DISTANCE AND PROPER MOTION MEASUREMENT OF THE RED SUPERGIANT, PZ CAS, IN VERY LONG BASELINE INTERFEROMETRY H₂O MASER ASTROMETRY

Initial calibration tasks were common for both the sources. First, the amplitude calibration was carried out by using the system noise temperatures (T_sys) and gain calibration data. The measurements of T_sys often have abrupt time variations, as shown in Figure 1, especially when rain falls happened during the observation. We consider such rapid and abrupt time variations to be caused by the absorption effect of water puddles on the top of the VERA antenna feedome. We carried out a polynomial fitting of the raw T_sys data after removing such abrupt changes. At epoch H, since T_sys was not recorded for either beams A or B of the ISHIGAKI station. Instead, T_sys of OGASA20 was used for ISHIGAKI, because the atmospheric environment (temperature and moisture) is similar. Note that minor amplitude calibration errors at this stage can be approximately corrected in the later self-calibration. The system noise temperatures for beams A and B were basically approximated to be identical, so the time variation of the system noise temperature must be mainly attributed to the common atmosphere.

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In the next step, the two-bit sampling bias in the analog-to-digital (A/D) conversion in VLBI signal processing was corrected. Earth-orientation-parameter (EOP) errors, ionospheric dispersive delays, and tropospheric delays were calibrated by applying precise delay tracking data calculated with an updated VLBI correlator model supplied by the NAOJ VLBI correlation center. The instrumental path differences between the two beams were corrected using the post-processing calibration data supplied by the correlation center (Honma et al. 2008). The phase-tracking centers for PZ Cas and J2339+6010 of the updated correlator model were set to the positions given in Table 3 for all the observing epochs.

Table 3. Phase Tracking Center Positions of the Observed Sources

	Right Ascension (J2000)	Declination (J2000)
PZ Cas	$23^{\mathrm{h}} 44^{\mathrm{m}} 03\buildrel{\mathrm{s}}\over{.}281900$	+61°47'22182000
J2339+6010	$23^{\mathrm{h}} 39^{\mathrm{m}} 21\buildrel{\mathrm{s}}\over{.}125210$	+60°10'11849000

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After the initial calibration, an image of the reference source, J2339+6010, was synthesized using the standard VLBI imaging method with the fringe fitting and self-calibration. We carried out the above process twice to make the image of J2339+6010: the first process started with a single point source model and the second with the reference source image that had been made in the previous process. After imaging, the phase-referencing calibration solutions were directly calculated from the reference source fringe data: the calibrating phase, Φ_cal(t), at time t was obtained from the following calculation with a vector average of all the spectral channels in all the BBCs:

$$\begin{eqnarray*} \mathrm{exp} [ i \Phi _{\mathrm{cal}}(t) ] &=& \sum ^{N}_{n=1} \sum ^{M}_{m=1} \mathrm{exp} \left\lbrace i \left[ \Phi ^{\mathrm{r}}_{\mathrm{raw}}(n, m, t)\right.\right. \nonumber \\ &&+ 2\pi \left[\nu ^{\mathrm{r}}_{n} + (m-1)\Delta \nu - \nu ^{\mathrm{t}}_{0}\right] \Delta \tau ^{\mathrm{r}}_{\mathrm{g}}(t) \nonumber \\ &&\left.\left.- \Phi ^{\mathrm{r}}_{\mathrm{v}}(U(t), V(t))\right] \right\rbrace, \nonumber \end{eqnarray*}$$

where $\Phi ^{\mathrm{r}}_{\mathrm{raw}}$ is the raw data fringe phase of the reference source and $\Phi ^{\mathrm{r}}_{\mathrm{v}}(U(t), V(t))$ is the visibility phase calculated from the CLEAN components of the reference source image for a baseline with U(t) and V(t). N = 14 is the BBC number, M = 64 is the spectral channel number in the single BBC, $\nu ^{\mathrm{t}}_{\mathrm{0}}$ is the RF frequency of the strongest H₂O maser emissions in beam A, $\nu ^{\mathrm{r}}_{n}$ is the RF frequency at the lower band edge of the nth BBC for the reference source as listed in Table 2, Δν = 250 kHz is the frequency spacing in the single BBC for the reference source, m is the integer number indicating the order of the frequency channel in the single BBC, and $\Delta \tau ^{\mathrm{r}}_{\mathrm{g}}(t)$ is the third-order polynomial fitting result of the temporal variation of the group delay obtained from the preceded fringe fitting for the reference source. We refer to the phase calibration solutions obtained with this method as direct phase-transfer (DPT) solutions. The DPT solutions were then smoothed by making a running mean with an averaging time of 16 s in order to reduce the thermal noise. The smoothed DPT solutions, as well as the one obtained with an ordinary AIPS phase-referencing analysis (AIPS CL table), are shown in Figure 2. Those two results are basically consistent with each other. Note that even when solutions could not be obtained with the fringe fitting or the following AIPS self-calibration because the signal-to-noise ratios (S/N) was below a cutoff value, the DPT solutions were created from the raw fringe data, as shown for the OGASA20 station in Figure 2. The final astrometric result with the smoothed DPT solutions is consistent with the AIPS CL table at epoch F.

