a $\chisq=562.6$, greatly exceeded the 277 degrees of freedom, owing to a number of sources with large residuals.

We iteratively removed the sources with the largest residuals. Using the outlier-tolerant Bayesian fitting approach (see §\[sect:Bayesian\]) minimizes potential bias, based on assumed “correct” parameter values, when editing data. However, to further guard against any residual bias, we first removed sources with $>6\sigma$ residuals, followed by re-fitting and removal of those with $>4\sigma$ residuals, and finally re-fitting and removal of those with $>3\sigma$ residuals (fits A2, A3 & A4, not listed here). In total, 15 sources[^4] were removed.

[lccccc]{} Parameter Estimates\
 (kpc) &$8.15\pm0.25$ &$8.34\pm0.16$ &$8.33\pm0.16$ &$8.30\pm0.19$ & $8.29\pm0.21$\
 () & $238\pm11\q$ &$240\pm8\q$ &$243\pm6\q$ &$239\pm8\q$ &$238\pm15$\
 ()&$-0.1\pm0.7\q$&$-0.2\pm0.4\q$ &$-0.2\pm0.4\q$ &$-0.1\pm0.4\q$ &$-0.1\pm0.4$\
\
 () &$10.4\pm1.8$ &$10.7\pm1.8$ &$10.7\pm1.8$ &$ 9.9\pm3.0$ &$ 9.6\pm3.9$\
 () &$15.1\pm7.3$ &$15.6\pm6.8$ &$12.2\pm2.0$ &$14.6\pm5.0$ &$16.1\pm13.5$\
 () &$8.2\pm1.2$ &$\p8.9\pm0.9$ &$\p8.7\pm0.9$ &$\p9.3\pm1.0$ &$\p9.3\pm1.0$\
\
 ()&$\p3.7\pm2.4$ &$\p2.9\pm2.1$ &$\p2.9\pm2.1$ &$\p2.2\pm3.0$ &$\p1.6\pm3.9$\
 ()&$-2.4\pm7.4$ &$-1.6\pm6.8$ &$-5.0\pm2.1  $ &$-2.4\pm5.0$ &$-1.2\pm13.6$\
\
Fit Statistics\
$\chisq$ & 562.6 & 224.9 & 225.1 & 224.7 & 224.1\
$N_{dof}$ & 277 & 232 & 232 & 232 & 232\
$N_{sources}$ & 95 & 80 & 80 & 80 & 80\
$r_{\Ro,\To}$ &0.61 &0.46 &0.74 &0.66 &0.44\
\[table:fits\]

Model A5
--------

With the resulting “clean” data set of 80 sources, we performed a least-squares fit (assuming a Gaussian PDF for the data uncertainties). We used the same loose priors (Set-A) as for model A1, namely solar motion components $\U = 11.1\pm1.2$, $\V = 15\pm10$, $\W = 7.2\pm1.1$  and average peculiar motions for HMSFRs of $\Usbar = 3\pm10$ and $\Vsbar =-3\pm10$ . This resulted in the parameter estimates listed under fit A5 in Table \[table:fits\]. This model produced a good $\chisq=224.9$ for 232 degrees of freedom and estimates of $\Ro=8.34\pm0.16$ kpc and $\To=240\pm8$ . We find $\Tdot=-0.2\pm0.4$ , indicating a very flat rotation curve for the Milky Way between radii of $\approx5$ and 16 kpc from the Galactic center.

Compared to the preliminary results of @Reid:09b based on 16 sources, where the Pearson product-moment correlation coefficient for  and  was high, $r_{\Ro,\To}=0.87$, with the larger number of sources and a better distribution across the Galaxy, these parameters are significantly less correlated, $r_{\Ro,\To}=0.46$. The joint and marginalized PDFs for these fundamental Galactic parameters are displayed in Figure \[fig:RoToPDF\].

The circular velocity parameters are still correlated (see §\[sect:correlations\]), but linear combinations of these parameters are well determined: $\To+\V=255.2\pm5.1$ and $\V-\Vsbar=17.1\pm1.0$. Also, the angular rotation rate for the Sun’s orbit about the Galactic center is constrained to $\pm1.4$% accuracy: $(\To+\V)/\Ro=30.57\pm0.43$ . This value is consistent with the reflex of the [*apparent*]{} motion of Sgr A\*, the assumed motionless supermassive black hole at the center of the Galaxy, which gives $30.26\pm0.12$  [@Reid:04].

The component of solar motion in the direction of Galactic rotation, $\V$, estimated to be $15.6\pm6.8$  is better constrained than the prior of $15\pm10$ . It is consistent with the [*local*]{} estimate (relative to Solar Neighborhood stars) of 12  [@Schoenrich:10] and the [*global*]{} estimate of @Bovy:12 of $26\pm3$  (relative to stars across the Milky Way).

Model B1
--------

In order to explore the sensitivity of the modeling to our priors, we fit the clean data set using the Set-B priors: adopting the latest Hipparcos measurement of the solar motion of $\U = 11.1\pm1.2$, $\V = 12.2\pm2.1$, $\W = 7.2\pm1.1$  [@Schoenrich:10] and no prior information on the average peculiar motion of the HMSFRs. This resulted in parameter estimates similar to those of model A5,  $\Ro=8.33\pm0.16$ kpc and $\To=243\pm6$ . The quality of fit, as measured by $\chisq=225.1$ for 232 degrees of freedom, was comparably good as for model A5. The average velocity lag of the HMSFRs relative to circular orbits, which was not constrained by priors, was $\Vsbar = -5.0 \pm 2.1$ . This is comparable to that found by @Reid:09b, after correcting for the 7  difference in the adopted solar motion values.

