Offsets of Masers with Respect to the Middle of the Perseus Arm and the Corotation Radius in the Milky Way

Most recent determinations of the distance from the Sun to the GC indicate around 8.0 ± 0.2 kpc (Vallée 2017a).

Below the Sun's orbit, for the inner spiral arms, one can look at the arm tangent from the Sun, in order to get the Galactic longitude where each tracer is observed at its maximum peak. Thus for the Sagittarius arm, its star-forming regions (protostars, dust lane) are seen from the Sun to be at the inner arm side: the arm tangent for the dust lane is at Galactic longitude +48°, while the arm tangent for the CO mid-arm is at +51° (see Table 1 in Vallée 2016). Its counterpart, the Carina arm, also shows the arm tangent for the dust lane to be at Galactic longitude −75°, while the arm tangent for the CO mid-arm is at −79° (see Table 1 in Vallée 2016).

For all of the inner arms, one finds an offset between the dust lane and the main CO mid-arm (see Table 4 and Figure 3 in Vallée 2014) with the dust lane on the inner side of an arm toward the GC. The mean separation from the dust lane to the CO mid-arm is about 315 pc (see Table 1 in Vallée 2016). Observationally, the Galactic longitudes of each spiral arm tracer (dust, cold CO) show a reversal on both sides of the Galactic longitude zero (Vallée 2016, 2017c).

Above the Sun's orbit, for the outer spiral arms, one cannot look at arm tangents from the Sun. However, one can look at the distance offsets of some tracers across a spiral arm, when measured through trigonometry.

Is the Perseus arm located inside or outside the corotation radius? Equivalently, for the Perseus arm, are the tracer offsets located on the inner edge or on the outer edge of the Perseus arm? Observationally, where is this corotation distance ${R}_{\mathrm{coro}}$ in the Milky Way disk?

Section 2 assembles recent observations of arm tracers having accurate distances, and compares them with a recent modeling of the spiral arm, in both space and velocity. Section 3 assembles and performs some statistics on the angular speed of the spiral pattern. Section 4 concludes on the likely location of the corotation radius in the Milky Way.

Various arm models have been proposed and compared (see Table 3 in Vallée 2017b). Here we employ the most recent spiral arm model, with an explanation for the GQ II and III in Vallée (2017b, Section 7) and for the inner GQ I and IV in Vallée (2017d, Section 3). Basically, this model fitted the CO 1–0 gas, peaking at the Galactic longitude of the tangent to the spiral arm, as seen from the Sun (given in Table 5 of Vallée 2016).

It was then found that these CO longitudes (in mid-arm) differ by about 315 pc from the Galactic longitudes where the dust lane and masers are seen tangentially (on the inside of the arm, toward the GC)—see Vallée (2016, his Figure 1 and Table 1). This CO-fitted model was then extended close to the GC (Vallée 2017d) and outward (here).

Below, we compare the distances and velocities of the masers and H ii regions to the CO-modeled Perseus arm, looking for distance offsets and velocity offsets.

Figure 1 shows the sources in Table 1 toward the anti-GC (filled circles), as well as a recent spiral arm model (curves), as explained in Vallée (2017b, his Figure 5(a)) and Vallée (2017d, his Figure 3).

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The model in Figure 1 employed the following characteristics: Sun to GC distance of 8.0 kpc (Vallée 2017a), mean arm pitch angle of −131 (Vallée 2017b), and logarithmic spiral shape (Vallée 2017c). The model fitted well the Galactic longitudes of the arm tangents in the CO tracers for the Sagittarius, Scutum, and Norma arms, as observed from the Sun and cataloged in Tables 3–10 in Vallée (2016), hence the CO separation from the maser lane is well obtained by different observed tangencies for the two tracers.

For the Perseus arm, it can be seen in Figure 1 that most (about 85%) of masers/H ii regions and in GQ II, with recently accurately determined distances, are located inward of the CO lane (middle of the arm)—at a mean offset of about 0.4 ± 0.1 kpc from the CO tracer. A roughly similar value is found in GQ III, but with much less data.

