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1.
A relationship between the energy gap (E G) and the density (ρ) over mean atomic weight (〈A〉) ratio for Fe-poor oxide and silicate minerals is derived from simple properties of their free atom-components. Theoretical considerations are based on the Lorentz electron theory of solids. The eigenfrequency ν 0 of elementary electron oscillators, in energy units h ν 0, is identified with the energy gap of a solid. The numerical relation is of the form $$(\langle U_0 \rangle ^2 - E_G^2 )\frac{{\langle A\rangle }}{\rho } = \frac{4}{3}\pi \hbar ^2 \frac{{e^2 }}{m}N = 276.79 eV^2 cm^3 /mol$$ where 〈U 0〉 is the average first ionization potential (per free atom), ? is crossed Planck's constant, e is the electron charge, m is the electron rest mass, and N is Avogadro's number. For several geophysically interesting oxide and silicate minerals which are in general composed of four different elements (O, Si, Mg and Al), we obtain from laboratory data that the mean value of $$\left\langle {[\langle U_0 \rangle ^2 - (E_G^{lab} )^2 ]\frac{{\langle A\rangle }}{\rho }} \right\rangle \approx 248.2 \pm 20.9eV^2 cm^3 /mol.$$ .  相似文献   

2.
Present work provides in-situ structural data at a fine temperature scale from RT to the melting point of nitratine, NaNO3. From the analysis of log e 33 versus log t plots, it is possible to prove that an univocal indication on the R \( \overline{3} \) c (low temperature, LT) → R \( \overline{3} \) m (high temperature, HT) transition mechanism cannot be obtained because of the relevant role played by the arbitrary assumptions required for defining the c 0 dependence from temperature of the HT phase. This is due to the occurrence of excess thermal expansion for the HT phase. A significantly better fit for an Ising-spin structural model over a non-Ising rigid-body one has been obtained for the LT phase. Moreover, the Ising model led to a smooth variation of the oxygen site x fractional coordinate throughout the transition. The structure of the HT polymorph has been successfully refined considering an oxygen site at x, 0, ½, with 50% occupancy. Such model was the only acceptable one from the crystal chemical point of view as the alternative model (oxygen site at x, y, z with 25% occupancy) led to unrealistically aplanar \( {\text{NO}}_{3}^{ - } \) groups.  相似文献   

3.
A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented:
  1. Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
  2. Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x 0 is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
  3. Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e?kt .
  4. Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
  5. Power laws with P(x)=x a lead to a rate law: ln〈Q〉=-(α + 1) ln t.
The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated.  相似文献   

4.
New results of UBV JHKLM photometry of the symbiotic Mira V407 Cyg performed in 1998–2002 are reported. In 2002, these observations were supplemented with RI observations and a search for rapid variability in the V band. The hot component of V407 Cyg experienced a strong flare in 1998, which was the second in the history of photometric observations of this star; this flare is still continuing. During the flare, the spectral energy distribution of the hot component can be approximated by blackbody radiation with a temperature of ~7200 K. At the maximum brightness, the bolometric flux from the hot component did not exceed 3% of the Mira's mean bolometric flux, while its bolometric luminosity was ~400L. Appreciable variations of the star's BV brightness \((\tilde0\mathop m\limits_. 7)\) on a timescale of several days have been observed. These variations are not correlated with variations of B-V. Flickering on a timescale of several minutes with an amplitude of \(\tilde0\mathop m\limits_. 2\) has been detected in the V band. The observations suggest that the hot component can be in three qualitatively different states. In a model with a rapidly rotating white dwarf, these states can be associated with (i) the quiescent state of the white dwarf (with a very low accretion rate), (ii) an ejection state, and (iii) an accretion state. The Mira pulsation period P is \( \approx 762\mathop d\limits_. 9\), with its infrared maximum occurring ~0.15P after the visual maximum. A “step” is observed on the ascending branch of the Mira infrared light curves. In 1998, the gradual increase of the mean K brightness of the Mira that had been observed since 1984 was interrupted by an unusually deep minimum, after which the mean level of the K brightness considerably decreased.  相似文献   

