Study of Anisotropic Compact Stars with Quintessence Field And

Eur. Phys. J. C (2019) 79:919 https://doi.org/10.1140/epjc/s10052-019-7427-7

Regular Article - Theoretical Physics

Study of anisotropic compact stars with quintessence ﬁeld and modiﬁed chaplygin gas in f (T) gravity

Pameli Sahaa, Ujjal Debnathb Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711 103, India

Received: 18 May 2019 / Accepted: 25 October 2019 / Published online: 12 November 2019 © The Author(s) 2019

Abstract In this work, we get an idea of the existence of 1 Introduction compact stars in the background of f (T ) modified gravity where T is a scalar torsion. We acquire the equations Recently, compact stars, the most elemental objects of the of motion using anisotropic property within the spherically galaxies, acquire much attention to the researchers to study compact star with electromagnetic field, quintessence field their ages, structures and evolutions in cosmology as well as and modified Chaplygin gas in the framework of modi- astrophysics. After the stellar death, the residue portion is fied f (T ) gravity. Then by matching condition, we derive formed as compact stars which can be classified into white the unknown constants of our model to obtain many phys- dwarfs, neutron stars, strange stars and black holes. Com- ical quantities to give a sketch of its nature and also study pact star is very densed object i.e., posses massive mass and anisotropic behavior, energy conditions and stability. Finally, smaller radius as compared to the ordinary star. In astro- we estimate the numerical values of mass, surface redshift physics, the study of neutron star and the strange star moti- etc from our model to compare with the observational data vates the researchers to explore their features and structures for different types of compact stars. very much. Massive neutron star again collapses into black hole but lower mass neutron star converts into quark star. Basically, neutron star consists of neutrons whereas strange star is made up with quarks or strange matters. After the Contents discovery of neutrons [1], the researchers have first imag- ined about the presence of neutron star. Then Hewish et al. 1 Introduction ...... 1 [2] has confirmed this prediction by observation of pulsars 2 An introduction of f (T ) gravity, electromagnetic (considered as rotating neutron stars later) like Her X-1, 4U field and quintessence field ...... 3 1820−30, RXJ 1856−37 and SAX J 1808.4−3658. In 1916, 3 Anisotropic compact star with electromagnetic field Karl Schwarzschild [3,4] has first given the interior stellar and quintessence field in f (T ) gravity ...... 3 solution which must be matched with the exterior solution. 4 Matching conditions ...... 5 We have noticed for a long time that the isotropic fluid (hav- 5 Physical analysis ...... 5 ing equal radial pressure (pr ) and transversal pressure (pt )) 5.1 Anisotropic effects ...... 7 is considered in the core of the stellar objects to study stellar 5.2 Energy conditions ...... 9 structure and stellar evolution. In 1972, Ruderman [5]forthe 5.3 Stability analysis ...... 10 first time observed that the interior geometry of the nuclear 15 / 5.4 Effective mass and compactness ...... 10 matter with a density of order 10 gm cc posses anisotropic = 5.5 Relation between mass and radius ...... 13 behavior (having pr pt ). This nature may come from dif- 5.6 Surface redshift ...... 13 ferent sources: the existence of solid core, in presence of type 6 Conclusions ...... 13 P superfluid, phase transition, rotation, magnetic field, mix- ture of two fluids, viscosity etc. Herrera et al. [6]havegiven References ...... 