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New models of charged anisotropic polytropes with radiation density

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Abstract

In this manuscript, new classes of polytropic models have been developed by using polytropic equation of state (PEoS) for spherically symmetric gravitating sources in isotropic coordinates. The inner fluid configuration is charged anisotropic and models are developed for different values of polytropic index \(n=1,~\frac{1}{2},~2,~\frac{2}{3}\). Mass and radii of eight stars 4U 1820-30, Cen X-3, EXO 1785-248, SMC X-4, LMC X-4, SAX J1808.4-3658, 4U 1538-52 and Her X-1 have been regained with the help of developed models. The stability of models is discussed by using speed of sound technique and graphical analysis of model parameters. It is concluded that all models are well behaved and physically acceptable.

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Correspondence to I. Noureen.

Appendix

Appendix

A.1 Polytropic index \(n=\frac{1}{2}\)

Equation (28) can be integrated with \(n=\frac{1}{2}\) to give

$$\begin{aligned} I(r)=K(a-br^2)^G(a+br^2)^H \mathrm{e}^{M(r)}, \end{aligned}$$
(A49)

where K is constant of integration and

$$\begin{aligned} G= & {} \frac{-1}{2}-18c^2\pi ^2\alpha -\frac{a^2(3b^2-4\mathrm{d}\pi ^2)^3\alpha }{8b^4\pi ^2} +\frac{3a(3b^2-4\mathrm{d}\pi ^2)^2\alpha (2c\pi +\epsilon )}{4b^3\pi }\nonumber \\&+\,\frac{2\mathrm{d}\pi ^2(1+12c^2\pi ^2\alpha +12c\pi \alpha \epsilon +3\alpha \epsilon ^2)}{b^2} -18c\pi \alpha \epsilon -\frac{9\alpha \epsilon ^2}{2}\nonumber \\&+\,\frac{\pi (8c^3\pi ^3\alpha +12c^2\pi ^2\alpha \epsilon +\alpha \epsilon (-1+\epsilon ^2)+2c(\pi +3\pi \alpha \epsilon ^2))}{a b}, \end{aligned}$$
(A50)
$$\begin{aligned} H= & {} \frac{1}{8b^4}\frac{a^2(3b^2+4\mathrm{d}\pi ^2)^3\alpha }{\pi ^2} -\frac{6ab\alpha (3b^2+4\mathrm{d}\pi ^2)^2(2c\pi +\epsilon )}{\pi }\nonumber \\&+\,4b^2(3b^2+4\mathrm{d}\pi ^2)\left( 1+12c^2\pi ^2\alpha +12c\pi \alpha \epsilon +3a\epsilon ^2\right) \nonumber \\&-\,\frac{8b^3\pi \left( 8c^3\pi ^3\alpha +12c^2\pi ^2\alpha \epsilon +\alpha \epsilon (-1+\epsilon ^2)+2c\left( \pi +3\pi \alpha \epsilon ^2\right) \right) }{a}, \end{aligned}$$
(A51)
$$\begin{aligned} M= & {} \frac{4d^2\pi ^2 r^2\alpha \left( -9ab+2\pi (6c\pi +\mathrm{d}\pi r^2+3\epsilon )\right) }{b^2}. \end{aligned}$$
(A52)

The degree of anisotropy becomes

$$\begin{aligned} \Delta= & {} 4b(-2a+br^2)-16\pi ^2(c+\mathrm{d}r^2) +\frac{\alpha (-3ab+2\pi (2c\pi +2\mathrm{d}\pi r^2+\epsilon ))^3}{\pi ^2}\nonumber \\&+\,4\left( -b(a(G-H)+b(G+H)r^2)+(a^2-b^2r^4)M^\prime \right) \nonumber \\&+\,\frac{1}{(a-br^2)^2}4r^2(b^2(a^2(G^2+(-1+H)H-G(1+2H))\nonumber \\&+\,2ab(-G+G^2+H-H^2)r^2\nonumber \\&+\,b^2(G^2+(-1+H)H+G(-1+2H))r^4)\nonumber \\&-\,2b(a(G-H)+b(G+H)r^2)(a^2-b^2r^4)M^\prime \nonumber \\&+\,(a^2-b^2r^4)^2(M^\prime )^2+(a^2-b^2r^4)M^{\prime \prime }). \end{aligned}$$
(A53)

