Image‐intensified video results from the 1998 Leonid shower: I. Atmospheric trajectories and physical structure

Abstract— Two‐station electro‐optical observations of the 1998 Leonid shower are presented. Precise heights and light curves were obtained for 79 Leonid meteors that ranged in brightness (at maximum luminosity) from +0.3 to +6.1 astronomical magnitude. The mean photometric mass of the data sample was 1.4 × 10^?6 kg. The dependence of astronomical magnitude at peak luminosity on photometric mass and zenith angle was consistent with earlier studies of faint sporadic meteors. For example, a Leonid meteoroid with a photometric mass of ～1.0 × 10‐⁷ kg corresponds to a peak meteor luminosity of about +4.5 astronomical magnitudes. The mean beginning height of the Leonid meteors in this sample was 112.6 km and the mean ending height was 95.3 km. The highest beginning height observed was 144.3 km. There is relatively little dependence of either the first or last heights on mass, which is indicative of meteoroids that have clustered into constituent grains prior to the onset of intensive grain ablation. The height distribution, combined with numerical modelling of the ablation of the meteoroids, suggests that silicate‐like materials are not the principal component of Leonid meteoroids and hints at the presence of a more volatile component. Light curves of many Leonid meteors were examined for evidence of the physical structure of the associated meteoroids: similar to the 1997 Leonid meteors, the narrow, nearly symmetric curves imply that the meteoroids are not solid objects. The light curves are consistent with a dustball structure.     

Atmospheric behavior and extreme beginning heights of the thirteen brightest photographic Leonid meteors from the ground‐based expedition to China

Abstract— Precise atmospheric trajectories including dynamic and photometric data on thirteen of the brightest Leonid fireballs have been determined from the double‐station photographic observations of Leonid meteors during the ground‐based expedition to China in 1998 November. the expedition was organized as a collaboration between the dutch and chinese academy of sciences and was supported by the leonid multi‐instrument aircraft campaign (mac) program (jenniskens and butow, 1999). All data presented here were taken at Xinglong Observatory and at a remote station, Lin Ting Kou near Beijing, on the night of 1998 November 16/17. At the Xinglong station, photographic cameras were accompanied by an all‐sky television camera equipped with an image intensifier and 15 mm fish‐eye objective in order to obtain precise timings for all observed meteors up to magnitude +2. Whereas beginning heights of photographed meteors are all lower than 130 km, those observed by the all‐sky television system are at ～160 km, and for three brightest events, even > 180 km. Such high beginnings for meteors have never before been observed. We also obtained a precise dynamic single‐body solution for the Leonid meteor 98003, including the ablation coefficient, which is an important material and structural quantity (0.16 s² km^?2). From this and from known photometry, we derived a density of this meteoroid of 0.7 g/cm³. Also, all PE coefficients indicate that these Leonid meteors belonged to the fireball group IIIB, which is typical for the most fragile and weak interplanetary bodies. From a photometric study of the meteor lightcurves, we found two typical shapes of light curves for these Leonid meteors.     

The luminosity functions of the 1969 Perseid and Orionid meteor showers

Visual counts of the 1969 Perseid and Orionid meteor showers are presented, comprising 288 Perseids and 56 Orionids. On the basis of the maximum-likelihood method of determining the power law luminosity function index, we derive s ≡ 1 + 2.5 log(r) = 1.56 ± 0.06 for the Perseids with m_v = +1 to ?5, and s ≈ 1.85 ± 0.1 for the Orionids with m_v = +2 to ?3. These values are somewhat lower than those found by other observers, but we confirm the approximate power law nature of the luminosity functions. Under the assumption that the masses of visual meteors are proportional to a power law function of the luminosities, this implies power law mass functions. If mass is directly proportional to luminosity, we have power law mass functions with the indices s given above.     

