Stellar Membership Research Articles

Stellar Membership of RCW 34 HII Star-Forming Region

The Vel OB1 association, located at a distance of 1.5-1.9 kpc, is one of the most actively studied star formation regions in the southern hemisphere. Our study focuses on an “Arclike” structure within the Vel OB1 association, which may have been formed from a shock wave caused by a supernova explosion about 2-2.5 Myr ago. Optical Gaia data revealed probable stellar members only in the southern part of the Arclike structure, where the RCW34 HII (IRAS 08546-4254) region is located. In contrast, the northeastern part of the Arclike structure, in the vicinity of the IRAS 08563-4225 source, hosts more embedded stellar population. Using near-infrared astrometric and photometric data from the 2MASS survey, we identified well-defined clusters of young stellar objects in the vicinity of both IRAS 08546-4254 and IRAS 08563-4225. The members of the cluster associated with IRAS 08563-4225 exhibit stronger infrared excess.

Observational bias and young massive cluster characterization − II. Can Gaia accurately observe young clusters and associations?

ABSTRACT Observations of clusters suffer from issues such as completeness, projection effects, resolving individual stars, and extinction. As such, how accurate measurements and conclusions are likely to be? Here, we take cluster simulations (Westerlund2- and Orion-type), synthetically observe them to obtain luminosities, accounting for extinction, and the inherent limits of Gaia, then place them within the real Gaia DR3 catalogue. We then attempt to rediscover the clusters at distances of between 500 and 4300 pc. We show the spatial and kinematic criteria that are best able to pick out the simulated clusters, maximizing completeness, and minimizing contamination. We then compare the properties of the ‘observed’ clusters with the original simulations. We looked at the degree of clustering, the identification of clusters and subclusters within the data sets, and whether the clusters are expanding or contracting. Even with a high level of incompleteness (e.g. $\lt 2~{{\ \rm per\ cent}}$ stellar members identified), similar qualitative conclusions tend to be reached compared to the original data set, but most quantitative conclusions are likely to be inaccurate. Accurate determination of the number, stellar membership, and kinematic properties of subclusters are the most problematic to correctly determine, particularly at larger distances due to the disappearance of cluster substructure as the data become more incomplete, but also at smaller distances where the misidentification of asterisms as true structure can be problematic. Unsurprisingly, we tend to obtain better quantitative agreement of properties for our more massive Westerlund2-type cluster. We also make optical style images of the clusters over our range of distances.

Albatross: a scalable simulation-based inference pipeline for analysing stellar streams in the Milky Way

ABSTRACT Stellar streams are potentially a very sensitive observational probe of galactic astrophysics, as well as the dark matter population in the Milky Way. On the other hand, performing a detailed, high-fidelity statistical analysis of these objects is challenging for a number of key reasons. First, the modelling of streams across their (potentially billions of years old) dynamical age is complex and computationally costly. Secondly, their detection and classification in large surveys such as Gaia renders a robust statistical description regarding e.g. the stellar membership probabilities, challenging. As a result, the majority of current analyses must resort to simplified models that use only subsets or summaries of the high quality data. In this work, we develop a new analysis framework that takes advantage of advances in simulation-based inference techniques to perform complete analysis on complex stream models. To facilitate this, we develop a new, modular dynamical modelling code sstrax for stellar streams that is highly accelerated using jax. We test our analysis pipeline on a mock observation that resembles the GD1 stream, and demonstrate that we can perform robust inference on all relevant parts of the stream model simultaneously. Finally, we present some outlook as to how this approach can be developed further to perform more complete and accurate statistical analyses of current and future data.

Collinder 135 and UBC 7: A physical pair of open clusters

Context. Given the closeness of the two open clusters Collinder 135 and UBC 7 on the sky, we investigate the possibility that the two clusters are physically related. Aims. We aim to recover the present-day stellar membership in the open clusters Cr 135 and UBC 7 (300 pc from the Sun) in order to constrain their kinematic parameters, ages, and masses and to restore their primordial phase space configuration. Methods. The most reliable cluster members are selected with our traditional method modified for the use of Gaia DR2 data. Numerical simulations use the integration of cluster trajectories backwards in time with our original high-order Hermite4 code φ−GRAPE. Results. We constrain the age, spatial coordinates, velocities, radii, and masses of the clusters. We estimate the actual separation of the cluster centres equal to 24 pc. The orbital integration shows that the clusters were much closer in the past if their current line-of-sight velocities are very similar and the total mass is more than seven times larger than the mass of the most reliable members. Conclusions. We conclude that the two clusters Cr 135 and UBC 7 might very well have formed a physical pair based on the observational evidence as well as numerical simulations. The probability of a chance coincidence is only about 2%.

