Compact stars

Compact stars on pseudo-spheroidal spacetime compatible with observational data

Abstract

A new class of solutions for Einstein’s field equations representing a static spherically symmetric anisotropic distribution of matter is obtained on the background of pseudo-spheroidal spacetime. We have prescribed the bounds of the model parameters and on the basis of the elementary criteria for physical acceptability, viz., regularity, stability and energy conditions. By taking the values of model parameters from the prescribed bounds, we have shown that our model is compatible with the observational data of a wide variety of compact stars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, SMC X-4 and Cen X-3.

General relativity; Exact solutions; Relativistic compact stars.
\synctex

=-1 \synctex=1

1 Introduction

The study of compact objects in agreement with observational data has received wide attention among researchers. A number of superdense star models, compatible with observational data, have appeared in literature in the recent past ( Murad (2013a),  Murad and Saba (2013b, c, 2014a),  Maurya et al. (2015) &  Sharma and Ratanpal (2013)). If spacetime admitting compact star models possess a definite three-space geometry, then it is a mathematically interesting problem also. The spheroidal spacetimes studied by  Vaidya and Tikekar (1982),  Tikekar (1990) and the paraboloidal spacetime studied by  Finch and Skea (1989),  Tikekar and Jotania (2005),  Sharma and Ratanpal (2013) are examples of spacetimes with definite 3-space geometry. The superdense star models developed by  Tikekar and Thomas (1998) has pseudo-spheroidal geometry. A number of researchers used this spacetime for developing physically viable models of compact stars under different assumptions on the physical content.

Theoretical investigations of  Ruderman (1972) and  Canuto (1974) suggest that matter may not isotropic in ultra high density regime. After the publication of the work of  Bowers and Liang (1974), there has been a large number of models devoted to the study of anisotropic distribution of matter.  Maharaj and Maartens (1989) developed an anisotropic model with uniform density and  Gokhroo and Mehra (1972) gave a more realistic anisotropic model with non-uniform density.  Tikekar (1999); Tikekar and Thomas (2005),  Thomas et al. (2005) developed superdense anisotropic distributions on pseudo-spheroidal spacetimes.  Thomas and Ratanpal (2007) studied non-adiabatic gravitational collapse of anisotropic distribution of matter accompanied by radial heat flux.  Dev and Gleiser (2002, 2003, 2004) have studied the impact of anisotropy on the stability of stellar configuration. Anisotropic distributions of matter incorporating linear equation of state have been studied by  Sharma and Maharaj (2007), &  Thirukkanesh and Maharaj (2008).  Komathiraj and Maharaj (2007) have studied charged distribution using linear equation of state.  Sunzu et al. (2014) studied charged anisotropic quark stars using linear equation of state. Anisotropic distributions of matter incorporating quadratic equation of state have been given by  Feroze and Siddiqui (2011) &  Maharaj and Takisa (2012).  Varela et al. (2010) used linear and non-linear equations of state for describing charged anisotropic distributions of matter.  Paul et al. (2011) have shown, in the MIT bag model of quark stars, that anisotropy can affect the bag constant. Polytropic equations of state has been used by  Thirukkanesh and Ragel (2012) &  Maharaj and Takisa (2013b).  Malaver (2013a, b, 2014) and  Thirukkanesh and Ragel (2014) have used modified Van der Waals equation of state for describing anisotropic charged compact stars.

Recently  Pandya et al. (2015) have developed anisotropic models of compact stars compatible with observational data by generalizing  Finch and Skea (1989) ansatz. The anisotropic stellar model given by  Sharma and Ratanpal (2013) is a subclass of the model of  Pandya et al. (2015). This model accommodates the observational data of a variety of compact objects recently studied by researchers. In the present article, we have obtained a new class of anisotropic stellar model of compact objects on the background of pseudo-spheroidal spacetimes. The physical parameter and geometric parameter appearing in the model are restricted as a result of various physical acceptability conditions imposed on the model. Another geometric parameters of the model plays the role of the radius of the spherical distribution of matter. It is found that our model yields values of different physical quantities that are in good agreement with the most recently available observational data of compact objects ( Gangopadhyay et al. (2013)) like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, SMC X-4 and Cen X-3.

We have organized the paper as follows: In section  2, we have solved the field equations and obtained restrictions on the model parameters using various physical requirements and energy conditions. The bounds for the model parameters and are obtained in section  3. In section  4, we have shown that our model is compatible with recent observational data of a number of compact objects ( Gangopadhyay et al. (2013)). The main results obtained in the present work is discussed in section  5.

