Outskirts of Galaxy Clusters A1139, A1314, A1656, A2040, A 2052, A2107: Star-Formation Rate

We investigate the variation of the fraction of galaxies with suppressed star formation (M_K < ?21 _. ^m 5) and early-type galaxies (frac_E) of the “red sequence” along the projected radius in six galaxy clusters:Coma (A1656), A1139, and A1314 in the Leo supercluster region (z ≈ 0.037) and A2040, A2052, A2107 in the Hercules supercluster region (z ≈ 0.036). According to SDSS (DR10) data, frac_E is the highest in the central regions of galaxy clusters and it is, on the average, equal to 0.62 ± 0.03, whereas in the 2–3R/R_200c interval and beyond the R_sp ≈ 0.95 ± 0.04 R_200m radius that we inferred from the observed profile frac_E is minimal and equal to 0.25 ± 0.02. This value coincides with the estimate frac_E = 0.24 ± 0.01 that we inferred for field galaxies located between the Hercules and Leo superclusters at the same redshifts. We show that the fraction of galaxies with suppressed star formation decreases continuously with cluster radius from 0.87 ± 0.02 in central regions down to 0.43 ± 0.03 in the 2–3 R/R_200c interval and beyond R_sp, but remains, on the average, higher than 26% than the corresponding fraction for field objects. This decrease is especially conspicuous in the galaxy mass interval log M_* [M_⊙] = 9.5–10. We found that galaxies with ongoing star formation have average clustercentric distances 1.5–2.5 R/R_200c and that their radial-velocity dispersions are higher than those of galaxies with suppressed star formation.     

Peculiar motions of galaxy clusters and groups in the Hercules and Leo supercluster regions

We present the results of our study of the peculiar motions for 41 galaxy clusters and groups in the regions of the Hercules and Leo superclusters (0.027 < z < 0.045). For this purpose, based on SDSS (Data Release 8) data, we compiled a sample of early-type galaxies in the investigated clusters and groups, constructed their fundamental planes (FPs), and determined independent distances and peculiar velocities. The Hubble law between the radial velocities galaxy clusters and the distances derived from the FPs holds for the rich Hercules supercluster as a whole. At the same time, however, significant peculiar motions along the line of sight with rms deviations of 736 ± 50 and 584 ± 50 km s^?1 are observed within this supercluster and in its immediate neighborhood. In the poor Leo supercluster, the rms deviation is also large, 625 ± 70 km s^?1. Its Hubble diagram exhibits the approach of galaxy clusters and groups along the line of sight relative to the most massive cluster A1185. In the immediate neighborhood of the Leo supercluster (virtually in its field), the rms deviation of the peculiar velocities is minimal, 287 ± 60 km s^?1.     

The Ursa Major supercluster of galaxies: I. The luminosity function

Catalogs of bright galaxies in the central regions of 11 clusters in the Ursa Major supercluster are presented. Absolute and relative coordinates and total B and R magnitudes are given for each galaxy. Plates taken with the 2-m Tautenburg Observatory telescope and CCD images obtained with the 6-m and 1-m SAO telescopes are used. The luminosity functions (LFs) for galaxies in the cluster nuclei (3 Mpc×3 Mpc) and the composite LF for the supercluster are constructed. The composite LF is well fitted by a Schechter function with $M_B^ * = - 20\mathop .\limits^m 91$ , α=?1.02 and with $M_R^ * = - 22\mathop .\limits^m 39$ , α=?1.06. A comparison with the LFs of field galaxies and of various samples of clusters and superclusters shows that the Ursa Major supercluster have LF parameters characteristic of the field and, thus, differ from those of the Corona Borealis supercluster, which is apparently at a later stage of dynamical evolution.     