Model C1
--------

Given the current uncertainty in the  component of solar motion, we fit the data with the Set-C priors, assuming no prior information for the solar motion, but using a stronger prior than for model A5 for the average peculiar motion of the HMSFRs: $\Usbar = 3\pm5$ and $\Vsbar =-3\pm5$ . As for model B1, we found most parameter estimates to be similar to model A5, eg, $\Ro=8.30\pm0.19$ kpc and $\To=239\pm8$ . For the solar motion, we find $\U = 9.9\pm2.0$, $\V =14.6\pm5.0$, and $\W = 9.3\pm1.0$ . The  value is consistent with revised @Schoenrich:10 [12 ] solar motion, but differs by $2\sigma$ from the @Bovy:12 estimate.

Model D1
--------

In order to facilitate the use of the results presented here with other Galactic parameter estimates, we perfomed a fit with essentially no informative priors. We did this by taking the A5 (Set-A) initial parameter values and assuming flat priors for all parameters except for  and . For these parameters we assumed equal probability for values within $\pm20$  of the initial values and zero probability outside this range in order to exclude unreasonable parameter values. The parameters that remain well determined include $\Ro=8.29\pm0.21$ kpc, $\To=238\pm15$ , $\Tdot=-0.1\pm0.4$ , $\U=9.6\pm3.9$ , $\W=9.3\pm1.0$ , and $\Usbar=1.6\pm3.9$ . The correlated velocity terms,  and  displayed nearly flat posteriori PDFs over their allowed ranges. However, linear combinations involving these parameters are very well constrained, $\To+\V = 253.8\pm6.4$  and $\V-\Vsbar = 17.2\pm1.2$ , as well as the angular rotation rate of the Sun about the Galactic center, $(\To+\V)/\Ro=30.64\pm0.41$ .

Rotation Curves {#sect:rotationcurves}
---------------

Next, we investigated the sensitivity of the fundamental Galactic parameters,  and , to alternative rotation curves. When fitting, we replaced the simple linear form, $\Theta(R)=\To+\Tdot~(R-\Ro)$, with the empirically determined functions of $\Theta(R)$ of @Clemens:85, the power-law parameterization of @Brand:93, a polynomial, and the “universal” rotation curve of @Persic:96. We adopted the Set-A priors in order to facilitate comparisons with the A5 fit. Table \[table:rotationcurves\] presents the fitting results for these rotation curves.

[lccccc]{} Parameter Estimates\
 (kpc) &$8.36\pm0.16$ &$8.12\pm0.14$ &$8.34\pm0.16$ &$8.34\pm0.17$ &$8.31\pm0.16$\
 () &$237\pm8\q$ &$221\pm8\q$ &$240\pm9\q$ &$241\pm9\q$ &$241\pm8\q$\
 () &$10.1\pm1.8~$ &$10.5\pm1.8\p$&$10.5\pm1.8\p$&$10.7\pm1.7\p$ &$10.5\pm1.7\p$\
 () &$19.4\pm6.8~$ &$25.0\pm6.8~$ &$15.5\pm6.8~$ &$14.7\pm6.8~$ &$14.4\pm6.8~$\
 () &$8.9\pm1.0$ &$8.9\pm1.0$ &$8.8\pm1.0$ &$8.8\pm0.9$ &$8.9\pm0.9$\
 ()&$2.4\pm2.1$ &$2.6\pm2.0$ &$2.8\pm2.0$ &$2.8\pm2.0$ &$2.6\pm2.1$\
 ()&$+3.4\pm6.8\p$ &$+8.5\pm6.8\p$&$-1.5\pm6.8$ &$-1.4\pm6.8\p$ &$-1.4\pm6.8\p$\
 () &... &... &$240\pm9~$ &$241\pm9~$ &$241\pm8~$\
&... &... &$~0.00\pm0.02$&$~~0.5\pm3.7$ &$~0.90\pm0.06$\
&... &... &... &$-15.1\pm8.4~$ &$~1.46\pm0.16$\
\
Fit Statistics\
$\chisq$ &229.7 & 248.1 & 225.2 & 221.9 &214.5\
$N_{dof}$ &233 & 233 & 231 & 230 &230\
$N_{sources}$&80 & 80 & 80 & 80 & 80\
$r_{\Ro,\To}$&0.46 &0.36 &0.48 &0.47 &0.47\
\[table:rotationcurves\]

@Clemens:85 supplied two curves with different shapes: one assuming the old IAU constants (C-10) of $\Ro=10$ kpc and $\To=250$  and the other assuming the revised constants (C-8.5) of $\Ro=8.5$ kpc and $\To=220$  currently in widespread use. The C-10 model has rotational speeds that rise faster with radius than the C-8.5 model. For either model, we fitted for different values of  (which we used to scale model radii) and  (which we used to scale rotation speeds).

