Searching for stars closely encountering with the solar system based on data from the Gaia DR1 and RAVE5 catalogues

We have searched for the stars that either encountered in the past or will encounter in the future with the Solar system closer than 2 pc. For this purpose, we took more than 216 000 stars with the measured proper motions and trigonometric parallaxes from the Gaia DR1 catalogue and their radial velocities from the RAVE5 catalogue. We have found several stars for which encounters closer than 1 pc are possible. The star GJ 710, for which the minimum distance is d _m = 0.063 ± 0.044 pc at time t _m = 1385 ± 52 thousand years, is the record-holder among them. Two more stars, TYC 8088-631-1 and TYC 6528-980-1, whose encounter parameters, however, are estimated with large errors, are of interest.     

OB stars in the Tycho-2 and 2MASS catalogues

The Tycho-2 proper motions and five-band Tycho-2 and 2MASS photometry for approximately 2.5 million common stars have been used to select OB stars and to determine the extinction and photometric distance for each of them. We have selected 37 485 stars and calculated their reddenings based on their positions in the two-color V _T -H, J-Ks diagrams relative to the zero-age main sequence and the theoretical reddening line for B5 stars. Tests confirm that the selected stars belong to the spectral types O-B with a small admixture of later types. We calculate the extinction coefficient R and its variations with Galactic longitude based on the positions of the selected stars in the two-color B _T -V _T , V _T -Ks diagram. The interstellar extinction for each star is calculated as the product of the reddening found and the coefficient R. The extinction and its variations with Galactic longitude agree well with the extinction based on the model by Arenou et al. (1992). Calibration of the relation between the absolute magnitude and reduced proper motion V _T − + 5 + 5 log μ for Hipparcos stars has allowed us to calculate the absolute magnitudes and photometric distances for the selected stars. The distances found agree with those derived from the Hipparcos parallaxes within 500 pc. The distribution of the stars and the extinction variations with distance found show that the selected stars form an almost complete sample of stars with spectral types earlier than B5 within about 750 pc of the Sun. The sample includes many noticeably reddened stars in the first and second Galactic quadrants that are absent from the Hipparcos and Tycho Spectral Types Catalogues. This slightly changes the pattern of the distribution of OB stars compared to the classical pattern based on Hipparcos. Original Russian Text ? G.A. Goncharov, 2008, published in Pis’ma v Astronomicheskiĭ Zhurnal, 2008, Vol. 34, No. 1, pp. 10–20.     

A Comprehensive Study of 94 Open Clusters Based on the Data from IPHAS,GAIA DR2, and Other Sky Surveys

We determine the color excesses, photometric distances, ages, astrometric parallaxes and proper motions for 94 open clusters in the northern part of the Milky Way. We estimate the color excesses and photometric distances based on the data from IPHAS photometric survey of the northern Galactic plane using individual total-to-selective extinction ratios R_r = A_r/E_r?i for each cluster computed via the color-difference method based on IPHAS r, i, and Hα-band, 2MASS J-, H-, and K_s-band, WISE W1-band, and Pan-STARRS i-, z-, and y-band data. The inferred R_r values vary significantly from cluster to cluster spanning the R_r = 3.1–5.2 interval with a mean and standard deviation equal to 〈R_r〉 = 3.99 and σR_r = 0.34, respectively.We identified cluster members using (1) absolute proper motions determined from individual-epoch positions of stars retrieved from IPHAS, 2MASS,URAT1, ALLWISE,UCAC5, and Gaia DR1 catalogs and positions of stars on individual Palomar Sky Survey plates reconstructed based on the data provided in USNO-B1.0 catalog and (2) absolute proper motions provided in Gaia DR2 catalog, and computed the averageGaia DR2 trigonometric parallaxes and propermotions of the clusters. Themean formal error of the inferred astrometric parallaxes of clusters is of about 7 μas, however, a comparison of astrometric and photometric parallaxes of our cluster sample implies that Gaia DR2 parallaxes are, on the average, systematically underestimated by 45 ± 9 μas. This result agrees with estimates obtained by other authors using other objects. At the same time, we find our photometric distance scale to be correct within the quoted errors (the inferred correction factor is equal to unity to within a standard error of 0.025).     

