Relativistic Models Of A Class Of Compact Objects
A class of general relativistic solutions in isotropic spherical polar coordinates are discussed which describe compact stars in hydrostatic equilibrium. The stellar models obtained here are characterized by four parameters, namely, , , and of geometrical significance related with inhomogenity of the matter content of the star. The stellar models obtained using the solutions are physically viable for a wide range of values of the parameters. The physical features of the compact objects taken up here are studied numerically for a number of admissible values of the parameters. Observational stellar mass data are used to construct suitable models of the compact stars.
PACS No(s). 04.20.Jb, 04.40.Dg, 95.30.Sf
Key Words: Relativistic Star, Compact object
The discovery of compact stellar objects, such as X-ray pulsars, namely Her X1, millisecond pulsar SAX J1808.43658, X-ray sources 4U 1820-30 and 4U 1728-34 which are regarded as the probable strange star candidates, has led to critical studies of relativistic models of such stellar configurations [1-10]. There are several such astrophysical as well as cosmological situations where one needs to consider the equation of state of matter involving matter densities of the order of or higher, exceeding the nuclear density. The conventional approach of obtaining models of relativistic stars in equilibirium heavily relies on the availibility of definite information about the equation of state of its matter content. Our knowledge about possible equation of state inside a superdense strange star at present is not known. In this context Vaidya-Tikekar  and Tikekar  have shown that in the absence of definite information about equation of state of matter content of stellar configuration, the alternative approach of prescribing suitable ansatz geometry for the interior physical 3-space of the configuration leads to simple easily tractable models of such stars which are physically viable. Relativistic models of superdense stars based on different solutions of Einstein’s field equations obtained by using Vaidya-Tikekar approach of assigning different geometries with physical 3-spaces of such objects have been studied by several workers [6, 7, 9, 10]. Pant and Sah  obtained a class of relativistic static non-singular analytic solutions in isotropic form describing space time of static spherically symmetric distribution of matter. The solution has been found to lead to a physically viable causal model of neutron star with a maximum mass .
In this paper we discuss a class of solution of relativistic field equations as obtained in Ref.  and examine physical plausibility of several models of a class of neutron stars using numerical procedures to explore the possibility of using it to describe interior of a compact star. It is also possible to estimate the radius of a star when its mass is known. It is also possible to determine the variation of matter density on its boundary surface and that at the center of a superdense star for the prescribed geometry. The plan of the paper is as follows : in sec 2. the relevant relativistic field equations have been set up and their solution is discussed. In sec 3. several features of physical relevance have been reported. In sec. 4, stellar models are discussed with the observational stellar mass data for different values of the parameters , , and . Finally in sec 5, we give a brief discussion.
2 Field Equation and Solution
The Einstein’s field equation is
where , , and are the metric tensor, Ricci scalar, Ricci tensor and energy momentum tensor respectively. We use the following form of the space time metric given by
using isotropic spherical polar coordinate. In the next section we use systems of units with , respectively.
The energy momentum tensor for a spherical distribution of matter in the form of perfect fluid in equilibrium is given by
where and are energy density and fluid pressure of matter respectively. Using the space time metric given by eq.(2), the Einstein’s field eq. (1) gives the following equations :
Now, pressure isotropy condition from eqs.(6) and (7) leads to the following relation between metric variables and :
It is a second order differential equation which permits a solution  as follows :
where , , and are arbitrary constants. In the above we denote
We observe that the geometry of that of the 3-space with metric
is that of a 3 sphere immersed in a 4-dimensional Euclidean space. Accordingly the geometry of physical space obtained at the section of the space time is given by
where, is given by eq.(10). Hence the geometry of the 3 space obtained at section of the space time of metric (12) is a deviation introduced in spherical 3 space and the parameter is a geometrical parameter measuring inhomogenity of the physical space. With , the space time metric (12) degenerates into that of Einstein’s static universe which is filled with matter of uniform density. The space time metric of Pant and Sah  is a generalization of the Buchdahl solution, the physical 3-space associated with which has the same feature. For , the solution reduces to that obtained by Buchdahl which is an analog of a classical polytrope of index 5. However, for , the solution corresponds to finite boundary models. Pant and Sah  obtained a class of non-singular analytic solution of the general relativistic field equations for a static spherically symmetric material distribution which is matched with Schwarzschild’s empty space time. In this paper we study physical properties of compact objects taking different values of , , and as permitted by the field equations. Using solution given by eq.(9) in eqs.(5)-(7), one obtains the explicit expressions for the energy density and fluid pressure as follows:
The exterior Schwarzschild line element is given by
where represents the mass of spherical object. The above metric can be expressed in an isotropic form 
using the transformation where is the radius of the compact object. This form of the Schwarzschild metric will be used here to match at the boundary with the interior metric given by eq. (12).