2 Spacetime metric

A three-pseudo spheroid immersed in four-dimensional Euclidean space has the Cartesian equation

The sections are spheres of real or imaginary radius according as or while the sections , and are respectively, hyperboloids of two sheets.
On taking the parametrization

(1)

the Euclidean metric

takes the form

(2)

where and . The metric (2) is regular for all points with and call pseudo-spheroidal metric (Tikekar and Thomas (1998)).
We take the interior metric describing the anisotropic matter distribution in the form

(3)

where, , are geometric parameters and . This spacetime, generally known as pseudo-spheroidal spacetime, has been studied by many researchers (Tikekar and Thomas (1998); Tikekar (1999); Tikekar and Thomas (2005); Thomas et al. (2005); Thomas and Ratanpal (2007); Paul et al. (2011); Chattopadhyay and Paul (2010); Chattopadhyay et al. (2012)).
Following  Maharaj and Maartens (1989), we write the energy-momentum tensor for anisotropic matter distribution in the form

(4)

where, and denote the proper density, fluid pressure and unit four-velocity of the fluid, respectively.
The anisotropic stress-tensor is given by

(5)

where, is a radial vector and denotes the magnitude of the anisotropic stress.
The non-vanishing components of the energy-momentum tensor are given by

(6)

Hence the radial and transverse pressures are given by

(7)
(8)

Then the magnitude of the anisotropic stress has the form

(9)

The physical and geometric variables, related through Einstein’s field equations

(10)

are to be determined from the following set of three equations:

(11)
(12)
(13)

where a prime denotes a differentiation with respect to . The equations (11) – (13) can be couched in the form

(14)
(15)
(16)

where

(17)

The energy-density and the mass within the radius have expressions

(18)
(19)

It can be easily obtained from equation (18) that

(20)

indicating that the density decreases radially outward.
In order to obtain the metric potential , we assume an expression for in equation (15), in the form

(21)

The radial pressure in the present form vanishes at and takes the value at the centre It is non-negative for all values of in the range Further, on differentiating equation (21) with respect to , we get

(22)

indicating that the pressure decreases radially outward. Since the geometric parameter takes the role of the boundary radius of the distribution. With this choice of , equation (15) can be integrated to obtain in the form

(23)

where is a constant of integration.
Therefore, the spacetime metric takes the explicit form

(24)

The constant of integration can be obtained by matching the interior spacetime metric (3) with the Schwarzschild exterior metric

(25)

across the boundary This gives

(26)

and

(27)

The expression for anisotropy is now readily available by substituting for , and in the equation (16).

(28)

where

(29)
(30)

It is easy to see that vanishes at origin which is a desired requirement for anisotropic distributions ( Murad (2013a),  Murad and Saba (2013b, c, 2014a) &  Bowers and Liang (1974)).
The expression for transverse pressure

(31)

can be obtained using equations (21) and (28).
Moreover, the condition will lead to the following inequality at

(32)

whereas at , the condition is evidently satisfied.
The expressions for and are given by

(33)
(34)

The conditions and at , respectively, give the inequalities

(35)

and

(36)

Similarly the above conditions at , respectively, give

(37)

and

(38)

The adiabatic index

(39)

has the explicit expression

(40)

The necessary condition for the model to represent a relativistic star is that throughout the star. at impose a condition on , viz.,

(41)

The strong energy condition at and respectively, give the following two inequalities

(42)

and

(43)

In order to obtain a valid range for the parameters and , we have to consider the inequalities (35) – (38) and (40) – (43) simultaneously.

3 Bounds for Model Parameters

The pseudo-spheroidal space-time model developed for anisotropic matter distribution contains a physical parameter related to the central pressure and two geometric parameters, viz., and . Since , the free parameter represents the radius of the distribution. The bounds for the other two parameters and are to be determined by the following requirements a physically acceptable model is expected to satisfy in its region of validity, .

    [label=0)]
  1. ;

  2. ;

  3. ;

  4. ;

  5. The adiabatic index

The conditions are automatically satisfied by equations (18), (21), (20), (22).
We have displayed in Table  1 the bounds on in terms of the parameter at the centre and on the boundary.

Physical requirements at at
Automatically satisfied
Table 1: Bounds for .

We have displayed the numerical values of the lower and upper bounds of for different values of in Table  2. We have considered the maximum of all lower limits of and minimum of all its upper limits. The admissible values of are those for which minimum of upper limit minus maximum of lower limit is positive. This condition restricts the values of in the range It is further observed that for and satisfies, respectively, the inequalities and . The shaded region in Figure  1 gives the permissible values of and Any values of and outside this region may violate one or other of the physical requirements of the model.