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Amplitude gain adjustment was performed using a third-order polynomial fitting for the amplitude self-calibration solutions for J2339+6010. Since S/N of J2339+6010 was unexpectedly low at epoch G, we could not obtain good solutions for J2339+6010 in the fringe fitting. We therefore obtained the group delay solutions from 3C 454.3 and set the amplitude gain to the unity for all the observing time. The DPT solutions were created from the J2339+6010 fringe data together with the group delay solutions determined with 3C 454.3. At epoch I, since IRIKI did not attend the observation, we used the remaining three stations in the data reduction.

The Doppler shift in PZ Cas spectra due to Earth's rotation and revolution was corrected after the phase referencing. The observed frequencies of the maser lines were converted to radial velocities with respect to the traditional LSR using the rest frequency of 22.235080 GHz for the H₂O 6₁₆–5₂₃ transition. Since the cross power spectra within the received frequency region showed a good flatness both in the phase and the amplitude, the receiver complex gain characteristics were not calibrated in this analysis. We inspected the cross power spectra for all the baselines in order to select a velocity channel of the maser emission which would be unresolved and not have rapid amplitude variation. At this step, fast phase changes with a typical timescale of a few minutes to a few tens of minutes can be removed by the phase referencing, but slow time variations including a bias component as shown in Figure 3 cannot. The maser image was distorted because of the residual time variations of the fringe phase on a timescale of several hours, which might be caused by uncertainties in the updated VLBI correlator model. We conducted the fringe fitting for the selected velocity channel with a solution interval of 20 minutes. We fitted the solutions to a third-order polynomial, as shown in Figure 3, and removed it from both of the fringe phases of PZ Cas and J2339+6010. This procedure removes the residual fringe phase error from the PZ Cas data while it is transferred to the J2339+6010 data. As a result, this leads to an improvement of the PZ Cas image quality, but it also creates a distortion of the J2339+6010 image. This distortion of the reference source causes an astrometric error in the relative position between the sources, as later discussed in Section 3.3.

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Figure 3 also shows the fringe phase residuals obtained with the AIPS CL table. Although the fringe-fitting results with the AIPS CL table can be almost identifiable to those with the DPT solutions, it is later noted that the image peak flux density at V_LSR = −42.0 km s⁻¹ with the DPT solutions is 2% lower than that with the AIPS CL table at epoch F. In addition, the image rms noise resulting from the DPT solutions is 5% higher than that resulting from the AIPS CL table. We can also see that the residual phases resulting from the DPT solutions in the fifth and sixth observing sessions in Figure 3 have a larger deviation than those resulting from the AIPS CL table. Although the number of the calibrated visibilities after applying the DPT solutions was larger than that with the AIPS CL table, especially in the last two sessions, the visibilities with lower S/N might degrade the final image quality. We have to admit that the AIPS CL table may lead to a slightly higher quality in the image synthesis because the fringe fitting and self-calibration take off visibility with low S/N. The problem is that there were some observations whose data showed unexpectedly low S/N for certain baselines. For such cases, it was hard to create the AIPS CL table with the fringe fitting and self-calibration. For the following three reasons, we used the DPT solutions instead of the AIPS CL table for all the observing epochs: (1) the DPT is useful even when the fringe fitting fails because of the low S/N, (2) the final astrometric results between the two methods have little difference, and (3) it is preferable to use a unified analysis scheme for obtaining scientific results from a data set with the same (U, V) distribution.

