The binary galaxies NGC 4038/39 have extended filamentary arms generated by tidal interactions (Toomre and Toomre, Ap. J. 178; 623, (1972)(TT)). The velocity field was determined by HI observations taken with the VLA (a facility operated by the NRAO under contract with the NSF), and the combined velocity and morphological information was used to constrain the allowed orbital parameters, halo characteristics, and dynamical friction. TT-type calculations were carried out with central masses and rings of test particles, and the calculated results compared with the data. Using disk orientations derived from optical data (Rubin et al., (1970) Ap. J. 160 81), and solving for the six remaining orbital parameters, central potential softening constant (representing the halo), and frictional relaxation time, a good fit between the model and the radio data was found. The best model is shown in Figure 1, and is superimposed on an HI column density map in Figure 2. The orbit is well-determined, and must be nearly parabolic; the pair are interacting for the first time, and if the galaxies have extensive massive halos much larger than their discs, then their tidal arms would be shorter and stubbier than observed. More limited halos are allowed; each galaxy could have up to 80% of its total mass in a halo, but the halos cannot be much larger than the discs. A halo several times larger than the disc, with 10 to 20 times the disc mass, is not permitted by the data.

Radial velocities good to 25 km/s have been measured for the remote globular clusters Eridanus, NGC 2419, Palomar 3, Palomar 4, Palomar 5, Palomar 14, and Palomar 15. Spectra with a resolution of 3.0 A were recorded for 4 to 10 stars at a time in each cluster, using an aperture plate with the KPNO 4m telescope, R-C spectrograph, Simmons camera, and baked IIIaJ plates. Radial velocities for each star were determined from the position of the night-sky emission feature near 4360A with respect to several strong atomic features in the vicinity.

For two remote systems, Eridanus and Palomar 14, the radial velocity transformed to the galactic rest frame is large enough to suggest a total mass for the Galaxy of 1012 M⊙, and a comparable mass is indicated from the average rest-frame velocities of the remote systems. A full presentation of the measurements and results will appear in the October 1, 1985, issue of the Astrophysical Journal.

The statistical analysis of magnitude and center-of-mass velocity data for binary galaxies will never provide an adequate estimate of the mass within the orbit, and will therefore never furnish useful information about the mass-to-light ratios of galaxies. There are three main reasons. Firstly, it is very difficult to find a sample which is both objectively selected and dynamically isolated, and in those cases with obvious tidal distortions it is not clear which two points of the confused systems can be taken to be the centers of mass. Secondly, there are several different ways to produce a mass estimator from the available data, and it is not clear which estimator corresponds most closely to what is normally thought of as the gravitational mass of the system. Finally, those mass estimates which include it as a variable depend quite strongly on the orbital eccentricity, and any analysis method is necessarily least sensitive to just this parameter. Therefore, variation even within a single estimator is both considerable and only weakly constrained.

The evidence for dark matter in binaries and groups of galaxies is very strong, and is seen in all recent observational studies. Measurements of mass in galactic systems is possible on scales ranging from 50 kpc using virial analysis of binary galaxies to 15 Mpc using Virgocentric infall analysis. The Ω estimates derived from these studies are generally consistent with Ω < 0.2, with a fairly weak trend toward larger Ω estimates on larger scales. However, measurements of the galaxy distribution in the IRAS catalog yields a dipole anisotropy consistent in direction with the microwave dipole anisotropy, suggesting that the local galaxy distribution is responsible for the microwave velocity. This will eventually provide the most reliable estimate of Ω, and is likely to result in a value somewhat larger than previous estimates on smaller scales. Study of the velocity field around large clusters in cosmological n-body experiments provides a useful guide for understanding the limitations of the spherically symmetric models of Virgocentric infall. We point out a number of biases that could affect the existing Virgocentric flow studies.

Data on the luminosity function (LF) of galaxies are reviewed and compared on a common magnitude system and with common assumptions. The result of Kirshner et al. (Astron. J. 88, 1285, 1983) is chosen as a best guess. Departures from this “standard LF” for specific galaxy types and environments (clusters, groups, field) are discussed briefly. If the Galactic absorption is AB = 0.2 csc|b| and the solar absolute magnitude is MB⊙ = 5.48, this LF leads to a mean luminosity density on the BT system, or about 1.4 × 10−2 h “Galaxies” Mpc−3. The mean M/L ratio needed to give critical cosmological density (Ω0 = 1) is then 920h in solar units on the BT0 (face-on) system. [The latter number can be larger by a factor ∼2 if a different LF is used, and larger still on different systems and with different assumptions (see Table).] The “weighed” M/L on this system is in clusters, but it is smaller in binaries and small groups. Estimates of the weighed (clumped) Ω0 vary, but it is definitely < 1. Comparison with constraints imposed by inflation and nucleosynthesis suggests that we distinguish at least two “dark-matter” problems: (1) What is the weighed mass, and how is it distributed? It contributes an Ω0 ∼ 0.1–0.5 and could be all baryonic if at the lower end of this range. (2) Is there additional matter, more smoothly distributed and probably nonbaryonic, which brings Ω0 up to unity? For details see Comments on Astrophys., in press.