The globular cluster M10: reassessment of stellar membership, distance, and age using its variable and HB stars

ABSTRACT Time-series VI CCD photometry of the globular cluster M10 (NGC 6254) is employed to perform a detailed identification, inspection of their light curves, their classification, and their cluster membership, of all the known variables reported up to 2018. The membership analysis is based on the $Gaia$-DR2 positions and proper motions. The metallicity of the cluster is estimated based on the sole RRc star known in the cluster. The Fourier decomposition of its light curve leads to [Fe/H]$_{\rm ZW}$ = $-1.59 \pm 0.23$ dex. The mean cluster distance, estimated by several independent methods, is $5.0 \pm 0.3$ kpc. A multiapproach search in a region of about 10$\times$10 arcmin$^2$ around the cluster revealed three new variables, one SX Phe (V35) and two sinusoidal variables on the red giant branch (RGB) of unclear classification (V36 and V37). Modelling the HB stars is very sensitive to the stellar hydrogen shell mass, which surrounds the 0.50 $\mathrm{ M}_{\odot }$ helium core. To match the full stretch of the HB population, a range of total mass of 0.56–0.62 $\mathrm{ M}_{\odot }$ is required. These models support a distance of 5.35 kpc and an age of about 13 Gyr, and hint to some individual variation of the mass-loss on the upper RGB, perhaps caused by the presence of closed magnetic field in red giants.

Blue straggler populations of seven open clusters with Gaia DR2

ABSTRACT Blue straggler stars (BSS) are well studied in globular clusters but their systematic study with secure membership determination is lacking in open clusters. We use Gaia DR2 data to determine accurate stellar membership for four intermediate-age open clusters, namely Melotte 66, NGC 2158, NGC 2506, and NGC 6819, and three old open clusters, namely, Berkeley 39, NGC 188, and NGC 6791, to subsequently study their BSS populations. The BSS radial distributions of five clusters, namely Melotte 66, NGC 188, NGC 2158, NGC 2506, and NGC 6791, show bimodal distributions, placing them with Family II globular clusters that are of intermediate dynamical ages. The location of minima, rmin, in the bimodal BSS radial distributions, varies from 1.5rc to 4.0rc, where rc is the core radius of the clusters. We find a positive correlation between rmin and Nrelax, the ratio of cluster age to the current central relaxation time of the cluster. We further report that this correlation is consistent in its slope, within the errors, with the slope of the globular cluster correlation between the same quantities, but with a slightly higher intercept. This is the first example in open clusters that shows BSS radial distributions as efficient probes of dynamical age. The BSS radial distributions of the remaining two clusters, Berkeley 39 and NGC 6819, are flat. The estimated Nrelax values of these two clusters, however, indicate that they are dynamically evolved. Berkeley 39 especially has its entire BSS population completely segregated to the inner regions of the cluster.

Vilnius photometry and Gaia astrometry of Melotte 105

Archival Vilnius CCD photometric observations are presented for the heavily reddened star cluster Melotte 105, resulting in colour-magnitude diagrams and spectral class estimates. There is considerable lack of agreement between studies for reddening, age, and distance for this cluster explaining why the archival data are being made available by this paper. The derived reddening E(B-V =0.34 +/- 0.04 mag and the distance V-M =12.9 +/- 0.3 mag directly from the Vilnius photometry. The Gaia Data Release 2 (DR2) and Vilnius photometric data of the cluster were used to estimate the structural parameters of the cluster, probability of stellar membership in the cluster, the distance modulus and the cluster age. Lack of $Y$ band observations prevented determination of metal abundance. The values of the colour excess and distance module are determined by two different methods (i.e., Q and Zero Age Main Sequence, or ZAMS, methods). A distance modulus of 12.85 +/- 0.07 mag was derived by ZAMS fitting, in good agreement with the above estimate. ZAMS fitting indicates a reddening of 0.403 +/- 0.02 mag, within two sigma of the estimate above. The cluster's metallicity and age are estimated to be 0.24 dex and 240 +/-25 Myr, respectively. The derived mass function is in good agreement with the Salpeter slope. The cluster space velocity components (U, V, W) were determined as (-3.90 +/- 3.34, -13.76 +/- 5.69, +3.45 +/- 0.41) km/s. Perigalactic and apogalactic distances were obtained as R_{p} = 6.85 and R_{a} = 7.44 kpc respectively. The maximum vertical distance from the Galactic plane was calculated as Z_{max} = 84 pc and the eccentricity of the orbit was determined as e = 0.042.

