Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state

Abstract

Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index $n$. The explicit form of the solution is presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and $n=1/5$, respectively. The case of $n=3$ is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven terms only. The power series representations of the geometric and physical properties of the linear barotropic and polytropic stars are also obtained.

Appendix: The first seven coefficients of the power series solution of the relativistic mass equation for arbitrary polytropic index \(n\)

The first seven coefficients $c_{2l+1}$, $l=1,2,\ldots,7$ describing the solution of the relativistic mass equation for a general relativistic polytropic star with arbitrary polytropic index $n\in \mathbf{R}$ are given by

$$\begin{aligned}& c_{3}=\frac{1}{3}, \end{aligned}$$

(135)

$$\begin{aligned}& c_{5}=-\frac{a(k+1)(3k+1)n}{30k(n+1)}, \end{aligned}$$

(136)

$$\begin{aligned}& c_{7}=\frac{a^{2}(k+1)(3k+1)n}{2520k^{2}(1+n)^{2}} \bigl[ k^{2}(30n+15)+k(18n-20)+8n-5 \bigr] , \end{aligned}$$

(137)

$$\begin{aligned}& c_{9} =-\frac{a^{3}(1+k)(1+3k)n}{408240k^{3}(1+n)^{3}} \bigl[315k^{4} \bigl( 6n^{2}+7n+2 \bigr) +6k^{3} \bigl( 288n^{2}-241n-140 \bigr) \\& \phantom{c_{9} =\,\,}{}+ 2k^{2} \bigl( 618n^{2}-809n+560 \bigr) +10k \bigl( 40n^{2}-123n+56 \bigr) +122n^{2}-183n+70 \bigr], \end{aligned}$$

(138)

$$\begin{aligned}& c_{11} =\frac{a^{4}(k+1)(3k+1)n}{179625600k^{4}(n+1)^{4}} \bigl[14175k^{6} \bigl( 24n^{3}+46n^{2}+29n+6 \bigr) +90k^{5} \bigl(4074n^{3}-1727n^{2} \\& \phantom{c_{11} =\,\,}{}- 4402n-1260 \bigr)+k^{4} \bigl( 323568n^{3}-412518n^{2}+330915n+160650 \bigr) +4k^{3} \\& \phantom{c_{11} =\,\,}{}\times \bigl( 38832n^{3}-139547n^{2}+106520n-50400 \bigr) +k^{2} \bigl( 71744n^{3}-256154n^{2}+418725n-154350 \bigr) \\& \phantom{c_{11} =\,\,}{}+ 18k \bigl( 942n^{3}-5847n^{2}+6490n-2100 \bigr) +5032n^{3}-12642n^{2}+10805n-3150 \bigr], \end{aligned}$$

(139)

$$\begin{aligned}& c_{13} =-\frac{a^{5}(k+1)(3k+1)n}{70053984000k^{5}(n+1)^{5}} \bigl[467775k^{8} \bigl( 120n^{4}+326n^{3}+329n^{2}+146n+24 \bigr) +1350k^{7} \\& \phantom{c_{11} =\,\,}{}\times \bigl( 47748n^{4}+2516n^{3}-77423n^{2}-55548n-11088 \bigr) +180k^{6} \bigl(354618n^{4}-343910n^{3} \\& \phantom{c_{11} =\,\,}{}+ 333441n^{2}+518150n+124740 \bigr)+6k^{5} \bigl(5942244n^{4}-22880944n^{3}+17820615n^{2} \\& \phantom{c_{11} =\,\,}{}- 11274200n-4851000 \bigr)+2k^{4} \bigl(9945804n^{4}-43963854n^{3}+95782105n^{2}-52438750n \\& \phantom{c_{11} =\,\,}{}+ 17740800 \bigr)+2k^{3} \bigl( 3358788n^{4}-27242948n^{3}+54777285n^{2}-60621700n+18711000 \bigr) \\& \phantom{c_{11} =\,\,}{}+ 20k^{2} \bigl( 138430n^{4}-873842n^{3}+2695909n^{2}-2399130n+679140 \bigr) +k \bigl(495624n^{4} \\& \phantom{c_{11} =\,\,}{}- 5988304n^{3}+11652870n^{2}-8520800n+2217600 \bigr)+183616n^{4}-663166n^{3} \\& \phantom{c_{11} =\,\,}{}+ 915935n^{2}-574850n+138600 \bigr], \end{aligned}$$

(140)

$$\begin{aligned}& c_{15} =\frac{a^{6}(k+1)(3k+1)n}{88268019840000k^{6}(n+1)^{6}} \bigl[42567525k^{10} \bigl(720n^{5}+2556n^{4}+3604n^{3}+2521n^{2}+874n \\& \phantom{c_{11} =\,\,}{}+ 120 \bigr)+28350k^{9} \bigl( 1249692n^{5}+686860n^{4}-2577633n^{3}-3421416n^{2}-1539028n-240240 \bigr) \\& \phantom{c_{11} =\,\,}{}+ 135k^{8} \bigl(278662344n^{5}-141122820n^{4}+282670258n^{3}+825130367n^{2}+466600470n \\& \phantom{c_{11} =\,\,}{}+ 79879800 \bigr)+36k^{7} \bigl(612790998n^{5}-2472456073n^{4}+1527229265n^{3}-1345472610n^{2} \\& \phantom{c_{11} =\,\,}{}- 1877749300n-399399000 \bigr)+6k^{6} \bigl(2273834244n^{5}-10928033994n^{4}+29030321910n^{3} \\& \phantom{c_{11} =\,\,}{}- 15039510755n^{2}+7751107700n+3006003000 \bigr)+4k^{5} \bigl(1272432138n^{5}-12749058409n^{4} \\& \phantom{c_{11} =\,\,}{}+32186860166n^{3}-50094341110n^{2}+21616077000n-5381376000 \bigr)+2k^{4} \bigl(1245531860n^{5} \\& \phantom{c_{11} =\,\,}{}- 10585913054n^{4}+47107629124n^{3}-68141956465n^{2}+56246234100n-14777763000 \bigr) \\& \phantom{c_{11} =\,\,}{}+ 4k^{3} \bigl(144913746n^{5}-2616936275n^{4}+9372735679n^{3}-19092417410n^{2}+14124980100n \\& \phantom{c_{11} =\,\,}{}- 3552549000 \bigr)+5k^{2} \bigl(58083800n^{5}-526446912n^{4}+2878464312n^{3}-4529906063n^{2} \\& \phantom{c_{11} =\,\,}{}+ 2903305230n-685284600 \bigr)+6k \bigl(5002696n^{5}-154497310n^{4}+462711169n^{3} \\& \phantom{c_{11} =\,\,}{}- 560651580n^{2}+316735300n-70070000 \bigr)+21625216n^{5}-103178392n^{4}+200573786n^{3} \\& \phantom{c_{11} =\,\,}{}- 199037015n^{2}+101038350n-21021000 \bigr]. \end{aligned}$$

(141)