@Brand:93 parameterize their rotation curve (BB) as a power law in Galactocentric radius, $R$, with potentially three adjustable parameters: $\Theta(R) = \aone (R/\Ro)^\atwo + \athr$. For a flat rotation curve ($\atwo=0$), parameters  and become degenerate. Since the Galaxy’s rotation curve is nearly flat over the range of radii we sample (see, eg, model A5 above), we held  at zero, solving only for  and . Indeed, we find the power law exponent, $\atwo=-0.01\pm0.01$, essentially flat. For this formulation, $\aone=\To$, and in Table \[table:rotationcurves\] we copy  to  to facilitate comparison with other models.

As an alternative to a power law rotation curve, we fitted a second-order polynomial (Poly) in $\rho=(R/\Ro)-1$: $\Theta(R) = \aone + \atwo \rho + \athr \rho^2$. The model fit parameters for this form of a rotation curve are similar to those from models C-10, BB and Univ.

The universal (Univ) rotation curve of @Persic:96 includes terms for an exponential disk and a halo. It can have three adjustable parameters: , the circular rotation speed at the radius enclosing 83% of the optical light ($R_{opt}$); $\atwo = R_{opt} / \Ro$; and , a core-radius parameter for the halo contribution, nominally 1.5 for an $L^*$ galaxy. With flat priors for the three rotation curve parameters, the posteriori PDF for $\atwo$ was bimodal, with the dominant peak at $\atwo=0.9$ and a second peak with 50% of the primary’s amplitude at $\atwo=0.1$. Since the secondary peak seems unlikely, we refit the data using a prior for $\atwo$ of $1.2\pm0.5$. We then obtained similar parameter values as other models (see Table \[table:rotationcurves\]), with the three adjustable rotation curve parameters of $\aone = 241\pm8$ , $\atwo = 0.90\pm0.06$, and $\athr = 1.46\pm0.16$.

All but one of the rotation curve models lead to similar values for the fundamental Galactic parameters  and  as our A5 fit. Only the Clemens “$\Ro=8.5$ kpc; $\To=220$ ” (C-8.5) rotation curve results in a marginally significant change in estimates of  and . However, this fit has a significantly poorer quality ($\chi^2=248.1$ for 233 degrees of freedom) than, for example, the A5 fit ($\chi^2=224.9$ for 232 degrees of freedom), and we do not consider this model further. We conclude that the fundamental Galactic parameters  and  are reasonably insensitive to a wide variety of rotation curve shapes.

With full 3-dimensional location and velocity information, we can transform our heliocentric velocities to a Galactocentric reference frame and calculate the tangential (circular) speed for each HMSFR. Figure \[fig:rotationcurve\] plots these speeds for [*all*]{} sources in Table \[table:parallaxes\]. Most published rotation curves for the Milky Way have come from only one component of velocity (radial), often using kinematic distances and [*assuming*]{} a value for . As such, the data in Fig \[fig:rotationcurve\] represent a considerable advance. See also the analysis of this data set by @Xin:13.

It is important to remember that the transformation from heliocentric to Galactocentric frames requires accurate values of , , and, most importantly, $\To+\V$, since the motion of the Sun has (by definition) been subtracted in the heliocentric frame. For most sources, increasing or decreasing the assumed value of $\To+\V$ would, correspondingly, move each data point up or down by about the same amount. Thus, the level of this, and essentially all published, rotation curves is determined mostly by $\To+\V$. Our results are the first to use fully 3-dimensional data to strongly constrain all three parameters: ,  and $\To+V$.

The dashed line in Fig \[fig:rotationcurve\] represent the linear rotation curve from the A5 fit, based only on sources with $R>4$ kpc. Sources used in the fit are plotted with filled symbols and the sources not used with open symbols. The dashed line indicates the expected rotation for sources in circular Galactic orbit ( $\Usbar=\Vsbar=0$). There are now sufficient data to clearly indicate that the rotation curve drops at Galactocentric radii $\lax4$ kpc. However, given the likelihood for a significant non-axisymmetric gravitational potential within $\approx4$ kpc of the center, more measurements are needed before extending a rotation curve to this region as azimuthal terms may be needed.

Peculiar Motions of HMSFRs
--------------------------

Figure \[fig:peculiarmotions\] shows the peculiar (non-circular) motions of all sources in Table \[table:parallaxes\] with motion uncertainties less than 20 . Similar results were described in the primary papers presenting the parallaxes and proper motions for each arm [@Sato:13; @Wu:13; @Xu:13; @Choi:13; @Zhang:13; @Hachi:13]. For uniformity, here the motions were calculated using the A5 fit parameters (see Table \[table:fits\]), but with zero correction for the average source peculiar motions. Typical peculiar motions are $\approx10$ , but some sources have much larger values. For example, many sources in the Perseus arm in the Galactic longitude range $\approx100^\circ$ to $\approx135^\circ$ display peculiar motions $\gax20$ . Many sources within $\approx4$ kpc of the Galactic Center display even larger peculiar motions, probably indicating that the rotation curve used here is inadequate to describe their Galactic orbits, especially in the presence of the Galactic bar(s).

