Non-circular Motions in the Outer Perseus Spiral Arm

TL;DR

This study measures parallax and proper motion of methanol masers in the Perseus arm, revealing its structure and kinematics, and providing insights into spiral arm formation models.

Contribution

It provides new astrometric measurements of masers in the Perseus arm, determining its pitch angle, width, and motion characteristics, which were previously unmeasured.

Findings

Spiral arm pitch angle of 9.2±1.5 degrees.

Arm width of 0.39 kpc.

Radial inward motion of 13.3±5.4 km/s.

Abstract

We report measurements of parallax and proper motion for five 6.7-GHz methanol maser sources in the outer regions of the Perseus arm as part of the BeSSeL Survey of the Galaxy. By combining our results with previous astrometric results, we determine an average spiral arm pitch angle of $9.2\pm1.5$ deg and an arm width of 0.39 kpc for this spiral arm. For sources in the interior side of the Perseus arm, we find on average a radial inward motion in the Galaxy of $13.3\pm5.4$ km s$^{-1}$ and counter to Galactic rotation of $6.2\pm3.2$ km s$^{-1}$. These characteristics are consistent with models for spiral arm formation that involve gas entering an arm to be shocked and then forming stars. However, similar data for other spiral arms do not show similar characteristics.

Tables

Table 1. Table 1: 6.7 GHz parallax and proper motion results

Source	Parallax	${\mu }_{\alpha }$cos$\delta$	${\mu }_{\delta }$
	(mas)	(mas yr-1)	(mas yr-1)
G094.60$-$1.79	0.179$\pm$0.048	$-$3.49$\pm$0.22	$-$2.65$\pm$0.23
G111.25$-$0.76	0.264$\pm$0.020	$-$2.69$\pm$0.28	$-$1.75$\pm$0.33
G136.84+1.16	0.442$\pm$0.123	0.43$\pm$0.48	$-$0.45$\pm$0.56
G173.48+2.44	0.594$\pm$0.014	0.62$\pm$0.63	$-$2.34$\pm$0.63
G188.94+0.88	0.465$\pm$0.042	$-$0.62$\pm$0.51	$-$1.87$\pm$0.53

Table 2. Table 2: Statistics for non-circular motion across the Perseus arm.

Interior side			Middle region			Exterior
$U$$>$	$V$$>$	$W$$>$	$U$$>$	$V$$>$	$W$$>$	$U$$>$	$V$$>$	$W$$>$
(km s-1)	(km s${}_{}{}^{-1})$	(km s-1)	(km s-1)	(km s-1)	(km s-1)	(km s-1)	(km s-1)	(km s-1)
(9 masers)			(10 masers)			(5 masers)
13.3$\pm$5.4	$-$2.5$\pm$2.8	0.0$\pm$2.4	5.9$\pm$2.1	$-$6.2$\pm$3.2	$-$1.0$\pm$3.0	2.7$\pm$7.6	3.4$\pm$2.8	$-$4.6$\pm$3.5

Table 3. Table 3: Non-circular motion for Perseus, Local and Sagittarius arms.

Spiral	Type	Interior side		$\#$	Exterior		$\#$
Arm
		$U$$>$	$V$$>$		$U$$>$	$V$$>$
		(km s-1)	(km s-1)		(km s-1)	(km s${}_{}{}^{-1})$
Perseus	Masers	13.3$\pm$5.4	$-$2.5$\pm$2.8	9	2.7$\pm$7.6	3.4$\pm$2.8	5
Local	Masers	2.3$\pm$2.8	$-$5.1$\pm$1.0	4	$-$4.4$\pm$4.1	$-$6.1$\pm$2.1	6
	OB stars	2.7$\pm$0.9	0.9$\pm$1.6	119	5.3$\pm$3.9	4.9$\pm$2.6	50
Sagittarius	Masers	3.1$\pm$4.6	2.6$\pm$3.5	3	11.1$\pm$6.8	15.5$\pm$4.6	3
	OB stars			0	1.1$\pm$3.5	7.2$\pm$1.3	32

Table 4. Table 4: Observational Information.