The study of the dynamics of the Coma Cluster is of interest for several reasons. First, there exists a great deal of observational information about the cluster, including data on morphology, magnitude, color and redshift for the galaxies, and reasonably detailed x-ray data for the hot gas. Second, the present dynamical state of the cluster is reasonably well-defined. In addition, the segregation of the more luminous (≡ massive) galaxies towards the cluster center shows that two-body relaxation effects are well-advanced (Capelato et al. 1980). The profile of velocity dispersion with radius shows that in the outer parts of the cluster the galaxy velocities are non-isothermal (des Forêts et al. 1984). There is, however, evidence of continuing dynamical evolution. The velocity field of the galaxies at large distances from the center of the cluster suggests continuing infall (Capelato et al. 1982), and two sub-condensations are located in the inner regions (Mazure and Proust 1986). A new dynamical analysis for the cluster is being carried out in two stages. First, a relaxed model with a wide mass spectrum (c.f. Inagaki 1980) is fitted to the data. The contribution of the intergalactic gas is taken into account. With HO = 75 km/sec/Mpc, the total mass within a 3° radius of the center is ∼ 1.5 × 1015 M⊙, of which ∼ 30% is in the intergalactic medium, and M/L ∼ 75 M⊙/L⊙. The ratio of specific energies of the galaxies and the gas is ∼ 1.1, i.e., there is no scale-height problem (these results are described more fully by Gerbal et al. 1986). A second “model independent” analysis using the profiles of the galactic density and velocity dispersion gives the radial dependence of the galactic mass, the gas mass and also gives the total mass, which is found to be ∼ 1.1 × 1015 M⊙ within 3° (Gerbal et al. 1984).

Group masses are almost always computed using the virial theorem. Bahcall and Tremaine (1981) studied the reliability of mass estimates based on the virial theorem for the case of test particles orbiting a central mass. They found that virial theorem estimates were both inefficient and biased. We have extended the work of Bahcall and Tremaine by exploring alternative mass estimators to the virial theorem for self-gravitating systems.

The knowledge of the masses of small groups of galaxies is important, because the timescales for collapse and virialization of groups depend on their mass-density. While the large inefficiency of the virial mass statistic is well known (Bahcall and Tremaine 1981, hereafter BT, and references therein), biases in the virial mass may produce wrong evolution timescales for small groups. These biases originate from group contamination by interlopers, and from the different galaxy and dark matter distributions inside the groups (caused by mass segregation). This second bias was first studied by Smith (1980, 1984) although already implicit in the work of Limber (1959). It formally arises because the ratio 2T/C where C = ΣFα·Rα (the Clausius virial) is not the same for the luminous and global matter distributions. We illustrate here some quantitative aspects of this Limber bias from the output of N-body simulations of groups of 8 galaxies described elsewhere (Mamon 1986).

The measurement of the amplitude of the Local Group infall velocity towards the Virgo Cluster is a crucial test for the value of the universal density parameter Ωo and the ratio of the universal Hubble constant to its local value. However, a very large discrepancy exists between total infall velocities derived from peculiar velocity field observations and those derived from “scaling” methods using standard candles in the Virgo and Coma clusters. The former have tended to produce high Virgocentric peculiar velocities (350 to 500 km s−1) whilst the latter give much lower values (-70 to 100 km s−1).

The intergalactic neutral hydrogen in the M96 group (Schneider et al. 1983) provides an unusual probe of the detailed mass distribution in a group of galaxies. Previous observations (Schneider 1985) tentatively suggested the existence of a large ring structure to the gas. Sensitive new observations made at Arecibo confirm that the intergalactic gas forms a 200 kpc diameter eccentric ring around the two central galaxies in the group, the giant elliptical galaxy NGC 3379 (M105) and the lenticular NGC 3384.

This discussion represents an update of an article submitted for publication in The Astrophysical Journal that describes the concentration of Abell clusters to the supergalactic equator. Since that article was written, our sample of Abell-class clusters has been augmented by 44% to now total 309 clusters within a redshift of 0.1c. The distribution of those clusters with distance from the supergalatic equator is illustrated in the accompanying figure. The normalized histogram includes an attempt to correct for the diminished unobscured area in planes off the supergalactic equator. Within 4 bins on the equator (i.e., within 80/h75 Mpc) there are 105 clusters, rather than the 48 clusters that would be expected with a random distribution. The 57 excess clusters represents a 4.6 σ signal.

What are the characteristic scale lengths and densities for the dark halos of galaxies, and the typical ratios of dark to luminous mass? For elliptical galaxies, the best estimates come from X-ray data which will be discussed in a later session. For spirals, the best estimates come from rotation curves. I will concentrate on the halo parameters for disk galaxies. At the end, there will be a few comments on stellar dynamical data for ellipticals, and on the unique information available for the dark halo of our Galaxy.