Parameter Correlations {#sect:correlations}
----------------------

[lrrrrrrrr]{} $\Ro$ &1.000 &0.465 &0.103 &0.452 &0.023 &$-$0.003&0.517 &$-$0.002 $\To$ &0.465 &1.000 &0.136 &0.243 &$-$0.796&$-$0.009&0.171 &$-$0.809 $\Tdot$ &0.103 &0.136 &1.000 &$-$0.124&$-$0.009&0.025 &$-$0.094 &$-$0.018 $\U$ &0.452 &0.243 &$-$0.124&1.000 &$-$0.014&$-$0.017&0.839 &0.025 $\V$ &0.023 &$-$0.796&$-$0.009&$-$0.014&1.000 &0.011 &$-$0.006 &0.990 $\W$ &$-$0.003&$-$0.009&0.025 &$-$0.017&0.011 &1.000 &$-$0.002 &0.010 $\Usbar$ &0.517 &0.171 &$-$0.094&0.839 &$-$0.006&$-$0.002&1.000 &0.028 $\Vsbar$ &$-$0.002&$-$0.809&$-$0.018&0.025 &0.990 &0.010 &0.028 &1.000 \[table:correlations\]

The Pearson product-moment correlation coefficients, $r$, for all parameters from fit A5 are listed in Table \[table:correlations\]. In the preliminary analysis of 16 HMSFRs with parallaxes and proper motions of @Reid:09b, the estimates of  and  were strongly correlated ($\rPpm=0.87$). However, with the much larger set of HMSFRs that covers a larger portion of the Galaxy, the correlation between  and   estimates is now moderate: $\rPpm=0.465$ for our reference A5 fit. However, there remains a significant anti-correlation between  and   ($r_{\To,\V}=-0.809$), as well as a strong correlation between  and   ($r_{\V,\Vsbar}=0.990$). As suggested by the fitted parameter values in Table \[table:fits\], our data strongly constrain the following combinations of these correlated parameters: $\To + \V=255.2\pm5.1$  and $\V - \Vsbar=17.1\pm1.0$ . Also, the combination of parameters that yield the angular orbital speed of the Sun about the Galactic center, $(\To+\V)/\Ro = 30.57 \pm 0.43$ , is more tightly constained than the individual parameters. Figure \[fig:3PDFs\] shows the marginalized PDFs for these combinations of parameters.

Comparison with Other Modeling Approaches
-----------------------------------------

Other groups have analyzed parallax and proper motion data sets from the BeSSeL Survey and the VERA project, focusing on different assumptions and results. @Bovy:09 confirmed the counter rotation of HMSFRs (assuming $\V=5$ ) noted by @Reid:09b and argued for a comparable value for  $(246\pm30$ ), but with considerably lower significance. Alternatively, @McMillan:10 found that the -component of solar motion of 5 , provided by @Dehnen:98, should be raised to $\approx12$ , thereby reducing the estimated counter-rotation of HMSFRs. @Bobylev:10, using 28 parallaxes available at that time and a Fourier analysis technique, estimated $\To=248\pm14$  and $\V=11.0\pm1.7$ , assuming $\Ro\equiv8.0$ kpc. Finally, @Honma:12, using 52 parallaxes, including some low-mass star forming regions, estimated $\Ro=8.05\pm0.45$ kpc and $\To=238\pm14$  for $\V\equiv12$ .

The @Bovy:09 re-analysis of our preliminary data employed a different approach than that of @Reid:09b. Bovy  treat the elements of the velocity dispersion tensor of the HMSFRs as free parameters. These parameters give the expected deviations (variances and covariances) of the velocity data from a smooth, axi-symmetric model of Galactic rotation and are used to adjust the weights applied to the different velocity components when fitting the data. However, while Bovy  found a significant trace for the tensor, the velocity dispersion parameters were only marginally constrained; formally none of the diagonal components had $>2.8\sigma$ formal significance. Also, their values for the radial and tangential components were nearly identical, suggesting that little is gained by making these free parameters versus adopting a single physically motivated value ($\sigma_{Vir}$) as we have done. Note that our value for $\sigma_{Vir}$ is comparable to the dispersion parameter ($\Delta_v$) values found by @McMillan:10, which range from about 6 to 10 , but is considerably smaller than those of @Bovy:09 of $\approx20$ . The reason for this difference is unclear, but might reflect different treatments of outlying data and/or increased parameter correlations associated with the 6 extra parameters used in solving for the tensor elements.

Discussion  {#sect:discussion}
===========

Solar Motion {#sect:solar_motion}
------------

If one adopts the theoretically motivated prior that HMSFRs have small peculiar motions (Set-C with no prior on the solar motion), then model fit C1 indicates $\V=14.6\pm5.0$ . This is a [*global*]{} measure of the peculiar motion of the Sun and, as such, is relative to a “rotational standard of rest” as opposed to a Local Standard of Rest (LSR), defined relative to stellar motions in the Solar Neighborhood. If Solar Neighborhood stars (extrapolated to a zero-dispersion sample) are, on average, stationary with respect to a circular orbit, then these two solar motion systems will be the same. Our estimate of $\V$ is consistent with the $12$  value of @Schoenrich:10, measured with respect to Solar neighborhood stars, but there is some tension between our global estimate of  and that of @Bovy:12 of $26\pm3$ , as these two estimates differ by about $2\sigma$. However, if one drops the prior that HMSFRs have small peculiar motions, then our result loses significance.

The large counter-rotation of HMSFRs, originally suggested by @Reid:09b, was based on the initial Hipparcos result of @Dehnen:98 that $\V=5$ . As the outcome of the @Schoenrich:10 re-analysis of Hipparcos data, which gives $\V=12$ , supersedes that lower $\V$ value, it now appears that any average counter-rotation of HMSFRs is $\lax5$ . Given that we strongly constrain $\V-\Vsbar=17.1\pm1.0$ , were one to independently constrain $\V$ with $\pm2$   accuracy, the issue of HMSFR counter rotation could be clarified.

