Explicit Analytical Expression for a Lanchester Attrition-Rate Coefficient for Bonder and Farrell’s m-Period Target-Engagement Policy

Working Paper #5, DTRA Project, July 9, 2001The purpose of this working paper is to give an explicit analytical expression for a Lanche s- ter-type attrition-rate coefficient for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets for Bonder and Farrell’s m-period target-acquisition policy1. It develops this result (its main result) from Taylor’s [2001d] new important general result (that does not depend on the target-engagement policy of a firer type or even on the particulars of the target-acquisition process) for a Lanchester attrition-rate coefficient for serial acquisition by developing explicit ana- lytical expressions for the two key intermediate quantities on which the coefficient depends: namely, (1) expected time to acquire a target that will be engaged, (2) next-target-type-to-be-engaged probability. An analytical expression for the former quantity (the expect value) was recently developed by one of the authors (Taylor [2001e]), while the paper at hand develops such an expression for the latter probability. These two new important intermediate results have allowed us to develop the explicit analytical expression for a Lanchester attrition-rate coefficient for Bonder and Farrell’s target- acquisition policy via Taylor’s general expression for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets. These analytical results are then verified against simulation results

Evolution of the Cluster Correlation Function

We study the evolution of the cluster correlation function and its richness-dependence from z = 0 to z = 3 using large-scale cosmological simulations. A standard flat LCDM model with \Omega_m = 0.3 and, for comparison, a tilted \Omega_m = 1 model, TSCDM, are used. The evolutionary predictions are presented in a format suitable for direct comparisons with observations. We find that the cluster correlation strength increases with redshift: high redshift clusters are clustered more strongly (in comoving scale) than low redshift clusters of the same mass. The increased correlations with redshift, in spite of the decreasing mass correlation strength, is caused by the strong increase in cluster bias with redshift: clusters represent higher density peaks of the mass distribution as the redshift increases. The richness-dependent cluster correlation function, presented as the correlation-scale versus cluster mean separation relation, R_0 - d, is found to be, remarkably, independent of redshift to z <~ 2 for LCDM and z <~ 1 for TCDM (for a fixed correlation function slope and cluster mass within a fixed comoving radius). The non-evolving R_0 - d relation implies that both the comoving clustering scale and the cluster mean separation increase with redshift for the same mass clusters so that the R_0 - d relation remains essentially unchanged. The evolution of the R_0 - d relation from z ~ 0 to z ~ 3 provides an important new tool in cosmology; it can be used to break degeneracies that exist at z ~ 0 and provide precise determination of cosmological parameters.Comment: AASTeX, 15 pages, including 5 figures, accepted version for publication in ApJ, vol.603, March 200

Dynamical Confirmation of SDSS Weak Lensing Scaling Laws

Galaxy masses can be estimated by a variety of methods; each applicable in different circumstances, and each suffering from different systematic uncertainties. Confirmation of results obtained by one technique with analysis by another is particularly important. Recent SDSS weak lensing measurements of the projected-mass correlation function reveal a linear relation between galaxy luminosities and the depth of their dark matter halos (measured on 260 \hinv kpc scales). In this work we use an entirely independent dynamical method to confirm these results. We begin by assembling a sample of 618 relatively isolated host galaxies, surrounded by a total of 1225 substantially fainter satellites. We observe the mean dynamical effect of these hosts on the motions of their satellites by assembling velocity difference histograms. Dividing the sample by host properties, we find significant variations in satellite velocity dispersion with host luminosity. We quantify these variations using a simple dynamical model, measuring \mtsd a dynamical mass within 260 \hinv kpc. The appropriateness of this mass reconstruction is checked by conducting a similar analysis within an N-body simulation. Comparison between the dynamical and lensing mass-to-light scalings shows reasonable agreement, providing some quantitative confirmation for the lensing results.Comment: 7 pages, 3 figures, accepted for publication in ApJ Letter

