Gravity field and interior of Rhea from Cassini data analysis

The Cassini spacecraft encountered Rhea on November 26, 2005. Analysis of the Doppler data acquired at and around closest approach yields the mass of Rhea and the quadrupole moments of its gravity field with unprecedented accuracy. We obtained which corresponds to a density of . Our results for J₂ and C₂₂ are (7.947±0.892)×¹⁰⁻⁴ and (2.3526±0.0476)×¹⁰⁻⁴, respectively. These values are consistent with hydrostatic equilibrium. From the value of C₂₂, we infer the non-dimensional moment of inertia C/MR²=0.3721±0.0036. Our models of Rhea's interior based on the gravity data favor an almost undifferentiated satellite. A discontinuity between a core and a mantle is possible but not required by the data. Models with a constant silicate mass fraction throughout the body cannot account for the determined quadrupole coefficients. The data exclude fully differentiated models in which the core would be composed of unhydrated silicates and the mantle would be composed of pure ice. If the mantle contains 10% in mass of silicates, the core extends to 630 km in radius and has a silicate mass fraction of 40%. A continuous model in which the silicates are more concentrated toward the center of the body than in the outer layers is allowed by the gravity data but excluded by thermal evolution considerations. The one model that fits the gravity data and is self-consistent when energy transport and ice melting are qualitatively considered is an “almost undifferentiated” Rhea, in which a very large uniform core is surrounded by a relatively thin ice shell containing no rock at all.     

A family of well behaved charge analogues of Durgapal’s perfect fluid exact solution in general relativity

This paper presents a new family of interior solutions of Einstein–Maxwell field equations in general relativity for a static spherically symmetric distribution of a charged perfect fluid with a particular form of charge distribution. This solution gives us wide range of parameter, K, for which the solution is well behaved hence, suitable for modeling of superdense star. For this solution the gravitational mass of a star is maximized with all degree of suitability by assuming the surface density equal to normal nuclear density, ρ _nm=2.5×10¹⁷ kg?m^?3. By this model we obtain the mass of the Crab pulsar, M _Crab, 1.36M _⊙ and radius 13.21 km, constraining the moment of inertia >?1.61×10³⁸ kg?m² for the conservative estimate of Crab nebula mass 2M _⊙. And M _Crab=1.96M _⊙ with radius R _Crab=14.38 km constraining the moment of inertia >?3.04×10³⁸ kg?m² for the newest estimate of Crab nebula mass, 4.6M _⊙. These results are quite well in agreement with the possible values of mass and radius of Crab pulsar. Besides this, our model yields moments of inertia for PSR J0737-3039A and PSR J0737-3039B, I _A =1.4285×10³⁸ kg?m² and I _B =1.3647×10³⁸ kg?m² respectively. It has been observed that under well behaved conditions this class of solutions gives us the overall maximum gravitational mass of super dense object, M _G(max)=4.7487M _⊙ with radius $R_{M_{\max}}=15.24~\mathrm{km}$ , surface redshift 0.9878, charge 7.47×10²⁰ C, and central density 4.31ρ _nm.     

The effect of ammonia ice on the outgoing thermal radiance from the atmosphere of Jupiter

We examine the effects of NH₃ ice particle clouds in the atmosphere of Jupiter on outgoing thermal radiances. The cloud models are characterized by a number density at the cloud base, by the ratio of the scale height of the vertical distribution of particles (H_p) to the gas scale height (H_g), and by an effective particle radius. NH₃ ice particle-scattering properties are scaled from laboratory measurements. The number density for the various particle radius and scale height models is inferred from the observed disk average radiance at 246 cm^?1, and preliminary lower limits on particle sizes are inferred from the lack of apparent NH₃ absorption features in the observed spectral radiances as well as the observed minimum flux near 2100 cm^?1. We find lower limits on the particle size of 3 μm if H_p/H_g = 0.15, or 10μmif H_p/H_g = 0.50 or 0.05. NH₃ ice particles are relatively dark near the far-infrared and 8.5-μm atmospheric windows, and the outgoing thermal radiances are not very sensitive to various assumptions about the particle-scattering function as opposed to radiances at 5 μm, where particles are relatively brighter. We examined observations in these three different spectral window regions which provide, in principle, complementary constraints on cloud parameters. Characterization of the cloud scale height is difficult, but a promising approach is the examination of radiances and their center-to-limb variation in spectral regions where there is significant opacity provided by gases of known vertical distribution. A blackbody cloud top model can reduce systematic errors due to clouds in temperature sounding to the level of 1K or less. The NH₃ clouds provide a substantial influence on the internal infrared flux field near the 600-mbar level.     

