Assumption Of Spherical Symmetry Research Articles

Structure and Evolution of the Envelopes of Deeply Embedded Massive Young Stars

The physical structure of the envelopes around a sample of 14 massive young stars is investigated using maps and spectra in submillimeter continuum and lines of C17O, CS, C34S, and H2CO. Nine of the sources are highly embedded luminous (103-105 L☉) young stellar objects that are bright near-infrared sources but weak in radio continuum; the other objects are similar but not bright in the near-infrared and contain "hot-core"-type objects and/or ultracompact H II regions. The data are used to constrain the temperature and density structure of the circumstellar envelopes on 102-105 AU scales, to investigate the relation between the different objects, and to search for evolutionary effects. The total column densities and the temperature profiles are obtained by fitting self-consistent dust models to submillimeter photometry. The calculated temperatures range from 300 to 1000 K at ~102 AU and from 10 to 30 K at ~105 AU from the star. Visual extinctions are a few hundred to a few thousand magnitudes, assuming a grain opacity at 1300 μm of ≈1 cm-2 g-1 of dust, as derived earlier for one of our sources. The mid-infrared data are consistent with a 30% decrease of the opacity at higher temperatures, caused by the evaporation of the ice mantles. The CS, C34S, and H2CO data as well as the submillimeter dust emission maps indicate density gradients n ∝ r-α. Assuming a constant CS abundance throughout the envelope, values of α = 1.0-1.5 are found, which is significantly flatter than the α = 2.0 ± 0.3 generally found for low-mass objects. This flattening may indicate that in massive young stellar objects, nonthermal pressure is more important for the support against gravitational collapse, while thermal pressure dominates for low-mass sources. We find α = 2 for two hot-core-type sources but regard this as an upper limit since, in these objects, the CS abundance may be enhanced in the warm gas close to the star. The assumption of spherical symmetry is tested by modeling infrared absorption line data of 13CO, CS emission-line profiles and near-infrared continuum. There is a distinct, but small deviation from spherical symmetry: the data are consistent with a decrease of the optical depth by a factor of ≈3 in the central ≲10''. The homogeneity of the envelopes is verified by the good agreement of the total masses in the power-law models with the virial masses. Modeling of C17O emission shows that ≈40%-90% of the CO is frozen out onto the dust. The CO abundances show a clear correlation with temperature, as expected if the abundance is controlled by freeze-out and thermal desorption. The CS abundance is 3 × 10-9 on average, ranging from (4-8) × 10-10 in the cold source GL 7009S to (1-2) × 10-8 in the two hot-core-type sources. Dense outflowing gas is seen in the CS and H2CO line wings; the predominance of blueshifted emission suggests the presence of dense, optically thick material within 10'' of the center. Interferometric continuum observations at 1300-3500 μm show compact emission, probably from a 03 diameter, optically thick dust component, such as a dense shell or a disk. The emission is a factor of 10-100 stronger than expected for the envelopes seen in the single-dish data, so that this component may be opaque enough to explain the asymmetric CS and H2CO line profiles. The evolution of the sources is traced by the overall temperature (measured by the far-infrared color), which increases systematically with the decreasing ratio of envelope mass to stellar mass. The observed anticorrelation of near-infrared and radio continuum emission suggests that the erosion of the envelope proceeds from the inside out. Conventional tracers of the evolution of low-mass objects do not change much over this narrow age range.

The origin and formation of cuspy density profiles through violent relaxation of stellar systems

It is shown that the cuspy density distributions observed in the cores of elliptical galaxies can be realized by dissipationless gravitational collapse. The initial models consist of power-law density spheres such as ρ∝r−1 with anisotropic velocity dispersions. Collapse simulations are carried out by integrating the collisionless Boltzmann equation directly, on the assumption of spherical symmetry. From the results obtained, the extent of constant density cores, formed through violent relaxation, decreases as the velocity anisotropy increases radially, and practically disappears for extremely radially anisotropic models. As a result, the relaxed density distributions become more cuspy with increasing radial velocity anisotropy. It is thus concluded that the velocity anisotropy could be a key ingredient for the formation of density cusps in a dissipationless collapse picture. The velocity dispersions increase with radius in the cores according to the nearly power-law density distributions. The power-law index, n, of the density profiles, defined as ρ∝r−n, changes from n≈2.1 at intermediate radii to a shallower power than n≈2.1 toward the centre. This density bend can be explained from our postulated local phase-space constraint that the phase-space density accessible to the relaxed state is determined at each radius by the maximum phase-space density of the initial state.