First results from video spectroscopy of 1998 Leonid meteors

Abstract— We report spectroscopic observations of meteors made from the FISTA aircraft on 1998 November 17 as a part of the Leonid multi-instrument aircraft campaign. Low-resolution spectra of 119 meteors of apparent visual magnitudes from +3 to ?4, corresponding to meteoroid masses from 10^?6 to 10^?3 kg, were obtained. After analyzing a representative sample of the spectra and comparing them to the spectra of Perseid meteors from the Ondrejov archive, the following conclusions were reached: Leonid meteoroids are very loose and disintegrate easily in the atmosphere. This leads to much faster evaporation of volatile Na than of other elements, an effect which is not observed in the Perseid meteors. Relative bulk abundances of Mg, Fe, Ca, and Na in Leonid meteors are nearly CI-chondritic within the uncertainty of the method (factor of 3). Smaller meteoroids tend to be poorer in Na, which is true also for Perseid meteors. Most meteoric vapor emissions could be reasonably well explained with the temperature of 4500 K. High-temperature meteoric emissions (Ca⁺, Mg⁺) are present only in bright meteors. Leonid spectra are very rich in atmospheric emissions of O, N, and N₂, even at high altitudes and in faint meteors. These emissions are therefore not connected with the meteor shock wave. Thermal continuum is also present in the spectra. Organic material was not revealed.     

Estimation of the galactic spiral pattern speed from Cepheids

To study the peculiarities of the Galactic spiral density wave, we have analyzed the space velocities of Galactic Cepheids with propermotions from the Hipparcos catalog and line-of-sight velocities from various sources. First, based on the entire sample of 185 stars and taking R ₀ = 8 kpc, we have found the components of the peculiar solar velocity (u _⊙, v _⊙) = (7.6, 11.6) ± (0.8, 1.1) km s^?1, the angular velocity of Galactic rotation Ω₀ = 27.5 ± 0.5 km s^?1 kpc^?1 and its derivatives Ω′₀ = ?4.12 ± 0.10 km s^?1 kpc^?2 and Ω″₀ = 0.85 ± 0.07 km s^?1 kpc^?3, the amplitudes of the velocity perturbations in the spiral density wave f _R = ?6.8 ± 0.7 and f _θ = 3.3 ± 0.5 km s^?1, the pitch angle of a two-armed spiral pattern (m = 2) i = ?4.6° ± 0.1° (which corresponds to a wavelength λ = 2.0 ± 0.1 kpc), and the phase of the Sun in the spiral density wave χ_⊙ = ?193° ± 5°. The phase χ_⊙ has been found to change noticeably with the mean age of the sample. Having analyzed these phase shifts, we have determined the mean value of the angular velocity difference Ω _p ? Ω, which depends significantly on the calibrations used to estimate the individual ages of Cepheids. When estimating the ages of Cepheids based on Efremov’s calibration, we have found |Ω _p ? Ω₀| = 10 ± 1_stat ± 3_syst km s^?1 kpc^?1. The ratio of the radial component of the gravitational force produced by the spiral arms to the total gravitational force of the Galaxy has been estimated to be f _r0 = 0.04 ± 0.01.     

Kinematics of the Gould Belt based on open clusters

We have redetermined the kinematic parameters of the Gould Belt using currently available data on the motion of nearby young (log t < 7.91) open clusters, OB associations, and moving stellar groups. Our modeling shows that the residual velocities reach their maximum values of ?4 km s^?1 for rotation (in the direction of Galactic rotation) and +4 km s^?1 for expansion at a distance from the kinematic center of ≈300 pc. We have taken the following parameters of the Gould Belt center: R ₀ = 150 pc and l ₀ = 128°. The whole structure is shown to move relative to the local standard of rest at a velocity of 10.7 ± 0.7 km s^?1 in the direction l = 274° ± 4° and b = ?1° ± 3°. Using the derived rotation velocity, we have estimated the virial mass of the Gould Belt to be 1.5 × 10⁶ M _⊙.     

