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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Measurement Uncertainty of Highly Asymmetrically Curved Elliptical Mirrors Using Multi-Pitch Slope Stitching Technique

Soft X-ray off-axis elliptical mirrors bring new challenges for X-ray mirror metrology. These highly asymmetrically curved elliptical cylindrical mirrors with a total slope range >10 mrad are extremely challenging to measure. Their total slope range exceeds the measuring range of most angular sensors used for X-ray mirror inspection. To overcome this problem, it is possible to stitch partial slope data by measuring the mirror at different pitch angles (multi-pitch angles). By revisiting the theory of the multi-pitch Nano-accuracy Surface Profiler (NSP), we derive the sampling position error on the mirror surface as a function of the mirror height profile and the measurement error of the pitch rotation center. When measuring “extreme”, highly asymmetrically curved, elliptical mirrors, the calculation of the mirror height profile with iterative reconstruction outperforms the classical “flat assumption” (i.e., assuming that the mirror sag is negligible). As demonstrated by our simulations, a proper tolerance evaluation on the measurement of pitch rotation center is needed to assess the measurement accuracy (systematic error) for these strongly aspherical mirrors using the multi-pitch NSP technique. Taking a real design of an “extreme” elliptical mirror as a case study, we conduct a Monte Carlo simulation to mimic the measurement and characterization process to analyze the impact of several error sources. With the measurement uncertainty of the pitch rotation center, the multi-pitch NSP measurement can estimate the grazing angle θ and the chief ray location x o with their uncertainties, as well as the slope residuals.

36 MATERIALS SCIENCE↗
NASA NTRS1978

Minimum film thickness in elliptical contacts for different regimes of fluid-film lubrication

The film-parameter equations are provided for four fluid-film lubrication regimes found in elliptical contacts. These regimes are isoviscous-rigid; viscous-rigid; elastohydrodynamic of low-elastic-modulus materials, or isoviscous-elastic; and elastohydrodynamic, or viscous-elastic. The influence or lack of influence of elastic and viscous effects is the factor that distinguishes these regimes. The film-parameter equations for the respective regimes come from earlier theoretical studies by the authors on elastohydrodynamic and hydrodynamic lubrication of elliptical conjunctions. These equations are restated and the results are presented as a map of the lubrication regimes, with film-thickness contours on a log-log grid of the viscosity and elasticity parameters for five values of the ellipticity parameter. The results present a complete theoretical film-parameter solution for elliptical contacts in the four lubrication regimes.

Hamrock, B. J.↗
NASA NTRS1978

On the rotation of elliptical galaxies

The tensor virial theorem is applied to models of early-type galaxies. First the theorem is applied to rotating elliptical galaxies whose constant-density surfaces are similar ellipsoids. A relationship is obtained between the observed rotations and the forms of generally triaxial galaxies. By applying the results of Robert (1962) to the evaluation of the components of the Chandrasekhar tensor which occurs in this relationship, it is found that the form of a galaxy that lacks global velocity anisotropy uniquely determines the ratio of its rotational and random kinetic energies independently of the radial density profile of that galaxy. A distribution of three-dimensional prolate spheroids is derived which accounts for the observed distribution of ellipticities reported by Sandage, Freeman, and Stokes (1970). This is then used to derive curves giving the expected frequency of occurrence of galaxies of given rotational velocities for each of a number of different apparent elongations on the sky. It is found that if elliptical galaxies are prolate, there should be little correlation between apparent ellipticity and rotation velocity.

Binney, J.↗
NASA NTRS1982

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies is introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The lineaar stabilities of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.

