In this paper, we study the existence of strange star in the background of f(T) modified gravity where T is a scalar torsion. In KB metric space, we derive the equations of motion using anisotropic property within the spherically strange star with modified Chaplygin gas in the framework of modified f(T) gravity. Then we obtain many physical quantities to describe the physical status such as anisotropic behavior, energy conditions, and stability. By the matching condition, we calculate the unknown parameters to evaluate the numerical values of mass, surface redshift, etc., from our model to make comparison with the observational data.
1. Introduction
In modern cosmology, cosmic acceleration is an interesting discovery. The observation of type Ia supernovae (SNeIa) together with the cosmic microwave background (CMB), large scale structure surveys (LSS), and Wilkinson Microwave Anisotropy Probe (WMAP) [1–4] ensures the presence of an exotic energy component dominating our universe which is entitled as dark energy (DE) having equation of state p=wρ with strong negative pressure. For accelerating expansion w must satisfy the range w<-1/3. If -1<w<-1/3 then it belongs to quintessence phase and if w<-1, then it belongs to phantom regime. In particular, when w=-1⇒p=-ρ then the equation of state of the interior region of a Gravastar (gravitationally vacuum condense star) is described in [5–10]. There are many investigations of this cosmic expansion and nature of DE based on different ways. These efforts can be classified as follows: (i) to modify the entire cosmic energy by including new components of DE and (ii) to modify Einstein-Hilbert action to get different types of modified theories of gravity such as f(R) gravity [11, 12], R being the Ricci scalar; f(T) gravity [13], T being the torsion; f(R,T) gravity [14], Gauss-Bonnet gravity, i.e., f(G) modified gravity [15], etc. Here we assume only f(T) gravity theory.
Since general relativity is similar to f(T), this theory could be a substitute form of the generalized general relativity, named as f(T) theory of gravity. The teleparallel equivalence of gravity (TEGR) gives the concept of this theory. There is defined Riemann-Cartan space-time together with Weitzenbock connections rather than Levi-Civita connections in f(T) theory. Here, nonzero torsion and zero curvature appear in the background space-time. Einstein gives this definition of space-time to give an idea of gravitation related to tetrad and torsion. Instead of metric field, tetrad field takes an important role in dynamic field in TEGR.
In f(T) gravity, equations of motions are second-order differential equations like GR whereas equations of motion are fourth-order inf(R) gravity. So, the former one is more convenient than the latter one. Recently, a wide interest has been seen to study the f(T) gravity [16–20]. There is no doubt of excellence of f(T) theory to explain the cosmic acceleration and analysis on large scale (clustering of galaxies) [21]. But GR must be a fantabulous agreement with solar system test and pulsar observation [22].
In theoretical astrophysics, f(T) version of BTZ black hole solutions has been calculated as f(T) theory was supported for examining the effects of f(T) models in 3 dimensions [23]. Later on [24], violation of Lorentz invariance made the first violation of black hole thermodynamics in f(T) gravity. Recently there are some static solutions which are spherically symmetric with charged source in f(T) theory [25]. The physical conditions have been studied [26] for the existence of astrophysical stars in f(T) theory after obtaining a large group of static perfect fluid solutions [27]. Capozziello et. al [28] have shown that, instead of f(R) gravity, f(T) removes the singularities for the exact black hole solution in D-Dimensions. Wormhole solution has been studied under f(T) gravity by Sharif and Rani [29]. They have also investigated f(T) gravity for static wormhole solution to verify energy conditions [30]. Again, for charged noncommutative wormhole solutions in f(T) gravity, Sharif and Rani [31, 32] have seen that this solution exists by violating energy conditions.
Generally, perfect fluid (isotropic fluid) inside the stellar object to study stellar structure and evolution is assumed because there exists isotropic pressure inside the fluid sphere. However, present observation shows that the fluid pressure of the highly compact astrophysical objects like X-ray pulsar, Her-X-1, X-ray buster 4U 1820-30, millisecond pulsar SAXJ1804.4-3658, etc. becomes anisotropy in nature which means the pressure can be rotten into two components such that one is radial pressure (pr) and the other is transverse pressure (pt). Now, Δ=pt-pr is known as the anisotropic factor. The anisotropy may arise for the different cases such as the existence of solid core, in presence of type P superfluid, phase transition, rotation, magnetic field, mixture of two fluids, and existence of external field. Generally, strange quark matter contains u, d, and s quarks. There are two ways to classify the formation of strange matter [33]. One way is the transformation of the quark hadron phase in the early universe and the other way is the reformation of neutron stars to strange matter at ultrahigh densities. A strange star is composed of the strange matter. Again the strange star can be classified into two types: Type I strange star withM/R>0.3 and Type II strange star with0.2<M/R<0.3. Depending on mass, radius, and energy density, the strange star is distinguished from the neutron star [34]. It has been the most interesting topic to study the models of anisotropic stars for the last periods in GR and modified theories of gravity [35]. There have been many discussions about anisotropic star models in [36–41]. It is becoming a scientific tool to discuss the compact star models with Krori-Barua metric [42–44]. It has been seen in [45] that neutron star solution in f(T) gravity model is possible if f(T) is a linear function of scalar torsion.
