Assumption Of Spherical Symmetry Research Articles

Mass of Clusters in Simulations

We show that dark matter halos in N-body simulations have a boundary layer (BL), which neatly separates dynamically bound mass from unbound materials. We define T(r) and W(r) as the differential kinetic and potential energy of halos and evaluate them in spherical shells. We notice that in simulated halos such differential quantities fulfill the following properties: (1) the differential ratio = -2T/W has at least one persistent (resolution-independent) minimum , such that, close to , (2) the function w = -d log W/d log r has a maximum, while (3) the relation () w() holds. BLs are set where these three properties are fulfilled, in halos found in simulations of tilted Einstein-de Sitter and ΛCDM models, run ad hoc, using the ART and GADGET codes; their presence is confirmed in larger simulations of the same models with a lower level of resolution. Here we find that ~97% of the ~300 clusters (per model) we have with M > 4.2 × 1014 h-1 M☉ own a BL. Those clusters that appear not to have a BL are seen to be undergoing major merging processes and to grossly violate spherical symmetry. The radius ≡ rc has significant properties. First of all, the mass Mc it encloses almost coincides with the mass Mdyn, evaluated from the velocities of all particles within rc, according to the theorem. Also, materials at r > rc are shown not be in equilibrium. Using rc we can then determine an individual density contrast Δc for each virialized halo, which we compare with the virial density contrast Δv 178Ω (where Ωm is the matter density parameter) obtained assuming a spherically symmetric and unperturbed fluctuation growth. As expected, for each mass scale, Δv is within the range of values Δc. However, the spread in Δc is wide, while the average Δc is ~25% smaller than the corresponding Δv. We argue that the matching of properties derived under the assumption of spherical symmetry must be a consequence of an approximate sphericity, after violent relaxation destroyed features related to ellipsoidal nonlinear growth. On the contrary, the spread of the final Δc is an imprint of the different initial three-dimensional geometries of fluctuations and of the variable environment during their collapse, as suggested by a comparison of our results with the Sheth & Tormen analysis.

MOCASSIN: a fully three-dimensional Monte Carlo photoionization code

The study of photoionized environments is fundamental to many astrophysical problems. Up to the present most photoionization codes have numerically solved the equations of radiative transfer by making the extreme simplifying assumption of spherical symmetry. Unfortunately very few real astronomical nebulae satisfy this requirement. To remedy these shortcomings, a self-consistent, three-dimensional radiative transfer code has been developed using Monte Carlo techniques. The code, mocassin, is designed to build realistic models of photoionized nebulae having arbitrary geometry and density distributions, with both the stellar and diffuse radiation fields treated self-consistently. In addition, the code is capable of treating one or more exciting stars located at non-central locations. The gaseous region is approximated by a cuboidal Cartesian grid composed of numerous cells. The physical conditions within each grid cell are determined by solving the thermal equilibrium and ionization balance equations. This requires a knowledge of the local primary and secondary radiation fields, which are calculated self-consistently by locally simulating the individual processes of ionization and recombination. The structure and the computational methods used in the mocassin code are described in this paper. mocassin has been benchmarked against established one-dimensional spherically symmetric codes for a number of standard cases, as defined by the Lexington/Meudon photoionization workshops: at Meudon in 1985 and at Lexington in 1995 and 2000. The results obtained for the benchmark cases are satisfactory and are presented in this paper. A performance analysis has also been carried out and is discussed here.

Aspherical galaxy clusters: Effects on cluster masses and gas mass fractions

We present an investigation of the effects of asphericity on the estimates of total mass and gas mass fraction in galaxy clusters from X-ray observations. We model the aspherical shape of galaxy clusters by a triaxial model and compare the true total mass and the true total gas mass fraction with the corresponding quantities obtained with the assumption of spherical symmetry. In the triaxial model we allow the extent along the line of sight to vary in order to describe elongated and compressed cluster shapes. Using a sample of 10 ROSAT clusters and a recent CHANDRA observation we find the following results. For prolate or oblate shapes the difference between triaxial and spherical model both in the mass and in the gas mass fraction are negligible (less than 3%). For more aspherical shapes the total mass is underestimated (overestimated) in the centre, if the cluster is compressed (elongated). The gas mass fraction is overestimated for compressed clusters and slightly underestimated for elongated clusters. Comparing X-ray masses with gravitational lensing estimates, we find that elongations along the line of sight can resolve discrepancies of masses determined by the two different methods of up to ∼30%. The combination of Sunyaev- Zel'dovich and X-ray observations is useful to measure the elongation of the cluster along the line of sight. As an application, we estimate the elongation of the cluster CL0016+16 with two different approaches, Sunyaev-Zel'dovich measurements and comparison of weak lensing and X-ray masses, and find reasonable agreement.