Project	Source	R.A.	Decl.	Epoch 1	Epoch 2	Epoch 3	Epoch 4
		(hh:mm:ss)	(dd:mm:ss)	(in 2012)	(in 2013)	(in 2013)	(in 2013)
BR149S	G136.84+1.16	02:49:33.609	+60:48:27.92	Dec 08	May 19	June 24	Nov 24
	J0244+6228	02:44:57.6966	+62:28:06.517
	J0248+6214	02:48:58.8920	+62:14:09.678
	J0306+6243	03:06:42.6595	+62:43:02.024
BR149T	G173.48+2.44	05:39:13.066	+35:45:51.28	Sep 22	March 09	April 05	Sep 13
	J0530+3723	05:30:12.5493	+37:23:32.620
	J0539+3308	05:39:09.6722	+33:08:15.496
	J0541+3301	05:41:49.4359	+33:01:31.890
	J0552+3754	05:52:17.9369	+37:54:25.281
	G188.94+0.88	06:08:53.341	+21:38:29.08
	J0603+2159	05:30:12.5493	+37:23:32.620
	J0607+2129	06:07:59.5657	+21:29:43.720
	J0607+2218	06:07:17.4360	+22:18:19.080
	J0608+2229	06:08:34.3109	+22:29:42.981
BR149U	G094.60$-$1.79	21:39:58.258	+50:14:21.02	Dec 03	May 12	June 06	Nov 23
	J2137+5101	21:37:00.9862	+51:01:36.129
	J2145+5147	21:45:07.6666	+51:47:02.243
	J2150+5103	21:50:14.2662	+51:03:32.264
	(J2139+5300)	21:39:53.6244	+53:00:16.599
	(G098.03+1.44)	21:43:01.431	+54:56:17.72
	J2123+5452	21:23:46.8349	+54:52:43.488
	(J2139+5300)	21:39:53.6244	+53:00:16.599
	J2139+5540	21:39:32.6175	+55:40:31.771
	J2145+5147	21:45:07.6666	+51:47:02.243
	G111.25$-$0.76	23:16:10.327	+59:55:28.66
	J2339+6010	23:39:21.1252	+60:10:11.849
	J2254+6209	22:54:25.2926	+62:09:38.723
	J2301+5706	23:01:26.6271	+57:06:25.499
	(J2314+5813)	23:14:19.0833	+58:13:47.647

Equations

$$\ln (R/R_{\mathrm{ref}})=-(\beta -{\beta }_{\mathrm{ref}})\tan \psi ,$$

Full text

Galaxy:kinematics and dynamics — ISM:individual objects (G094.60-1.79, G098.03+1.44, G111.25-0.76, G136.84+1.16, G173.48+2.44, G188.94+0.88) — techniques:interferometric — VLBA

1 Introduction

While spiral patterns can be prominent in disk galaxies, their formation mechanism and the dynamical evolution of spiral arms remain under discussion. Two general mechanisms for spiral arm formation have dominated the discussion in the literature:

1. density-wave theories (Lin & Shu 1964; Roberts 1969) and
2. dynamic theories (Goldreich & Lynden-Bell 1965; Toomre 1969). In density-wave theories, spiral structures are long-lived and rotate nearly uniformly, while stars and gas rotate differentially and pass through the arms. In dynamic theories, arms are short-lived and reform as open structures. They are seen in $N$-body simulations of multi-arm spirals (e.g., Sellwood & Carlberg 1984; Sellwood 2000, 2010; Sellwood & Binney 2002; Fuchs et al. 2005; Fujii et al. 2011), unbarred grand-design spirals (Sellwood 2011), and barred spirals (e.g., Baba et al. 2009; Grand et al. 2012; Roca-Fàbrega et al. 2013; Baba 2015).