Recently, Abbas and his collaborations [46–50] have discussed the anisotropic compact star models in GR, f(R), f(G), and f(T) theories in diagonal tetrad case with Krori and Barua (KB) metric. Abbas et al. [49] have studied anisotropic strange star which corresponds to quintessence dark energy model with the help equation of state p=αρ, where 0<α<1. A study of strange star with MIT bag model in the framework of f(T) gravity has been done by Abbas et al. [51]. Here, our main motivation of this paper is to study the anisotropic strange star models in the framework of f(T) gravity with diagonal tetrad in presence of electric field and modified Chaplygin gas. In Section 2, we give a brief idea of f(T) gravity. In Section 3, we study anisotropic quintessence strange star in f(T) gravity with the help of modified Chaplygin gas. In Section 4, we analyze many physical phenomenon of this whole system. By matching of two metrics, the unknown constants are found out. We also make stability analysis. In Section 5, we calculate the mass function, compactness, and surface redshift function from our model to compare with observational data and finally, in Section 6, we give the summarization.
2. fT Gravity: Fundamentals
In this section, we briefly overview the basics of f(T) gravity. We define the torsion and the con-torsion tensor as follows [51]:(1)Tμνα=Γνμα-Γμνα=eiα∂μeνi-∂νeμi(2)Kαμν=-12Tαμν-Tανμ-Tαμνand the components of the tensor Sαμν are defined as(3)Sαμν=12Kαμν+δαμTββν-δανTββμ;one can write the torsion scalar as(4)T=TμναSαμν
Now, one can define the modified teleparallel action by replacing T with a function of T, in analogy to f(R) gravity [52, 53], as follows:(5)S=∫d4xe116πfT+LMatterΦAwhere we used G=c=1 and ΦA is matter fields.
The ordinary matter is an anisotropic fluid so that the energy-momentum tensor is given by(6)Tμν=ρ+ptuμuν-ptδμν+pr-ptvμvνwhere uμ is the four-velocity, vμ is radial four vectors, ρ is the energy density, pr is the radial pressure, and pt is transverse pressure. Further, the energy-momentum tensor for electromagnetic field is given by(7)Eμν=14πgδωFμδFων-14gμνFδωFδωwhere Fμν is the Maxwell field tensor defined as(8)Fμν=Φν,μ-Φμ,νand Φμ is the four potential.
3. Anisotropic Strange Quintessence Star in f(T) Gravity
We consider the KB metric [42] describing the interior space-time of a strange star(9)ds2=-eardt2+ebrdr2+r2dθ2+sin2θdϕ2where we assume a(r) and b(r) are(10)ar=Br2+Cr3,br=Ar3where A, B, and C are arbitrary constants. For the charged fluid source with density ρ(r), radial pressure pr(r), and tangential pressure pt(r), the Einstein-Maxwell (EM) equations reduce to the form (G=c=1) [51](11)Tr=2e-bra′+1r(12)T′r=2e-bra″+1r2-Tb′+1rwhere the prime ′ denotes the derivative with respect to the radial coordinate r.
Now the equations of motion for anisotropic fluid are [51](13)4πρ+E2=f4-T-1r2-e-bra′+b′fT2(14)4πpr-E2=T-1r2fT2-f4(15)4πpt+E2=T2+e-ba″2+a′4+12rfT2-f4(16)cotθ2r2T′fTT=0(17)Er=1r∫0r4πr2σeλ/2dr=qrr2where q(r) is the total charge within a sphere of radius r.
We introduce the modified Chaplygin gas (MCG) having equation of state [54](18)pr=ξρ-ζραwhere ξ, α, and ζ are free parameters of the model.
From (16) we get(19)fT=βT+β1where β and β1 are integration constants and we assume β1=0 for simple case.
