Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power $p>1$ loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.Comment: 18 pages. We included an example for the loss of convexity and pinching. In the third version we dropped the concavity assumption on F. Comments are welcom

Counts and Sizes of Galaxies in the Hubble Deep Field - South: Implications for the Next Generation Space Telescope

Science objectives for the Next Generation Space Telescope (NGST) include a large component of galaxy surveys, both imaging and spectroscopy. The Hubble Deep Field datasets include the deepest observations ever made in the ultraviolet, optical and near infrared, reaching depths comparable to that expected for NGST spectroscopy. We present the source counts, galaxy sizes and isophotal filling factors of the HDF-South images. The observed integrated galaxy counts reach >500 galaxies per square arcminute at AB<30. We extend these counts to faint levels in the infrared using models. The trend previously seen that fainter galaxies are smaller, continues to AB=29 in the high resolution HDF-S STIS image, where galaxies have a typical half-light radius of 0.1 arcseconds. Extensive Monte Carlo simulations show that the small measured sizes are not due to selection effects until >29mag. Using the HDF-S NICMOS image, we show that galaxies are smaller in the near infrared than they are in the optical. We analyze the isophotal filling factor of the HDF-S STIS image, and show that this image is mostly empty sky even at the limits of galaxy detection, a conclusion we expect to hold true for NGST spectroscopy. At the surface brightness limits expected for NGST imaging, however, about a quarter of the sky is occupied by the outer isophotes of AB<30 galaxies. We discuss the implications of these data on several design concepts of the NGST near-infrared spectrograph. We compare the effects of resolution and the confusion limit of various designs, as well as the multiplexing advantages of either multi-object or full-field spectroscopy. We argue that the optimal choice for NGST spectroscopy of high redshift galaxies is a multi-object spectrograph (MOS) with target selection by a micro electro mechanical system (MEMS) device.Comment: 27 pages including 10 figures, accepted for publication in the Astronomical Journal, June 2000, abridged abstrac

Heating the bubbly gas of galaxy clusters with weak shocks and sound waves

Using hydrodynamic simulations and a technique to extract the rotational component of the velocity field, we show how bubbles of relativistic gas inflated by AGN jets in galaxy clusters act as a catalyst, transforming the energy carried by sound and shock waves to heat. The energy is stored in a vortex field around the bubbles which can subsequently be dissipated. The efficiency of this process is set mainly by the fraction of the cluster volume filled by (sub-)kpc scale filaments and bubbles of relativistic plasma.Comment: Accepted for publication in ApJ Letters after minor wording changes, 4 figures, 4 page

Multiwavelength study of Cygnus A III. Evidence for relic lobe plasma

We study the particle energy distribution in the cocoon surrounding Cygnus A, using radio images between 151 MHz and 15 GHz and a 200 ks Chandra ACIS-I image. We show that the excess low frequency emission in the the lobe further from Earth cannot be explained by absorption or excess adiabatic expansion of the lobe or a combination of both. We show that this excess emission is consistent with emission from a relic counterlobe and a relic counterjet that are being re-energized by compression from the current lobe. We detect hints of a relic hotspot at the end of the relic X-ray jet in the more distant lobe. We do not detect relic emission in the lobe nearer to Earth as expected from light travel-time effects assuming intrinsic symmetry. We determine that the duration of the previous jet activity phase was slightly less than that of the current jet-active phase. Further, we explain some features observed at 5 and 15 GHz as due to the presence of a relic jet.Comment: Accepted for publication with MNRAS, 10 pages with 10 figure

Shocks and sonic booms in the intracluster medium: X-ray shells and radio galaxy activity

Motivated by hydrodynamic simulations, we discuss the X-ray appearance of radio galaxies embedded in the intracluster medium (ICM) of a galaxy cluster. We distinguish three regimes. In the early life of a powerful source, the entire radio cocoon is expanding supersonically and hence drives a strong shock into the ICM. Eventually, the sides of the cocoon become subsonic and the ICM is disturbed by the sonic booms of the jet's working surface. In both of these regimes, X-ray observations would find an X-ray shell. In the strong shock regime, this shell will be hot and relatively thin. However, in the weak shock (sonic-boom) regime, the shell will be approximately the same temperature as the undisturbed ICM. If a cooling flow is present, the observed shell may even be cooler than the undisturbed ICM due to the lifting of cooler material into the shell from the inner (cooler) regions of the cluster. In the third and final regime, the cocoon has collapsed and no well-defined X-ray shell will be seen. We discuss ways of estimating the power and age of the source once its regime of behavior has been determined.Comment: 4 pages, submitted for publication in Astrophysical Journal. Full paper (including figure) can be obtained from http://rocinante.Colorado.EDU/~chris/papers/xray_hydro.p