While our estimate of  has a large uncertainty (owing to correlations with  and ), we find  and  are well constrained. In fit D1, in which no informative prior was used for the components of motion either toward the Galactic center or perpendicular to the Galactic plane, we find that $\U=9.6\pm3.9$ and $\W=9.3\pm1.0$ , respectively. Our estimate of the Sun’s motion toward the Galactic center is in agreement with most other estimates,  $11.1\pm1.2$  by @Schoenrich:10 and $10\pm1$  by @Bovy:12; see also the compilation of estimates by @Coskunoglu:11.

The solar motion component perpendicular to the Galactic plane, $\W$, is generally considered to be straight forwardly determined and recent estimates typically range between 7.2  [@Schoenrich:10] (relative to local stars within $\sim0.2$ kpc) and 7.6  [@Feast:97] (relative to stars within $\sim3$ kpc), with uncertainties of about $\pm0.5$ . We find a slightly larger value of $\W=9.3\pm1.0$  (for model D1 which used no informative priors for the solar motion), which may be significant; the difference between the locally and our globally measured value ( relative to stars across the Galaxy) is $2.1\pm1.1$ . Note that one might expect a small difference between measurements with respect to a local and a global distribution of stars were the disk of the Galaxy to precess owing to Local Group torques. Simulations of galaxy interactions in a group suggest that a disk galaxy can complete one precession cycle over a Hubble time. Were the Milky Way to do this, one would expect a vertical pecessional motion at a Galactocentric radius of the solar neighborhood of order $\Ro\Ho~\sim0.6$ . It is possible that the differences in the local and global estimates of $\W$ can, in part, be explained in this manner.

Galactic Rotation Curve and Disk Scale Length
---------------------------------------------

Among the various forms of rotation curves that we fit to the data, the universal curve advocated by @Persic:96 to apply to most spiral galaxies yielded the best fit (see discussion in §\[sect:rotationcurves\], Table \[table:rotationcurves\] and Fig. \[fig:rotationcurve\]). This rotation curve matches the flat to slightly declining run of velocity with Galactocentric radius from $R\approx5\rightarrow16$ kpc, as well as reasonably tracing the decline in orbital velocity for $R\lax5$ kpc. However, many of the sources near the Galactic bar(s) cannot be well modeled with any axi-symmetric rotation curve.

The best fit value for our $\atwo$ parameter ($R_{opt}/\Ro$), coupled with our estimate of $\Ro=8.34\pm0.16$, locates $R_{opt}$ at $7.5\pm0.52$ kpc. The $\atwo$ parameter is sensitive to the slope of the rotation curve (near $R_{opt}$) and the radius at which it turns down toward the Galactic center. For example, setting $\atwo=0.7$ steepens the rotation curve at large radii and moves the turn down radius to $\approx3.5$ kpc, while setting $\atwo=1.1$ flattens the rotation curve and increases the turn down radius to $\approx6.5$ kpc. Given that the (thin) disk scale length, $R_D = R_{opt} / 3.2$ [@Persic:96], we estimate $R_D=2.44\pm0.16$ kpc. Estimates of $R_D$ in the literature range from $\approx 1 \rightarrow 6$ kpc [@Kent:91; @Chang:11; @McMillan:11], with most consistent with a value between $2\rightarrow3$ kpc. Our estimate is also consistent with that of @Porcel:98 who modeled the positions and magnitudes of 700,000 stars in the Two Micron Galactic Survey database and found $R_D=2.3\pm0.3$ kpc and, more recently, @Bovy:13, who modeled the dynamics of $\approx16,000$ stars from the SEGUE survey and concluded that $R_D=2.14\pm0.14$ kpc.

The Distance to the Galactic Center: 
-------------------------------------

Models A5, B1 and C1, which used different combinations of solar motion and/or average source peculiar motion priors, have comparable $\chi^2$ values and all parameter estimates are statistically consistent. Because the priors for Model A5 are the least restrictive in keeping with current knowledge, we adopt those parameters as representative. Specifically, we find $\Ro=8.34\pm0.16$ kpc, $\To=240\pm8$  and $\Tdot=-0.2\pm0.4$ . As noted in §\[sect:correlations\] and §\[sect:rotationcurves\], with the much larger data set now available, estimates of  and  are no longer strongly correlated and appear fairly insensitive to the assumed nature of the rotation curve. These parameter estimates are consistent with, but significantly better than, the preliminary values of $\Ro=8.4\pm0.6$ kpc, $\To=254\pm16$  and a nearly flat rotation curve reported in @Reid:09b, based on parallaxes and proper motions of 16 HMSFRs and assuming $\V=5$ , and $\Ro=8.05\pm0.45$ kpc and $\To=238\pm14$  from @Honma:12, based on a sample of 52 sources and assuming $\V=12$ .  