Comparing the spatial distributions of stars and gas in a spiral arm may help to distinguish between the two mechanisms (e.g., Dobbs & Baba 2014; Sakai et al. 2015; Baba et al. 2018; Egusa et al. 2017). Density-wave theories predict a spatial offset between the gravitational potential minimum of a spiral arm (traced by the distribution of old stars) and the peak of gas density (Roberts 1969). On the other hand, dynamic (non-stationary) spiral-arm models predict that both star and gas accumulate into a minimum in the spiral potential and hence are not separated (e.g., Dobbs & Bonnell 2008; Wada et al. 2011; Baba et al. 2015). In the future one could test these theories by comparing the 3-dimensional (3-d) positions of older stars measured by Gaia with that of masers from newly formed stars measured by Very Long Baseline Interferometry (VLBI).

If gas entering a spiral arm is shocked prior to the formation of stars, the resulting stars should display the kinematic signature of that shock. By measuring 3-d velocity fields, one could therefore determine if such shocks occur and how strong they are. Such as the Bar and Spiral Structure Legacy (BeSSeL) Survey and VLBI Exploration of Radio Astrometry (VERA) have yielded precise distances and 3-d velocity fields for high-mass star-forming regions (HMSFRs) associated with spiral arms (e.g., Reid et al. 2009b; Honma et al. 2012; Reid et al. 2014). Optical astrometric results from $Gaia$ DR2 typically have parallax uncertainties larger than 20 $\mu$as (e.g. see Fig. 7 in Gaia Collaboration et al. 2018) and are starting to become significant for this type of study.

Recently, Xu et al. (2006), Sakai et al. (2012) and Choi et al. (2014) showed evidence for systematic (radially) inward motion for HMSFRs in the Perseus arm. Here, we report new astrometric results obtained with the National Radio Astronomy Observatory (NRAO)111NRAO official HP:

https://public.nrao.edu/ Very Long Baseline Array (VLBA), which more clearly reveal the structure and kinematics of the Perseus arm. In section 2, we describe our VLBA observations. In section 3, we outline the data reduction. In section 4, we show new astrometric results for five 6.7-GHz CH3OH masers. In section 5, we discuss the structure and kinematics of the Perseus arm, based on our new results and Gaia DR2 results for OB-type stars (taken from Xu et al. 2018) and compare those with spiral arm models. In section 6, we summarize the paper.

2 Observation

We observed a total of six methanol masers (ie, the CH3OH ($J_{K}$ = 5${}_{1}-$ 6${}_{0}A^{+}$) transition at a rest frequency of 6.668519 GHz) under VLBA programs BR149S, T and U (see Table 4 in the appendix)222Please also see the BeSSeL survey HP:

http://bessel.vlbi-astrometry.org/observations. Each set of observations was optimized to sample the peaks of the sinusoidal parallax signature in right ascension over one year, as described for previous BeSSeL Survey observations (e.g., Xu et al. 2016; Reid et al. 2017). Each maser source, listed in Table 4, was observed with three or four background quasars (QSOs). A single observation involved (i) four half-hour “geodetic blocks” spaced by about 2 hours for clock and atmospheric delay calibration, (ii) “manual phase-calibration” scans of a bright quasar (QSO) every 2 hours, iii) fast switching between a target maser and each QSO, used for relative position determination.

Observational data was recorded on the Mark5A system at 512 Mbps. Geodetic block data was taken in left circular polarization with four 16 MHz bands spanning 496 MHz centered at both 4.3 and 7.3 GHz (8 IFs in total). Fast switching data was taken in dual circular polarization with four adjacent 16-MHz bands spanning 64 MHz. The data were correlated with the DiFX software correlator (Deller et al. 2011) in Socorro, NM. The fast switching data were correlated in two passes: for the maser (line) data the central 8 MHz of the third IF band was correlated with 1000 channels, giving a frequency (velocity) spacing of 8 kHz (0.36 km s*-1*) at the rest frequency. The continuum data for all IFs were correlated with 32 spectral channels.

3 Data reduction

The VLBA data reduction was conducted with the NRAO Astronomical Image Processing System (AIPS) and a ParselTongue pipeline described in previous BeSSeL Survey papers (e.g., Reid et al. 2009a). Details of the techniques employed to determine parallax and proper motion for 6.7-GHz CH3OH masers are described in Reid et al. (2017). Here, we briefly outline the data reduction.