Direct Analysis of Spectra of the Type Ic Supernova SN 1994I

Synthetic spectra generated with the parameterized supernova synthetic-spectrum code SYNOW are compared to observed photospheric-phase spectra of the Type Ic supernova SN 1994I. The observed optical spectra can be well matched by synthetic spectra that are based on the assumption of spherical symmetry. We consider the identification of the infrared absorption feature observed near 10250 ?, which previously has been attributed to He I ?10830 and regarded as strong evidence that SN 1994I ejected some helium. We have difficulty accounting for the infrared absorption with He I alone. It could be a blend of He I and C I lines. Alternatively, we find that it can be fitted by Si I lines without compromising the fit in the optical region. In synthetic spectra that match the observed spectra, from 4 days before to 26 days after the time of maximum brightness, the adopted velocity at the photosphere decreases from 17,500 to 7000 km s-1. Simple estimates of the kinetic energy carried by the ejected mass give values that are near the canonical supernova energy of 1051 ergs. The velocities and kinetic energies that we find for SN 1994I in this way are much lower than those that we find elsewhere for the peculiar Type Ic SNe 1997ef and 1998bw, which therefore appear to have been hyperenergetic.

Neutrino transport in core collapse supernovae

Core collapse supernovae occupy a special place in the cosmic hierarchy for their role in disseminating and producing most elements in the Universe heavier than hydrogen and helium, without which life as we know it would not be possible. These catastrophic events mark the end of a massive star's life and the birth of a neutron star or a black hole, and are the most energetic explosions in the Cosmos. Core collapse supernovae result when the iron core of a massive star becomes unstable at the end of the star's evolution, collapses on itself, rebounds at ultra-high densities, and produces a shock wave that will ultimately be responsible for disrupting the star. As infalling core material passes through the shock, it is compressed and heated, and the core nuclei are dissociated (broken up) at the expense of thermal, pressure-producing energy behind the shock, thereby weakening it. In addition to this energy loss, energy is carried away from the shocked region by massless particles know as “neutrinos”. The shock stalls, and is later thought to be revived by a “neutrino heating” mechanism. At the time the shock stalls, the core consists of an inner “neutrinosphere” radiating neutrinos and “antineutrinos” of three “flavors”: “electron”, “muon”, and “tau” neutrinos and their antineutrinos. This inner core will ultimately radiate away its thermal energy, cool, and go on to form a neutron star or a black hole. Revival of the stalled shock above the neutrinosphere is mediated by the absorption of electron neutrinos and antineutrinos emerging from the radiating proto-neutron star. This heating depends sensitively on the neutrino luminosities, spectra, and distribution of neutrino direction cosines in the region behind the shock. In turn, this depends on the neutrino transport through three regions: the “neutrino-thick”, diffusion region deep within the core below the neutrinosphere, the “semitransparent” region encompassing the neutrinosphere, and the “neutrino-thin”, streaming region at larger radii. Sufficient accuracy for a definitive simulation of the supernova outcome can be obtained only via a solution of the neutrino Boltzmann transport equations and their coupling to the hydrodynamics equations governing the evolution of the core material. In this article we present a numerical method to solve the neutrino Boltzmann equations coupled to the core hydrodynamics. Spherical symmetry is assumed, but our methods extend to multidimensional Boltzmann transport simulations. (With the assumption of spherical symmetry, radius is the only spatial variable, rendering the simulation, at least as far as the spatial dimensions are concerned, one-dimensional. However, as we will discuss, the Boltzmann equation is a “phase space” equation in radius and neutrino direction cosine and energy, and therefore, inherently multidimensional even when spherical symmetry is assumed.) We also present the results of comparisons of “multigroup flux-limited diffusion” (approximate) neutrino transport and Boltzmann (exact) neutrino transport in post-core bounce supernova environments, with an eye toward the quantities central to the neutrino-heating, shock-revival mechanism. Multigroup flux-limited diffusion is the most sophisticated transport approximation implemented thus far in core collapse supernova simulations, and will be described in some detail in this article. Our results demonstrate that differences significant to shock revival are obtained with the two transport schemes, supporting the claim that accurate neutrino transport is paramount in simulations of these important cosmic events. We discuss the ramifications our results have for our ongoing simulations of core collapse supernovae with exact Boltzmann neutrino transport.