Radial and rotational velocities for a sample of magnetic CP stars

The spectra taken with the Main Stellar Spectrograph (MSS) of the 6-m telescope with a resolution of R ～ 15000 and a signal-to-noise ratio of 200–300 are used to determine the radial velocities and projected rotational velocities (υ _e sin i) for 32 magnetic CP stars. Measured υ _e sin i values range from 18 km/s (the lower boundary determined by the instrumental profile) to 65 km/s. Measurements of standard stars demonstrate the absence of systematic differences between our and published data. Eight of the 32 magnetic stars are found or confirmed to be binary and binarity is suspected for another four stars. The components of tangential velocity are determined for 27 stars with known parallaxes.     

Systematic error of the Gaia DR1 TGAS parallaxes from data for the red giant clump

Based on the Gaia DR1 TGAS parallaxes and photometry from the Tycho-2, Gaia, 2MASS, andWISE catalogues, we have produced a sample of ~100 000 clump red giants within ~800 pc of the Sun. The systematic variations of the mode of their absolute magnitude as a function of the distance, magnitude, and other parameters have been analyzed. We show that these variations reach 0.7 mag and cannot be explained by variations in the interstellar extinction or intrinsic properties of stars and by selection. The only explanation seems to be a systematic error of the Gaia DR1 TGAS parallax dependent on the square of the observed distance in kpc: 0.18R ² mas. Allowance for this error reduces significantly the systematic dependences of the absolute magnitude mode on all parameters. This error reaches 0.1 mas within 800 pc of the Sun and allows an upper limit for the accuracy of the TGAS parallaxes to be estimated as 0.2 mas. A careful allowance for such errors is needed to use clump red giants as “standard candles.” This eliminates all discrepancies between the theoretical and empirical estimates of the characteristics of these stars and allows us to obtain the first estimates of the modes of their absolute magnitudes from the Gaia parallaxes: mode(M _H ) = ?1.49 ^m ± 0.04 ^m , mode(M _Ks ) = ?1.63 ^m ± 0.03 ^m , mode(M _W1) = ?1.67 ^m ± 0.05 ^m mode(M _W2) = ?1.67 ^m ± 0.05 ^m , mode(M _W3) = ?1.66 ^m ± 0.02 ^m , mode(M _W4) = ?1.73 ^m ± 0.03 ^m , as well as the corresponding estimates of their de-reddened colors.     

New orbits of wide visual double stars

Based on photographic and CCD observations with the Pulkovo 26-inch refractor, radial velocity measurements with the 1.5-m RTT-150 telescope (TUBITAK National Observatory, Turkey), and highly accurate observations published in the WDS catalog, we have obtained the orbits of ten wide visual double stars by the apparent motion parameter method. The orientation of the orbits in the Galactic coordinate system has been determined. For the outer pair of the multiple star HIP 12780 we have calculated a family of orbits with a minimum period P = 4634 yr. Two equivalent solutions with the same period have been obtained for the stars HIP 50 (P = 949 yr) and HIP 66195 (P = 3237 yr). We have unambiguously determined the orbits of six stars: HIP 12777 (P = 3327 yr), HIP 15058 (P = 420 yr), HIP 33287 (P = 1090 yr), HIP 48429 (P = 1066 yr), HIP 69751 (P = 957 yr), and HIP 73846 (P = 1348 yr). The orbit of HIP 55068 is orientated perpendicularly to the plane of the sky, P >1000 yr. The star HIP 48429 is suspected to have an invisible companion.     

