Singular Isothermal Sphere Model Research Articles

HSTlarge-field weak lensing analysis of MS 2053−04: study of the mass distribution and mass-to-light ratio of X-ray luminous clusters at 0.22<z<0.83

We have detected the weak lensing signal induced by the cluster of galaxies MS 2053-04 (z =0.58) from a two-colour mosaic of six Hubble Space Telescope (HST ) WFPC2 images. The best-fitting singular isothermal sphere model to the observed tangential distortion yields an Einstein radius r (E) =6.2+/-1.8 arcsec, which corresponds to a velocity dispersion of (Omega (m) =0.3, Omega(Lambda) =0.0). This result is in good agreement with the observed velocity dispersion of 817+/-80 km s(-1) from cluster members. The observed average rest-frame mass-to-light ratio within a radius aperture is 184+/-56 h (50) M-./L (B.). After correction for luminosity evolution to z =0, this value changes to 291+/-89+/-19 h (50) M-. /L (B .) (where the first error indicates the statistical uncertainty in the measurement of the mass-to-light ratio, and the second error is due to the uncertainty in luminosity evolution). MS 2053 is the third cluster we have studied using mosaics of deep WFPC2 images. For all three clusters we find good agreement between dynamical and weak lensing velocity dispersions, in contrast to weak lensing studies based on single WFPC2 pointings on cluster cores. This result demonstrates the importance of wide-field data. We have compared the ensemble-averaged cluster profile of the clusters in our sample with the predicted Navarro, Frenk & White (NFW) profile, and find that an NFW profile can fit the observed lensing signal well. The best-fitting concentration parameter is found to be (68 per cent confidence) times the predicted value from an open cold dark matter (CDM) model. The observed mass-to-light ratios of the clusters in our sample evolve with redshift, and are inconsistent with a constant, non-evolving, mass-to-light ratio at the 99 per cent confidence level. The evolution is consistent with the results derived from the evolution of the fundamental plane of early-type galaxies. The resulting average mass-to-light ratio for massive clusters at z =0 is found to be 239+/-18+/-9 h (50) M-. /L (B .).

The first detection of weak gravitational shear in infrared observations: Abell 1689

We present the first detection of weak gravitational shear at infrared wavelengths, using observations of the lensing cluster Abell 1689, taken with the SofI camera on the ESO-NTT telescope. The imprint of cluster lenses on the shapes of the background galaxy population has previously been harnessed at optical wavelengths, and this gravitational shear signal enables cluster mass distributions to be probed, independent of whether the matter is luminous or dark. At near-infrared wavelengths, the spectrophotometric properties of galaxies facilitate a clean selection of background objects for use in the lensing analysis. A finite-field mass reconstruction and application of the aperture mass ( M ap ) statistic are presented. The probability that the peak of the M ap detection ($\frac{S}{N}\sim 5$), arises from a chance alignment of background sources is only ~$4.5\times 10^{-7}$. The velocity dispersion of the best-fit singular isothermal sphere model for the cluster is $\sigma_{\rm 1D}=1030^{+70}_{-80}\,{\rm km}\,{{\rm s}^{-1}}$, and we find a K -band mass-to-light ratio of ~$40\,M_{\odot}/L_{\odot}$ inside a $0.44\,{\rm Mpc}$ radius.

Modelling submillimetre spectra of the protostellar infall candidates NGC 1333-IRAS 2 and Serpens SMM4

We present a radiative transfer model, which is applicable to the study of submillimetre spectral line observations of protostellar envelopes. The model uses an exact, non-LTE, spherically symmetric radiative transfer ‘Stenholm’ method, which numerically solves the radiative transfer problem by the process of ‘Λ-iteration’. We also present submillimetre spectral line data of the Class 0 protostars NGC 1333–IRAS 2 and Serpens SMM4. We model the data using the Stenholm code. We examine the physical constraints which can be used to limit the number and range of parameters used in protostellar envelope models, and identify the turbulent velocity and tracer molecule abundance as the principal sources of uncertainty in the radiative transfer modelling. We explore the trends in the appearance of the predicted line profiles as key parameters in the models are varied, such as infall velocity profile, turbulence and rotation. The formation of the characteristic asymmetric double-peaked line profile in infalling envelopes is discussed. We find that the separation of the two peaks of a typical infall profile is dependent not on the evolutionary status of the collapsing protostar, but on the turbulent velocity dispersion in the envelope. We also find that the line shapes can be significantly altered by rotation. Fits are found for the observed line profiles of IRAS 2 and SMM4 using plausible infall model parameters. The density and velocity profiles in our best-fitting models are inconsistent with a singular isothermal sphere model (SIS), since for both objects modelled, the infall velocities appear further advanced than a SIS model would predict, given the density profile. We find better agreement with a form of collapse which assumes non-static initial conditions, in agreement with other recent findings. We also find some evidence that the infall velocities are retarded from free-fall towards the centre of the cloud, probably by rotation, and that the envelope of SMM4 is rotationally flattened.