The DPT solutions and the long-term phase calibration data were applied to the remaining frequency channels to make image cubes for the LSR velocity range of −23.0 to −60.8 km s⁻¹ with a step of 0.2107 km s⁻¹. Image synthesis with 60 μas pixel for a region of 4096 × 4096 pixels, roughly a 246 × 246 square milliarcsecond (mas) region, was made by using IMAGR. To initially pick up maser emissions in the maps, we used the AIPS two-dimensional (2D) Gaussian component survey task, SAD, in each velocity channel map in order to survey emissions. We visually inspected the surveyed Gaussian components one by one. We stored the image pixels that were larger than a 5σ noise level for a visually confirmed maser emission. Hereafter we define a maser "spot" as an emission in a single velocity channel, and a maser "feature" as a group of spots observed in at least two consecutive velocity channels at a coincident or very closely located positions.

In the phase-referencing astrometric analysis, positions of the PZ Cas maser spots were obtained with respect to that of the J2339+6010 image. The positional error includes the effect of a peak position shift due to uncertainties in the VLBI correlator model (atmospheric excess path lengths [EPLs], antenna positions, EOP, and so on) for the sources. Hereafter we refer to the relative position error of a pair of point sources using the phase referencing due to uncertainties in a VLBI correlator model as an astrometric error.

We conducted Monte Carlo simulations of phase-referencing observations for the pair of PZ Cas and J2339+6010 to estimate the astrometric error. For generating simulated phase-referencing fringes, we used an improved version of a VLBI observation simulator, ARIS (Astronomical Radio Interferometer Simulator; Asaki et al. 2007), with new functions simulating station clocks (Rioja et al. 2012) and amplitude calibration errors. In the simulations, we assumed flux densities of PZ Cas and J2339+6010 of equivalently 10 Jy and 0.2 Jy for 15.6 kHz and 224 MHz bandwidths, respectively, which were determined from our observations. The sources were assumed to be point sources. The observation schedule and observing system settings in the simulations, such as the recording bandwidth and A/D quantization level, were adopted from the actual VERA observations. A tropospheric zenith EPL error of 2 cm and other parameters were set as suggested for VERA observations (Honma et al. 2010). For simplification of the simulations, reference source position errors were not considered. The simulation results from 200 trials are shown in Figure 4. They show that the 1σ errors in the relative position are 72 and 33 μas in right ascension and declination, respectively. We used these standard deviations as the astrometric error in the following analysis.

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At epoch I, the IRIKI station geographically located at the center of the VERA could not attend the observation, so the astrometric accuracy may become worse with the remaining three stations. We conducted another simulation series for the three-station case, and the resultant astrometric accuracy is 101 and 39 μas for right ascension and declination, respectively. We therefore used this astrometric accuracy at epoch I in the following analysis.

In the first step of the annual parallax analysis, we performed a Levenberg–Marquardt least-squares model fitting of the maser motions in the same way as that described by Asaki et al. (2010) using the peak positions of the selected maser spots. We adopted the astrometric error as mentioned in Section 3.3 and a tentative value of a morphology uncertainty (a positional uncertainty due to maser morphology variation) of 50μas in the fitting, equivalent to 0.1 AU by assuming the distance of 2 kpc to the source. Figures 6 and 7 show the images and the fitting results of all the selected maser spots. The stellar annual parallax, π*, was determined by the combined fitting with the Levenberg–Marquardt least-squares analysis for all the selected spots. The obtained stellar annual parallax was 0.380 ± 0.011 mas at this stage.