GPS radio holography as a tool for remote sensing of the atmosphere, mesosphere, and terrestrial surface from space

GPS radio occultation (RO) signals are highly coherent and precise, and thus sufficient for holographic investigation of the atmosphere, ionosphere, and the Earth's surface from space. In principle, three-dimensional radio-holographic remote sensing is possible by using new radio holographic equations to retrieve the radio field within the atmosphere from a radio field known at some interface outside the atmosphere. A simplified two-dimensional form of the radio-holographic equations which are developed under an assumption of local spherical symmetry can be used to obtain two-dimensional radio images of the atmosphere and terrestrial surface. To achieve this, radio holograms recorded by a GPS receiver onboard a low earth orbit (LEO) satellite at two GPS frequencies can be used and focused synthetic aperture principle applied. Analysis of GPS/MET RO data is presented to show the effectiveness of a radio-holographic approach. It is shown that the amplitude of GPS radio signals (in addition to phase data) can be used to obtain detailed altitude profiles of the vertical gradient of refractivity in the atmosphere and electron density in the mesosphere. The results demonstrate the applicability of GPS radio holography for a detailed global study of the natural processes in the atmosphere and mesosphere.

Singularity threshold of the nonlinear sigma model using 3D adaptive mesh refinement

Numerical solutions to the nonlinear sigma model, a wave map from $3+1$ Minkowski space to ${S}^{3},$ are computed in three spatial dimensions using adaptive mesh refinement. For initial data with compact support the model is known to have two regimes: one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states.

Gravitating mass profiles of nearby galaxy clusters and relations with X-ray gas temperature, luminosity and mass

We consider a sample of 22 nearby clusters of galaxies observed with the Medium Energy Concentrator Spectrometer (MECS) on board of BeppoSAX. They cover the range in gas temperature between 3 and 10 keV, with bolometric X-ray luminosity between 2e44 erg/s and 6e45 erg/s. Using the de-projected gas temperature and density profiles resolved in a number of bins between 5 and 7 and obtained from this dataset only, we recover the total gravitating mass profiles for 20 objects just applying the (i) spherical symmetry and (ii) hydrostatic equilibrium assumptions. We investigate the correlations between total mass, gas temperature, luminosity at several overdensities values and find that the slopes of these relations are independent from the considered overdensity and consistent with what is predicted from the cluster scaling laws. The best-fit results on the normalization of the M-T relation are slightly lower, but still consistent considering the large errors that we measure, with hydrodynamical simulations. A segregation between relaxed and non-relaxed systems is present in each plane of these relations pointing out to a significant component in their intrinsic scatter. This segregation becomes more evident at higher overdensities and when physical quantities, like M_gas and L, that are direct function of the amount of gas observed, are considered.

Sensitivity analysis for airborne sounding of the troposphere by GNSS radio occultation

The usual geometry for radio occultation sounding using global navigation satellite system (GNSS) signals has the receiver placed on a low earth orbit (LEO) satellite. We investigate a new geometric approach, assuming an airborne rather than a spaceborne receiver. Information on the refractivity structure and hence the pressure, temperature, and humidity can be retrieved from accurate airborne measurements of amplitude and phase delay of the signals occulted by the troposphere. We present some advantages and disadvantages for the concept of making measurements from commercial aircraft equipped with proper GNSS receivers and antennae compared to the spaceborne case. We simulated realistic airborne occultation observations and assessed the characteristics of their geometry and sampling. We also compared the dynamic range of the signal with the magnitude of error sources that affect the measurements. Findings include that an airborne system has the potential to provide more profiles per unit area below 10 km height than a constellation of up to 25 satellites over the North Atlantic (though with inferior global coverage), and that the signal to noise ratio (SNR) should be better below 5 km than in the LEO case. Though the receiver velocity error is larger than for the LEO system, it is still small enough relative to the signal level to retrieve useful information. The estimated sensitivity of the technique is better than 0.1% refractivity at 3 km altitude increasing to 0.5% refractivity at 11 km. Because of the large horizontal drift of the tangent point of up to 450 km, the assumption of spherical symmetry in the existence of significant 3D variations in structure is expected to be a major error source, in addition to the airplane velocity uncertainty.