Computation of Jupiter interior models from gravitational inversion theory

A method for deriving a planetary interior model which exactly satisfies a set of N gravitational constraints is implemented. For Jupiter, recent spacecraft measurements provide the mass, radius at a standard pressure level, rotation law, multipole moments of the internal mass distribution, and constraints on the internal composition and temperature distribution. By appropriate iterations, interior models are found which exactly satisfy these constraints. The models are assumed to have constant chemical composition and constant specific entropy in the hydrogenic envelope. The derived pressure-density relation in the outer envelope depends sensitively on the observational uncertainty in the mass multipole moment J₄. Models are not forced to fit the more indirectly derived constraints, which are instead used as consistency checks. For a helium mass fraction in the envelope (Y) equal to 0.20, the inferred pressure at a mass density ≈ 0.2 g/cm³ is about a factor of 2 higher than would be indicated by experimental hydrogen shock compression data in the relevant pressure range of 10⁵ to 10⁶ bar. The inferred pressure distribution is in much better agreement with the shock data for a nominal Y = 0.30 ± 0.05. This value of Y is interpreted in terms of an enhancement in the envelope, by a factor of order 5 over solar abundance, of species primarily consisting of CH₄, NH₃, and possibly H₂O. The same method is applied to Saturn, but existing uncertainties in Saturn's gravitational parameters are still too large to allow useful conclusions about the composition of its envelope.     

A limit for gravitational collapse

Earlier, under certain simplifying assumptions, on the basis of the General Theory of Relativity, it has been concluded by many authors that when the radius of a gravitationally collapsing spherical object of massM reaches the critical value of the Scharzschild radiusR _s=2GM/c ², then, in a co-moving frame, the object collapses catastrophically to a point. However, in drawing this conclusion due consideration has not been given to the nuclear forces between the nucleons. In particular, the very strong ‘hard-core’ repulsive interaction between the nucleons which has the range ～0.4×10^?13 cm has been totally ignored. On taking into account this ‘hard-core’ repulsive interaction, it is found that no spherical object of massM g can collapse to a volume of radius smaller thanR _min=(1.68×10^?6)M ^1/3 cm or to a density larger than ρ_max=5.0 × 10¹⁶ g cm^?3. It has also been pointed out that objects of mass smaller thanM _c～1.21×10³³ g can not cross the Schwarzschild barrier and gravitationally collapse. The only course left to the objects of mass less thanM _cis to reach the equilibrium as either a white dwarf or a neutron star.     

A new well behaved class of charge analogue of Adler’s relativistic exact solution

The paper presents a new class of parametric interior solutions of Einstein–Maxwell field equations in general relativity for a static spherically symmetric distribution of a charged perfect fluid with a particular form of electric field intensity. This solution gives us wide range of parameter, K (0.69≤K≤7.1), for which the solution is well behaved hence, suitable for modeling of superdense star. For this solution the gravitational mass of a superdense object is maximized with all degree of suitability by assuming the surface density of the star equal to the normal nuclear density ρ _nm=2.5×10¹⁷kg?m^?3. By this model we obtain the mass of the Crab pulsar M _Crab=1.401M _⊙ and the radius, R _Crab=12.98 km constraining the moment of inertia I _NS,38>1.61 for the conservative estimate of Crab nebula mass 2M _⊙ and M _Crab=2.0156M _⊙ with radius, R _Crab=14.07 km constraining the moment of inertia I _NS,38>3.04 for the newest estimate of Crab nebula mass 4.6M _⊙ which are quite well in agreement with the possible values of mass and radius of Crab pulsar. Besides this, our model yields the moments of inertia for PSR J0737-3039A and PSR J0737-3039B are I _A,38=1.4624 and I _B,38=1.2689 respectively. It has been observed that under well behaved conditions this class of parametric solution gives us the maximum gravitational mass of causal superdense object 2.8020M _⊙ with radius 14.49 km, surface redshift z _R =0.4319, charge Q=4.67×10²⁰ C, and central density ρ _c =2.68ρ _nm.     

