Structure of the Star with Ideal Gases

In this paper, we provide a simplified stellar structure model for ideal gases, in which the particles are only driven by gravity. According to the model, the structural information of the star can be roughly solved by the total mass and radius of a star. To get more accurate results, the model should be modified by introducing other interaction among particles and rotation of the star.

The spherical symmetric metric for a static star is described by Schwarzschild metric g µν = diag b(r), −a(r), −r 2 , −r 2 sin 2 θ . (1.1) For the energy momentum tensor of perfect fluid T µν = (ρ + P )U µ U ν − P g µν = diag bρ, aP, r 2 P, r 2 sin 2 θP , (1.2) where ρ(r), P (r) are proper mass energy density and pressure, U µ = ( √ b, 0, 0, 0), we have the independent equations as follow To determine the stellar structure of an irrotational star, we solve these equations. However (1.3)-(1.4) is not a closed system, the solution depends on an equation of state(EOS) P = P (ρ). For polytropes, we take P = P 0 ρ γ [1]. For the compact stars, there are a lot of EOS derived from particle models [2]- [7], which provide the structural information and parameters such as the maximum mass for neutron stars. However, in these EOS the boosting effects of the gravity on particles seem to be overlooked or adopted unconsciously. As shown in [8], the static equilibrium equation (1.5) is insufficient to describe such dynamical effects on fluid, and a Gibbs' type law should be included. In this paper, according to this Gibbs' type law, we derive the stellar structure model, which shows that the boosting effects of gravity play an important role to the stellar structure.
Before expanding the model, we make a few simple calculations and examine the behavior of the metric and particles inside a star to get some intuition. The first phenomenon is that, the temporal singularity and spatial singularity occur at different time and place if the spacetime becomes singular, and the temporal one seems to occur firstly.
Denoting the mass distribution by where R is the radius of the star, and the Schwarzschild radius becomes For any normal star with R > R s . From the above solution we learn a(r) is a continuous function and So the spatial singularity a → ∞ does not appear at the center of the star when the singularity begins to form.
On the other hand, by (1.4) and P ≥ 0, we find b ′ (r) is a continuous function satisfying (1.10) (1.9) shows b(r) is a monotone increasing function of r with smoothness at least C 1 ([0, ∞)).
Consequently, the temporal singularity b → 0 should take place at the center.  The second phenomenon is that, the particles near the center of the star are unbalanced, and violent explosion takes place inside the star before the temporal singularity occurs.
According to fluid mechanics, −∂ r P corresponds to the radial boosting force, so (1.12) means violent explosion.
More clearly, we examine the motion of a particle inside the star. Solving the geodesic in the orthogonal subspace (t, r, θ), we get[9, 10] where C 1 , C 2 are constants. The normal velocity of the particle is given by 14) The sum of the speeds provides an equality similar to the energy conservation law (1.15) holds for all particles with v ϕ = 0 due to the symmetry of the spacetime.
From (1.15) we learn v → 1 when b → 0, this means all particles escape at light velocity when the temporal singularity occurs. So instead of a final collapse, the fate of a star with heavy mass may be explosion and disintegration. The gravity of a star drives the inside particles to move rapidly and leads to high temperature. How the particles to react to the collapse of a star needs further research with dynamical models. A heuristic computation for axisymmetrical collapse is presented in [11], which reveals that the fate of a collapsing star sensitively depends on the parameters in the EOS.

II. THE EQUATIONS FOR STELLAR STRUCTURE
In this paper, we simply take the star as a ball of ideal gases, which satisfies the following assumptions: (A1) All particles are classical ones only driven by the gravity, namely, they are characterized by 4-vector momentum p µ k and move along geodesic. (A2) The collisions among particles are elastic, and then they can be ignored in statistical sense [9,10].
(A3) The nuclear reaction and radiation are stable and slowly varying process in a normal star, in contrast with the mass energy, the energy related to this process is small noise, so we treat all photons as particles and omit the process of its generation and radiation.
These are some usual assumptions suitable for fluid stars. However, together with (1.3)-(1.4), they are enough to give us a simplified self consistent stellar structure theory. In [8], we derived the EOS for such system as follows

2)
where N is the number density of particles, N 0 = N 0 (m, σ) is related to property of the particles but independent of J, c = 2.99 × 10 8 m/s the light velocity, σ= 2 5 a factor reflecting the energy distribution function,m the mean static mass of all particles, J dimensionless temperature, which is used as independent variable, (ρ, P ) are usual energy density and pressure, ̺ a mass density defined by ̺ ≡ N 0m .
(2.4) By (2.2) and (2.3), we get the polytropic index γ is not a constant for large range of T satisfying 1 < γ < 5 3 , and the velocity of sound where R is the radius of the star, and R s = 2M tot the Schwarzschild radius. Substituting where χ is a constant length scale, The exact solution to (2.8) and (2.9) seems not easy to be obtained. However, they are dimensionless equations convenient for numeric resolution. If we take χ = 1 as the unit of length, the solution can be uniquely determined by the following boundary conditions  Then by (2.2) and (2.3) we solve the mass density and pressure at the center How to determine the concrete function N 0 (m, σ) is an interesting problem.
The dimensionless equations (2.8) and (2.9) simplify the relations between parameters.
This function is similar to that of the similarity theory [12]. However in these equations the information of the interaction among particles is ignored, so it can not provide the critical data such as the maximum density, the largest mass. To get such data the potentials and interactive fields should be introduced to the energy momentum tensor [8].