Effect of pressure anisotropy on Buchdahl-type relativistic compact stars

Abstract

We consider exact models for dense relativistic stars with anisotropic pressures and containing Buchdahl-type spacetime geometry. The Buchdahl condition can be transformed to an Euler–Cauchy equation for the gravitational potentials. We solve this condition to find a new exact solution to the Einstein field equations with anisotropic matter distribution. We show that the exact solution produces a realistic model of a compact relativistic star satisfying all physical requirements. The regularity, equilibrium, casuality, stability, energy conditions and compactness limits for a well behaved compact sphere are satisfied.

Appendix

In the Appendix we outline the method for finding the solution of field equations and anisotropy factor. Use the transformation in Eq. (11) to get

$$\begin{aligned} X=\sqrt{\frac{K}{(K-1)}}\sqrt{1+\frac{Cr^{2}}{K}} \,\,\,\, and \,\,\,\, p_t(r)-p_r(r)=\varDelta , \end{aligned}$$

$$\begin{aligned}&\frac{d^2Y}{dX^2}+\frac{X}{(1-X^2)}\,\frac{dY}{dX} \nonumber \\&\quad +\frac{1}{(1-X^2)}\,\Bigg (1-K+\kappa \, \varDelta \, K \frac{[(1-K)\,(1-X)]^2}{C\,[\,X^2\,(K-1)-K\,]} \Bigg )\,Y=0. \end{aligned}$$

(44)

We compare Eq. (44) with the standard form

$$\begin{aligned} \frac{d^2Y}{dX^2}+P(X)\,\frac{dY}{dX}+Q(X)\,Y= R(X), \end{aligned}$$

(45)

where

$$\begin{aligned} P(X)= & {} \frac{X}{(1-X^2)}, \\ Q(X)= & {} \frac{1}{(1-X^2)}\,\Bigg (1-K+\kappa \, \varDelta \, K \frac{[(1-K)\,(1-X)]^2}{C\,[\,X^2\,(K-1)-K\,]} \Bigg ), \\ R(X)= & {} 0. \end{aligned}$$

To solve (45) we choose the integrating factor $u=e^{-\frac{1}{2}\,\int {P(X)}dX}$. Then (45) leads to the following complete solution

$$\begin{aligned} Y(X) = \Bigg \{ \begin{array}{rcl} (1-X^2)^{1/4}\,\,Z(X)~~~ \text{ for } &{} X \le 1, ~~~K\ge 0 \\ (1-X^2)^{1/4}\,\,Z(X)~~~\text{ for } &{} X\le 1, ~~~K\ge 0 \end{array} \end{aligned}$$

(46)

where Z(X) can be determined by the following differential equation

$$\begin{aligned}&\frac{d^2Z}{dX^2}+ \Bigg [ \frac{4\,(1-X^2)\,(1-K)-(3X^2+2)}{4\,(1-X^2)^2} \nonumber \\&\quad + \frac{\kappa \, \varDelta \, K\,(1-K)^2\,(1-X)}{C\,[\,X^2\,(K-1)-K\,]} \Bigg ]Z=0. \end{aligned}$$

(47)

In order to solve Eq. (47) we choose the expression for anisotropy factor $\varDelta$ as

$$\begin{aligned} \varDelta =\frac{\beta \,C\,[\,4\,(1-X^2)\,(1-K)-(3X^2+2)\,]\,(X^2\,K-X^2-K)}{4\,\kappa \,K\,(1-X^2)^3\,(K-1)^2}. \end{aligned}$$

(48)

We can study the behavior of the model for small r close to the centre of the star. The analytical expressions of the anisotropic factor $\varDelta$ using a series expansion leads to

$$\begin{aligned} \varDelta= & {} \frac{\beta \,C^2\,r^2}{4\,\kappa \,(K-1)^5}\,\big [(-2+9 K- 15 K^2- 10\, K^3) \nonumber \\&-(1- 12 K + 105 K^2- 40 K^3) Cr^2 \nonumber \\&+(9- 123\,K+ 60 K^2)\,C^2r^4 - (42- 24 K) C^3r^6+O(r^8) \big ]. \end{aligned}$$

(49)

The expressions of physical parameters $p_r$, $p_r$, $\rho$, $V^2_r$ and $V^2_t$ at the centre become

$$\begin{aligned} (\,p_{r}\,)_{r=0}= & {} (\,p_{t}\,)_{r=0}= \frac{1}{\kappa }\,\left[ \frac{92\,C(501F+13001)}{125(141F+1479)}-\frac{3\,C}{7}\right] , \end{aligned}$$

(50)

$$\begin{aligned} (\,\rho \,)_{r=0}= & {} \frac{9\,C}{7\,\kappa }, \end{aligned}$$

(51)

$$\begin{aligned} (\,V^{2}_{r}\,)_{r=0}= & {} \frac{-7}{30\,C}\Bigg [ \frac{1}{(0.383F+4.017)^2}\Big ((0.383F+4.017)\bigg ( -\frac{12C}{7}(1.728F+10.378)\nonumber \\&+2C(2.866F+10.664) \bigg )-2\,C\,F_1 \Big )-\frac{6C}{7}\Bigg ], \end{aligned}$$

(52)

$$\begin{aligned} (\,V^{2}_{t}\,)_{r=0}= & {} \frac{-7}{30\,C}\Bigg [ \frac{1}{(0.383F+4.017)^2}\Big ((0.383F+4.017)\bigg ( -\frac{12C}{7}(1.728F+10.378)\nonumber \\&+2C(2.866F+10.664) \bigg )-2C\,F_1 \Big )+\frac{6C}{7}+\frac{15\beta \,C}{11}\Bigg ], \end{aligned}$$

(53)

where $F=\frac{B}{A}$ and $F_1=(1.728F+10.378)(0.951F+2.081)$.