Messier 87 - The galaxy of greatest known mass.
Radio galaxy Messier 87 mass determined using virial theorem for nucleus velocity dispersion
Brandt, J. C.↗
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X-rays from the Coma cluster of galaxies
X ray source from Coma cluster of galaxies, interpreting observed flux on basis of missing mass suggested by Virial theorem
Felten, J. E.↗
Velocity dispersions in galaxies: 1: The SO galaxy NGC 7332
A Coude spectrum of the SO galaxy NGC 7332 with 0.9 A resolution from 4186 to 4364 A was obtained with the SEC vidicon television camera and the Hale telescope. Comparisons with spectra of G and K giant stars, numerically broadened for various Maxwellian velocity distributions, give a dispersion velocity in the line of sight of 160 + or - 20 km/sec with the best fit at G8III. The dispersion appears to be constant within + or - 35 km/sec out to 1.4 kpc (H = 100 km/sec/mpc). After correction for projection, the rotation curve has a slope of 0.16 km/sec/pc at the center and a velocity of 130 km/sec at 1.4 kpc where it is still increasing. For an estimated effective radius of 3.5 kpc enclosing half the light, the virial theorem gives a mass of 1.4 x 10 to the 11th power solar masses if the mass-to-light ratio is constant throughout the galaxy. The photographic luminosity is 8.3 x 10 to the 9th power solar luminosities so that the M/L ratio is 17.
Morton, D. C.↗
Velocity dispersions in galaxies. I - The E7 galaxy NGC 7332.
A coude spectrum of the E7 galaxy NGC 7332 with 0.9 A-resolution from 4186 to 4364 A was obtained with the Princeton SEC vidicon television camera and the Hale telescope. Comparisons with spectra of G and K giant stars, numerically broadened for various Maxwellian velocity distributions, give a dispersion velocity in the line of sight of 160 (plus or minus 20) km/sec with the best fit at G8 III. The dispersion appears to be constant within plus or minus 35 km/sec out to 1.4 kpc. After correction for projection, the rotation curve has a slope of 0.18 km/sec per pc at the center and a velocity of 130 km/sec at 1.4 kpc where it is still increasing. For an estimated effective radius of 3.5 kpc enclosing half the light, the virial theorem gives a mass of 140 billion solar masses if the mass-to-light ratio is constant throughout the galaxy.
Morton, D. C.↗
The estimation of masses of individual galaxies in clusters of galaxies.
Three different methods of estimating masses are discussed. The 'density method' is based on the analysis of the density distribution of galaxies around the object whose mass is to be found. The 'bound-galaxy method' gives estimates of the mass of a double, triple, or quadruple system from analysis of the orbital motion of the components. The 'virial method' utilizes the formulas derived for the second method to obtain estimates of the virial-theorem masses of whole clusters, and thus to obtain upper limits on the mass of an individual galaxy in a cluster. The analytic formulas are developed and compared with computer experiments, and some applications are given.
Wolf, R. A.↗
On the rotation of elliptical galaxies
The tensor virial theorem is applied to models of early-type galaxies. First the theorem is applied to rotating elliptical galaxies whose constant-density surfaces are similar ellipsoids. A relationship is obtained between the observed rotations and the forms of generally triaxial galaxies. By applying the results of Robert (1962) to the evaluation of the components of the Chandrasekhar tensor which occurs in this relationship, it is found that the form of a galaxy that lacks global velocity anisotropy uniquely determines the ratio of its rotational and random kinetic energies independently of the radial density profile of that galaxy. A distribution of three-dimensional prolate spheroids is derived which accounts for the observed distribution of ellipticities reported by Sandage, Freeman, and Stokes (1970). This is then used to derive curves giving the expected frequency of occurrence of galaxies of given rotational velocities for each of a number of different apparent elongations on the sky. It is found that if elliptical galaxies are prolate, there should be little correlation between apparent ellipticity and rotation velocity.
