Abstract
This paper explored the physical acceptability conditions for anisotropic matter configurations in General Relativity. The study considered a generalized polytropic equation of state P=kappa {rho }^{gamma }+alpha rho -beta for a heuristic anisotropy. We integrated the corresponding Lane–Emden equation for several hundred models and found the parameter-space portion ensuring the physical acceptability of the configurations. Polytropes based on the total energy density are more viable than those with baryonic density, and small positive local anisotropies produce acceptable models. We also found that polytropic configurations where tangential pressures are greater than radial ones are also more acceptable. Finally, convective disturbances do not generate cracking instabilities. Several models emerging from our simulations could represent candidates of astrophysical compact objects.
Highlights
The polytropic equation of state (EoS) is one of the most common assumptions for modelling self-gravitating matter distributions in Newtonian and relativistic astrophysical scenarios
Page 3 of 22 176 two barotropic equations of state, P = P(ρ) and P⊥ = P⊥(P(ρ), ρ) ≡ P⊥(ρ). These two EoS involving the radial and tangential pressures, together with the matching conditions – i.e. initial conditions for the system of first-order differential equations, P(R) = Pb = 0 and m(R) = mb = M, lead to a system of differential equations for ρ(r ) which can be solved to obtain the inner structure of a self-gravitating relativistic compact object
To apply the master polytropic equation of state (9) in more realistic astrophysical scenarios, we integrated numerically the system of structure Eqs. (7) and (8) assuming the equation of state (9)
Summary
The polytropic equation of state (EoS) is one of the most common assumptions for modelling self-gravitating matter distributions in Newtonian and relativistic astrophysical scenarios. The initial approach of Bowers and Liang [11], followed by other schemes like: proportional to gravitation [12]; quasilocal [22]; covariant [23]; Karmarkar embedding class I [24,25]; gravitational decoupling [26,27]; double polytrope [28]; conditioning the complexity factor [29] and one of the most popular proposals: providing both, a particular barotropic equation of state P = P(ρ) and a density profile (or equivalently a metric function) [30–37] Various of these strategies may lead to viable astrophysical models [18, 38, 39]. In a recent work [37], we considered the latter of the above approaches, i.e. introducing local anisotropy providing a polytropic EoS and an ansatz on the energy density profile We found that this type of anisotropic matter distribution has a singular tangential sound velocity at the surface when the polytropic index is n > 1, and is commonly overlooked in the literature (see, for example, references [35,40–46]).
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