5.
Using the four-channel automatic photoelectric photometer of the Sternberg Astronomical Institute’s Tien Shan Mountain Observatory, we have acquired accurate (σobs≈0.004m) W BV R brightness measurements for the eclipsing binary AR Cas during selected phases before eclipse ingress and after egress, as well as at the center of minima. A joint analysis of these measurements with other published data has enabled us to derive for the first time a self-consistent set of physical and geometrical parameters for the star and the evolutionary age of its components, t=(60±3)×106 years. We have found the period of the apsidal motion (Uobs=1100±160 years, \(\dot \omega _{obs} = 0^\circ .327 \pm 0^\circ .049\) years?1) and the apsidal parameter of the primary, logk 2,1 obs =?2.41±0.08, with the apsidal parameter being in good agreement with current models of stellar evolution. There is an ultraviolet excess in the primary’s radiation, Δ(U?B)=?0.12m and Δ(B?V)=?0.06m, possibly due to a metal deficiency in the star’s atmosphere.  相似文献   

6.
A new mineral, mendigite (IMA no. 2014-007), isostructural with bustamite, has been found in the In den Dellen pumice quarry near Mendig, Laacher Lake area, Eifel Mountains, Rhineland-Palatinate (Rheinland-Pfalz), Germany. Associated minerals are sanidine, nosean, rhodonite, tephroite, magnetite, and a pyrochlore-group mineral. Mendigite occurs as clusters of long-prismatic crystals (up to 0.1 × 0.2 × 2.5 mm in size) in cavities within sanidinite. The color is dark brown with a brown streak. Perfect cleavage is parallel to (001). D calc = 3.56 g/cm3. The IR spectrum shows the absence of H2O and OH groups. Mendigite is biaxial (–), α = 1.722 (calc), β = 1.782(5), γ = 1.796(5), 2V meas = 50(10)°. The chemical composition (electron microprobe, mean of 4 point analyses, the Mn2+/Mn3+ ratio determined from structural data and charge-balance constraints) is as follows (wt %): 0.36 MgO, 10.78 CaO, 37.47 MnO, 2.91 Mn2O3, 4.42 Fe2O3, 1.08 Al2O3, 43.80 SiO2, total 100.82. The empirical formula is Mn2.00(Mn1.33Ca0.67) (Mn0.50 2+ Mn0.28 3+ Fe0.15 3+ Mg0.07)(Ca0.80 (Mn0.20 2+)(Si5.57 Fe0.27 3+ Al0.16O18). The idealized formula is Mn2Mn2MnCa(Si3O9)2. The crystal structure has been refined for a single crystal. Mendigite is triclinic, space group \(P\bar 1\); the unit-cell parameters are a = 7.0993(4), b = 7.6370(5), c = 7.7037(4) Å, α = 79.58(1)°, β = 62.62(1)°, γ = 76.47(1)°; V = 359.29(4) Å3, Z = 1. The strongest reflections on the X-ray powder diffraction pattern [d, Å (I, %) (hkl)] are: 3.72 (32) (020), 3.40 (20) (002, 021), 3.199 (25) (012), 3.000 (26), (\(01\bar 2\), \(1\bar 20\)), 2.885 (100) (221, \(2\bar 11\), \(1\bar 21\)), 2.691 (21) (222, \(2\bar 10\)), 2.397 (21) (\(02\bar 2\), \(21\bar 1\), 203, 031), 1.774 (37) (412, \(3\bar 21\)). The type specimen is deposited in the Fersman Mineralogical Museum, Russian Academy of Sciences, Moscow, registration number 4420/1.  相似文献   

