1 Introduction

Study of compact objects takes much attention to the astrophysicists for the last few years. The possible compact objects are Neutron Star, Pulsar, Strange Star etc. Neutron stars are created almost entirely of neutrons, while strange stars are of quark matter, or strange matter. Generally Pulsar is a subclass of Neutron stars. Pulsar in a compact relativistic binary system has become an interesting compact object to the astrophysicists for the last few years. In 1974, Joseph Hooton Taylor, Jr. and Russell Hulse discovered for the first time a pulsar PSR B1913+16 in a binary system. PSR B1913+16 is a pulsar (a radiating neutron star) which together with another neutron star is in orbit around a common center of mass, thus forming a binary star system. This pulsar orbits another neutron star with an orbital period of just eight hours. In 2013 Antoniadis et al. (Antoniadis et al. 2013) measured the mass of the Pulsar J0348+0432 in a compact relativistic binary by radio-timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion, which is in a 2.46-hour orbit. They used these to derive the component masses and orbital parameters, infer the system’s motion. Antoniadis et al. find that the white dwarf has a mass of \(0.172 \pm0.003 M_{\odot}\) which, combined with orbital velocity measurements, yields a pulsar mass of \(2.01 \pm0.04 M_{\odot}\). Additionally, over a span of 2 years, they observed a significant decrease in the orbital period, \(\dot{P}^{obs} _{b} = -8.6 \pm1.4~\upmu \mbox{s}\,\mbox{year}^{-1}\) in their radio timing data. Many researchers studied (Aziz et al. 2016; Bronnikov and Fabris 2006; Bhar et al. 2016; Bhar and Murad 2016; Dymnikova 2002; Deb et al. 2012; Egeland 2007; Hossein et al. 2012; Kalam et al. 2012, 2013a, 2013b, 2014a, 2014b, 2016a, 2016b, 2016c, 2017; Lobo 2006; Maurya et al. 2015, 2016, 2017; Maharaj et al. 2014, 2016; Ngubelanga and Maharaj 2015; Paul et al. 2015; Pant et al. 2016; Pradhan and Pant 2014; Rahaman et al. 2012a, 2012b) compact objects in various directions. Scientists used different techniques such as computational, observational or theoretical analysis to study astrophysical objects. Finch and Skea (1989) proposed an isotropic static, spherically symmetric perfect fluid spacetime which results from an ansatz proposed by Duorah and Ray (1987). Finch and Skea had given the correct solution following from the ansatz of Duorah and Ray. They analysis throughout the star and shows that the solution is physical within its Schwarzschild radius. This solution may be useful to study the isotropic star since the equation of state which connects the density of the compact object with its pressure is relatively simple. Motivating with this fact, we want to explain the physical behavior of a massive pulsar J0348+0432 (Antoniadis et al. 2013) in a compact relativistic binary. Our main objective is to estimate the radius of the pulsar.

Therefore, considering the isotropic model, we want to explain the physical behavior of the compact object (Pulsar J0348+0432).

We have organized the article as follows: In Sect. 2, we have discussed the interior space-time of the compact object. In Sect. 3, we have studied some special features of the compact object namely, density and pressure behavior of the compact object, energy conditions, matching conditions, TOV equation, stability, adiabatic index, mass-radius relation, compactness, surface redshift in different sub-sections. The article is concluded with a short discussion with numerical data in Sect. 4.

2 Interior spacetime of the compact object

In spherically symmetric matter distribution the interior space time is:

$$\begin{aligned} ds^{2} = -e^{\nu(r)}\,dt^{2} + e^{\lambda(r)}\,dr^{2} + r^{2} \bigl(d\theta ^{2} +\sin^{2} \theta d\phi^{2}\bigr) \end{aligned}$$
(1)

Einstein’s field equations for the metric (1) accordingly are obtained as (\(c=1,G=1\))

$$\begin{aligned} 8\pi\rho =& e^{-\lambda} \biggl[ \frac{\lambda^{\prime}}{r}-\frac{1}{r ^{2}} \biggr] +\frac{1}{r^{2}}, \end{aligned}$$
(2)
$$\begin{aligned} 8\pi p =& e^{-\lambda} \biggl[ \frac{\nu^{\prime}}{r}+\frac{1}{r^{2}} \biggr] - \frac{1}{r ^{2}}. \end{aligned}$$
(3)