Lower Limit for Max Upper Limit for Min Min - Max
\bigstrut
2 0.33 -2.50 0.11 0.33 0.31 1 2.5 2.33 1.63 3.30 0.31 -0.02
2.05 0.35 -2.61 0.12 0.35 0.35 1.05 2.625 2.49 1.72 3.54 0.35 0.00
2.1 0.37 -2.71 0.12 0.37 0.39 1.1 2.75 2.65 1.82 3.78 0.39 0.02
2.4 0.47 -3.28 0.18 0.47 0.66 1.4 3.5 3.66 2.40 5.32 0.66 0.19
2.8 0.60 -3.83 0.26 0.60 1.17 1.8 4.5 5.17 3.21 7.40 1.17 0.57
3 0.67 -4.00 0.30 0.67 1.50 2 5 6.00 3.63 8.40 1.50 0.83
3.1 0.70 -4.06 0.32 0.70 1.68 2.1 5.25 6.43 3.84 8.88 1.68 0.98
3.2 0.73 -4.09 0.35 0.73 1.88 2.2 5.5 6.87 4.05 9.35 1.88 1.14
3.4 0.80 -4.10 0.39 0.80 2.30 2.4 6 7.79 4.48 10.23 2.30 1.50
3.47 0.82 -4.08 0.40 0.82 2.47 2.47 6.175 8.12 4.63 10.53 2.47 1.64
3.8 0.93 -3.81 0.48 0.93 3.33 2.8 7 9.75 5.34 11.79 2.80 1.87
4 1.00 -3.50 0.52 1.00 3.94 3 7.5 10.80 5.78 12.46 3.00 2.00
4.2 1.07 -3.07 0.57 1.07 4.61 3.2 8 11.89 6.22 13.05 3.20 2.13
4.4 1.13 -2.52 0.62 1.13 5.35 3.4 8.5 13.02 6.66 13.58 3.40 2.27
4.8 1.27 -0.99 0.71 1.27 7.04 3.8 9.5 15.41 7.55 14.45 3.80 2.53
5 1.33 0.00 0.76 1.33 8.00 4 10 16.67 8.00 14.80 4.00 2.67
5.2 1.40 1.15 0.81 1.40 9.04 4.2 10.5 17.97 8.45 15.10 4.20 2.80
5.24 1.41 1.40 0.82 1.41 9.26 4.24 10.6 18.23 8.54 15.15 4.24 2.83
5.4 1.47 2.46 0.85 2.46 10.16 4.4 11 19.31 8.90 15.35 4.40 1.94
5.67 1.56 4.52 0.92 4.52 11.82 4.67 11.675 21.18 9.52 15.63 4.67 0.15
5.69 1.56 4.69 0.92 4.69 11.95 4.69 11.725 21.32 9.56 15.64 4.69 0.00
5.71 1.57 4.85 0.93 4.85 12.08 4.71 11.775 21.46 9.61 15.66 4.71 -0.14
Table 2: Permissible values of and
Figure 1: Permissible values of and

4 Compact Star Models

In order to validate the model, we examine our model with observational data. We have considered the pulsar 4U 1820-30 whose estimated mass and radius are and 9.1 km ( Gangopadhyay et al. (2013)). If we set these values for mass and radius then from equation (26) we obtain the value of which is well inside the valid range for . Similarly assuming masses of some well studied compact stars like PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, SMC X-4 and Cen X-3, we have obtained the same radius calculated by  Gangopadhyay et al. (2013) for values of in the valid range. The values of mass, radius, and other relevant quantities like central density density at the boundary , the compactification parameter and at the centre for are shown in Table  3.

STAR
(Km) (MeV fm) (MeV fm)
4U 1820-30 3.100 1.58 9.1 2290.97 277.12 0.256 0.206
PSR J1903+327 3.176 1.667 9.438 2129.82 257.62 0.261 0.199
4U 1608-52 3.458 1.74 9.31 2188.78 267.75 0.276 0.176
Vela X-1 3.407 1.77 9.56 2075.80 251.08 0.273 0.179
PSR J1614-2230 3.997 1.97 9.69 2020.48 244.39 0.300 0.144
SMC X-4 2.514 1.29 8.831 2432.67 294.25 0.215 0.285
Cen X-3 2.838 1.49 9.178 2252.20 272.42 0.239 0.235
Table 3: Estimation of physical values based on observational data for .