Chemical abundances of open clusters from high-resolution infrared spectra – II. NGC 752

ABSTRACT We present a detailed near-infrared chemical abundance analysis of 10 red giant members of the Galactic open cluster NGC 752. High-resolution (R ≃ 45000) near-infrared spectral data were gathered with the Immersion Grating Infrared Spectrograph, providing simultaneous coverage of the complete H and K bands. We derived the abundances of H-burning (C, N, O), α (Mg, Si, S, Ca), light odd-Z (Na, Al, P, K), Fe-group (Sc, Ti, Cr, Fe, Co, Ni), and neutron-capture (Ce, Nd, Yb) elements. We report the abundances of S, P, K, Ce, and Yb in NGC 752 for the first time. Our analysis yields solar-metallicity and solar abundance ratios for almost all of the elements heavier than the CNO group in NGC 752. O and N abundances were measured from a number of OH and CN features in the H band, and C abundances were determined mainly from CO molecular lines in the K band. High-excitation $\rm{C\,\small {I}}$ lines present in both near-infrared and optical spectra were also included in the C abundance determinations. Carbon isotopic ratios were derived from the R-branch band heads of first overtone (2−0) and (3−1) 12CO and (2−0) 13CO lines near 23 440 Å and (3−1) 13CO lines at about 23 730 Å. The CNO abundances and 12C/13C ratios are all consistent with our giants having completed ‘first dredge-up’ envelope mixing of CN-cyle products. We independently assessed NGC 752 stellar membership from Gaia astrometry, leading to a new colour–magnitude diagram for this cluster. Applications of Victoria isochrones and MESA models to these data yield an updated NGC 752 cluster age (1.52 Gyr) and evolutionary stage indications for the programme stars. The photometric evidence and spectroscopic light element abundances all suggest that the most, perhaps all of the programme stars are members of the helium-burning red clump in this cluster.

The Stellar Membership of the Taurus Star-forming Region*

Abstract The high-precision astrometry from the second data release of the Gaia mission has made it possible to greatly improve the census of members of nearby clusters and associations. I have applied the Gaia data to the Taurus star-forming region, refining the sample of known members and identifying candidates for undiscovered members. The resulting samples of members and candidates provide the best constraints to date on the distribution of ages and the initial mass function (IMF) in Taurus. Several studies over the last 30 years have proposed the existence of a population of older stars (≳10 Myr) that is associated with the Taurus clouds. The data from Gaia demonstrate that such a population does not exist. Meanwhile, previous IMF estimates for small fields surrounding the Taurus aggregates have exhibited a surplus of K7–M0 stars (0.7–0.8 M ⊙) relative to star-forming clusters such as IC 348 and the Orion Nebula Cluster. However, that difference disappears when the new census of the entire region is considered, which should be complete for spectral types earlier than M6–M7 at A J < 1. Thus, there is little variation in the stellar IMF across the 3–4 orders of magnitude in stellar density that are present in nearby star-forming regions. Finally, I note that the proper motions of two previously known members, KPNO 15 and 2MASS J04355209 + 2255039, indicate that they may have been ejected from the same location within the L1536 cloud ∼7200 years ago.

