Polytropic spheres with isothermal cores

The paper deals with massive fluid spheres with an isothermal core (having finite central density) and a polytropic envelope in terms of the General Relativity. The matching of the solutions in the core and envelope has been done using Bondi's condition,H=0 and also without it. The study reveals that since this condition does not correspond to any particular physical situation the maximum values of fractional core size, fractional core mass and the redshift do not occur atH=0, but that they occur at some other point. Within the permissible physical conditions (dP/dρ≤1) the maximum values ofM _core/M,R _core/R and the surface redshift, for an isothermal coreP=ρ/3, are respectively 0.473, 0.554, and 0.565. Using the conditionH=0, it has been shown that for isothermal cores corresponding to the equation of the stateP≥0.6ρ, the configurations are pulsationally unstable.     

Coastal dune conservation on an Irish commonage: community-based management or tragedy of the commons?

In Ireland 'commonage' refers to lands jointly owned by several individuals who have grazing rights. Commonage can provide the low-intensity grazing regime regarded as optimal for habitat conservation, and it is also unlikely to suffer the negative impacts of building development or coastal engineering. Today, however, the traditional control systems of coastal commonage are generally moribund, leading to habitat degradation. The only viable future management model is likely to be one based on local community control. Community management would have the legitimacy to counter the negative perceptions of external authority that generate environmental degradation.     

Neutron stars and general equation of nuclear matter

It has been shown that the mass of neutron stars obtained from equations of state based on nuclear theory depend upon the number of baryons assembled in it but not on the type of interactions considered. On examining the behaviour of different equations of state based on nuclear theories, a simple polytropic equation of state,P = (K/N)(p –p _s)^N is proposed. The results obtained forN=1.75 cover the entire range of neutron star masses obtained from the equations of state based on nuclear theories and give a maximum mass of 2.8M . Depending upon various mechanisms for energy output the mass of Crab pulsar is estimated to range from 0.32M to 1.5M . The relation connecting the coordinate mass,M, and the rest mass,M ₀, may be written asM/M 0.93 (M ₀/M)^0.9.     

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.     

Equation of state of dense nuclear matter and an upper bound on neutron star masses

We have discussed, in general, the important physical parameters, likemaximum mass, radius, and the minimum rotation period of self-bound,causally consistent, and pulsationally stable neutron stars (Q-starmodels) by using a realistic stiff EOS (such that, the speed of sound,v P ^N, or nP=K(E-E _a ) ⁿ, where K 1 and n =1/(1-2N);where P and E represent respectively, the pressure and theenergy-density, and E _a is the value of E at the surface (r = a) of the configuration) within the two constraints imposed by: (i) The minimumrotation period, P _rot, for the pulsar known to date corresponds to1.558 ms, and (ii) The maximum number density anywhere inside thestructure for the models described as Q-stars cannot exceed 1nucleon/fm³. By using the empirical formula given by Koranda,Stergioulas and Friedman (1997) (KSF-formula), and imposing constraint(i), we have obtained an upper bound of M _max 7.76 M radius a 32.5 km, and the central energy-density around 2.17 ×10¹⁴ g cm^-3 (for n =1.01). Constraint (ii) provides the minimumrotation period, P _rot 0.489 ms for the maximum mass M _max 2.4 M, and the central energy-density around 2.20 ×10¹⁵ g cm^-3 (for n =1.01). The speed of sound at the centre ofthese models approaches 99% of the speed of light `c' (in thevacuum) and vanishes at the surface of the configuration together withpressure. If we relax the maximum Kepler frequency imposed by the fastestrotating pulsar known to date (constraint (i)), in view of certainobservational effects and theoretical evidences, and allow the present EOSto produce larger rotation rates than the 1.558 ms pulsar, the maximummass of the non-rotating model drops down to a value 7.2 M .The higher values of masses ( 7 M ) and radii (31-32 km) obtained in this study imply that these models may representthe massive compact objects like Cyg X-1, Cyg XR-1, LMC X-3, and otherswhich are known as black hole candidates (BHCs). This study also suggestthat the strongest contender for black hole at present might be recurrentnova V404 Cyg (mass estimate 8 -12 M ).     

Isothermal neutron star cores

By considering the relativistic expression for isothermal NS cores,T·e ^/2 = constant, we have shown that some of the standard equations of state, when applied to NS cores, correspond to constancy of some adiabatic exponents. It has been shown that the equation of state,P=KE, corresponds to ₁ = to ₂ = ₃ 1 +K and the equation of state, dP/dE=K, corresponds to ₃ 1 +K. The conditions under which different equations of state represent isothermal cores have been obtained: For isothermal NS, the local temperatureT, can be expressed in terms of pressureP, energy densityE, and rest mass density . For example: (a)P =KE :T = constant × (E/); (b)P=KE :T = constant × (P/); (c) dP/dE =K :T ^K ; (d) = ₂ :T = constant × (P/E); and (e) = ₃ :T = constant × (P/)^1/2. Equation of state corresponding to = ₂ is obtained as:P=E/ln(K/E) and the equation corresponding to = ₃ comes out as:E=P ln(K/P). Core-envelope models can be developed for these two cases. When core equation corresponding to = ₂ or = ₃ is used in the core, we can ensure the continuity of dP/dE at the core-envelope boundary, along with the continuity ofP, E, , and . The parameters of isothermal NS cores corresponding to the cases = ₂ and = ₃, have been obtained. The maximum mass of these NS cores comes out to be 2.7 .     

Relativistic polytropic spheres in general relativity

Spherical-symmetric massive structures with a relativistic polytropic equation of state have been studied in detail. Thevarious static parameters have been calculated under all possible variations of ₀ (=P ₀/E ₀0 pertaining to the central value) and the polytropic indexn. In particular, the values of the static parameters have been identified under the extreme relativistic conditions and for the most bound structures. The structures have been used to model neutron stars and the various neutron parameters have been calculated. Radial pulsation and rotation of these configurations have also been discussed.