This mean offset, and its direction (inward), are similar to that found earlier for the other inner arms (Vallée 2014). The implication from the density wave theory of this maser offset from the cold CO mid-arm and inward arm location is that the Perseus arm lies inside the corotation radius.

In the GQ III, the Perseus spiral arm gets farther away from the Sun, and observational parallaxes get smaller and difficult to observe.

Figure 2 shows the same sources from Table 1, and the radial velocities from a recent spiral arm model (Vallée 2017b, Figure 5(b); Vallée 2017d, his Figure 4). The CO tracer for the Perseus arm is shown in yellow.

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The model spiral arm model in Figure 2 used the latest values for the orbital circular speed around the GC of 230 km s⁻¹ (Vallée 2017a), and for the start of each spiral arm near the GC of 2.2 kpc (Vallée 2017b). This latest velocity model compares favorably with earlier published but incomplete models (Table 3 and Section 7 in Vallée 2017b).

For the Perseus arm, it can be seen in Figure 2 that most (about 85%) masers and H ii regions in GQ II (at left) have a more negative velocity than the CO tracer of the Perseus arm—at a mean offset of about 9 ± 2 km s⁻¹ from the CO tracer. The mean velocity offset, and its direction (more negative), follow the prediction of the density wave theory.

In the third GQ, the Perseus spiral arm gets much more distant, and the signal strengths get weaker. A roughly similar value is found in GQ III, but with much less accuracy.

The observations shown in Figure 1, notably the location of the masers and H ii regions being closer to the GC than the CO mid-arm, and the observations shown in Figure 2, notably the velocity of the masers and H ii regions being more negative than that of the CO mid-arm in quadrant II, together make a strong suggestion that the angular speed of the stars and gas there is larger than the density wave's angular speed near the Perseus arm.

Therefore, the density wave's corotation radius must be beyond the Perseus arm—thus beyond a Galactic radius ${R}_{\mathrm{coro}}$ of 10.8 kpc (the Perseus arm distance at longitude $180^\circ$ in Figure 1).

At that possible ${R}_{\mathrm{coro}}$ value of 10.8 kpc, past the Perseus arm and closer than the Cygnus arm, the circular speed of gas and stars there (230 km s⁻¹; see Vallée 2017a) and the Perseus arm distance to the GC (10.8 kpc) give an angular velocity of gas and stars there of ${{\rm{\Omega }}}_{\mathrm{gas}}$ = ${V}_{\mathrm{rot}}$/R = 21.3 km s⁻¹ kpc⁻¹.

If this Galactic distance is at the corotation point, then the density wave spiral pattern is at ${{\rm{\Omega }}}_{\mathrm{sp}}$ = 21.3 km s⁻¹ kpc⁻¹ for the Milky Way. If not, then ${R}_{\mathrm{coro}}$ > 10.8 kpc, and there ${{\rm{\Omega }}}_{\mathrm{sp}}$< 21.3 km s⁻¹ kpc⁻¹.

The observations shown in Figure 2, notably the velocity offsets of masers and H ii regions with respect to the CO mid-arm, are similar to the streaming predicted in the density wave theory. Roberts (1975, his Figure 12) predicts an offset of around 30 km s⁻¹ along a circular orbit.

Here in Figure 2 we observe a radial velocity offset near 0 km s⁻¹ at l = 180°, then near 15 km s⁻¹ at l = 140°, near 25 km s⁻¹ at l = 115°, and we can extrapolate it near 35 km s⁻¹ at l = 90°. For that longitude of $90^\circ$, the masers have a reduced orbital speed (230–35 = 195 km s⁻¹), at a Galactic radius of 9.4 kpc, giving an ${{\rm{\Omega }}}_{\mathrm{sp}}$ = ${V}_{\mathrm{maser}}$/${R}_{\mathrm{Perseus}}$< 20.7 km s⁻¹ kpc⁻¹. This corresponds to ${R}_{\mathrm{coro}}$ > 11.1 kpc.

CO arm tracers employed in the model are the diffuse CO emissions with a broad angular size (about 9 arcmin; Vallée 2016) coming from a large chunk of the spiral arm, but not the narrow CO cloud tracers harboring the maser emission from local star-forming regions (and at a different radial velocity than the mid-arm's velocity).