The line element Eq. (1) has the form

$$\begin{aligned} \mathrm{d}s^2= & {} -K(a-br^2)^{2G}(a+br^2)^{2(H-1)}\mathrm{e}^{2M(r)}\mathrm{d}t^2\nonumber \\&+(a+br^2)^{-2}\left[ \mathrm{d}r^2+r^2(\mathrm{d}\theta ^2+sin^2\theta \mathrm{d}\phi ^2)\right] . \end{aligned}$$
(A54)

A.2 Polytropic Index \(n=2\)

Equation (28) can be integrated with \(n=2\) to give

$$\begin{aligned} I(r)=K(a-br^2)^G(a+br^2)^H\mathrm{e}^M [S(r)]^Q[V(r)]^W, \end{aligned}$$
(A55)

where K is the constant of integration and

$$\begin{aligned} G= & {} \frac{-1}{2} +\frac{2\mathrm{d}\pi ^2}{b^2}+\frac{\pi (2c\pi -\alpha \epsilon )}{ab}, \end{aligned}$$
(A56)
$$\begin{aligned} H= & {} \frac{3}{2}+ \frac{2\mathrm{d}\pi ^2}{b^2}+\frac{\pi (-2c\pi +\alpha \epsilon )}{ab}, \end{aligned}$$
(A57)
$$\begin{aligned} Q= & {} \frac{-\alpha \left( a(3b^2+4\mathrm{d}\pi ^2) -2b\pi (2c\pi +\epsilon )\right) ^{\frac{3}{2}}}{2ab^{\frac{5}{2}}\sqrt{2\pi }}, \end{aligned}$$
(A58)
$$\begin{aligned} W= & {} \frac{-\alpha \left( a(-3b^2+4\mathrm{d}\pi ^2) +2b\pi (2c\pi +\epsilon )\right) ^{\frac{3}{2}}}{2ab^{\frac{5}{2}}\sqrt{2\pi }}, \end{aligned}$$
(A59)
$$\begin{aligned} f(r)= & {} 3ab-2\pi (2c\pi +2\mathrm{d}\pi r^2+\epsilon ) \end{aligned}$$
(A60)
$$\begin{aligned} S= & {} \frac{\sqrt{a(3b^2+4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon )} +\sqrt{b f(r)}}{\sqrt{a(3b^2+4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon )}-\sqrt{b f(r)}}, \end{aligned}$$
(A61)
$$\begin{aligned} V= & {} \frac{\sqrt{a(-3b^2+4\mathrm{d}\pi ^2)+2b\pi (2c\pi +\epsilon )} +\sqrt{b f(r)}}{\sqrt{a(-3b^2+4\mathrm{d}\pi ^2)+2b\pi (2c\pi +\epsilon )}-\sqrt{b f(r)}}, \end{aligned}$$
(A62)
$$\begin{aligned} M= & {} \frac{4\mathrm{d}\pi ^{\frac{3}{2}}\alpha \sqrt{2f(r)}}{b^2}. \end{aligned}$$
(A63)