In our stellar model, we consider the interior space-time as:

$$\begin{aligned} ds^{2} =& -D^{2}\bigl[ \bigl( B-A \sqrt{1+Cr^{2}} \bigr) \cos\sqrt{1+Cr ^{2}} \\ &{}+ \bigl( A+B\sqrt{1+Cr^{2}} \bigr) \sin\sqrt{1+Cr^{2}} \bigr]^{2}dt^{2} \\ &{}+ \bigl( 1+Cr^{2} \bigr) dr^{2} +r^{2}d \varOmega^{2} \end{aligned}$$
(4)

Finch and Skea (1989) proposed such type of metric (4) where \(A\), \(B\), \(C\) and \(D\) are constants.

We assume that the energy-momentum tensor for the interior of the compact object has the standard form

$$ T_{\nu}^{\mu}= ( -\rho, p, p, p) $$
(5)

where \(\rho\) and \(p\) are energy-density and isotropic pressure respectively.

Now from the above metric (4) we can write,

$$\begin{aligned}& e^{-\lambda} = \bigl(1+Cr^{2}\bigr)^{-1} \end{aligned}$$
(6)
$$\begin{aligned}& \lambda^{\prime} = \frac{2Cr}{(1+Cr^{2})} \end{aligned}$$
(7)
$$\begin{aligned}& \nu^{\prime} = 2Cr \bigl[ B\cos\sqrt{1+Cr^{2}}+A\sin\sqrt{1+Cr^{2}} \bigr] \\& \phantom{\nu^{\prime} =}{}\times\bigl( \bigl( B-A\sqrt{1+Cr^{2}} \bigr) \cos\sqrt{1+Cr^{2}} \\& \phantom{\nu^{\prime} =}{}+ \bigl( A+B\sqrt{1+Cr^{2}} \bigr) \sin\sqrt{1+Cr^{2}} \bigr)^{-1} \end{aligned}$$
(8)

Einstein’s field equations for the metric (4) accordingly are obtained as (\(c=1,G=1\))

$$\begin{aligned}& 8\pi\rho = \frac{C(3+Cr^{2})}{(1+Cr^{2})^{2}} \end{aligned}$$
(9)
$$\begin{aligned}& 8\pi p = \\& \frac{[ \frac{2C [ B\cos\sqrt{1+Cr^{2}}+A\sin\sqrt{1+Cr ^{2}} ] }{ [ ( B-A\sqrt{1+Cr^{2}} ) \cos\sqrt{1+Cr ^{2}}+ ( A+B\sqrt{1+Cr^{2}} ) \sin\sqrt{1+Cr^{2}} ] } +\frac{1}{r^{2}} ]}{(1+Cr^{2})} \\& \quad-\frac{1}{r^{2}} \end{aligned}$$
(10)

3 Exploration of physical properties

In this section we will investigate the following features of the compact object:

3.1 Density and pressure behavior of the compact object

To check whether at the center, the matter density and pressure dominates or not, we see the following:

$$\begin{aligned}& \frac{d\rho}{dr} = - \frac{C^{2} r(5+Cr^{2})}{4\pi(1+Cr^{2})^{3}} < 0, \\& \frac{d\rho}{dr} (r=0) = 0, \\& \frac{d^{2} \rho}{dr^{2}} = \frac{C^{2}(-5+Cr^{2}(22+3Cr^{2}))}{4 \pi(1+Cr^{2})^{4}} \\& \frac{d^{2} \rho}{dr^{2}}(r=0) = -\frac{5C^{2}}{4\pi} < 0. \end{aligned}$$