15 a review to analyze local anisotropic nature for self gravi- tating systems. A stable structure of stellar objects has been found in the context of anisotropic nature by Hossein et al. a e-mail: [email protected] [7]. Kalam et al. [8] have investigated anisotropic neutron b e-mail: [email protected] 123 919 Page 2 of 16 Eur. Phys. J. C (2019) 79 :919 star with quintessence dark energy. A new exact solution for itation with tetrad and torsion. Many authors have revealed compact star has been discovered by Paul et al. [9] to preserve wide interest in [26Ð38]. the hydrostatic equilibrium. Many astrophysical phenomena In theoretical astrophysics, Dent [39] has derived the solu- of quark star and neutron star have been discussed in ref. tion of BTZ black hole in f (T ) version in 3-dimensions. [10,11]. Rahaman et al. [12] have observed the existence of Later, first violation of black hole thermodynamics in f (T ) strange star using MIT bag model to study mass and redshift gravity has been seen through violation of Lorentz invariance functions. Again, a stable anisotropic quintessence strange [40]. For the existence of astrophysical stars in f (T ) the- star model has been proposed by Bhar [13]. Murad [14] has ory, the physical conditions have been investigated [41] with also studied anisotropic charged strange star with MIT bag very much attention after acquiring a large group of static model to find out the radial pressure and energy density. A perfect fluid solutions [42]. Some static solutions for spheri- stability of strange star with the influence of anisotropic con- cally symmetric case with charged source in f (T ) theory has text using MIT bag model has been investigated by Arban˜il been obtained [43]. Capozziello et al. [44] have observed the and Malheiro [15]. removal of singularity of the exact black hole by f (T ) grav- After publishing of Einstein’s General Relativity (GR) ity instead of f (R) gravity in D dimensions. Sharif and Rani in 1915, our mysterious universe has been sketched more [45] have studied wormhole under f (T ) gravity. They have clearly to people. Recently, we have come to know from many studied static wormhole solution in f (T ) gravity to investi- observational evidences like Large Scale Structure (LSS), gate energy conditions [46] and also proved that this solution Cosmic Microwave Background (CMB) radiation etc. that exists by violating energy conditions for charged noncommu- our universe is expanding with acceleration which asserts that tative wormhole [47,48]. the cosmic expansion is going due to some peculiar source of The study of anisotropic stars in the background of Gen- energy having a massive negative pressure, known as Dark eral Relativity (GR) and modified gravity has drawn much Energy (DE). Similarly, there exists also a mysterious com- interest to many researchers [49Ð54]. Krori-Barua (KB) met- ponent, known as Dark Matter (DM). A telescope is unable to ric [55,56] is the most useful metric to discuss the compact detect dark matter but its gravitational effects on visible mat- star models. Abbas and his collaborations [57Ð61]havealso ter and gravitational lensing of the background radiation are studied this type of stars in GR, f (R), f (G) and f (T ) with giving us the evidence of its existence. On the other hand the the help of (KB) metric. Abbas et al. [60] have investigated Equation of State (EoS) of DE is given by w = p/ρ where anisotropic strange star taking p = αρ where 0 <α<1 w is called the EoS parameter lying in the range w<−1/3. which plays as a quintessence dark energy model. A study of If −1