Astrometric and photometric study of Dias 4, Dias 6, and other five open clusters using ground-based and Gaia DR2 data

We present a study of 7 southern open clusters based on UBVRI CCD photomety (Johnsons-Cousins system) and Gaia DR2 data. Dias 4, Dias 6 and four other clusters had UBVRI photometric observations determined for the first time. From the observational UBVRI data we obtained photometric membership probability estimates and, using the proper motions from the UCAC5 catalog, we also determined the kinematic membership. From Gaia DR2 astrometric data we determine the stellar membership using proper motions and parallaxes, taking into account the full covariance matrix. For both independent sets of data and membership we apply our non subjective multidimensional global optimization tool to fit theoretical isochrones to determine the distance, age, reddening, metallicity and binary fraction of the clusters. The results of the mean proper motions, distances and ages are in agreement, but the ones obtained from Gaia DR2 data are more precise in both membership selection and estimated parameters. In the case of NGC 6087, the Cepheid S Nor, member of the open cluster, was used to obtain an independent distance estimate, confirming the one determined by our fitting method. We also report a serendipitous discovery of two new clusters in the extended field near what was originally Dias 4.

Near-infrared study of open clusters Teutsch 10 and Teutsch 25

The astrophysical parameters have been estimated for two unstudied open star clusters Teutsch 10 and Teutsch 25 using the Two Micron All Sky Survey (2MASS) database. Radius is estimated as 4.5 arcmin for both clusters using radial density profiles. We have estimated proper motion values in both RA and DEC directions as 2.28±0.3 and -0.38±0.11 mas yr−1 for Teutsch 10 and 0.48±0.3 and 3.35±0.16 mas yr−1 for Teutsch 25 using PPMXL1http://vizer.cfa.harvard.edu/vizbin/VizieR?-source=I/317.1 catalog. By estimating the stellar membership probabilities, we have identified 30 and 28 most likely members for Teutsch 10 and Teutsch 25 respectively. We have estimated the reddening as E(B-V)=0.96±0.3 mag for Teutsch 10 and 0.58±0.2 mag for Teutsch 25, while the corresponding distances are 2.4±0.2 and 1.9±0.1 kpc. Ages of 70±10 Myr for Teutsch 10 and 900±100 Myr for Teutsch 25 are estimated using the theoretical isochrones of metallicity Z = 0.019. The mass function slopes are derived as 1.23±0.30 and 1.09±0.35 for Teutsch 10 and Teutsch 25 respectively. Estimated mass function slope for both the clusters are close to the Salpeter value (x=1.35) within the errors. Estimated values of dynamical relaxation time are found to be less than cluster’s age for these objects. This concludes that both objects are dynamically relaxed. The possible reason for relaxation may be due to dynamical evolution or imprint of star formation or both.

Membership Determination of van den Bergh Open Clusters vdB92, vdB146 (NGC 7129) and vdB150

In this paper we study some van den Bergh open clusters combining photometry and astrometry. A model which analyses the proper motion distribution and the stellar density is applied to find the kinematic parameters and stellar membership in the region of the open clusters vdB92, vdB146 (NGC 7129) and vdB150. The astrometric data are obtained from UCAC4 catalogue. The centre coordinates, the components of mean proper motion, the angular diameter and the astrometric members are reported, taking the following values: for vdB92: α=105.97281°±0.13113°, δ=−11.57814°±0.10575°, μαcosδ=−3.46±0.19mas/yr,μδ=1.27±0.19mas/yr, 34', 60 members; for vdB146: α=325.78423°±0.15297°, δ=66.13575°±0.02907°, μαcosδ=−2.71±0.25mas/yr, μδ=−3.32±0.25mas/yr, 9', 5 members; and for vdB150: α=332.22519°±0.06074°, δ=73.40232°±0.0.07789°, μαcosδ=3.07±0.90mas/yr, μδ=4.65±0.90mas/yr, 15', 7 members. The incidence of the proper motion errors in the determination of the cluster parameters and of the stellar membership is analysed and it is found that they do not significantly change. We finally compare the astrometric members with the photometric ones given in the literature.

Be STARS IN THE OPEN CLUSTER NGC 6830

ABSTRACT We report the discovery of two new Be stars, and re-identify one known Be star in the open cluster NGC 6830. Eleven Hα emitters were discovered using the Hα imaging photometry of the Palomar Transient Factory Survey. Stellar membership of the candidates was verified with photometric and kinematic information using 2MASS data and proper motions. The spectroscopic confirmation was carried out by using the Shane 3 m telescope at the Lick observatory. Based on their spectral types, three Hα emitters were confirmed as Be stars with Hα equivalent widths greater than −10 Å. Two objects were also observed by the new spectrograph spectral energy distribution-machine (SED-machine) on the Palomar 60-inch Telescope. The SED-machine results show strong Hα emission lines, which are consistent with the results of the Lick observations. The high efficiency of the SED-machine can provide rapid observations for Be stars in a comprehensive survey in the future.