The Geometry of HD 165763: A Polarization Study of a WC Star

We have obtained spectropolarimetric data of HD 165763 (WR 111, WC 5) with a spectral resolution of 1.24 A, covering the wavelength range from 4950 to 6200 A. The continuum is polarized at a level of 0.39% at 5805 A, but there is no polarization variation across the emission lines. The latter indicates that most of the polarization arises from the interstellar medium. It further suggests that any global deviation of the atmosphere from spherical symmetry, if it exists, is small. Radiative transfer calculations of axisymmetric stellar wind models are used to predict polarization changes across the very strong C IV (λ5805) emission line. We fitted the observational data with the models by using the continuum polarization as a constraint and by treating the interstellar polarization as a free parameter instead of using unreliable values of interstellar polarization estimated from analysis of field stars. The results from the χ2 testing of the model suggest that the global deviation from spherical symmetry of this object is no larger than 20%, and it is probably less than 10%. In our formulation, the ratio of the equatorial density and the polar density (ρeq/ρpole) corresponding to the 20% upper limit is about 1.25. A similar conclusion is obtained from comparison of continuum-minus-line polarization of the observations with that of our models. None of the single WC stars (except for WR 103) with spectropolarimetric data show a variation in polarization across emission lines. Therefore, global deviations from spherical symmetry of WC stars are expected to be small in general. The relatively low value of the upper limit for WR 111 indicates that mass-loss enhancement due to rotation is unlikely to explain the difference between the observed and the predicted WC mass-loss rates. It also suggests that a significant amount of angular momentum is removed by mass loss during the pre-WC star stage of stellar evolution. A low value for the upper limit of deviation from spherical geometry is important, since it validates the assumption of spherical symmetry used in many current evolutionary and atmospheric models of single WC stars.

X‐Ray Mass Estimates atz∼ 0.3 for the Canadian Network for Observational Cosmology Cluster Sample

Results are presented from the analysis of ROSAT High-Resolution Imager (HRI) and Position-Sensitive Proportional Counter (PSPC) observations of the Canadian Network for Observational Cosmology (CNOC) subsample of the Extended Medium-Sensitivity Survey (EMSS) high-redshift galaxy clusters. X-ray surface brightness profiles of 14 clusters with 0.17 < z < 0.55 are constructed and fit to isothermal β models. Where possible, we use both the HRI and PSPC data to constrain the fit. Under the assumptions of isothermality, hydrostatic equilibrium, and spherical symmetry, we derive total X-ray masses within a range of radii from 141 to 526 h-1100 kpc. These masses are compared with both the dynamical masses obtained from galaxy velocities and the projected masses from published gravitational lensing studies. We find no systematic bias between X-ray and dynamical methods across the sample, with an average MDyn/MX=1.04 ± 0.07, although individual clusters exhibit mass discrepancies up to a factor of 2. We estimate that the systematic effects due to cooling flows, nonequilibrium systems, and temperature gradients affect the average mass ratio by no more than 15%-20%. Weak gravitational lensing masses appear to be systematically higher than X-ray results by factors of ~50%, while strong-lensing estimates show larger discrepancies (factors of ~2.5). However, these comparisons are complicated by the need to extrapolate the X-ray data to larger or smaller radii. We calculate X-ray-derived cluster gas masses, from which we obtain a cluster baryon fraction of ~5% h-3/2100, yielding Ω0~0.3 h-1/2100.