α β γ Now, the modified teleparallel action is defined as follows a(r) = Br + Cr , b(r) = Ar (17) [70,71] where A, B and C are arbitrary constants. Here, α ≥ 2, β ≥ 2 4 1 S = d xe f (T ) + LMatter( A) (9) γ ≥ α = β α = β = γ = 16π and 2 are constants but .For 2, 3, the unknown functions a(r) and b(r) reduce to the forms in = = ( ) where G c 1 and LMatter A is the matter Lagrangian. which anisotropic quintessence star has been studied in [69]. Now, we consider our model containing quintessence field For this metric, the torsion scalar T and its derivative are and electromagnetic field along with anisotropic pressure. given as [62] So, the Einstein equations can taken as −b Matter q EM 2e 1 Gμν = 8πG(Tμν + Tμν + Tμν ). (10) T (r) = a + , (18) r r −b Here, the ordinary matter corresponding to anisotropic fluid 2e a 1 2 T (r) = a − − b a + − (19) has energy-momentum tensor as r r r r 2 Matter Tμν = (ρ + pt )uμuν − pt gμν + (pr − pt )vμvν (11) where the prime denotes the derivative with respect to the radial coordinate r. 123 919 Page 4 of 16 Eur. Phys. J. C (2019) 79 :919

Now, the equations of motion for anisotropic fluid along ζ p = ξρ − (26) with quintessence field and magnetic field in the framework r ρα of f (T ) gravity are as follows [62]: where ξ, α and ζ are free parameters of our model. −b −b e f 1 e Now, solving equation (23) we get (assuming T = 0) − T f + − T − − (a + b ) f TT 2 T r 2 r r ( ) = β + β , 2 f T 1T 2 (27) = 8π(ρ + ρq ) + E , (20) where β and β being integration constants and we assume 1 f 2 1 2 (T − ) fT − = 8π(pr − ρq ) − E , (21) β = r 2 2 2 0 for simple case. a 1 Now from equations (14), (15), (17), (18), (22) and (23) −b e + T fTT α = ρ 2 r (taking 1) we obtain the equation in : γ 2 2 −Ar α β γ T −b a a 1 8πr ρ (1 + ξ)− β1e ρ(αBr + βCr + γ Ar ) + + e + + (a − b ) fT 2 2 4 2r −8πr 2ζ = 0. (28) f 1 2 − = 8π pt + (3wq + 1)ρq + E , (22) 2 2 Solving this equation, we get the value of energy density as

γ β −Ar (α α + β β + γ γ ) + β2 −2Arγ (α α + β β + γ γ )2 + π 2 4ζ( + ξ) 1e Br Cr Ar 1 e Br Cr Ar 256 r 1 ρ = . (29) 16πr 2(1 + ξ)

Then we found the expressions of the radial pressure, transverse pressure and density for quintessence ﬁeld as

γ ξβ −Ar (α α + β β + γ γ ) + ξ β2 −2Arγ (α α + β β + γ γ )2 + π 2 4ζ( + ξ) 1e Br Cr Ar 1 e Br Cr Ar 256 r 1 pr = 16πr 2(1 + ξ) 16πζr 2(1 + ξ) − . (30) −Arγ α β γ 2 −2Arγ α β γ 2 2 4 β1e (αBr + βCr + γ Ar ) + β e (αBr + βCr + γ Ar ) + 256π r ζ(1 + ξ) 1 1 −Arγ 1 α−2 β−2 1 α−1 β−1 α−1 β−1 pt = β e (α(α − 1)Br + β(β − 1)Cr ) + (αBr + βCr )(αBr + βCr 8π 1 2 4 β 2 β −Arγ β β γ −1 1 α−2 β−2 γ −2 2 q 1 1 1e γ 1 2 −γ Ar )+ (αBr +βCr −γ Ar ) − − − (3wq + 1) (γ Ar − 1)+ + 2 2 r 4 2 8π r 2 r 2 2

− γ α β γ γ 2 β Ar (α + β + γ ) + β2 −2Ar (α α + β β + γ γ )2 + π 2 4ζ( + ξ) q 1e Br Cr Ar 1 e Br Cr Ar 256 r 1 − − . (31) r 4 16πr 2(1 + ξ)

− b −Arγ 2 e 2 cot θ 1 β e γ β β q T f = 0, (23) ρ = 1 (γ Ar − 1) + 1 + 2 − 2 TT q π 2 2 4 2r 8 r r 2 r r ( ) −Arγ α β γ 1 2 b q r β1e (αBr + βCr + γ Ar ) E(r) = 4πr σ(r)e 2 dr = (24) − 2 2 πr2( + ξ) r 0 r 16 1 ( ) β2 −2Arγ (α α+β β +γ γ )2+ π2 4ζ( +ξ) where q r is the total charge within a sphere of radius r. 1 e Br Cr Ar 256 r 1 ( ) + . Here, we take the total charge q r as in the power law 16πr2(1+ξ) form: (32) m q(r) = q0r (25)

The equation of state parameters wr and wt along radial where q0 > 0, m > 0. Here we consider the ﬂuid source behaves as modiﬁed Chaplygin gas (MCG) whose equation and transversal directions of our model are given by of state is [74] 123 Eur. Phys. J. C (2019) 79 :919 Page 5 of 16 919