Analysis of the properties of clusters of galaxies in the region of the Ursa Major supercluster

We analyze the properties of the clusters of galaxies in the region of the Ursa Major (UMa) supercluster using observational data from SDSS and 2MASS catalogs. The region studied includes a supercluster (with a galaxy and cluster overdensity of 3 and 15, respectively) and field clusters inside the 150-Mpc diameter surrounding region. The total dynamical mass of 10 clusters of galaxies in UMa is equal to 2.25 × 10¹⁵ M _⊙, and the mass of 11 clusters of galaxies in the UMa neighborhood is equal to 1.70 × 10¹⁵ M _⊙. The fraction of early-type galaxies brighter than M _K ^* + 1 in the virialized regions of clusters is, on the average, equal to 70%, and it is virtually independent on the mass of the cluster. The fraction of these galaxies and their average photometric parameters are almost the same both for UMa clusters and for the clusters located in its surroundings. Parameters of the clusters of galaxies, such as infrared luminosities up to a fixed magnitude, the mass-to-luminosity ratio, and the number of galaxies have almost the same correlations with the cluster mass as in other samples of galaxies clusters. However, the scatter of these parameters for UMa member clusters is twice smaller than the corresponding scatter for field clusters, possibly, due to the common origin of UMa clusters and synchronized dynamical evolution of clusters in the supercluster.     

Analysis of the properties of galaxy clusters in the Leo supercluster region

We analyze the properties of galaxy clusters in the region of the Leo supercluster using observational data from the SDSS and 2MASS catalogs. We have selected 14 galaxy clusters with a total dynamical mass of 1.77 × 10¹⁵ M _⊙ in the supercluster region 130 by 60 Mpc in the plane of the sky (z ≃ 0.037). The composite luminosity function of the supercluster is described by a Schechter function with parameters that, within the error limits, correspond to field galaxies and does not differ from the luminosity function of the richer Ursa Major (UMa) supercluster for the same luminosity range (the bright end). The luminosity functions of early-type and late-type galaxies in Leo at the faint end are characterized by a sharp decrease (α = −0.60±0.08) and a steep increase (α = −1.44± 0.10) in the number of galaxies, respectively. In the virialized cluster regions, the fraction of early-type galaxies selected by the u-r color, bulge contribution, and concentration index among the galaxies brighter than M _K ^* + 1 is, on average, 62%. This fraction is smaller than that in the UMa supercluster at a 2–3σ level. The near-infrared luminosities of galaxy clusters down to a fixed absolute magnitude correlate with their masses almost in the same way as for other samples of galaxy clusters (L _200,K ∝ M ₂₀₀^0.63±0.11)).     

Evolution of galaxy groups

We study the variations of the properties of groups of galaxies with dynamical masses of 10¹³ M _⊙<M ₂₀₀<10¹⁴ M _⊙, represented by two samples: one has redshifts of z < 0.027 and is located in the vicinity of the Coma cluster, the other has z > 0.027, and is located in the regions of the following superclusters of galaxies: Hercules, Leo, Bootes, Ursa Major, and Corona Borealis. Using the archived data of the SDSS and 2MASX catalogs, we determined the concentration of galaxies in the systems by measuring it as the inner density of the group within the distance of the fifth closest galaxy from the center brighter than M _K = ?23_. ^m 3. We also measured the magnitude gap between the first and the fourth brightest galaxies ΔM ₁₄ located within one half of the selected radius R ₂₀₀, the fraction of early-type galaxies, and the ratio of bright dwarf galaxies (Mr = [?18_. ^m 5,?16_. ^m 5]) to giant galaxies (M _r < ?18_. ^m 5) (DGR) within the radius R ₂₀₀. The main aim of the investigation is to find among these characteristics the ones that reflect the evolution of groups of galaxies.We determined that the ratio of bright dwarf galaxies to early-type giant galaxies on the red sequence depends only on the x-ray luminosity: the DGR increases with luminosity. The fraction of early-type galaxies in the considered systems is equal, on average, to 0.65 ± 0.01, and varies significantly for galaxies with σ₂₀₀ < 300 kms^?1. Based on the luminosity of the brightest galaxy, the magnitude gap between the first and the fourth brightest galaxies in the groups, and on model computations of these parameters, we selected four fossil group candidates: AWM4, NGC0533, NGC0741, and NGC6098 (where the brightest galaxy is a double).We observe no increase in the number of faint galaxies (the α parameter of the Schechter function is less than 1) in our composite luminosity function (LF) for galaxy systems with z < 0.027 in the M _K = [?26m,?21_. ^m 5] range, whereas earlier we obtained α > 1 for the LF of the Hercules and Leo superclusters of galaxies.     