These spatial offset and velocity offsets are definite, and show that the Perseus arm is definitely inside the corotation radius, since here the orbiting gas and stars must hit the inside of the arm to produce the inward spatial offset (Figure 1) and the more negative velocity offset (Figure 2). This is a main result of this work, thanks to the precise parallax distances to masers near the Perseas arm.

To get a rough estimate of the location of the corotation radius, we turn to the recent literature.

Some past reviews have pegged the angular rotation of the spiral arm's pattern over a wide range of values, from 17 to 30 km s⁻¹ kpc⁻¹ (Gerhard 2011) or even 12–30 km s⁻¹ kpc⁻¹ (Foster & Cooper 2010), yet each prediction had a small published error bar of around 1 km s⁻¹ kpc⁻¹. Thus the range of the predicted pattern speed ${{\rm{\Omega }}}_{\mathrm{sp}}$ is much larger than the mean quoted error.

The prediction is often obtained after a complex model is employed, along with some hypotheses. These values transform to a corotation radius estimate from 7.7 to 19 kpc (for ${V}_{\mathrm{gas}}$= 230 km s⁻¹).

Table 2 shows a summary of recently predicted values for ${R}_{\mathrm{coro}}$, since 2010, as based on modeling of some observational data, excluding those with results having the corotation radius below 10.0 kpc (see below). It is expected that later predictions would have somewhat smaller error bars, hence our listing of predictions published since the year 2010.

Table 2.  Published Recent (Since 2010) Angular Rotation of the Spiral Pattern Speed ${{\rm{\Omega }}}_{\mathrm{sp}}$ and Deduced Corotation Radius ${R}_{\mathrm{coro}}$ (10 kpc and Above)

${{\rm{\Omega }}}_{\mathrm{sp}}$	Observed Data	${R}_{\mathrm{coro}}$	References
(km s−1 kpc−1)	used in Model	(kpc);
		(a)
16 ± 1	terminal H i velocity	14.3	Section 3.2.2 in Foster & Cooper (2010)
17 ± 0.6	stellar kinematics	13.5	Section 12 in Antoja et al. (2011)
18.6 ± 0.3	radial velocity of stars	12.4	Section 3.2 in Siebert et al. (2012)
19.3 ± 0.8	peculiar vel. V = 0	11.9	Section 4.2 in Sakai et al. (2015); see (b)
20 ± 5	H i and CO data	11.5	Table 2 in Koda et al. (2016)
20 ± 2	CO data	11.5	Section 5 in Pettitt et al. (2014)
20.3 ± 0.5	early-type stars	11.3	Section 3.2 in Silva & Napilowotzki (2013)
<20.7	maser veloc. offset	>11.1	This paper (Section 2.3)
<21.3	maser offset in space	>10.8	This paper (Section 2.2)
23 ± 0.5	open star clusters	10.0	Table 4 in Junqueira et al. (2015)
23 ± 0.5	H i gas flow	10.0	Section 2.2.3 in Li et al. (2016)

Note. (a) We deduce ${R}_{\mathrm{coro}}$ from the relation ${R}_{\mathrm{coro}}\,.\,{{\rm{\Omega }}}_{\mathrm{sp}}={{\boldsymbol{V}}}_{\mathrm{gas}}$, with ${V}_{\mathrm{gas}}$ = 230 km s⁻¹ (Vallée 2017a). (b) Their ${R}_{\mathrm{coro}}$ = 12.4 kpc value for their ${R}_{\mathrm{Sun}}$ = 8.33 is corrected here for our 8.0 kpc value (giving 11.9), and then the ${{\rm{\Omega }}}_{\mathrm{sp}}$ value is deduced from our ${V}_{\mathrm{gas}}$ value above.

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Many such values were obtained using tracers whose distance estimates have a large error bar; distance estimates near 10% are obtained from various techniques, except the trigonometric parallax technique.

Large distance errors do not allow us to see a tracer (maser, say) being offset from the CO mid-arm by 0.3 kpc. Outward from the Sun's orbit, one needs a linear resolution of at least 100 pc toward the Perseus arm; only the parallax method provides error bars at the level of about 1%.