While there are numerous estimates of the distance to the Galactic center in the literature ( @Reid:93), here we only compare those based on direct distance measurements. A parallax for the water masers in Sgr B2, a star forming region projected less than 0.1 kpc from the Galactic center, indicates $\Ro=7.9\pm0.8$ kpc [@Reid:09c], consistent with, but considerably less accurate than, our current result. More competitive estimates of  come from the orbits of “S-stars” about the supermassive blackhole Sgr A\*. Combining the nearly two decades of data from the ESO NTT/VLT [@Gillessen:09a] and Keck [@Ghez:08] telescopes that trace more than one full orbit for the star S2 (a.k.a. S0-2), @Gillessen:09b conclude that $\Ro=8.28\pm0.33$ kpc. Recently the Keck group, extending their time sequence of observations by only a few years, announced a value of $\Ro=7.7\pm0.4$ kpc [@Morris:12], in mild tension both with the @Gillessen:09b analysis and our parallax-based result. However, in the latest publication of the Keck group, @Do:13 combined modeling of the distribution and space velocities of stars within the central 0.5 pc of the Galactic center with the stellar orbital result for star S0-2 [@Ghez:08] and conclude that $\Ro=8.46^{+0.42}_{-0.38}$ kpc, removing any tension with our estimate and that of the ESO group. We conclude that our estimate of $\Ro=8.34\pm0.16$ kpc is consistent with that from the Galactic center stellar orbits and is likely the most accurate to date.

The Circular Rotation Speed at the Sun: 
----------------------------------------

Over the last four decades there have been many estimates of  ranging from $\sim170 \rightarrow 270$  [@Kerr:86; @Olling:98]. Focussing the discussion to the more direct measurements, two recent studies favor a lower and one a higher value of  than our estimate of $\To=240\pm8$ . @Koposov:10 model the orbit of the GD-1 stream from a tidally disrupted stellar cluster in the Milky Way halo and estimate $\To+\V=221\pm18$, where the @Dehnen:98 solar motion component of $\V=5$  was adopted. Recently, @Bovy:12 modeled line-of-sight velocities of 3365 stars from APOGEE and find $\To=218\pm6$ , but with a large value for the solar motion component in the direction of Galactic rotation, $\V=26\pm3$ . Their full tangential speed $\To+\V=242^{+10}_{-3}$ is consistent with our value of $252.2\pm4.8$ , suggesting the discrepancy between the Bovy   and our results are probably caused by differences in the solar motion. However, another recent study by @Carlin:12, modeling the Sagittarius tidal stream, yields $\To$ estimates from $232 \rightarrow 264$ .

Our data also strongly constrain the angular rotation of the Sun about the Galactic center, $(\To+\V)/\Ro=30.57\pm0.43$ . This value can be compared with an independent and direct estimate based on the proper motion of Sgr A\*, interpreted as the reflex motion from the Sun’s Galactic orbit, of $30.24\pm0.12$  [@Reid:04]. For $\Ro=8.34\pm0.16$ kpc, the proper motion of Sgr A\* translates to $\To+\V=252.2\pm4.8$ , in good agreement with the parallax results. We conclude that $\To$ exceeds the IAU recommended value of 220   with $>95$% probability provided that $\V\lax23$ . Clearly, independent [*global*]{} measures of $\V$ are critical to establish $\To$ and $\Vsbar$ with high accuracy.

Changing the value of  would have widespread impact in astrophysics. For example, increasing  by 20  with respect to the IAU recommended value of 220  reduces kinematic distances by about 10%, leading to a decrease of 20% in estimated young star luminosities, a corresponding decrease in estimated cloud masses, and a change in young stellar object ages. Estimates of the total mass of the dark matter halo of the Milky Way scale as $V_{max}^2~R_{Vir}$. Since the maximum in the rotation curve ($V_{max}$) and the Virial radius ($R_{Vir}$) scale linearly with , the mass of the halo scales as $\Theta_0^3$, leading to a 30% increase in the estimate of the Milky Way’s (dark-matter dominated) mass. This, in turn, affects the expected dark-matter annihilation signal [@Finkbeiner:09], increases the “missing satellite” problem [@Wang:12], and increases the likelihood that the Magellanic Clouds are bound to the Milky Way [@Shattow:09].

The Hulse-Taylor Binary Pulsar and Gravitation Radiation
--------------------------------------------------------

An interesting example of the effects of Galactic parameters on fundamental physics comes from the Hulse-Taylor binary pulsar. The dominant uncertainty in measuring the gravitational radiation damping of the binary’s orbit comes from the need to correct for the effects of the Galactic accelerations of the Sun and the binary [@Damour:91; @Weisberg:10]. These accelerations contribute $\approx1$% to the [*apparent*]{} orbital period decay. In 1993 when the Nobel Prize was awarded in part for this work, the IAU recommended values were $\Ro=8.5\pm1.1$ kpc and $\To=220\pm20$  [@Kerr:86]. Using these Galactic parameters, the formalism of Damour & Taylor, improved pulsar timing data of Weisberg, Nice, & Taylor, and a pulsar distance of 9.9 kpc, the binary’s orbital period decays at a rate of $0.9994\pm0.0023$ times that prediction from general relativity (GR). Using the improved Galactic parameters from the A5 fit ($\Ro=8.34\pm0.16$ kpc and $\To=240\pm8$ ), gives a GR test value of $0.9976\pm0.0008$. This provides a three-fold improvement in accuracy. Both of these examples assumed a distance to the binary pulsar of 9.9 kpc [@Weisberg:08]. Given the improvement in the Galactic parameter values, the dominant uncertainty in the GR test now is the uncertain pulsar distance. A pulsar distance of 7.2 kpc would bring the GR test value to 1.0000 and a trigonometric parallax accurate to $\pm8$%, which is possible with in-beam calibration with the VLBA, would bring the contribution of distance uncertainty down to that of the current Galactic parameter uncertainty. Alternatively, if one assumes GR is correct, the current improvement in Galactic parameters suggests that the Hulse-Taylor binary pulsar’s distance is $7.2\pm0.5$ kpc.