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If we carefully look at Figure 7, the maser spots do not seem to be properly identified with the image peak positions because multiple maser spots are spatially blended. This is caused when the multiple maser spots, whose size is typically 1 AU (Reid & Moran 1981), are closely located comparing with the synthesized beam size of ∼1 mas in this case, equivalent to 2.6 AU at the distance of PZ Cas. To identify an individual maser spot properly, especially for blended structures, we performed AIPS JMFIT for the 2D multiple Gaussian component fitting of the emissions. Here we refer to a maser spot that can be identified with the image peak position as an isolated maser spot, and to a maser spot that cannot be identified with a single peak position because of the insufficient spatial resolution as a blended component. If we can identify 2D Gaussian components using JMFIT, instead of SAD that automatically searches multiple spots, in the brightness distribution, we refer to such spots as identified spots. Figure 8 provides an example of the annual parallax least-squares fitting for a maser spot using either the image peak position or the 2D Gaussian peak. Table 4 lists our combined fitting results using the image peak only for the isolated spots, only for the blended components, and for all those involved in the two groups. We also tried to make another combined fitting using the Gaussian peak only for the isolated spots, only for the blended but identified spots with JMFIT, and all of them, as listed in the bottom row in Table 4. The combined fitting results for the isolated spots show little difference between the cases of image peak and Gaussian peak. However, the fitting result for the blended spots with JMFIT peak positions is definitely changed from that with the image peak for the blended components and becomes closer to that obtained with the isolated spots. If a maser spot is recognized as an isolated spot, in other words, if a maser cloud is spatially resolved into multiple spots in the image, the image peak position can be treated as the maser position. Because this cannot be the case for the maser spots of PZ Cas, we adopted the Gaussian peak positions in our least-squares model fitting. The model fitting results obtained from the individual spots are listed in Table 5. From the combined fitting analysis by using all the selected maser spots, we obtained the stellar annual parallax of 0.356 ± 0.011 mas with assumptions of the above astrometric error and morphology uncertainty at this stage.

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Although H₂O maser features around RSGs are good tracers for astrometry (Richards et al. 2012), the morphology of the maser spots observed with the high-resolution VLBI is no longer negligible for the precise maser astrometry. These monitoring observations of PZ Cas have been conducted so often for two years that we can investigate the contribution of the morphology uncertainty to the maser astrometry. Figure 9 shows the position residuals of the 16 selected maser spots after removing the stellar annual parallax of 0.356 mas, proper motion, and initial position estimated for each of the spots. The standard deviation of the residuals is 93 and 110 μas for right ascension and declination, respectively. Assuming that those errors have the statistical characteristics of a Gaussian distribution, the morphology uncertainty is 59(=$\sqrt{93^2\hbox{--}72^2}$) and 105(=$\sqrt{110^2\hbox{--}33^2}$) μas for right ascension and declination, respectively, where 72 and 33 μas are the astrometric errors discussed in Section 3.3. It is unlikely that the morphology uncertainty has different values between right ascension and declination, so we adopted 105 μas as the morphology uncertainty to be on the safer side. Based on the trigonometric parallax of 0.356 mas, 105 μas is 0.29 AU at PZ Cas. We conducted the combined fitting with all the maser spots whose positions were determined with JMFIT and obtained π* of 0.356 ± 0.011 mas at this stage.

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Note that the above error of the annual parallax is a result of considering the random statistical errors from the image S/N and the randomly changing maser spot position, as well as a common positional shift to all the maser spots at a specific epoch due to the astrometric error. Because the majority of the maser spots and features in the current parallax measurement are associated with the maser group a, the obtained stellar annual parallax could be affected by a time variation of the specific maser features at a certain level. Assuming that all the 13 maser spots belonging to group a have an identified systematic error, the annual parallax error can be evaluated to be $0.011\times \sqrt{13}=0.040$ mas. However, if we assume that maser spots belonging to a certain maser feature have a common systematic error but that some of them have their own time variations, the annual parallax error should be estimated to be ranging somewhere between 0.011 and 0.040 mas. To evaluate the contributions of such a systematic error, we carried out Monte Carlo simulations introduced by Asaki et al. (2010) for our astrometric analysis. First we prepared the imitated data set of the 16 maser spots with position changes added by considering the fixed stellar annual parallax and the maser proper motions. Second we added an artificial positional offset due to the image S/N, the astrometric accuracy, and the morphology uncertainty to each of the maser spot positions. The positional offset caused by the image S/N was given to each of the maser spots at each epoch independently. The positional offset due to the astrometric error was common to all the maser spots at a specific epoch but randomly different between the epochs. In this paper, we assumed that the positional offset due to the morphology uncertainty was common to all the maser spots belonging to a specific maser feature at a specific epoch, but randomly different between the features and the epochs. We assumed the standard deviation of the maser morphology uncertainty was 105 μas, equivalent to the intrinsic size of 0.29 AU. In a single simulation trial, we obtained the stellar annual parallax for the imitated data set. Thirdly we carried out 200 trials in order to obtain the standard deviation of the stellar annual parallax solutions. The resultant standard deviation was 0.026 mas which can be treated as the annual parallax error in our astrometric analysis. The resultant estimate of the stellar annual parallax was 0.356 ± 0.026 mas, corresponding to $2.81 ^{+0.22}_{-0.19}$ kpc.