Constraints on the composition of the martian south polar cap from gravity and topography

The polar caps of Mars have long been acknowledged to be composed of unknown proportions of water ice, solid CO₂ (dry ice), and dust. Gravity and topography data are here analyzed over the southern cap to place constraints on its density, and hence composition. Using a localized spectral analysis combined with a lithospheric flexure model of ice cap loading, the best fit density of the volatile-rich south polar layered deposits is found to be 1271 kg m⁻³ with 1-σ limits of 1166 and 1391 kg m⁻³. The best fit elastic thickness of this geologically young deposit is 140 km, though any value greater than 102 km can fit the observations. The best fit density implies that about 55% dry ice by volume could be sequestered in these deposits if they were completely dust free. Alternatively, if these deposits were completely free of solid CO₂, the dust content would be constrained to lie between about 14 and 28% by volume. The bulk thermal conductivity of the polar cap is not significantly affected by these maximum allowable concentrations of dust. However, even if a moderate quantity of solid CO₂ were present as horizontal layers, the bulk thermal conductivity of the polar cap would be significantly reduced. Reasonable estimates of the present day heat flow of Mars predict that dry ice beneath the thicker portions of the south polar cap would have melted. Depending on the quantity of solid CO₂ in these deposits today, it is even possible that water ice could melt where the cap is thickest. If independent estimates for either the dust or CO₂ content of the south polar cap could be obtained, and if radar sounding data could determine whether this polar cap is presently experiencing basal melting or not, it would be possible to use these observations to place tight constraints on the present day heat flow of Mars.     

Cratering of icy targets by different impactors: Laboratory experiments and implications for cratering in the Solar System

Studies of impacts (impactor velocity about 5 km s⁻¹) on icy targets were performed. The prime goal was to study the response of solid CO₂ targets to impacts and to find the differences between the results of impacts on CO₂ targets with those on H₂O ice targets. The crater dimensions in CO₂ ice were found to scale with impact energy, with little dependence on projectile density (which ranged from nylon to copper, i.e., 1150-8930 kg m⁻³). At equal temperatures, craters in CO₂ ice were the same diameter as those in water ice, but were shallower and smaller in volume. In addition, the shape of the radial profiles of the craters was found to depend strongly on the type of ice and to change with impact energy. The impact speed of the data is comparable to that for impacts on many types of icy bodies in the outer Solar System (e.g., the satellites of the giant planets, the cometary nuclei and the Kuiper Belt objects), but the size and thus energy of the impactors is lower. Scaling with impact energy is demonstrated for the impacts on CO₂ ice. The issue of impact disruption (rather than cratering) is discussed by analogy with that on water ice. Expressions for the critical energy density for the onset of disruption rather than cratering are established for water ice as a function of porosity and silicate content. Although the critical energy density for disruption of CO₂ ice is not established, it is argued that the critical energy to disrupt a CO₂ ice body will be greater than that for a (non-porous) water ice body of the similar mass.     

A class of well-behaved generalized charged analogues of Vaidya-Tikekar type fluid sphere in general relativity

In this paper first ever we have developed a class of well behaved charged fluid spheres expressed by a space time with its hypersurfaces $t = \operatorname {const}$ . as spheroid for the case 0<K<1 with surface density 2×10¹⁴ gm/cm³. The same utilized to construct a superdense star and seen that star satisfies all well behaved condition for 0<K≤0.038. The maximum mass occupied and the corresponding radius are found to be 4.830982M _Θ and 20.7612 km respectively. The redshift at the center and on the surface is given z ₀=0.425367 and z _a =0.240901.     

Roles of statistical mechanics in determining the density profile of mild relaxation

The r ^?4 law of cold collapse has been explained in other work. Here we try to explain the density profile of mild relaxation by statistical mechanics. In this paper we first generate many kinds of initial conditions with the same mass and energy to see whether there are other initial factors that can change the density profile of an isolated equilibrium self-gravitating system; then for a more general initial condition we discuss the role of mass and energy in determining the final density profile. Next we use our previous results obtained from statistical mechanics to fit these simulations, and find that when the masses of the particles in clumps are less than 5 % of the total mass, or the initial density is shallower than r ^?2, the whole virialized density profile (VDF) can be fitted well by our equation of state with three parameters, and some other cases can be explained by the theory with the r ^?4 law. We conclude that statistical physics may play an important role in determining the shape of VDF in the mild relaxation, mass and energy can control the values of the central density and the system’s radius, but there are still other initial configurations that can affect the VDF.     