0.5truein This work was partially funded by the ERC Advanced Investigator Grant GLOSTAR (247078). The work was supported in part by the National Science Foundation of China (under grants 10921063, 11073046, 11073054 and 11133008) and the Key Laboratory for Radio Astronomy, Chinese Academy of Sciences. AB acknowledges support by the National Science Centre Poland through grant 2011/03/B/ST9/00627.

0.5truein [*Facilities:*]{} , ,

Ando, K., Nagayama, T., Omodaka, T.,  2011, , 63, 45 Asaki, Y., Deguchi, S., Imai, H., Hachisuka, K., Miyoshi, M. & Honma, M. 2010, , 721, 267 Bartkiewicz, A., Brunthaler, A., Szymczak, M. van Langevelde, H. J & Reid, M. J. 2008, , 490, 787 Benjamin, R. A. 2008, in “Massive Star Formation: Observations Confront Theory,” ASP Conference Series, Vol. 387, eds. H. Beuther, H. Linz & Th. Henning, p. 375 Benjamin, R. A. 2005, , 630, L149 Blaauw, A. 1985, in “The Milky Way Galaxy: Proceedings of IAU Symp. 106” H. van Woerden et al. eds., (Dordrecht, D. Reidel Pub. Co.), p. 335 Blitz, L. & Spergel,, D. N. 1991, , 379, 631 Bobylev, V. V. & Bajkova, A. T. 2010, , 408, 1788 Bovy, J., Hogg, D. W. & Rix, H.-W. 2009, , 704, 1704 Bovy, J., Prieto, C. A., Beers, T. C.,  2012, , 759, 131 Bovy, J. & Rix, H.-W. 2013, , 779, 115 Brand, J. & Blitz, L. 1993, , 275, 67 Brunthaler, A., Reid, M. J., Menten, K. M., Zheng, X. W., Moscadelli, L. & Xu, Y. 2009, , 693, 424 Burkert, A., Genzel, R., Bouché, N.   2010, , 725, 2324 Carlin, J. L., Majewski, S. R., Casetti-Dinescu, D. I., Law, D. R., Girard, T. M. & Patterson, R. J. 2012, , 744, 25 Chang, C.-K., Ko, C.-M. & Peng, T.-H. 2011, , 740, 34 Choi, Y. K. , 2008, , 60, 1007 Choi, Y. K., Hachisuka, K., Reid, M. J.,  2014, submitted to Clemens, D. P. 1985, , 295, 422 Coskunoglu, B, Ak, S., Bilir, S.  2011, , 412, 1237 Damour, T. & Taylor, J. H. 1991, , 366, 501 Dehnen, W., & Binney, J. J., , 1998, 298,387 Do, T., Martinez,, G. D., Yelda, S.,  2013, , 779, 6 Eyer, L., Holl, B., Pourbaix, D.  2013, CEAB, 37, 115 Feast, M. & Whitelock, P. 1997, , 291, 683 Finkbeiner, D. P., Slatyer, T. R., Weiner, N., & Yavin, I. 2009, [*JCAP*]{}, 9, 37 Ghez, A. M., Salim, S., Weinberg, N. N.,  2008, , 689, 1044 Gillessen, S., Eisenhauer, F. Trippe, S.,  2009a, , 692, 1075 Gillessen, S., Eisenhauer, F., Fritz, T. K.,  2009b, , 707, L114 Hachisuka, K.  2006, , 645, 337 Hachisuka, K. Brunthaler, A., Menten, K. M.,  2009, , 696, 1981 Hachisuka, K., Choi, Y. K., Reid, M. J.,  2014, submitted to Hammersley, P. L., Garzón, F., Mahoney, T. J., López-Corredoira, M., Torres, M. A. P. 2000, , 317, 45 Hirota, T, Ando, K., Bushimata, T.,  2008, , 60, 961 Honma, M., Bushimata, T., Choi, Y. K.,  2007, , 59, 889 Honma, M., Hirota, T., Kan-Ya, Y.,  2007, , 63, 17 Honma,M. 2012, , 64, 136 Immer, K., Reid, M. J., Menten, K. M., Brunthaler, A., & Dame, T. M. 2013, , 553, 117 Kennicutt, R. C. Jr. 1981, , 86, 1847 Kent, S. M., Dame, T. M. & Fazio, G. 1991, , 378, 131 Kerr, F. J. & Lynden-Bell, D. 1986, , 221, 1023 Kim, M. K., Hirota, T., Honma, M.,  2008, , 60, 991 Koposov, S. E., Rix, H.