Binney, J.↗
Gravitational potential energy of the earth - A spherical harmonic approach
A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic expansion agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the crust and mantle of -2.77 x 10 to the 29th ergs, an order of magnitude below McKenzie's (1966) estimate. McKenzie's result stems from mathematical error. Our figure is almost identical with Kaula's (1963) estimate of the minimum shear strain energy in the mantle, a not unexpected result on the basis of the virial theorem. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the 20th P is found by assuming that the total geothermal flux is due to viscous dissipation of energy. This number is almost six orders of magnitude below MacDonald's (1966) estimate of the viscosity and removes his objection to convection. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at 1% efficiency, then the viscosity is 10 to the 22nd P, a number preferred by Cathles (1975) and Peltier and Andrew (1976) as the viscosity of the mantle.
Rubincam, D. P.↗
The relation between velocity dispersion and central galaxy density in clusters of galaxies
A correlation between cluster velocity dispersion and average central galaxy density is reported. The correlation covers the range from rich clusters to small groups of galaxies, or, in terms of velocity dispersion, from v sub r approximately 1500 to approximately 100 km/s. This result is useful for estimating unknown velocity dispersions in clusters with the aid of the relatively easily determined 0.5 Mpc central galaxy density parameter. When combined with the virial theorem, the above relation also suggests that the mass-to-light ratio of galaxy systems increases with the system's velocity dispersion.
Bahcall, N. A.↗
Seven poor clusters of galaxies
The measurement of 83 new redshifts from galaxies in the region of seven of the poor clusters of galaxies identified by Morgan et al (1975) and Albert et al (1977) has been followed by an estimation of cluster masses through the application of both the virial theorem and the projected mas method. For each system, these two estimates are consistent. For the two clusters with highest X-ray luminosities, the line-of-sight velocity dispersions are about 700 km/sec, while for the five other clusters, the dispersions are of the order of less than about 370 km/sec. The D or cD galaxy in each poor cluster is at the kinematic center of each system.
Beers, T. C.↗
Acoustic-radiation stress in solids. I - Theory
The general case of acoustic-radiation stress associated with quasi-compressional and quasi-shear waves propagating in infinite and semiinfinite lossless solids of arbitrary crystalline symmetry is studied. The Boussinesq radiation stress is defined and found to depend directly on an acoustic nonlinearity parameter which characterizes the radiation-induced static strain, a stress-generalized nonlinearity parameter which characterizes the stress nonlinearity, and the energy density of the propagating wave. Application of the Boltzmann-Ehrenfest principle of adiabatic invariance to a self-constrained system described by the nonlinear equations of motion allows the acoustic-radiation-induced static strain to be identified with a self-constrained variation in the time-averaged product of the internal energy density and displacement gradient. The time-averaged product is scaled by the acoustic nonlinearity parameter and represents the first-order nonlinearity in the virial theorem. Finally, the relationship between the Boussinesq and the Cauchy radiation stress is obtained in a closed three-dimensional form.
Cantrell, J. H., Jr.↗
Dark matter near the sun
The amount of dark matter in the disk of the Galaxy at the solar position is determined by comparing the observed distributions of tracer stars with the predictions obtained from different assumptions of how the unseen matter is distributed. The major uncertainties, observational and theoretical, are estimated. For all the observed samples, typical models imply that about half of the mass in the solar vicinity must be in the form of unobserved matter. The volume density of unobserved material near the sun is about 0.1 solar mass/cu pc; the corresponding column density is about 30 solar masses/cu pc. This, so far unseen, material must be in a disk with an exponential scale height of less than 0.7 kpc. All the existing observations are consistent with the unseen disk material being in the form of stars not massive enough to burn hydrogen. It is suggested that the unseen material that is required to hold up the rotation curves of galaxies and to satisfy the virial theorem for clusters of galaxies might also be in the form of low-mass stars.
Bahcall, J. N.↗
Linear clusters of galaxies - A999 and A1016
Redshifts have been measured for galaxies in two of the 'linear' clusters of the sample of Adams, Strom, and Strom (1980), including 44 redshifts in A999 and 40 in A1016. From the data, it is concluded that the galaxies in A999 are probably drawn from a spherically symmetric distribution, while those in A1016 probably are not. Both A999 and A1016 have mass-to-light ratios lower than typical of other clusters. The effect of anisotropy on the determination of cluster masses from the virial theorem is examined, and it is found that if the shortest axes of these clusters are close to the line of sight, the mass-to-light ratio may be underestimated by about 50 percent. No significant evidence is found for alignments of individual cluster members with the cluster axis in the convincing linear cluster A1016. There is similarly no evidence of segregation by luminosity morphological type in A1016.