Flux of very faint Leonid meteors observed with a 3 meter liquid mirror telescope intensified charge‐coupled device system

Abstract— We have used a 3.0 m diameter liquid mirror telescope (LMT) coupled to a microchannel plate image‐intensified charge‐coupled device (CCD) detector to study the 1999 Leonid meteor shower. This is the largest aperture optical instrument ever utilized for meteor detection. While the observing system is sensitive down to stars of +18 astronomical magnitude under optimum conditions, when corrections for meteor motion are applied the majority of the meteors collected fall in the absolute magnitude range from +5 to +10, corresponding to photometric masses from about 10^?7 to 10^?9 kg. This is largely due to the fact that the field of view of the LMT was only 0.28°, so that only a small portion of the luminous meteor trail was recorded. While the flux of these small (1.4 times 10^?9 kg) Leonid meteors is low (on the order of one Leonid meteor per hour per square kilometer perpendicular to the Leonid), we do have clear evidence that the Leonid stream contains particles in the mass range studied here. The data showed a possibly significant peak in Leonid flux (9.3 ± 3.5) for the 1 h period from 11:00 to 12:00 u.t. 1999 November 17 (solar longitude 234.653 to 234.695, epoch 2000.0), although the main trend of these results is a broad low‐level Leonid activity. There is evidence that small meteoroids are more widely distributed in the Leonid stream, as would be expected from cometary ejection stream models. As would be expected from an extrapolation of mass distribution indices for brighter meteors, the vast majority of meteors at this size are sporadic. The LMT is a powerful detector of sporadic meteors, with an average non‐Leonid detection rate of more than 140 meteor events per hour.     

Airborne intensified charge-coupled device observations of the 1998 Leonid shower

Abstract— We have used dual coaxial microchannel plate image-intensified monochrome charge-coupled device (CCD) detectors run at standard NTSC frame rates (30 frames per second, fps) to study the Leonid meteor shower on 1998 November 17 from an airborne platform at an altitude of ～13 km. These observations were part of NASA's 1998 Leonid multi-instrument aircraft campaign (MAC). The observing systems had fields of view (width) of 16.3° and 9.5°, and limiting stellar sensitivities of +8.3^m and +8.9^m. During 12 h of recording, 230 meteors were detected, of which 65 were Leonid meteors. Light curves are presented for 53 of these meteors. The magnitudes at peak brightness of the meteors investigated were generally in the range from +4.0^m to +6.0^m. The mass distribution indices for the two samples are 1.67 and 1.44, with the former being based on the wider field of view dataset. The light curves were skewed with the brightest point towards the beginning of the meteor trail. The F parameter for points one magnitude below maximum luminosity had a mean value of 0.47 for the wider field system and 0.37 for the more sensitive narrower field system. We provide leading and trailing edge light curve slopes for each meteor as another indication of light curve shape. There were few obvious flares on the light curves, indicating that in-flight fragmentation into a large number of grains is not common. There is variability in light curve shape from meteor to meteor. The light curves are inconsistent with single, compact body meteor theory, and we interpret the data as indicative of a two-component dustball model with metal or silicate grains bonded by a lower boiling point, possibly organic, substance. The variation in light curve shape may be indicative of differences in mass distribution of the constituent grains. We provide trail length vs. magnitude data. There is only a slight hint of a bend at +5^m in the data, representing the difference between meteors that have broken into a cluster of grains prior to grain ablation, and those that continue to fragment during the grain ablation phase. Two specific meteors show interesting light curve features. One meteor is nebulous in appearance, with significant transverse width. The apparent light production region extends for 450 m from the center of the meteor path. Another meteor has several main fragments, and evidence of significant separated fragments. We offer several suggestions for improvements for the 1999 Leonid MAC light curve experiment.     