Semianalytical Models for Lensing by Dark Halos. I. Splitting Angles

We use the semianalytical approach to analyze gravitational lensing of remote quasars by foreground dark halos in various cold dark matter (CDM) cosmologies in order to determine the sensitivity of the predictions for probabilities of image separations to the input assumptions regarding the properties of halos and cosmological models. The power spectrum of primordial fluctuations is normalized by the cluster abundance constraints. The mass function of dark halos is assumed to be given by the Press-Schechter function. The mass density profile of dark halos is alternatively taken to be the singular isothermal sphere (SIS), the Navarro-Frenk-White (NFW) profile, or the generalized NFW profile. The cosmologies being considered include the Einstein-de Sitter model (SCDM), the open model (OCDM), and the flat Λ-model (LCDM). As expected, we find that the lensing probability is extremely sensitive to the mass density profile of lenses (dark halos) and somewhat less so to the mean mass density in the universe and the amplitude of primordial fluctuations. NFW halos are very much less effective in producing multiple images than SIS halos. For NFW lenses, the SCDM model produces fewer lensing events than the OCDM/LCDM models by 2 orders of magnitude. For SIS lenses, the SCDM model produces more lensing events with small splitting angles and fewer lensing events with large splitting angles than the OCDM/LCDM models, which is due to the fact that for large-mass halos, the Press-Schechter function is very sensitive to the amplitude of primordial fluctuations. In all cases the difference between the OCDM model and the LCDM model is not dramatic. None of these models are consistent with current observations: the SIS models predict too many large splitting lenses, while the NFW models predict too few small splitting lenses. Essentially, the observed high ratio of small splitting to large splitting lenses is not predicted correctly. This indicates that there must be at least two populations of halos in the universe: small-mass halos with a steep inner density slope and large-mass halos with a shallow inner density slope. A combination of SIS and NFW halos can reasonably reproduce the current observations if we choose the mass for the transition from SIS to NFW to be ~1013 M☉, as might plausibly occur because of baryonic cooling and contraction in lower mass systems. Additionally, there is a tendency for CDM models to have too much power on small scales, i.e., too much mass concentration. From our sensitivity studies it appears that the cures proposed for other apparent difficulties of CDM would help here as well, an example being the warm dark matter variant, which is shown to produce fewer large splitting lenses than the corresponding CDM model by 1 order of magnitude.

Constraints on the Warm Dark Matter Model from Gravitational Lensing

Formation of sub-galactic halos is suppressed in warm dark matter (WDM) model due to thermal motion of WDM particles. This may provide a natural resolution to some puzzles in standard cold dark matter (CDM) theory such as the cusped density profiles of virialized dark halos and the overabundance of low mass satellites. One of the observational tests of the WDM model is to measure the gravitationally lensed images of distant quasars below sub-arcsecond scales. In this Letter, we report a comparison of the lensing probabilities of multiple images between CDM and WDM models using a singular isothermal sphere model for the mass density profiles of dark halos and the Press-Schechter mass function for their distribution and cosmic evolution. It is shown that the differential probability of multiple images with small angular separations down to 10 milliarcseconds should allow one to set useful constraints on the WDM particle mass. We discuss briefly the feasibility and uncertainties of this method in future radio surveys (e.g. VLBI) for gravitational lensing.