The largest source of relative position error for 6.7 GHz astrometric data is uncertainty in the ionospheric delay calibration. For the ionospheric delay calibration, we firstly applied the Global Ionospheric Maps obtained from NASA’s ftp server333ftp://cddis.gsfc.nasa.gov/gps/products/ionex/. However, at our observing frequency of 6.7 GHz, tropospheric and ionospheric delay residuals can still be significant, with residual path-delays of $\sim$5 cm for both components. Using the geodetic-block observations, tropospheric (non-dispersive) delays were estimated by differencing delays at 4.3 and 7.3 GHz and subtracting these from the total delays. These were modeled as owing to a zenith delay for each observations block. To better calibrate the ionospheric delay residual, the delay differences between 4.3 and 7.3 GHz bands were scaled to the 6.7 GHz CH3OH band and a residual zenith dispersive delay could also be determined.

After applying the geodetic-block calibrations to the phase-reference data, we used a bright maser spot as the phase reference for the associated QSOs. In cases where the maser displayed significant structure, we self-calibrated the maser data and applied these solutions to both maser and QSO data. All sources were imaged and the positions of compact components were determined by fitting elliptical Gaussian brightness distributions. The variations the positions of maser spots relative to background QSOs were then modeled as owing to parallax and proper motion components.

Delay residuals at 6.7 GHz are generally dominated by the ionospheric miscalibration and can cause a systematic position shift across the sky (so called ionospheric wedges). We used our multiple QSO data to account for these effects. As discussed in Reid et al. (2017), one can generate an “artificial QSO” at the position of the target maser to remove most of the effects of the ionospheric wedges. In this paper we incorporated an improved procedure that solved for the wedge effects at each epoch, while at the same time estimating the parallax and proper motion components as described in Wu et al. (2019).

4 Results

We estimated trigonometric parallaxes and proper motions for five of the six sources we observed. For G098.03+1.44 the brightest maser spot was too faint ($<0.5$ Jy) to use as a phase reference. In order to estimate a single proper motion for each source, we averaged the values for all spots. We adopted these values for the proper motion of the central star, but added $\pm 5$ km s*-1* in quadrature to the fitted error estimates for each motion coordinate in order to allow for uncertainty in this step. Note that class II CH3OH masers generally have internal motions of about 5 km s*-1* (e.g., Moscadelli et al. 2010). The parallax and proper motions results are summarized in Table 1 (see also Fig. 1).

Some sources have published parallax and proper motions for 22-GHz water or 12.2-GHz methanol masers as discussed below. When combining parallaxes, we used variance weighting. However, for combined proper motion estimates of 6.7-GHz methanol and 22-GHz water masers motions, we adopted the methanol values, since water masers typically form in outflows of tens of km s*-1*, and transferring the maser motions to that of the central star is less certain than from methanol masers.

We now briefly discuss some individual sources.

4.1 G094.60$-$1.79

All background QSOs were northward of the maser, rather than surrounding it. When accounting for the effects of ionospheric wedges, this requires extrapolation of the fitted planar position tilt instead of interpolation, and we expect increased parallax uncertainty. To allow for this, we added $\pm 0.05$ mas per degree of offset times the offset of the nearest QSO in quadrature with the formal parallax uncertainty. This parallax gradient error source is a rough estimate based on BeSSeL Survey experience fitting for 6.7 GHz data for many sources.

Oh et al. (2010) used the VERA array and estimated a parallax of $0.326\pm 0.031$ mas based on three 22-GHz water maser spots. In order to be conservative, we have inflated their uncertainty by $\sqrt{3}$ to $\pm 0.054$ mas in order to allow for correlated systematic errors caused by similar differential atmospheric delay differences between maser spots and a background QSO. Generally, residual atmospheric delay errors dominate cm-wave VLBI parallax uncertainty. Another result for this source comes from Choi et al. (2014), who obtained a 22-GHz parallax of $0.253\pm 0.024$ mas using the VLBA. In order to assess if the three parallax results are statistically consistent we calculated parallax differences of $0.147\pm 0.072$, $0.074\pm 0.054$, and $0.073\pm 0.059$ mas. Only the first difference is marginally statistically significant, while the other two are statistically insignificant. We conclude that these could reasonably have come from random differences and combine all three by variance-weighting to give a best parallax for G094.60$-$1.79 of $0.250\pm 0.020$ mas.