7.
Isotope fractionation during the evaporation of silicate melt and condensation of vapor has been widely used to explain various isotope signals observed in lunar soils, cosmic spherules, calcium–aluminum-rich inclusions, and bulk compositions of planetary materials. During evaporation and condensation, the equilibrium isotope fractionation factor (α) between high-temperature silicate melt and vapor is a fundamental parameter that can constrain the melt’s isotopic compositions. However, equilibrium α is difficult to calibrate experimentally. Here we used Mg as an example and calculated equilibrium Mg isotope fractionation in MgSiO3 and Mg2SiO4 melt–vapor systems based on first-principles molecular dynamics and the high-temperature approximation of the Bigeleisen–Mayer equation. We found that, at 2500 K, δ25Mg values in the MgSiO3 and Mg2SiO4 melts were 0.141?±?0.004 and 0.143?±?0.003‰ more positive than in their respective vapors. The corresponding δ26Mg values were 0.270?±?0.008 and 0.274?±?0.006‰ more positive than in vapors, respectively. The general \(\alpha - T\) equations describing the equilibrium Mg α in MgSiO3 and Mg2SiO4 melt–vapor systems were: \(\alpha_{{{\text{Mg}}\left( {\text{l}} \right) - {\text{Mg}}\left( {\text{g}} \right)}} = 1 + \frac{{5.264 \times 10^{5} }}{{T^{2} }}\left( {\frac{1}{m} - \frac{1}{{m^{\prime}}}} \right)\) and \(\alpha_{{{\text{Mg}}\left( {\text{l}} \right) - {\text{Mg}}\left( {\text{g}} \right)}} = 1 + \frac{{5.340 \times 10^{5} }}{{T^{2} }}\left( {\frac{1}{m} - \frac{1}{{m^{\prime}}}} \right)\), respectively, where m is the mass of light isotope 24Mg and m′ is the mass of the heavier isotope, 25Mg or 26Mg. These results offer a necessary parameter for mechanistic understanding of Mg isotope fractionation during evaporation and condensation that commonly occurs during the early stages of planetary formation and evolution.  相似文献   

8.
Widely extended, cation stacking faults in experimentally deformed Mg2GeO4 spinel have been studied using transmission electron microscopy (TEM). The faults lie on {110} planes. The displacement vector is of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) and is normal to the fault plane. The partial dislocations which bound the stacking fault have colinear Burgers vectors of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) which are normal to the fault plane.  相似文献   

9.
The unnamed mineral CuFe2S4 has been found from sulfide Cu–Ni ores of the Lovnoozero deposit in the Kola Peninsula, Russia. It occurs in norite composed of orthopyroxene (bronzite), Ca-rich plagioclase (66% An), pargasite, and phlogopite. The last two minerals are replaced by talc, chlorite and carbonates. Monoclinic pyrrhotite, pentlandite, chalcopyrite, and pyrite are associated ore minerals. Phase CuFe2S4 is enclosed predominantly in chalcopyrite, probably replacing it, and occurs in later carbonate veinlets together with redeposited sulfides. It is light yellow with a brownish tint and metallic luster. The Mohs hardness is 5–5.5; VHN 654 ± 86 kgs/mm2. Density (calc.) = 4.524 g/cm3. The mineral is anisotropic, internal reflections are absent. Reflectance values (λ, nm R g and R p %) are: 440 30.3 29.5, 500 43.7 42.8, 560 50.9 49.6, 620 52.4 51.2, 640 52.6 51.4, 680 52.8 51.6, 700 52.7 51.4. CuFe2S4 is monoclinic, a = 6.260(4), b = 5.39(1), c = 13.19(1) Å, β = 94.88(7)°, V = 443(1) Å3, Z = 4. The strongest reflections in the powder diffraction pattern are [d, Å (I) (hkl)]: 4.150 (10) (012), 3.559 (4) (\(11\bar 2\)), 3.020 (4) (\(10\bar 4\)), 2.560 (3) (\(21\bar 2\)), 2.500 (3) (\(10\bar 5\)), 2.340 (3) (\(12\bar 2\)), 1.817 (3) (215), 1.489 (3) (402). The chemical composition is as follows, wt %: 20.44 Cu, 35.85 Fe, 0.65 Ni, 0.14 Co, 43.15 S, total is 100.23. The empirical formula calculated on the basis of 7 atoms is Cu0.969(Fe1.934Ni0.034Co0.007)1.975S4.056. According to its mode of occurrence, the mineral was formed as a result of low temperature processes involving metamorphic hydrothermal solutions.  相似文献   

10.
We have analyzed optical and infrared light curves of GU Mus obtained during the system's quiescent state and carried out computations for “hot-line” and “hot-spot” models. The hot-line model describes the optical variability of GU Mus better than the hot-spot model. Season-to-season variations of the shape, amplitude, and mean levels of the optical and infrared light curves of GU Mus are due to changing parameters of the hot line and, to a lesser degree, of the accretion disk. Taking into account the contribution of the variability of the disk + hot line system to the variability of the system as a whole, we are able to reliably estimate the orbital inclination, \(i = 54^\circ \pm 1^\circ .3\), and the mass of the black hole, M X =(6.7–7.6)M.  相似文献   