Weak-Lensing Study of Low-Mass Galaxy Groups: Implications for Ω[TINF][ITAL]m[/ITAL][/TINF

We report on the first measurement of the average mass and mass-to-light ratio of galaxy groups by analyzing the weak-lensing signal induced by these systems. The groups, which have velocity dispersions of 50-400 km s(-1), have been selected from the Canadian Network for Observational Cosmology Field Galaxy Redshift Survey (CNOC2). This survey allows the identification of a large number of groups with redshifts ranging from z = 0.12 to 0.55, ideal for a weak-lensing analysis of their mass distribution. For our analysis we use a sample of 50 groups that are selected on the basis of a careful dynamical analysis of group candidates. We detect a signal at the 99% confidence limit. The best-fit singular isothermal sphere model yields an Einstein radius. r(E) = 0.72 +/- 0.29. This corresponds to a velocity dispersion of [sigma (2)](1/2) = 274(-59)(+48) km s(-1) (using photometric redshift distributions for the source galaxies), which is in good agreement with the dynamical estimate. Under the assumption that the light traces the mass, we find an average mass-to-light ratio of 191 +/- 83 h (M)./L-B. in the rest-frame B band. Unlike dynamical estimates, this result is insensitive to problems associated with determining group membership. After correction of the observed mass-to-light ratio for luminosity evolution to z = 0, we find 254 +/- 110 h M./L-B., lower than what is found for rich clusters. We use the observed mass-to-light ratio to estimate the matter density of the universe, for which we find Omega (m) = 0.19 +/- 0.10 (Omega (Lambda) = 0), in good agreement with other recent estimates. For a closed universe (Omega (m) + Omega (Lambda) = 1), we obtain Omega (m) = 0.13 +/- 0.07.

CTQ 839: Candidate for the Smallest Projected Separation Binary Quasar

We report the discovery of the new double quasar CTQ 839. This B = 18.3, radio-quiet quasar pair is separated by 21 in BRI and H filters, with magnitude differences of ?mB = 2.5, ?mR = ?mI = 1.9, and ?mH = 2.3. Spectral observations reveal both components to be z = 2.24 quasars, with relative redshifts that agree at the 100 km s-1 level but exhibit pronounced differences in the equivalent widths of related emission features, as well as an enhancement of blue continuum flux in the brighter component as compared with the fainter component longward of the Ly? emission feature. In general, similar redshift double quasars can be the result of a physical binary pair or of a single quasar multiply imaged by gravitational lensing. Empirical point-spread function subtraction of R and H band images of CTQ 839 reveal no indication of a lensing galaxy and place a detection limit of R = 22.5 and H = 17.4 for a third component in the system. For an Einstein-de Sitter cosmology and singular isothermal sphere model, the R band detection limit constrains the characteristics of any lensing galaxy to zl 1 with a corresponding luminosity of L 5 L*, while an analysis based on the redshift probability distribution for the lensing galaxy argues against the existence of a zl 1 lens at the 2 ? level. A similar analysis for a ?-dominated cosmology, however, does not significantly constrain the existence of any lensing galaxy. The broadband flux differences, spectral dissimilarities, and failure to detect a lensing galaxy make the lensing hypothesis for CTQ 839 unlikely. The similar redshifts of the two components would then argue for a physical quasar binary. At a projected separation of 8.3 h-1 kpc (?m = 1), CTQ 839 would be the smallest projected separation binary quasar currently known.

Weak Lensing Analysis of Cl 1358+62 UsingHubble Space TelescopeObservations

We report on the detection of weak gravitational lensing of faint, distant background objects by Cl 1358+62, a rich cluster of galaxies at a redshift of z = 0.33. The observations consist of a large, multicolor mosaic of Hubble Space Telescope WFPC2 images. The number density of approximately 50 background objects arcmin-2 allows us to do a detailed weak lensing analysis of this cluster. We detect a weak lensing signal out to~1.5 Mpc from the cluster center. The observed distortion is consistent with a singular isothermal sphere model with a velocity dispersion of 780 ± 50 km s-1. The total projected mass within a radius of 1 Mpc corresponding to this model is (4.4 ± 0.6) × 1014 M☉. The errors given here represent the random error due to the ellipticities of the background galaxies. The uncertainty in the redshift distribution introduces an additional systematic error of ~10% in the weak lensing mass. The weak lensing mass is slightly lower than dynamical estimates and agrees well with X-ray mass estimates. The mass distribution is elongated in a similar way as the light. The axis ratio of 0.30 ± 0.15 and position angle of -21° ± 7° were measured directly from the observations and agree very well with a previous strong lensing determination. A two-dimensional reconstruction of the cluster mass surface density shows that the peak of the mass distribution coincides with the peak of the light distribution. We find a value of (90 ± 13)h50 M☉ L for the mass-to-light ratio, consistent with being constant with radius. The point-spread function of HST is highly anisotropic at the edges of the individual chips. This systematically perturbs the shapes of objects, and we present a method for applying the appropriate correction.