Galactic rotation parameters from data on open star clusters

Currently available data on the field of velocities V _r , V _l , V _b for open star clusters are used to perform a kinematic analysis of various samples that differ by heliocentric distance, age, and membership in individual structures (the Orion, Carina-Sagittarius, and Perseus arms). Based on 375 clusters located within 5 kpc of the Sun with ages up to 1 Gyr, we have determined the Galactic rotation parameters ω ₀ = ?26.0 ± 0.3 km s^?1 kpc^?1, ω′₀ = 4.18 ± 0.17 km s^?1 kpc^?2, ω″₀ = ?0.45 ± 0.06 km s^?1 kpc^?3, the system contraction parameter K = ?2.4 ± 0.1 km s^?1 kpc^?1, and the parameters of the kinematic center R ₀ = 7.4 ± 0.3 kpc and l ₀ = 0° ± 1°. The Galactocentric distance R ₀ in the model used has been found to depend significantly on the sample age. Thus, for example, it is 9.5 ± 0.7 and 5.6 ± 0.3 kpc for the samples of young (≤50 Myr) and old (>50 Myr) clusters, respectively. Our study of the kinematics of young open star clusters in various spiral arms has shown that the kinematic parameters are similar to the parameters obtained from the entire sample for the Carina-Sagittarius and Perseus arms and differ significantly from them for the Orion arm. The contraction effect is shown to be typical of star clusters with various ages. It is most pronounced for clusters with a mean age of ≈100 Myr, with the contraction velocity being Kr = ?4.3 ± 1.0 km s^?1.     

Video Observations of the 2006 Leonid Outburst

We carried out double station observations of the Leonid meteor shower outburst, which occurred in the morning hours of November 19, 2006. Using image-intensified cameras we recorded approximately 100 Leonid meteors. As predicted, the outburst was rich especially in fainter meteors. The activity profile shows that the peak of the outburst occurred at 4:40 ± 0:05 UT. The maximum reached flux was 0.03 meteoroids km⁻² hod⁻¹ for meteors brighter than +6.5 magnitude.     

Video Observations Of Leonids 1999

We analyse data obtained by different ground-based video camera systems during the 1999 Leonid meteor storm. We observe similar activity profiles at nearby observing sites, but significant differences over distances in the order of 4,000 km. The main peak occured at 02:03 UT (λ_⊙=235.286, J2000, corrected for the time of the topocentric stream encounter). At the Iberian peninsula quasi-periodic activity fluctuations with a period of about 7 min were recorded. The camera in Jordan detected a broad plateau of activity at 01:39–01:53 UT, but no periodic variations. The Leonid brightness distribution derived from all cameras shows a lack of faint meteors with a turning point close to +3^m, which corresponds to meteoroids of approximately 10^-3 g. We find a pin-point radiant at αalpha=153.65 ±0.1, δ=21.80 ±0. (λ_⊙=235.290). The radiant positionis identical before and after the storm, and also during the storm no driftis observed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Infrared photometry of Nova Delphini 2013 (=V339 Del) in the first sixty days after its outburst

The results of JHKLM photometry for Nova Delphini 2013 obtained in the first sixty days after its outburst are analyzed. Analysis of the energy distribution in a wide spectral range (0.36–5 µm) has shown that the source mimics the emission of normal supergiants of spectral types B5 and A0 for two dates near its optical brightness maximum, August 15.94 UT and August 16.86 UT, respectively. The distance to the nova has been estimated to be D ≈ 3 kpc. For these dates, the following parameters have been estimated: the source’s bolometric fluxes ～9 × 10^?7 and ～7.2 × 10^?7 erg s^?1 cm^?2, luminosities L ≈ 2.5 × 10⁵ L _⊙ and ≈2 × 10⁵ L _⊙, and radii R ≈ 6.3 × 10¹² and ≈1.2 × 10¹³ cm. The nova’s expansion velocity near its optical brightness maximum was ～700 km s^?1. An infrared (IR) excess associated with the formation of a dust shell is shown to have appeared in the energy distribution one month after the optical brightness maximum. The parameters of the dust component have been estimated for two dates of observations, JD2456557.28 (September 21, 2013) and JD2456577.18 (October 11, 2013). For these dates, the dust shell parameters have been estimated: the color temperatures ≈1500 and ≈1200 K, radii ≈6.5 × 10¹³ and 1.7 × 10¹⁴ cm, luminosities ～4 × 10³ L _⊙ and ～1.1 × 10⁴ L _⊙, and the dust mass ～1.6 × 10²⁴ and ～10²⁵ g. The total mass of the material ejected in twenty days (gas + dust) could reach ～1.1 × 10^?6 M _⊙. The rate of dust supply to the nova shell was ～8 × 10^?8 M _⊙ yr^?1. The expansion velocity of the dust shell was about 600 km s^?1.     