⎧ ⎫ 2 ⎨ 2 ⎬ pr 16πζr (1+ξ) wr = = ξ− (33) ρ ⎩ γ γ ⎭ β −Ar (α α+β β +γ γ )+ β2 −2Ar (α α+β β +γ γ )2+ π 2 4ζ( +ξ) 1e Br Cr Ar 1 e Br Cr Ar 256 r 1 and We consider three different compact stars as Vela X- p w = t 1 (CS1), SAXJ1808.4-3658 (CS2) and 4U1820-30 (CS3) t ρ (34) with their masses and radii in Table 1. With respect to Table 1, where pt and ρ are given by equations (31) and (29). we take three sets of values of α, β and γ to get three different The above solutions are viable if ξ =−1. sets of numerical values of A, B and C for CS1, CS2 and CS3 in Tables 2, 3, 4 respectively. Now we have drawn different figures for physical quanti- 4 Matching conditions ties ρ, pr , ρ + 3pr , ρq and pt using Eqs. (29)Ð(32) with the values of different values of A, B and C taking from Tables 2, By matching condition, many researchers have compared the 3, 4.Figs.1, 2, 3 ensure that our proposed model is a candi- ρ> ,ρ + < w < − 1 exterior solution with the interior solution [24,58,60,62]. We date of dark energy as 0 3pr 0 i.e., r 3 make correspondence between our interior solution and the (See equation (33)). Again, from Fig. 3, we can conclude that exterior solution evoked by Reissner-Nordstrm metric whose our model corresponds to quintessence dark energy model, line element is not phantom dark energy candidate due to ρ + 3pr < 0 i.e., w < − 1 and ρ + p > 0 i.e., w > −1 (See Figs. 10, 11, 2M Q2 2M Q2 −1 r 3 r r ds2 =− 1 − + dt2 + 1 − + dr2 12). We have taken quintessence field in transverse direction r r 2 r r 2 of anisotropic compact star in f (T ) gravity with modified +r 2(dθ 2 + sin2 θdφ2). (35) Chaplygin gas. So, Figs. 4, 5 indicate that for quintessence where m, r and Q are mass, radius and charge respectively. field ρq > 0 and pt < 0. We assume the boundary of interior and exterior regions occurs at r=R. So on the boundary, we have m(r)=M and − + 5 Physical analysis − + − + ∂g ∂g g = g , g = g , tt = tt , (36) tt tt rr rr ∂r ∂r The central density ρ0 and central radial pressure p0 are given where − and + indicate to interior and exterior solutions. by Now, using (36) and the metrics (16), (35), we can obtain

⎧ 2 2 2 ⎪ Bβ1+ B β +64π ζ(1+ξ) ⎨⎪ 1 α = ,β= γ = , 8π(1+ξ) when 2 3

ρ0 = lim ρ = r→0 ⎪ Cβ + C2β2+64π2ζ(1+ξ) ⎩⎪ 1 1 α = ,β= ,γ= α = ,β= ,γ= 8π(1+ξ) when 3 2 4or 4 2 3 ⎧ ⎪ −32π2ζ(1+ξ)+Bβ ξ{Bβ + B2β2+64π2ζ(1+ξ)} ⎪ 1 1 1 ⎪ when α = 2,β= γ = 3, ⎪ 4π(1+ξ){Bβ + B2β2+64π2ζ(1+ξ)} ⎪ 1 1 ⎨ − π2ζ( +ξ)+ β ξ{ β + 2β2+ π2ζ( +ξ)} p0 = lim pr = 32 1 C 1 C 1 C 1 64 1 r→0 ⎪ when α = 3,β= 2,γ= 4, ⎪ 4π(1+ξ){Cβ + C2β2+64π2ζ(1+ξ)} ⎪ 1 1 ⎪ ⎩ or α = 4,β= 2,γ= 3.

⎫ 1 2M Q2 A =− γ ln 1− + , ⎪ R R R2 ⎪ ⎬⎪ In the following subsections, we investigate the nature of −1 1 2M Q2 M Q2 2M Q2 B = α −β ln 1− + +2 − 1− + , the anisotropic compact star as follows: R (α−β) R R2 R R2 R R2 ⎪ ⎪ − ⎪ 2 2 2 1 ⎭⎪ C = 1 α − 2M + Q − M − Q − 2M + Q Rβ (α−β) ln 1 R R2 2 R R2 1 R R2 (37)

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Table 1 Different compact stars M Q2 M(M) R( )μ= μ = are taken with their masses and Compact stars Km M R c R2 radii Vela X-1 (CS1) 1.77 9.56 0.273091 0.0133624 SAXJ1808.4-3658 (CS2) 1.435 7.07 0.299 0.0266898 4U1820-30 (CS3) 2.25 10 0.332 0.0133208

