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NASA NTRS1989

Two-point correlation function of cosmic-string loops

The two-point correlations of cosmic-string loops are studied with numerical simulations of the evolution of a cosmic-string network in an expanding universe. It is found that the initial positions of loops that are chopped off the network have a correlation function that is quite similar to the highest estimates of the Abell-cluster correlation function, but these correlations are rapidly washed out by the motion of the loops. The implications for the cosmic-string galaxy-formation scenario are briefly discussed.

Bennett, David P.↗
NASA NTRS1989

The Shane Wirtanen counts - Observability of the galaxy correlation function

An explicit test of the ability of the galaxy two-point correlation function to be recovered from the Lick counts of galaxies (Shane and Wirtanen, 1967) is reported. High-fidelity simulations are created, starting with the distribution of galaxies in space. These distributions are projected onto a set of simulated Lick plates which are reduced to a uniform limiting magnitude with the method of Seldner et al. (1977). The simulated catalogs are then analyzed with the methods of Groth and Peebles (1977). The resulting correlation functions are found to reproduce accurately the correlation functions that were built in during the first step of the simulations.

Brown, Michael E.↗
NASA NTRS1981

Correlation function studies for snow and ice

The random medium model is used to characterize snow and ice fields in the interpretation of active and passive microwave remote sensing data. A correlation function is used to describe the random permittivity fluctuations with the associated mean and variance and correlation lengths; and several samples are investigated to determine typical correlation functions for snow and ice. It is shown that correlation functions are extracted directly from appropriate ground truth data, and an exponential correlation function is observed for snow and ice with lengths corresponding to the actual size of ice particles or air bubbles. Thus, given that a medium has spatially stationary statistics and a small medium, the random medium model can interpret remote sensing data where theoretical parameters correspond to actual physical parameters of the terrain.

Vallese, F.↗
NASA NTRS1977

The velocity correlation function in cosmic-ray diffusion theory

It is shown that Earl's (1973) eigenvalue sum for the cosmic-ray spatial diffusion coefficient parallel to the mean magnetic field is precisely equivalent to the time integral of the particle-velocity correlation function parallel to the mean field. A derivation due to Kubo (1957) is applied to cosmic-ray pitch-angle scattering, and it is proven that all nine components of the cosmic-ray diffusion tensor can be expressed as integrals over the velocity correlation function. A pitch-angle correlation function is derived, and the effect of long-wavelength turbulence on the velocity correlation function and spatial diffusion coefficients is examined. Application of the velocity-correlation method to a realistic case involving both pitch-angle scattering and appreciable fluctuation in the direction of the local field indicates that long-wavelength turbulence in the local field reduces the parallel diffusion coefficient and places an upper limit on the ratio of the perpendicular to parallel diffusion coefficients.

Forman, M. A.↗
NASA NTRS1988

The mean density and two-point correlation function for the CfA redshift survey slices

The effect of large-scale inhomogeneities on the determination of the mean number density and the two-point spatial correlation function were investigated for two complete slices of the extension of the Center for Astrophysics (CfA) redshift survey (de Lapparent et al., 1986). It was found that the mean galaxy number density for the two strips is uncertain by 25 percent, more so than previously estimated. The large uncertainty in the mean density introduces substantial uncertainty in the determination of the two-point correlation function, particularly at large scale; thus, for the 12-deg slice of the CfA redshift survey, the amplitude of the correlation function at intermediate scales is uncertain by a factor of 2. The large uncertainties in the correlation functions might reflect the lack of a fair sample.

De Lapparent, Valerie↗
NASA NTRS1998

A Kinematically Consistent Two-Point Correlation Function

A simple kinematically consistent expression for the longitudinal two-point correlation function related to both the integral length scale and the Taylor microscale is obtained. On the inner scale, in a region of width inversely proportional to the turbulent Reynolds number, the function has the appropriate curvature at the origin. The expression for two-point correlation is related to the nonlinear cascade rate, or dissipation epsilon, a quantity that is carried as part of a typical single-point turbulence closure simulation. Constructing an expression for the two-point correlation whose curvature at the origin is the Taylor microscale incorporates one of the fundamental quantities characterizing turbulence, epsilon, into a model for the two-point correlation function. The integral of the function also gives, as is required, an outer integral length scale of the turbulence independent of viscosity. The proposed expression is obtained by kinematic arguments; the intention is to produce a practically applicable expression in terms of simple elementary functions that allow an analytical evaluation, by asymptotic methods, of diverse functionals relevant to single-point turbulence closures. Using the expression devised an example of the asymptotic method by which functionals of the two-point correlation can be evaluated is given.