Kwok, J. H.↗
NASA NTRS1983

Elastohydrodynamics of elliptical contacts for materials of low elastic modulus

The influence of the ellipticity parameter k and the dimensionless speed U, load W, and materials G parameters on minimum film thickness for materials of low elastic modulus was investigated. The ellipticity parameter was varied from 1 (a ball-on-plane configuration) to 12 (a configuration approaching a line contact); U and W were each varied by one order of magnitude. Seventeen cases were used to generate the minimum- and central-film-thickness relations. The influence of lubricant starvation on minimum film thickness in starved elliptical, elastohydrodynamic configurations was also investigated for materials of low elastic modulus. Lubricant starvation was studied simply by moving the inlet boundary closer to the center of the conjunction in the numerical solutions. Contour plots of pressure and film thickness in and around the contact were presented for both fully flooded and starved lubrication conditions. It is evident from these figures that the inlet pressure contours become less circular and closer to the edge of the Hertzian contact zone and that the film thickness decreases substantially as the serverity of starvation increases. The results presented reveal the essential features of both fully flooded and starved, elliptical, elastohydrodynamic conjunctions for materials of low elastic modulus.

Hamrock, B. J.↗
NASA NTRS1985

Downwash in the plane of symmetry of an elliptically loaded wing

A closed-form solution for the downwash in the plane of symmetry of an elliptically loaded line is given. This theoretical result is derived from Prandtl's lifting-line theory and assumes that: (1) a three-dimensional wing can be replaced by a straight lifting line, (2) this line is elliptically loaded, and (3) the trailing wake is a flat-sheet which does not roll up. The first assumption is reasonable for distances greater than about 1 chord from the wing aerodynamic center. The second assumption is satisfied by any combination of wing twist, spanwise camber variation, or planform that approximates elliptic loading. The third assumption is justified only for high-aspect-ratio wings at low lift coefficients and downstream distances less than about 1 span from the aerodynamic center. It is shown, however, that assuming the wake to be fully rolled up gives downwash values reasonably close to those of the flat-sheet solution derived in this paper. The wing can therefore be modeled as a single horseshoe vortex with the same lift and total circulation as the equivalent ellipticity loaded line, and the predicted downwash will be a close approximation independent of aspect ratio and lift coefficient. The flat-sheet equation and the fully rolled up wake equation are both one-line formulas that predict the upwash field in front of the wing, as well as the downwash field behind it. These formulas are useful for preliminary estimates of the complex aerodynamic interaction between two wings (i.e., canard, tandem wing, and conventional aircraft) including the effects of gap and stagger.

Phillips, J. D.↗
NASA NTRS1986

Regularity estimates up to the boundary for elliptic systems of difference equations

Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

Strikwerda, J. C.↗
NASA NTRS1985

Boxy isophotes, discs and dust lanes in elliptical galaxies

CCD images of 42 elliptical and S0 galaxies are examined for low-contrast structures or subtle distortions of the isophotes from perfect ellipses. 75 percent of the galaxies have isophotes completely describable as concentric ellipses to within the photometry errors. 'Boxy' isophotes, stellar discs, and dust lanes are detected in the remaining 25 percent of the sample. The boxy elliptical galaxies appear dynamically indistinguishable from normal ellipticals and are therefore different from boxy bulges, which rotate rapidly. Most of the galaxies with faint discs, however, appear dynamically similar to S0 galaxies. Nearly edge-on dust lanes are found in four galaxies, which suggests that dust lanes may commonly occur in elliptical galaxies.

Lauer, T. R.↗
NASA NTRS1986

Effects of initial condition on circular and elliptic jet noise

Noise produced by cold elliptic jets was studied experimentally using a 4 cm air jet in an anechoic chamber by varying the aspect ratio, nozzle exit boundary layer, and Mach number. Elliptic geometry is found to accentuate the sound produced by pairing of the shear layer vortices in the laminar boundary jet. Calculation of the pressure field produced by the pairing of elliptic vortex rings shows that the ratio of sound intensities produced by the pairing of inviscid elliptic vortex filament rings should be equal to their aspect ratio, the minor axis producing greater sound. Tripping the nozzle boundary layer changes the instability of the shear layer and removes the laminar vortices. If the sound produced by pairing and/or breakdown is considered to be the difference between laminar and turbulent jet noise, the pairing mechanism is seen to be very significant in laminar boundary layer jets.