New Features of Parenago’s Discontinuity from Gaia DR1 Data

The velocity field of main-sequence stars and red giants from the TGAS catalogue with heliocentric distances up to 1.5 kpc has been analyzed for various spectral types. To estimate the influence of a low accuracy of stellar parallax measurements on the results of a kinematic analysis of distant stars, first we have studied in detail how the kinematic parameters derived with 1/π distances are shifted when these distances are replaced by three other versions of distances from Astraatmadja et al. (2016b). We have obtained detailed tables in which the ranges of these shifts in the Ogorodnikov–Milne and Bottlinger model parameters are shown for the stars of each spectral type. We have the smallest shifts in the case of determining the Oort coefficients A and B, for which there are 10% shifts only for main-sequence stars of spectral type B. In the remaining cases, these shifts are 0–3%. For the remaining parameters the shifts do not exceed 30%. Thus, we have shown that using the 1/π distance scale in estimating the Ogorodnikov–Milne and Bottlingermodel parameters (except for the parameter Ω″₀) yields reliable results even when using parallaxes with large relative errors (up to 60%). To study Parenago’s discontinuity, we have investigated the dependence of the Ogorodnikov–Milne and Bottlinger model parameters on color for 1 260 071 mainsequence stars and 534 387 red giants. As far as we know, such a data set is used for the first time to investigate Parenago’s discontinuity. The main result is the detection of maximum points at B ? V = 0.75 after which the solar velocity component V and the Oort coefficient B decrease when moving from blue stars to red ones. This fact is a new feature of Parenago’s discontinuity, because the component V does not change in the classical case at B ? V >0.6. We have made an attempt to represent the well-known Parenago’s discontinuity as a special case of the more complex effect of a gradual change in a number of kinematic parameters as the mean age and composition of the group of stars under study changes.     

Search for Stellar Streams Based on Data from the RAVE5 and Gaia TGAS Catalogues

We have analyzed the space velocities of stars with the proper motions and trigonometric parallaxes from the Gaia TGAS catalogue in combination with the line-of-sight velocities from the RAVE5 catalogue. In the V, \(\sqrt {{U^2} + 2{V^2}} \) velocity plane we have identified three clumps, BB17-1, BB17-2, and BB17-3, in the region of large velocities (V<?150 km s^?1). The stars of the BB17-1 and BB17-2 clumps are associated with the kinematic groups VelHel-6 and VelHel-7 detected previously by Helmi et al. We give the greatest attention to the BB17-3 clump. The latter is shown to be most closely linked with the debris of the globular cluster ω Cen. In the BB17-3 clump we have identified 28 stars with a low velocity dispersion with respect to the center of their distribution. All these stars have very close individual age estimates: log t ≈ 10. The distribution of metallicity indices in this sample is typical for the stars of the globular cluster ω Cen. In our opinion, the BB17-3 clump can be described as a homogeneous stream in the debris of the cluster ω Cen.     

Spectroscopic and photometric characteristics of stars in the field of the cluster vdB 130

The observational data for 24 stars toward the young cluster vdB 130 are analyzed. The spectroscopic and photometric BV RIJHK observations have been carried out with the following telescopes: 6-m at the Special Astrophysical Observatory of the Russian Academy of Sciences, 60-cm at the Southern Station of the Moscow State University, and 2.5-m at the Caucasus Observatory of the Sternberg Astronomical Institute of theMoscow State University. Nine stars previously selected as cluster members have been found to belong to different subtypes of type B. A minimum color excess toward the cluster, E(B ? V) = 0.9 mag, has been revealed for the vdB 130 stars lying outside the molecular cloud. Maximum color excesses, E(B ? V) = 1.3?1.4 mag, have been found in the spectra of cluster stars 1r and 5r observed in dust blobs. Inside the cluster R _v is shown to differ from the standard one. The overwhelming majority of the remaining investigated stars belong to late types and have minor color excesses (≤0.3) typical of close distances.     

Testing the Distance Scale of the Gaia TGAS Catalogue by the Kinematic Method

We have studied the simultaneous and separate solutions of the basic kinematic equations obtained using the stellar velocities calculated on the basis of data from the Gaia TGAS and RAVE5 catalogues. By comparing the values of Ω'₀ found by separately analyzing only the line-of-sight velocities of stars and only their proper motions, we have determined the distance scale correction factor p to be close to unity, 0.97 ± 0.04. Based on the proper motions of stars from the Gaia TGAS catalogue with relative trigonometric parallax errors less than 10% (they are at a mean distance of 226 pc), we have found the components of the group velocity vector for the sample stars relative to the Sun (U, V,W)_⊙ = (9.28, 20.35, 7.36) ± (0.05, 0.07, 0.05) km s^?1, the angular velocity of Galactic rotation Ω⁰ = 27.24 ± 0.30 km s?1 kpc?1, and its first derivative Ω'₀ = ?3.77 ± 0.06 km s^?1 kpc^?2; here, the circular rotation velocity of the Sun around the Galactic center is V₀ = 218 ± 6 km s^?1 kpc (for the adopted distance R₀ = 8.0 ± 0.2 kpc), while the Oort constants are A = 15.07 ± 0.25 km s^?1 kpc^?1 and B = ?12.17 ± 0.39 km s^?1 kpc^?1, p = 0.98 ± 0.08. The kinematics of Gaia TGAS stars with parallax errors more than 10% has been studied by invoking the distances from a paper by Astraatmadja and Bailer-Jones that were corrected for the Lutz–Kelker bias. We show that the second derivative of the angular velocity of Galactic rotation Ω'₀ = 0.864 ± 0.021 km s^?1 kpc^?3 is well determined from stars at a mean distance of 537 pc. On the whole, we have found that the distances of stars from the Gaia TGAS catalogue calculated using their trigonometric parallaxes do not require any additional correction factor.     

