Dynamical analysis of radiating spherical collapse in Palatini f(R) gravity

Abstract

We discuss dynamical instability of non-adiabatic anisotropic collapse of spherical self-gravitating systems through collapse equation in Palatini f(R) gravity. We take R+ϵRⁿ model and assume hydrostatic equilibrium of celestial object at large past time such that T(−∞)=0. Considering perturbation from hydrostatic phase, and linearizing the dynamical as well as field equations, we evaluate instability constraints at Newtonian as well as post-Newtonian approximations. We conclude that pressure and heat flow assist collapsing phenomenon while Palatini f(R) dark source terms affect dynamics of self-gravitating object due to its non-attractive nature through stiffness parameter.

Appendices

Appendix A

The dark source terms D₀ and D₁ of Eqs. (25) and (26) are

$$\begin{aligned} &D_0=\frac{(-1)}{A^2} \biggl\{ \biggl( \frac{f}{R}-f_R \biggr) \frac{R}{2}- \frac{f''_R}{B^2} \\ &\phantom{D_0=\,}+\frac{\dot{f_R}}{A^2} \biggl(\frac{ \dot{B}}{B}+ \frac{9\dot{f_R}}{4f_R}+\frac{2\dot{C}}{C} \biggr) \\ &\phantom{D_0=\,}-\frac{f'_R}{B^2} \biggl( \frac{B'}{B} +\frac{f'_R}{4f_R}-\frac{2C'}{C} \biggr) \biggr\} _{,0} \\ &\phantom{D_0=\,}+\frac{\dot{f_R}}{f_RA} \biggl\{ \frac{3\ddot{f_R}}{ 2A^2}- \frac{R}{2} \biggl(f_R-\frac{f}{R} \biggr)+ \frac{3f''_R}{ 2B^2} \\ &\phantom{D_0=\,}-\frac{\dot{f_R}}{A^2} \biggl(\frac{3\dot{B}}{2B} +\frac{3\dot{A}}{2A}+\frac{5\dot{C}}{C} + \frac{6\dot{f_R}}{f_R} \biggr) \\ &\phantom{D_0=\,}-\frac{f'_R}{B^2} \biggl(\frac{3A'}{ 2A}+ \frac{3B'}{2B}-\frac{3C'}{C}+\frac{3f'_R}{2f_R} \biggr) \biggr\} \\ &\phantom{D_0=\,}+\frac{\dot{B}}{AB} \biggl\{ \frac{f''_R}{B^2} +\frac{\ddot{f_R}}{A^2}-\frac{\dot{f_R}}{A^2} \biggl( \frac{5\dot{f_R}}{2f_R}+\frac{\dot{A}}{A}+\frac{4\dot{C}}{C} +\frac{\dot{B}}{B} \biggr) \\ &\phantom{D_0=\,}-\frac{f'_R}{B^2} \biggl(\frac{A'}{A} +\frac{5f'_R}{2f_R}+ \frac{B'}{B} \biggr) \biggr\} \\ &\phantom{D_0=\,}+\frac{(-1)}{B^2A} \biggl(\dot{f'_R}- \frac{5}{2}\frac{\dot{ f_R}f'_R}{f_R}-\frac{A'}{A}\dot{f_R}- \frac{\dot{B}}{B}f'_R \biggr) \\ &\phantom{D_0=\,}\times\biggl( \frac{3A'}{A}+\frac{B'}{B}+\frac{3f'_R}{f_R}+\frac{2C'}{C} \biggr) \\ &\phantom{D_0=\,}+{A} \biggl[\frac{1}{A^2B^2} \biggl\{ \dot{f'_R} -\frac{A'}{A}\dot{f_R}- \frac{\dot{B}}{B}f'_R-\frac{5}{2} \frac{ \dot{f_R}f'_R}{f_R} \biggr\} \biggr]_{,1}, \end{aligned}$$

