1. Introduction

The behavior of particles inside compact stars like white dwarfs, formed by matter
with density lower than the nuclear density, can be experimentally reproduced in
some cases by means of nuclear accelerators [^{1}^{,}^{2}]. In adittion, complementary theoretical models that help
describing the interactions in the interior of compact stars, even for low energy
interactions, the dynamics can be explored as well through general relativity or non
relativistic effective theories [^{3}^{,}^{4}]. For compact objects like neutron or quark stars that have
a density larger than the nuclear density it is necessary to consider theoretical
models using gravitational theories like general relativity. On the other hand, even
though experiments have been carried out to explore the quark reactions that could
happen inside the stars, the development of experiments for these order of densities
is not possible at laboratory level yet. However, phenomenological models
approaching compact stars through equations of state with parameters determined by
comparison with observations have been proposed [^{5}^{,}^{6}].

Among the different kinds of stars are the low mass ones, whose behavior can be
different from those with mass on the order of the sun mass. On this subject, there
is still no consensus about the equations of state that could describe the interior
of these stars, although different models are being currently used, from those that
assume polytropic equations [^{7}] to
others like PARSEC models [^{8}].
While the description of the interior of static stars is carried out through the
construction of solutions to the Einstein equations, some of which assume that the
interior is described by a perfect fluid [^{9}^{-}^{11}], anisotropic [^{12}^{,}^{13}], and charged [^{14}^{-}^{17}], these last two models are characterized by
compactness values that are larger than the ones in the case described by a perfect
fluid [^{18}^{,}^{19}].

Some other ways to build interior solutions is by means of imposing some geometric
condition, one of which is to suppose four-dimensional space-time embedded in a
five-dimensional pseudo Euclidean space [^{20}^{,}^{21}] known as Class I solutions for a spherically
symmetric space-time there is a differential equation that relates the metric
functions *g*_{rr} and
*g*_{tt} that, under suitable
choices of one of these functions allows integrating the system, generating
physically acceptable models [^{22}].

The type of solutions Class I is useful when stellar models without charge and
anisotropic pressures are proposed [^{21}^{-}^{24}] or with anisotropic pressures and electrically charged
[^{25}], and has been used to
describe the interior of compact objects such as Cen X-3, PSR J0348+0432, PSR
B0943+10 and XTE J1739-285. However for the case of stellar models associated with
perfect fluid, the condition that the radial pressure
*P*_{r} and tangential pressure
*P*_{t} be the same, implies a
differential equation between the metric potentials
*g*_{rr} and
*g*_{tt}, different from solutions
of the Class I. and in such case a there is no guarantee that these functions
satisfy both equations.

In our case we propose a model described by a perfect fluid. Therefore, the
construction of our solution, obtained from the differential equation emerging from
the equality of radial and tangential pressures
*P*_{t} =
*P*_{r} and from imposing a new
gravitational potential *g*_{tt} motivated
by recent work [^{26}^{,}^{27}] in which new physically acceptable stellar solutions have
been built with a different form to those previously proposed [^{28}]. An advantage of the choice of our potential
*g*_{tt} is that it generates a model
with a speed of sound that is monotonically decreasing as a function of the radial
coordinate. We also show that the model is physically acceptable and that it can be
applicable to the star PSR B0943 + 10.

Physically acceptable solutions for a perfect fluid in a spherically symmetric space
are required to satisfy the following conditions [^{28}]:

The solution must be regular,

*i.e*. the geometry and the physical quantities inside the star have to be non-singular.The pressure is null on the star surface.

The pressure and density must be positive and monotonic decreasing functions of

*r*.The causality condition must be satisfied, the sound speed must be lower than the speed of light.

The gravitational redshift function must be positive, finite and monotonically decreasing in

*r*.

Among the set of stars of low mass there are some compact stars, like the PSR B0943 +
10 with mass 0.02 M_{ʘ} and radius 2.6 km, a quark star candidate [^{29}] with compactness ratio equal to
0.06093. In this work, we intend to describe this object using the low mass star
model that is introduced in what follows.

This paper is organized in the following manner: In Sec. 2 we work out the solution considering that the interior of the star is described by a perfect fluid; Section 3 focuses on obtaining the validity intervals for the parameters involved in the model, stemming from the physical conditions mentioned above. In Sec. 4 we show that this solution applies to low mass compact stars and that it describes the behavior of PSR B0943 + 10. Finally, we end with the conclusions of this analysis.