4.2 G111.25$-$0.76

Choi et al. (2014) derived a parallax of $0.299\pm 0.022$ mas for water masers and the difference between this and our result is not statistically significant ($0.035\pm 0.030$ mas). Thus we variance-weighted them to obtain a best parallax of 0.280$\pm$0.015 mas.

4.3 G136.84+1.16

All background QSOs were northward of the maser, and, as discussed above for G094.60$-$1.70, we inflated the parallax uncertainty to account for a likely $\pm 0.05$ (mas/degree) parallax gradient. However, since the maser’s structure was fairly extended, the final parallax uncertainty is quite large ($\pm 0.123$ mas).

4.4 G188.94+0.88

Oh et al. (2010) using VERA obtained a parallax of $0.569\pm 0.068$ mas for water masers, while Reid et al. (2009a) using the VLBA found a value of $0.476\pm 0.006$ mas based on 12-GHz methanol masers. The differences among the three parallax measurements are not statistically significant ($0.104\pm 0.080$, $0.011\pm 0.042$, and $0.093\pm 0.068$ mas), and we variance weighted them to obtain a best parallax of 0.476$\pm$0.006 mas. For a combined proper motion, we use our 6.7-GHz and the published 12-GHz methanol maser result, yielding ($\mu_{\alpha}\rm{cos}\delta$, $\mu_{\delta}$) = ($-0.30\pm 0.60$, $-1.95\pm 0.51$) mas yr*-1*, where we have added in quadrature $\pm 5$ km s*-1* for each component uncertainty.

5 Discussion

5.1 Pitch angle and arm width of the Perseus arm

Using the five astrometric results discussed above, along with other sources in the literature, we now evaluate the pitch angle and width of the Perseus arm based on 27 sources. Following Reid et al. (2014), we fitted a logarithmic spiral-arm model to the locations of the Perseus arm sources using the following equation:

[TABLE]

where $R_{{\rm ref}}$ and $\beta_{{\rm ref}}$ are a Galactocentric radius (kpc) at a reference azimuth (radians). Azimuth ($\beta$) is defined as zero toward the Sun as viewed from the Galactic center and increases with Galactic longitude, and $\psi$ is the spiral pitch angle. $\beta_{{\rm ref}}$ = 13.2 degrees was chosen to be near the midpoint of the azimuth values. Our best fitting values are $R_{{\rm ref}}$ = 9.93 $\pm$ 0.09 kpc and $\psi$ = 9.2 $\pm$ 1.5 degrees. These results are consistent with those in Reid et al. (2014) within errors. The arm’s width, defined as the $1\sigma$ scatter in the sources perpendicular to the fitted arm, is 0.39 kpc.

5.2 Non-circular motion in the Perseus arm

We now assess the three-dimensional non-circular (peculiar) motions of Perseus arm sources, based on the parallaxes, proper motions and LSR velocities. The LSR velocity of the central star is estimated from the masers and from observations of thermal line emission (e.g., CO) from the parent cloud. Peculiar motions are referenced to a model Galactic rotation curve, using Galactic parameters ($R_{0}$ and $\Theta_{0}$), and solar motion ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$). Note that $U$ values are positive directed toward the Galactic center, $V$ is in the direction of the Galactic rotation, and $W$ is toward the north Galactic pole. In the following, we use 24 of the 27 sources available, removing three outliers (G043.16+0.01, owing to its large motion uncertainty of $>20$ km s*-1*, and G108.20+0.58 and G229.57+0.15 owing to their large deviation of $>$2.2$\sigma$ from the spiral arm fit).