Similar to the cases seen in other RSGs (VY CMa, Choi et al. 2008; S Per, Asaki et al. 2010; VX Sgr, Kamohara et al. 2005; Chen et al. 2006; NML Cyg, Nagayama et al. 2008; Zhang et al. 2012b), the maser features around PZ Cas are widely distributed over a 200 × 200 square mas (corresponding to 562 × 562 square AU at a distance of 2.81 kpc) while the velocity of the features (−52 to −37 km s⁻¹) is distributed more closely to the stellar systemic velocity than those found in other RSGs are. It is likely that the detected maser groups are located at the tangential parts of the shell-like circumstellar envelope, whose 3D velocity has a small line-of-sight velocity. We determined the proper motions of all detected maser spots were identified at least in two epochs. Table 6 lists maser spots whose proper motions and positions were calculated with the stellar annual parallax determined above. Using the proper motion data listed in Tables 5 and 6, we then conducted the least-squares fitting analysis to obtain the position of PZ Cas and the stellar proper motion based on the spherically expanding flow model (Imai et al. 2011). The obtained stellar proper motion in right ascension and declination, $\mu ^{*}_\alpha \cos {\delta }= -$3.7 ± 0.2 mas yr⁻¹ and $\mu ^{*}_\delta =-$2.0 ± 0.3 mas yr⁻¹, respectively. The astrometric analysis result for PZ Cas is listed in Table 7. The proper motions of the detected maser spots together with the stellar position indicated by the cross mark are shown in Figure 10.

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Table 7. Astrometry Analysis Results of PZ Cas

Annual Parallax	π* = 0.356 ± 0.026 mas (2.81$^{+0.22}_{-0.19}$ kpc)
Position (J2000)a	Right Ascension	Declination
(2006 Apr 20)	$23^{\mathrm{h}} 44^{\mathrm{m}} 03\buildrel{\mathrm{s}}\over{.}2816$ ± 00004	+61°47'22187 ± 0003
Stellar proper motion	$\mu ^{*}_{\alpha }\cos {\delta }= -3.7 \pm 0.2$ mas yr−1	$\mu ^{*}_{\delta }= -2.0 \pm 0.3$ mas yr−1

Note. ^a The position errors of J2339+6010 are not taken into consideration. This position is located at the origin of the PZ Cas image of Figure 5.

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We obtained the trigonometric parallax to PZ Cas, $D=2.81^{+0.22}_{-0.19}$ kpc as described in Section 4.1. PZ Cas is thought to be a member of the Cas OB5 association in the Perseus spiral arm. It is of interest to compare the distance to PZ Cas determined in this paper to distances that have been accepted by other methods. Mel'nik & Dambis (2009) estimated the trigonometric parallax to Cas OB5 to be 2.0 kpc from the median of trigonometric parallaxes of the member stars obtained with Hipparcos. Since a typical statistical error of Hipparcos annual parallaxes is ∼1 mas, we cannot fully rely on this statistical value for the object. Another estimate of the distance to Cas OB5 is a distance modulus of 12, corresponding to the photometric parallax of 2.5 kpc (Humphreys 1978). VLBI phase-referencing astrometry has revealed that its measured trigonometric parallaxes for RSGs are well consistent with photometric parallaxes of the related star clusters within 10%–20% (S Per, Asaki et al. 2010; VY CMa, Zhang et al. 2012a; NML Cyg, Zhang et al. 2012b). Therefore, it is highly plausible that PZ Cas is involved in Cas OB5 from the geometrical viewpoint.

However, the difference between the distances derived from PZ Cas's annual parallax and the photometric parallax of Cas OB5 is too large to be explained from the viewpoint of its extent in depth, compared with the distribution on the sky of the member stars. One of the plausible reasons for the inconsistency is the overestimation of A_V to Cas OB5. If we attribute the inconsistency to the A_V value, the difference is 0.16 mag, which is comparable to a typical statistical error of A_V. The Gaia mission is expected to yield measurements of accurate trigonometric distances for much more sampled stars than ever.