and

$$\begin{aligned}& \frac{dp}{dr} = -C^{2} r \bigl[ B\cos\sqrt{1+Cr^{2}}\\& \phantom{\frac{dp}{dr} =}{}+ A \sin\sqrt{1+Cr ^{2}} \bigr] \bigl(- \bigl[ A+ACr^{2} \\& \phantom{\frac{dp}{dr} =}{}-B\sqrt{1+Cr^{2}} \bigl(2+Cr^{2}\bigr) \bigr] \cos\sqrt{1+Cr ^{2}} \\& \phantom{\frac{dp}{dr} =}{} + \bigl[ B+BCr^{2} +A\sqrt{1+Cr^{2}} \bigl(2+Cr^{2} \bigr) \bigr] \\& \phantom{\frac{dp}{dr} =}{}\times\sin\sqrt{1+Cr ^{2}} \bigr) \bigl(4\pi\bigl(1+Cr^{2}\bigr)^{5/2} \\& \phantom{\frac{dp}{dr} =}{}\times\bigl( \bigl( B-A \sqrt{1+Cr^{2}} \bigr) \cos\sqrt{1+Cr^{2}} \\& \phantom{\frac{dp}{dr} =}{} + \bigl( A+B\sqrt{1+Cr^{2}} \bigr) \sin\sqrt{1+Cr^{2}} \bigr)^{2}\bigr)^{-1} < 0, \\& \frac{dp}{dr} (r=0) = 0, \\& \frac{d^{2} p}{dr^{2}}(r=0) = < 0. \end{aligned}$$

Clearly, at the center, the density and pressure of the star is maximum and it decreases radially outward.

Thus, the energy density and the pressure are well behaved in the interior of the stellar structure.

Variations of the energy-density and pressure in our proposed model have been shown in Figs. 1 and 2, respectively. Here also energy-density and pressure gradients is negative (see Figs. 3 and 4). Equation of state (relation between pressure and density) of the compact object is shown in Fig. 5.

Fig. 1
figure 1

Variation of the energy-density (\(\rho\)) at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 2
figure 2

Variation of the pressure (\(p\)) at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 3
figure 3

Variation of the energy-density gradients (\(\frac{d\rho}{dr}\)) at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 4
figure 4

Variation of the pressure gradients (\(\frac{dp}{dr}\)) at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 5
figure 5

Equation of state (EOS) of the Pulsar J0348+0432 in our model shown above figure, where \(\alpha\) \((=-8.61631\times10^{-5})\) and \(\beta\) \((=0.31938)\) are constant (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km} ^{-2}\) respectively)

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3.2 Energy conditions

In our model, all the energy conditions, namely null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC), are satisfied at the center (\(r=0\)). It is evident from Figs. 1, 2 the following energy conditions holds good:

  1. (i)

    NEC: \(p_{0}+\rho_{0}\geq0\),

  2. (ii)

    WEC: \(p_{0}+\rho_{0}\geq0\), \(\rho_{0}\geq0\),

  3. (iii)

    SEC: \(p_{0}+\rho_{0}\geq0\), \(3p_{0}+\rho_{0}\geq0\),

  4. (iv)

    DEC: \(\rho_{0} > |p_{0}| \).

Where \(\rho_{0} =0.00095493~\mbox{km}^{-2}\) and \(p_{0}=0.00022828~\mbox{km}^{-2}\) (calculated using MAPLE software).

3.3 Matching conditions

Interior metric of the compact object will be matched to the Schwarzschild exterior solution at the boundary i.e., at \(r=b\).

$$\begin{aligned} ds^{2} =& - \biggl( 1-\frac{2M}{r} \biggr) dt^{2} + \biggl( 1-\frac {2M}{r} \biggr) ^{-1}dr^{2} \\ &{}+ r^{2} \bigl(d\theta^{2} +\sin^{2}\theta d \phi^{2}\bigr) \end{aligned}$$
(11)

For the continuity of the metric functions \(g_{tt}\), \(g_{rr} \) and \(\frac{\partial g_{tt}}{\partial r}\) at the boundary, we get

$$\begin{aligned}& D^{2}\bigl[ \bigl( B-A\sqrt{1+Cr^{2}} \bigr) \cos \sqrt{1+Cr^{2}} \\& \qquad+ \bigl( A+B\sqrt{1+Cr^{2}} \bigr) \sin\sqrt{1+Cr^{2}} \bigr]^{2} \\& \quad= 1 - \frac{2M}{b} \end{aligned}$$
(12)
$$\begin{aligned}& \bigl(1+Cr^{2}\bigr)= \biggl( 1-\frac{2M}{b} \biggr) ^{-1} . \end{aligned}$$
(13)

Now from Eq. (13), we get the compactification factor as

$$ u = \frac{M}{b} = \frac{Cb^{2}}{2+2Cb^{2}} $$
(14)