Dwarf-to-giant galaxy ratio in superclusters and in the field at 0.02 &lt; <Emphasis Type="Italic">z</Emphasis> &lt; 0.05

We report the results of the study of red-sequence (RS) galaxies in 47 galaxy clusters (0.023 < z < 0.047) located in different environments: in the superclusters Hercules and Leo, and in the field, based on the SDSS catalog data. In the

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interval, the number of bright RS dwarf galaxies in galaxy clusters increaseswith the X-ray luminosity of the cluster as logN ∝ log _X ^0.64 . The dwarf-to-giant ratio (DGR) does not depend on the surroundings, mass, or richness of the cluster. This ratio is seen to increase for galaxy clusters with log L _X > 43.5 erg/s or σ > 520 km/s. The compositeDGR of galaxy clusters, determined both from the membership in different structures and the X-ray luminosity along the radius R ₂₀₀, is minimum in the central regions of the clusters (about 0.6 ± 0.06), reaches a maximum within 0.3–0.9R ₂₀₀ (about 0.9 ± 0.10), and decreases approximately to 0.7 ± 0.03 upon reaching the radius 1.4 R ₂₀₀.

     

The formation of ring galaxies

The conditions under which a head-on collision between a disk galaxy and a spherical galaxy can lead to ring formation are investigated, using the impulsive approximation. The spherical galaxy is modeled as a polytrope of indexn=4 and radiusR _S and the disk galaxy as an exponential disk whose surface density is given by \(\sigma (r) = \sigma _c e^{ - 4r/R_D } \) , where σ _c is the central density andR _D is the radius of the disk. The formation and properties of the rings are closely related to the fractional change in binding energy of the disk galaxy, given by ΔU/?U?=γ _D β _D , where (GM _S ² R _D )/(V ² M _D R _S ² ),M _S andM _D being the masses of the spherical and disk galaxies, respectively, and β _D ≡β _D (n, σ, ?,i) is a function of the models of the two galaxies, the ratio of the radii of the two galaxies ?=R _S /R _D , and the angle of inclinationi, of the disk to the direction of relative motion of the two galaxies. Calculations are made for the caseR _S =R _D . Since practically the entire mass of the spherical galaxy, for the chosen model, lies within 1/3 of its radius, the radius of the spherical galaxy is effectively \(\tfrac{1}{3}\) that of the disk galaxy. It is found that as a result of the collision, the innermost and the outer parts of the disk galaxy are not much affected, but the intermediate region expands and gets evacuated, leading to the crowding of stars in a preferential region forming a ring structure. The rings are best formed for a normal, on-axis collision. For this case, rings form when ΔU/|U| lies between \(\tfrac{1}{2}\) and 2, while they are very sharp and bright when ΔU/|U| lies between \(\tfrac{1}{2}\) and 1. Within this range, as ΔU/|U| increases, the rings become sharper and their positions shift outwards with respect to the centre of the disk galaxy. The relationship $$\gamma _D = 0.0016 + 0.045s_{{\text{max}}}^2 ,$$ wheres _max is the radial distance of the density maximum of the ring from the centre of the disk galaxy (measured in terms of the radius of the disk galaxy as unit) enables us to finds _max from γ _D and vice versa, and interpret some prominent ring galaxies. The effect of introducing a bulge to the disk is to distribute the tidal disruptive effects more evenly and, hence, reduce the sharpness of the ring.     

Detailed study of the Ursa Major supercluster of galaxies using the 2MASS and SDSS Catalogs

We study the infrared (K _s band) properties of clusters of galaxies in the Ursa Major supercluster using data from 2MASS (Two-Micron All-Sky Survey) and SDSS (Sloan Digital Sky Survey). We identified three large filaments with mean redshifts of z = 0.051, 0.060, and 0.071. All clusters of the supercluster are located in these filaments. We determined the total K _s-band luminosities and masses for 11 clusters of galaxies within comparable physical regions (within a radius R ₂₀₀ close to the virial radius) using a homogeneous method. We constructed a combined luminosity function for the supercluster in this region, which can be described by the Schechter function with the following parameters: M _K ^* = ?24.50 and α = ?0.98. The infrared luminosities of the clusters of galaxies correlate with their masses; the M/L _K ratios of the systems increase with their masses (luminosities), with most of the Ursa Major clusters of galaxies (particularly the richer ones) closely following the relations derived previously for a large sample of clusters and groups of galaxies. The total mass-to-infrared-luminosity ratio is 52 M _⊙/L _⊙ for six Abell clusters and 49M _⊙/L _⊙ for all of the clusters, except Anon2.     

