Anisotropic stellar structures in the f(T) theory of gravity with quintessence via embedding approach

  • This work suggests a new quintessence anisotropic compact stars model in the $f(T)$ gravity by using the off-diagonal tetrad and the power-law as $f(T)=\beta T^n$, where T being the scalar torsion and $\beta$ and n are some real constants. The acquired field equations incorporating the anisotropic matter source along with the quintessence field, in the $f(T)$ gravity, are investigated by making use of the specific character of the scalar torsion T for the observed stars ${\rm{PSRJ1614}}-2230$, $4U 1608-52$, ${\rm{Cen}} X-3$, ${\rm{EXO1785}}-248$, and $SMC X-1$. It is suggested that all the stellar structures under examination are advantageously independent of any central singularity and are stable. Diverse physical features which are crucially important for the emergence of the stellar structures are conferred with comprehensive graphical analysis.
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Allah Ditta, Mushtaq Ahmad, Ibrar Hussain and G. Mustafa. Anisotropic stellar structures in the f(T) theory of gravity with quintessence via embedding approach[J]. Chinese Physics C.
Allah Ditta, Mushtaq Ahmad, Ibrar Hussain and G. Mustafa. Anisotropic stellar structures in the f(T) theory of gravity with quintessence via embedding approach[J]. Chinese Physics C. shu
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Anisotropic stellar structures in the f(T) theory of gravity with quintessence via embedding approach

    Corresponding author: Allah Ditta, mradshahid01@gmail.com
    Corresponding author: Mushtaq Ahmad, mushtaq.sial@nu.edu.pk
    Corresponding author: Ibrar Hussain, ibrar.hussain@seecs.nust.edu.pk
    Corresponding author: G. Mustafa, gmustafa3828@gmail.com
  • 1. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
  • 2. National University of Computer and Emerging Sciences, Chiniot-Faisalabad Campus, Pakistan
  • 3. School of Electrical Engineering and Computer Science, National University of Sciences and Technology, H-12, Islamabad, Pakistan
  • 4. Department of Mathematics, Shanghai University, Shanghai, 200444, Shanghai, People’s Republic of China

Abstract: This work suggests a new quintessence anisotropic compact stars model in the $f(T)$ gravity by using the off-diagonal tetrad and the power-law as $f(T)=\beta T^n$, where T being the scalar torsion and $\beta$ and n are some real constants. The acquired field equations incorporating the anisotropic matter source along with the quintessence field, in the $f(T)$ gravity, are investigated by making use of the specific character of the scalar torsion T for the observed stars ${\rm{PSRJ1614}}-2230$, $4U 1608-52$, ${\rm{Cen}} X-3$, ${\rm{EXO1785}}-248$, and $SMC X-1$. It is suggested that all the stellar structures under examination are advantageously independent of any central singularity and are stable. Diverse physical features which are crucially important for the emergence of the stellar structures are conferred with comprehensive graphical analysis.

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    I.   INTRODUCTION
    • Einstein's General Relativity (GR) has proven to be the most captivating success of the previous century. Supported by the observations [1], GR enlightens several problems connected not only to the solar system scale but to the cosmological scales as well. Numerous observational pieces of evidence from the Supernova type Ia [2, 3], the high-redshift of supernova [4], Planck data [5], the large-scale structure [610], etc., designate the accelerated expanding universe. An astonishing and contentious result from GR expects that a matter-dominated Universe (or radiation) negatively accelerates due to the existence of gravitational attraction. The accelerated expression of our Universe originates due to the dark energy (DE) [11], a mysterious galactic fluid containing a uniform density distribution, and a negative pressure, which GR cannot explain. The ambiguous behavior of the DE has stimulated several cosmologists to disclose its apparent attributes. Modified theories of gravity are viewed as a good and a hopeful decision to explain its nature.

      The DE is well thought-out to be repulsive exhibiting the negative pressure. The equation of state $ (EOS) $ describing the DE is $ p = \omega_q\rho $ such that $ \omega_{q}<0 $. The parameter $ \omega_q $ denotes the DE. For the expanding universe, the value of $ \omega_q $ must be restricted to $ \omega_{q}<-1/3 $. If $ \omega_q $ attains the bound $ -1<\omega_{q}<-1/3 $ then it is classified as the quintessence scalar field. The quintessence in the gravitational Physics, is a theoretical approach for the explanation of the DE. More precisely a scalar field, hypothesized as a description of observing an accelerating rate of expanding our Universe. The dynamical concept of the quintessence is quite different from the explanation of the DE as given by the cosmological constant in the Einstein field equations (EFEs), which is constant by definition i.e., that does not change with time. Quintessence can both behave as either attractive or repulsive liable on the proportion of its kinetic and the potential energy. The investigators working out this claim believe that the quintessence turns into repulsive around ten billion years ago, that is 3.5 billion years later the Big Bang had emerged. For the representation of the escalating Universe, many theories have been structured but the GR remains the source of unparalleled aspiration. In addition to its beautiful approach of enlightening diverse epochs of the Universe evolution, it has also broadened our emerging concepts of structuring gravity in the cosmos. But still, there exist some weaknesses in the GR which remain unaddressed. Buchdal [12], gave a modest concept of replacing the Ricci scalar R by a function $ f(R) $ in the EFEs owing to the emergence of modified theories of gravity, naming a few as, $ f(R) $, $ f(T) $, the Teleparallel theory of gravity, here T being the torsion scalar, $ f(R,{\cal{T}}) $ with $ {\cal{T}} $ as the trace of the energy-momentum tensor, $ f(G) $, and $ f(R,G) $ gravity, where G represents the Gauss-Bonnet $ (GB) $ invariant [13, 14, 1618], and has the representation $ G_{} = R^2+4R_{\mu\theta\phi\nu}R^{\mu\theta\phi\nu}-4R_{\mu\nu}R^{\mu\nu} $. These theories have enlightened the resolution of tackling the complexities involving the quantum gravity and to deliver the investigators with various platforms through which the reasons behind the accelerating expansion of our Universe have been discussed.