The bolometric luminosity, L = 4πD²F_bol, of PZ Cas was estimated to be 2.0 × 10⁵L_☉, 2.1 × 10⁵L_☉, and 1.9 × 10⁵L_☉ by Josselin et al. (2000), Levesque et al. (2005), and Mauron & Josselin (2011), respectively, with the distance modulus of Cas OB5 of 12 (Humphreys 1978). To derive the mass and age of PZ Cas, we first rescaled the luminosity values with our trigonometric parallax to 2.5 × 10⁵L_☉, 2.7 × 10⁵L_☉, and 2.4 × 10⁵L_☉, respectively. Levesque et al. (2005) estimated the effective temperature of PZ Cas to be 3600 K from the MARCS stellar atmosphere models and the absolute spectro-photometry. Using this effective temperature, we ploted PZ Cas in the H-R diagram reported by Meynet & Maeder (2003), as shown in Figure 11. In this figure the filled circles represent the re-estimated luminosities based on Josselin et al. (2000), Levesque et al. (2005), and Mauron & Josselin (2011), while the open circles represent the originally reported ones. The luminosity previously estimated by using the distance modulus yields the initial mass of around 20 M_☉ while, using our trigonometric parallax, it is estimated to be ∼25 M_☉.

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According to evolutionary tracks shown in Figures 1 and 2 of Meynet & Maeder (2003), the lifetime is estimated to be ∼8 Myr for an initial mass of 25 M_☉ and ∼10 Myr for an initial mass of 20 M_☉. Since the ages of four open clusters in Cas OB5 have been reported to be 120 ± 20 Myr for NGC 7790 (Gupta et al. 2000), and 30.2, 45.7, and 13.2 Myr for NGC 7788, Frolov 1, and King 12, respectively (Kharchenko et al. 2005), it is expected that PZ Cas was born after the first generation in Cas OB5.

The stellar proper motion and the radial velocity of PZ Cas enabled us to discuss its 3D motion in the Milky Way. In the analysis, we adopted the Galactocentric distance to the Sun of 8.5 kpc, and the flat Galactic rotation curve with the rotation velocity of 220 km s⁻¹. For the solar motion, we adopted the latest value obtained by Schönrich et al. (2010), 11.10 ± 0.75 km s⁻¹ toward the direction of the Galactic center (U_☉ component), 12.24 ± 0.47 km s⁻¹ toward the direction of the Galactic rotation (V_☉ component), and 7.25 ± 0.37 km s⁻¹ toward the north pole (W_☉ component). Since H₂O maser features around RSGs are spherically distributed, it is hardly plausible that an uncertainty in the stellar proper motion was caused by the anisotropic distribution, as those in the case of outflows of star-forming regions has to be considered (e.g., Sanna et al. 2009). Therefore, we did not introduce an additional uncertainty of the PZ Cas stellar motion other than the statistical error described in Section 4.2 in this error estimation. We followed the methods described by Johnson & Soderblom (1987) and Reid et al. (2009) to derive the peculiar motion and its error. The resultant peculiar motion of PZ Cas was (U_s, V_s, W_s)= (22.8 ± 1.5, 7.1 ± 4.4, −5.7 ± 4.4) km s⁻¹.

Different from peculiar motions of Galactic high-mass star-forming regions (HMSFRs) typically showing negative V_s components (−10 to −20 km s⁻¹; e.g., Reid et al. 2009), PZ Cas has a large peculiar motion not in the V_s component but in U_s. At first, in order to discuss whether PZ Cas's peculiar motion is due to the internal motion inside Cas OB5, we investigated proper motions of the member stars of Cas OB5 using the Hipparcos catalog (Høg et al. 2000). Positions and velocities of Cas OB5 member stars (OB stars and RSGs) cataloged in Garmany & Stencel (1992) with compiled information from previous literatures (Evans 1967; Famaey et al. 2005; Fehrenbach et al. 1996; Humphreys 1970; Valdes et al. 2004; Van Leeuwen 2007; Wilson 1953) are listed in Table 8. Figure 12 shows the positions and proper motions of the member stars listed in Table 8 as well as stars with annual parallax <5 mas and proper motion <8 mas yr⁻¹ in order to roughly exclude foreground stars. In Figure 12, the J = 1 → 0 CO contour map of a Galactic plane survey (Dame et al. 2001) taken from SkyView (Internet's Virtual Telescope)⁵ is superimposed. We obtained the unweighted average of those proper motions (μ_αcos δ = −2.8 ± 1.2 and μ_δ = −1.9 ± 0.8 mas yr⁻¹) which is well consistent with Cas OB5's peculiar motion obtained by Mel'nik & Dambis (2009) (μ_αcos δ = −3.8 ± 1.2 and μ_δ = −1.4 ± 0.8 mas yr⁻¹). Figure 13 shows 3D views of the relative positions and peculiar motions of PZ Cas and the member stars of Cas OB5 whose radial velocities can be found in previous literatures as listed in Table 8. Here we adopted PZ Cas's trigonometric distance for Cas OB5. The averaged peculiar motion of those member stars was ($\bar{U}$, $\bar{V}$, $\bar{W}$)=(8 ± 14, −1 ± 18, −9 ± 18) km s⁻¹ whose U component was prominent comparing with the other two. Therefore, we concluded that the peculiar motion of PZ Cas with the large U_s component reflects Cas OB5's peculiar motion.