(A.1)

$$\begin{aligned} &D_1=B \biggl\{ \frac{-1}{(BA)^2} \biggl( \dot{f'_R}-\frac{5 \dot{f_R}f'_R}{2f_R}-\frac{A'}{A} \dot{f_R}-\frac{\dot{B}}{B} f'_R \biggr) \biggr\} _{,0} \\ &\phantom{D_1=\,}+\frac{1}{B} \biggl\{ \frac{\ddot{f_R}}{ A^2}- \frac{R}{2} \biggl(f_R-\frac{f}{R} \biggr)-\frac{\dot{f_R}}{ A^2} \biggl(\frac{\dot{A}}{A} \\ &\phantom{D_1=\,}+\frac{\dot{f_R}}{4f_R}+ \frac{2 \dot{C}}{C} \biggr) \\ &\phantom{D_1=\,}-\frac{f'_R}{B^2} \biggl(\frac{2C'}{C}+ \frac{ 9f'_R}{4f_R}+\frac{A'}{A} \biggr) \biggr\} _{,1} \\ &\phantom{D_1=\,}+\frac{A'}{BA} \biggl\{ \frac{f''_R}{B^2}+ \frac{\ddot{f_R}}{ A^2}-\frac{\dot{f_R}}{A^2} \biggl(\frac{5\dot{f_R}}{2f_R}+ \frac{ \dot{A}}{A}+\frac{4\dot{C}}{C}+\frac{\dot{B}}{B} \biggr) \\ &\phantom{D_1=\,}-\frac{f'_R}{ B^2} \biggl(\frac{5f'_R}{2f_R}+\frac{A'}{A} +\frac{B'}{B} \biggr) \biggr\} \\ &\phantom{D_1=\,}+\frac{f'_R}{f_RB} \biggl\{ \biggl(\frac{f}{R}-f_R \biggr) \frac{R}{2}+\frac{3\ddot{f_R}}{ 2A^2}+\frac{3f_R''}{2B^2} \\ &\phantom{D_1=\,}-\frac{3\dot{f_R}}{2A^2} \biggl(\frac{ \dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{f_R}}{f_R} +\frac{14\dot{C}}{3C} \biggr) \\ &\phantom{D_1=\,}-\frac{f'_R}{B^2} \biggl(\frac{3C'}{C}+\frac{3A'}{2A}+\frac{3B'}{2B}+ \frac{6f'_R}{ f_R} \biggr) \biggr\} \\ &\phantom{D_1=\,}+\frac{2C'}{CB} \biggl\{ \frac{f''_R}{B^2} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{3C}}{C}+\frac{\dot{B}}{B} \biggr) \\ &\phantom{D_1=\,}-\frac{f'_R}{B^2} \biggl(\frac{C'}{C}+\frac{5f'_R}{2f_R} + \frac{B'}{B} \biggr) \biggr\} \\ &\phantom{D_1=\,}+\frac{(-1)}{BA^2} \biggl(- \frac{A'}{ A}\dot{f_R}+\dot{f'_R} -\frac{5\dot{f_R}f'_R}{2f_R}-\frac{\dot{B}}{B}f'_R \biggr) \\ &\phantom{D_1=\,}\times\biggl(\frac{\dot{A}}{A}+\frac{3\dot{B}}{B}+\frac{3\dot{C}}{C}+ \frac{3\dot{f_R}}{f_R} \biggr). \end{aligned}$$

(A.2)