Now from (10), (11), (13), (14), (18), and (19) we obtain the equation in ρ(20)8π1+ξρα+1-βe-Ar32B+3Cr+3Arρα-8πζ=0Here we take α=1; then (20) reduces to the quadratic equation in ρ(21)8π1+ξρ2-βe-Ar32B+3Cr+3Arρ-8πζ=0Solving this equation we get the value of energy density as(22)ρ=2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξand corresponding components are(23)pr=ξ2Bβ+3Cβr+3Aβr+ξ256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξ-ζ2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr2/16πeAr31+ξ(24)ρ+3pr=1+3ξ2Bβ+3Cβr+3Aβr+1+3ξ256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξ-3ζ2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr2/16πeAr31+ξ(25)E2=β2r2eAr3-1+3Ar3+β2r2-2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr24eAr31+ξ(26)pt=βe-Ar38π2B+3C-3Ar+32Br32C+A+34rC-A3Cr3+2+1r2-β8πr2+2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξ
Now from Figures 1 and 2, we conclude that anisotropic strange star in f(T) gravity with modified Chaplygin gas acts as a dark energy candidate due to ρ>0,pr<0. Again with the help of Figures 3 and 4, we notice that the equation of state w=pr/ρ lies between -1/3 and -1; i.e., the corresponding model belongs to quintessence phase not phantom phase.
This figure represents the variation of ρ versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of pr versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of ρ+3pr versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of ρ+pr versus r (km) for the strange star taking β=1, β=2, and β=3.
The amount of net charge inside a sphere having radius r is(27)q=r2β2r2eAr3-1+3Ar3+β2r2-2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr24eAr31+ξ
4. Physical Analysis
The central density ρ0 and central radial pressure p0 are given by (28)ρ0=ρr=0=2Bβ+256ζπ21+ξ+4B2β216π1+ξand (29)p0=prr=0=2Bβξ+ξ256ζπ21+ξ+4B2β216π1+ξ-ζ2Bβ+256ζπ21+ξ+4B2β2/16π1+ξ
In this section, we investigate the nature of the anisotropic compact star as the following subsection.
Figures 1–5 represent the plots by taking B=5, C=1, A=2, ξ=2, and ζ=1.
This figure represents the variation of pt versus r (km) for the strange star taking β=1, β=2, and β=3.
4.1. Anisotropic Behavior
Now we take the derivatives of (22) and (23) with respect to r, given by(30)dρdr=3βC+A+768ζe2Ar3Aπ21+ξr2+768Aζe2Ar3π21+ξr2+23Cβ+3Aβ2Bβ+3Crβ+3Arβ/2256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ216eAr3π1+ξ-24eAr3Aπr21+ξβ2B+3Cr+3Ar+256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ216eAr3π1+ξ2and(31)dprdr=ξ3βC+A+1536ζe2Ar3Aπ21+ξr2+23Cβ+3Aβ2Bβ+3Crβ+3Arβ/2256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ216eAr3π1+ξ+16ζeAr3π1+ξ+3βC+A+1536ζe2Ar3Aπ21+ξr2+23Cβ+3Aβ2Bβ+3Crβ+3Arβ/2256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ2β2B+3Cr+3Ar+256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ22-48ζeAr3Aπr21+ξβ2B+3Cr+3Ar+256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ2-24ξeAr3Aπr21+ξβ2B+3Cr+3Ar+256ζe2Ar3π21+ξ+2Bβ+3Crβ+3Arβ2128e2Ar3π21+ξ2
Now we present the evolution of dρ/dr and dpr/dr by Figures 6 and 7. Figure 6 shows that dρ/dr decreases keeping dρ/dr<0 (as energy density decreases) and Figure 7 shows that dpr/dr decreases keeping dpr/dr<0 (as for dark energy pressure is negatively very high; i.e., pressure decreases negatively). From Figures 6, 7, 8, and 9 we notice that, at r=1.46,(32)dρdr=0,dprdr=0,d2ρdr2<0,d2prdr2<0.This points out that the energy density and radial pressure have maximum value at r=1.46 of the quintessence strange star model in f(T) gravity.
This figure represents the variation of dρ/dr versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of dpr/dr versus r (km) for strange the star taking β=1, β=2, and β=3.
This figure represents the variation of d2ρ/dr2 versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of d2pr/dr2 versus r (km) for the strange star taking β=1, β=2, and β=3.
Now the anisotropic stress (Δ=pt-pr) is as follows [51]:(33)Δ=βe-Ar38π2B+3C-3Ar+32Br32C+A+C-A94Cr4+32r+1r2-β8πr2+2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξ-ξ2Bβ+3Cβr+3Aβr+ξ256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr216πeAr31+ξ-ζ2Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr2/16πeAr31+ξ
From Figure 10, we notice that Δ>0 for β=1,-6 which imply that the anisotropic stress is outwardly directed and there exists repulsive gravitational force for the strange star and for β=-15, Δ<0 in somewhere implying the existence of attractive gravitational force and Δ>0 in the remaining part implying the existence of repulsive gravitational force of the strange star.