The measure of anisotropy is

$$\begin{aligned} \Delta= & {} 4b(-2a+br^2)-16\pi ^2(c+\mathrm{d}r^2)\nonumber \\&-\,\frac{\alpha (12ab-8\pi (2c\pi +2\mathrm{d}\pi r^2 +\epsilon ))^{\frac{3}{2}}}{2\sqrt{2\pi }}\nonumber \\&+\, \frac{4((a^2-b^2r^4)QVS^\prime +S(W(a^2-b^2r^4)V^\prime }{SV}\nonumber \\&+\,\frac{(-b(a(G-H)+b(G+H)r^2)+(a^2-b^2r^4)M^\prime )))}{S}\nonumber \\&+\,\frac{1}{(a-br^2)^2S^2V^2}\Big (4r^2((a^2-b^2r^4)^2(-1+Q)QV^2 (S^\prime )^2\nonumber \\&+\,(a^2-b^2r^4)Q S V(2W(a^2-b^2r^4)S^\prime V^\prime \nonumber \\&+\,V(2S^\prime (-b(a(G-H)+b(G+H)r^2)\nonumber \\&+\,(a^2-b^2r^4)M^\prime )+(a^2-b^2r^4)S^{\prime \prime }))\nonumber \\&+\,S^2((-1+W)W(a^2-b^2r^4)^2 (V^\prime )^2\nonumber \\&+\,W(a^2-b^2r^4)V(2V^\prime (-b(a(G-H)\nonumber \\&+\,b(G+H)r^2)+(a^2-b^2r^4)M^\prime )\nonumber \\&+\,(a^2-b^2r^4)V^{\prime \prime })\nonumber \\&+\,V^2(b^2(a^2(G^2+(-1+H)H-G(1+2H))\nonumber \\&+\,2ab(-G+G^2+H-H^2)r^2\nonumber \\&+\,b^2(G^2+(-1+H)H+G(-1+2H))r^4)\nonumber \\&-\,2b(a(G-H)+b(G+H)(a^2-b^2r^4)M^\prime \nonumber \\&+\,(a^2-b^2r^4)^2(M^\prime )^2+(a^2-b^2r^4)^2M^{\prime \prime }))\Big ). \end{aligned}$$
(A64)

Here the constants and function of r are given in above equations. The barotropic equation of state has the form \(P_r=\alpha \rho ^{\frac{3}{2}}\) gives the line element Eq. (1) as

$$\begin{aligned} \mathrm{d}s^2= & {} -K(a-br^2)^{2G}(a+br^2)^{2(H-1)} \times [S(r)]^{2Q}[V(r)]^{2W}\mathrm{e}^{2M(r)}\mathrm{d}t^2\nonumber \\&+(a+br^2)^{-2}[\mathrm{d}r^2+r^2(\mathrm{d}\theta ^2+sin^2\theta \mathrm{d}\phi ^2)]. \end{aligned}$$
(A65)

A.3 Polytropic index \(n=\frac{2}{3}\)

Equation (28) can be integrated with \(n=\frac{2}{3}\) to give

$$\begin{aligned} I(r)=K(a-br^2)^G(a+br^2)^H\mathrm{e}^M [S(r)]^Q[V(r)]^W, \end{aligned}$$
(A66)

where K is the constant of integration and

$$\begin{aligned} G= & {} \frac{-1}{2} +\frac{2\mathrm{d}\pi ^2}{b^2}+\frac{\pi (2c\pi -\alpha \epsilon )}{ab}, \end{aligned}$$
(A67)
$$\begin{aligned} H= & {} \frac{3}{2}+ \frac{2\mathrm{d}\pi ^2}{b^2}+\frac{\pi (-2c\pi +\alpha \epsilon )}{ab}, \end{aligned}$$
(A68)
$$\begin{aligned} Q= & {} \frac{-\alpha (a(3b^2+4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon ))^{\frac{5}{2}}}{4\sqrt{2}ab^{\frac{7}{2}}\pi ^{\frac{3}{2}}}, \end{aligned}$$
(A69)
$$\begin{aligned} W= & {} \frac{-\alpha (a(3b^2-4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon ))^3}{4ab^{\frac{7}{2}} \pi ^{\frac{3}{2}}\sqrt{a(-6b^2+8\mathrm{d}\pi 2) +4b\pi (2c\pi +\epsilon )}}, \end{aligned}$$
(A70)
$$\begin{aligned} f(r)= & {} 3ab-2\pi (2c\pi +2\mathrm{d}\pi r^2+\epsilon ) \end{aligned}$$
(A71)
$$\begin{aligned} S= & {} \frac{\sqrt{a(3b^2+4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon )} +\sqrt{b f(r)}}{\sqrt{a(3b^2+4\mathrm{d}\pi ^2)-2b\pi (2c\pi +\epsilon )}-\sqrt{b f(r)}}, \end{aligned}$$
(A72)
$$\begin{aligned} V= & {} \frac{\sqrt{a(-3b^2+4\mathrm{d}\pi ^2)+2b\pi (2c\pi +\epsilon )} +\sqrt{b f(r)}}{\sqrt{a(-3b^2+4\mathrm{d}\pi ^2)+2b\pi (2c\pi +\epsilon )}-\sqrt{b f(r)}}, \end{aligned}$$
(A73)
$$\begin{aligned} M= & {} \frac{-2d\alpha \sqrt{2\pi f(r)} (-21ab+2\pi (14c\pi +2\mathrm{d}\pi r^2+7\epsilon ))}{3b^2}. \end{aligned}$$
(A74)

