The physical properties of an analytic model for a relativistic star

An exact solution of Einstein's equations for a static isentropic perfect fluid sphere is examined in detail. The analysis yields a strong indication that the model isstable with respect to infinitesimal radial pulsations. This means that the temperature is decreasing outwards. We prove that the adiabatic speed of sound is everywhere less than the speed of light if and only if the radius of the sphere is larger than 1.61 times its Schwarzschild radius. We further show that the strong energy condition is fulfilled everywhere if and only if the radius is larger than 1.76 times the Schwarzschild radius. The necessary and sufficient condition for the speed of sound to be decreasing outwards is given, and we find that this criterion is fulfilled if the fluid is causal. Taking the values of the pressure and the density to be somewhere given by the maximum values from Baymet al.'s equation of state, i.e., ₀=5.1×10¹⁴ g cm^–3 andp ₀=7.4×10³³ dyne cm^–2, we find the maximum mass of the fluid sphere to be 2.5 solar masses.Dedicated to the memory of the late George Cunliffe McVittie (1904–1988).     

Some physical properties and stability of an exact model of a relativistic star

An exact solution of Einstein's field equations for an isentropic fluid sphere is examined. It turns out that the crucial factor for the physical properties and the stability of this model is the degree of incompressibility. Necessary and sufficient conditions are given for the weak and the strong energy conditions to be fulfilled and for the speed of sound to be less than the speed of light. The speed of sound always has a minimum at the center of the fluid sphere. But two possibilities exist: either the speed of sound is increasing all the way outwards to the surface of the sphere, or the speed of sound is first increasing, then reaching a maximum when still inside the fluid sphere, and thereafter decreasing outwards to the surface. The adiabatic index is investigated and is found to be increasing outwards for the actual degrees of compressibility. This adiabatic index is always greater than unity, and the temperature is thus decreasing throughout the sphere. The necessary and sufficient condition for the adiabatic index to be greater than 4/3 is also given. (This is a necessary condition for the fluid sphere to be stable.) Chandrasekhar's pulsation equation with boundary conditions is investigated, and the fluid sphere is found to bestable if and only if the degree of incompressibility is greater than a certain value.Dedicated to the memory of the late Bronislaw Kuchowicz.     

A class of exact isotropic solutions of Einstein’s equations and relativistic stellar models in general relativity

In this paper we have studied a particular class of exact solutions of Einstein’s gravitational field equations for spherically symmetric and static perfect fluid distribution in isotropic coordinates. The Schwarzschild compactness parameter, GM/c ² R, can attain the maximum value 0.1956 up to which the solution satisfies the elementary tests of physical relevance. The solution also found to have monotonic decreasing adiabatic sound speed from the centre to the boundary of the fluid sphere. A wide range of fluid spheres of different mass and radius for a given compactness is possible. The maximum mass of the fluid distribution is calculated by using stellar surface density as parameter. The values of different physical variables obtained for some potential strange star candidates like Her X-1, 4U 1538–52, LMC X-4, SAX J1808.4?3658 given by our analytical model demonstrate the astrophysical significance of our class of relativistic stellar models in the study of internal structure of compact star such as self-bound strange quark star.     

Some new exact solutions with finite central parameters and?uniform radial motion of sound

We present three new categories of exact and spherically symmetric Solutions with finite central parameters of the general relativistic field equations. Two well behaved solutions in curvature coordinates first category are being studied extensively. These solutions describe perfect 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 for these solutions. Keeping in view of well behaved nature of these solutions, one of the solution (I₁) is studied extensively. The solution (I₁) gives us wide range of Schwarzschild parameter u (0.138≤u≤0.263), for which the solution is well behaved hence, suitable for modeling of Neutron star. For this solution the mass of Neutron star is maximized with all degree of suitability and by assuming the surface density ρ _b =2×10¹⁴ g/cm³. Corresponding to u=0.263, the maximum mass of Neutron star comes out to be 3.369 M _Θ with linear dimension 37.77 km and central and surface redshifts are 4.858 and 0.4524 respectively. We also study some well known regular solutions (T-4, D-1, D-2, H, A, P) of Einstein’s field equations in curvature coordinates with the feature of constant adiabatic sound speed. We have chosen those values of Schwarzschild parameter u for which, these solutions describe perfect fluid balls realistic equations of state. However, except (P) solution, all these solutions have monotonically non-decreasing feature of adiabatic sound speed. Hence (P) solution is having a well behaved model for uniform radial motion of sound. Keeping in view of well behaved nature of the solution for this feature and assuming the surface density; ρ _b =2×10¹⁴ g/cm³, the maximum mass of Neutron star comes out to be 1.34 M _Θ with linear dimension 28.74 km. Corresponding central and surface redshifts are 1.002 and 0.1752 respectively.     