Saturn's rings: Particle size distributions for thin layer models

The small physical thickness of Saturn's rings requires that radio occultation observations be interpreted using scattering models with limited amounts of multiple scatter. A new model in which the possible order of near-forward scatter is strictly limited allows for the small physical thickness, and can be used to relate Voyager 1 observations of 3.6-and 13-cm wavelength microwave scatter from Saturn's rings to the ring particle size distribution function n(a), for particles with radius 0.001 ≤ a ≤ 20 m. This limited-scatter model yields solutions for particle size distribution functions for eight regions in Saturn's rings, which exhibit approximately inverse-cubic power-law behavior, with large-size cutoffs in particle radius ranging from about 5 m in ring C to about 10 m in parts of ring A. The power-law index is about 3.1 in ring C, about 2.8 in the Cassini division, and increases systematically with radial location in ring A from 2.7 at 2.10R_s to slightly more than 3.0 at 2.24R_s. Corresponding mass densities are 32–43 kg/m² in ring C, 188 kg/m² in the Cassini division, and 244–344 kg/m² in ring A, under the assumption that the material density of the particles is 0.9 g/cm³. These values are a factor of 1 to 2 lower than first-order mass loading estimates derived from resonance phenomena. In view of the uncertainties in the measurements and in the linear density wave model, and the strong arguments for icy particles with specific gravity not greater than about 1, we interpret this discrepancy as being indicative of possible differences in the regions studied, or systematic errors in the interpretation of the scattering results, the density wave phenomena, or some combination of the above.     

Seeable universe and its accelerated expansion: an observational test

From the equivalence principle, one gets the strength of the gravitational effect of a mass M on the metric at position r from it. It is proportional to the dimensionless parameter β ²=2GM/rc ², which normally is ?1. Here G is the gravitational constant, M the mass of the gravitating body, r the position of the metric from the gravitating body and c the speed of light. The seeable universe is the sphere, with center at the observer, having a size such that it shall contain all light emitted within it. For this to occur one can impose that the gravitational effect on the velocity of light at r is zero for the radial component, and non zero for the tangential one. Light is then trapped. The condition is given by the equality R _g =2GM/c ², where R _g represents the radius of the seeable universe. It is the gravitational radius of the mass M. The result has been presented elsewhere as the condition for the universe to be treated as a black hole. According to present observations, for the case of our universe taken as flat (k=0), and the equation of state as p=?ρc ², we prove here from the Einstein’s cosmological equations that the universe is expanding in an accelerated way as t ², a constant acceleration as has been observed. This implies that the gravitational radius of the universe (at the event horizon) expands as t ². Taking c as constant, observing the galaxies deep in space this means deep in time as ct, linear. Then, far away galaxies from the observer that we see today will disappear in time as they get out of the distance ct that is <R _g . The accelerated expanding vacuum will drag them out of sight. This may be a valid test for the present ideas in cosmology. Previous calculations are here halved by our results.     

The mass,gravity field,and ephemeris of Mercury

This paper represents a final report on the gravity analysis of radio Doppler and range data generated by the Deep Space Network (DSN) with Mariner 10 during two of its encounters with Mercury in March 1974 and March 1975. A combined least-squares fit to Doppler data from both encounters has resulted in a determination of two second degree gravity harmonics, J₂ = (6.0 ± 2.0) × 10⁻⁵ and C₂₂ = (1.0 ± 0.5) × 10⁻⁵, referred to an equatorial radius of 2439 km, plus an indication of a gravity anomaly in the region of closest approach of Mariner 10 to Mercury in March 1975 amounting to a mass deficiency of about GM = −0.1 km³sec⁻². An analysis is included that defends the integrity of previously published values for the mass of Mercury (H. T. Howard et al. 1974, Science 185, 179–180; P. B. Esposito, J. D. Anderson, and A. T. Y. Ng 1978, COSPAR: Space Res. 17, 639–644). This is in response to a published suggestion by R. A. Lyttleton (1980, Q. J. R. Astron. Soc. 21, 400–413; 1981, Q. J. R. Astron. Soc. 22, 322–323) that the accepted values may be in error by more than 30%. We conclude that there is no basis for being suspicious of the earlier determinations and obtain a mass GM = 22,032.09 ± 0.91 km³sec⁻² or a Sun to Mercury mass ratio of 6,023,600 ± 250. The corresponding mean density of Mercury is 5.43 ± 0.01 g cm⁻³. The one-sigma error limits on the gravity results include an assessment of systematic error, including the possibility that harmonics other than J₂and C₂₂ are significantly different from zero. A discussion of the utility of the DSN radio range data obtained with Mariner 10 is included. These data are most applicable to the improvement of the ephemeris of Mercury, in particular the determination of the precession of the perihelion.     