SEARCHING FOR Be STARS IN THE OPEN CLUSTER NGC 663

We present Be star candidates in the open cluster NGC\,663, identified by H$\alpha$ imaging photometry with the Palomar Transient Factory Survey, as a pilot program to investigate how the Be star phenomena, the emission spectra, extended circumstellar envelopes, and fast rotation, correlate with massive stellar evolution. Stellar membership of the candidates was verified by 2MASS magnitudes and colors, and by proper motions. We discover 4 new Be stars and exclude one known Be star from being a member due to its inconsistent proper motions. The fraction of Be stars to member stars [N(Be)/N(members)] in NGC\,663 is 3.5\%. The spectral type of the 34 Be stars in NGC\,663 shows bimodal peaks at B0--B2 and B5--B7, which is consistent with the statistics in most star clusters. Additionally, we also discover 23 emission-line stars of different types, including non-member Be stars, dwarfs, and giants.

The implementation of binned Kernel density estimation to determine open clusters’ proper motions: validation of the method

Stellar membership determination of an open cluster is an important process to do before further analysis. Basically, there are two classes of membership determination method: parametric and non-parametric. In this study, an alternative of non-parametric method based on Binned Kernel Density Estimation that accounts measurements errors (simply called BKDE-e) is proposed. This method is applied upon proper motions data to determine cluster's membership kinematically and estimate the average proper motions of the cluster. Monte Carlo simulations show that the average proper motions determination using this proposed method is statistically more accurate than ordinary Kernel Density Estimator (KDE). By including measurement errors in the calculation, the mode location from the resulting density estimate is less sensitive to non-physical or stochastic fluctuation as compared to ordinary KDE that excludes measurement errors. For the typical mean measurement error of 7 mas/yr, BKDE-e suppresses the potential of miscalculation by a factor of two compared to KDE. With median accuracy of about 93%, BKDE-e method has comparable accuracy with respect to parametric method (modified Sanders algorithm). Application to real data from The Fourth USNO CCD Astrograph Catalog (UCAC4), especially to NGC 2682 is also performed. The mode of member stars distribution on Vector Point Diagram is located at $\mu_{\alpha}\cos\delta=-9.94\pm0.85$ mas/yr and $\mu_{\delta}=-4.92\pm0.88$ mas/yr. Although the BKDE-e performance does not overtake parametric approach, it serves a new view of doing membership analysis, expandable to astrometric and photometric data or even in binary cluster search.

A revisit to the regions of some van den Bergh open clusters using photometric and astrometric parameters

We present results of a study that combines photometry and astrometry for the open clusters vdB80, vdB85 and vdB130. We apply a model which analyses the proper motion distribution and the stellar density to find the kinematic parameters and stellar membership in the region of the mentioned open clusters. The astrometric data are obtained from UCAC4 catalogue. For each cluster, we report the centre coordinates, the components of mean proper motion, the angular diameter and the astrometric members. They are: vdB80: α=97°.73938±0°.00846,δ=-9°.66953±0°.01177,μαcosδ=-2.13±0.47mas/yr,μδ=-0.95±0.47mas/yr,12′, 15 members; vdB85: α=101°.71670±0°.00808,δ=1°.34392±0°.01253,μαcosδ=0.89±0.43mas/yr,μδ=3.24±0.43mas/yr,8′, 9 members; vdB130: α=304°.44001±0°.01407,δ=39°.32745±0°.00726,μαcosδ=-4.14±0.25mas/yr,μδ=-5.15±0.25mas/yr,9′, 9 members. We analyse the incidence of the proper motion errors in the determination of the cluster parameters and of the stellar membership and find that they are not significantly changed. We finally compare the astrometric members with the photometric ones given in the literature.

Proper motions of the optically visible open clusters based on the UCAC4 catalog

We present a catalog of mean proper motions and membership probabilities of individual stars for optically visible open clusters, which have been determined using data from the UCAC4 catalog in a homogeneous way. The mean proper motion of the cluster and the membership probabilities of the stars in the region of each cluster were determined by applying the statistical method in a modified fashion. In this study, we applied a global optimization procedure to fit the observed distribution of proper motions with two overlapping normal bivariate frequency functions, which also take the individual proper motion errors into account. For 724 clusters, this is the first determination of proper motion, and for the whole sample, we present results with a much larger number of identified astrometric member stars. Furthermore, it was possible to estimate the mean radial velocity of 364 clusters (102 unpublished so far) with the stellar membership using published radial velocity catalogs. These results provide an increase of 30% and 19% in the sample of open clusters with a determined mean absolute proper motion and mean radial velocity, respectively.