A Statistical Estimate of Errors in the Calculation of Radio-Occultation Bending Angles Caused by a 2D Approximation of Ray Tracing and the Assumption of Spherical Symmetry of the Atmosphere

Abstract Atmospheric data from National Centers for Environmental Prediction (NCEP) analyses and orbital parameters from 133 real Global Positioning System (GPS) meteorological data soundings are used to compute the “true” bending angle profiles using an accurate 3D ray-tracing procedure. They are then compared with approximate profiles obtained using the spherical symmetry assumption and an efficient 2D ray-tracing model. The average fractional error of the bending angles due to the spherical symmetry assumption is less than 0.15%. The average fractional error due to the use of the 2D ray-tracing model is slightly greater than that due to the spherical symmetry assumption. The vertical error correlations due to the spherical symmetry assumption are sharp between 6 and 8 km and broader above and below this layer. The vertical error correlations associated with the 2D forward model show a nearly diagonal structure below 15 km with high correlation confined to a 2-km layer centered at the observation level.

Improved Mapping of Tropospheric Delays

The authors compare several methods to map the a priori tropospheric delay of global positioning system (GPS) signals from the zenith direction to lower elevations. This is commonly achieved with so-called mapping functions. Dry mapping functions are applied to the hydrostatic delay; wet mapping functions are used to map the zenith wet delay to lower elevation angles. The authors compared the following mapping techniques against raytraced delays computed for radiosonde profiles under the assumption of spherical symmetry: (a) the Niell mapping function; (b) mapping through the COSPAR International Reference Atmosphere with added water vapor climatology; (c) the same as b with added use of surface meteorological temperature, pressure, and humidity; and (d) use of the numerical reanalysis model of the National Centers for Environmental Prediction–National Center for Atmospheric Research. Based on comparisons with all available global radiosondes (∼1000 per day), for every fifth day of 1997 (73 days), the authors found that dry mapping based on method d performs 2–3 times better than a for elevations 15° and below. The authors further report that b and c perform better dry mapping than a, with an improvement of ∼50%. Smaller improvements are also shown for wet delay mapping by b, c, and d as compared to a. At 5° and below, the Niell dry mapping function has biases that vary with season by 1%, and it displays significant systematic errors (2%–4% at 5° elevation) between 30° and 90° southern latitude during the northern winter months. It is concluded that the most demanding meteorological and geodetic GPS applications should use location- and time-specific “direct” mapping functions such as b, c, or d rather than parameterized functions, especially if low elevation observations are used. The authors describe how this improved mapping can be implemented in GPS analysis software.

Radio occultation bending angle and impact parameter errors caused by horizontal refractive index gradients in the troposphere: A simulation study

Radio occultation (RO) bending angle and impact parameter values are derived from a Doppler shift measurement, assuming spherical symmetry. The purpose of this work is to illustrate the errors that arise when this assumption is not valid. Doppler shift values have been simulated for ray paths through a three‐dimensional refractive index field derived from a mesoscale model forecast, which has a horizontal grid of 12 km by 12 km, and includes water vapor. These have then been inverted, making the spherical symmetry assumption. It is demonstrated that refractive index gradients perpendicular to the ray path can cause errors in both the bending angle and impact parameter values, but the latter is the more significant. It is shown that the impact parameter value at the tangent point can differ by around ∼100 m from the derived value. This can cause an effective bending angle error exceeding ∼10% near the surface. A statistical analysis of the errors caused by horizontal gradients for simulations through 54 mesoscale forecasts, using fixed spacecraft trajectories and tangent point locations, is presented. In general, the bending angle errors are found to be ∼3% near the surface. A new set of analytical expressions for errors has been derived. These are based on integrating the horizontal gradients along the ray path and are found to be in good agreement with the simulation results. The implications of this work for the assimilation of RO data into numerical weather prediction models are discussed and areas of future work are outlined.