3.4 TOV equation

For fluid distribution, the generalized TOV equation has the form

$$ \frac{dp}{dr} +\frac{1}{2} \nu^{\prime} ( \rho+ p ) = 0. $$
(15)

The modified TOV equation describes the equilibrium condition for the compact object subject to effective gravitational (\(F_{g}\)) and effective hydrostatic (\(F_{h}\)) force nature of the stellar object as

$$ F_{h}+ F_{g} = 0, $$
(16)

where,

$$\begin{aligned} F_{g} =& -\frac{1}{2} \nu^{\prime} ( \rho+p ) \end{aligned}$$
(17)
$$\begin{aligned} F_{h} =& -\frac{dp}{dr} \end{aligned}$$
(18)

Therefore, the static equilibrium configurations do exist in the presence of gravitational and hydrostatic forces. Figure 6 shows the equilibrium condition under gravitational and hydrostatic forces.

Fig. 6
figure 6

The Behaviors of gravitational (\(F_{g}\)) and hydrostatic (\(F _{h}\)) forces at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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3.5 Stability

For a physically acceptable model, velocity of sound should be within the range \(0 \leq v^{2}=(\frac{dp}{d\rho})\leq1\) (Herrera 1992; Abreu et al. 2007). In our isotropic model, \(v^{2} \leq1\). We plot the sound speed of the Pulsar J0348+0432 in Fig. 7 and observed that it satisfies well the inequalities \(0\leq v^{2} \leq1\). Therefore our model is well stabled.

Fig. 7
figure 7

Variation of the sound speed at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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3.6 Adiabatic index

The dynamical stability of the stellar model against the infinitesimal radial adiabatic perturbation was introduced by Chandrasekhar (1964). Later this stability condition was developed and applied to astrophysical cases by Bardeen et al. (1966), Knutsen (1988), Mak and Harko (2013), gradually. Since the adiabatic index should be \(\gamma= \frac{\rho+p}{p} \frac{dp}{d\rho}> \frac{4}{3}\) within the isotropic stable star, we plot the adiabatic index for our compact object in Fig. 8 and observe that these parameter satisfy the condition \(\gamma> \frac{4}{3}\) everywhere within the star.

Fig. 8
figure 8

Variation of the adiabatic index \(\gamma\) at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km} ^{-2}\) respectively)

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3.7 Mass-radius relation and surface redshift

Here, we studied the maximum allowable mass-radius ratio. According to Buchdahl (1959), for a static spherically symmetric perfect fluid, allowable mass-radius ratio is given by \(\frac{ Mass}{Radius} < \frac{4}{9}\). In our model, the gravitational mass (\(M\)) in terms of the energy density \(\rho\) can be expressed as

$$ M=4\pi \int^{b}_{0} \rho r^{2} dr = \frac{Cb^{3}}{2+2Cb^{2}} $$
(19)

The compactness, \(u\) is given by

$$ u= \frac{ M(b)}{b}= \frac{Cb^{2}}{2+2Cb^{2}} $$
(20)

The nature of the mass and compactness of the pulsar are shown in Figs. 9 and 10.

Fig. 9
figure 9

Nature of the mass function at the stellar interior of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 10
figure 10

Variation of \(u\) against radial parameter \(r\) of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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The surface redshift (\(Z_{s}\)) corresponding to the above compactness (\(u\)) is as follows:

$$ 1+Z_{s}= \bigl[ 1-(2 u ) \bigr] ^{-\frac{1}{2}} , $$
(21)

where

$$ Z_{s}= \frac{1}{\sqrt{1+Cb^{2}}}-1 $$
(22)

Therefore the maximum surface redshift for Pulsar J0348+ 0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively) is 0.43 (Fig. 11).

Fig. 11
figure 11

Variation of the red-shift function (\(Z_{s}\)) against radial parameter \(r\) of the Pulsar J0348+0432 (taking \(A=0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km} ^{-2}\) respectively)

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4 Conclusion

In this article, we have investigated the nature of the massive pulsar J0348+0432 (Antoniadis et al. 2013) in a compact relativistic binary by considering it as isotropic in nature and the space-time of it to be described by Finch and Skea (1989) metric.