We have made a rotation curve analysis of a sample of spiral galaxies for which both photometric and kinematical data of reasonable quality are available in the literature. From the photometric radial luminosity profile, assuming constant mass-to-light ratios for bulge and disk separately, we calculate a rotation curve due to the luminous mass in a galaxy. Comparison with the observed rotation velocities allows us to derive a halo rotation curve, which can be used to derive characteristic halo parameters. The decomposition into luminous and dark matter is not unique, with as extremes a “minimum” disk (M/L = 0) and a “maximum” disk (M/L as high as possible while requiring a realistic halo mass distribution without a hollow core).

Using both photometric and gas kinematic data of three galaxies of different type (NGC 7814 SO, NGC 2841 Sb, NGC 300 Sd), we estimate the density of the dark halo and visible components which gives the best fit with the observed rotation curves. We then compute the ratio of each component force to the total force as function of radius. Finally, we derive from the potential energy tensor the quantitative effect of each component on bulge, disk and halo dynamics.

The study of the ratio of dark-to-luminous matter in spiral galaxies has been the subject of several recent studies (Bahcall 1983; Casertano 1983; Carignan and Freeman 1985; van Albada et al. 1985; Carignan 1985). These studies have been possible because of the large number of high sensitivity HI observations which became available in the last few years, allowing to probe the halo potential to very large galactocentric distances. In the case of NGC 3198 (van Albada et al. 1985), it was even possible to derive the rotation curve out to 11 disc scale lengths. One of the important questions, forming the motivation for this type of work, is whether the ratio of dark-to-luminous matter is a function of galaxy mass and/or luminosity.

We discuss available information on the distribution of luminous and dark matter in eight galaxies. The galaxies have been chosen according to the following criteria: 1) existence of a good rotation curve, extending well beyond the optical radius; 2) a mass model has been published; 3) valuable constraints can be put on the amount of dark matter inside the optical radius. A full description of the data and reduction procedures is in Bahcall and Casertano (1985).

Luminosity profiles and rotation curves for 37 Sb and Sc galaxies have been combined to derive mass/light ratios for the stellar component and scale parameters for a dark halo component.

Mass-to-light ratios M/LH−0.5 and M/LB are plotted against color B-H−0.5 in Fig. 1 for 82 nearby spirals from the catalog of Aaronson et al. (1982, Ap. J. Suppl. 50, 241), with BT magnitudes from the RC2, |b| > 20°, 45° ≤ i ≤ 80°, and HI mass estimates. Total masses and infrared H−0.5 magnitudes are measured within the blue isophotal radii R25 and R25/3 respectively, which depend on galaxy color. This color bias is corrected for by replacing R25 by R'25, the radius a galaxy would have at a standard color B-H−0.5 = 2.17. Stellar masses M* and LHC luminosities within R'25 are obtained by substracting twice the HI mass and by extrapolation, respectively. Corrected ratios M*/LHC and M*/LB versus corrected color B-HC are shown in Fig. 1 together with theoretical model predictions. The corrected observed ratios are systematically larger for bluer galaxies than predicted so that bluer spirals seem to have relatively more massive halos, in agreement with earlier results (Tinsley, B.M. 1981 M.N.R.A.S. 194, 63; Vader, J. P. 1984, in Formation and Evolution of Galaxies and Large Structures in the Universe, eds. J. Audouze and J. T. Thanh Van, p. 227).

This paper reviews the observational evidence for dark matter (DM) in dwarf spiral (dS) and dwarf spheroidal (dE) galaxies. The most secure detection of DM in dwarf galaxies is given by HI rotation curves. They provide estimates of DM halo parameters, i. e., isothermal core radii rc, central densities ρ0 and one-dimensional velocity dispersions σ. The smallest DM halo measured so far is in DDO 127 (MB = −14.5 for H0 = 75 km s−1 Mpc−1, rc ≃ 2.3 kpc, σ ≃ 27 km s−1). If this halo is made of neutrinos of mass mv, then phase-space constraints imply that mv > 110 eV. This is difficult to reconcile with cosmological upper limits giving Ω ≤ 1. Ultimately, dE galaxies will provide the strongest constraints on DM in dwarf galaxies; a detailed look at present results shows that they are not yet conclusive.

We report the cumulative results of an on-going effort to measure the stellar velocity dispersion in two nearby dwarf spheroidal galaxies. Radial velocities having an accuracy ≲ 2 km s−1 have now been secured for ten stars in Ursa Minor and eleven stars in Draco (including 16 K giants and 5 C types). Most objects have been observed at two or more epochs. Stars having non-variable velocities yield in both dwarfs a large (∼ 10 km s−1) dispersion. These results cannot be explained by atmospheric motions, and circumstantial evidence suggests that the effects of undetected binaries are also not likely to be important. Instead, it seems that both spheroidals contain a substantial dark matter component, which therefore must be “cold” in form.