The portions of first and second dynamical equations (41) and (42) are

$$\begin{aligned} &D_2=\epsilon\beta \biggl[\frac{en(n-1)R_0^{n-1}}{2\beta}+ \frac{ 1}{B_0^2} \biggl(\omega_3-\frac{2b\omega_2}{B_0} \biggr) \\ &\phantom{D_2=\,}+\frac{1}{ B_0^2} \biggl(\omega_1-\frac{2bR'_0}{B_0} \biggr) \\ &\phantom{D_2=\,}\times\biggl\{ \frac{ B'_0}{B_0}+\frac{\epsilon\beta}{4} +\frac{{R_0'}}{1+{\epsilon}nR_0^{n-1}}-\frac{2}{r} \biggr\} \biggr] \\ &\phantom{D_2=\,}+\frac{b\epsilon\beta}{B_0^3} \biggl[\omega_3 -R'_0 \biggl(\frac{A_0'}{A_0}+\frac{5\epsilon\beta{R_0'}}{2(1+{ \epsilon}nR_0^{n-1})}+\frac{B_0'}{B_0} \biggr) \\ &\phantom{D_2=\,}+\frac{2c}{r} \frac{\epsilon\beta{R_0'}}{B_0^2} \biggl(\frac{1}{r} - \frac{A_0'}{A_0} \biggr) \biggr] \\ &\phantom{D_2=\,}+\frac{\epsilon^2\beta^2e}{ 1+{\epsilon}nR_0^{n-1}} \biggl[ \frac{3\omega_2}{2B_0^2}+\frac {(n-1)R_0^n}{2\beta} \\ &\phantom{D_2=\,}-\frac{R_0'}{B_0^2} \biggl\{ \frac{3A_0'}{ 2A_0}+\frac{3B_0'}{2B_0} -\frac{3}{r}+\frac{3\epsilon\beta{R_0'}}{2(1 +{\epsilon}nR_0^{n-1})} \biggr\} \biggr] \\ &\phantom{D_2=\,}+{\epsilon}A_0^2 \biggl[\frac{\beta}{A_0^2B_0^2} \biggl\{ \omega_1-\frac{bR_0'}{B_0} -\frac{eA_0'}{A_0} \\ &\phantom{D_2=\,}-\frac{5\epsilon\beta{e}R_0'}{2(1+{\epsilon} nR_0^{n-1})} \biggr\} \biggr]' \\ &\phantom{D_2=\,}-\frac{\epsilon\beta}{B_0^2} \biggl[\omega_1- \frac{bR_0'}{B_0} -\frac{eR_0'}{A_0}-\frac{5\epsilon\beta{e}R_0'}{2(1+{\epsilon} nR_0^{n-1})} \biggr] \\ &\phantom{D_2=\,}\times\biggl[ \frac{3A_0'}{A_0}+\frac{3\epsilon \beta{R_0'}}{1+{\epsilon}nR_0^{n-1}}+\frac{2}{r}+\frac{B_0'}{ B_0} \biggr], \end{aligned}$$

(A.3)