3 Physical properties of a compact star
The solution given by eq.(9) is useful to study physical features of a compact star in a general way which are outlined as follows:
(1) In this model, and are determined using eqs.(13) and (14). We note that is obviously positive for any positive and , while leads to two different cases: (i) with and (ii) with .
(2) At the boundary of the star (), the interior solution should be matched with the isotropic form of Schwarzschild exterior solution,i.e.,
(3) The physical radius of a star , is determined knowing the radial distance where the pressure at the boundary vanishes (i.e., at ). The physical radius is related to the radial distance () through the relation .
(4) the ratio is determined using eqs. (9) and (16), which is given by
(5) The density inside the star should be positive i.e., .
(6) Inside the star the stellar model should satisfy the condition, for the sound propagation to be causal.
The usual boundary conditions are that the first and second fundamental forms be continuous across the boundary . Applying the boundary conditions we determine which is given by
Equating eqs.(9) and (16) at the boundary , we get a eighth order polynomial equation in (here is replaced by ):
where , and are constants. Imposing the condition that pressure at the boundary vanishes in eq.(14), we determine which is given by,
Thus, the size of a star is determined by and . It is evident that a real is permitted when (i) with , or (ii) with . Using eqs.(20) and (21), a polynomial equation in , and is obtained. Although the eq.(20) is a polynomial of degree eight we note that only one realistic solution for is obtained for different domains of the values of any pair of parameters namely, , and . Subsequently the other parameters may be determined. For example, (i) when , we found that and satisfy the following inequalities and , (ii) when , the range of permitted values are and . However, for a given , e.g., (i) , we note that the permitted values of lies in the range , and (ii)that for , one obtains realistic solution for .
The square of the acoustic velocity takes the form :
A variation of for and is displayed in table 1. It is evident that is maximum at the center and gradually decreases outward. It is also found that inside the star the constrain is always maintained which ensures causality. In table 2, variation of from the centre to the boundary for different values of and are presented. It is evident that as increases decreases at the centre. The variation of the central density with and are displayed in tables (3) and (4) for and respectively. It is evident that the central density () decreases with an increase in . Thus stellar models with larger accommodate a denser compact object compare to that for lower values of and . The variation of pressure and density with radial distance are drawn employing eqs.(13) and (14) which are shown in figs.(1)-(4). Since it is not possible to express pressure in terms of density we study the behaviour of pressure and density inside the curve numerically.In fig.(5) a variation of pressure with density is plotted for different model parameters.
|in the unit of|
|in the unit of||&||&||&|
|in the unit of|
|in the unit of|
||M (mass) in||(radius) in km|
||M (mass) in||(radius) in km|
4 Physical Analysis :
In this section we analyze the physical properties of compact objects numerically. For given values of and , the radial coordinate at which the pressure vanishes may be determined from eq.(14). The mass to radial distance is estimated from eq.(18), which in turn determines the physical size of the compact star (). For a given set of values of the parameters , and , the mass () and radius of a compact object is obtained in terms of the model parameter . Thus for a known mass of a compact star is determined which in turn determines the corresponding radius. As the equation to determine the parameters in the model is highly non-linear and intractable in known functional form, we adopt numerical technique in the next section.
The radial variation of pressure and density of compact stars for different parameters are shown in figs. (1)-(5). It is evident that as is increased both the pressure and density at the centre is found to decrease and at the same time it corresponds to a smaller size accommodating more mass.
For a given mass of a compact star , it is possible to estimate the corresponding radius in terms of the parameter . We note that for a given mass of a compact star known from observation, the radius of the star may be estimated from a given . However, as the radius of a neutron star is , it is possible to obtain a class of stellar model taking different so that the size of the star is satisfies the upper bound. In the next section we consider a few such stars whose masses are known from observations.
Model 1 : We consider X-ray pulsar Her X-1 [12, 15, 16] which is characterized by mass , where = the solar mass and found that it permits a star with radius km, for km. The compactness of the star in this case is . The ratio of density at the boundary to that at the centre for the star is 0.0003 which is possible for the set of parameters and . Taking different values of we get different models but a physically realistic model is obtained which accommodates a compact star with radius 10 km. For example, if km, one obtains a compact object with radius km. In the later case we note that the ratio of density at the boundary to that at the centre is very high (0.99). The compactness of the star is 0.189 which is permitted for the set of parameters and with .