Ristorcelli, J. R.↗
NASA NTRS1983

The spatial correlation function of rich clusters of galaxies

A series of objective statistical estimators is applied to directly study the three-dimensional distribution of rich clusters, using a recently completed redshift sample of 104 Abell clusters out to distance class four and galactic latitude of 30 deg or more. The sample and the relevant positional and redshift selection effects are discussed, and the two-point angular correlation function and nearest-neighbor distribution of clusters in the sample are determined, as well as in the larger D = 5+6 sample and in several subsamples of different richness classes and latitude zones. The redshift correlation function of clusters for various angular separations on the sky is examined as a test for the reality of the observed angular correlations. The two-point spatial correlation function of clusters in the sample is determined, and the consistency among the three observed correlation functions is tested. A model that describes all the observed correlations is presented.

Bahcall, N. A.↗
NASA NTRS1993

Omega from the anisotropy of the redshift correlation function in the IRAS 2 Jansky survey

Peculiar velocities distort the correlation function of galaxies in redshift space. In the linear regime, the distortion has a characteristic quadrupole plus hexadecapole form, with amplitude depending on the cosmological density parameter Omega. I report here measurements of the anisotropy of the correlation function in the IRAS 2 Jy redshift survey. The inferred value of Omega is Omega = 0.5 + 0.5 or - 0.25.

Hamilton, A. J. S.↗
NASA NTRS2014

Two Dimensional Symmetric Correlation Functions of the S Operator and Two Dimensional Fourier Transforms: Considering the Line Coupling for P and R Lines of Linear Molecules

The refinement of the Robert-Bonamy (RB) formalism by considering the line coupling for isotropic Raman Q lines of linear molecules developed in our previous study [Q. Ma, C. Boulet, and R. H. Tipping, J. Chem. Phys. 139, 034305 (2013)] has been extended to infrared P and R lines. In these calculations, the main task is to derive diagonal and off-diagonal matrix elements of the Liouville operator iS1 − S2 introduced in the formalism. When one considers the line coupling for isotropic Raman Q lines where their initial and final rotational quantum numbers are identical, the derivations of off-diagonal elements do not require extra correlation functions of the ^S operator and their Fourier transforms except for those used in deriving diagonal elements. In contrast, the derivations for infrared P and R lines become more difficult because they require a lot of new correlation functions and their Fourier transforms. By introducing two dimensional correlation functions labeled by two tensor ranks and making variable changes to become even functions, the derivations only require the latters' two dimensional Fourier transforms evaluated at two modulation frequencies characterizing the averaged energy gap and the frequency detuning between the two coupled transitions. With the coordinate representation, it is easy to accurately derive these two dimensional correlation functions. Meanwhile, by using the sampling theory one is able to effectively evaluate their two dimensional Fourier transforms. Thus, the obstacles in considering the line coupling for P and R lines have been overcome. Numerical calculations have been carried out for the half-widths of both the isotropic Raman Q lines and the infrared P and R lines of C2H2 broadened by N2. In comparison with values derived from the RB formalism, new calculated values are significantly reduced and become closer to measurements.

infrared radiation↗
NASA NTRS2012

Causal Correlation Functions and Fourier Transforms: Application in Calculating Pressure Induced Shifts

By adopting a concept from signal processing, instead of starting from the correlation functions which are even, one considers the causal correlation functions whose Fourier transforms become complex. Their real and imaginary parts multiplied by 2 are the Fourier transforms of the original correlations and the subsequent Hilbert transforms, respectively. Thus, by taking this step one can complete the two previously needed transforms. However, to obviate performing the Cauchy principal integrations required in the Hilbert transforms is the greatest advantage. Meanwhile, because the causal correlations are well-bounded within the time domain and band limited in the frequency domain, one can replace their Fourier transforms by the discrete Fourier transforms and the latter can be carried out with the FFT algorithm. This replacement is justified by sampling theory because the Fourier transforms can be derived from the discrete Fourier transforms with the Nyquis rate without any distortions. We apply this method in calculating pressure induced shifts of H2O lines and obtain more reliable values. By comparing the calculated shifts with those in HITRAN 2008 and by screening both of them with the pair identity and the smooth variation rules, one can conclude many of shift values in HITRAN are not correct.