Table 2 The values of A, B and Compact stars A (Km−2) B (Km−2) C (Km−2) C have been obtained with α = β = γ = 2, 3 from Table 1 Vela X-1 (CS1) 0.0008710312 −0.037147216 0.003014661038 using equation (37) SAXJ1808.4-3658 (CS2) 0.0023968248 −0.07625293 0.008388596829 4U1820-30 (CS3) 0.0010517646 −0.049798583 0.003928093861

Table 3 The values of A, B and Compact stars A (Km−2) B (Km−2) C (Km−2) C have been obtained with α = β = γ = 3, 2and 4 from Vela X-1 (CS1) 0.00009111205 0.0074533913 −0.037147217 Table 1 using equation (37) SAXJ1808.4-3658 (CS2) 0.00033901341 0.0083885968 −0.076252931 4U1820-30 (CS3) 0.00010517646 0.0039280939 −0.049798584

Table 4 The values of A, B and Compact stars A (Km−2) B (Km−2) C (Km−2) C have been obtained with α = β = γ = 4, 2and 3 from Vela X-1 (CS1) 0.00087103116 0.00015767056 −0.022737137 Table 1 using equation (37) SAXJ1808.4-3658 (CS2) 0.0023968248 0.00059325296 −0.046599241 4U1820-30 (CS3) 0.0010517646 0.000196404693 −0.030158115

Fig. 1 Variations of energy density ρ (MeV/fm3) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

3 Fig. 2 Variations of radial pressure pr (MeV/fm ) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

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3 Fig. 3 Variations of ρ + 3pr (MeV/fm ) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

3 Fig. 4 Variations of ρq (MeV/fm ) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

3 Fig. 5 Variations of transversal pressure pt (MeV/fm ) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

5.1 Anisotropic effects which implies that the anisotropic force is attractive like quintessence ﬁeld. Since, our model consists of ordinary mat- ( − ) = 2 pt pr The anisotropic force ( r ) of our anisotropic ter and quintessence ﬁeld so the effective anisotropic force is quintessence compact object is attractive and that is why our model is very much compatible.

1 − γ 1 α− β− 1 α− β− α− β− γ − = β e Ar (α(α − 1)Br 2 + β(β − 1)Cr 2) + (αBr 1 + βCr 1)(αBr 1 + βCr 1 − γ Ar 1) 8πr 1 2 4 β 2 β −Arγ β β 2 1 α−2 β−2 γ −2 2 q 1 1 1e γ 1 2 q + (αBr + βCr − γ Ar ) − − − (3wq + 1) (γ Ar − 1) + + − 2 2 r 4 2 8πr r 2 r 2 2 r 4

γ −Ar α β γ 2 −2Arγ α β γ 2 2 4 β1e (αBr + βCr + γ Ar ) + β e (αBr + βCr + γ Ar ) + 256π r ζ(1 + ξ) − 1 16πr 3 16πζr(1 + ξ) − . (38) β −Arγ (α α + β β + γ γ ) + β2 −2Arγ (α α + β β + γ γ )2 + π 2 4ζ( + ξ) 1e Br Cr Ar 1 e Br Cr Ar 256 r 1

From Fig. 6, we have noticed that is always nega- Now, we take derivatives of energy density and radial pres- tive i.e., pt < pr for three compact stars (CS1, CS2, CS3) sure with respect to radius to see where the energy density and

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Fig. 6 Variations of versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