      The induction of the advanced technological approaches made numerous researchers to study the nature of the stellar compact objects by exploring their physical attributes [1924]. Typically, it is well-thought-out that these stellar bodies are made up of some perfect fluid. However, recent observations confirmed that the fluid pressure of massive celestial objects such as $ 4U1820-30 $, $ PSR J 1614-2230 $, and $ SAXJ1808.4-3658(SS1) $, etc. no more remains isotropic and rather behaves as anisotropic. Herreraa and Santos [25] have discussed the possible existence of the anisotropic fluid within the framework of self-gravitation by taking into consideration the examples of Newtonian theory and also of the GR. Herrera [26] has discussed the conditions for the stability of the isotropic pressure in the background of collapsing spherically symmetric, dissipative fluid distributions. Capozziello et al. [27], have presented the stellar compact structures possessing the hydrostatic equilibrium through the Lane-Emden equation formulated for the $ f(R) $ theory of gravity. Bowers and Liang [28], have investigated the locally anisotropic relativistic compact spheres through hydrostatic equilibrium and induced that the massive compact structures might be anisotropic in the presence of the fluxidity-superconductivity interaction. Capozziello et al. [29], have also studied the spherically symmetric solutions using the notion of the Noether Symmetries in the $ f(R) $ theory of gravity. Abbas et al.[30], have investigated the dynamical expressions by modeling anisotropic compact stars in the presence of the quintessence scalar field. Moreover, for their work, they have employed Krori-Barua and Starobinsky model in the $ f(R) $ theory of gravity. Bhar [31], has structured an exclusive model about anisotropic strange stars in comparison to the Schwarzschild exterior geometry. Further, he has evaluated the EFEs by including the quintessence scalar field. From the implementation of the Krori-Barua metric he has obtained some exact solutions for the compact stellar objects. Capozziello et al. [32], have worked out gravitational waves in the $ f(T,B) $ theory of gravity, produced by the corresponding compact objects. To examine a compact stellar object in the presence of the quintessence field, Kalam et al. [33], have proposed a relativistic model of compact stellar object with the anisotropic pressure and the normal matter. Nojiri and Odintsov [34], have established that ultimately any evolution of the Universe might be recreated for the theories under investigation. Harko and Lobo [35], have explored the possibility of mixing of two different perfect fluids with the diverse four-velocity vectors and some special parameters. Capozziello and Laurentis [36], have debated the geometrical explanation of the modified gravity theory to indicate particular suppositions in GR. It is more valuable to point out here that the $ f(T) $ gravity is simpler to understand as compared to $ f(R) $ gravity as its field equations are of second-order while $ f(R) $ gravity field equations are of fourth-order. However, in literature [37, 38], it has been argued that the Platini version of $ f(R) $ gravity produce a system of second-order field equations. While comparing to GR, it is examined [39] that the $ f(T) $ gravity shows an extra degree freedom under Lorentz transformation and hence always remained non-variant. Importantly, as $ f(T) $ gravity is invariant under the Lorentz transformation, good or bad tetrad's selection plays a defining role in this particular theory. The reality of the strange stellar leftover in teleparallel $ f(T) $ gravity has been presented by Saha and his collaborators [40]. They have formulated the equation of motion by incorporating the anisotropic environment with chaplygin gas inside. Atazadeh and Darabi [41], have explored the viable nature of a modified gravitational theory, namely $ f(R, G) $ by imposing some energy conditions. Sharif and Ikram [42], have studied the warm inflation scenario in the context of the Gauss-Bonnet $ f(G) $ gravity by inducting the scalar fields in the $ FRW $ spacetime. Maurya and Govender [43], have discussed the Einstein-Maxwell equations and have presented their exact solutions for the spherically symmetric stellar objects. Shamir and Zia [44], have investigated the anisotropic compact structures in the $ f(R, G) $ theory of gravity. A viable approach to deriving the solutions of the field equations in the background of stellar objects has been discussed, known as the Karmarkar condition. This condition was first anticipated by Karmarkar [45], and it is considered a compulsory requirement for a spherically symmetric space-time to be of embedding class I. It essentially supports us to combine the gravitational metric components. Maurya and Maharaj [46], have attained anisotropic embedding solution by employing the spherically symmetric geometry by implementing the Karmarkar condition. Odintsov and Oikonomou [47], have analysed the evolving inflation and DE in the $ f(R,G) $ theory of gravity.

      For the last few years, the perception of parallelism as an equivalent formulation of GR is getting much consideration as an alternate gravitational theory, well acknowledged as the teleparallel equivalent of GR (TEGR) [4850]. This formalism corresponds to the generalized manifold which takes into account a quantity, known as torsion. Ferraro and Fiorini [51, 52], have investigated the TEGR modifications with consideration of cosmology, acknowledged as the $ f(T) $ theory of gravity. The fascinating fragment of $ f(T) $ gravity is to exhibit the second-order field equations and it is structured with a generic Lagrangian quite dissimilar to the $ f(R) $ and several other theories gravity [53, 54]. As for the theoretical or observational cosmology is concerned, numerous researchers effectively implemented the $ f(T) $ gravity in their research [39, 5564]. Deliduman and Yapiskan [65] and also Wang [66], have employed $ f(T) $ gravity to work out the static and spherically symmetric exact solutions describing relativistic compact objects. Deliduman and Yapiskan [67], have constructed the standard relativistic conserved equation, that relativistic compact structures do not exist in $ f(T) $ theory of gravity. However, Bohmer et al. [67], have presented that they actually do exist. In the same line, several other investigations on $ f(T) $ gravity may be referred for more study [6871].

      Enthused by the above discussions, in the present study we investigate the quintessence strange compact stars in the $ f(T) $ theory of gravity by incorporating the observational statistics of the stars $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $. The plan of this manuscript is done as: The Section II provides the fundamental concepts about the $ f(T) $ theory of gravity. In Section III, the exclusive expressions for the physical quantities such as energy density, pressure terms, quintessence density etc have been worked out. Section IV is devoted to the matching conditions through the induction of Schwarzschild outer metric along with the comparison with the interior metric. In Section V, detailed analysis of the physical stellar features has been presented. Section VI concludes our work.

    II.   BASICS OF THE ${{f}}{\bf{(}}{ T}{\bf {)}}$ THEORY OF GRAVITY
    • The action integral of the for $ f(T) $ theory is [7274]

      $ I = \int dx^{4} e\left\{ \frac{1}{2k^{2}} f(T)+{\cal{L}}_{(M)}\right\}, $

      (1)

      where $ e = det\left( e_{\mu}^{A}\right) = \sqrt{-g} $ and $ k^{2} = 8\pi G = 1 $. The variation of the above action results the field equations in the general form as

      $ e_{i} {}^{\alpha}S_{\alpha}{}^{\mu\nu}f_{TT}\partial_{\mu}T+e^{-1}\partial_{\mu}(ee_{i}{}^{\alpha}S_{\alpha}{}^{\mu\nu})f_{T}- e_{\nu}{}^{i}T^{\alpha}{}_{\mu i}S_{\alpha}{}^{\nu\mu}f_{T}-\frac{1}{4}e_{i}{}^{\nu}f = -4\pi e_{\nu}{}^{i}\overset{ e-m}{{\cal{T}}}_{i}^{\nu}, $

      (2)

      where $ \overset{ e-m}{{\cal{T}}}_{i}^{\nu} $ is the energy momentum tensor. Where $ f_{T} $ is the derivative of $ f(T) $ w.r.t T and $ f_{TT} $ is the double derivative w.r.t T. where $ \overset{ e-m}{{\cal{T}}}_{i}^{\nu} = \overset{ matter}{{\cal{T}}}_{i}^{\nu}+\overset{ q}{{\cal{T}}}_{i}^{\nu} $, $ \overset{ q}{{\cal{T}}}_{i}^{\nu} $ is the energy-momentum for the quintessence field equations with the energy density $ \rho_{q} $ and equation of state parameter $ w_{q} $$\left( -1 \!\!<\!\! w_{q}\!\!<\!\! -\dfrac{1}{3} \right)$. Here the components of $ \overset{ q}{{\cal{T}}}_{i}^{\nu} $ are defined as