Metric fluctuation corrections to Hawking radiation

We study how fluctuations of the black hole geometry affect the properties of Hawking radiation. Even though we treat the fluctuations classically, we believe that the results so obtained indicate what might be the effects induced by quantum fluctuations in a self-consistent treatment. To characterize the fluctuations, we use the model introduced by York in which they are described by an advanced Vaidya metric with a fluctuating mass. Under the assumption of spherical symmetry, we solve the equation of null outgoing rays. Then, by neglecting the greybody factor, we calculate the late time corrections to the s-wave contributions of the energy flux and the asymptotic spectrum. We find three kinds of modifications. First, the energy flux fluctuates around its average value with amplitudes and frequencies determined by those of the metric fluctuations. Secondly, this average value receives two positive contributions, one of which can be reinterpreted as due to the ``renormalization'' of the surface gravity induced by the metric fluctuations. Finally, the asymptotic spectrum is modified by the addition of terms containing thermal factors in which the frequency of the metric fluctuations acts as a chemical potential.

Conference Summary: The Demise of Spherical and Stationary Winds

AbstractThe observational evidence and the theoretical work presented at this colloquium demonstrate that significant non-spherical effects and time variations are very common phenomena in the winds from hot stars. Hence, the assumptions of spherical symmetry and stationarity, while highly successful for the theory of stellar structure, appear to be inadequate for describing and understanding the winds of hot stars. This conclusion has significant consequences for the appearance and evolution of hot stars.

Finite time and asymptotic behaviour of the maximal excursion of a random walk

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed form and is an approximation of the exact distribution comparable to that obtained by real space renormalization methods. Then we focus on the early time behaviour of this quantity. The instantaneous diffusion exponent $\nu_n$ exhibits a systematic overshooting of the long time exponent. Exact results are obtained in one dimension up to third order in $n^{-1/2}$. In two dimensions, on a regular lattice and on the Sierpi\'nski gasket we find numerically that the analytic scaling $\nu_n \simeq \nu+A n^{-\nu}$ holds.

Spherically symmetric static solution for colliding null dust

The Einstein equations are completely integrated in the presence of two (incoming and outgoing) streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is also written in double null and in radiation coordinates and it is reinterpreted as an anisotropic fluid. Interior matching with a static fluid and exterior matching with the Vaidya solution along null hypersurfaces is discussed. The connection with two-dimensional dilaton gravity is established.

X‐Ray Detection of the Primary Lens Galaxy Cluster of the Gravitational Lens System Q0957+561

Analysis of several recent ROSAT HRI observations of the gravitationally lensed system Q0957+561 has led to the detection at the 3sigma level of the cluster lens containing the primary galaxy G1. The total mass was estimated by applying the equation of hydrostatic equilibrium to the detected hot intracluster gas for a range of cluster core radii, cluster sizes and for different values of the Hubble constant. X-ray estimates of the lensing cluster mass provide a means to determine the cluster contribution to the deflection of rays originating from the quasar Q0957+561. The present mass estimates were used to evaluate the convergence parameter kappa, the ratio of the local surface mass density of the cluster to the critical surface mass density for lensing. The convergence parameter, kappa, calculated in the vicinity of the lensed images, was found to range between 0.07 and 0.21, depending on the assumed cluster core radius and cluster extent. This range of uncertainty in kappa does not include possible systematic errors arising from the estimation of the cluster temperature through the use of the cluster luminosity-temperature relation and the assumption of spherical symmetry of the cluster gas. Applying this range of values of kappa to the lensing model of Grogin & Narayan (1996) for Q0957+561 but not accounting for uncertainties in that model yields a range of values for the Hubble constant:67<H_0<82 km s^-1 Mpc^-1, for a time delay of 1.1 years.