$$\begin{aligned} &D_3=\frac{-\epsilon}{A_0^2} \biggl[\omega_1- \frac{bR_0'}{ B_0}-\frac{eA_0'}{A_0}-\frac{5\epsilon\beta{e}R_0'}{2 (1+{\epsilon}nR_0^{n-1})} \biggr]\ddot{T} \\ &\phantom{D_3=\,}+T \biggl[-\epsilon \beta \biggl\{ \frac{en(n-1)R_0^{n-1}}{2\beta} \\ &\phantom{D_3=\,}+\frac{R_0'}{B_0^2} \biggl\{ \frac{9\epsilon \beta}{4(1+{\epsilon}nR_0^{n-1})} \biggl( \omega_1-\frac{ e\epsilon\beta{R_0'}}{1+{\epsilon}nR_0^{n-1}} \biggr) \\ &\phantom{D_3=\,}+ \biggl(\frac{a}{A_0} \biggr)'+ \biggl(\frac{c}{r} \biggr)' \biggr\} - \frac{1}{B_0^2} \biggl(\omega_1 -\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_3=\,}\times\biggl\{ \frac {A_0'}{A_0}+\frac{9\epsilon\beta{R_0'}}{4(1+{\epsilon}nR_ 0^{n-1})}+\frac{2}{r} \biggr\} \biggr\} \biggr]' \\ &\phantom{D_3=\,}+ \biggl(\frac{ e\beta}{A_0^2} \biggr)'\epsilon \ddot{T}+\frac{e\epsilon{ \beta}A_0'}{A_0^3}\ddot{T}+\frac{{\epsilon}\beta{T}{A'_0 }}{A_0B_0^2} \\ &\phantom{D_3=\,}\times \biggl[\omega_3-\frac{2b\omega_2}{B_0}- \biggl( \omega_1 -\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_3=\,}\times\biggl\{ \frac{B_0'}{B_0}+ \frac{5{ \epsilon}\beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+\frac{A_0'}{ A_0} \biggr\} \\ &\phantom{D_3=\,}-R_0'\biggl\{ \biggl(\frac{a}{A_0} \biggr)' +\frac{5\epsilon\beta}{2(1+{\epsilon}nR_0^{ n-1})} \\ &\phantom{D_3=\,}\times\biggl(\omega_1- \frac{e\epsilon\beta{R_0'}}{1+{ \epsilon}nR_0^{n-1}} \biggr)+ \biggl(\frac{b}{B_0} \biggr)' \biggr\} \biggr] \\ &\phantom{D_3=\,}+\frac{\epsilon\beta{T}}{B_0^2} \biggl( \frac{a}{A_0} \biggr)' \biggl[\omega_2-R_0' \\ &\phantom{D_3=\,}\times \biggl\{ \frac{B_0'}{B_0}+\frac{5{\epsilon} \beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+ \frac{A_0'}{A_0} \biggr\} \biggr] \\ &\phantom{D_3=\,}+\frac{2\epsilon\beta{T}}{rB_0^2} \biggl[ \omega_3-\frac{2b\omega_2}{B_0}- \biggl(\omega_1- \frac{2b R_0'}{B_0} \biggr) \\ &\phantom{D_3=\,}\times\biggl\{ \frac{B_0'}{B_0} +\frac{5{\epsilon}\beta{R_0'}}{2(1+{\epsilon} nR_0^{n-1})}+\frac{1}{r} \biggr\} \\ &\phantom{D_3=\,}-R_0' \biggl\{ \biggl(\frac{b}{ B_0} \biggr)'+\frac{5\epsilon\beta}{2(1+{\epsilon}nR_0^{n -1})} \\ &\phantom{D_3=\,}\times\biggl(\omega_1- \frac{e\epsilon\beta{R_0'}}{1+{\epsilon} nR_0^{n-1}} \biggr) + \biggl(\frac{c}{r} \biggr)' \biggr\} \biggr]+\frac{2\epsilon\beta{T}}{B_0^2} \\ &\phantom{D_3=\,}\times \biggl(\frac{c}{r} \biggr)' \biggl[\omega_2-R_0' \biggl\{ \frac{B_0'}{B_0}+\frac{5\epsilon \beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+\frac{1}{r} \biggr\} \biggr] \\ &\phantom{D_3=\,}+\frac{\epsilon^2\beta^2T}{1+{\epsilon}nR_0^{n-1}} \biggl(\omega_1-\frac{e\epsilon\beta{R_0'}}{1+{ \epsilon}nR_0^{n-1}} \biggr) \\ &\phantom{D_3=\,}\times\biggl[-\frac{(n-1)R_0^n}{ 2\beta}+\frac{3\omega_2}{2B_0^2} \\ &\phantom{D_3=\,}-\frac{R_0'}{B_0} \biggl\{ \frac{3A_0'}{2A_0}+\frac{6\epsilon\beta{R_0'}}{ 1+{\epsilon}nR_0^{n-1}} +\frac{3B_0'}{2B_0}+\frac{3}{r} \biggr\} \biggr] \\ &\phantom{D_3=\,}+\frac{\epsilon^2\beta^2R_0'T}{1+{\epsilon}n R_0^{n-1}} \biggl[-\frac{en(n-1)}{2}R_0^{n-1} \\ &\phantom{D_3=\,}+\frac{3}{2B_0^2} \biggl(\omega_3-\frac{2b\omega_2}{B_0} \biggr) -\frac{1}{B_0^2} \biggl(\omega_1-\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_3=\,}\times\biggl\{ \frac{3A_0'}{2A_0}+\frac{6\epsilon\beta{R_0'}}{ 1+{\epsilon}nR_0^{n-1}}+\frac{3B_0'}{2B_0}+ \frac{3}{r} \biggr\} \\ &\phantom{D_3=\,}-\frac{R'_0}{B_0} \biggl\{ \frac{3}{2} \biggl( \frac{a}{A_0} \biggr)'+\frac{3}{2} \biggl(\frac{b}{B_0} \biggr)'+3 \biggl(\frac{c}{r} \biggr)' \\ &\phantom{D_3=\,}+\frac{6\epsilon\beta}{1+{\epsilon} nR_0^{n-1}} \biggl(\omega_1-\frac{e\epsilon\beta{R_0'}}{ 1+{\epsilon}nR_0^{n-1}} \biggr) \biggr\} \biggr] \\ &\phantom{D_3=\,}+\frac{3\epsilon^2\beta^2eR'_0\ddot{T}}{2A_0^2(1+{\epsilon}nR_0^{n-1})}. \end{aligned}$$

(A.4)