Spectroscopy of the optical Einstein ring 0047 - 2808

We present optical and near-infrared spectroscopic observations of the optical Einstein ring 0047 – 2808. We detect both [O III] lines λλ4959, 5007 near ∼ 2.3 μm, confirming the redshift of the lensed source as z = 3.595. The Lyα line is redshifted relative to the [OIII] line by 140 ± 20 km s−1. Similar velocity shifts have been seen in nearby starburst galaxies. The [OIII] line is very narrow, 130 km s−1 FWHM. If the ring is the image of the centre of a galaxy, the one-dimensional stellar velocity dispersion σ = 55 km s−1 is considerably smaller than the value predicted by Baugh et al. for the somewhat brighter Lyman-break galaxies. The Lyα line is significantly broader than the [O III] line, probably due to resonant scattering. The stellar central velocity dispersion of the early-type deflector galaxy at z = 0.485 is 250 ± 30 km s−1. This value is in good agreement both with the value predicted from the radius of the Einstein ring (and a singular isothermal sphere model for the deflector), and with the value estimated from the Dn−σ relation.

The Cosmological Constant and Statistical Lensing of Giant Arcs

Using a singular isothermal sphere model for the matter distribution of foreground clusters of galaxies, we study the statistics of giant ares in flat cosmologies with and without a cosmological constant. We find that the relative number of arcs predicted within z = 1 in a universe with {OMEGA}_0_ = 0.3 and λ_0_ = {DELTA}/(3H_0_^2^) = 0.7 is a factor of ~2 greater than that predicted in the Einstein-de Sitter universe ({OMEGA}_0_ = 1, λ_0_ = 0). For a luminosity-dependent evolution model of the number density of background galaxies that accounts for the overdensity of faint blue galaxies at z_s_ ~ 0.4, the Einstein-de Sitter cosmological model predicts that ~5% of clusters of galaxies with X-ray luminosity L_x_ > 2 x 10^44^ ergs s^-1^ should have giant arcs, with length-to-width ratio greater than 10. This is a factor of ~4 lower than the observed fraction in the gravitational-lensing survey of distant X-ray-selected Einstein Extended Medium-Sensitivity Survey clusters of galaxies, indicating a significant deviation of the matter distribution of clusters of galaxies from simple isothermal spheres and/or the presence of a significant cosmological constant. It will be profitable to further study the constraints on the cosmological constant from giant arcs with more realistic cluster models.

Analytic results for the gravitational lens statistics of singular isothermal spheres in general cosmologies

The natural cosmological distance for the analysis of gravitational lens statistics with a singular isothermal sphere model for the lens potential is the proper motion distance. By using the proper motion distance we derive analytic expressions for the cross-sections, averages of the image separations, and averages of the distances to the lens for both point-like and resolved sources that depend only on the total density of the cosmological model and the proper motion distance to the source. The ratio of the square of the mean image separation to the mean square image separation for the lenses in a survey is a robust measure of survey completeness

Statistics of gravitational lenses - The uncertainties

The assumptions in the analysis of gravitational lensing statistics are examined. Special emphasis is given to the uncertainties in the theoretical predictions. It is shown that a simple redshift cutoff model, which may result from galaxy evolution, can significantly reduce the lensing probability and explain the large mean separation of images in observed gravitational lenses. This effect may affect the constraint on the contribution of the cosmological constant to producing a flat universe from the number counts of the observed lenses. For the Omega(0) = 1 (filled beam) model, the lensing probability of early-type galaxies with finite core radii is reduced roughly by a factor of 2 for high-redshift quasars as compared with the corresponding singular isothermal sphere model. The finite core radius effect is about 20 percent for a lambda-dominated flat universe. It is also shown that the most recent galaxy luminosity function gives lensing probabilities that are smaller than previously estimated roughly by a factor of 3.