Comparisons of GPS/MET retrieved ionospheric electron density and ground based ionosonde data

The Global Positioning System/Meteorology (GPS/MET) mission has been the first experiment to use a low Earth orbiting (LEO) satellite (the MicroLab-1) to receive multi-channel Global Positioning System (GPS) carrier phase signals and demonstrate active limb sounding of the Earth’s atmosphere and ionosphere by radio occultation technique. Under the assumption of spherical symmetry at the locality of the occultation, the dual-band phase data have been processed to yield ray-path bending angle profiles, which have then been used to yield profiles of refractive index via the Abel integral transform. The refractivity profiles can then, in turn, yield profiles of ionospheric electron density and other atmospheric variables such as neutral atmospheric density, pressure, and temperature in the stratosphere and upper troposphere, and water vapor in the lower troposphere with the aid of independent temperature data. To approach a near real-time process, electron density profiles can also be derived by the Abel transform through the computation of total electron content (TEC) assuming straight-line propagation (neglecting bending). In order to assess the accuracy of the GPS/MET ionospheric electron density retrievals, coincidences of ionosonde data with GPS/MET occultations have been examined. The retrieved electron density profiles from GPS/MET TEC observations have been compared with ionogram inversion results derived from digital ionospheric sounders operated by the National Central University (the Chung-Li digisonde; 24.6°N, 121.0°E) and by Utah State University (the Bear-Lake dynasonde; 41.9°N, 111.4°W). A fuzzy classification method for the automatic identification and scaling of ionogram traces has been applied to recorded ionograms, and then bottomside ionospheric electron density profiles are determined from true-height analysis. The comparison results show better agreement for both of the derived electron density profiles and the F2-layer critical frequency ( foF2) at mid-latitude observations than at low-latitude observations. The rms foF2 differences from the GPS/MET retrievals are 0.61 MHz to the Bear-Lake dynasonde measurements and 1.62 MHz to the Chung-Li digisonde measurements.

Ionospheric tomography using Ørsted GPS measurements - preliminary results

Data from the Turbo-Rogue GPS receiver onboard the Danish Ørsted Satellite are used to derive satellite-to-satellite Total Electron Content (TEC). We present preliminary results obtaining Electronic Density fields from ionospheric tomography using TEC Ørsted data. TEC is based on single frequency measurements, since only the L1 signal is of good quality. The C/A pseudo range (C1) and the L1 phase are used to obtain the TEC measurement. Electron density profiles from individual occultations are derived, based on the Abel transform, and compared with the tomographical solution. The assumption of spherical symmetry in the Abel transform limits the accuracy of the profiles and often results in a bias. We select occultations observed along the orbital plane to minimize the spherical assymetry effects. The tomographic grid is also confined to a narrow torus close to the orbital plane. The TEC calculations are based on data from 6 hour (3.6 orbits) periods. Two days have been selected for this study, December 5, 1999 and February 12, 1999.

Mid‐Infrared Interferometry on Spectral Lines. III. Ammonia and Silane around IRC +10216 and VY Canis Majoris

Using the University of California Berkeley Infrared Spatial Interferometer with a radio frequency (RF) filter bank, the first interferometric observations of mid-infrared molecular absorption features of ammonia (NH3) and silane (SiH4) with very high spectral resolution (λ/Δλ ~ 105) were made. Under the assumptions of spherical symmetry and uniform outflow, these new data permitted the molecular stratification around carbon star IRC +10216 and red supergiant VY CMa to be investigated. For IRC +10216, both ammonia and silane were found to form in the dusty outflow significantly beyond both the dust formation and gas acceleration zones. Specifically, ammonia was found to form before silane in a region of decaying gas turbulence (20R*), while the silane is produced in a region of relatively smooth gas flow much farther from the star (80R*). The depletion of gas-phase SiS onto grains soon after dust formation may fuel silane-producing reactions on the grain surfaces. For VY CMa, a combination of interferometric and spectral observations suggest that NH3 is forming near the termination of the gas acceleration phase in a region of high gas turbulence (~40R*).