The extra curvature term of the collapse equation (48) is

$$\begin{aligned} &D_4=\frac{-\omega^2\epsilon}{A_0^2} \biggl[\omega_1- \frac{bR_0'}{ B_0}-\frac{eA_0'}{A_0}-\frac{5\epsilon\beta{e}R_0'}{2(1+ {\epsilon}nR_0^{n-1})} \biggr] \\ &\phantom{D_4=\,}+ \biggl[- \epsilon\beta \biggl\{ \frac{en(n-1)R_0^{n-1}}{2\beta} \\ &\phantom{D_4=\,}+\frac{R_0'}{B_0^2} \biggl\{ \frac{9\epsilon\beta}{ 4(1+{\epsilon}nR_0^{n-1})} \biggl( \omega_1-\frac{e\epsilon\beta {R_0'}}{1+{\epsilon}nR_0^{n-1}} \biggr) \\ &\phantom{D_4=\,}+ \biggl(\frac{a}{A_0} \biggr)'+ \biggl(\frac{c}{r} \biggr)' \biggr\} - \frac{1}{B_0^2} \biggl(\omega_1 -\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_4=\,}\times\biggl\{ \frac{A_0'}{A_0}+\frac{9\epsilon\beta{R_0'}}{4(1+{\epsilon}nR_0^{n-1})} +\frac{2}{r} \biggr\} \biggr\} \biggr]' \\ &\phantom{D_4=\,}+ \biggl(\frac{e\beta}{A_0^2} \biggr)'\epsilon \omega^2+\frac{e\epsilon{\beta}A_0'}{A_0^3}\omega^2 +\frac{{\epsilon}\beta{T}{A'_0}}{A_0B_0^2} \\ &\phantom{D_4=\,}\times \biggl[\omega_3-\frac{2b\omega_2}{B_0}- \biggl( \omega_1 -\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_4=\,}\times\biggl\{ \frac{B_0'}{B_0}+ \frac{5{ \epsilon}\beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+\frac{A_0'}{ A_0} \biggr\} \\ &\phantom{D_4=\,}-R_0'\biggl\{ \biggl(\frac{a}{A_0} \biggr)' +\frac{5\epsilon\beta}{2(1+{\epsilon}nR_0^{n-1} )} \\ &\phantom{D_4=\,}\times\biggl(\omega_1- \frac{e\epsilon\beta{R_0'}}{1+{\epsilon} nR_0^{n-1}} \biggr)+ \biggl(\frac{b}{B_0} \biggr)' \biggr\} \biggr]+\frac{\epsilon\beta}{B_0^2} \\ &\phantom{D_4=\,}\times \biggl(\frac{a}{A_0} \biggr)' \biggl[\omega_2-R_0' \biggl\{ \frac{B_0'}{B_0}+\frac{5{\epsilon} \beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+ \frac{A_0'}{A_0} \biggr\} \biggr] \\ &\phantom{D_4=\,}+\frac{2\epsilon\beta}{rB_0^2} \biggl[ \omega_3-\frac{2b\omega_2}{B_0}- \biggl(\omega_1- \frac{ 2bR_0'}{B_0} \biggr) \\ &\phantom{D_4=\,}\times\biggl\{ \frac{B_0'}{B_0} +\frac{5{\epsilon}\beta{R_0'}}{2(1+{\epsilon} nR_0^{n-1})}+\frac{1}{r} \biggr\} \\ &\phantom{D_4=\,}-R_0' \biggl\{ \biggl(\frac{b}{ B_0} \biggr)'+\frac{5\epsilon\beta}{2(1+{\epsilon}nR_0^{n -1})} \\ &\phantom{D_4=\,}\times\biggl(\omega_1- \frac{e\epsilon\beta{R_0'}}{ 1+{\epsilon}nR_0^{n-1}} \biggr) + \biggl(\frac{c}{r} \biggr)' \biggr\} \biggr] +\frac{2\epsilon\beta}{B_0^2} \\ &\phantom{D_4=\,}\times \biggl(\frac{c}{r} \biggr)' \biggl[\omega_2-R_0' \biggl\{ \frac{B_0'}{B_0}+\frac{5\epsilon \beta{R_0'}}{2(1+{\epsilon}nR_0^{n-1})}+\frac{1}{r} \biggr\} \biggr] \\ &\phantom{D_4=\,}+\frac{\epsilon^2\beta^2}{1+{\epsilon} nR_0^{n-1}} \biggl(\omega_1-\frac{e\epsilon\beta{R_0'}}{1+ {\epsilon}nR_0^{n-1}} \biggr) \\ &\phantom{D_4=\,}\times\biggl[-\frac{(n-1)R_0^n}{ 2\beta}+\frac{3\omega_2}{2B_0^2} \\ &\phantom{D_4=\,}-\frac{R_0'}{B_0} \biggl\{ \frac{3A_0'}{2A_0}+\frac{6\epsilon\beta{R_0' }}{1+{\epsilon}nR_0^{n-1}} +\frac{3B_0'}{2B_0}+\frac{3}{r} \biggr\} \biggr] \\ &\phantom{D_4=\,}+\frac{\epsilon^2\beta^2R_0'}{1+{\epsilon}nR_0^{ n-1}} \biggl[-\frac{en(n-1)}{2}R_0^{n-1} \\ &\phantom{D_4=\,}+\frac{3}{2B_0^2} \biggl(\omega_3-\frac{2b\omega_2}{B_0} \biggr) - \frac{1}{B_0^2} \biggl(\omega_1-\frac{2bR_0'}{B_0} \biggr) \\ &\phantom{D_4=\,}\times\biggl\{ \frac{3A_0'}{2A_0}+\frac{6\epsilon\beta{R_0'}}{ 1+{\epsilon}nR_0^{n-1}}+\frac{3B_0'}{2B_0}+ \frac{3}{r} \biggr\} \\ &\phantom{D_4=\,}-\frac{R'_0}{B_0} \biggl\{ \frac{3}{2} \biggl( \frac{a}{A_0} \biggr)'+\frac{3}{2} \biggl(\frac{b}{B_0} \biggr)'+3 \biggl( \frac{c}{r} \biggr)' \\ &\phantom{D_4=\,}+\frac{6\epsilon\beta}{1+{\epsilon} nR_0^{n-1}} \biggl(\omega_1-\frac{e\epsilon\beta{R_0'}}{1 +{\epsilon}nR_0^{n-1}} \biggr) \biggr\} \biggr] \\ &\phantom{D_4=\,}+\frac{3\epsilon^2\beta^2eR'_0\omega^2}{2A_0^2(1+{\epsilon}nR_0^{n-1})}. \end{aligned}$$

