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**Announcement from Editorial Board **
** **
The following article has been retracted due to plagiarism. This paper published in Vol.3 No.4, April 2012, has been removed from this site.
Title: On the Relativistic Stars
Author: Silvia Morales, Roberto Aquilano
It has been brought to our attention that this paper appears exactly as it was published in Astroparticle Physics, vol. 32, 2009, 153. ("On the nature of relativistic stars" by Silvia Morales, Alejandra Zorzi, Roberto Aquilano)
Consequently, this paper has been retracted by Journal of Modern Physics.
JMP Editorial Board
April 30, 2015

The X-ray bursters are binaries stars with a neutron star and a normal companion star of mass approximately to a solar mass. Due to their weak magnetic fields and high accretion rates, the neutron star equilibrium spin periods are expected to be of order of milliseconds. Detection of coherent millisecond X-ray pulsations would constitute a strong evidence for the evolution of millisecond radio pulsars [1-3]. It would support the currently favored beat frequency model of horizontal-branch QPO formation [4, 5], in which quasi-periodic oscillations occur at different frequencies between the neutron star rotation frequency and the Kepler frequency of the accreting matter, as it falls onto the neutron star. It would provide a precise means to measure orbital periods and masses and to study accretion torques and neutron star dynamics on very short time scales.

Instead of ordinary matter, the possibility that strange matter was the true ground state of strong interactions has attracted considerable attention [

In this work a simple model of non-rotating stars is studied and modelled by a spherically symmetric distribution, to neutron and strange matter stars.

The hydrostatic equilibrium equation for relativistic stars in the Tolmann-Oppenheimer-Volkoff equation is

where m(r) is the mass enclosed in a sphere with radius r and local density G is the gravitational constant and P(r) the pressure with radious r, and this equation in the Newtonian limit is as follows

and

The differential form of this equation is

with constant density

if

also, the total mass of star is

The Equation (2.5) in function of the total mass will be

Then, with (2.7) in (2.2), we have and putting (2.5) in (2.2) we have _{}

where, with radial integrating, we obtain

But, , and in the equilibrium is

or in function of other constant

We define the new constant also, we can write the Equation (2.8) in this form

Considering small perturbations around the equilibrium, we will obtain the frequency. For a non relativistic fluid, no viscous and without dissipative effects, the moment equation is

where P is the pressure, is the gravitational potential and v is the velocity of fluid. The oscillation dynamic will be governed for the Lagrangian perturbations of this equation

If in the moment Equation (2.11), we consider, we obtain the hydrostatic equilibrium equation

The non perturbative configuration is static, thus we write

And we have

1)

By the application of the perturbative properties, we can write the integrals in this form

2) For the state equation (: density, s: entropy)

For adiabatic perturbations

where is the adiabatic index that governs the perturbations (2.16).

With the Equations (2.14) and (2.15), and replacing in the Equation (2.13), and for the adiabatic equation we will utilize for and we obtain this

Here, the quantities are expressed in terms of and non perturbative variables. The perturbation in the gravitational potential is deduced of the Poissón equations,

or

For the Equations (2.14) and (2.19)

The Equation (2.18) has this solution

To radial perturbations of the spherical star

Replacing the Equation (2.21) in (2.17)

and considering radial oscillations of this form

the Equation (2.22) is reduced to

United with appropriate outline conditions it is a lineal self-value problem. The outline conditions are

•

•

As we say that is finite in.

The Equation (2.25) is obvious because of its spherical symmetry. The Equation (2.26) informs us that an element of fluid in the non perturbative surface is moving toward the perturbative surface. In the equilibrium configuration (Equation (2.9)) this equation is valid

To make simpler the calculus, it has been defined a constant “a” ((7)), then and we can write the Equation (2.24) in this form

In order to simplify, we will omit the sub-index 0 (equilibrium values) (7) and propose a potential with solution in series

The coefficient of is the initial equation

Also, or. We choose to carry out the outline condition (2.25), the Equation (2.27) results

with

We obtain that, then

constant

These series diverge, and can not satisfy the outline condition (2.26), safe that the series finish. For this

Then

The solution in the minor mode with k = 0 is

• Si.

• Si is real.

• Si is imaginary and the star is unstable.

These conclusions are equivalent to the Chandrasekhar limit, which should be correct only for stars with (e.g. normal neutron stars and white dwarfs), but not for stars with non-zero surface density.

A method to solve the self-valued Equation (2.24) for strange matter stars is the variational principle

where satisfies the outline conditions in (2.25) and (2.26). The absolute minimum of this expression is the square of the angular frequency of fundamental mode pulsation. In the Newtonian limit an approximate expression is

where is the adiabatic index to average according to the pressure

Then, the Equation (3.1) will be

where is the self-gravity energy of the star,

and

Then, the equation of state to strange matter is

where B = 60 MeV/fm^{3}

If and:

If the density is constant, with the outline conditions only for quark stars not very massive, but not suitable for normal neutron stars,

with the Equations (3.4) and (3.5) the expressions， and are

Replacing these results in the Equation (3.1), we obtain the expression to angular frequency of fundamental mode as

In this work we used a strange matter star not very massive with a general relativistic formalism in order to

study radial oscillations and to compare with normal neutron stars. In this case, we use different values to and confirm the previous results obtained [