The Mass and Mass‐to‐Light Ratio for Clusters of Galaxies at Large Radii

We construct models describing the velocity field in the infall regions of clusters of galaxies. In all models the velocity field is the superposition of a radial systematic component that is assumed to be spherically symmetric and a component of random nature. The latter accounts for the combined effects of small-scale substructure and observational errors. The effect of the noise term is to smear out the caustic envelopes searched for by previous groups, resulting in models that resemble the observations quite well. When the systematic component is known, the infall velocity and the mass profile of the infall region of the clusters can be determined. Given a particular model, it is possible to calculate the distribution function f(cz, θ, m) at all points in the observable (cz, θ, m) phase space outside the virialized region. It is demonstrated, on simulated data, that the use of a maximum likelihood estimator enables identification of the correct model, even at realistic noise levels. To minimize systematic effects caused by deviations from spherical infall, observations from a number of clusters should be used when applying the method to a real data set. Combining data from different clusters is a straightforward procedure. In the models the mass-to-light ratio is allowed to vary freely with radius. Furthermore, the models do not require any knowledge of the virial region. Rather, such knowledge can be used to test the models' ability to produce realistic results by comparing the mass estimated by the models at small radii to the mass estimates obtained for the inner region using the virial theorem. As a corollary we show how distance information of the quality obtained using the Tully-Fisher relation enhances the likelihood signal. Such distance information might be used in the future either to strengthen the results for this type of model or to allow more advanced models to be used (e.g., models breaking the assumption of spherical symmetry). The method relies on observations of redshifts of galaxies at large angular separations from the centers of the clusters. Currently, such data exist only for Coma, to which we apply the method as an example. The estimated mass at r = 2.5 h-1 Mpc falls in the range 1.1-2.4 × 1015 h-1 M☉, depending on the model and on the level of the noise term. We find rturn to be of the order of 10 h-1 Mpc, a result that is rather robust against changes in the model and subdivision of the data in various ways. We find 0.6<Ω0.60/b0.75<0.8, where b is the bias parameter, when using the luminosity profile adopted in the models. Interpreting this result, one must recall that the luminosity function of Coma is not well known in the infall region, and that ideally the method should be used on several clusters simultaneously to minimize the effects of deviations from the spherical model. The method provides a promising way of measuring the infall velocities in the outer regions of clusters. However, a robust determination of cluster properties must await future redshift observations of galaxies in the outer regions of a number of clusters.

The Thermal Response of a Pulsar Glitch: The Nonspherically Symmetric Case

We study the thermal evolution of a pulsar after a glitch in which the energy is released from a relatively compact region. A set of relativistic thermal transport and energy balance equations is used to study the thermal evolution, without making the assumption of spherical symmetry. We use an exact cooling model to solve this set of differential equations. Our results could differ significantly from those obtained under the assumption of spherical symmetry. Even for young pulsars with a hot core like the Vela pulsar, a detectable hot spot could be observed after a glitch if a large amount of energy is released in a small region close to the surface of the star. The results suggest that the intensity variation and the relative phases of hard X-ray emissions in different epochs may provide important information on the equation of state.

Analysis of ionospheric electron density distribution from GPS/MET occultations

The authors present results from analysis of GPS/MET occultation data obtained from the University Corporation for Atmospheric Research database. They have used high-rate occultation data to study the performance of an earlier 4D ionospheric electron content model-obtained tomographically with the use of the GPS constellation-with satisfactory results. In addition, they have processed UCAR level-one medium-rate occultation data by using the GIPSY package to obtain aligned phases. Under the assumption of local spherical symmetry, this phase information has been processed to yield ray-path bending angles through Doppler-shift analysis, which have then been used to yield profiles of electronic density via the Abel transform. These profiles give important information for preparing future tomographic work, although the limitations imposed by the working assumptions in this approach cannot be ignored.