(A.5)

Appendix B

The quantity D_1S appearing in Eq. (40) is given as follows

$$\begin{aligned} &D_{1S}=\frac{\epsilon}{B_0} \biggl\{ -\frac{(n-1)}{2} R_0^n \\ &\phantom{D_{1S}=\,}-\frac{{\beta}R'_0}{B_0^2} \biggl(\frac{A'_0}{A_0} + \frac{9\epsilon{\beta}R'_0}{4(1+{\epsilon}nR_0^{n-1})} +\frac{2}{r} \biggr) \biggr\} _{,1}+ \frac{A'_0}{A_0B^3_0} \\ &\phantom{D_{1S}=\,}\times\epsilon{\beta} \biggl[\omega_2-R'_0 \biggl(\frac{ A'_0}{A_0}+\frac{5\epsilon{\beta}R'_0}{2(1+{\epsilon} nR_0^{n-1})}+\frac{B'_0}{B_0} \biggr) \biggr] \\ &\phantom{D_{1S}=\,}+\frac{2{ \epsilon}{\beta}}{rB_0^3} \biggl[\omega_2-R'_0 \biggl( \frac{B'_0}{B_0} +\frac{1}{r}+\frac{5\epsilon{\beta}R'_0}{ 2(1+{\epsilon}nR_0^{n-1})} \biggr) \biggr] \\ &\phantom{D_{1S}=\,}+\frac{ \epsilon^2\beta^2R'_0}{B_0(1+{\epsilon}nR_0^{n-1})} \biggl[\frac{-(n-1)R_0^n}{2\beta}+\frac{3\omega_2}{ 2B_0^2} \\ &\phantom{D_{1S}=\,}-\frac{R'_0}{B_0^2} \biggl(\frac{3A'_0}{2A_0} + \frac{3B'_0}{2B_0}+\frac{6\epsilon{\beta}R'_0}{(1 +{\epsilon}nR_0^{n-1})}+\frac{3}{r} \biggr) \biggr]. \end{aligned}$$