Fourier analysis of light curves of eclipsing variable stars

The photometric perturbationsB _h ^(l) arising from both tidal and rotational distortion of a close eclipsing binary have been given in two previous papers (Livaniou, 1977; Rovithis-Livaniou, 1977). The aim of the present paper will be to find the eclipse perturbationsB _2m =B _{2m, tid} +B _{2m, rot} of a close binary exhibiting partial eclipses. This will be done giving the suitable combinations of theB _h ^(l) 's and will make easier the application to real stars. After a very brief introduction, Section 2 gives both theB _{2m, tid} andB _{2m, rot} for uniformly bright discs; while in Sections 3 and 4 they are given for linear and quadratic limb-darkening, respectively. Finally, Section 5 gives a brief discussion of the results.     

Infrared luminosities of local-volume galaxies

Based on data from the Two-Micrometer All-Sky Survey (2MASS), we analyzed the infrared properties of 451 Local-Volume galaxies at distances D ≤ 10 Mpc. We determined the K-band luminosity function of the galaxies in the range of absolute magnitudes from ?25^m to ?11^m. The local luminosity density within 8 Mpc is 6.8 × 10⁸L_⊙ Mpc^?3, a factor of 1.5 ± 0.1 higher than the global mean K-band luminosity density. We determined the ratios of the virial mass to the K-band luminosity for nearby groups and clusters of galaxies. In the luminosity range from 5 × 10¹⁰ to 2 × 10¹³L_⊙, the dependence log(M/L_K) ∝ (0.27 ± 0.03) log L_K with a dispersion of ～0.1 comparable to the measurement errors of the masses and luminosities of the systems of galaxies holds for the groups and clusters of galaxies. The ensemble-averaged ratio, 〈M/L_K〉 ? (20–25) M_⊙/L_⊙, was found to be much smaller than the expected global ratio, (80–90)M_⊙/L_⊙, in the standard model with Ω_m = 0.27. This discrepancy can be eliminated if the bulk of the dark matter in the Universe is not associated with galaxies and their systems.     

Variability and possible rapid evolution of the hot post-AGB stars Hen 3-1347, Hen 3-1428, and LSS 4634

We present the results of spectroscopic and photometric observations for three hot southern-hemisphere post-AGB objects, Hen 3-1347 = IRAS 17074-1845, Hen 3-1428 = IRAS 17311-4924, and LSS 4634 = IRAS 18023-3409. In the spectrograms taken with the 1.9-m telescope of the South African Astronomical Observatory (SAAO) in 2012, we have measured the equivalent widths of the most prominent spectral lines. Comparison of the new data with those published previously points to a change in the spectra of Hen 3-1428 and LSS 4634 in the last 20 years. Based on ASAS data, we have detected rapid photometric variability in all three stars with an amplitude up to 0 _· ^m 3-0 _· ^m 4 in the V band. A similarity between the patterns of variability for the sample stars and other hot protoplanetary nebulae is pointed out. We present the results of UBV observations for Hen 3-1347, according to which the star undergoes rapid irregular brightness variations with maximum amplitudes ΔV = 0 _· ^m 25, ΔB = 0 _· ^m 25, and ΔU = 0 _· ^m 30 and shows color-magnitude correlations. Based on archival data, we have traced the photometric history of the stars over more than 100 years. Hen 3-1347 and LSS 4634 have exhibited a significant fading on a long time scale. The revealed brightness and spectrum variations in the stars, along with evidence for their enhanced mass, may be indicative of their rapid post-AGB evolution.     