Results of Magnetic-Field Measurements with the 6-m Telescope. V. Observations in 2011

The paper presents results of measurements of magnetic-field longitudinal components B _e , radial velocities V _r , and projections of the rotation velocity in the line of sight v _e sin i for 74 objects, mainly main-sequence chemically peculiar stars and standard stars. Observations were carried out in 2011 at the 6-m BTA telescope using the Main Stellar Spectrograph (MSS) with a Zeeman analyzer. Seven new magnetic stars were discovered: HD38129, HD47152, HD50341, HD63347, HD188501, HD191287, and HD260858. Three more stars were suspected to have magnetic fields. Observations of magnetic standard stars and non-magnetic stars confirm the absence of any systematic errors capable of introducing distortions into the B _e longitudinal-field measurement results. The paper gives comments on the research results for each of the 74 stars.     

Kinematics of B-F Stars as a Function of Their Dereddened Color from Gaia and PCRV Data

Parallaxes with an accuracy better than 10% and proper motions from the Gaia DR1 TGAS catalogue, radial velocities from the Pulkovo Compilation of Radial Velocities (PCRV), accurate Tycho-2 photometry, theoretical PARSEC, MIST, YaPSI, BaSTI isochrones, and the most accurate reddening and interstellar extinction estimates have been used to analyze the kinematics of 9543 thin-disk B-F stars as a function of their dereddened color. The stars under consideration are located on the Hertzsprung–Russell diagram relative to the isochrones with an accuracy of a few hundredths of a magnitude, i.e., at the level of uncertainty in the parallax, photometry, reddening, extinction, and the isochrones themselves. This has allowed us to choose the most plausible reddening and extinction estimates and to conclude that the reddening and extinction were significantly underestimated in some kinematic studies of other authors. Owing to the higher accuracy of TGAS parallaxes than that of Hipparcos ones, the median accuracy of the velocity components U, V, W in this study has improved to 1.7 km s^?1, although outside the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5 ^m the kinematic characteristics are noticeably biased due to the incompleteness of the sample. We have confirmed the variations in the mean velocity of stars relative to the Sun and the stellar velocity dispersion as a function of their dereddened color known from the Hipparcos data. Given the age estimates for the stars under consideration from the TRILEGAL model and the Geneva–Copenhagen survey, these variations may be considered as variations as a function of the stellar age. A comparison of our results with the results of other studies of the stellar kinematics near the Sun has shown that selection and reddening underestimation explain almost completely the discrepancies between the results. The dispersions and mean velocities from the results of reliable studies fit into a ±2 km s^?1 corridor, while the ratios σ _V /σ _U and σ _W /σ _U fit into ±0.05. Based on all reliable studies in the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5^m, i.e., for an age from 0.23 to 2.4 Gyr, we have found: W_⊙ = 7.15 km s^?1, \({\sigma _U} = 16.0{e^{1.29({B_T} - {V_T})o}}\), \({\sigma _V} = 10.9{e^{1.11({B_T} - {V_T})o}}\), \({\sigma _W} = 6.8{e^{1.46({B_T} - {V_T})o}}\), the stellar velocity dispersions in km s^?1 are proportional to the age in Gyr raised to the power β _U = 0.33, β _V = 0.285, and β _W = 0.37.     

Catalog of red clump candidates in Tycho-2

We present the technique we used to compile a catalog of about 61 000 local stars brighter than K _s = 8.2 ^m which were identified as most likely red clump candidates on the basis of their reduced proper motions in the K _s band. The catalog was compiled from the combined Tycho-2 and 2MASS data for the stars with color indices J-K _s ranging from 0.5 ^m to 0.8 ^m . It includes the equatorial coordinates, the proper motions, the magnitudes B _T , V _T , J, H, and K _s , and the probabilities for the stars to be red clump giants.     