(B.1)

The quantities ζ _j are

$$\begin{aligned} &\zeta_1=\tilde{\Delta}_1 \biggl(1- \frac{2m_0}{r}-\frac{ \tilde{R}r^2}{12} \biggr)^2+(\mu_0+P_{r0}) \\ &\phantom{\zeta_1=\,}\times\biggl\{ \frac{ \epsilon\tilde{\beta}\tilde{e}}{2(1+\epsilon{n}\tilde{R }_0^{n-1})}+b \biggl(1- \frac{2m_0}{r} -\frac{\tilde{R}r^2}{12} \biggr) \biggr\} \\ &\phantom{\zeta_1=\,}+( \mu_0+P_{\bot0}) \biggl(\frac{2c}{r}+ \frac{\epsilon\tilde{ \beta}\tilde{e}}{1+\epsilon{n}\tilde{R}_0^{n-1}} \biggr) \\ &\phantom{\zeta_1=\,}+\frac{\epsilon\tilde{\beta}\tilde{e}\mu_0}{1+\epsilon{n} \tilde{R}_0^{n-1}}+\tilde{\Delta}_1 \biggl(1-\frac{2m_0}{r}-\frac{\tilde{R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_1=\,}\times\biggl\{ \frac{4(12m_0-\tilde{R}r^3)}{r(12r-24m_0-r^3\tilde{R})} +\frac{2}{r} \biggr\} + \frac{D_5}{\kappa}, \end{aligned}$$

(B.2)

$$\begin{aligned} &\zeta_2=\tilde{\Delta}_1 \biggl(1- \frac{2m_0}{r}-\frac{\tilde{R} r^2}{12} \biggr) \\ &\phantom{\zeta_2=\,}+(\mu_0+P_{\bot0}) \biggl(\frac{2c}{r}+\frac{ \epsilon\tilde{\beta}\tilde{e}}{1+\epsilon{n}\tilde{R}_0^{n -1}} \biggr) \\ &\phantom{\zeta_2=\,}+\tilde{ \Delta}_1 \biggl(1-\frac{2m_0}{r}-\frac{\tilde{R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_2=\,} \biggl\{ \frac{4(12m_0-\tilde{R}r^3)}{r(12r-24m_0-r^3\tilde{R})} +\frac{2}{r} \biggr\} \\ &\phantom{\zeta_2=\,}+ \frac{D_5}{\kappa}+\frac{\epsilon \tilde{\beta}\tilde{e}\mu_0}{1+\epsilon{n}\tilde{R}_0^{n-1}}, \end{aligned}$$

(B.3)

$$\begin{aligned} &\zeta_3=\tilde{\Delta}_1 \biggl(1- \frac{2m_0}{r}-\frac{ \tilde{R}r^2}{12} \biggr)^2+( \mu_0+P_{r0}) \\ &\phantom{\zeta_3=\,}\times\biggl\{ \frac{ \epsilon\tilde{\beta}\tilde{e}}{2(1+\epsilon{n}\tilde{R }_0^{n-1})}+b \biggl(1- \frac{2m_0}{r} -\frac{\tilde{R}r^2}{12} \biggr) \biggr\} \\ &\phantom{\zeta_3=\,}+\tilde{ \Delta}_1 \biggl(1-\frac{2m_0}{r}-\frac{\tilde{ R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_3=\,}\times\biggl\{ \frac{4(12m_0-\tilde{R}r^3)}{ r(12r-24m_0-r^3\tilde{R})}+\frac{2}{r} \biggr\} \\ &\phantom{\zeta_3=\,}+\frac{\epsilon\tilde{\beta}\tilde{e}\mu_0}{1+\epsilon{n} \tilde{R}_0^{n-1}}+\frac{D_5}{\kappa}. \end{aligned}$$

(B.4)