In order to examine the nature of various physical quantities throughout the distribution, we have considered a particular star 4U 1820-30 for which mass radius the physical parameter and the geometric parameter . We have shown the variation of density and pressures in Figure  2 and Figure  3, respectively. It is observed that the transverse pressure is less than the radial pressure for in the range Subsequently dominates in the region . The radial pressure vanishes at In Figure  4, we have shown the variation of anisotropy throughout the distribution. The variations of sound speed in the radial and transverse directions are shown in Figure  5. From Figure  6, it is evident that the strong energy condition, is satisfied throughout the distribution. Though we have not assumed any explicit expression for the EOS in our model, we have shown the nature of variation of pressures and against density in Figure  7. For a relativistic model to be stable in its region of validity, we must have the adiabatic index The variation against radius is shown in Figure  8. It is clear from Figure 8 that throughout the star. The variation of gravitational red shift, in the radial direction is shown in Figure 9. It is easy to note that the red shift is monotonically decreasing function from the centre to boundary. Further, the red shift at the centre and on the boundary are both positive and finite.

Figure 2: Density Vs Radius
Figure 3: Radial and Transverse Pressures Vs Radius
Figure 4: Anisotropy Vs Radius
Figure 5: Vs Radius
Figure 6: Strong Energy Condition Vs Radius
Figure 7: Equation of State
Figure 8: Adiabatic Index Vs Radius
Figure 9: Gravitational Red Shift

5 Discussion

Spherical distribution of matter on pseudo-spheroidal spacetimes have been studied by a number of researchers in the recent past  Tikekar and Thomas (1998); Tikekar (1999); Tikekar and Thomas (2005); Thomas et al. (2005); Thomas and Ratanpal (2007); Paul et al. (2011); Chattopadhyay and Paul (2010); Chattopadhyay et al. (2012). In this paper, we have obtained a new class of solutions to Einstein’s field equations for a spherically symmetric anisotropic distribution of matter and have shown that our model can fit to the observational data of a number of well studied pulsars ( Gangopadhyay et al. (2013)). On assuming a particular form of radial pressure and on the basis of elementary criteria for physical acceptability of a compact spherically symmetric distribution of matter, we have obtained the bounds for the physical as well as geometric parameters of the model. It is found that our model can accommodate a number of pulsars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, SMC X-4 and Cen X-3. We also have studied, in detail, a particular pulsar 4U 1820-30, and have shown graphically the profile of different physical quantities throughout the distribution. In short, study of compact stars on the background of pseudo-spheroidal spacetime is highly interesting in the sense that it generates models compatible with observational data and at the same time having a definite 3-space geometry, namely, pseudo-spheroidal geometry which many other spacetimes may not possess.

Acknowledgements

The authors would like to thank IUCAA, Pune for the facilities and hospitality provided them for carrying out this work.

