Anisotropic strange stars through embedding technique in massive Brans–Dicke gravity

Abstract

This paper investigates the existence and properties of anisotropic strange quark stars in the context of massive Brans–Dicke theory. The field equations are constructed in Jordan frame by assuming a suitable potential function with MIT bag model. We employ the embedding class-one approach as well as junction conditions to determine the unknown metric functions. Radius of the strange star candidate, LMC X-4, is predicted through its observed mass for different values of the bag constant. We analyze the effects of coupling parameter as well as mass of scalar field on state determinants and execute multiple checks on the stability and viability of the spherical system. It is concluded that the resulting stellar structure is physically viable and stable as it satisfies the energy conditions as well as essential stability criteria.

Appendix A

The components of $T^{\gamma \Phi }_{\delta }$ are obtained as

$$\begin{aligned} T_0^{0\Phi }= & {} e^{-\lambda }\left[ \Phi ''+\left( \frac{2}{r}-\frac{\lambda '}{2} \right) \Phi '+\frac{\omega _\mathrm{BD}}{2\Phi }\Phi '^2-e^\lambda \frac{V(\Phi )}{2}\right] ,\end{aligned}$$

(A1)

$$\begin{aligned} T_1^{1\Phi }= & {} e^{-\lambda }\left[ \left( \frac{2}{r}+\frac{\nu '}{2}\right) \Phi '-\frac{\omega _\mathrm{BD}}{2\Phi }\Phi '^2-e^\lambda \frac{V(\Phi )}{2})\right] ,\end{aligned}$$

(A2)

$$\begin{aligned} T_2^{2\Phi }= & {} e^{-\lambda }\left[ \Phi ''+\left( \frac{1}{r}-\frac{\lambda '}{2}+\frac{\nu '}{2}\right) \Phi '+\frac{\omega _\mathrm{BD}}{2\Phi }\Phi '^2 -e^\lambda \frac{V(\Phi )}{2} \right] . \end{aligned}$$

(A3)

Energy density and pressure components take the following form

$$\begin{aligned} \rho= & {} \frac{1}{2r^2}\left\{ \xi ^2\Phi ^2\left[ 2R^2(2M-R)\left( M\left( r^2-2R^2\right) +R^3\right) \left( r\Phi '(r)+2\Phi (r)\right) \right] \right. \nonumber \\&-\left. \xi \left[ r^2R^2\omega _\mathrm{BD}(R-2M)\Phi '^2(r) +2r\Phi (r)\left( \left( R^3-M\left( r^2+2R^2\right) \right) \Phi '(r)\right. \right. \right. \nonumber \\&+\left. \left. \left. rR^2 (R-2M)\Phi ''(r)\right) -2\Phi ^2(r)\left( 2M(r-R)(r+R)+R^3\right) \right] \right. \nonumber \\&+\,r^2 V(\Phi )+\left. 2\Phi (r)\right\} , \end{aligned}$$

(A4)

$$\begin{aligned} p_r= & {} \frac{\xi }{4r^2}r^2R^2\omega _\mathrm{BD}(2M-R)\Phi '^2(r)+2\Phi ^2(r)\left( 2M(r-R)(r+R)+R^3\right) \nonumber \\&+r\Phi (r)\left( \left( 2M(r-R)(r+R)+R^3\right) \Phi '(r)+rR^2(2 M-R)\Phi ''(r)\right) \nonumber \\&+\frac{\xi ^2\Phi ^2}{4r}(R^2(2M-R)\left( M\left( r^2-2 R^2\right) +R^3\right) \left( r\Phi '(r)+2\Phi (r)\right) )-\mathcal {B}, \end{aligned}$$

(A5)

$$\begin{aligned} p_\perp= & {} \frac{\Phi \xi }{4r^2}\left[ \left( M\left( r^2-2R^2\right) +R^3\right) \left( 2\Phi (r)\left( 2M(r-R)(r+R)+R^3\right) \right. \right. \nonumber \\&+\left. \left. 3rR^2(R-2M)\Phi '(r)\right) \right] +\xi \left[ 3 r^2R^2\omega _\mathrm{BD}(R-2M)\Phi '^2(r)+r\Phi (r)\right. \nonumber \\&\times \left. \left( \left( -2Mr^2+14M R^2-7R^3\right) \Phi '(r)+3rR^2(R-2M)\Phi ''(r)\right) -2\Phi ^2(r)\right. \nonumber \\&\times \left. \left( 2M\left( r^2-3 R^2\right) +3R^3\right) \right] -\mathcal {B}+\frac{\Phi (r)}{r^2}, \end{aligned}$$

(A6)

where $\xi =\frac{\Phi ^{-1}(r)}{\left( 2Mr^2e^{\frac{M(r-R)(r+R)}{R^2(R-2 M)}}+R^2(R-2M)\right) }$.