The quantity D₅(r) in the above equations is

$$\begin{aligned} {D}_{5} =&\frac{\epsilon\tilde{e}n}{2}(n-1)\tilde{R}_0^{n-1} +\frac{{\epsilon}^2\tilde{\beta}\tilde{e}(n-1)\tilde{ R}_0^n}{2(1+{\epsilon}n\tilde{R}_0^{n-1})} \\ &{}+\epsilon \biggl(1 -\frac{4m_0}{r}-\frac{r^2\tilde{R}}{12} \biggr) \\ &{}\times\biggl[\frac{ 2\tilde{\beta}\tilde{e}}{r} \biggl(1 -\frac{4m_0r\tilde{R}}{3}-\frac{r^4\tilde{R}^2}{ 36} \biggr) \\ &{}\times\frac{(12m_0-r^3\tilde{R})}{(12r-24m_0-r^3\tilde{R})} \biggr]. \end{aligned}$$

(B.5)

The quantities ζ _jG are

$$\begin{aligned} &\zeta_{1G}=\Delta_{1G} \biggl(1- \frac{2m_0}{r}-\frac{\tilde{R} r^2}{12} \biggr)^2 \\ &\phantom{\zeta_{1G}=\,}+b( \mu_0+P_{r0}) \biggl(1-\frac{2m_0}{r}- \frac{ \tilde{R}r^2}{12} \biggr)+\frac{2c}{r} (\mu_0+P_{\bot0}) \\ &\phantom{\zeta_{1G}=\,}+\Delta_{1G} \biggl(1-\frac{2m_0}{r}- \frac{\tilde{R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_{1G}=\,}\times \biggl\{ \frac{4(12m_0-\tilde{R} r^3)}{r(12r-24m_0-r^3\tilde{R})}+\frac{2}{r} \biggr\} , \end{aligned}$$

(B.6)

$$\begin{aligned} &\zeta_{2G}=\Delta_{1G} \biggl(1- \frac{2m_0}{r}-\frac{\tilde{R} r^2}{12} \biggr)^2+\frac{2c}{r}( \mu_0+P_{\bot0}) \\ &\phantom{\zeta_{2G}=\,}+\Delta_{1G} \biggl(1- \frac{2m_0}{r}-\frac{\tilde{R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_{2G}=\,}\times \biggl\{ \frac{4(12m_0-\tilde{R}r^3)}{r(12r-24m_0-r^3 \tilde{R})}+\frac{2}{r} \biggr\} , \end{aligned}$$

(B.7)

$$\begin{aligned} &\zeta_{3G}=\Delta_{1G} \biggl(1- \frac{2m_0}{r}-\frac{\tilde{R} r^2}{12} \biggr)^2 \\ &\phantom{\zeta_{3G}=\,}+b( \mu_0+P_{r0}) \biggl(1-\frac{2m_0}{r}- \frac{ \tilde{R}r^2}{12} \biggr) \\ &\phantom{\zeta_{3G}=\,}+\Delta_{1G} \biggl(1-\frac{2m_0}{r}-\frac{\tilde{R}r^2}{12} \biggr)^2 \\ &\phantom{\zeta_{3G}=\,}\times\biggl\{ \frac{4(12m_0-\tilde{R}r^3)}{r(12r-24m_0-r^3\tilde{R})} +\frac{2}{r} \biggr\} . \end{aligned}$$

(B.8)

The coefficients of the differential equation (43) are given by

$$\begin{aligned} &\psi_1=\frac{2c}{rA_0^2}, \qquad \psi_2=\frac{\kappa\tilde{\Delta}}{1+{\epsilon}n\tilde{R }_0^{n-1}}, \end{aligned}$$

(B.9)

$$\begin{aligned} &\psi_3=- \frac{A'_0}{A_0} \biggl(\frac{c}{r} \biggr)'- \frac{1}{r} \biggl(\frac{a}{A_0} \biggr)'+ \frac{2}{rB_0^2} \biggl\{ \biggl( \frac{c}{r} \biggr)'- \frac{b}{B_0} \biggr\} \\ &\phantom{\psi_3=\,}-\frac{e{\epsilon}n (n-1)\tilde{R}_0^{n-2}}{(1+{\epsilon}n\tilde{R}_0^{n-1})^2} \biggl\{ \frac{2A'_0}{A_0B_0^2r}- \frac{1}{r^2}+\frac{1}{B_0^2 r^2} \biggr\} \\ &\phantom{\psi_3=\,}-\frac{2c}{r^3}+ \frac{2bA'_0}{rA_0B_0}. \end{aligned}$$

(B.10)