      $ \overset{ q}{{\cal{T}}}_{t}^{t} = \overset{ q}{{\cal{T}}}_{r}^{r} = -\rho_{q}, $

      (3)

      $ \overset{ q}{{\cal{T}}}_{\theta}^{\theta} = \overset{ q}{{\cal{T}}}_{\phi}^{\phi} = \frac{(3 w_{q}+1)\rho_{q}}{2}. $

      (4)

      Torsion and the super potential tensors used in Eq. (2) are given in general as

      $ T^{\lambda}_{\mu\nu} = e_{B}{}^{\lambda}(\partial_{\mu}e^B{}_\nu-\partial_{\nu}e^B{}_{\mu}), $

      (5)

      $ K^{\mu\nu}_{\lambda} = -\frac{1}{2}\left(T^{\mu\nu}{}_{\lambda}-T^{\nu\mu}{}_{\rho}-T_{\lambda} {}^{\mu\nu}\right), $

      (6)

      $ S_{\lambda}{}^{\mu\nu} = \frac{1}{2}\left(K^{\mu\nu}{}_{\lambda}+\delta^{\mu}{}_{\lambda}{T^{\alpha\mu}{}_{\alpha}}-\delta^{\nu}{}_{\lambda} {T^{\alpha\mu}{}_{\alpha}}\right). $

      (7)

      The density of the teleparallel Lagrangian is defined by the torsion scalar as

      $ T = T^{\lambda}{}_{\mu\nu}S_{\lambda}{}^{\mu\nu}. $

      (8)

      For the present investigation, the character of the scalar torsion T is of crucial importance.

      Due to the flatness of the manifold, the Riemann curvature tensor turns out to be zero. Containing the two fragments, the one part of the curvature tensor defines the Levi-Civita connection, while the second part provides the Weitzenböck connection. Similarly, Ricci scalar R also delivers two dissimilar geometrical entities. Keeping this in view, the torsion-less Ricci scalar R, in the Einstein- Hilbert action, in the shape of the torsion which may be viewed as an expression of T, as given above, can be reproduced. It should be noted that the teleparallel theory of gravity has been proclaimed alike to GR under the two separate contexts of the local Lorentz transformation and the arbitrary transformation coordinates. Part one is non-trivial to be observed and the second Lorentz part adequately delivers the geometry in such way that the construction of the teleparallel action of the GR fluctuates from its metric formulation because of its surface expression. Likewise, one can envision that the modified $ f(R) $ and $ f(T) $ theories of gravity display resemblance to their surface geometries, which are due the local Lorentz invariance, affected by the $ f(R) $ theory of gravity.

      Here we built stellar structures by taking the spherically symmetric spacetime

      $ ds^{2} = e^{a(r)}dt^{2}-e^{b(r)} dr^{2}-r^{2} d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2}, $

      (9)

      where $ a(r) $ and $ b(r) $ solely depend on the radial coordinate r. We will deal with these metric potentials by the Karmarkar conditions in the later part of this work. The Nicola and Bohmer [75] have shown some reservations by declaring as the incorrect choice to the diagonal tetrad in torsion base theories of gravity, as this bad tetrad raises certain solar system limitations. They have also mentioned in their study that good tetrad has no restrictions on the choice of the model of $ f(T) $ being linear or non-linear, while the diagonal tetrad restricts the $ f(T) $ model to a linear one. The off diagonal tetrad choice is a corrected choice due to its boosted, and rotated behavior [39]. Here we calibrate field equations by using the off diagonal tetrad matrix

      $ e_\mu ^\nu = \left( {\begin{array}{*{20}{c}} {{e^{\frac{{a(r)}}{2}}}}&0&0&0\\ 0&{{e^{\frac{{b(r)}}{2}}}\sin \theta \cos \phi }&{r\cos \theta \cos \phi }&{ - r\sin \theta \sin \phi }\\ 0&{{e^{\frac{{b(r)}}{2}}}\sin \theta \sin \phi }&{r\cos \theta \sin \phi }&{ - \sin \theta \cos \phi }\\ 0&{{e^{\frac{{b(r)}}{2}}}\cos \theta }&{ - r\sin \theta }&0 \end{array}} \right). $

      (10)

      Here e is the determinant of $ e_{\mu}^{\nu} $ given as $ e^{a(r)+b(r)}r^{2} \sin\theta $. Energy Momentum Tensor for anisotropic fluid defining the interior of compact star is

      $ \overset{ e-m}{{\cal{T}}}_{\gamma\beta} = (\rho+p_{t})u_{\gamma}u_{\beta}-p_{t}g_{\gamma\beta}+(p_{r}-p_{t})v_{\gamma}v_{\beta}, $

      (11)

      where $ u_{\gamma} = e^{\frac{\mu}{2}}\delta_{\gamma}^{0} $ and $ v_{\gamma} = e^{\frac{\nu}{2}}\delta_{\gamma}^{1} $, $ \rho $, $ P_{r} $ and $ P_{t} $ are respectively the energy density, radial and tangential pressures.

    III.   GENERALIZED SOLUTION FOR COMPACT STAR
    • Manipulating Eqs. (2-11), we have following important expressions

      $ \begin{aligned}[b]&\rho +\rho_q = -\frac{e^{-\frac{b(r)}{2}}}{r}\left(e^{-\frac{b(r)}{2}}-1\right)F' T'- \left(\frac{T(r)}{2}-\frac{1}{r^2}-\frac{e^{-b(r)}}{r^2} \left(1-r b'(r)\right)\right)\frac{F}{2}+\frac{f}{4}, \\ &p_r-\rho_q = \left(\frac{T(r)}{2}-\frac{1}{r^2}-\frac{e^{-b(r)}}{r^2}\left(1+r a'(r)\right)\right)\frac{F}{2}-\frac{f}{4}, \end{aligned} $

      (12)

      $ \begin{aligned}[b] p_t+\frac{1}{2} (3w_q+1)\rho_q =& \frac{e^{-b(r)}}{2}\left(\frac{a'(r)}{2}+\frac{1}{r}-\frac{e^{\frac{b(r)}{2}}}{r}\right)F' T'+ \\ &\left(e^{-b(r)} \left(\frac{a''(r)}{2}+\left(\frac{a'(r)}{4}+\frac{1}{2 r}\right) \left(a'(r)-b'(r)\right)\right)\right. + \left.\frac{T(r)}{2}\right)\frac{F}{2}-\frac{f}{4} . \end{aligned} $

      (13)

      In above equations F is derivative of f with respect to torsion scalar $ T(r) $ and prime on F again is the derivative of F with respect to $ T(r) $. Torsion $ T(r) $ and its derivative with respect to the radial coordinate r is also given as