On the equilibrium figures of an ideal rotating fluid in the post-newtonian approximation of general relativity

A stability criterion is given for the equilibrium form of an ideal rotating fluid in the post-Newtonian approximation. This generalizes the known Lyapunov criterion in classical dynamics. The sphere stability is also investigated and it is shown that it is stable only whenR>22.2R _g (R is the relativistic sphere radius,R _g the Schwarzschild radius).     

The Pioneer anomaly and a Machian universe

We discuss astronomical and astrophysical evidence, which we relate to the principle of zero-total energy of the Universe, that imply several relations among the mass M, the radius R and the angular momentum L of a “large” sphere representing a Machian Universe. By calculating the angular speed, we find a peculiar centripetal acceleration for the Universe. This is an ubiquituous property that relates one observer to any observable. It turns out that this is exactly the anomalous acceleration observed on the Pioneers spaceships. We have thus shown that this anomaly is to be considered a property of the Machian Universe. We discuss several possible arguments against our proposal.     

Blazar天体射电爆发的持续注入喷流模型

为了解释Blazar天体射电爆发的普遍演化特性，本文提出一个持续注入的喷流模型．假定在喷流基底上以相对论性速度持续地注入相对记等离子体（由磁场和相对论电子组成）．这一等离子体在沿喷流向外运动时经受绝热膨胀损耗．理论计算表明，射电爆发的频谱演化具有Valtaoja等人所建议的典型的3阶段演化形式.它们非常好地重现了Blazar天体中观测到的射电爆发的普遍行为．     

Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

The dynamics of small global perturbations in the form of a linear combination of a finite number of non‐axisymmetric eigenmodes is studied in the two‐dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with adiabatic index γ = 5/3 rotating with a power law angular velocity distribution Γ ∝ r^–q , 1.5 < q < 2.0, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth G is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber m and increases for higher values of m. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order ∼10² over ∼6 typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spherical Magnetic Vortex in an External Potential Field: A Dissipative Contraction

We consider the dissipative evolution of a spherical magnetic vortex with a force-free internal structure, located in a resistive medium and held in equilibrium by the potential external field. The magnetic field inside the sphere is force-free (the model of Chandrasekhar in Proc. Natl. Acad. Sci. 42, 1, 1956). Topologically, it is a set of magnetic toroids enclosed in spherical layers. A new exact MHD solution has been derived, describing a slow, uniform, radial compression of a magnetic spheroid under the pressure of an ambient field, when the plasma density and pressure are growing inside it. There is no dissipation in the potential field outside the sphere, but inside the sphere, where the current density can be high enough, the magnetic energy is continuously converted into heat. Joule dissipation lowers the magnetic pressure inside the sphere, which balances the pressure of the ambient field. This results in radial contraction of the magnetic sphere with a speed defined by the conductivity of the plasma and the characteristic spatial scale of the magnetic field inside the sphere. Formally, the sphere shrinks to zero within a finite time interval (magnetic collapse). The time of compression can be relatively small, within a day, even for a sphere with a radius of about 1 Mm, if the magnetic helicity trapped initially in the sphere (which is proportional to the number of magnetic toroids in the sphere) is quite large. The magnetic system is open along its axis of symmetry. On this axis, the magnetic and electric fields are strictly radial and sign-variable along the radius, so the plasma will be ejected along the axis of magnetic sphere outwards in both directions (as jets) at a rate much higher than the diffusive one, and the charged particles will be accelerated unevenly, in spurts, creating quasi-regular X-ray spikes. The applications of the solution to solar flares are discussed.     