11.
The pressure–volume–temperature (PVT) relation of CaIrO3 post-perovskite (ppv) was measured at pressures and temperatures up to 8.6 GPa and 1,273 K, respectively, with energy-dispersive synchrotron X-ray diffraction using a DIA-type, cubic-anvil apparatus (SAM85). Unit-cell dimensions were derived from the Le Bail full profile refinement technique, and the results were fitted using the third-order Birth-Murnaghan equation of state. The derived bulk modulus \( K_{T0} \) at ambient pressure and temperature is 168.3 ± 7.1 GPa with a pressure derivative \( K_{T0}^{\prime } \) = 5.4 ± 0.7. All of the high temperature data, combined with previous experimental data, are fitted using the high-temperature Birch-Murnaghan equation of state, the thermal pressure approach, and the Mie-Grüneisen-Debye formalism. The refined thermoelastic parameters for CaIrO3 ppv are: temperature derivative of bulk modulus \( (\partial K_{T} /\partial T)_{P} \) = ?0.038 ± 0.011 GPa K?1, \( \alpha K_{T} \) = 0.0039 ± 0.0001 GPa K?1, \( \left( {\partial K_{T} /\partial T} \right)_{V} \) = ?0.012 ± 0.002 GPa K?1, and \( \left( {\partial^{2} P/\partial T^{2} } \right)_{V} \) = 1.9 ± 0.3 × 10?6 GPa2 K?2. Using the Mie-Grüneisen-Debye formalism, we obtain Grüneisen parameter \( \gamma_{0} \) = 0.92 ± 0.01 and its volume dependence q = 3.4 ± 0.6. The systematic variation of bulk moduli for several oxide post-perovskites can be described approximately by the relationship K T0  = 5406.0/V(molar) + 5.9 GPa.  相似文献   

12.
Laboratory measurements are required to study geophysical properties of the subsurface because of lacking direct observation of Earth’s crust. In this research, compressional (P) and shear (S) wave velocity measurements have been conducted on cylindrical specimens of Quartz-micaschist cored using rock blocks taken from the zinc and lead Angouran mine, Zanjan, northwest of Iran. Cylindrical rock specimens were prepared from the blocks by coring in 0°, 30°, 45°, 60°, and 90° into the foliation direction. P- and S-wave velocities were measured along the cylindrical specimens with different foliation orientations. Percent variations of the P- and S-wave velocities (Thomsen’s anisotropic parameters ε and γ) and constant dynamic modulus of test results have been determined. Percent variations of the P-wave velocity (ε) increase with an increase of the foliation angle with respect to the propagating waves direction by a parabolic function as it shows P-wave velocity differences up to a maximum value of 50 %. Thomsen’s anisotropic parameter of γ has also the same function with the foliation angle. Meanwhile, foliation orientation has a much greater influence on ε than γ for foliation angle from 45° to 90° as \( \frac{\varepsilon }{\gamma } \) ratio increases with an increase of foliation angle. Values of dynamic elastic modulus (E), Poisson’s ratio (ν), shear modulus (μ), bulk modulus (K), and Lamé’s constant (λ) increase with the increase of foliation angle with the parabolic function. The results show that dynamic elastic modulus, Poisson’s ratio, shear modulus, bulk modulus, and Lamé’s constant have anisotropic behavior in relation with the foliation orientation.  相似文献   