Bridges, J. E.↗
NASA NTRS1990

The evolution of the interstellar medium in elliptical galaxies. I - The early wind phase

The evolving mass loss rates from type I and II SNe, stellar winds, and PN and the associated evolution of the mass-averaged temperature of the stellar ejecta are calculated. At early times, the mass-averaged temperature of the ejecta is found to be much greater than the central escape temperature from even very luminous elliptical galaxies due to the high frequency of type II SNe. The amount of oxygen and iron that can be injected into the intracluster medium from type II SN-driven winds in elliptical galaxies is calculated and the results are compared with observations. The gas ejected during this early wind phase is very hot and can heat the cluster gas to temperatures greater than that produced by gravitational heating alone. Two numerical simulations concerning the evolution of the ISM in an elliptical galaxy model with L(B) = 10 to the 11th solar are presented which confirm the existence of an early type II SN-driven wind phase and reproduce the observed X-ray properties of bright elliptical galaxies in the present epoch.

David, L. P.↗
NASA NTRS1992

Molecular gas in elliptical galaxies with dust lanes

We have searched for CO(1-0) line emission in eight dust lane elliptical and lenticular galaxies using the Nobeyama 45 m telescope. Five of the eight galaxies, including the well-studied elliptical NGC 1052, have CO emission at above the 5-sigma level, with inferred molecular gas masses ranging from 10 exp 8 to a few times 10 exp 9 solar masses. Our selection criterion differs from previous surveys in that it does not depend on the FIR fluxes, and thus is less sensitive to the sizes and distances of the host galaxies or to the degree to which dust is heated. The relatively high detection rate of CO in these ellipticals suggests a close correlation between molecular mass and cold dust. Compared with previously studied samples of FIR selected early-type galaxies, our sample has on average four times more CO emission per unit FIR (40-120 microns) luminosity. If the intrinsic gas-to-dust ratio of these galaxies as similar to that of the Milky Way, then only about 5 percent of the dust mass in dust lane ellipticals radiates substantially at 60 and 100 microns, and the remaining dust must be colder than about 30 K.

Wang, Zhong↗
NASA NTRS1992

The noise from supersonic elliptic jets

This paper presents calculations of the noise radiated by a supersonic elliptic jet. The large scale structures in the jet, that are the predominant source of noise in the downstream direction, are modeled as instability waves. The evolution of the instability waves is determined by a local, linear, inviscid analysis. An expression is derived for the acoustic field outside the jet and the far field directivity associated with each instability wave. Calculations are performed for a Mach 1.5 elliptic jet with aspect ratio 2:1 and a Mach 2.0 elliptic jet with aspect ratio 2:1 and a Mach 2.0 elliptic jet with aspect ratio 3:1. The mean flow development is taken from experimental results. Comparisons are made with far field acoustic measurements.

Morris, Philip J.↗
NASA NTRS1994

Modeling near wall effects in second moment closures by elliptic relaxation

The elliptic relaxation model of Durbin (1993) for modeling near-wall turbulence using second moment closures (SMC) is compared to DNS data for a channel flow at Re(sub t) = 395. The agreement for second order statistics and even the terms in their balance equation is quite satisfactory, confirming that very little viscous effects (via Kolmogoroff scales) need to be added to the high Reynolds versions of SMC for near-wall-turbulence. The essential near-wall feature is thus the kinematic blocking effect that a solid wall exerts on the turbulence through the fluctuating pressure, which is best modeled by an elliptic operator. Above the transition layer, the effect of the original elliptic operator decays rapidly, and it is suggested that the log-layer is better reproduced by adding a non-homogeneous reduction of the return to isotropy, the gradient of the turbulent length scale being used as a measure of the inhomogeneity of the log-layer. The elliptic operator was quite easily applied to the non-linear Craft & Launder pressure-strain model yielding an improved distinction between the spanwise and wall normal stresses, although at higher Reynolds number (Re) and away from the wall, the streamwise component is severely underpredicted, as well as the transition in the mean velocity from the log to the wake profiles. In this area a significant change of behavior was observed in the DNS pressure-strain term, entirely ignored in the models.

