1. INTRODUCTION

Isothermal models play an important role in many astrophysical problems, in particular in stellar structure and galactic dynamics (^{Chandrasekhar 1939}; ^{Binney & Tremaine 1987}; ^{Rose 1998}; ^{Chavanis 2002}, ²⁰⁰⁸⁾. Regarding stellar structure and evolution theory, one can obtain the behavior of the physical variables through the isothermal self-gravitating spheres. In the equation of state P = Kρ ^1+1/n (where P is the pressure, ρ is the density and K is the polytropic constant), the polytropic index n ranges from 0 to ∞ (^{Liu 1996}; ^{Mirza 2009}). In galactic dynamics, the polytropic index n is larger than 0.5 and the Lane-Emden equation for isothermal configurations is considered as a generating function of potential models for flattened galactic systems (^{Binney & Tremaine 1987}).

In the framework of Newtonian mechanics (nonrelativistic), isothermal spheres have been much investigated by many authors (^{Milgrom 1984}; ^{Liu 1996}; ^{Natarajan & Lynden-Bell 1997}; ^{Roxburgh & Stockman 1999}; ^{Hunter 2001}; ^{Nouh 2004}; ^{Mirza 2009}; ^{Raga et al. 2013}). In the framework of the theory of general relativity, numerical studies on the relativistic isothermal spheres were performed by several authors (^{Edward & Merilan 1981a},^b; ^{Zhang et al. 1983}; ^{Chavanis 2002}, ²⁰⁰⁸; ^{Chau et al. 1984}). (^{Chau et al. 1984}) extended the work done by ^{Edward & Merilan (1981a},^b) to determine the static structure of general relativistic, partially degenerate, isothermal (with arbitrary temperature) configurations. ^{Chavanis (2002}, ²⁰⁰⁸⁾ investigated the structure and stability of isothermal gas spheres in the framework of general relativity with a linear equation of state P = qє (є: the mass-energy density, P: pressure and q → 0 for Newtonian gravity). ^{Sharma (1990)} introduced approximate analytical solutions to the TOV equation of hydrostatic equilibrium in a concise and simple form using a Pad´e approximation technique. His results show that general relativity isothermal configurations are of finite extent. The geometrical size derived by ^{Sharma (1990)}, ξ = ξ₁, for the configurations is limited to ξ₁ = 2.

Due to the lack of a full analytical solution to the TOV equation of hydrostatic equilibrium and the importance of the study of relativistic isothermal configurations for various astronomical objects, we introduce a new approximate analytical solution to the Tolman-Oppenheimer-Volkoff (TOV) equation based on a combination of the two techniques: the Euler-Abel transformation and the Pad´e approximation (^{Nouh & Saad 2013}). The constructed analytical solution is in a general form such that we can increase the number of terms of the series solution for the desired accuracy. We display the solution curves in the ξ-θ and ξ-ν phase planes for three general relativistic effects σ = 0.1, 0.2 and 0.3. The proposed method is promising and can examine efficiently the behavior of the physical parameters of stars, such as density, pressure, temperature, etc. from the center towards the surface.

2. FORMULATIONS

The Tolman-Oppenheimer-Volkoff equation of the isothermal gas sphere could be given by ^{Sharma (1990)}.

where M_r is the total mass energy or “effective mass” of the star of radius r including its gravitational field, that is,

Define the variables, ξ, θ, ν and the relativistic parameter σ by the following equations.

where ρ _c and P_c define the central density and central pressure of the star respectively. ν is the mass function of radius ξ , and c is the speed of light. Equations (1) and (2) can be transformed into the dimensionless forms in the (ξ, θ) plane as:

which satisfy the initial conditions

If the pressure is much smaller than the energy density at the center of a star (i.e. σ tends to zero), the non-relativistic case, then the systems (5) and (6) reduce to yield the classical non degenerate isothermal structure equations (^{Chandrasekhar 1939})