Cluster Ellipticities as a Cosmological Probe

We investigate the dependence of ellipticities of clusters of galaxies on cosmological parameters using large-scale cosmological simulations. We determine cluster ellipticities out to redshift unity for LCDM models with different mean densities $\Omega_m$ and amplitudes of mass fluctuation $\sigma_{8,0}$. The mean ellipticity increases monotonically with redshift for all models. Larger values of $\sigma_{8,0}$, i.e., earlier cluster formation time, produce lower ellipticities. The dependence of ellipticity on $\Omega_m$ is relatively weak in the range $0.2 \leq \Omega_m \leq 0.5$ for high mass clusters. The mean ellipticity $\bar{e}(z)$ decreases linearly with the amplitude of fluctuations at the cluster redshift $z$, nearly independent of $\Omega_m$; on average, older clusters are more relaxed and are thus less elliptical. The distribution of ellipticities about the mean is approximated by a Gaussian, allowing a simple characterization of the evolution of ellipticity with redshift as a function of cosmological parameters. At $z=0$, the mean ellipticity of high mass clusters is approximated by $\bar{e}(z=0) = 0.248-0.069 \sigma_{8,0} + 0.013 \Omega_{m,0}$. This relation opens up the possibility that, when compared with future observations of large cluster samples, the mean cluster ellipticity and its evolution could be used as a new, independent tool to constrain cosmological parameters, especially the amplitude of mass fluctuations, $\sigma_{8,0}$.Comment: 16 pages, 4 figure

A Map of the Universe

We have produced a new conformal map of the universe illustrating recent discoveries, ranging from Kuiper belt objects in the Solar system, to the galaxies and quasars from the Sloan Digital Sky Survey. This map projection, based on the logarithm map of the complex plane, preserves shapes locally, and yet is able to display the entire range of astronomical scales from the Earth's neighborhood to the cosmic microwave background. The conformal nature of the projection, preserving shapes locally, may be of particular use for analyzing large scale structure. Prominent in the map is a Sloan Great Wall of galaxies 1.37 billion light years long, 80% longer than the Great Wall discovered by Geller and Huchra and therefore the largest observed structure in the universe.Comment: Figure 8, and additional material accessible on the web at: http://www.astro.princeton.edu/~mjuric/universe

Exploratory Chandra Observations of the Three Highest Redshift Quasars Known

We report on exploratory Chandra observations of the three highest redshift quasars known (z = 5.82, 5.99, and 6.28), all found in the Sloan Digital Sky Survey. These data, combined with a previous XMM-Newton observation of a z = 5.74 quasar, form a complete set of color-selected, z > 5.7 quasars. X-ray emission is detected from all of the quasars at levels that indicate that the X-ray to optical flux ratios of z ~ 6 optically selected quasars are similar to those of lower redshift quasars. The observations demonstrate that it will be feasible to obtain quality X-ray spectra of z ~ 6 quasars with current and future X-ray missions.Comment: 15 pages, ApJL, in press; small revisions to address referee Comment

The overdensities of galaxy environments as a function of luminosity and color

We study the mean environments of galaxies in the Sloan Digital Sky Survey as a function of rest-frame luminosity and color. Overdensities in galaxy number are estimated in $8 h^{-1} \mathrm{Mpc}$ and $1 h^{-1} \mathrm{Mpc}$ spheres centered on $125,000$ galaxies taken from the SDSS spectroscopic sample. We find that, at constant color, overdensity is independent of luminosity for galaxies with the blue colors of spirals. This suggests that, at fixed star-formation history, spiral-galaxy mass is a very weak function of environment. Overdensity does depend on luminosity for galaxies with the red colors of early types; both low-luminosity and high-luminosity red galaxies are found to be in highly overdense regions.Comment: submitted to ApJ

The Angular Clustering of Galaxy Pairs

We identify close pairs of galaxies from 278 deg^2 of Sloan Digital Sky Survey commissioning imaging data. The pairs are drawn from a sample of 330,041 galaxies with 18 < r^* < 20. We determine the angular correlation function of galaxy pairs, and find it to be stronger than the correlation function of single galaxies by a factor of 2.9 +/- 0.4. The two correlation functions have the same logarithmic slope of 0.77. We invert Limber's equation to estimate the three-dimensional correlation functions; we find clustering lengths of r_0= 4.2 +/- 0.4 h^{-1} Mpc for galaxies and 7.8 +/- 0.7 h^{-1} Mpc for galaxy pairs. These results agree well with the global richness dependence of the correlation functions of galaxy systems.Comment: 12 pages. ApJ, in pres