      $ \begin{aligned}[b] T(r) =& \frac{1}{r^2}\left(2 e^{-b(r)} \left(e^{\frac{b(r)}{2}}-1\right) \left(e^{\frac{b(r)}{2}}-1-r a'(r)\right)\right. ,\\ T' =& \frac{e^{-\frac{b(r)}{2}} b'(r) \left(-r a'(r)+e^{\frac{b(r)}{2}}-1\right)}{r^2}-\frac{2 e^{-b(r)} \left(e^{\frac{b(r)}{2}}-1\right) b'(r) \left(-r a'(r)+e^{\frac{b(r)}{2}}-1\right)}{r^2} \\ &-\frac{4 e^{-b(r)} \left(e^{\frac{b(r)}{2}}-1\right) \left(-r a'(r)+e^{\frac{b(r)}{2}}-1\right)}{r^3}+\frac{2 e^{-b(r)} \left(e^{\frac{b(r)}{2}}-1\right) \left(-r a''(r)-a'(r)+\dfrac{1}{2} e^{\frac{b(r)}{2}} b'(r)\right)}{r^2}. \end{aligned} $

      (14)

      Diagonal tetrad provides linear algebraic form of $ f(T) $ function. But off-diagonal tetrad does not result any parameter to restrict the construction of a consistent model in $ f(T) $ gravity. The following extended teleparallel $ f(T) $ power law viable model [76] is given as:

      $ f(T) = \beta T^{k}, $

      (15)

      where $ \beta $, and k are any real constants. For the power-law model, if we put $ k = 1 $, we get the teleparallel gravity. If we put $ k>1 $, we get the generalized teleparallel gravity. However, in this study, we take $ k = 2 $, which is a well fitted value with the off diagonal tetrad choice. For $ f(T) $ gravity, the underlying scenario gives the realistic stellar objects solution threads by normal matter except a particular range of radial coordinate with observed data. Now we discuss the Karmarkar condition, which is an integral tool for the current study. The groundwork in the regard of the Karmarkar condition is established on the class-I space-time. Eisenhart [77] provided a sufficient condition for the symmetric tensor of rank two as well as on Riemann Christoffel tensor, and it is defined as

      $ \begin{array}{l} \Sigma(\Lambda_{\mu\eta}\beta_{\upsilon\gamma} - \Lambda_{\mu\gamma}\Lambda_{\nu\eta}) = R_{\mu\upsilon\eta\gamma}, \\ \;\;\;\;\;\;\;\; \Lambda_{\mu\nu};n-\Lambda_{\nu\eta};\nu = 0. \end{array} $

      Here; stands for covariant derivative whereas $ \Sigma = \pm1 $. These values signify a space-like or time-like manifold relying on the sign considered as $ - $ or $ + $. Now, by taking into account Riemann curvature components, which are non-zero for the geometry of the space-time and by also conferring non-zero components of the symmetric tensor $ \Lambda_{\nu\eta} $ which is of order two, we incorporate a relation as follows. Now the relation for Karmarkar condition is defined as:

      $ R_{0101}R_{2323} = R_{0202}R_{1313}-R_{1202}R_{1303}, $

      (16)

      we have the following Riemannian non-zero components read as

      $ \begin{aligned}[b] &R_{0101} = -\frac{1}{4}e^{a(r)}\left(-a'(r)b'(r)+a'^{2}(r)+2a''(r)\right),\\ &R_{2323} = -r^{2}\sin^{2}\theta\left(1-e^{-b(r)}\right), R_{0202} = -\frac{1}{2}ra'(r)e^{a(r)-b(r)},\\ &R_{1313} = -\frac{1}{2}b'(r) r\sin^{2}\theta, R_{1202} = 0, R_{1303} = 0. \end{aligned} $

      (17)

      Fitting the above values of Riemannian components in relation (16) give rise to a differential equation having form

      $ a'(r)+\frac{2 a''(r)}{a'(r)} = \frac{e^{b(r)} b'(r)}{e^{b(r)}-1} $

      (18)

      Embedding class one solutions are achieved from Eq.(18) as they can be embedded in 5-dimensional Euclidean Space. By the integration of equation Eq.(18), we have

      $ e^{a(r)} = \left(A+B \int \sqrt{e^{b(r)}-1} \, dr\right)^2 , $

      (19)

      $ b(r) = \log \left(a r^2 e^{b r^2+c r^4}+1\right), $

      (20)

      or exclusively

      $ a(r) = \log \left[\left(\frac{B \sqrt{a r^2 e^{b r^2+c r^4}} {\rm{Dawson}}F\left(\dfrac{2 c r^2+b}{2 \sqrt{2} \sqrt{c}}\right)}{\sqrt{2} \sqrt{c} r}+A\right)^2\right], $

      (21)

      where A and B are the integration constants. The final expressions for energy density and pressure components are calculated as:

      $ \begin{aligned}[b] \rho = &-\frac{1}{(\gamma +1) \left(f_1 r-1\right) \left(\sqrt{2} a B f_3 r r \left(f_1 r-1\right) e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7 r} \left(f_1 r-1\right)\right)\right){}^2}\Biggr[\beta f_4^2 f_5 2^{\kappa -1} \kappa r \\ &\times \Biggr[\frac{1}{\sqrt{f_7} f_2^2}\Biggr[2 \left(f_1-1\right) r \Biggr[2 c \left(2 a B f_4 (\kappa -1) r^3 e^{b r^2+c r^4}+A^2 \sqrt{f_7} \left(2 \left(f_1-1\right) \kappa -2 f_1+1\right)\right)+2 \sqrt{2} a B \sqrt{c} f_3 r e^{b r^2+c r^4} \\ &\times \left(A \left(2 \left(f_1-1\right) \kappa -2 f_1+1\right)+B \sqrt{f_7} (\kappa -1) r \left(b r^2+2 c r^4+1\right)\right)+\frac{B^2 f_3^2 f_7^{3/2} \left(2 \left(f_1-1\right) \kappa -2 f_1+1\right)}{r^2}\Biggr]\Biggr] \\ &-\frac{1}{\left(f_7+1\right) f_2}\Big[\sqrt{f_7} \left(b r^2+2 c r^4+1\right) \Big[2 \sqrt{c} \left(2 a B r^3 \left(\left(f_1-2\right) \kappa -f_1+3\right) e^{b r^2+c r^4}+A \sqrt{f_7} \left(f_1-1\right) (2 \kappa -3)\right) \\ &+\sqrt{2} a B \left(f_1-1\right) f_3 (2 \kappa -3) r e^{b r^2+c r^4}\Big]\Big]+\frac{8 a B^2 c r^5 e^{b r^2+c r^4}}{f_2^2 r}-\frac{2 a B \sqrt{c} r^2 \left(2 \left(f_1-1\right) \kappa -f_1-1\right) e^{b r^2+c r^4}}{f_4}\Biggr]\Biggr], \end{aligned} $

      (22)