Chapman, G. N. F.↗
Linear clusters of galaxies - A194
New measurements for 160 redshifts and previous measurements for 108 other redshifts are presented for galaxies within 5 deg of A194. The galaxy distribution in A194 is shown to be inconsistent with a spherically symmetric King model. A mass-to-light ratio is derived using the virial theorem which is lower than the mean for the groups in the CfA redshift survey (Huchra and Geller, 1982; Geller, 1984). A nonparametric test for galaxy-cluster alignment and a Chi-squared test are used to search for alignment of galaxy major axes with the axis of A194. Evidence for neither luminosity segregation nor significant differences in the velocity or surface distributions of galaxies as a function of morphological type is found.
Chapman, G. N. F.↗
Magnetic confinement of cosmic clouds
The role of the magnetic field in the confinement or compression of interstellar gas clouds is reconsidered. The virial theorem for an isolated magnetized cloud in the presence of distant magnetic sources is reformulated in terms of moments of the internal and external currents, and an equilibrium condition is derived. This condition is applied to the interaction between isolated clouds for the simple- and artificial-case in which the field of each cloud is a dipole. With the simplest of statistical assumptions, the probability of any given cloud being compressed is calculated as about 10 percent, the magnetic field acting as a medium which transmits the kinetic pressure between clouds. Even when compression occurs the magnetic pressure 1/2 B-squared may decrease on leaving the cloud surface.
Azar, Michel↗
Galaxy clustering and the origin of large-scale flows
Peebles's 'cosmic virial theorem' is extended from its original range of validity at small separations, where hydrostatic equilibrium holds, to large separations, in which linear gravitational stability theory applies. The rms pairwise velocity difference at separation r is shown to depend on the spatial galaxy correlation function xi(x) only for x less than r. Gravitational instability theory can therefore be tested by comparing the two up to the maximum separation for which both can reliably be determined, and there is no dependence on the poorly known large-scale density and velocity fields. With the expected improvement in the data over the next few years, however, this method should yield a reliable determination of omega.
Juszkiewicz, R.↗
The shape of the mass distribution in M31 from its globular cluster system
The velocity dispersion and rotation velocity of the M31 globular cluster system depend on the relative division of mass between the flat disk and a spherically symmetric halo. Using the tensor virial theorem, it is shown in detail how the mass ratio can be constrained. Radial velocities have been collected for 149 globular clusters in M31. With no assumptions about the isotropy of the velocity distribution, the globular cluster kinematics are consistent with the mass distribution inferred by the rotation curve but otherwise place no strong constraints on the relative division of the mass. If the velocity distribution is isotropic, models with the disk mass ranging between 1/2 and 1 times its maximum possible value are marginally favored.
Kent, Stephen M.↗
Aerospace applications of high temperature superconductivity
A review is presented of all the applications that are part of the NASA program to develop space technology capitalizing on the potential benefit of high temperature superconducting materials. The applications in three major areas are being pursued: sensors and cryogenic systems, space communications, and propulsion and power systems. This review places emphasis on space communications applications and the propulsion and power applications. It is concluded that the power and propulsion applications will eventually be limited by structural considerations rather than by the availability of suitable superconductors. A cursory examination of structural limitations implied by the virial theorem suggested that there is an upper limit to the size of high field magnetic systems that are feasible in space.
Connolly, D. J.↗
Galactic oscillations
Long-lived oscillations that act like normal modes are described. The total kinetic energy is found to vary with time by amounts far in excess of the fluctuations expected from the virial theorem, and the variation shows periodic patterns that suggest oscillations. Experimental results indicate that oscillation amplitudes depend on the nature of the model. It is noted that it is difficult to answer questions about likely amplitudes in real galaxies with any confidence at the present time.
Miller, R. H.↗