Warped discs and the directional stability of jets in Active Galactic Nuclei

Warped accretion discs in Active Galactic Nuclei (AGN) exert a torque on the black hole that tends to align the rotation axis with the angular momentum of the outer disc. We compute the magnitude of this torque by solving numerically for the steady state shape of the warped disc, and verify that the analytic solution of Scheuer and Feiler (1996) provides an excellent approximation. We generalise these results for discs with strong warps and arbitrary surface density profiles, and calculate the timescale on which the black hole becomes aligned with the angular momentum in the outer disc. For massive holes and accretion rates of the order of the Eddington limit the alignment timescale is always short (less than a Myr), so that jets accelerated from the inner disc region provide a prompt tracer of the angular momentum of gas at large radii in the disc. Longer timescales are predicted for low luminosity systems, depending on the degree of anisotropy in the disc's hydrodynamic response to shear and warp, and for the final decay of modest warps at large radii in the disc that are potentially observable via VLBI. We discuss the implications of this for the inferred accretion history of those Active Galactic Nuclei whose jet directions appear to be stable over long timescales. The large energy deposition rate at modest disc radii during rapid realignment episodes should make such objects transiently bright at optical and infrared wavelengths.Comment: MNRAS, in press. Revised to match accepted version, with one new figure showing alignment timescale as a function of black hole mas

The distribution and cosmic evolution of massive black hole spins

We study the expected distribution of massive black hole (MBH) spins and its evolution with cosmic time in the context of hierarchical galaxy formation theories. Our model uses Monte Carlo realizations of the merger hierarchy in a LCDM cosmology, coupled to semi-analytical recipes, to follow the merger history of dark matter halos, the dynamics of the MBHs they host, and their growth via gas accretion and binary coalescences. The coalescence of comparable mass holes increases the spin of MBHs, while the capture of smaller companions in randomly-oriented orbits acts to spin holes down. We find that, given the distribution of MBH binary mass ratios in hierarchical models, binary coalescences alone do not lead to a systematic spin-up or spin-down of MBHs with time: the spin distribution retains memory of its initial conditions. By contrast, because of the Bardeen-Petterson effect, gas accretion via a thin disk tends to spin holes up even if the direction of the spin axis changes randomly in time. In our models, accretion dominates over black hole captures and efficiently spins holes up. The spin distribution is heavily skewed towards fast-rotating Kerr holes, is already in place at early epochs, and does not change much below redshift 5. If accretion is via a thin disk, about 70% of all MBHs are maximally rotating and have radiative efficiencies approaching 30% (assuming a "standard'' spin-efficiency conversion). Even in the conservative case where accretion is via a geometrically thick disk, about 80% of all MBHs have spin parameters a/m > 0.8 and accretion efficiencies > 12%. Rapidly spinning holes with high radiative efficiencies may satisfy constraints based on comparing the local MBH mass density with the mass density inferred from luminous quasars (Soltan's argument).Comment: 15 pages, 9 figures, accepted for publication in the Astrophysical Journa

Young stars and non-stellar emission in the aligned radio galaxy 3C 256

We present ground-based images of the z=1.824 radio galaxy 3C 256 in the standard BVRIJHK filters and an interference filter centered at 8800A, a Hubble Space Telescope image in a filter dominated by Ly-alpha emission (F336W), and spectra covering rest-frame wavelengths from Ly-alpha to [O III] 5007. Together with published polarimetry observations, we use these to decompose the overall spectral energy distribution into nebular continuum emission, scattered quasar light, and stellar emission. The nebular continuum and scattered light together comprise half (one third) of the V-band (K-band) light within a 4-arcsec aperture, and are responsible for the strong alignment between the optical/near-infrared light and the radio emission. The stellar emission is dominated by a population estimated to be 100-200 Myr old (assuming a Salpeter IMF), and formed in a short burst with a peak star formation rate of 1-4x10^3 Msun/yr. The total stellar mass is estimated to be no more than 2x10^{11} Msun, which is far less than other luminous radio galaxies at similar redshifts, and suggests that 3C 256 will undergo further star formation or mergers.Comment: 35 pages including 10 figures; to appear in Nov 10 Ap