The results are quite interesting: Density and pressure of the pulsar J0348+0432 are well behaved (Figs. 1, 2, 3 and 4). Our model satisfies stellar equation (TOV) and all the energy conditions namely null energy conditions (NEC), weak energy conditions (WEC), strong energy conditions (SEC) and dominated energy conditions (DEC) (Figs. 1, 2 and 6). We also obtain the mass function (\(M\)), compactness (\(u\)) and surface redshift (\(Z_{s}\)). Also our model is well stable according to Herrera stability condition (Herrera 1992). Antoniadis et al. (2013) measured the mass of the pulsar J0348+0432 by radio-timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion, which is in a 2.46-hour orbit. They used these to derive the component masses and orbital parameters, infer the system’s motion, and constrain its age and they find that the white dwarf has a mass of \(0.172 \pm0.003 M_{\odot}\), which, combined with orbital velocity measurements, yields a pulsar mass of \(2.01 \pm0.04 M_{\odot}\). Taking into account, the mass of the pulsar obtained by Antoniadis et al., we consider the values of the constants \(A = 0.01307 \), \(B = 0.012 \) and \(C = 0.008~\mbox{km}^{-2} \), such that the pressure drops from its maximum value (at center) to zero at the boundary. From mass-radius relation (Fig. 9), any interior features of the pulsar can be evaluated. According to our model, the possible radius of the pulsar J0348+0432 is found to be \(11.4~\mbox{km} \leq \mathit{Radius}\ (b) \leq11.64~\mbox{km}\) (Fig. 12) which is quite significant. The equation of state of the pulsar is found to be soft (Fig. 5). The EoS would be like \(p = \alpha+ \beta\rho\), where \(\alpha\) and \(\beta\) are constants having the numerical values \(\alpha=-8.61631\times10^{-5}\) and \(\beta=0.31938\). The surface redshift (\(Z_{s}\)) for the pulsar is also found to be 0.43 (Radius \(b = 11.52~\mbox{km}\), \(A = 0.01307\), \(B = 0.012\) and \(C = 0.008~\mbox{km}^{-2}\)) which is \(\leq0.8\) that implies the validity of our model (Fig. 11) (Haensel et al. 2000; Buchdahl 1959). It is to be mentioned here that while solving Einstein’s field equations, we set \(c=G=1\). Now, plugging \(G\) and \(c\) into the relevant equations, the values of the central density, surface density and central pressure of our pulsar turn out to be \(\rho_{0} = 1.2891 \times10^{15}~\mbox{gm}/\mbox{cm}^{3}\), \(\rho_{b} = 0.4106 \times10^{15}~\mbox{gm}/\mbox{cm}^{3}\) and \(p_{0} = 5.6013 \times10^{35}~\mbox{dyne}/\mbox{cm}^{2}\) for the numerical values of the parameters \(b = 11.52~\mbox{km}\), \(A = 0.01307 \), \(B = 0.012 \) and \(C = 0.008~\mbox{km}^{-2} \). It is also worthy to be mention here that our model is well applicable to other Pulsars also such as PSR J1614-2230, PSR J1903+0327, PSR J0045-7319 and PSR J1023+0038 (pl. see Fig. 13).

Fig. 12
figure 12

Possible radius of the Pulsar J0348+0432 (taking \(A= 0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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Fig. 13
figure 13

Application of our model for other Pulsars (taking \(A= 0.01307\), \(B=0.012\) and \(C=0.008~\mbox{km}^{-2}\) respectively)

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We actually considering a special metric to describe the pulsar J0348+0432, where the involved metric parameters \(A\), \(B\), \(C\) are evaluated by considering all kinds of required conditions. When the values of these parameters are known, the equation of state as well as the central density are fixed. Conventionally, the mass-radius curve are calculated under a given equation of state for various values of central density; by a given value of the central density, the mass and radius of a compact star are fixed. But our model is different and theoretically interesting. According to our model, J0348+0432 and some other pulsars such as J1614-2230, J1903+0327, J0045-7319 and J1023+0038 etc. have the same values of \(A\), \(B\), \(C\), and consequently the same central density and the same equation of state. More interestingly, if we starts from the center with a certain central density, the structure of a pulsar can be determined by stopping at any radius where pressure becomes zero.