About the Algebraic Solutions of Smallest Enclosing Cylinders Problems

Given n points in Euclidean space E^d, we propose an algebraic algorithm to compute the best fitting (d-1)-cylinder. This algorithm computes the unknown direction of the axis of the cylinder. The location of the axis and the radius of the cylinder are deduced analytically from this direction. Special attention is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius enclosing cylinder is computed algebrically from constrained minimization of a quartic form of the unknown direction of the axis. For the latter, an analytical condition of existence of the circumscribed cylinder is given, and the algorithm reduces to find the zeroes of an one unknown polynomial of degree at most 6. In both cases, the other parameters of the cylinder are deduced analytically. The minimal radius enclosing cylinder is computed analytically for the regular tetrahedron and for a trigonal bipyramids family with a symmetry axis of order 3.Comment: 13 pages, 0 figure; revised version submitted to publication (previous version is a copy of the original one of 2010

The Nature of Associated Absorption and the UV-X-ray Connection in 3C 288.1

We discuss new Hubble Space Telescope spectroscopy of the radio-loud quasar, 3C 288.1. The data cover ~590 A to ~1610 A in the quasar rest frame. They reveal a wealth of associated absorption lines (AALs) with no accompanying Lyman-limit absorption. The metallic AALs range in ionization from C III and N III to Ne VIII and Mg X. We use these data and photoionization models to derive the following properties of the AAL gas: 1) There are multiple ionization zones within the AAL region, spanning a factor of at least ~50 in ionization parameter. 2) The overall ionization is consistent with the ``warm'' X-ray continuum absorbers measured in Seyfert 1 nuclei and other QSOs. However, 3) the column densities implied by the AALs in 3C 288.1 are too low to produce significant bound-free absorption at any UV-X-ray wavelengths. Substantial X-ray absorption would require yet another zone, having a much higher ionization or a much lower velocity dispersion than the main AAL region. 4) The total hydrogen column density in the AAL gas is log N_H (cm-2)= 20.2. 5) The metallicity is roughly half solar. 6) The AALs have deconvolved widths of ~900 km/s and their centroids are consistent with no shift from the quasar systemic velocity (conservatively within +/-1000 km/s). 7) There are no direct indicators of the absorber's location in our data, but the high ionization and high metallicity both suggest a close physical relationship to the quasar/host galaxy environment. Finally, the UV continuum shape gives no indication of a ``blue bump'' at higher energies. There is a distinct break of unknown origin at ~1030 A, and the decline toward higher energies (with spectral index alpha = -1.73, for f_nu ~ nu^alpha) is even steeper than a single power-law interpolation from 1030 A to soft X-rays.Comment: 27 pages with figures and tables, in press with Ap

Temperature and Kinematics of CIV Absorption Systems

We use Keck HIRES spectra of three intermediate redshift QSOs to study the physical state and kinematics of the individual components of CIV selected heavy element absorption systems. Fewer than 8 % of all CIV lines with column densities greater than 10^{12.5} cm^{-2} have Doppler parameters b < 6 km/s. A formal decomposition into thermal and non-thermal motion using the simultaneous presence of SiIV gives a mean thermal Doppler parameter b_{therm}(CIV) = 7.2 km/s, corresponding to a temperature of 38,000 K although temperatures possibly in excess of 300,000 K occur occasionally. We also find tentative evidence for a mild increase of temperature with HI column density. Non-thermal motions within components are typically small (< 10 km/s) for most systems, indicative of a quiescent environment. The two-point correlation function (TPCF) of CIV systems on scales up to 500 km/s suggests that there is more than one source of velocity dispersion. The shape of the TPCF can be understood if the CIV systems are caused by ensembles of objects with the kinematics of dwarf galaxies on a small scale, while following the Hubble flow on a larger scale. Individual high redshift CIV components may be the building blocks of future normal galaxies in a hierarchical structure formation scenario.Comment: submitted to the ApJ Letters, March 16, 1996 (in press); (13 Latex pages, 4 Postscript figures, and psfig.sty included