Leonid outburst activity 1996: A broad structure and a first occurrence of a narrow peak of fainter meteors

Abstract— In 1996, a broad outburst structure of bright Leonid meteors similar to the 1995 and the 1994 displays (Jenniskens, 1996; Langbroek, 1996b) was observed. In addition, a second narrow outburst structure of fainter meteors, which will be reported and discussed in this paper, has with certainty been observed. This observation marks the first detection of such a narrow structure in the new series of Leonid outbursts. It has a similar exponential activity behaviour and similar emphasis on fainter meteors as shown by the 1866 and 1966 Leonid storm structures. Similar narrow peaks have been observed in 1965 and 1969 (Jenniskens, 1995, 1996). The broad 1996 structure of bright meteors peaked at November 17.31 ± 0.04 (λ 235°.28 ± 0.04 (2000.0)). The additional narrow structure peaked at November 17.20 ± 0.01 (λ 235°.172 ± 0.007). The occurrence of the narrow peak can best be explained as a first modest sign of presence of the meteoroid structure that should be responsible for the expected meteor storm activity of the Leonids in 1998–1999. The appearance 0.°085 before the node of 55P/Tempel-Tuttle suggests that the expected 1998–1999 Leonid storms might peak just before passage through the node of the comet.     

Kinematic analysis of solar-neighborhood stars based on RAVE4 data

We consider stars with radial velocities, proper motions, and distance estimates from the RAVE4 catalogue. Based on a sample of more than 145 000 stars at distances r < 0.5 kpc, we have found the following kinematic parameters: \({\left( {U,{\kern 1pt} V,{\kern 1pt} W} \right)_ \odot }\) = (9.12, 20.80, 7.66) ± (0.10, 0.10, 0.08) km s^?1, Ω₀ = 28.71 ± 0.63 km s^?1 kpc^?1, and Ω₀ ^’ = ?4.28 ± 0.11 km s^?1 kpc^?2. This gives the linear rotation velocity V ₀ = 230 ± 12 km s^?1 (for the adopted R ₀ = 8.0 ± 0.4 kpc) and the Oort constants A = 17.12 ± 0.45 km s^?1 kpc^?1 and B = ?11.60 ± 0.77 km s^?1 kpc^?1. The 2D velocity distributions in the UV, UW, and VW planes have been constructed using a local sample, r < 0.25 kpc, consisting of ~47 000 stars. A difference of the UV velocity distribution from the previously known ones constructed from a smaller amount of data has been revealed. It lies in the fact that our distribution has an extremely enhanced branch near the Wolf 630 peak. A previously unknown peak at (U, V) = (?96, ?10) km s^?1 and a separate new feature in the Wolf 630 stream, with the coordinates of its center being (U, V) = (30, ?40) km s^?1, have been detected.     

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.     