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Why does PZ Cas, or Cas OB5, have such a large peculiar motion in the U component? Since the member stars including PZ Cas are located on the western side of a large shell-like structure molecular gas (H_I super shell) with the center position of a Galactic coordinate of (118°, −1°), one may consider that supernovae in precedent giant molecular clouds triggered the formation of those stars and that they have an expanding motion (Sato et al. 2008). We investigated Hipparcos proper motions of the member stars of Cas OB5, as well as those of Cas OB4 and Cas OB7 located on the eastern side of the shell-like structure, as shown in Figure 12. We also investigated 3D relative positions and peculiar motions of those of Cas OB4 and Cas OB7, as shown in Table 9. We adopted the photometric parallaxes for Cas OB4 and Cas OB7 reported by Garmany & Stencel (1992). Figure 13 shows three-dimensional views of the positions and proper motions of the member stars of Cas OB5 as well as Cas OB4 and Cas OB7. We could hardly find clear evidence of an expanding motion of the member stars, either in the proper motions or the 3D motions between those three OB associations. One of the member stars of Cas OB7, HD 4842, has a much larger proper motion than those of the others, even if the Hipparcos trigonometric parallax of 1.46 ± 1.21 mas is considered. Therefore, this star could be a runaway star (Fujii & Zwart 2011).

VLBI astrometric observations of Galactic maser sources on Perseus spiral arm have revealed large peculiar motions of Galactic HMSFRs. According to Sakai et al. (2012), 33 sources whose peculiar motions have been measured in the Perseus and Outer arms show that these peculiar motions are almost located in the fourth quadrant in the U–V plane (U > 0 and V < 0). The peculiar motion of Cas OB5 is located in the fourth quadrant while that of PZ Cas is located in the first quadrant. Such a peculiar motion of the OB association may indicate the standard density wave theory (Lin & Shu 1964). However, it is hard to explain the velocity dispersion of the peculiar motions of the HMSFRs from the density wave theory.

Such a large velocity dispersion in peculiar motions can be seen for the Galactic OB associations within 3 kpc from the Sun (Mel'nik & Dambis 2009). The peculiar motion of Cas OB5 hits upon an interesting idea of "time-dependent potential spiral arms'. Baba et al. (2009) reported on results of large N-body and smoothed particle hydrodynamics (SPH) simulations that calculated the motions of multi-phase, self-gravity gas particles in self-induced stellar spiral arms in a galactic disk. In their simulations, spiral arms are developed by the self-gravity but cannot keep their forms even in one galactic rotation evolution. Other simulation results by Wada et al. (2011) suggest that a massive cloud falls into a spiral arm's potential from both sides, that is, there are flows of cold gas with various velocities. The flows converge into condensations of cold gas near the bottom of the potential well. The relative velocities of the gas to the stellar spirals are comparable to the random motion of the gas. Another interesting point reported by Wada et al. (2011) is that supernovae feedback is not essential to change the velocity field of the gas near the spiral potential. Each of the cold massive clouds and stars with ages younger than 30 Myr may have a peculiar motion with a velocity dispersion of ∼30 km s⁻¹, which is not predicted by the density wave theory. Although we have not had samples large enough to discuss the whole structure of the Milky Way, the self-consistent gravity spiral galaxy model can explain those peculiar motions of HMSFRs in the Milky Way. In other words, a live spiral structure model can fit to the recent sample of the HMSFRs' spatial motions including Cas OB5. A deep understanding of the dynamics of the Galactic spiral arms and the origin of the peculiar motions of HMSFRs will be possible in the near future when massive numerical simulations can be combined with much larger VLBI astrometric samples which only can be conducted toward the Galactic plane obscured by dense gas.