Photometric study of the open cluster – II. Stellar population and dynamical evolution in NGC 559

We present UBVRI photometry of stars in the field of the intermediate-age open cluster NGC 559. By determining the stellar membership probabilities derived through a photometric and kinematic study of the cluster, we identify the 22 most probable cluster members. These are used to obtain robust cluster parameters. The mean proper motion of the cluster is \mu_x = -3.29+/-0.35, \mu_y = -1.24+/-0.28 mas/yr. The radial distribution of the stellar surface density gives a cluster radius of 4'.5+/-0'.2 (3.2+/-0.2 pc). By fitting solar metallicity stellar isochrones to the colour-colour and colour-magnitude diagrams, we find a uniform cluster reddening of E(B-V) = 0.82+/-0.02. The cluster has an age of 224+/-25 Myr and is at a distance of 2.43+/-0.23 kpc. From the optical and near-infrared two-colour diagrams, we obtain colour excesses in the direction of the cluster E(V-K) = 2.14+/-0.02, E(J-K) = 0.37+/-0.01, and E(B-V)= 0.76+/-0.04. A total-to-selective extinction of Rv=3.5+/-0.1 is found in the direction of the cluster which is marginally higher than the normal value. We derive the luminosity function and the mass function for the cluster main sequence. The mass function slope is found to be -2.1+/-0.3. We find evidence of mass segregation in this dynamically relaxed cluster.

An Investigation of the Loss of Planet-Forming Potential in Intermediate-Sized Young Embedded Star Clusters

ABSTRACTA large fraction of stars forming in our galaxy are born within clusters embedded in giant molecular clouds. In these environments, the background UV radiation fields impinging upon circumstellar disks can often dominate over the radiation fields produced by each disk’s central star. As a result, this background radiation can drive the evaporation of circumstellar disks and lead to the loss of planet-forming potential within a cluster. This article presents a detailed analysis of this process for clusters whose stellar membership falls within the range of 100 ≤ N ≤ 1000. For these intermediate-sized clusters, the background UV field is often dominated by the most massive stellar member. Due to the steep slope of the initial mass function, the amount of background UV light that bathes clusters of similar size displays significant variance. As a result, we perform a statistical analysis of this problem by calculating distributions of FUV flux values impinging upon star/disk systems for several cluster scenarios. We find that in the absence of dust attenuation, giant planet formation would likely be inhibited in approximately half of the systems forming within intermediate-sized clusters, regardless of stellar membership. In contrast, the presence of dust can significantly lower this value, with the effect considerably more pronounced in more populated clusters.

KINEMATIC SIGNATURES OF SUBVIRIAL INITIAL CONDITIONS IN YOUNG CLUSTERS

Motivated by kinematic observations of young embedded clusters, this paper explores possible kinematic signatures produced by asphericity and departures from initial virial equilibrium in these systems. Specifically, the kinematic quantity that is measured and calculated in this study is the distribution of the line-of-sight velocities as a function of position along the cluster. Although clusters are found within a wide range of sizes, we focus on the regime with stellar membership $N \sim 10^3$. The gravitational potential of these young clusters is dominated by the gas, and the geometry of the gas distribution is generalized to include axisymmetric (and triaxial) forms. With this loss of symmetry, the kinematic results thus depend on viewing angle. This work also considers the stars to begin their trajectories with subvirial speeds, as indicated by observations of core motions in such clusters. Our results determine the conditions necessary for the kinematic signature to display interesting structure, i.e., a non-spherical potential, a viewing angle that is not along one of the principal axes, and subvirial starting conditions. We characterize the effects on this signature due to projection angle, initial stellar velocities, cluster elongation, and star formation efficiency. Finally, we compare our theoretical results to recent kinematic observations of the Orion Nebula Cluster; we find that the observations can be explained provided that the cluster is non-spherical, starts with subvirial initial velocities, and is not viewed along a principal axis.