Model 2 : We consider X-ray pulsar 4U 1700- 37 which is characterized by mass . We note that for , and , the corresponding radius of the above star is km with km. The ratio of density at the boundary to that at the centre for the star in this case is 0.820. However, for the set of values , and , a compact object is permitted with radius km when km. The ratio of density at the boundary to that at the centre for the star in this case is 0.0003. Another stellar model is obtained for a set of values with , and , where the ratio of density at the boundary to that at the centre is 0.99. In the later case the values is more compare to that one obtains taking . However both the cases permits a star with compactness factor .
Model 3 : We consider a neutron star J1518+4904 which is characterized by mass . For , and , the radius of the star estimated here is km with km. The ratio of density at the boundary to that at the centre for the star is 0.82. In this case the compactness factor of the star is . For we note the following : (i) when and , it admits a star with radius km for km and (ii) when and , it admits a star with radius for km for km. The ratio of density at the boundary to that at the centre for the star in the first case is 0.0003 and that in the later case is 0.988. However, the compactness factor for the former is 0.3 which is higher than that in the second case (0.189).
Model 4 : We consider a neutron star J1748-2021 B which is characterized by mass . For , and , a star of radius km with km ids permmited . The ratio of density at the boundary to that at the centre for the star is 0.856. The compactness factor is . In the other case one obtains a star with radius km with km when and . A star of smaller size is thus permitted in the later case with compactness factor (0.32) than that of the formal model.
For , stellar model admits a star with radius km for km, and . However a smaller star with radius km is permitted here when , km with and . The ratio of density at the boundary to that at the centre in the first case is 0.0017 which is higher than the later (0.0015). The compactness factor in the former model is 0.20 which is lesser than the later case 0.32.
5 Discussions :
In this paper, we present general relativistic solution for a class of compact stars which are in hydrostatic equilibrium considering the isotropic form for a static spherically symmetric matter distribution. The general relativistic solution obtained by Pant and Sah  is employed here to study compact objects. We use isotropic form of the exterior Schwarzschild solution to match at the boundary of the compact object. The stellar models discussed here contains four parameters , , and . The observed mass of a star determines for known values of , , .
We note the following: (i) In fig. 1, variation of pressure with radial distance is plotted for different for given values of and . The figures show that as increases pressure decreases inside the star. (ii) In fig. 2, radial variation of density is plotted for different . We note higher density for lower . (iii) The variation of inside the star for a given set of values of and are shown in table 1. The causality condition is obeyed inside the star and is maximum at the center which however found to decrease monotonically radially outward. For different and , values of is also shown in table 2. It is evident that decreases for an increase in and values. (iv) Variation of central density for different values of and with and are presented separately in tables (3) and (4) respectively. We note that the central density decreases as the value for the pair ( and ) increases. From tables (3) and (4) similar tendency for central density is found to exist when is increased. As the isotropic Schwarzschild metric is singular at , the model considered here may be useful to represent a strange star with or . (v) In tables (5) and (6), the mass of a star with its maximum size is shown for different values of and taking density of a star at the boundary. We obtain here a class of relativistic stars for different values of , , and . (vi) The density profile of a given star with different values of and is shown in table 7. As increases the ratio of density at the boundary to that at the center is found to increase accommodating more compact stars. (vii) In fig. 3, variation of pressure with radial distance is plotted for different values of . It is evident that as increases pressure decreases. (viii) In fig. 4, variation of density with radial distance is plotted for different . We note that as is increased both the density and the pressure decreases. But the size of a star increases with an increase in thereby accommodating more compact stars. (ix) In fig. 5, variation of pressure with density is plotted for different . We note that for a given density pressure is more for higher , this leads to a star with higher central density.
In sec. 4, we present models of the neutron stars that are tested for some known compact objects. As the equation of state is not known we analyze the star for known geometry considered here. The radii of the compact stars namely, neutron stars are also estimated here for known mass with a given . The parameter permits a class of compact objects, some of which are relevant observationally. Considering observed masses of the compact objects namely, X-ray pulsars Her X-1, 4U 1700-37 and neutron stars J1518+4904, J1748-2021 B we analyze the interior of the star. We obtain a class of compact stars models for various with given values of , and . The stellar models obtained here can accomodate highly compact objects. However a detail study of the stellar composition at high pressure and density will be taken up elsewhere.
BCP would like to acknowledge fruitfull discussion with Mira Dey and Jisnu Dey while visiting IUCAA, Pune. Authors would like to thank IUCAA, Pune and IRC, Physics Department, North Bengal University (NBU) for providing facilities to complete the work. BCP would like to thank University Grants Commission, New Delhi for financial support. RT is thankfull to UGC for its award of Emiritus Fellowship. The authors would like to thank the referee for constructive criticism.
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