Figure 2 shows the non-circular motions of Perseus-arm sources with uncertainties less than 20 km s*-1*. The solid curve in Fig. 2 represents a logarithmic spiral-arm fit and the dashed lines indicate ($\pm 1\sigma$) width (see section 5.1). Figure 3 plots the non-circular motions as a function of distance perpendicular to the arm, $\rho_{\perp}$, defined positive outward from the arm. Interestingly, we see a significant velocity gradient of $-25\pm 8$ km s*-1* kpc*-1* in $U$ vs. $\rho_{\perp}$. Note that excluding the two high points with $U>25$ km s*-1* and $\rho_{\perp}<-0.2$ kpc as potential outliers still yields a significant gradient of $-16\pm 7$ km s*-1* kpc*-1*. The other peculiar motion components ($V,W$) do not show a statistically significant gradient across the spiral arm.

As an alternative approach to examining systematics in the peculiar motions, Table 2 presents unweighted means of (<$$U$$>, <$$V$$>, <$$W$$>) in three $\rho_{\perp}$ bins: the interior given by $-\frac{3\sigma_{\rm{Per}}}{2}\leqq\rho_{\perp}\rm{(kpc)}<-\frac{\sigma_{\rm{Per}}}{2}$), the middle given by ($-\frac{\sigma_{\rm{Per}}}{2}$ $\leqq\rho_{\perp}$ (kpc) $<\frac{\sigma_{\rm{Per}}}{2}$), and the exterior given by ($\frac{\sigma_{\rm{Per}}}{2}\leqq\rho_{\perp}\rm{(kpc)}<\frac{3\sigma_{\rm{Per}}}{2}$). In the above, $\sigma_{\rm{Per}}$ is a Gaussian $1\sigma$ width for the arm, which we estimate to be 0.39 kpc. For uncertainties, we adopt the standard error of the mean, because the scatter evident in Figures 3(Left) is much larger than would be suggested by the measurement uncertainties, indicating there is significant “astrophysical” noise.

As anticipated by the negative gradient of $U$ vs $\rho_{\perp}<0$, the results in Table 2 show that sources toward the interior side of the arm are moving radially inward with <$$U$$> $=13.3\pm 5.4$ km s*-1* (2.5$\sigma$ for 9 masers), while sources exterior to the arm show a small average $U$ motion of $=2.7\pm 7.6$ km s*-1* (for 5 masers). Regarding the <$$V$$> component of peculiar motion, the result in Table 2 provides marginally significant evidence for a small average motion counter to Galactic rotation of <$$V$$> $=-6.2\pm 3.2$ km s*-1* (1.9$\sigma$ for 10 maser) in the middle region of the Perseus arm. We investigated the sensitivity of the above results to the value of the value of the pitch angle used to define the trace of the Perseus arm. Changing the pitch angle by $\pm 2\sigma$ and recalculated the average peculiar motions on the interior, middle and exterior of the arm yielded no significant changes.

We now compare our observational results with basic predictions from various models for spiral arm formation.

5.2.1 Density wave model without shock

Linear density-wave theories which rely purely on gravity have difficulty explaining the radially inward motion at the interior side of the Perseus spiral arm as observed in the VLBI astrometric data. This is because gas entering an overdense arm is accelerated gravitationally and should show radially outward motion at the interior side of the arm as shown in Fig. 10 of Sakai et al. (2015).

5.2.2 Density wave model with a shock

Our finding of a large positive <$$U$$> value, corresponding to radially inward motion, for the interior side of the Perseus arm is consistent with density wave theories which include a shock as gas in circular Galactic orbits encounters slower rotating spiral arms (Roberts 1969, 1972) triggering the formation of stars. The hydrodynamic shock model of Roberts (1972) with a pitch angle, $\psi$, of $12^{\circ}$, a pattern speed, $\Omega_{{\rm p}}$, of 12.5 km s*-1* kpc*-1*, a gaseous dispersion speed, $a$, of 8 km s*-1*, and a spiral potential with an enhancement, $F$, of 7.5% compared to the axisymmetric potential, predicts a velocity jump in front of the gravitational potential minimum as shown by Fig. 1 of Roberts (1972).