Particle size distributions in Saturn's rings from voyager 1 radio occultation

Observations of microwave opacity τ[λ] and near forward scatter from Saturn's rings at wavelengths λ of 3.6 and 13 cm from the Voyager 1 ring occultation experiment contain information regarding ring particle sizes in the range of about a = 0.01 to 15 m radius. The opacity measurements τ[3.6] and τ[13] are sufficient to constrain the scale factor n(a₀) and index q of a power law incremental size distribution n(a) = n(a₀)[a₀/a]^q, assuming known minimum and maximum sizes and a many-particle-thick model. The families of such distributions are highly convergent in the centimeter-size range. Forward scatter at 3.6 cm can be used to solve for a general distribution over the radius range $\text{1 ? a ? 15}\text{m}$ by integral inversion and inverse scattering methods, again assuming a many-particle-thick slab-type radiative transfer model. Distributions n(a) valid over $\text{0.01 ? a ? 15}\text{m}$ are obtained by combining the results from the two types of measurements above. Mass distributions may be computed directly from n(a). Such distributions, partly measured and partly synthesized, have been obtained for four features in the ring system centered at 1.35, 1.51, 2.01, and 2.12 Saturn radii (R_s). The size and mass distributions both cut off sharply at a ? 4–5 m; the mass distribution peaks over the narrow size range $\text{3 ? a ? 4}\text{m}$ for all four locations. No single power law distribution is consistent with the data over the entire interval $\text{0.01 ? a ? 5}\text{m}$, although a power law-type model is consistent with the data over a limited size range of $\text{0.01 ? a ? 1}\text{m}$, where the indices q = 3.4 and 3.3 are obtained from the slab model for the features located at 1.51 and 2.01 R_s. The fractional contribution of the suprameter particles to the microwave opacity in each feature appears to be about $\text{1}\text{3}\text{,}\text{1}\text{3}\text{,}\text{2}\text{3}\text{,}\text{and}\text{1}$, respectively, with the fraction at 2.12 R_s being the least certain. The cumulative surface mass per unit area obtained for the classical slab model is approximately 11, 16, 41, and 132 g/cm² for the four features, respectively, if the particles are solid H₂O ice. Both the fractional opacity and the mass density estimates represent upper bounds implied by the assumption of a uniformly mixed set of particles in a many-particle-thick vertical profile; lower estimates would result if the rings were assumed to be nearly a monolayer or if the vertical distribution of particles were size dependent.     

A well-behaved class of charged analogue of Durgapal solution

We present a well behaved class of charged analogue of M.C. Durgapal (J. Phys. A, Math. Gen. 15:2637, 1982) solution. This solution describes charged fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one and causality condition is obeyed at the centre. The outmarch of pressure, density, pressure-density ratio and the adiabatic speed of sound is monotonically decreasing, however, the electric intensity is monotonically increasing in nature. This solution gives us wide range of parameter for every positive value of n for which the solution is well behaved hence, suitable for modeling of super dense stars. Keeping in view of well behaved nature of this solution, one new class of solution is being studied extensively. Moreover, this class of solution gives us wide range of constant K (0≤K≤2.2) for which the solution is well behaved hence, suitable for modeling of super dense stars like strange quark stars, neutron stars and pulsars. For this class of solution the mass of a star is maximized with all degree of suitability, compatible with quark stars, neutron stars and pulsars. By assuming the surface density ρ _b =2×10¹⁴ g/cm³ (like, Brecher and Capocaso, Nature 259:377, 1976), corresponding to K=0 with X=0..235, the resulting well behaved model has the mass M=4.03M _Θ , radius r _b =19.53 km and moment of inertia I=1.213×10⁴⁶ g?cm²; for K=1.5 with X=0.235, the resulting well behaved model has the mass M=4.43M _Θ , radius r _b =18.04 km and moment of inertia I=1.136×10⁴⁶ g?cm²; for K=2.2 with X=0.235, the resulting well behaved model has the mass M=4.56M _Θ , radius r _b =17.30 km and moment of inertia I=1.076×10⁴⁶ g?cm². These values of masses and moment of inertia are found to be consistent with the crab pulsars.     