      $ \begin{aligned}[b] \rho_q =& \frac{\beta f_5 2^{\kappa -2} }{\gamma +1}\left[\frac{8 \gamma \sqrt{f_7} (\kappa -1) \kappa}{\left(f_7+1\right) \left(f_1-1\right) \left(\sqrt{2} a B \left(f_1-1\right) f_3 r e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7} \left(f_1-1\right)\right)\right){}^2}\right. \\ &\times \Biggr[2 c \Big[a^2 r^4 e^{2 r^2 \left(b+c r^2\right)} \left(A^2 \sqrt{f_7}+A B r \left(b r^2+2 c r^4-f_1+2\right)-B^2 \sqrt{f_7} \left(f_1-1\right) r^2\right)+f_7 \Big[A^2 \sqrt{f_7} \\ &\times \left(-b \left(f_1-1\right) r^2-2 c \left(f_1-1\right) r^4-3 f_1+4\right)+A B \left(f_1-1\right) r^3 \left(b+2 c r^2\right)-B^2 \sqrt{f_7} \left(f_1-1\right) r^2\Big]-2 A^2 \sqrt{f_7} \\ &\times \left(f_1-1\right)\Big]+\sqrt{2} a B \sqrt{c} f_3 r \Biggr[2 A e^{b r^2+c r^4} \Big[a^2 r^4 e^{2 r^2 \left(b+c r^2\right)}-f_7 \left(b \left(f_1-1\right) r^2+2 c \left(f_1-1\right) r^4+3 f_1-4\right)-2 f_1 \\ &\left.+2\Big]+\frac{B f_7^{3/2} r \left(-a e^{b r^2+c r^4} \left(-2 c r^4+f_1-2\right)+b \left(f_1+f_7-1\right)+2 c \left(f_1-1\right) r^2\right)}{a}\right]-\frac{B^2 f_3^2 f_7^{3/2}}{r^2} \\ &\times\Big[-a^2 r^4 e^{2 r^2 \left(b+c r^2\right)}+f_7 \left(b \left(f_1-1\right) r^2+2 c \left(f_1-1\right) r^4+3 f_1-4\right)+2 f_1-2\Big]\Biggr] \\ &+\frac{\kappa}{\left(f_7+1\right) \left(f_1-1\right) \left(\sqrt{2} a B \left(f_1-1\right) f_3 r e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7} \left(f_1-1\right)\right)\right)} \Big[2 \sqrt{c} \\ &\times \left(2 a^2 B r^5 e^{2 r^2 \left(b+c r^2\right)}+f_7 \left(A \sqrt{f_7} \left(2 b \gamma r^2+4 c \gamma r^4+\gamma +1\right)+2 B r\right)-A (\gamma -1) \sqrt{f_7}\right)+\sqrt{2} a B f_3 r e^{b r^2+c r^4} \\ &\left.\times \left(f_7 \left(2 b \gamma r^2+4 c \gamma r^4+\gamma +1\right)-\gamma +1\right)\Big]+(\gamma +1) \kappa -(\gamma +1) \left(\frac{\sqrt{f_7} \left(f_7+1\right) \kappa {\rm{f}}_2}{f_6}+1\right)\right], \end{aligned}$

      (23)

      $ \begin{aligned}[b] p_r =& -\frac{\beta \gamma f_4^2 f_5 2^{\kappa -1} \kappa r}{(\gamma +1) \left(f_1-1\right) \left(\sqrt{2} a B \left(f_1-1\right) f_3 r e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7} \left(f_1-1\right)\right)\right){}^2} \left[\frac{2 \left(f_1-1\right) r}{\sqrt{f_7} f_2^2} \right.\\ &\times \Biggr[2 c \left(2 a B f_4 (\kappa -1) r^3 e^{b r^2+c r^4}+A^2 \sqrt{f_7} \left(2 \left(f_1-1\right) \kappa -2 f_1+1\right)\right)+2 \sqrt{2} a B \sqrt{c} f_3 r e^{b r^2+c r^4} \Big[A \Big[2 \left(f_1-1\right) \\ &\times\kappa -2 f_1+1\Big]+B \sqrt{f_7} (\kappa -1) r \left(b r^2+2 c r^4+1\right)\Big]+\frac{B^2 f_3^2 f_7^{3/2} \left(2 \left(f_1-1\right) \kappa -2 f_1+1\right)}{r^2}\Biggr] \\ &-\frac{2 a B \sqrt{c} r^2 \left(2 \left(f_1-1\right) \kappa -f_1-1\right) e^{b r^2+c r^4}}{\sqrt{2} a B {\rm{rf}}_3 e^{b r^2+c r^4}+2 A \sqrt{c} \sqrt{f_7}}-\frac{\sqrt{f_7} \left(b r^2+2 c r^4+1\right)}{\left(f_7+1\right) f_2} \Big[2 \sqrt{c} \Big[2 a B r^3 \left(\left(f_1-2\right) \kappa -f_1+3\right) \\ &\left.\times e^{b r^2+c r^4}+A \sqrt{f_7} \left(f_1-1\right) (2 \kappa -3)\Big]+\sqrt{2} a B \left(f_1-1\right) f_3 (2 \kappa -3) r e^{b r^2+c r^4}\Big]+\frac{8 a B^2 c r^5 e^{b r^2+c r^4}}{f_2^2 r}\right], \end{aligned} $

      (24)

      $\begin{aligned}[b] p_t =& \frac{2^{\kappa -4} \beta f_5 }{\gamma +1}\Biggr[-6 (w_q+1) (\gamma +1)+\kappa \Biggr[6 (w_q+1) (\gamma +1)\\ &+\frac{4 f_7}{\left(f_7+1\right) \left(f_1-1\right) \left(\sqrt{2} a B e^{c r^4+b r^2} r \left(f_1-1\right) f_3-2 \sqrt{c} \left(2 a B e^{c r^4+b r^2} r^3-A \sqrt{f_7} \left(f_1-1\right)\right)\right)} \Big[2 \Big[B r \Big[2 c r^4+2 c \gamma r^4\\ &+b (\gamma +1) r^2+3 w_q+2 \gamma +(3 w_q+\gamma +2) f_7+3\Big]+A \left(2 c r^4+b r^2+1\right) (3 w_q \gamma -1) \sqrt{f_7}\Big] \sqrt{c}+a B e^{c r^4+b r^2} r \\ &\times \left(2 c r^4+b r^2+1\right) (3 w_q \gamma -1) \sqrt{2} f_3\Big] \\ &-\frac{16 r (\kappa -1)}{\left(f_7+1\right) \left(f_1-1\right){}^2 f_2 \left(\sqrt{2} a B e^{c r^4+b r^2} r \left(f_1-1\right) f_3-2 \sqrt{c} \left(2 a B e^{c r^4+b r^2} r^3-A \sqrt{f_7} \left(f_1-1\right)\right)\right){}^2} \Big[2 \sqrt{c} \Big[a B e^{c r^4+b r^2} \\ &\times r^3 (\gamma +1)-A \sqrt{f_7} \left(f_1-1\right) ((3 w_q+2) \gamma +1)\Big]-\sqrt{2} a B e^{c r^4+b r^2} r \left(f_1-1\right) ((3 w_q+2) \gamma +1) f_3\Big] \\ &\times\Biggr[-\frac{B^2 \left(-a^2 e^{2 r^2 \left(c r^2+b\right)} r^4+2 f_1+\left(2 c \left(f_1-1\right) r^4+b \left(f_1-1\right) r^2+3 f_1-4\right) f_7-2\right) f_3^2 f_7^{3/2}}{r^2}+2 c \Big[a^2 e^{2 r^2 \left(c r^2+b\right)} \\ &\times\left(\sqrt{f_7} A^2+B r \left(2 c r^4+b r^2-f_1+2\right) A-B^2 r^2 \sqrt{f_7} \left(f_1-1\right)\right) r^4-2 A^2 \sqrt{f_7} \left(f_1-1\right)+\Big[A B \left(2 c r^2+b\right)\end{aligned} $