Note on aTheorem of Relativistic Hydrodynamics

A well known theorem of relativistic hydrodynamics states that the streamlines of an isentropic perfect fluid are the future-pointing timelike (FPT) curves extremizing the integral J = ∫ _S1 ^S2 fds, where f is the so-called index function and s the proper time on the world line of the fluid particle. The integral is taken over all possible FPT curves with regular representations xⁱ = xⁱ (s) joining the fixed end events E₁, E₂. The purpose of this note is to show that the streamlines of an adiabatic perfect fluid can likewise be regarded as extremizing curves of the functional J provided the class of admissible curves consists of those FPT curves satisfying the side condition uⁱ∂_iS = 0, uⁱ unit 4-velocity and S the specific proper entropy of the fluid, with the first end point fixed and the second being the end point variable. __________ Published in Astrofizika, Vol. 48, No. 4, pp. 641–647 (October–December, 2005).     

Direct GRB fireball Lorentz factor measurements through REM early afterglow observations

The huge energies involved in gamma-ray bursts (GRBs) coupled with the short emission time scales unavoidably imply that the emitting source is moving relativistically, with a speed close to that of light. Here we present the REM telescope observations of the early-time near-infrared light curves of the GRB 060418 and GRB 060607A afterglows. The detection of the afterglow peak provides for the first time a direct measurement of the initial Lorentz factor Γ₀ of the radiating material. We find that the emitting region was indeed highly relativistic in the first seconds after the explosions, with Γ₀∼400. Comparison with the Lorentz factor as determined at later epochs provides direct evidence that the emitting shell is decelerating and confirms that the afterglow emission is powered by the dissipation of bulk kinetic energy. The deceleration radius was inferred to be R _dec≈10¹⁷ cm. This is much larger than the internal shocks radius (believed to power the prompt emission), thus providing further evidence for a different origin of the prompt and afterglow stages of the GRB. Susanna D. Vergani on behalf of the REM collaboration.     

Gravitational collapse of a massive sphere of constant energy density

Gravitational collapse of a massive sphere of constant density has been studied from the point of view of a Keplerian observer. The asymptotic nature of collapse is attributed to the development of negative gravitational pressure acting radially outwards within the structure. The region of negative pressure asymptotically covers the entire interior asu=mass/radius tends to half.     

Physical characteristics ofN-dimensional,radially-symmetric polytropes

The physical characteristics radius, mass, mean density, gravitational potential and acceleration, gravitational and internal energy are presented with the aid of the gamma function forN-dimensional, radially-symmetric polytropes. The virial theorem with external pressure is derived in the relativistic limit, with Newtonian gravitation still valid. The gravitational energy of polytropes obeying the generalized Schuster—Emden integral is shown to be finite. Finiteness of mass and radius is discussed for the cases of practical interestN=1 (slab),N=2 (cylinder), andN=3 (sphere). Uniform contraction or expansion ofN-dimensional polytropes is considered in the last section.     

On the stability of ultrarelativistic stars

This paper present two new theorems on the theory of the stability of highly relativistic stars. Thefirst theorem states that a highly relativistic, spherical star is stable if and only if its adiabatic index (assumed to be constant in the interior regions) is greater than a certain critical value, _crit which depends in a specified way on the high-density equation of state. This critical value is analogous to the Newtonian value , but because of relativistic effects it is typically somewhat larger than . Thesecond theorem shows that at high central densities, the curves of —(binding energy) vs. radius, —E _B (R) for certain hot, isentropic sequences of stellar models must exhibit damped clockwise spirals. This spiraling reflects the onset of instability in one radial mode of pulsation after another as the central density increases along the sequence.     