The Substellar Population in the Young σ Orionis Cluster, Spatial Distribution

We review different surveys, in the optical and infrared, conducted in the very young (age 1–8 Myr), nearby (d ～ 350 pc) σ Orionis cluster aimed to characterize the substellar population. We describe spectral characteristics of very low mass stars, brown dwarfs and planetary mass objects in the cluster with spectral types from K7 to T6. We study the spatial distribution of the substellar population detected in a IZJ survey covering an area of 1.12 deg.² We find that the radial distribution of substellar objects can be well fitted by an exponential law (ρ = ρ_o e ^?r/ro ), with a central density (ρ _o ) of 0.26 ± 0.03 objects/arcmin² and a characteristic radius (r _o) of 8.8 arcmin ± 0.6 (equivalent to 0.90 ± 0.06 pc at the distance of the cluster). We discuss the presence of possible inhomogeneities in this distribution due to the existence of subclustering. We also compare the spatial distribution of the substellar population with previously known stars in the cluster. We report the initial mass spectrum in the substellar domain.     

Mass-radius curve for extrasolar Earth-like planets and ocean planets

By comparison with the Earth-like planets and the large icy satellites of the Solar System, one can model the internal structure of extrasolar planets. The input parameters are the composition of the star (Fe/Si and Mg/Si), the Mg content of the mantle (Mg# = Mg/[Mg + Fe]), the amount of H₂O and the total mass of the planet. Equation of State (EoS) of the different materials that are likely to be present within such planets have been obtained thanks to recent progress in high-pressure experiments. They are used to compute the planetary radius as a function of the total mass. Based on accretion models and data on planetary differentiation, the internal structure is likely to consist of an iron-rich core, a silicate mantle and an outer silicate crust resulting from magma formation in the mantle. The amount of H₂O and the surface temperature control the possibility for these planets to harbor an ocean. In preparation to the interpretation of the forthcoming data from the CNES led CoRoT (Convection Rotation and Transit) mission and from ground-based observations, this paper investigates the relationship between radius and mass. If H₂O is not an important component (less than 0.1%) of the total mass of the planet, then a relation (R/REarth)=ab(M/MEarth) is calculated with (a,b)=(1,0.306) and (a,b)=(1,0.274) for ¹⁰⁻²MEarth<M<MEarth and MEarth<M<10MEarth, respectively. Calculations for a planet that contains 50% H₂O suggest that the radius would be more than 25% larger than that based on the Earth-like model, with (a,b)=(1.258,0.302) for ¹⁰⁻²MEarth<M<MEarth and (a,b)=(1.262,0.275) for MEarth<M<10MEarth, respectively. For a surface temperature of 300 K, the thickness of the ocean varies from 150 to 50 km for planets 1 to 10 times the Earth's mass, respectively. Application of this algorithm to bodies of the Solar System provides not only a good fit to most terrestrial planets and large icy satellites, but also insights for discussing future observations of exoplanets.     

A calculation of structural parameters of spherical stellar systems by means of characteristic functions method. I. Theory

To calculate structural parameters of stellar systems such as an effective radius and central space (or surface) density, the method of characteristic functions is suggested. The characteristic function of the system is a Fourier image of their normalized space density profile f₃(r). In the case of spherical symmetry the probability distribution of r (Q₃(r) = (3/a³)r²f₃(r)) and its orthogonal projections have the same characteristic functions. This fact is used to calculate the effective radii of a few star cluster models (King law, Plummer model and Gausian profile). It is shown, that the characteristic function for King law clusters tends to a finite generalised function if the concentration parameter c is large. The expression for the effective radius (at c ≫ 1) is given. The formula of the effective radius in the Plummer model as well as the relation between the one-dimensional central velocity dispersion and the root mean square velocity are obtained. It is shown, that in the Gaussian model and for King law clusters the effective radius (half-mass visual radius) can differ from the effective (harmonic) radius a few times. This fact should be taken into account in estimating the mass-to-light ratio from the virial mass of such systems using the King radius.     

A class of well behaved charged analogues of Vaidya–Tikekar type super-dense star

In the present article a model of well behaved charged superdense star with surface density 2×10¹⁴ gm/cm³ is constructed by considering a static spherically symmetric metric with t=const hypersurfaces as hyperboloid. So far well behaved model described by such metric could not be obtained. Maximum mass of the star is found to be 0.343457M _⊙ and the corresponding radius is 9.57459 km. The red shift at the centre and on the surface are given as 0.068887 and 0.031726 respectively.