2. Perfect fluid solution

According to the Jebsen-Birkhoff theorem for stellar models, the exterior of a star
is described by the Schwarzschild metric [^{30}^{-}^{32}]

where *M* is the mass of the star, *G* is the
gravitational constant and *c* is the speed of light. This metric and
its second fundamental form have to be continuous over the boundary [^{33}^{,}^{34}]. On the other hand, the interior geometry
*r* ≤ *R*, for a perfect fluid, is determined by
the solution of the Einstein equations

where *G*_{μv} are the Einstein tensor
components, P is the pressure, ρ is the density of the energy-matter measured by an
observer with a four-velocity *u*^{μ} and
*g*_{μv} the metric components, which
for a static and spherically symmetric spacetime can be described by the line
element

Considering this metric in Eq. (2), one obtains the conditions

And from the Bianchi identity *∇*_{μ}
*G*^{μv} = 0, Eq. (2) implies

where ' denotes the derivative with respect to the *r* coordinate.
Substituting the form of the density and the pressure given by Eqs. (4) and (5) in Eq. (6), it follows that Eq. (6) is satisfied, so we only need to consider the system formed by
Eqs. (4)-(6). To build a model that is applicable
to a compact low mass star, we assume a particular form of
*Y*(*r*)given by

Although this function is new, the functional form of this has recently been proposed
[^{26,}^{27}] to generate stellar models that describe compact
objects. Therefore, guided by the previous results, we assume this functional form.
If we use this in the expressions for the pressure given by Eqs. (5) and (6) and subtracting one from the other,
the following differential equation for *B* results

This can be solved analytically and allows us to obtain

with

The functions *Y* and *B*, given by Eqs. (8) and (10) respectively, determine the metric
and allow us to get the form of the hydrostatical variables. From *B*
and Eq. (4), the density follows

where *S*_{1} (*r*) is given by:

The pressure is obtained from Eq. (5)
using *Y* and the function *B*

Another relevant parameter in the construction of a stellar model is the speed of sound defined as:

from where we get:

with:

Having solved the system of equations for the perfect fluid model, we now proceed to determine the validity intervals of the parameters in order to have a physically acceptable solution.

3. Behavior of the solution

The determination of the physical values of the model parameters requires to fulfil the conditions listed in Sec 1. The first derivative of the pressure, density and speed of sound at the origin vanish. On the other hand, evaluation of the functions and their second derivatives at the origin, results in the following set of inequalities after imposing the conditions enunciated in Sec. 1

The sign of *a* is determined by

Hence *a* < 0and using the previous set of inequalities, we find
that these are satisfied if

The integration constant *A* is related to *a* and the
radius of the object, *R*, since the pressure must be null on the
surface, that is *P*(*R*) = 0. Imposing this condition
we get with *y* = *aR*^{2} > 0. From here
we see that the model depends only on the parameter *y*. Using this
form of *A* in Eq.
(15) and the fact that *A*(*y*) is a
monotonic decreasing function for *y* > 0 in the range given by
the inequality (15), the maximum
acceptable value for *y* = *y*_{max} takes
place when *A*(*y*_{max}) = (387/256), from
where we get *y* ∈ [0,0.2149]. In this *y* interval,
the forms of the density, pressure and speed of sound functions inside the star are
described by the Figs. 1-3. For the graphic representation of these functions we define
dimensionless variables *r* → *x* =
*r*/*R*, *ρ* →
*kR*^{2}*c*^{2}*ρ*,
*P* → *kR*^{2}
*P* and *v*^{2} →
*v*^{2}/*c*^{2}.

Figure 1 shows that the density is a monotonic
decreasing function and that the difference between the density in the core and on
the surface lowers as the value of *y* decreases. The maximal
difference between the density in the surface and in the core happens for
*y* = 0.02149, when we have ρ_{c} =
1.0394ρ_{b}.

The behavior observed in Fig. 2 for the pressure is the expected one for physically acceptable solutions in the case of a perfect fluid, that is, a monotonic decreasing behavior and the existence of a region, the surface, where the pressure is null.

As we can see from Fig. 3, the speed of sound at
the core for the maximal value of *y* is the same as the speed of
light in vacuum and decreases as the distance to the surface decreases. Moreover,
when *y* has lower values, the speed of sound in the interior
decreases. From the figures presented here, we have that the behavior of the
density, pressure and speed of sound correspond to monotonic positive decreasing
regular functions, which makes this model physically acceptable.