If we assume the full jump velocity is in the line-of-sight direction, the line-of-sight vector of Roberts (1972) model can be decomposed into $U$ (and $V$) jumps by subtracting the rotation curve model of Roberts (1972). This assumption is reasonable for the Perseus arm at the Galactic longitude range $l$ = 110$-$140 (deg) since the non-circular motion vectors in Fig. 2 are aligned in the line-of-sight direction at the section. The $U$ jumps with amplitudes of 20$-$30 km s*-1*, shown by cyan and green curves in Fig. 4, indicate large positive <$$U$$> values for the interior side of the Perseus arm and no significant <$$U$$> values for the exterior of the arm, which are consistent with the observational results. We also investigated the shock model of Roberts (1972) with $\psi$ = 8*∘* and $F$ = 5% (taken from his Figures 7 and 8) and found similar characteristics, but with shock velocities decreased to 10$-$20 km s*-1* and the shock locations shifted by $<$ 150 pc.

5.2.3 Dynamic spiral arm formation

We now compare the observational results with a dynamic spiral-arm model proposed by Baba et al. (2018). The dynamic spiral-arm model is a barred spiral galaxy generated from $N-$body$/$hydrodynamics simulations, and amplitudes, pitch angles, and pattern speeds of spiral arms change within a few hundred million years. Baba et al. (2018) picked spiral arms in growth and disruption phases, respectively, from the model. The growth phase has negative <$$U$$> value for the interior side of an arm and thus is inconsistent with our observational results. While the disruption phase has positive <$$U$$> for the interior side of the arm, it has negative values for the exterior of the arm, and thus also does not agree with our observational results.

5.3 Universality of non-circular arm motions

We confirm that the shock model of Roberts (1972) can explain the observed radially inward motions in the interior side of the Perseus arm. In order to investigate the universality of these motions, we examine the non-circular motions for other spiral arms using the same procedure applied to the Perseus arm and the VLBI astrometric results compiled in Reid et al. (2014). We also examine $Gaia$ DR2 results for OB-type stars taken from Fig. 2(a) of Xu et al. (2018). Table 3 displays the non-circular motion components (<$$U$$>, <$$V$$>) for the interior and exterior of the spiral arms. While there are some statistically significant average motions, no clear trend is evident for these spiral arms, suggesting a more complex picture than expected from the basic models discussed above.

6 Summary

We presented parallaxes and proper motions for five 6.7-GHz methanol masers associated with HMSFRs in the outer portion of the Perseus spiral arm as part of the BeSSeL Survey of the Galaxy (see Figure 1 and Table 1). Combining these new and previous VLBI results, we determined a spiral-arm pitch angle of 9.2 $\pm$ 1.5 deg and an arm width of 0.39 kpc (see Section 5.1).

We divided the sources into interior, middle and exterior regions of the Perseus arm and averaged the non-circular motion components ($<U>$, $<V>$, $<W>$) for each region. For nine sources in the interior of the arm, we found a radially inward motion of $<U>$ = 13.3 $\pm$ 5.4 km s*-1*; for 10 sources in the middle of the arm, we obtained a marginal detection of motion slower than Galactic rotation of $<V>$ = $-$6.2 $\pm$ 3.2 km s*-1*; and for 5 sources in the exterior of the arm, we found no statistically significant non-circular motion (see Section 5.2 and Table 2). These characteristics are consistent with predictions of models for spiral arm formation that involve gas entering an arm to be shocked and then forming stars as shown by Fig. 4.

We performed a similar analysis on previous VLBI astrometric data, as well as on $Gaia$ DR2 results for OB-type stars, for stars in other spiral arms. While some statistically significant non-circular motions are found in other arms, no clear pattern among arms was found (see Table 3). This suggests a more complex picture than expected from basic spiral-arm models.

We acknowledge anonymous referee for valuable comments, which improved the manuscript.

VLBA.

Appendix A Appendix

Here, we show supplemental materials to further document observations (Table 4) and detailed maser maps for (Fig. 5).