ρ 2ρ 2 Fig. 7 d dpr d d pr r A B C Variations of derivatives of dr , dr , dr2 and dr2 with respect to (km) with the numerical values of , and from Table 2 radial pressure achieve their maximum values. With respect From Data of Table 3: dρ dp 2ρ 2 to three Tables 2, 3 and 4,ifwemake = 0 and r = 0 d < d pr < r = . dr dr dr2 0 and dr2 0at 10 66 for CS1 model, at then r = 7.69 for CS2 model and at r = 10.38 for CS3 model From Data of Table 1: respectively. So, the energy density and radial pressure take 2ρ 2 maximum values at r = 10.66 for CS1 model, at r = 7.69 d < 0 and d pr < 0atr = 10.11 for CS1 model, dr2 dr2 for CS2 model and at r = 10.38 for CS3 model respectively. at r = 7.31 for CS2 model and at r = 9.9 for CS3 model However, we conclude for Table 2: with respect to Fig. 7, respectively. So, the energy density and radial pressure take ρ d > 0 always, so the energy density is increasing and dpr maximum values at r = 10.11 for CS1 model, at r = 7.31 dr dr is positive at first then changes its sign to negative, so the for CS2 model and at r = 9.9 for CS3 model respec- radial pressure is increasing first and then decreases; Table 3: tively. dρ dpr with respect to Fig. 8, dr and dr both are positive from From Data of Table 2: beginning then turn to negative sign, so consequently, the 2ρ 2 d < 0 and d pr < 0atr = 8.29 for CS1 model, at energy density and the radial pressure both are increasing first dr2 dr2 ρ = . = . d dpr r 7 06 for CS2 model and at r 9 63 for CS3 model and then decrease; Table 4: with respect to Fig. 9, dr and dr respectively. So, the energy density and radial pressure take both are positive from beginning then turn to negative sign maximum values at r = 8.29 for CS1 model, at r = 7.06 like the previous case, so the energy density and the radial for CS2 model and at r = 9.63 for CS3 model respec- pressure both are also increasing first and then decrease in tively. this case.

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ρ 2ρ 2 Fig. 8 d dpr d d pr r A B C Variations of derivatives of dr , dr , dr2 and dr2 with respect to (km) with the numerical values of , and from Table 3

ρ 2ρ 2 Fig. 9 d dpr d d pr r A B C Variations of derivatives of dr , dr , dr2 and dr2 with respect to (km) with the numerical values of , and from Table 4

5.2 Energy conditions parts: Null Energy Condition (NEC), WeakEnergy Condition (WEC) and Strong Energy Condition (SEC). These are very The most crucial physical properties are energy conditions useful in general relativity and modiﬁed gravity [24,59,62]. to verify the existence of realistic matter distribution in this The energy conditions are: stellar model. These energy conditions are divided into three

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Fig. 10 Variations of energy conditions versus r (km) with the numerical values of A, B and C from Table 2

: ρ + E2 ≥ , We can check stability of our proposed model by adiabatic NEC 8π 0 : ρ + ≥ ,ρ+ + E2 ≥ , index [77Ð80] which depicts the stiffness of the EoS param- WEC pr 0 pt 4π 0 E2 eter for our model. By this theory, many researchers have SEC : ρ + pr + 2pt + π ≥ 0. 4 examined the dynamical stability against infinitesimal radial From Figs. 10, 11, 12, we can conclude that our model perturbation of the realistic as well as non-realistic stellar satisfies first two conditions for arbitrary values of parameters objects. For anisotropic fluid, the adiabatic index is defined but the last condition is satisfied only when β2 < 0. as ρ dp 5.3 Stability analysis = 1 + r . pr dρ

To analyze stability of stellar structure against external fluc- According to [77], the evaluated value of this adiabatic tuation plays an essential key role for any physically con- index should be greater than 4/3. Fig. 17 ensures that our sistent model. According to Herrera’s cracking condition proposed model satisfies this condition i.e., our anisotropic v2 = / ρ [24,59,62,75], the sound speed square ( s dp d )must quintessence compact star model in f (T ) gravity with mod- [ , ] lie in the interval 0 1 to be a physically stable stellar object. ified Chaplygin gas maintains stability range also. For our anisotropic quintessence compact star model, we ≤ v2 ≤ ≤ v2 ≤ have 0 sr 1 and 0 st 1 always (See Figs. 13, 14) v2 v2 where sr and st are sound speeds for radial and transversal 5.4 Effective mass and compactness components. Next, we calculate the difference of two speeds of our According to [24], the mass function within the radius r is model to examine whether transversal sound speed square is defined by greater than the radial one or not. Actually, Herrera’s cracking condition [75] has prospected a different way to find out the r 2 potentially stable or unstable model. If the radial sound speed m(r) = 4πr ρdr 0 r square is greater than the transversal sound speed square 1 − γ α β γ = β e Ar (αBr + βCr + γ Ar ) then the model is potentially stable otherwise it is poten- 4(1 + ξ) 1 0 tially unstable. From Fig. 15, we can decide that our model − γ α β γ |v2 − v2 |≤ + β2e 2Ar (αBr +βCr +γ Ar )2+256π2r4ζ(1+ξ) dr. is potentially stable. Clearly, st sr 1 is verified [76] 1 from Fig. 16. (39) 123 Eur. Phys. J. C (2019) 79 :919 Page 11 of 16 919