      $ \begin{aligned}[b] \qquad\qquad &\times\left(f_1-1\right) r^3-B^2 \sqrt{f_7} \left(f_1-1\right) r^2+A^2 \sqrt{f_7} \left(-2 c \left(f_1-1\right) r^4-b \left(f_1-1\right) r^2-3 f_1+4\right)\Big] f_7\Big]+a B r \sqrt{2} \sqrt{c} \\ &\times\Biggr[\frac{B r \left(2 c \left(f_1-1\right) r^2-a e^{c r^4+b r^2} \left(-2 c r^4+f_1-2\right)+b \left(f_1+f_7-1\right)\right) f_7^{3/2}}{a}+2 A e^{c r^4+b r^2} \Big[a^2 e^{2 r^2 \left(c r^2+b\right)} r^4 \\ &\times-f_7 \left(2 c \left(f_1-1\right) r^4+b \left(f_1-1\right) r^2+3 f_1-4\right)-2 f_1+2\Big]\Biggr] f_3\Biggr]\Biggr]-\frac{2 \sqrt{f_7} (3 w_q+1) \left(2 \gamma +(\gamma +1) f_7\right) \kappa {\rm{f}}_2}{f_6}\Biggr] \end{aligned}$

      (25)

      $ \Delta = p_t-p_r, $

      (26)

      Figure 1.  Evolution of metric potentials versus. Here in we fix $ k=2 $, $b=0.000015$ $\gamma =0.333$, $w_q=-1.00009$, $c=0.000015$ and $\beta =-4$.

      where

      $ \begin{aligned}[b]& f_1 = \sqrt{f_7+1},\;\;\;\;f_2 = 2 A \sqrt{c} r+\sqrt{2} B \sqrt{f_7} f_3,\;\;\;\;f_3 = F\left(\frac{2 c r^2+b}{2 \sqrt{2} \sqrt{c}}\right),\;\;\;\;f_4 = A b r^2+2 A c r^4+A-B \sqrt{f_7} r,\\ &f_5 = \left(\frac{\left(f_1-1\right) \left(\sqrt{2} a B \left(f_1-1\right) f_3 r e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7} \left(f_1-1\right)\right)\right)}{\sqrt{f_7} \left(f_7+1\right) f_2 r}\right){}^{\kappa },\\ &f_6 = \left(f_1-1\right) r \left(\sqrt{2} a B \left(f_1-1\right) f_3 r e^{b r^2+c r^4}-2 \sqrt{c} \left(2 a B r^3 e^{b r^2+c r^4}-A \sqrt{f_7} \left(f_1-1\right)\right)\right),\;\;\;\;f_7 = a r^2 e^{b r^2+c r^4}. \end{aligned} $

    IV.   MATCHING CONDITIONS
    • No matter what remains the geometrical structure of the star whether exterior or from the interior, the inner boundary metric does not change. This emerging situation authorizes nevertheless of the referential frame that the metric components will have to remain continuous along the entire boundary. In the $ GR $, while investigating the Schwarzschild solution associated to the stellar remnants is well-thought-out to be the top choice from all the available diverse options of the matching conditions. Besides, the suitable choice in case of working with the theories of modified gravity is to consider for the non-zero pressure and the energy density. Several researchers [7879] have produced a great work about the boundary conditions. Goswami et al. [80] worked out the matching conditions while investigating the modified gravity by incorporating some special limitations to stellar compact structures along with the thermodynamically associated properties. Many investigators [8184] have effectively employed the Schwarzschild geometry while working out the diverse stellar solutions. To have the expressions for the field equations a few restrictions are applied at the boundary $ r = R $, that is $ p_r(r = R) = 0 $. We also here intend to match the Schwarzschild exterior geometry with the interior geometry to obtain the favorable results.

      $ ds^2 = -\bigg(1-\frac{2M}{r}\bigg)dt^2+\bigg(\frac{1}{1-2M/r}\bigg)dr^2+r^{2}\bigg(d{\theta}^2+{\sin}^2{\theta}d{\phi}^2\bigg), $

      (27)

      where M represents the total stellar mass and R is the total radius of the star. Taking into account the metric potentials, following relations are employed at the boundary $ r = R $:

      $ g_{tt}^- = g_{tt}^+,\;\;\;\;\;\;\;g_{rr}^- = g_{rr}^+,\; \; \; \; \frac{\partial g_{tt}^- }{\partial r} = \frac{\partial g_{tt}^+}{\partial r}. $

      (28)

      The signature of the intrinsic geometry and extrinsic geometry are taken as (-,+,+,+) and (+,-,-,-), respectively. The desired restrictions are achieved by making comparison of the interior and the exterior geometry as they are, and are worked out as following:

      $\begin{aligned}[b] &A = -\frac{1}{R^2}log_e\bigg(1-\frac{2M}{r}\bigg),\; \; \; \; B = -\frac{M}{R^2}log_e\bigg(1-\frac{2M}{r}\bigg)^{-1} ,\\ &C = log_e\bigg(1-\frac{2M}{r}\bigg)-\frac{M}{R}\bigg(1-\frac{2M}{r}\bigg)^{-1}. \end{aligned}$

      (29)

      The approximated values of the mass M and the radius R of the stellar objects, $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $ are considered to determine the unknowns as given in the table

    V.   PHYSICAL ANALYSIS
    • This section is dedicated to the exploration of some critical properties connected to the virtually massive stars. These comprise the energy density $ \rho $, radial pressure $ p_r $, the tangential pressure $ p_t $, and the discussions on the quintessence field along with their physical interpretation under $ f(T) $. This discussion also includes the energy conditions, anisotropic pressure, compactness factor, and the sound speed of the star with reference to the radial and tangential components.

      Star nameObserved mass($ M_{o}$)Predicted Radius ($R km$)aAB
      PSRJ1614-22301.9712.1820.00230990.7341470.0233421
      4U 1608-521.7411.7510.002284320.7584280.0231551
      Cen X-31.4911.2240.002257230.7852030.0229539
      EXO1785-2481.310.7750.002234010.806020.0227945
      SMC X-11.0410.0670.002199440.8353740.0225762

      Table 1.  Values of constants of compact Stars by fixing $ k=2 $, $b=0.000015$ $\gamma =0.333$, $w_q=-1.00009$, $c=0.000015$ and $\beta =-4$.

    • A.   Energy density, quintessence density, and pressure profiles

    • The most important stellar environment responsible for the emergence of the compact stars comprises the corresponding profiles of the energy density along with the radial and tangential pressures. For our ongoing investigations, we have investigated the profiles of the energy density, quintessence density and the pressure terms. It is quite apparent from the plots of the corresponding graphs as shown in Figs. (3-4) that the energy density acquires the highest value at the center of the star depicting the ultra-dense nature of the star. Moreover, the tangential and radial pressure terms are positive and acquire the maximum values at the surface of the compact stars. The profiles of the stars also indicate the presence of anisotropic matter configuration free from any singularities for our model under $ f(T) $ gravity.