The fundamental plane and other scaling relations for galaxy groups and clusters

In this paper we study the relations between the main characteristics of groups and clusters of galaxies using the archival data of the SDSS and 2MASX catalogs. We have developed and implemented a new method of determining the size of galaxy systems and their effective radius which contains half of the galaxies and not half the luminosity, since the luminosity of the brightest galaxy in a group can account for over 50% of the total luminosity of the group. The derived parameters (log L_K, logR_e, and log σ₂₀₀) for 94 systems of galaxies (0.0038 < z < 0.09) determine the Fundamental Plane (FP), which, with a scatter of 0.15, is similar in form to the FP of galaxy clusters obtained by Schaeffer et al. (1993) and D’Onofrio et al. (2013) with other methods and for different bands. We show that the FP in the near-infrared region (NIR) for 94 galaxy systems has the form of L_K ∝ \(R_e^{0.70 \pm {{0.13}_\sigma }1.34 \pm 0.13}\), whereas in x-rays it has the form of—L_X ∝ \(R_e^{1.15 \pm {{0.39}_\sigma }2.56 \pm 0.40}\). The form of the FP for groups and clusters is consistent with the FP for early-type galaxies determined in the same way. The form of the FP for galaxy systems deviates from the shape that one would expect from virial predictions. Adding the mass-to-light ratio as a fourth independent parameter has little effect on this deviation, but decreases the scatter of the FP for a sample of rich galaxy clusters by 12%.     

Fourier analysis of the light curves of eclipsing variables,XXIII

The method of evaluating the photometric perturbationsB _2m of eclipsing variables in the frequency domain, developed by Kopal (1959, 1975e, 1978) for an interpretation of mutual eclipses in systems whose components are distorted by axial rotation and mutual tidal action. The aim of the present paper has been to establish explicit expressions for the photometric perturbationB _2m in such systems, regardless of the kind of eclipses and non-integral values ofm. Recently, Kopal (1978) introduced two different kinds of integrals with respect to associated α-functions andI-integrals which have been expressed in terms of certain general types of series that can be easily programmed for automatic computation within seconds of real time on highspeed computers. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Section 3 we shall deduce the integral $$\int_0^{\theta \prime } {\tfrac{{\alpha _n^\prime }}{\delta }} d(sin^{2m} \theta )$$ in terms of a certain general type of series and also β-function, which should enable us to evaluate explicit expressions forf _* ^(h) ,f ₁ ^(h) ,f ₂ ^(h) as well asB _2m .     

New activity cycle of the classical symbiotic star V1413 Aql

We present the results of our photometric UBV JHKLM observations in 2008?C2011 for the classical symbiotic star V1413 Aql. At the end of 2008, the hot component of V1413 Aql experienced the next strong outburst (??V > 2 _· ^m 5). According to the photometric criterion (B-V ?? 0 _· ^m 9 ± 0 _· ^m 2), the star was in an active state even in the period preceding the strong 2008 outburst. Two eclipsing minima of the same amplitude were observed for V1413 Aql in 2010 and 2011. Our analysis of the 2011 eclipse has allowed us to estimate the sizes of the components with respect to the orbital semimajor axis if the system is seen edge-on: the radius of the cool component is R _g/a = 0.28 and the radius of the hot component is R _h/a = 0.17. However, judging by the B-V color index, the eclipse may be noncentral.     

Broadband spectrum of the total X-ray emission from the galaxy M31

We present the results of measurements of the total X-ray flux from the Andromeda galaxy (M31) in the 3-100 keV band based on data from the RXTE/PCA, INTEGRAL/ISGRI, and SWIFT/BAT space experiments. We show that the total emission from the galaxy has a multicomponent spectrum whose main characteristics are specified by binaries emitting in the optically thick and optically thin regimes. The galaxy’s luminosity at energies 20–100 keV gives about 6% of its total luminosity in the 3–100 keV band. The emissivity of the stellar population in M31 is L _{2–20 keV} ～ 1.1 × 10²⁹ erg s^?1 M _⊙ ^?1 in the 2–20 keV band and L _{20–100 keV} ～ 8 × 10²⁷ erg s^?1 M _⊙ ^?1 in the 20–100 keV band. Since low-mass X-ray binaries at high luminosities pass into a soft state with a small fraction of hard X-ray emission, the detection of individual hard X-ray sources in M31 requires a sensitivity that is tens of times better (up to 10^?13 erg s^?1 cm^?2) than is needed to detect the total hard X-ray emission from the entire galaxy. Allowance for the contribution from the hard spectral component of the galaxy changes the galaxy’s effective Compton temperature approximately by a factor of 2, from ～1.1 to ～2.1 keV.     

