Extending PT symmetry from Heisenberg algebra to E2 algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.Comment: 8 pages, 7 figure

Spatially resolved spectroscopy of Coma cluster early-type galaxies IV. Completing the dataset

The long-slit spectra obtained along the minor axis, offset major axis and diagonal axis are presented for 12 E and S0 galaxies of the Coma cluster drawn from a magnitude-limited sample studied before. The rotation curves, velocity dispersion profiles and the H_3 and H_4 coefficients of the Hermite decomposition of the line of sight velocity distribution are derived. The radial profiles of the Hbeta, Mg, and Fe line strength indices are measured too. In addition, the surface photometry of the central regions of a subsample of 4 galaxies recently obtained with Hubble Space Telescope is presented. The data will be used to construct dynamical models of the galaxies and study their stellar populations.Comment: 40 pages, 7 figures, 6 tables. Accepted for publication in ApJ

The Dynamical Fingerprint of Core Scouring in Massive Elliptical Galaxies

The most massive elliptical galaxies have low-density centers or cores that differ dramatically from the high-density centers of less massive ellipticals and bulges of disk galaxies. These cores have been interpreted as the result of mergers of supermassive black hole binaries, which depopulate galaxy centers by gravitationally slingshotting central stars toward large radii. Such binaries naturally form in mergers of luminous galaxies. Here, we analyze the population of central stellar orbits in 11 massive elliptical galaxies that we observed with the integral field spectrograph SINFONI at the European Southern Observatory Very Large Telescope. Our dynamical analysis is orbit-based and includes the effects of a central black hole, the mass distribution of the stars, and a dark matter halo. We show that the use of integral field kinematics and the inclusion of dark matter is important to conclude upon the distribution of stellar orbits in galaxy centers. Six of our galaxies are core galaxies. In these six galaxies, but not in the galaxies without cores, we detect a coherent lack of stars on radial orbits in the core region and a uniform excess of radial orbits outside of it: when scaled by the core radius, the radial profiles of the classical anisotropy parameter beta are nearly identical in core galaxies. Moreover, they match quantitatively the predictions of black hole binary simulations, providing the first convincing dynamical evidence for core scouring in the most massive elliptical galaxies.Comment: 8 pages, 3 figures, accepted by Ap

TOPEX orbit determination using GPS signals plus a sidetone ranging system

The GPS orbit determination was studied to see how well the radial coordinate for altimeter satellites such as TOPEX could be found by on board measurements of GPS signals, including the reconstructed carrier phase. The inclusion on altimeter satellites of an additional high accuracy tracking system is recommended. It is suggested that a sidetone ranging system is used in conjunction with TRANET 2 beacons

Does the complex deformation of the Riemann equation exhibit shocks?

The Riemann equation $u_t+uu_x=0$, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is \cP\cT symmetric. A one-parameter \cP\cT-invariant complex deformation of this equation, $u_t-iu(iu_x)^\epsilon= 0$ ($\epsilon$ real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless $\epsilon$ is an odd integer.Comment: latex, 8 page

Internal Dynamics, Structure and Formation of Dwarf Elliptical Galaxies: II. Rotating Versus Non-Rotating Dwarfs