Hydrodynamic Evolution of GRB Afterglow

1 INTRODUCTIONMom the detection at X-rap optical and radio wavelengths of ganuna-ray bursts (GRBs)since 1997 (Costa et al. 1997; van Paradijs et al. 1997; Sahu et al. 1997; Djorgovsld et al. 1997;Metzger et al. 1997; Frail et al. 1997; Taylor et al. 1997; Kulforni et al. 1998; Halpern et al.1998; Castro-Tirado et al. 1999; Kulkalni et al. 1999; Galama et al. 1999), we have come toknow that the GRBs can release 1051 ~ 1054 ergs in a few seconds and that the fireball modelcan describ…     

The flow of a two-component plasma model in a porous rotating hot sphere

The paper studies the flow of a two-component hot plasma in a porous rotating sphere. Asymptotic solutions are derived for small rotation Reynolds number (Re). The overall analysis of the study shows that the temperature distribution of the gas inside the sphere has a minimum value, for the various values of the radiation parameterN ² at a point where the radius of the solar sphere is 0.5r (wherer ₀ is the dimensional radius of the sphere). The general result of the studies are discussed quantitatively. The problem has a lot of application in the understanding of the interior of astrophysical bodies.     

Newtonian and post Newtonian expansionfree fluid evolution in f(R) gravity

We consider a collapsing sphere and discuss its evolution under the vanishing expansion scalar in the framework of f(R) gravity. The fluid is assumed to be locally anisotropic which evolves adiabatically. To study the dynamics of the collapsing fluid, Newtonian and post Newtonian regimes are taken into account. The field equations are investigated for a well-known f(R) model of the form R+δR ² admitting Schwarzschild solution. The perturbation scheme is used on the dynamical equations to explore the instability conditions of expansionfree fluid evolution. We conclude that instability conditions depend upon pressure anisotropy, energy density and some constraints arising from this theory.     

Radiative transfer equation in spherically-symmetric non-scattering media

We solved the equation of radiative transfer in spherically-symmetric shells with arbitrary internal sources. We integrated the equation of transfer on the discrete grid of angle and radius given by [_j–1, _j] [r_i–1, r_i]. The size in the angle coordinates is determined by the roots of a quadrature formula where as the size in the radial coordinate is determined by the non-negativity of the reflection and transmission operators. We considered two cases of variation of the Planck function. (1) Constant throughout the medium and (2) varying as 1/r ². We find that in the inner shells, the radiation directed toward the centre of the sphere is more than that directed away from the centre of the sphere. In the outer shells the converse is true.     

A simple way to classify supermassive black holes

We propose a classification of supermassive black holes (SMBHs) based on their efficiency in the conversion of infalling mass in emitted radiation. We use a theoretical model that assumes a conservation of angular momentum between the gas falling inside the hole and the photons emitted outwards, and suggests the existence of the scaling relation M_•‐R_eσ³, where M_• is the mass of the central SMBH, whereas R_e and σ are the effective radius and velocity dispersion of the host galaxies (bulges), respectively. We apply our model on a data set of 57 galaxies of different morphological types and with M_• measurements, obtained through the analysis of Spitzer /IRAC 3.6‐µ m images. In order to find the best fit of the corresponding scaling law, we use the FITEXY routine to perform a least‐squares regression of M_• on R_eσ³ for the considered sample of galaxies. Our analysis shows that the relation is tight and our theoretical model allows to easily estimate the efficiency of mass conversion into radiation of the central SMBHs. Finally we propose a new appealing way to classify the SMBHs in terms of this parameter. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

First ever closed form solution for charged fluid sphere expressed by a space time with its hypersurfaces t= constant as spheroid is obtained for the case 0<K<1. The same is utilized to construct a superdense star with surface density 2×10¹⁴ gm/cm³. The star is seen to satisfy the reality and causality conditions for 0<K≤0.045 and possesses maximum mass and radius to be 0.065216M _Θ and 1.137496 km respectively. Moreover the interior of the star satisfy strong energy condition. However in the absence of the causality condition, the reality conditions are valid for a wider range 0<K≤0.13. The maximum mass and radius for the later case are 1.296798M _Θ and 2.6107 km respectively for the strong energy condition, while the said parameters for the weak energy condition read as 1.546269M _Θ and 2.590062 km respectively.