Kinematics of Hot Subdwarfs from the Gaia DR2 Catalogue

We have studied the kinematic properties of the candidates for hot subdwarfs (HSDs) selected by Geier et al. from theGaiaDR2 catalogue. We have used a total of 12 515 stars with relative trigonometric parallax errors less than 30%. The HSDs are shown to have different kinematics, depending on their positions on the celestial sphere. For example, the sample of low-latitude (|b| < 20°) HSDs rotates around the Galactic center with a linear velocity V0 = 221 ± 5 km s^?1. This suggests that they belong to the Galactic thin disk. At the same time, they lag behind the local standard of rest by ΔV_☉ ~ 16 km s^?1 due to the asymmetric drift. The high-latitude (|b| ≥ 20°) HSDs rotate with a considerably lower velocity, V_☉ = 168 ^± 6 km s^?1. Their lagging behind the local standard of rest is already ΔV_☉ ～ 40 km s^?1. Based on the entire sample of 12 515 HSDs, we have found a positive rotation around the x axis significantly differing from zero with an angular velocity ω₁ = 1.36±0.24 km s^?1 kpc^?1. We have studied the samples of HSDs that are complete within r < 1.5 kpc. Based on them, we have traced the evolution of the parameters of the residual velocity ellipsoid as a function of both latitude |b| and coordinate |z|. The following vertical disk scale heights have been found: h = 180 ± 6 and 290 ± 10 pc from the low- and high-latitude HSDs, respectively. A new estimate of the local stellar density Σ_out = 53 ± 4 M☉ kpc^?2 has been obtained for z_out = 0.56 kpc from the high-latitude HSDs.

     

Trigonometric parallaxes and a kinematically adjusted distance sale for OB associations

By directly comparing the photometric distances of Blaha and Humphreys (1989) (BH) to OB associations and field stars with the corresponding Hipparcos trigonometric parallaxes, we show that the BH distance scale is overestimated, on average, by 10–20%. This result is independently corroborated by applying the rigorous statistical-parallax method and its simplified analog (finding a kinematically adjusted rotation-curve solution from radial velocities and proper motions) to a sample of OB associations. These two methods lead us to conclude that the BH distance scale for OB associations should be shrunk, on average, by 11±6 and 24±10%, respectively. Kinematical parameters have been determined for the system of OB associations: u ₀ = 8.2 ± 1.3 km s^?1, v ₀ = 11.9 ± 1.1 km s^?1, w ₀ = 9.5 ± 0.9 km s^?1, σ _u = 8.2 ± 1.1 km s^?1, σ _v = 5.8 ± 0.8 km s^?1, σ _w = 5.0 ± 0.8 km s^?1, Ω₀ = 29.1 ± 1.0 km s^?1 kpc^?1, Ω₀′ = ?4.57 ± 0.20 km s^?1 kpc^?2, and Ω₀″ = 1.32 ± 0.14 km s^?1 kpc^?3. The distance scale for OB associations reduced by 20% matches the short Cepheid distance scale (Berdnikov and Efremov 1985; Sitnik and Mel’nik 1996). Our results are a further argument for the short distance scale in the Universe.     

Galactic kinematics from data on open star clusters from the MWSC catalogue

Open star clusters from the MWSC (Milky Way Star Clusters) catalogue have been used to determine the Galactic rotation parameters. The circular rotation velocity of the solar neighborhood around the Galactic center has been found from data on more than 2000 clusters of various ages to be V ₀ = 236 ± 6 km s^?1 for the adopted Galactocentric distance of the Sun R ₀ = 8.3 ± 0.2 kpc. The derived angular velocity parameters are Ω ₀ = 28.48 ± 0.36 km s^?1 kpc^?1, Ω’₀ = ?3.50 ± 0.08 km s^?1 kpc_?2, and Ω″₀ = 0.331 ± 0.037 km s^?1 kpc^?3. The influence of the spiral density wave has been detected only in the sample of clusters younger than 50 Myr. For these clusters the amplitudes of the tangential and radial velocity perturbations are f ^θ = 5.6 ± 1.6 km s^?1 and f _R = 7.7 ± 1.4 km s^?1, respectively; the perturbation wavelengths are λ _θ = 2.6 ± 0.5 kpc (i _θ = ^?11? ± 2?) and λ _R = 2.1 ± 0.5 kpc (i _R = ?9? ± 2?) for the adopted four-armed model (m = 4). The Sun’s phase in the spiral density wave is (χ_⊙)_θ = ?62? ± 9? and (χ_⊙)_R = ?85? ± 10? from the residual tangential and radial velocities, respectively.