The measure of anisotropy is

$$\begin{aligned} \Delta= & {} 4b(-2a+br^2)-16\pi ^2(c+\mathrm{d}r^2)-\frac{\sqrt{2} \alpha (3ab-2\pi (2c\pi +2\mathrm{d}\pi r^2+\epsilon ))^{\frac{5}{2}}}{\pi ^{\frac{3}{2}}}\nonumber \\&+\, \frac{4((a^2-b^2r^4)QVS^\prime +S(W(a^2-b^2r^4)V^\prime }{SV}\nonumber \\&+\,\frac{(-b(a(G-H)+b(G+H)r^2)+(a^2-b^2r^4)M^\prime )))}{S}\nonumber \\&+\,\frac{1}{(a-br^2)^2S^2V^2}\Big (4r^2((a^2-b^2r^4)^2(-1+Q)QV^2 (S^\prime )^2\nonumber \\&+\,(a^2-b^2r^4)Q SV( 2W(a^2-b^2r^4)S^\prime V^\prime \nonumber \\&+\,V(2S^\prime (-b(a(G-H)+b(G+H)r^2)\nonumber \\&+\,(a^2-b^2r^4)M^\prime )+(a^2-b^2r^4)S^{\prime \prime })) +S^2((-1+W)W(a^2-b^2r^4)^2 (V^\prime )^2\nonumber \\&+\,W(a^2-b^2r^4)V(2V^\prime (-b(a(G-H)+b(G+H)r^2)+(a^2-b^2r^4)M^\prime )\nonumber \\&+\,(a^2-b^2r^4)V^{\prime \prime })+V^2(b^2(a^2(G^2+(-1+H)H-G(1+2H))\nonumber \\&+\,2ab(-G+G^2+H-H^2)r^2+b^2(G^2+(-1+H)H+G(-1+2H))r^4)\nonumber \\&-\,2b(a(G-H)+b(G+H)(a^2-b^2r^4)M^\prime \nonumber \\&+\,(a^2-b^2r^4)^2(M^\prime )^2+(a^2-b^2r^4)^2M^{\prime \prime }))\Big ). \end{aligned}$$
(A75)

Here the constants and function of r are given in above equations. The barotropic equation of state has the form \(P_r=\alpha \rho ^{\frac{5}{2}}\) gives the line element Eq. (1) as

$$\begin{aligned} \mathrm{d}s^2= & {} -K(a-br^2)^{2G}(a+br^2)^{2(H-1)} \times [S(r)]^{2Q}[V(r)]^{2W}e^{2M(r)}\mathrm{d}t^2\nonumber \\&+(a+br^2)^{-2}[\mathrm{d}r^2+r^2(\mathrm{d}\theta ^2+\sin ^2\theta \mathrm{d}\phi ^2)]. \end{aligned}$$
(A76)

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Mardan, S.A., Siddiqui, A.A., Noureen, I. et al. New models of charged anisotropic polytropes with radiation density. Eur. Phys. J. Plus 135, 3 (2020). https://doi.org/10.1140/epjp/s13360-019-00077-0

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