13.
Owing to the lack of consistent spatial time series data on actual evapotranspiration (ET), very few studies have been conducted on the long-term trend and variability in ET at a national scale over the Indian subcontinent. The present study uses biome specific ET data derived from NOAA satellite’s advanced very high resolution radiometer to investigate the trends and variability in ET over India from 1983 to 2006. Trend analysis using the non-parametric Mann–Kendall test showed that the domain average ET decreased during the period at a rate of \(0.22\,\hbox {mm year}^{-1}\). A strong decreasing trend (\(m = -1.75\, \hbox {mm year}^{-1}\), \(F = 17.41\), \(P\) 0.01) was observed in forest regions. Seasonal analyses indicated a decreasing trend during southwest summer monsoon (\(m= -0.320\, \hbox {mm season}^{-1}\,\hbox {year}^{-1})\) and post-monsoon period (\(m= -0.188\, \hbox {mm season}^{-1 }\,\hbox {year}^{-1})\). In contrast, an increasing trend was observed during northeast winter monsoon (\(m = 0.156 \,\hbox {mm season}^{-1 }\,\hbox {year}^{-1})\) and pre-monsoon (\(m = 0.068\, \hbox {mm season}^{-1 }\,\hbox {year}^{-1})\) periods. Despite an overall net decline in the country, a considerable increase ( \(4 \,\hbox {mm year}^{-1}\)) was observed over arid and semi-arid regions. Grid level correlation with various climatic parameters exhibited a strong positive correlation (\(r \!>\!0.5\)) of ET with soil moisture and precipitation over semi-arid and arid regions, whereas a negative correlation (\(r\) \(-0.5\)) occurred with temperature and insolation in dry regions of western India. The results of this analysis are useful for understanding regional ET dynamics and its relationship with various climatic parameters over India. Future studies on the effects of ET changes on the hydrological cycle, carbon cycle, and energy partitioning are needed to account for the feedbacks to the climate.  相似文献   

14.
Ab initio calculations of thermo-elastic properties of beryl (Al4Be6Si12O36) have been carried out at the hybrid HF/DFT level by using the B3LYP and WC1LYP Hamiltonians. Static geometries and vibrational frequencies were calculated at different values of the unit cell volume to get static pressure and mode-γ Grüneisen’s parameters. Zero point and thermal pressures were calculated by following a standard statistical-thermodynamics approach, within the limit of the quasi-harmonic approximation, and added to the static pressure at each volume, to get the total pressure (P) as a function of both temperature (T) and cell volume (V). The resulting P(V, T) curves were fitted by appropriate EoS’, to get bulk modulus (K 0) and its derivative (K′), at different temperatures. The calculation successfully reproduced the available experimental data concerning compressibility at room temperature (the WC1LYP Hamiltonian provided K 0 and K′ values of 180.2 Gpa and 4.0, respectively) and the low values observed for the thermal expansion coefficient. A zone-centre soft mode \( P6/mcc \to P\bar{1} \) phase transition was predicted to occur at a pressure of about 14 GPa; the reduction of the frequency of the soft vibrational mode, as the pressure is increased, and the similar behaviour of the majority of the low-frequency modes, provided an explanation of the thermal behaviour of the crystal, which is consistent with the RUM model (Rigid Unit Model; Dove et al. in Miner Mag 59:629–639, 1995), where the negative contribution to thermal expansion is ascribed to a geometric effect connected to the tilting of rigid polyhedra in framework silicates.  相似文献   

15.
We analyze the general 2D isosceles three-body problem for various ratios ? of the mass of the central body to the mass of each of the other two bodies. We set the initial conditions using two parameters: the virial coefficient k and the parameter \(\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }\), where \(\dot r\) is the relative velocity of the two outer bodies and \(\dot R\) is the velocity of the central body relative to the center of mass of the outer bodies. We compare statistical dependences between evolutionary parameters of triple systems with various values of ?, and analyze the k and μ dependences of the number of crossings of the center of mass of the triple system by the central body and the lifetime of the system. We construct the functions Rmax(rmax), where rmax and Rmax are the maximum achievable distances between the outer bodies, and between the central body and the center of mass of the outer bodies in the triple system. The parameter ? proves to be the most important parameter of the problem, and determines the relationship between the measures of the regular and stochastic trajectories. However, there exist “seeds” of stochasticity, even at small ?~10?2. The measure of the stochastic orbits increases with ?; when ?≥10, virtually the entire region of the initial conditions corresponds to stochastic trajectories.  相似文献   