We present spatially-resolved internal kinematics and stellar chemical abundances for a sample of dwarf elliptical (dE) galaxies in the Virgo Cluster observed with Keck/ESI. We find that 4 out of 17 dEs have major axis rotation velocities consistent with rotational flattening, while the remaining dEs have no detectable major axis rotation. Despite this difference in internal kinematics, rotating and non-rotating dEs are remarkably similar in terms of their position in the Fundamental Plane, morphological structure, stellar populations, and local environment. We present evidence for faint underlying disks and/or weak substructure in a fraction of both rotating and non-rotating dEs, but a comparable number of counter-examples exist for both types which show no evidence of such structure. Absorption-line strengths were determined based on the Lick/IDS system (Hbeta, Mgb, Fe5270, Fe5335) for the central region of each galaxy. We find no difference in the line-strength indices, and hence stellar populations, between rotating and non-rotating dE galaxies. The best-fitting mean age and metallicity for our 17 dE sample are 5 Gyr and Fe/H = -0.3 dex, respectively, with rms spreads of 3 Gyr and 0.1 dex. The majority of dEs are consistent with solar alpha/Fe abundance ratios. By contrast, the stellar populations of classical elliptical galaxies are, on average, older, more metal rich, and alpha-enhanced relative to our dE sample. The local environments of both dEs types appear to be diverse in terms of their proximity to larger galaxies in real or velocity space within the Virgo Cluster. Thus, rotating and non-rotating dEs are remarkably similar in terms of their structure, stellar content, and local environments, presenting a significant challenge to theoretical models of their formation. (abridged)Comment: 33 pages, 12 figures. To appear in the October 2003 Astronomical Journal. See http://www.ucolick.org/~mgeha/geha_dE.ps.gz for version with high resolution figure

The Stellar Kinematic Fields of NGC 3379

We have measured the stellar kinematic profiles of NGC 3379 along four position angles using the MMT. The data extend 90" from the center, at essentially seeing-limited resolution out to 17". The mean velocities and dispersions have total errors better than 10 km/s (frequently better than 5 km/s) out to 55". We find very weak (3 km/s) rotation on the minor axis interior to 12", and no detectable rotation above 6 km/s from 12" to 50" or above 16 km/s out to 90" (95% confidence). However, a Fourier reconstruction of the mean velocity field from all 4 sampled PAs does indicate a 5 degree twist of the kinematic major axis, opposite to the known isophotal twist. The h_3 and h_4 parameters are small over the entire observed region. The azimuthally-averaged dispersion profile joins smoothly at large radii with the dispersions of planetary nebulae. Unexpectedly, we find sharp bends in the major-axis rotation curve, also visible (though less pronounced) on the diagonal position angles. The outermost bend coincides in position with other sharp kinematic features: an abrupt flattening of the dispersion profile, and local peaks in h_3 and h_4. All of these features are in a region where the surface brightness profile departs significantly from a de Vaucouleurs law. Features such as these are not generally known in ellipticals owing to a lack of data at comparable resolution; however, very similar behavior is seen the kinematics of the edge-on S0 NGC 3115. We discuss the suggestion that NGC 3379 could be a misclassified S0; preliminary results from dynamical modeling indicate that it may be a flattened, weakly triaxial system seen in an orientation that makes it appear round.Comment: 31 pages incl. 4 tables, Latex, AASTeX v4.0, with 17 eps figures. To appear in The Astronomical Journal, February 199

The evolution of the color gradients of early-type cluster galaxies

We investigate the origin of color gradients in cluster early-type galaxies to probe whether pure age or pure metallicity gradients can explain the observed data in local and distant (z approx 0.4) samples. We measure the surface brightness profiles of the 20 brightest early-type galaxies of CL0949+44 (hereafter CL0949) at redshift z=0.35-0.38 from HST WF2 frames taken in the filters F555W, F675W, F814W. We determine the color profiles (V-R)(r), (V-I)(r), and (R-I)(r) as a function of the radial distance r in arcsec, and fit logarithmic gradients in the range -0.2 to 0.1 mag per decade. These values are similar to what is found locally for the colors (U-B), (U-V), (B-V) which approximately match the (V-R), (V-I), (R-I) at redshift approx 0.4. We analyse the results with up to date stellar population models. We find that passive evolution of metallicity gradients (approx 0.2 dex per radial decade) provides a consistent explanation of the local and distant galaxies' data. Invoking pure age gradients (with fixed metallicity) to explain local color gradients produces too steep gradients at redshifts z approx 0.4. Pure age gradients are consistent with the data only if large present day ages (>=15 Gyr) are assumed for the galaxy centers.Comment: 23 pages, 19 figures, Accepted for publication in A&