Systematic Errors in the Hubble Constant Based upon Measurement of the Sunyaev‐Zeldovich Effect

Values of the Hubble constant reported to date that are based upon measurement of the Sunyaev-Zeldovich (SZ) effect in clusters of galaxies are systematically lower than those derived by other methods (e.g., Cepheid variable stars or the Tully-Fisher relation). We investigate the possibility that systematic errors may be introduced into the analysis by the generally adopted assumptions that observed clusters are in hydrostatic equilibrium, are spherically symmetric, and are isothermal. We construct self-consistent theoretical models of merging clusters of galaxies, using hydrodynamic/N-body simulations. We then compute the magnitude of H0 derived from the SZ effect at different times and at different projection angles, both from first principles and by applying each of the standard assumptions used in the interpretation of observations. Our results indicate that the assumption of isothermality in the evolving clusters can result in H0 being underestimated by 10%-30%, depending upon both epoch and projection angle. Moreover, use of the projected, emission-weighted temperature profile under the assumption of spherical symmetry does not significantly improve the situation except in the case of more extreme mergers (i.e., those involving relatively gas-rich subclusters). Although less significant, we find that asphericity in the gas density can also result in a 15% error in H0. If the cluster is prolate (as is generally the case for on-axis, or nearly on-axis, mergers) and viewed along its major axis, H0 will be systematically underestimated. More extreme off-axis mergers may result in oblate merger remnants, which, when viewed nearly face-on, may result in an overestimation of H0. A similar effect is noted when viewing a prolate distribution along a line of sight that is nearly perpendicular to its major axis. In both cases the potential overestimation occurs only when the remnant is viewed within 15°-30° of face-on. Bulk gas motions and the kinematic SZ effect do not appear to be significant except for a brief period during the very early stages of a merger. Our study shows that the most meaningful SZ measurement will be accompanied by high-resolution temperature data and a detailed dynamical modeling of the observed system. In lieu of this, a large sample selected to avoid dynamically evolving systems is preferred.

Interpretation of Ultraviolet Absorption Lines in SN 1006

We present a theoretical interpretation of the broad silicon and iron ultraviolet absorption features observed with the Hubble Space Telescope (HST) in the spectrum of the Schweizer-Middleditch star behind the remnant of SN 1006. These features are caused by supernova ejecta in SN 1006. We propose that the redshifted Si II 1260 A feature consists of both unshocked and shocked Si II. The sharp red edge of the line at 7070 km s-1 indicates the position of the reverse shock, while its Gaussian blue edge reveals shocked Si with a mean velocity of 5050 km s-1 and a dispersion of 1240 km s-1, which implies a reverse shock velocity of 2860 km s-1. The measured velocities satisfy the energy jump condition for a strong shock, provided that all the shock energy goes into ions, with little or no collisionless heating of electrons. The line profiles of the Si III and Si IV absorption features indicate that they arise mostly from shocked Si. The total mass of shocked and unshocked Si inferred from the Si II, Si III, and Si IV profiles is MSi = 0.25 ± 0.01 M☉ on the assumption of spherical symmetry. Unshocked Si extends upward from 5600 km s-1. Although there appears to be some Fe mixed with the Si at lower velocities 7070 km s-1, the absence of Fe II absorption with the same profile as the shocked Si II suggests little Fe mixed with Si at higher (before being shocked) velocities. The column density of shocked Si II is close to that expected for Si II undergoing steady state collisional ionization behind the reverse shock, provided that the electron to Si II ratio is low, from which we infer that most of the shocked Si is likely to be of a fairly high degree of purity, unmixed with other elements. We propose that the ambient interstellar density on the far side of SN 1006 is anomalously low compared to the density around the rest of the remnant. This would simultaneously explain the high velocity of the redshifted Si absorption, the absence of blueshifted Si absorption, and the low density of the absorbing Si compared to the high Si density required to produce the observed Si X-ray line emission. We have reanalyzed the Fe II absorption lines and have concluded that the earlier evidence for high-velocity blueshifted Fe II extending to ~-8000 km s-1 is not compelling. We interpret the blue edge on the Fe II profiles at -4200 km s-1 as the position of the reverse shock on the near side of SN 1006. The mass of Fe II inferred from the red edge of the Fe II profile is MFe II = 0.029 ± 0.004 M☉ up to 7070 km s-1, if spherical symmetry is assumed. The low ionization state of unshocked Si inferred from our analysis of the silicon features, Si II/Si = 0.92 ± 0.07, suggests a correspondingly low ionization state of unshocked iron, with Fe II/Fe = 0.66 -->+ 0.29−0.22. If this is correct, then the total mass of Fe up to 7070 km s-1 is MFe = 0.044 -->−0.013+0.022 M☉ with a 3 σ upper limit of MFe < 0.16 M☉. Such a low ionization state and mass of iron is consistent with the recent observation of Fe III 1123 A with the Hopkins Ultraviolet Telescope (HUT), which indicates Fe III/Fe II = 1.1 ± 0.9 but conflicts with the expected presence of several tenths of a solar mass of iron in this suspected Type Ia supernova remnant. However, the inference from the present HST data is too indirect, and the HUT data are too noisy, to rule out a large mass of iron. Reobservation of the Fe III 1123 A line at higher signal-to-noise ratio with Far Ultraviolet Space Explorer will be important in determining the degree of ionization and hence mass of iron in SN 1006.