4. Compact stars of low mass

From the condition of continuity between the internal and the external metrics, we
obtain the constant *C*^{2} = (1-2*GM* /
*c*^{2}*R*) /
*Y*(*R*) and the compactness ratio

which together with the form of the density profile determine the kind of objects
that can be described by the model. In this case, the value of the compactness
*u* = *u*(*y*) as a function of
*y* is monotonic and decreasing, so its minimal value is obtained
for *y* = 0 and its maximal value takes place for *y*
= 0.02149, being *u* = 0.06093. For this value of maximal
compactness, the speed of sound at the origin is the same as the speed of the light
and the value on the surface is 0.91404*c*, while the maximal value
of the density ratio core/surface is ρ_{c} /
ρ_{b} = 1.0394. This allows us to propose that the
model presented in this analysis is applicable to compact low-mass stars. One of
these stars is the quark star candidate PSR B0943 + 10, with observed mass 0.02
M_{ʘ} and radius *R* = 2.6 km.

If we consider the maximum compactness value, there is still room for a variety of
different stars. For instance, if the mass is M = 0.02 M_{ʘ}, its radius
will be *R* = 4.8464 km, with central density
ρ_{c} = 8.5338x10^{17} kg/m^{3} and
surface density ρ_{b} = 8.2107x10^{17}
kg/m^{3}. While for a star with radius 2.6 km, the mass would be
*M* = 0.1073 M_{ʘ}, with central density
ρ_{c} = 2.9651x10^{18} kg/m^{3} and
surface density ρ_{c} = 2.8529x10^{18}
kg/m^{3}. In these two cases, the object with higher density,
approximately ten times the nuclear density, is the one with lower mass and
radius.

For the case of PSR B0943 + 10, with compactness *u* = 0.01136, our
analysis delivers the values for the parameter space in the interior of the star
kisted in Table I.

r(m) |
ρ(10^{17} kg/m^{3}) |
P(10^{32} Pa) |
v_{s}(c) |

0 | 5.4266 | 2.8301 | 0.9103 |

260 | 5.4262 | 2.8016 | 0.9102 |

520 | 5.4251 | 2.7163 | 0.9098 |

780 | 5.4232 | 2.5740 | 0.9092 |

1040 | 5.4205 | 2.3751 | 0.9083 |

1300 | 5.4170 | 2.1195 | 0.9072 |

1560 | 5.4128 | 1.8075 | 0.9058 |

1820 | 5.4078 | 1.4393 | 0.9042 |

2080 | 5.4020 | 1.0151 | 0.9023 |

2340 | 5.3954 | 0.5352 | 0.9003 |

2600 | 5.3881 | 0 | 0.8979 |

Hence, even though the density is a monotonic decreasing function, its variation is
very slow since both the central density and the density at the surface are of the
same order ρ_{c} = 1.0072ρ_{b} ,
and higher than the nuclear density, which is characteristic of the quark stars. A
similar pattern is followed by the speed of sound, that satisfies
*v*_{s} (0) =
1.0138*v*_{s} (*R*). The
evolution of these two parameters can be seen in Figs.
1 and 3 and correspond to the curves
at the bottom of each panel, with *y* = 0.0036. From here we see that
the model is adequate and physically acceptable to represent the star PSR B0943 +
10.

5. Conclusions

Assuming a specific form for the metric function
*Y*(*r*), we derive a relativistic stellar
solution for compact stars considering a perfect fluid to model its interior. The
regular and monotonic decreasing evolution of the density, pressure and speed of
sound makes the solution physically acceptable. On the other hand, we show that the
speed of sound grows with density and equals the speed of light in the center of the
star for the highest density case. When we apply this model to the quark star
candidate PSR B0943 + 10, with mass 0.02 M_{ʘ} and radius *R*
= 2.6 km, it comes out that the density at the surface is
ρ_{b} = 5.3881x10^{17} kg/m^{3}, which
is larger than the nuclear saturation density, and its central density is slightly
higher than the surface density ρ_{c} =
1.0072ρ_{b} , which is a characteristic density for
compact stars. Although we have chosen the star PSR B0943 + 10 to apply our model,
it can be useful as well to analyse stars with compactness ratio *u*
≤ 0.06093.