16.
17.
In this study, a series of natural dam overtopping laboratory tests are reported. In these tests, the effect of seven different sediment mixtures on the breaching process was investigated. According to the test results, three stages of the breaching process of natural dams made of different materials were observed. Backward erosion was the primary cause for the incising slopes. The effects of backward erosion became stronger with the larger fines contents of the materials. With an increase in the median diameter (d 50) of particles, the breaching time became longer. However, the peak discharge became smaller. With an increase in the fines contents (p), the median diameter of the particles and the void ratio were changed, which resulted in a decrease in the breaching time and an increase in the peak discharge. The breaching time and peak discharge were more sensitive to the median diameter than to the fines contents. The relation between breach width and depth was found to follow a logistic function \( W\kern0.5em =\kern0.5em \frac{\zeta }{1\kern0.5em +\kern0.5em {e}^{\left(-k\left(D\kern0.5em -\kern0.5em {D}_0\right)\right)}} \). The parameters ζ, k, and D 0 are defined by a linear relationship with the median diameter and fines content. A breach of the side slope occurred as a tensile failure when the fines contents of the materials were large; otherwise, shear failure occurred. Furthermore, when the materials had fewer fines contents, the volume of the collapsed breach side slope became larger.  相似文献   

18.
It is shown that the approximation of the complex, tidally distorted shape of a star as a circular disc with local line profiles and a linear limb-darkening law, which is usually applied when deriving equatorial stellar rotation velocities from line profiles, leads to overestimation of the equatorial velocity V rot sin i and underestimation of the component mass ratio q = M x /M v . A formula enabling correction of the effect of these simplifying assumptions on the shape of a star is used to re-determine the mass ratios q and the masses of the black holes M x and visual components M v in low-mass X-ray binary systems containing black holes. Taking into account the tidal–rotational distortion of the stellar shape can significantly increase the mass ratios q = M x /M v , reducing M v , while M x changes only slightly. The resulting distribution of M v attains its maximum near M v ? 0.35M , in disagreement with the results of population synthesis computations realizing standard models for Galactic X-ray novae with black holes. Possible ways to overcome this inconsistency are discussed. The derived distribution of M x also differs strongly from the mass distribution for massive stars in the Galaxy.  相似文献   

19.
It is shown that a model with accretion in a “quasi-propeller” mode can explain the observed spindown of pulsars with periods P<0.1 s. The mean accretion rate for 39 selected objects is \(\dot M = 5.6 \times 10^{ - 11} M_ \odot /year\). If \(\dot M\) is constant during the pulsar’s lifetime, the neutron star will stop rotating after 107 years. The mean magnetic field at the neutron-star surface calculated in this model, \(\bar H_0 = 6.8 \times 10^8 G\), is consistent to an order of magnitude with the values of H0 for millisecond pulsars from known catalogs. However, the actual value of H0 for particular objects can differ from the catalog values by appreciable factors, and these quantities must be recalculated using more adequate models. The accretion disk around the neutron star should not impede the escape of the pulsar’s radiation, since this radiation is generated near the light cylinder in pulsars with P<0.1 s. Pulsars such as PSR 0531+21 and PSR 0833-45 have probably spun down due to the effect of magnetic-dipole radiation. If the difference in the braking indices for these objects from n=3 is due to the effect of accretion, the accretion rate must be of the order of 1018 g/s.  相似文献   

20.
A new class of figures of equilibrium for a rotating gravitating fluid located inside a gravitating ring or torus is studied. These figures form a family of sequences of generalized oblate spheroids, in which there is for any value of the tidal parameter α in the interval 0 ≤ \(0 \leqslant \frac{\alpha }{{\pi G\rho }} \leqslant 0.1867\) ≤ 0.1867 a sequence of spheroids with oblatenesses emin (α) ≤ eemax (α). A series of classicalMaclaurin spheroids from a sphere to a flat disk is obtained for α = 0. At intermediate values 0 < ααmax, there are two limiting non-rotating spheroids in each sequence. When α = αmax, the sequence degenerates into a single non-rotating spheroid with ecr ≈ 0.9600, corresponding to the maximum oblateness of E7 elliptical galaxies. The second part of the paper considers the influence of rings of dark matter on the dynamics of elliptical galaxies. It is proposed that the equilibrium of an oblate isolated non-rotating galaxy is unstable, and it cannot be supported purely by anisotropy of the stellar velocity dispersion. A ring of dark matter can stabilize a weakly rotating galaxy, supplementing standard dynamical models for such stellar systems. In order for a galaxy to acquire appreciable oblateness, the mass of the ring must be an order of magnitude higher than the mass of the galaxy itself, consistent with the ratios of the masses of dark and baryonic matter in the Universe. The influence of massive external rings could shed light on the existence of galaxies with the critical oblateness E7.  相似文献   

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