-W. & Hogg, D. W. 2010, , 712, 260 Kurayama, T., Nakagawa, A., Sawada-Satoh, S,  2011, , 63, 513 Liszt, H. S. & Burton, W. B. 1980, , 236, 779 McMillan, P. J. 2011, , 414, 2446 McMillan, P. J. & Binney J. J. 2010, , 402, 934 Menten, K. M., Reid, M. J., Forbrich J. & Brunthaler, A. 2007, , 474, 515 Moellenbrock, G. A., Claussen, M. J. & Goss, W. M. 2009, , 694, 192 Morris, M. R., Meyer, L & Ghez, A. 2012, [*RAA*]{}, 12, 995 Moscadelli, L., Reid, M. J., Menten, K. M., Brunthaler, A., Zheng, X. W. & Xu, Y. 2009, , 693, 406 Moscadelli, L., Cesaroni, R., Rioja, M. J., Dodson, R., Reid, M. J., 2011, , 526, 66 Nagayama, T., Omodaka, T., Nakagawa, A,  2011, , 63, 23 Niinuma, K. Nagayama, T., Hirota, T.,  2011, , 63, 9 Oh, C. S., Kobayashi, H., Honma, M., Hirota, T., Sato, K., Ueno, Y., 2010, , 62, 101 Olling, R. P. & Merrifield, M. R. 1998, , 297,943 Persic, M., Salucci, P. & Stel, F. 1996, , 281, 27 Porcel, C., Garzon, F., Jimenez-Vicente, J. & Battaner, E. 1998, , 330, 136 Reid, M. J. 1993, , 31, 345 Reid, M. J. & Brunthaler, A., 2004, , 616, 872 Reid, M. J., Menten, K. M., Brunthaler, A., Zheng, X. W., Moscadelli, L. & Xu, Y. 2009, , 693, 397 Reid, M. J., Menten, K. M., Zheng, X. W.  2009, , 700, 137 Reid, M. J., Menten, K. M., Zheng, X. W., Brunthaler, A., & Xu, Y. 2009, , 705, 1548 Roberts, W. W. & Yuan, C. 1970, , 161, 877 Rygl, K. L. J., Brunthaler, A., Reid, M. J., Menten, K. M., van Langevelde, H. J., Xu, Y., 2010, , 511, 2 Rygl, K. L. J., Brunthaler, A., Sanna, A.,  2012, , 539, 79 Sandstrom, K. M., Peek, J. E. G., Bower, G. C., Bolatto, A. D. & Plambeck, R. L., 2007, , 667, 1161 Sanna, A., Reid, M. J., Moscadelli, L.,  2009, , 706, 464 Sanna, A., Reid, M. J., Dame, T.,  2012, , 745, 82 Sanna, A., Reid, M. J., Menten, K. M.,  2014, , 781, 108 Sato, M. 2008, , 60, 975 Sato, M., Hirota, T., Reid, M.,   2010a, , 62, 287 Sato, M., Reid, M. J., Brunthaler, A. & Menten, K. M. 2010b, , 720, 1055 Sato, M., Wu, Y. W., Immer, K.,  2014, submitted to Savchenko, S. S. & Reshetnikov, V. P. 2013, , 436, 1074 Schoenrich, R., Binney, J. & Dehnen, W. 2010, , 403, 1829 Shattow, G. & Loeb, A. 2009, , 392, 21 Shiozaki, S., Imai, H., Tafoya, D.,  2011, , 63, 1219 Sivia, D. & Skilling, J. 2006, Data Analysis: A Bayesian Tutorial (2nd ed.; New York, Oxford Univ. Press), 168 Wang, J., Frenk, C. S., Navarro, J. F., Gao, L., Sawala, T. 2012, , 424, 2715 Weisberg, J. M., Stanimirović, S., Xilouris, K., 2008, , 674, 286 Weisberg, J. M., Nice, D. J., & Taylor, J. H. 2010, , 722, 1030 Wu, Y. W., Sato, M., Reid, M. J.,  2014, submitted to Xin, X.-S. & Zheng, X.-W. 2013, [*Res. Astron. Astroph.*]{}, 13, 849 Xu, Y., Reid, M. J., Zheng, W. W. & Menten, K. M. 2006, [*Science*]{}, 311, 54 Xu, Y., Reid, M. J., Menten, K. M., Brunthaler, A., Zheng, X. W. & Moscadelli, L. 2009, , 693, 413 Xu, Y., Moscadelli, L., Reid, M. J.,  2011, , 733, 25 Xu, Y., Li, J. J., Reid, M. J.,  2013, , 769,15 Zhang, B., Zheng, X. W., Reid, M. J.,   2009, , 693, 419 Zhang, B., Reid, M. J., Menten, K. M., & Zheng, X. W., 2012a, , 744, 23 Zhang, B., Reid, M. J., Menten, K. M., Zheng, X. W., Brunthaler, A., 2012b, , 544, 42 Zhang, B., Reid, M. J., Menten, K. M.,  2013a, , 775, 79 Zhang, B., Moscadelli, L., Sato, M.,  2014, , 781, 89

[^1]: http://bessel.vlbi-astrometry.org

[^2]: http://veraserver.mtk.nao.ac.jp

[^3]: Removing sources for which $R<4$ kpc: G$000.67-00.03$, G$009.62+00.19$, G$010.47+00.02$, G$010.62-00.38$, G$012.02-00.03$, G$023.43-00.18$, G$023.70-00.19$, G$027.36-00.16$

[^4]: Removing outlying sources: G012.68$-$00.18, G016.58$-$00.05, G023.65$-$00.12, G025.70$+$00.04, G028.86$+$00.06, G029.95$-$00.01, G031.28$+$00.06, G033.64$-$00.22, G034.39$+$00.22, G078.12$+$03.63, G108.59$+$00.49, G111.54+0.77, G122.01$-$07.08, G133.94+01.06, G176.51$+$00.20