Foster & Brunt (2015, Figure 5(a)) investigated the photometric/spectroscopic/luminosity distance (with large error bars) to H ii regions in the Perseus arm, noting conflicting distance estimates with parallactic distances (with small error bars) from the literature.

Our results above (Section 2) indicate that the Perseus arm has a maser lane which is definitely offset from the main mid-arm CO, thus the Perseus arm is definitely inside the corotation radius, so then the ${R}_{\mathrm{coro}}$ is definitely greater than 10.8 kpc. This definiteness is important, as it follows here that one can set aside all the predictions from the literature having smaller ${R}_{\mathrm{coro}}$ values (based on nontrigonometric, nonparallax distances, and various assumptions).

Thus, one can perform standard statistics for all data predicting a corotation radius beyond 10.8 kpc. These basic statistics yield a mean ${R}_{\mathrm{coro}}$ of 12.0 ± 1.2 kpc and ${{\rm{\Omega }}}_{\mathrm{sp}}$ of 19.2 ± 1.7 km s⁻¹ kpc⁻¹.

This research assumes a global spiral pattern, with a single pattern speed. This assumption follows the work of many theoretical pioneers, reviewed already in Shu (2016). This implicitly assumes the usual quasi-stationary galaxy-wide spiral density wave (global SDW) with a single spiral pattern speed. This assumption is backed by predictions of a spatial offset between the hot dust/maser density peak and the cold molecular/stellar spiral arm, as observed with telescopes, and of a mirror image of this offset as one crosses the Galactic meridian (e.g., Vallée 2016). Over a whole Galactic disk, the global SDW has a dust offset from the molecular arm, being inward below the corotation radius, and outward above the corotation. Inside corotation, the observations show a spatial separation between the CO mid-arm and the dust/maser lane, of about 315 pc, over the Galactic disk, leading to a mirror image of this separation as one goes across the Galactic meridian (Vallée 2016).

Other theories assumed a local transient spiral pattern. Thus, Baba et al. (2016) employed a recurrent dynamic spiral theory (RDS), in which the gas does not flow through a spiral arm, but flows into the arm from both inner and outer sides (their Figure 5). The RDS may have some local offsets, but not in a global systemic way over a whole Galactic disk; thus the dynamic spiral theory denies the existence of a global tracer offset (e.g., Baba et al. 2016), but it can be local. The transient-recurrent dynamic spiral reconnection theory (Dobbs & Baba 2014, Section 2.2) has arms breaking and reconnecting, and a high pitch angle from $20^\circ$ to $40^\circ$ (Dobbs & Baba 2014, Figure 10), unlike the mid $13^\circ$ pitch in the Milky Way. Also, Baba et al. (2018) proposed disruptions in the Perseus arms, using Cepheid distances; these distance estimates are based on photometric period–luminosity techniques, not on trigonometric parallaxes, and have error bars too large for comparisons (near 10%; Vallée 2017c). The RDS has no global, systemic radial dust offset from the cold molecular mid-arm (Section 4 in Baba et al. 2015).

The theory of stochastic, self-propagating star formation (SSP) produces an irregular, flocculent arm (Dobbs & Baba 2014, Section 2.5), not the regular pattern observed in the Milky Way.

The theory of tides between two nearby galaxies via kinetic density waves (KDW) predicts only two tidal arms, not four as seen in the Milky Way, and no spatial offset between tracers. The induced spiral arms will rotate slower than the galaxy rotates, but there is no global galaxy-wide offset of hot dust versus the cold molecular arm at most times (Dobbs et al. 2010; Pettitt & Wadsley 2018).

This research also assumes a constant circular speed for the gas and stars, from a minimum Galactic radius of 3 kpc going outward. This assumption has been verified observationally out to a Galactic radius of 15 kpc (e.g., a review by Foster & Cooper 2010, Figure 2). Here we concentrate with the Perseus arm, around the Galactic radius of 10 kpc (e.g., Foster & Brunt 2015, Figure 5(b)), where the constant circular speed is verified.