      Figure 3.  Evolution of energy density $\rho$ (Left) and quintessence density $\rho_q$(Right).

      Figure 4.  Evolution of radial pressure $p_r$ (Left) and tangential pressure $p_t$(Right).

    • B.   Energy conditions

    • The role of the energy constraints among the other physical features in describing the existence of anisotropic compact stars has been greatly acknowledged in the literature, as they analyse the environment for the presence of the matter distribution. Moreover, their role also provides the analysis for the distribution of normal and exotic matter contained within the core of the stellar structure. Several fruitful conclusions have been made due to these energy constraints. The expressions corresponding to the Null energy constraints $ (NEC) $, Strong energy constraints $ (SEC) $, the Dominant energy constraint $ (DEC) $, and the Week energy constraint $ (WEC) $ are given below.

      $ \begin{aligned}[b] NEC:\rho+p_r\geqslant0,\rho+p_t\geqslant 0,&\\ WEC:\rho\geqslant 0,\rho+p_r\geqslant0,\rho+p_t\geqslant0,&\\ SEC:\rho+p_r\geqslant0,\rho+p_t\geqslant0\,\rho+p_r+2p_t\geqslant0,&\\ DEC:\rho>|p_r|,\rho>|p_t|.&\\ \end{aligned} $

      (30)

      The graphical evolution of the energy constraints has been depicted in Figs. (5). It is clear from the positive profiles of the energy conditions for all the stars, $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}- 248 $, and $ SMC X-1 $, that our obtained solutions are physically favorable under $ f(T) $ gravity.

      Figure 5.  Evolution of energy conditions (Left) and forces (Right).

      Figure 6.  Evolution of EOS $w_t$.

    • C.   Anisotropic constraints

    • The expression $\dfrac{d\rho}{dr}$, $\dfrac{dp_{r}}{dr}$ and $\dfrac{dp_{t}}{dr}$ are meant for the total derivatives of the energy density, the radial pressure, and the tangential pressure with respect to the radius r of the compact star, respectively. The graphical description of these radial derivatives is provided in the right plots of Fig. (2) which favorably suggest that the first order derivative gives the negatively increasing evolution.

      Figure 2.  Evolution of anisotropy $\Delta$ (Left) and gradients(Right).

      $ \frac{d\rho}{dr}<0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{dp_{r}}{dr}<0. $

      (31)

      It may be noted that $\dfrac{d\rho}{dr}$ and $\dfrac{dp_{r}}{dr}$ at the core $ r = 0 $ of the star it with the following observation

      $ \frac{d\rho}{dr} = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{dp_{r}}{dr} = 0. $

      This confirms the maximum bound of the radial pressure $ p_r $ along with the central density $ \rho $.Hence, the maximal value is attained at $ r = 0 $ by $ \rho $ and $ p_r $.

    • D.   Equilibrium under various forces

    • The generalized TOV equation in anisotropic matter distribution is given as

      $ \frac{d p_r}{dr}+\frac{a^{'}(\rho+p_r)}{2}-\frac{2(p_t-p_r)}{r} = 0,\\ $

      (32)

      where the Eq.(32) provides an important information about the stellar hydrostatic-equilibrium under the total effect of the diverse three forces, namely $ F_{a} $, the anisotropic, $ F_{h} $, the hydrostatic force, and $ F_{g} $, the gravitational force. The null effect of the combined forces depicts the equilibrium condition such that

      $ F_g+F_h+F_a = 0, $

      with

      $ F_g = -\frac{a^{'}(\rho+p_r)}{2},\; \; F_h = -\frac{d p_r}{dr},\; {\rm{and}}\; \; F_a = \frac{2(p_t-p_r)}{r}. $

      (33)

      From the right plot of the Fig. (5), it may be induced that under the combined effect of the forces $ F_{g} $, $ F_{h} $ and $ F_{a} $ the hydrostatic compact equilibrium could be achieved. It is pertinent to mention here that somewhere if $ p_{r} = p_{t} $ then the force $ F_{a} $ vanishes, which simply conveys that the equilibrium turns independent of the anisotropic force $ F_{a} $.

    • E.   Stability analysis

    • The stability is constituted by the sound speeds associated to the radial and transversal components having representation of $ v^{2}_{sr} $ and $ v^{2}_{st} $, respectively. They must validate the constraints i.e., $ 0\leqslant{v^{2}_{st}}\leqslant 1 $ and $0\leqslant{v^{2}_{sr}}\leqslant1$ [85], such that $= v^{2}_{sr} = \dfrac{dp_{r}}{d\rho}$ and $v^{2}_{st} = \dfrac{dp_{t}}{d\rho}$. A comprehensive study of the stability of anisotropic spheres has been done by Chan and his coauthors [86]. They have discussed Newtonian and post-Newtonian approximations in the background of anisotropy distribution. The corresponding plots of the sound speeds as depicted in Figs. (8) confirm that the evolution of the radial and transversal sound speeds for strange star candidates $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $ remain within the desired constraints of stability as discussed. For all the candidates of the strange stars the bounds of both, the radial and the transversal sound speeds are justified. Within the anisotropic matter distribution, the approximation of the theoretically firm and unstable epochs may be obtained from the modifications of the sound speeds propagations which has the expression $ v^{2}_{st}-v^{2}_{sr} $ approving the constraint $ 0<|v^{2}_{st}-v^{2}_{sr}|<1 $. One may confirm this from the Fig.(9). Therefore, the total stability may be obtained for compact stars modelled under $ f(T) $ gravity.

      Figure 8.  Evolution of $\mid v_{r}^{2}$-$v_{t}^{2}\mid$.

    • F.   EoS parameter and the measurement of anisotropy

    • For the case of anisotropic matter distribution, the EoS parameter incorporating radial and transversal components may be expressed as

      $ \omega_{r} = \frac{p_{r}}{\rho}\; \; \; \; {\rm{and}}\; \; \; \; \omega_{t} = \frac{p_{t}}{\rho}. $

      (34)

      The analysis of the EoS parameters is provided with respect to the increasing stellar radius and is graphically represented in Fig.(7) which clearly demonstrates that for all strange star candidates $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $, the conditions $ 0<\omega_{r}<1 $ and $ 0<\omega_{t}<1 $ have been obtained. Hence, our stellar model in $ f(T) $ gravity is truly viable. Now, the anisotropy here is expressed by the symbol$ \Delta $, and is measured as

      Figure 7.  Evolution of Sound speeds $v_{r}^{2}$ (Left) and $v_{t}^{2}$ (Right).

      $ \Delta = \frac{2}{r}{(p_t-p_r)}, $

      (35)

      which provides the information regarding the anisotropic conduct of the model under discussion. The term $ \Delta $ has to be positive if $ p_t>p_r $, displaying that the anisotropy going outward, and when, $ p_r>p_t $, the $ \Delta $ becomes negative showing that it will be directed inward. For our model incorporating all the stars $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $, the evolution of the $ \Delta $ when plotted against the radii r depicts the positive increasing behavior(as shown in the left plot of Fig. 2) proposing some repelling anisotropic force followed by highly-dense matter source.