correlation function↗
NASA NTRS1990

Direct measurement of correlation functions in a lattice Lorentz gas

Simulations of a two-dimensional ballistic Lorentz gas on a lattice are reported. A moment-propagation technique allows direct measurements of the velocity correlation function and its moments with low relative errors for all times. The predicted 1/t-sq algebraic tails in the velocity correlation function are observed at all studied scatterer densities, unlike what has been reported for continuous systems. In the square lattice a fast oscillation is observed, consistent with the existence of staggered density modes. For the second-rank tensor correlation function, an extremely slow approach to the expected 1/t exp 3 tail is found.

Binder, P.-M.↗
NASA NTRS1988

The discrete correlation function - A new method for analyzing unevenly sampled variability data

A method for measuring correlation functions without interpolating in the temporal domain is proposed which provides an assumption-free representation of the correlation measured in the data and allows meaningful error estimates. Physical interpretation of the cross-correlation function of two series believed to be related by a convolution is shown to require knowledge of the input function's fluctuation power spectrum. Application of the method to two systems reveals no correlation for the optical data of Akn 120, but a strong correlation for the UV data of NGC 4151, placing bounds of between 1.2 and 20 light days on the size of the line-emitting region.

Edelson, R. A.↗
NASA NTRS1986

The Shane-Wirtanen counts - Plate correction factors and correlation functions

It has recently been argued that the break in the galaxy correlation function observed in the Lick counts of Shane and Wirtanen may be an artifact of plate variations in limiting magnitude. This has stimulated us to generalize our analysis of the effect of errors in the plate depths to include a possible systematic term, and to develop tests for the random and systematic errors. The tests show that neither can have appreciably affected the distinctive feature found in the galaxy two-point correlation function for the Lick sample.

Groth, Edward J.↗
NASA NTRS1971

Causality of the nonequilibrium correlation functions.

It is shown that the correct lowest-order two-particle correlation function solution to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy is given via a Green's function whose Fourier transform satisfies the analyticity requirements for causality. The Bogoliubov adiabatic approximation is shown to imply a solution which is acausal. Extension to the s-particle correlation function is discussed.

Murphy, F. X.↗
NASA NTRS1990

Recognition and characterization of hierarchical interstellar structure. I - Correlation function

The problem of the quantitative description of multiscale structure in interstellar cloud complexes and gravitational collapse calculations is considered, emphasizing the recognition and characterization of hierarchical fragmentation structure. The response of the two-point correlation function to a variety of analytical models for density structure is discussed for simple clustering of pointlike clouds to more complex models involving clouds with a distribution of sizes and densities and hierarchical substructure. By expressing the density distribution as the superposition of individual clouds, it is shown that the correlation function generates two types of terms: those involving each cloud's density convolved with itself and those involving pairs of different clouds. Major distortion of the correlation function are introduced by the presence of any image features with size scales a significant fraction of the image size.

Houlahan, Padraig↗
NASA NTRS1993

The two-point correlation function of randomly distributed Lyman alpha clouds

It is often assumed that Ly-alpha forest clouds are randomly distributed, intergalactic objects that are highly ionized by the UV background produced by quasars. If these assumptions are true, fluctuations in the UV background should produce a nonzero two point correlation function in the Ly-alpha forest. This effect, which is really just a generalization of the proximity effect, is more significant at high redshift (z is approximately 3-4) because the mean free path for UV photons is smaller there, and the fluctuations correspondingly larger. This effect was studied using both the semi-analytic techniques of Zuo's recent papers and Monte Carlo simulations. The correlation function is expected to have a small yet potentially measurable amplitude that is consistent with current upper limits. Furthermore, the signature of this effect is distinctive because the nonzero correlation function extends over the photon mean free path, which is larger than the expected scale of large-scale structure. Observations or upper limits on this effect could provide information about the source of the ionizing background at high redshifts and the nature of the Ly-alpha forest clouds.

Fardal, Mark A.↗