References

  1. Murad M. H., Astrophys. Space Sci. 343 (2013) 187. doi: 10.1007/s10509-012-1258-4.
  2. Murad M. H. and Saba F., Astrophys. Space Sci. 343 (2013) 587. doi: 10.1007/s10509-012-1277-1.
  3. Murad M. H. and Saba F., Astrophys. Space Sci. 344 (2013) 69. doi: 10.1007/s10509-012-1320-2.
  4. Murad M. H. and Saba F., arXiv:1408.5126v2 (2014).
  5. Maurya S. K., Gupta Y. K., Ray S. and Chowdhury S. R. arXiv:1506.02498v1 (2015).
  6. Sharma R. and Ratanpal B. S., Int. J. Mod. Phys. D 13 (2013) 1350074.doi: 10.1142/S0218271813500740.
  7. Vaidya P. C. and Tikekar R. , J. Astrophys. Astron. 3, (1982) 325. doi: 10.1007/BF02714870.
  8. Tikekar R. S., Journal of Mathematical Physics 31 (1990) 2454. doi: 10.1063/1.528851.
  9. Finch M. R. and Skea J. E. F., Class. Quantum Grav. 6 (1989) 467. doi: 10.1088/0264-9381/6/4/007.
  10. Tikekar R. and Jotania K., Int. J. Mod. Phys. D 14 (2005) 1037. doi: 10.1142/S021827180500722X.
  11. Tikekar R. and Thomas V. O., Pramana J. Phys. 50 (1998) 95. doi: 10.1007/BF02847521.
  12. Ruderman R., Astro. Astrophys. 10 (1972) 427. doi: 10.1146/annurev.aa.10.090172.002235
  13. Canuto V., Annu. Rev. Astron. Astrophys. 12 (1974) 167. doi: 10.1146/annurev.aa.12.090174.001123.
  14. Bowers R. and Liang E., Astrophys. J. 188 (1974) 657. doi: 10.1086/152760.
  15. Maharaj S. D. and Maartens R., Gen. Relativ. Grav. 21 (1989) 899. doi: 10.1007/BF00769863.
  16. Gokhroo M. K. and Mehra A. L., Gen. Rel. Grav 26 (1994) 75. doi: 10.1007/BF02088210.
  17. Tikekar R. and Thomas V. O., Pramana J. Phys. 52 (1999) 237. doi: 10.1007/BF02828886.
  18. Tikekar R. and Thomas V. O., Pramana J. Phys. 64 (2005) 5. doi: 10.1007/BF02704525.
  19. Thomas V. O., Ratanpal B. S. and Vinodkumar P. C., Int. J. Mod. Phys. D 14 (2005) 85. doi: 10.1007/s12043-012-0268-7.
  20. Thomas V. O. and Ratanpal B. S., Int. J. Mod. Phys. D 16 (2007) 9. doi: 10.1142/S0218271805005852.
  21. Dev K. and Gleiser M., Gen. Rel. Grav. 34 (2002) 1793. doi: 10.1023/A:1020707906543.
  22. Dev K. and Gleiser M., Gen. Rel. Grav. 35 (2003) 1435.10. doi: 1023/A:1024534702166.
  23. Dev K. and Gleiser M., Int. J. Mod. Phys. D 13 (2004) 1389. doi: 10.1142/S0218271804005584.
  24. Sharma R. and Maharaj S. D., Mon. Not. R. Astron. Soc. 375 (2007) 1265. doi: 10.1111/j.1365-2966.2006.11355.x.
  25. Thirukkanesh S. and Maharaj S. D., Class. Quantum Grav. 25 (2008) 235001. doi: 10.1088/0264-9381/25/23/235001.
  26. Komathiraj K. and Maharaj S. D., Intenational Journal of Modern Physics D 16 (2007) 1803. doi: 10.1142/S0218271807011103.
  27. Sunzu J. M., Maharaj S. D., Ray S., Astrophys. Space Sci. 352 (2014) 719. doi: 10.1007/s10509-014-1918-7.
  28. Feroze T. and Siddiqui A. A., Gen. Relativ. Grav. 43 (2011) 1025. doi: 10.1007/s10714-010-1121-2.
  29. Maharaj S. D. and Takisa P. M., Gen. Relativ. Grav. 44 (2012) 1419. doi: 10.1007/s10714-012-1347-2
  30. Varela V., Rahaman F., Ray S., Chakraborty K. and Kalam M., Phys. Rev. D 82 (2010) 044052. doi: 10.1103/PhysRevD.82.044052.
  31. Paul B. C., Chattopadhyay P. K., Karmakar S. and Tikekar R., Mod. Phys. Lett. A 26 (2011) 575. doi: 10.1142/S0217732311034943.
  32. Thirukkanesh S., Ragel F. S., Pramana J. Phys. 78 (2012) 687. doi: 10.1007/s12043-012-0268-7.
  33. Maharaj S. D. and Takisa P. M., Gen. Relativ. Grav. 45 (2013b) 1951. doi: 10.1007/s10714-013-1570-5.
  34. Malaver M., American Journal of Astronomy and Astrophysics 1 (2013) 41. doi: 10.11648/j.ajaa.20130104.11.
  35. Malaver M., World Applied Programming 3 (2013) 309.
  36. Malaver M., Frontiers of Mathematics and its Applications 1 (2014) 9. doi: 10.12966/fmia.03.02.2014.
  37. Thirukkanesh S., Ragel F. S., Pramana J. Phys. 83 (2014) 83. doi: 10.1007/s12043-014-0766-x.
  38. Pandya D. M., Thomas V. O., Sharma R., Astrophys. Space Sci. 356 (2015) 285. doi: 10.1007/s10509-014-2207-1.
  39. Gangopadhyay T., Ray S., Li X-D., Dey J. and Dey M., Mon. Not. R. Astron. Soc. 431 (2013) 3216. doi: 10.1093/mnras/stt401.
  40. Chattopadhyay P. C. and Paul B. C., Pramana- j. of phys. 74 (2010) 513. doi: 10.1007/s12043-010-0046-3.
  41. Chattopadhyay P. C., Deb R. and Paul B. C., International Journal of Modern Physics D 21 (2012) 1250071. doi:10.1142/S021827181250071X.
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