Laurence, D.↗
NASA NTRS1994

Geometrical evidence for dark matter: X-ray constraints on the mass of the elliptical galaxy NGC 720

We describe (1) a new test for dark matter and alternate theories of gravitation based on the relative geometries of the X-ray and optical surface brightness distributions and an assumed form for the potential, of the optical light, (2) a technique to measure the shapes of the total gravitating matter and dark matter of an ellipsoidal system which is insensitive to the precise value of the temperature of the gas and to modest temperature gradients, and (3) a new method to determine the ratio of dark mass to stellar mass that is dependent on the functional forms for the visible star, gas and dark matter distributions, but independent of the distance to the galaxy or the gas temperature. We apply these techniques to X-ray data from the ROSAT Position Sensitive Proportional Counter (PSPC) of the optically flattened elliptical galaxy NGC 720; the optical isophotes have ellipticity epsilon approximately 0.40 extending out to approximately 120 sec. The X-ray isophotes are significantly elongated, epsilon = 0.20-0.30 for semimajor axis a approximately 100 sec. The major axes of the optical and X-ray isophotes are misaligned by approximately 30 deg +/- 15 deg. Spectral analysis of the X-ray data reveals no evidence of temperature gradients or anisotropies and demonstrates that a single-temperature plasma (T approximately 0.6 keV) having subsolar heavy element abundances and a two-temperature model having solar abundances describe the spectrum equally well. Considering only the relative geometries of the X-ray and optical surface brightness distributions and an assumed functional form for the potential of the optical light, we conclude that matter distributed like the optical light cannot produce the observed ellipticities of the X-ray isophotes, independent of the gas pressure, the gas temperature, and the value of the stellar mass; this comparison assumes a state of quasi-hydrostatic equilibrium so that the three-dimensional surfaces of the gas emissivity trace the three-dimensional isopotential surfaces -- we discuss the viability of this assumption for NGC 720. Milgrom's Modification of Newtonian Dynamics (MOND) cannot dispel this manifestation of dark matter. Hence, geometrical considerations require, without mention of pressure or temperature, the presence of an extended, massive dark matter halo in NGC 720. Employing essentially the technique of Buote & Canizares (1992; Buote 1992) we use the shape of the X-ray surface brightness to constrain the shape of the total gravitating matter. The total matter is modeled as either an oblate or prolate spheriod of constant shape and orientation having either a Ferrers (rho approximately r(exp -n)) or Hernquist density. Assuming the X-ray gas is in hydrostatic equilibrium, we construct a model X-ray gas distribution for various temperature profiles. We determine the ellipticity of the total gravitating matter to be epsilon approximately 0.50-0.70. Using the single-temperature model we estimate a total mass approximately (0.41-1.4) x 10(exp 12) h(sub 80) solar mass interior to the ellipsoid of semimajor axis 43.6 h(sub 80) kpc. Ferrers densities as steep as r(exp -3) do not fit the data, but the r(exp -2) and Hernquist models yield excellent fits. We estimate the mass distributions of the stars and the gas and fit the dark matter directly. For a given gas equation of state and functional forms for the visible stars, gas, and dark matter, these models yield a distance-independent and temperature-independent measurement of the ratio of dark mass to stellar mass M(sub DM)/M(sub stars). We estimate a minimum M(sub DM)/M(sub stars) greater than or equal to 4 which corresponds to a total mass slightly greater than that derived from the single-temperature models for distance D = 20h(sub 80) Mpc.

Buote, David A.↗
NASA NTRS1995

Neutral hydrogen associated with shells and other fine structure in NGC 2865: A dynamically young elliptical?