Magnetic-field dependence of chemical anomalies in CP stars

The dependence of the degree of anomaly of parameter Z of Geneva photometry (Z0 = Z _CP ?Z _norm.) on the average surface magnetic field Bs is analyzed. The Z0 value is proportional to the degree of anomaly of chemical composition. It was found that Bs → 0 corresponds Z0 → ?0.010÷ ?0.015, i.e., part of CP stars are virtually devoid of magnetic field, but exhibit chemical anomalies. This effect may be due to selection whereby only objects with strong chemical anomalies are classified as CP stars, thereby producing a deficit of stars with relatively weak anomalies. Moreover, CP stars have other sources of stabilization of their atmospheres besides the magnetic field, e.g., slow rotation. Formulas relating Z0 to Bs are derived.     

Kinematics of the galaxy from OB stars with proper motions from the Gaia DR1 catalogue

We consider two samples of OB stars with different distance scales that we have studied previously. The first and second samples consist of massive spectroscopic binaries with photometric distances and distances determined from interstellar calcium lines, respectively. The OB stars are located at heliocentric distances up to 7 kpc. We have identified them with the Gaia DR1 catalogue. Using the proper motions taken from the Gaia DR1 catalogue is shown to reduce the random errors in the Galactic rotation parameters compared to the previously known results. By analyzing the proper motions and parallaxes of 208 OB stars from the Gaia DR1 catalogue with a relative parallax error of less than 200%, we have found the following kinematic parameters: (U, V)_⊙ = (8.67, 6.63)± (0.88, 0.98) km s^?1, Ω₀ = 27.35 ± 0.77 km s^?1 kpc^?1, Ω′₀ = ?4.13 ± 0.13 km s^?1 kpc^?2, and Ω″₀ = 0.672 ± 0.070 km s^?1 kpc^?3, the Oort constants are A = ?16.53 ± 0.52 km s^?1 kpc^?1 and B = 10.82 ± 0.93 km s^?1 kpc^?1, and the linear circular rotation velocity of the local standard of rest around the Galactic rotation axis is V ₀ = 219 ± 8 km s^?1 for the adopted R ₀ = 8.0 ± 0.2 kpc. Based on the same stars, we have derived the rotation parameters only from their line-of-sight velocities. By comparing the estimated values of Ω′₀, we have found the distance scale factor for the Gaia DR1 catalogue to be close to unity: 0.96. Based on 238 OB stars of the combined sample with photometric distances for the stars of the first sample and distances in the calcium distance scale for the stars of the second sample, line-of-sight velocities, and proper motions from the Gaia DR1 catalogue, we have found the following kinematic parameters: (U, V, W)_⊙ = (8.19, 9.28, 8.79)± (0.74, 0.92, 0.74) km s^?1, Ω₀ = 31.53 ± 0.54 km s^?1 kpc^?1, Ω′₀ = ?4.44 ± 0.12 km s^?1 kpc^?2, and Ω″₀ = 0.706 ± 0.100 km s^?1 kpc^?3; here, A = ?17.77 ± 0.46 km s^?1 kpc^?1, B = 13.76 ± 0.71 km s^?1 kpc^?1, and V ₀ = 252 ± 8 km s^?1.     

Results of magnetic field measurements of CP-stars performed with the 6-m telescope. III. Observations in 2009

We present the results of measuring longitudinal magnetic fields (B_e), rotation velocities (v_e sin i), and radial velocities (V_r) of 44 stars observed with the Main Stellar Spectrograph (MSS) of the 6-m BTA telescope of the Special Astrophysical Observatory in 2009. For the first time, magnetic fields were detected for the stars HD5441, HD199180, HD225627, and BD+00° 4535. We show that for the same stars, the longitudinal fields B_e measured from the Hβ hydrogen line core and from metal lines can differ by 10% and up to a factor of 2–3. Except in rare cases, magnetic fields measured from the metal lines are stronger. We believe that this phenomenon is of a physical nature and depends on the magnetic field topology and the physical conditions inside a specific star. Observations of standard stars without magnetic fields confirm the absence of systematic errors capable of introducing distortions into the longitudinal-field measurement results. In this work we comment on the results for each of the stars.     