Fig. 11 Variations of energy conditions versus r (km) with the numerical values of A, B and C from Table 3

Fig. 12 Variations of energy conditions versus r (km) with the numerical values of A, B and C from Table 4

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v2 Fig. 13 Variations of sr versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

v2 Fig. 14 Variations of st versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

v2 − v2 Fig. 15 Variations of st sr versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

|v2 − v2 | Fig. 16 Variations of st sr versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

Fig. 17 Variations of versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

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Table 5 Calculated values of mass, energy density and pressure of our model from Table 2

2 Compact stars Mass standard Mass from ρ0 (gm/cc)ρR (gm/cc) p0 (dyne/cm ) data (in km) model (in km)

Vela X − 1 (CS1) 2.61075 2.64376 2.403810507 × 1015 2.403792768 × 1015 −2.160060323 × 1036 SAXJ1808.4 − 3658 (CS2) 2.11662 2.10657 1.431916338 × 1015 1.431894946 × 1015 −1.285972818 × 1036 4U1820 − 30 (CS3) 3.31875 3.30465 1.739754757 × 1015 1.739733365 × 1015 −1.562934990 × 1036

Table 6 Calculated values of mass, energy density and pressure of our model from Table 3

2 Compact stars Mass standard Mass from ρ0 (gm/cc)ρR (gm/cc) p0 (dyne/cm ) data (in km) model (in km)

Vela X − 1 (CS1) 2.61075 2.60238 1.245098292 × 1015 1.245174948 × 1015 −1.118544296 × 1036 SAXJ1808.4 − 3658 (CS2) 2.11662 2.12999 1.563552689 × 1015 1.563723828 × 1015 −1.404243397 × 1036 4U1820 − 30 (CS3) 3.31875 3.31554 1.328755055 × 1015 1.328878061 × 1015 −1.193639739 × 1036

Table 7 Calculated values of mass, energy density and pressure of our model from Table 4

2 Compact stars Mass standard Mass from ρ0 (gm/cc)ρR (gm/cc) p0 (dyne/cm ) data (in km) model (in km)

Vela X − 1 (CS1) 2.61075 2.64586 1.455574550 × 1015 1.455603073 × 1015 −1.307867618 × 1036 SAXJ1808.4 − 3658 (CS2) 2.11662 2.10746 1.627389393 × 1015 1.627433961 × 1015 −1.461994045 × 1036 4U1820 − 30 (CS3) 3.31875 3.31585 1.662385577 × 1015 1.662423014 × 1015 −1.493684884 × 1036

Now, we draw the mass functions with respect to radius from it lies between 1/4 and 1/2[81]. According to [82], twice of the numerical values of Tables 2, 3 and 4 respectively. As the compactification factor (2M/R) takes maximum allowed r → 0, m(r) → 0inFig.17 for three different compact value 8/9 for our model (See Tables 8, 9, 10). stars so we can conclude that the mass function is regular at origin. We also notice that the mass function is monotonic 5.6 Surface redshift increasing with respect to radius. Again, the form of compactness of the star is defined by The redshift function is defined as [24,59,62] u(r) [24] 1 ( ) zs = − 1. (41) ( ) = m r ( ) u r 1 − 2m r r r r 1 − γ α β γ = β e Ar (αBr + βCr + γ Ar ) 4(1 + ξ)r 1 We calculate the values of redshift for CS1, CS2 and CS3 0 γ + β2e−2Ar (αBrα+βCrβ +γ Arγ )2+256π2r4ζ(1+ξ) dr. in Tables 8, 9, 10. According to Bohmer and Harko [83], 1 the surface redshift can be arbitrarily large, it must be less (40) than ≤ 5 for an anisotropic star in the appearance of a cos- mological constant. Though our model does not contain cos- Now, we evaluate the values of mass, central energy den- mological constant but the maximum surface redshift from ≤ sity, central radial pressure and radial pressure at the bound- our model is always 5 (See Tables 8, 9, 10 and Fig. 20). ary for the different three compact stars from our model to So, our anisotropic quintessence compact star model is quite compare with observational data (See Tables 5, 6, 7). reasonable.