    • G.   Mass-Radius relation, compactness, and redshift analysis

    • The stellar mass as a function of radii r is defined by the following integral

      $ m(r) = 4\int_{0}^{r}\pi\acute{r^2}\rho{d}\acute{r}. $

      (36)

      It is evident from the mass-radius graph as shown in Fig. (9) that the mass is directly proportional to the radius r such that as $ r\rightarrow0 $, the $ m(r)\rightarrow0 $, showing that mass function remains continuous at the core of the star. Also, the mass-radius ratio, must remain $\dfrac{2M}{r}\leqslant\dfrac{8}{9}$ as determined by Buchdahl [87], which in our case is inclined within the desired range.

      Figure 9.  Evolution of mass function (Left) and compactness parameter (Right).

      Figure 10.  Evolution of redshift function.

      Now, the following integral defines the compactness $ \mu(r) $ (plotted in Fig. 9) of the stellar structure as

      $ \mu(r) = \frac{4}{r}\int_{0}^{r}\pi\acute{r^2}\rho{d}\acute{r}. $

      (37)

      Providing the redshift function $ Z_{S} $ as

      $ Z_{S}+1 = [1-2\mu(r)]^{\frac{-1}{2}}. $

      (38)

      The graphical representation is provided in the Fig. (9). The numerical estimate of $ Z_{S} $ remains within the desired condition of $ Z_{S}\leqslant 2 $, favoring the viability of our model under investigation.

    VI.   CONCLUSION
    • As an equivalent structuring of the GR, the notion of the parallelism, is largely being attracted as an alternate theory of gravity for the last few years and has been well acknowledged as the teleparallel equivalent of GR (TEGR). The concept behind this pedagogy is the existence of even more standard manifold which takes into account the curvature besides a quantity called torsion. A big number of renown investigators has explored the modifications of TEGR with reference to the cosmology, recognized as $ f(T) $ theory of gravity. The most attractive part of the $ f(T) $ gravity is to possess the second-order field equations dissimilar to those of the $ f(R) $ gravity and it is built with a comprehensive Lagrangian.

      In our present work, we have employed a general model for the possible existence of the static and anisotropic compact structures in the spherically symmetric metric and by using a power law model in the background of the $ f(T) $-modified gravity. To our best of knowledge, this is the first attempt to study stellar objects in $ f(T) $ theory of gravity with quintessence via embedding approach. The statement just validates the results of our study. In fact, our theoretical calculations support realistic models of stars $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO 1785}}-248 $, and $ SMC X-1 $. The stability and singularity free nature of these realistic models is physically important, and our results are in good agreement in this scenario. Moreover, through some manipulations, the corresponding field equations are solved for the compact stars. We have established our calculations under the assumptions of the statistics corresponding to the $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $, as the strange star candidates with the appropriate choice of the values of the parameter n. Our work here administers the investigation of the possible existence of the quintessence compact stars possessing anisotropic nature due to the extremely dense structure in the framework of the $ f(T) $ theory of gravity. For the evolving Universe in different epochs, particularly the gravitational stellar collapse has been explored by incorporating the spacetime symmetries along exclusive matching of the Schwarzschild vacuum solution. Graphical illustration of some exclusive features of the quintessence stellar structures in the $ f(T) $ gravity is presented. The energy density $ \rho $, the transversal pressure $ p_t $, the radial pressure $ p_r $, anisotropy limitations and the quintessence energy density $ \rho_q $ have been analysed in the context of the $ f(T) $ gravity by using off diagonal tetrad and power law given as $ f(T) = \beta T^n $. Here are some of the key features which we have found during our investigation with our focus on the energy density, radial and tangential pressures and the quintessence field along with their physical interpretation under $ f(T) $ gravity. Besides, the other interesting features include the energy restrictions, anisotropy, compactness and the sound speed of the stellar remnants referring to the radial and tangential components.

      ● The crucial physical aspects for the existence of stellar structures comprise energy density and the radial and tangential pressures. It is advantageously clear from the respective plots in the Figs. (3-4) that the energy density at the stellar core attains the highest value showing the highly dense character of the star. Also, the tangential and radial pressure terms are positive and attain the maximum values at star surface. These profiles also offer the existence of anisotropic matter distribution independent of singularities for $ f(T) $ model under investigation. Furthermore, the profiles of the quintessence density $ \rho_q $ reflects the negative behavior, favoring our stellar $ f(T) $ gravity model.

      ● The role of the energy constraints is quite obvious in the literature telling about the compact stellar remnants. The plots of the corresponding energy conditions has been presented in the Figs. (5). It is evident from their positive profiles for all the stars, $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $, that our acquired solutions are physically viable in the $ f(T) $ gravity.

      ● The profiles of $\dfrac{d\rho}{dr}$, $\dfrac{dp_{r}}{dr}$ and $\dfrac{dp_{t}}{dr}$ depicting the total derivatives of $ \rho $, $ p_r $, and $ p_t $ with respect to the stellar radii r, respectively have been provided in the Fig. (2) which tells that the first derivative shows negatively acquiring evolution. This validates the highest bound of the radial pressure $ p_r $ with the central density $ \rho $. Therefore, the highest value is achieved by $ \rho $ and $ p_r $ at $ r = 0 $.

      ● As for the hydrostatic equilibrium is concerned, it is evident from the right panel of the Fig. (5), that under the total effect of the acting forces $ F_{g} $, $ F_{h} $ and $ F_{a} $ the stellar equilibrium is achieved. It is worth mentioning that at certain situation like $ p_{r} = p_{t} $, the force $ F_{a} $ is vanished, hinting that the equilibrium is free from the effect of anisotropic force $ F_{a} $.

      ● The corresponding plots of the sound speeds as depicted in the Figs.(8) suggesting that the radial and transversal sound speeds for all the stars $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $ are bounded within the desired constraints of stability. One may confirm from the Fig.(9) that the constraint $ 0<|v^{2}_{st}-v^{2}_{sr}|<1 $ for the stability of the compact star is achieved. Therefore, the over all stability may be obtained for compact stars modeled under $ f(T) $ gravity.

      ● The constraint parameter EoS is expressed by $ 0<\omega_{t}<1 $ and is plotted in the Fig.(7). It is easy to see that it favors the corresponding matter distribution under $ f(T) $ gravity.

      ● For the stars $ {\rm{PSRJ1614}}-2230 $, $ 4U 1608-52 $, $ {\rm{Cen}} X-3 $, $ {\rm{EXO1785}}-248 $, and $ SMC X-1 $, the anisotropy $ \Delta $ with respect to r gives the positive increasing behavior (as shown in the left plot of the Fig. 2) suggesting the repelling anisotropic forces incorporated by highly-dense matter source.

      ● The Fig. (9) provides graphical representation of the red-shift function. The approximate value of $ Z_{S} $ falls within the desired condition of $ Z_{S}\leqslant 2 $, supporting our model under investigation.

      It is worth mentioning here that our obtained solutions in this study represent more dense stellar structures as compared to past related works on compact objects in extended theories of gravity [88]-[91].

    ACKNOWLEDGMENTS
    • We are grateful to the anonymous referees for their valuable comments, which improved the presentation of this paper.

Reference (91)

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