Mass‐to‐Light Ratios of Groups and Clusters of Galaxies

We constrain the mass-to-light ratios, gas mass fractions, baryon mass fractions and the ratios of total to luminous mass for a sample of eight nearby relaxed galaxy groups and clusters: A262, A426, A478, A1795, A2052, A2063, A2199 and MKW4s. We useASCA spatially resolved spectroscopic X-ray observations and ROSAT PSPC images to constrain the total and gas masses of these clusters. To measure cluster luminosities we use galaxy catalogs resulting from the digitization and automated processing of the second generation Palomar Sky Survey plates calibrated with CCD images in the Gunn-Thuan g, r, and i bands. Under the assumption of hydrostatic equilibrium and spherical symmetry, we can measure the total masses of clusters from their intra-cluster gas temperature and dens ity profiles. Spatially resolved ASCA spectra show that the gas temperature decreases with increasing distance from the center. By comparison, the assumption that the gas is isothermal results in an underestimate of the total ma ss at small radii, and an overestimate at large cluster radii. We have obtained luminosity functions for all clusters in our sample. After correcting for background and foreground galaxies, we estimate the total cluster luminosity using Schechter function fits to the galaxy catalogs. In the three lowest redshift clusters where we can sample to fainter absolute magnitudes, we have detected a flattening of the luminosity function at intermediate magni tudes and a rise at the faint end. These clusters were fitted with a sum of two Schechter functions. The remaining cl usters were well fitted with a single Schechter function. Assuming H0 = 50 h50 km s -1 Mpc -1 , the measured mass-to-light ratios are ∼ 100 h50 M⊙/L⊙. This, along with a high baryonic fraction, is indicative of a low density universe with 0 ∼ 0.15 - 0.2. Subject headings: Cosmology — galaxies: clusters: individual — intergalactic medium — X-rays: galaxies

Radio holographic principle for observing natural processes in the atmosphere and retrieving meteorological parameters from radio occultation data

The radio holographic principle is briefly described and tested by using radio occultation data of the GPS/MET and MIR/GEO experiments. Sub-Fresnel spatial resolution ∼12 m/pixel was achieved using focused synthetic aperture radio holographic approach, and direct evidence of multibeam propagation effects in the atmosphere was obtained. The achieved instrumental accuracy in angular distance measurements was near 0.004 milliradian/pixel, and observed angular distance between different rays was equal to 0.3 milliradians. The angular resolution of the radio holographic method depends on the wavelength as λ1 compared to λ1/2 in conventional methods. In general case the principal limit of the vertical resolution may be determined using focused synthetic aperture antenna theory and may achieve a value ∼20–40 m under assumptions of spherical symmetry and quiet atmospheric conditions. Wave structures were discovered in the altitude distribution of the gradient electron density at a height interval of 60–95 km with spatial period 1–2 km and vertical resolution 300–500 m. Good correspondence was found between the temperature profiles revealed by radio holographic analysis and those obtained by traditional retrieval using UCAR GPS/MET data.

Improving the Abel inversion by adding ground GPS data to LEO radio occultations in ionospheric sounding

GPS radio occultations allow the sounding of the Earth's atmosphere (i.e. troposphere and ionosphere). The basic observable of this technique is the additional delay, due to the refractivity index, of a radio signal when passing through the atmosphere. This additional delay is proportional to the integrated refractivity, in such a way that we can obtain an estimation of the vertical refractivity profiles using observations at different elevation angles by solving an inverse problem. Traditionally, the solution of this inverse problem is obtained by using the Abel inversion algorithm assuming a refractivity index that only depends on the altitude. In this paper we present a modified Abel inversion algorithm for ionospheric sounding that overcomes the spherical symmetry assumption of the traditional Abel inversion algorithm. Processing a set of simulated data and 1 day of real data with this algorithm, a clear improvement over the traditional one can be obtained when comparing the derived critical frequencies with the ionosonde measurements. It is also shown that this improvement is sufficient to measure critical frequencies associated with the ionospheric E layer.