We report the discovery of neutral hydrogen in a second elliptical galaxy with shells, NGC 2865. Very Large Array (VLA) images reveal an association between the neutral hydrogen (H I) and the fine structure (shells, tails, and loops) in the galaxy. Similar to what we previously observed in NGC 5128 (Centaurus A), most of the 6 x 10(exp 8)/h(exp 2) solar mass of cold gas is found in a broken ring in the outer regions of NGC 2865 (beyond 0.5D(sub 25)) and is displaced to the outside of the shells and loops. The measured velocities cover a range of 500 km/s around the systematic velocity. The velocity field of the outer H I has the same sense and magnitude (and line of nodes) as that of the stars in the elliptical body. Although NGC 2865 appears to be a relaxed elliptical galaxy, deep images, photometry, and spectroscopy suggest that the galaxy might be the recent (less than 7 Gyr) product of a major disk-disk merger -- a 'dynamically young elliptical.' Our H I data support this hypothesis. Nevertheless, the association between gas and stellar fine structure, with gas displaced outward from the stars in projected position, implies gas motions not predicted by any of the current merger scenarios. Using the H I ring and assuming nearly circular motion, we measure M/L(sub B) at large radii (4 x 0.5D(sub 25)). We find M/L(sub B) = 33 +/- 4 h, a factor of 5 greater than the value of M/L(sub B) found for the central regions, indicating the presence of a dark halo.

Schiminovich, D.↗
DOE OSTI2023

A Preferential Growth Channel for Supermassive Black Holes in Elliptical Galaxies at z ≲ 2

The assembly of stellar and supermassive black hole (SMBH) mass in elliptical galaxies since z ~ 1 can help to diagnose the origins of locally observed correlations between SMBH mass and stellar mass. We therefore construct three samples of elliptical galaxies, one at z ~ 0 and two at 0.7 ≲ z ≲ 2.5, and quantify their relative positions in the M BH -M * plane. Using a Bayesian analysis framework, we find evidence for translational offsets in both stellar mass and SMBH mass between the local sample and both higher-redshift samples. The offsets in stellar mass are small, and consistent with measurement bias, but the offsets in SMBH mass are much larger, reaching a factor of 7 between z ~ 1 and z ~ 0. The magnitude of the SMBH offset may also depend on redshift, reaching a factor of ~20 at z ~ 2. The result is robust against variation in the high- and low-redshift samples and changes in the analysis approach. The magnitude and redshift evolution of the offset are challenging to explain in terms of selection and measurement biases. We conclude that either there is a physical mechanism that preferentially grows SMBHs in elliptical galaxies at z ≲ 2, or that selection and measurement biases are both underestimated, and depend on redshift.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1935

Strength Tests of Thin-walled Duralumin Cylinders of Elliptic Section

This report is the fifth of a series presenting the results of strength tests of thin-walled cylinders and truncated cones of circular and elliptic section; it includes the results obtained from torsion tests on 30 cylinders, pure bending tests on 30 cylinders, and combined transverse shear and bending tests on 60 cylinders. All the cylinders tested were elliptic section with the ends clamped to rigid bulkheads. In the pure bending and combined transverse shear and bending tests the loads were applied in the plane of the major axis. The results of the tests on elliptic cylinders are correlated with the results of corresponding tests on circular cylinders and are presented in charts suitable for use in design.

STRESSED SKIN - CYLINDERS, ELLIPTIC - STRENGTH↗
NASA NTRS2013

Dusty Feedback from Massive Black Holes in Two Elliptical Galaxies

Far-infrared dust emission from elliptical galaxies informs us about galaxy mergers, feedback energy outbursts from supermassive black holes and the age of galactic stars. We report on the role of AGN feedback observationally by looking for its signatures in elliptical galaxies at recent epochs in the nearby universe. We present Herschel observations of two elliptical galaxies with strong and spatially extended FIR emission from colder grains 5-10 kpc distant from the galaxy cores. Extended excess cold dust emission is interpreted as evidence of recent feedback-generated AGN energy outbursts in these galaxies, visible only in the FIR, from buoyant gaseous outflows from the galaxy cores.

Galaxies↗