5.5 Relation between mass and radius 6 Conclusions In this section, we study the relation between mass and radius for three different compact stars to check whether all data are This work has given out the quintessence dark energy behav- lying in the desired range or not. The factor “M/R” is called ior of the anisotropic compact star model in f (T ) gravity compactiﬁcation factor. Weconclude from Tables 8, 9, 10 that with modiﬁed Chaplygin gas consisting of ordinary matter 123 919 Page 14 of 16 Eur. Phys. J. C (2019) 79 :919

Table 8 Calculated values of the desired parameters of our model from Table 2 ρ Compact stars M (standard data) M from model 2M < 8 0 z R R R 9 ρR s Vela X − 1 (CS1) 0.273091 0.276544 0.553088 1.000007416 0.495853 SAXJ1808.4 − 3658 (CS2) 0.299381 0.297989 0.595918 1.000014940 0.573132 4U1820 − 30 (CS3) 0.331875 0.330465 0.660930 1.000012296 0.717336

Table 9 Calculated values of the desired parameters of our model from Table 3 ρ Compact stars M (standard data) M from model 2M < 8 0 z R R R 9 ρR s Vela X − 1 (CS1) 0.273091 0.272215 0.544431 0.999938437 0.481572 SAXJ1808.4 − 3658 (CS2) 0.299381 0.301272 0.602543 0.999890556 0.586189 4U1820 − 30 (CS3) 0.331875 0.331554 0.663108 0.999907436 0.722878

Table 10 Calculated values of the desired parameters of our model from Table 4 ρ Compact stars M (standard data) M from model 2M < 8 0 z R R R 9 ρR s Vela X − 1 (CS1) 0.273091 0.276764 0.553527 0.999980404 0.496589 SAXJ1808.4 − 3658 (CS2) 0.299381 0.298085 0.596170 0.999972614 0.573622 4U1820 − 30 (CS3) 0.331875 0.331585 0.663170 0.999977480 0.723037

Fig. 18 Variations of m(r) versus r (km) with the numerical values of A, B and C from Tables 1, 2 and 3 respectively

together with quintessence field along tangential component. highly negative, ρ + 3pr < 0, energy density corresponding Using the diagonal tetrad field we have obtained the equations to quintessence field (ρq ) is positive and transversal pres- of motion by taking of quintessence field and electromagnetic sure (pt ) is negative due to considering quintessence field field with respect to spherically symmetric metric. We have along tangential component of our model. Anisotropic force taken the forms of a(r) and b(r) as a(r) = Brα + Crβ and has been calculated to see whether this is positive or negative. b(r) = Ar γ where we have chosen three sets values of these From Fig. 6, we can say that <0 always for CS1, CS2 and parameters i.e., α = 2,β = γ = 3; α = 3,β = 2,γ = 4 CS3 i.e., there exists attractive force like quintessence field to and α = 4,β = 2,γ = 3. Here, we have assumed the total ensure our model to be realistic. Next, we have also noticed m charge in the form q(r) = q0r . Next, we have solved all from Figs. 7, 8, 9 that the both the energy density (ρ) and of these equations to get ρ, pr , ρq and pt in terms of r and radial pressure (pr ) are monotonic increasing in some region some constants. Using matching condition, we have evalu- and monotonic decreasing in some places with respect to r ated the values of A, B and C in Tables 2, 3, 4 with respect and they attain various maximum values for CS1, CS2 and to Table 1 for those three cases of the parameters α, β and γ . CS3 with respect to three sets values of α, β, γ . By these values of constants we have plotted the figures for From Figs. 10, 11, 12, we have noticed that all energy the above mentioned physical quantities for CS1, CS2 and conditions except SEC are satisfied for all values of param- CS3 (See Figs. 1, 2, 3, 4, 5). From these figures we can con- eters for our proposed model. The last condition is satisfied clude that our model corresponds to quintessence dark energy only for β2 < 0. By stability analysis given on the basic of ρ

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Fig. 19 Variations of u(r) versus r (km) with the numerical values of A, B and C from Tables 2, 3 and 4 respectively

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