A 450 Day Light Curve of the Radio Afterglow of GRB 970508: Fireball Calorimetry

We report on the results of an extensive monitoring campaign of the radio afterglow of GRB 970508, lasting 450 days after the burst. The spectral and temporal radio behavior indicate that the fireball has undergone a transition to subrelativistic expansion at t ~ 100 days. This allows us to perform calorimetry of the explosion. The derived total energy, E0 ~ 5 × 1050 ergs, is well below the ~5 × 1051 ergs inferred under the assumption of spherical symmetry from gamma-ray and early afterglow observations. A natural consequence of this result, which can also account for deviations at t < 100 days from the spherical relativistic fireball model predictions, is that the fireball was initially a wide-angle jet of opening angle ~30°. Our analysis also allows determination of the energy fractions carried by the electrons and magnetic field and the density of the ambient medium surrounding the fireball. We find that during the subrelativistic expansion, the electrons and magnetic field are close to equipartition and the density of the ambient medium is ~1 cm -3. The inferred density rules out the possibility that the fireball expands into a strongly nonuniform medium, as would be expected, e.g., in the case of a massive star progenitor.

Stationary and static stellar dynamic models with axial symmetry

We study the existence of stationary solutions of the Vlasov-Poisson system with finite radius and finite mass in the stellar dynamics case. So far, the existence of such solutions is known only under the assumption of spherical symmetry. Using the implicit function theorem we show that certain stationary, spherically symmetric solutions can be embedded in one parameter families of stationary, axially symmetric solutions with finite radius and finite mass. In general, these new steady states have non-vanishing average velocity field, but they can also be constructed such that their velocity field does vanish, in which case they are called static.

Biases in Expansion Distances of Novae Arising from the Prolate Geometry of Nova Shells

ABSTRACTExpansion distances (or expansion parallaxes) for classical novae are based on comparing a measurement of the shell expansion velocity, multiplied by the time since outburst, with some measure of the angular size of the shell. We review and formalize this method in the case of prolate spheroidal shells. For such shells there is no unique angular size except when the shell is seen pole‐on, and several different measures of angular size have been used in the literature. We present expressions for the maximum line‐of‐sight velocity from a complete, expanding shell and for its projected major and minor axes, in terms of the intrinsic axis ratio and the inclination of the polar axis to the line of sight. For six distinct definitions of “angular size” we tabulate the error in distance that is introduced under the assumption of spherical symmetry (i.e., without correcting for inclination and axis ratio). The errors can be significant and systematic and can affect studies of novae whether considered individually or statistically. Each of the six estimators overpredicts the distance when the polar axis is close to the line of sight, and most underpredict the distance when the polar axis is close to the plane of the sky. Use of the straight mean of the projected semimajor and semiminor axes gives the least distance bias for an ensemble of randomly oriented prolate shells, and we recommend this method when individual inclinations and axis ratios cannot be ascertained. The best individual expansion distances, however, result from a full spatio‐kinematic modeling of the nova shell. We discuss several practical complications that affect expansion distance measurements of real nova shells. We recommend that nova shell expansion distances be based on velocity and angular size measurements made contemporaneously if possible and that the same ions and transitions be used for the imaging and velocity measurements. We emphasize the need for complete and explicit reporting of measurement procedures and results, regardless of the specific method used.

Extrusion movements of a tunnel head reinforced by finite length bolts—a closed-form solution using homogenization approach

An analytical model on the general behaviour of a tunnel head, reinforced by finite length bolts is proposed. This model is based on the homogenization method and spherical symmetry assumption. Despite its simplicity, and in consequence its limits of validity, it does allow a quick estimation of the key design parameters: frontal displacement, extension of decompressed zone, ground stresses and bolt tension, and constitutes thereby a very useful and handy tool for design engineers. In particular, the influence of the reinforcement length, as well as other important design parameters, are studied by the proposed model. The charts resulting from the parametric studies are directly applicable. Otherwise, the comparison to a 3D numerical model is also presented in this paper. The first results provide the validation of the analytical solution, at least in terms of average extrusion movements. Copyright © 2000 John Wiley & Sons, Ltd.