3. SERIES SOLUTION

Equation (5) can be written in the form

Consider a series expansion near the origin ξ = 0 , of the form

that satisfies the initial conditions (7); at θ(0) = 0 yields α₀ = 0; then

Multiplying equation (11) by ξ² yields

With the help of some algebraic operations on series (^{Nouh and Saad 2013}), we get:

Inserting Equation (9) in Equation (6) yields

Integrating both sides of the last equation gives

Multiplying both sides of equation (12) by ν and using equation (16)we obtain:

Let

then

Implementing the formula of multiplication of two series(^{Nouh & Saad 2013}), Equation (19) becomes:

where

Substituting equations (13),(14),(16) and (20) in equation (9) yields:

that is,

Then, the series coefficients formula has the form:

where

For example, the coefficient a₁ can be computed from

Considering σ = 0 in equation (18), we get α1=16 which is the first series term in the case of the nonrelativistic Lane-Emden solution.

4. RESULTS

The analytical solution of equations (5) and (6) with the given initial conditions in equation (7) determines the relativistic structure of the configuration. This solution is represented by equation (10) together with equations (23) to (25). The radius of convergence ξ₁ of the power series solution without applying any acceleration techniques is limited. Tables 1, 2 and 3 show the solutions θ(ξ₁), the mass function ν(ξ₁) and the errors between analytical and numerical solutions ε₁ and ε₂ respectively. It is found that the maximum radii of convergence for the relativistic effects σ = 0.1,0.2 and 0.3 are 3.58, 3.23 and 2.8 respectively. Beyond the mentioned values, the power series solution is either slowly convergent or divergent. It is worth noting that we have used the fourth order Runge-Kutta method for the numerical solution of the TOV equation.

Therefore, it was necessary to improve the power series solution of the TOV relativistic equation, which in turn contributes to obtain a better approximation to the physical parameters of stars. A combination of the two techniques for the Euler-Abel transformation and the Pad´e approximation (^{Nouh 2004}; Nouh & Saad 2013) has been utilized. Figures 1, 2 and 3 represent the solution curves of the TOV equation in the ξ-θ phase plane for the general relativistic effects σ = 0.1,0.2 and 0.3, respectively. Each figure displays both analytical (dashed line) and numerical (solid line) solutions and their relative error. The maximum relative error was of order 10⁻³ for σ = 0.1,0.2, while for σ = 0.3 the relative error was of order 10⁻². In the same way, Figures 4, 5 and 6 show the solution curves in the ξ-ν phase plane for the general relativistic effects σ = 0.1,0.2 and σ = 0.3 respectively. When comparing the accelerated solution with the numerical one, we found that the series tends to converge slowly for ξ > 20. The most important finding is that the physical range of the convergent power series solutions extended to around ten times that of the classical procedures.

5. CONCLUSIONS

We have constructed general analytical formulations for solving the hydrostatic equilibrium equation (TOV equation) in the framework of relativistic isothermal gas spheres. Traditional procedures for its solution were not effective and the radius of convergence of the power series solution was limited, as shown in Tables 1, 2 and 3. That is, the maximum radii of convergence for the relativistic effects σ = 0.1,0.2 and 0.3 were ξ₁ = 3.58, ξ₁ = 3.23 and ξ₁ = 2.8, respectively. When we improved the power series solution utilizing a combination of the two techniques: an Euler-Abel transformation and a Padé approximation (^{Nouh 2004}; ^{Nouh & Saad 2013}), the radius of convergence increased. As a result, the physical range of the convergent power series solutions was extended to about ten times that of the traditional procedures (ξ₁ ≈ 35), as displayed in Figures 1, 2 and 3. The implementation of the proposed analytical solution contributed to improve the accuracy of the calculation of the physical parameters of the stars. As shown in Figures 7 to 12 for the density and mass profiles of the star, the maximum relative error reached 10⁻³, which establishes the validity and accuracy of the suggested procedure. The analytical solution derived in the present paper may be a good contribution to the field of stellar structure.