Physical properties of X-ray gas in galaxy cluster CL0024+17

We analyzed the X-ray data obtained by the Chandra telescope for the galaxy cluster CL0024+17 (z = 0.39). The mean temperature of the cluster is estimated (kT = 4.35 _?0.44 ^+0.51 keV) and the surface brightness profile is derived. We generated the mass and density profiles for dark matter and gas using numerical simulations and the Navarro-Frenk-White dark matter density profile (Navarro et al., 1995) for a spherically symmetric cluster in which gas is in hydrostatic equilibrium with the cluster field. The total mass of the cluster is estimated to be M ₂₀₀ = 3.51 _?0.47 ^+0.38 × 10 _Sun ¹⁴ within a radius of R ₂₀₀ = 1.24 _?0.17 ^+0.12 Mpc of the cluster center. The contribution of dark matter to the total mass of the cluster is estimated as ${{M_{200_{DM} } } \mathord{\left/ {\vphantom {{M_{200_{DM} } } {M_{tot} }}} \right. \kern-0em} {M_{tot} }} = 0.89$ .     

A scaling relation between the SZ decrement and the Thomson depth in clusters of galaxies

In a recent study of dark mater N-body simulations, a scaling relation between the SZ decrement and the Thomson depth of a cluster of galaxies of the form ΔT _r ∝τ _T ² has been found (Diaferio et al. 2000). In this paper, it will be shown that such a scaling relation arises if the intracluster gas is distributed similar to the dark matter density described by the NFW-profile and the finite spatial resolution of the numerical simulation is taken into account. It is furthermore investigated whether the ΔT _r ∝τ _T ² relation holds for analytical models of an isothermal gas sphere in the gravitational potential of a dark matter halo distributed according to the NFW-profile, the available experimental data of SZE observations, and recent results from cosmological gas-dynamical simulations of clusters of galaxies. Combining such a relation with temperature estimates from X-ray observations would provide information about a dependence of T _e on τ_T. The Thomson depth might therefore emerge as another important scaling parameter in studies of clusters of galaxies.     

Photometric elements and apsidal motion of the eclipsing binary NSV 18773

The photometric elements of the eclipsing binary NSV 18773 (HD 99898) have been determined for the first time by analyzing its V-and I-band light curves from the ASAS-2 and ASAS-3 catalogs. Based on these elements and using other published spectroscopic and photometric data, we constructed a consistent system of geometrical and physical parameters for the system that consists of two stars (M ₁ = 20M _⊙, Sp₁=B0V, R ₁ = 5.0R _⊙ and M ₂ = 14M _⊙, Sp₂ = B1V, R ₂ = 6.5R _⊙) in elliptical orbits (P = 5 _. ^d 049, e = 0.365, a = 40.1R _⊙). The distance to the system is d = 3.3 kpc, the interstellar extinction is A _V = 2 _. ^m 0, and the age is t = 2.8 × 10⁶ yr. NSV 18773 is a visual binary with components V _A = 9 _. ^m 9 and V _B = 10 _. ^m 3 separated by 0 _. ^" 8. The third light (L ₃ = 0.61) that we found by analyzing the light curves shows that the eclipsing binary is the system’s fainter component B. We confirmed the rapid apsidal motion of the star detected by Otero and Wils (2006) and refined its observed period: U _obs = 150 ± 6 yr. Our photometric elements and physical parameters allowed the apsidal parameter $\bar k_2^{obs} = 0.0135(14)$ , which reflects the density distribution along the radii of the component stars, to be determined. Within the error limits, the derived parameter agrees with its theoretically expected value, $\bar k_2^{th} = 0.0119(8)$ , from current evolutionary models of stars of the corresponding masses and ages.     

The structure of clusters with bimodal distributions of galaxy radial velocities,I. A1035

The structure of the A1035 cluster of galaxies (10^h32^m + 40°13′, cz ～ 22000 kms^?1), which exhibits a bimodal distribution of galaxy radial velocities (ΔV ≈ 3000 kms^?1), is analyzed using three methods of determining the relative distances to clusters from early-type galaxies: the Kormendy relation corrected for the dependence of residuals on galaxy magnitude, the photometric plane, and the fundamental plane. We use the data obtained with the l-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences and SDSS (DR5) data to show that A1035 consists of two gravitationally unbound independent clusters. These clusters with the velocity dispersions of 566 km s^?1 and 610 km s^?1 and masses within R ₂₀₀ equal to 2.7 · 10¹⁴ and 3.5 · 10¹⁴ M _⊙, respectively, obey the Hubble law.