Analysis of the Z distribution of young objects in the Galactic thin disk

We have obtained new estimates of the Sun’s distance from the symmetry plane Z _⊙ and the vertical disk scale height h using currently available data on stellar OB associations, Wolf–Rayet stars, HII regions, and Cepheids. Based on individual determinations, we have calculated the mean Z _⊙ = ?16 ± 2 pc. Based on the model of a self-gravitating isothermal disk for the density distribution, we have found the following vertical disk scale heights: h = 40.2 ± 2.1 pc from OB associations, h = 47.8 ± 3.9 pc from Wolf–Rayet stars, h = 48.4 ± 2.5 pc from HII regions, and h = 66.2 ± 1.6 pc from Cepheids. We have estimated the surface, Σ = 6 kpc^?2, and volume, D(Z _⊙) = 50.6 kpc^?3, densities from a sample of OB associations. We have found that there could be ～5000 OB associations in the Galaxy.     

Evolution of low-mass helium stars in semidetached binaries

We systematically investigate the evolution of low-mass (0.35, 0.40, and 0.65M _⊙) helium donors in semidetached binaries with white-dwarf accretors. The initial periods of the binaries are chosen in such a way that the helium abundance in the center of the models at the time of Roche lobe overflow varies between Y _c = 0.98 and Y _c ? 0.1. The results of our calculations can be used to analyze the formation scenarios and evolutionary status of AM CVn stars. We show that the minimum orbital periods of the semidetached binaries depend weakly on the total mass of the components and the evolutionary phase of the donor at the time of Roche lobe overflow and are 9–10 min. The differences in the mass transfer rates after P _orb reaches its minimum in the range P _orb ≈ 10–40 min do not exceed a factor of ～2.5. For P _orb ? 20 min, the mass-losing stars are weakly degenerate homogeneous cooling objects; the He, C, N, O, and Ne abundances depend on the evolutionary phase at which Roche lobe overflow occurred. For the binaries that are currently believed to be the most probable candidates for AM CVn stars with helium donors, Y ? 0.4, X _C ? 0.3, X _O ? 0.25, and X _N ? 0.5 × 10^?2. In the binaries under consideration, once P _orb ≈ 40 min has been reached, the mass loss time scale begins to exceed the thermal time scale of the donors, the latter begin to contract, their matter becomes degenerate, and the populations of AMCVn stars with white-dwarf and helium-star progenitors of their donors probably merge together.     

Kinematics of the Galaxy from OB Stars with Data from the Gaia DR2 Catalogue

We have selected and analyzed a sample of OB stars with known line-of-sight velocities determined through ground-based observations and with trigonometric parallaxes and propermotions from the Gaia DR2 catalogue. Some of the stars in our sample have distance estimates made from calcium lines. A direct comparison with the trigonometric distance scale has shown that the calcium distance scale should be reduced by 13%. The following parameters of the Galactic rotation curve have been determined from 495 OB stars with relative parallax errors less than 30%: (U, V,W)_⊙ = (8.16, 11.19, 8.55)± (0.48, 0.56, 0.48) km s^?1, Ω₀ = 28.92 ± 0.39 km s^?1 kpc^?1, Ω'₀ = ?4.087 ± 0.083 km s^?1 kpc^?2, and Ω″ ₀ = 0.703 ± 0.067 km s?1 kpc^?3, where the circular velocity of the local standard of rest is V0 = 231 ± 5 km s^?1 (for the adopted R₀ = 8.0 ± 0.15 kpc). The parameters of the Galactic spiral density wave have been found from the series of radial, V_R, residual tangential, ΔV_circ, and vertical, W, velocities of OB stars by applying a periodogram analysis. The amplitudes of the radial, tangential, and vertical velocity perturbations are f_R = 7.1± 0.3 km s^?1, f_θ = 6.5 ± 0.4 km s^?1, and f_W = 4.8± 0.8 km s^?1, respectively; the perturbation wavelengths are λ_R = 3.3 ± 0.1 kpc, λ_θ = 2.3 ± 0.2 kpc, and λ_W = 2.6 ± 0.5 kpc; and the Sun’s radial phase in the spiral density wave is (χ_⊙)_R = ?135? ± 5?, (χ_⊙)θ = ?123? ± 8?, and (χ_⊙)_W = ?132? ± 21? for the adopted four-armed spiral pattern.