Sample variance and Lyman α forest transmission statistics

We compare the observed probability distribution function (PDF) of the transmission in the H I Lyman α forest, measured from the Ultraviolet and Visual Echelle Spectrograph (UVES) ‘Large Programme’ sample at redshifts z = [2, 2.5, 3], to results from the GIMIC cosmological simulations. Our measured values for the mean transmission and its PDF are in good agreement with published results. Errors on statistics measured from high-resolution data are typically estimated using bootstrap or jackknife resampling techniques after splitting the spectra into chunks. We demonstrate that these methods tend to underestimate the sample variance unless the chunk size is much larger than is commonly the case. We therefore estimate the sample variance from the simulations. We conclude that observed and simulated transmission statistics are in good agreement; in particular, we do not require the temperature–density relation to be ‘inverted’

Discovery of a Perseus-like cloud in the early Universe: HI-to-H2 transition, carbon monoxide and small dust grains at zabs=2.53 towards the quasar J0000+0048

We present the discovery of a molecular cloud at zabs=2.5255 along the line of sight to the quasar J0000+0048. We perform a detailed analysis of the absorption lines from ionic, neutral atomic and molecular species in different excitation levels, as well as the broad-band dust extinction. We find that the absorber classifies as a Damped Lyman-alpha system (DLA) with logN(HI)(cm^-2)=20.8+/-0.1. The DLA has super-Solar metallicity with a depletion pattern typical of cold gas and an overall molecular fraction ~50%. This is the highest f-value observed to date in a high-z intervening system. Most of the molecular hydrogen arises from a clearly identified narrow (b~0.7 km/s), cold component in which CO molecules are also found, with logN(CO)~15. We study the chemical and physical conditions in the cold gas. We find that the line of sight probes the gas deep after the HI-to-H2 transition in a ~4-5 pc-size cloud with volumic density nH~80 cm^-3 and temperature of only 50 K. Our model suggests that the presence of small dust grains (down to about 0.001 {\mu}m) and high cosmic ray ionisation rate (zeta_H a few times 10^-15 s^-1) are needed to explain the observed atomic and molecular abundances. The presence of small grains is also in agreement with the observed steep extinction curve that also features a 2175 A bump. The properties of this cloud are very similar to what is seen in diffuse molecular regions of the nearby Perseus complex. The high excitation temperature of CO rotational levels towards J0000+0048 betrays however the higher temperature of the cosmic microwave background. Using the derived physical conditions, we correct for a small contribution (0.3 K) of collisional excitation and obtain TCMB(z = 2.53)~9.6 K, in perfect agreement with the predicted adiabatic cooling of the Universe. [abridged]Comment: 24 pages, 24 figures, accepted for publication in A&

The UVES Large Program for testing fundamental physics - III. Constraints on the fine-structure constant from 3 telescopes

Large statistical samples of quasar spectra have previously indicated possible cosmological variations in the fine-structure constant, $\alpha$. A smaller sample of higher signal-to-noise ratio spectra, with dedicated calibration, would allow a detailed test of this evidence. Towards that end, we observed equatorial quasar HS 1549$+$1919 with three telescopes: the Very Large Telescope, Keck and, for the first time in such analyses, Subaru. By directly comparing these spectra to each other, and by `supercalibrating' them using asteroid and iodine-cell tests, we detected and removed long-range distortions of the quasar spectra's wavelength scales which would have caused significant systematic errors in our $\alpha$ measurements. For each telescope we measure the relative deviation in $\alpha$ from the current laboratory value, $\Delta\alpha/\alpha$, in 3 absorption systems at redshifts $z_{\mathrm{abs}}=1.143$, 1.342, and 1.802. The nine measurements of $\Delta\alpha/\alpha$ are all consistent with zero at the 2-$\sigma$ level, with 1-$\sigma$ statistical (systematic) uncertainties 5.6--24 (1.8--7.0) parts per million (ppm). They are also consistent with each other at the 1-$\sigma$ level, allowing us to form a combined value for each telescope and, finally, a single value for this line of sight: $\Delta\alpha/\alpha=-5.4 \pm 3.3_{\mathrm{stat}} \pm 1.5_{\mathrm{sys}}$ ppm, consistent with both zero and previous, large samples. We also average all Large Programme results measuring $\Delta\alpha/\alpha=-0.6 \pm 1.9_{\mathrm{stat}} \pm 0.9_{\mathrm{sys}}$ ppm. Our results demonstrate the robustness and reliability at the 3 ppm level afforded by supercalibration techniques and direct comparison of spectra from different telescopes.Comment: 24 pages, 11 figures, 9 table