A Phase‐Space Approach to Collisionless Stellar Systems Using a Particle Method

A particle method for reproducing the phase space of collisionless stellar systems is described. The key idea originates in Liouville's theorem which states that the distribution function (DF) at time t can be derived from tracing necessary orbits back to t=0. To make this procedure feasible, a self-consistent field (SCF) method for solving Poisson's equation is adopted to compute the orbits of arbitrary stars. As an example, for the violent relaxation of a uniform-density sphere, the phase-space evolution which the current method generates is compared to that obtained with a phase-space method for integrating the collisionless Boltzmann equation, on the assumption of spherical symmetry. Then, excellent agreement is found between the two methods if an optimal basis set for the SCF technique is chosen. Since this reproduction method requires only the functional form of initial DFs but needs no assumptions about symmetry of the system, the success in reproducing the phase-space evolution implies that there would be no need of directly solving the collisionless Boltzmann equation in order to access phase space even for systems without any special symmetries. The effects of basis sets used in SCF simulations on the reproduced phase space are also discussed.

A New Generation of Photoionization Codes: Three‐dimensional Models. The Bipolar Planetary Nebula IC 4406

A three-dimensional self-consistent photoionization code is developed in order to build more realistic models for asymmetric and/or inhomogeneous photoionized nebulae. With these models the assumption of spherical or plane-parallel symmetry can be dropped and models with various geometries can be treated. The gaseous region is divided into numerous cubic cells. The physical conditions in each cell are obtained taking into account the effect of the optical depth and the diffuse radiation, which depend on the other cells. In order to compare to observational data, line intensities are calculated for the whole nebula, as well as for regions covered by a given slit. Isophotal maps are also obtained.

Impossibility of the Cylindrically Symmetric Einstein-Straus Model

The classical Einstein-Straus problem treats the influence of the expansion of the universe on the static vacuum surrounding a spherically symmetric object. In this Letter we study the next simplest step by dropping the assumption of spherical symmetry and considering the case of cylindrical symmetry. Our main result is that the cylindrically symmetric analog of the Einstein-Straus model is impossible, even without any restriction on the matter content of the static cavity. This fact forbids the embedding of some static objects (say strings or similar objects) into the standard cosmological models.

Effect of periodic horizontal gradients on the retrieval of atmospheric profiles from occultation measurements

Spherical symmetry is generally assumed during the retrieval process of atmospheric profiles from occultation measurements. The existence of periodic horizontal gradients, occurring on scales comparable to the distances traveled by the rays around their tangent point, is produced by gravity waves. These waves can introduce density perturbations of up to 1 or 2% in amplitude and affect the retrieved parameters accordingly. We show the consequences of ignoring these gradients in the retrieved refractivity profiles when spherical symmetry is assumed. We find that only the waves, with horizontal wavelengths close to the horizontal distance that rays travel in their final 6 or 8 km in the vertical before they are tangent in the atmosphere, will have an influence on the retrieved profiles by introducing a phase delay in the local profile of the retrieved refractivity. The horizontal wavelength of these waves corresponds to the minimum horizontal resolution associated with the retrieved profiles. We also find that smaller scale waves do not have any significant impact on the retrieved profiles, as their contribution cancels out by averaging through the periodic perturbation, while waves with very long horizontal wavelengths are in good agreement with a local spherical symmetry assumption.