This figure represents the variation of Δ versus r (km) for the strange star taking β=1, β=-6, and β=-15.
Figure 11 shows that E2 is decreasing with the increment of the radial coordinate.
This figure represents the variation of E2 versus r (km) for the strange star taking β=1, β=2, and β=3.
4.2. Energy Conditions
Energy conditions are very useful tools to discuss cosmological geometry in general relativity and modified gravity [10, 48, 51]. These include null energy condition (NEC), weak energy condition (WEC), and strong energy condition (SEC), given as(34)NEC:ρ+E28π≥0,WEC:ρ+pr≥0,ρ+pt+E24π≥0,SEC:ρ+pr+2pt+E24π≥0. Figures 6, 7, 8, and 9 represent the plots of dρ/dr, dpr/dr, d2ρ/dr2, and d2pr/dr2 with respect to r to show the maximality of density and radial pressure at r=1.46 of the strange star taking B=1, C=7, A=0.1, ξ=9, and ζ=10.
Figure 10 represents the plots of Δ with respect to r to show the presence of repulsive and attractive force of the strange star and Figure 11 represents the plot of E2 with respect to r taking B=10, C=1, A=10, ξ=2, and ζ=1.
From Figures 12, 4, 13, and 14, we observe that the interior of our proposed strange star model satisfies all energy conditions.
This figure represents the variation of ρ+E2/8π versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of ρ+pt+E2/4π versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of ρ+pr+2pt+E2/4π versus r (km) for the strange star taking β=1, β=2, and β=3.
4.3. Matching Conditions
Many authors have worked on the matching condition to compare the exterior solution with the interior solution [10, 47, 49, 51]. We correspond the exterior geometry with our interior solution, evoked by the Schwarzschild solution which is given by the line element(35)ds2=-1-2Mrdt2+1-2Mr-1dr2+r2dθ2+sin2θdϕ2.
The continuity of the metric components gtt, grr, and ∂gtt/∂r at the boundary surface r=R yields(36)gtt-=gtt+,grr-=grr+,∂gtt-∂r=∂gtt+∂r,where − and + indicate interior and exterior solutions. Now, using (36) and the metrics (9) and (35), we have(37)A=-1R3ln1-2MR,B=3R2ln1-2MR-2MR31-2MR-1,C=2MR41-2MR-1-2R3ln1-2MR.For the values of M and R for a strange stars, we compute the constants A, B, and C, specified as in Table 1.
The values of A, B, and C have been obtained using (37).
Compact Stars
M(M⊙)
R(Km)
A(Km-2)
B(Km-2)
C(Km-2)
SAXJ1808.4-3658(SS1)
1.435
7.07
0.001473644346
-0.044926791
0.004880923098
4U1820-30
2.25
10
0.0005978370008
-0.026116928
0.00201385582
VelaX-12
1.77
9.99
0.0004388196046
-0.018169038
0.001428126229
PSRJ1614-2230
1.97
10.3
0.000441203995
-0.019472555
0.00144933537
4.4. Stability
Now we calculate the two sound speed squares vsr2, vst2 for the radial and transverse coordinate, respectively. Herrera [55] introduced cracking concept and developed a new technique to examine potential stability for the matter. If we investigate the sign of the difference vst2-vsr2 then we can conclude whether our strange star is potential stable or not; i.e., if the radial speed sound is greater than the transverse speed sound, then there exists potentially stable region; otherwise, the region will be potentially unstable [10, 47, 49, 51]. It is clear from Figures 15 and 16 that 0<vsr2≤1 and 0<vst2≤1 always within the stellar objects. From Figure 17, we see that the corresponding difference is negative which means the radial speed sound is greater than the transverse speed sound which implies that our proposed strange star model is potentially stable in the framework of f(T) gravity. Again Figure 18 shows that vst2-vsr2≤1 is satisfied [56].
This figure represents the variation of vsr2 versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of vst2 versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of vst2-vsr2 versus r (km) for the strange star taking β=1, β=2, and β=3.
This figure represents the variation of vst2-vsr2 versus r (km) for the strange star taking β=1, β=2, and β=3.
Figures 12, 13, and 14 represent the plots to understand the validation of the energy conditions taking B=10, C=1, A=10, ξ=2, and ζ=1.
Figures 15, 16, 17, and 18 represent the plots to show the stability of our proposed model taking B=5, C=1, A=2, ξ=2, and ζ=1.
5. Some Fundamental Calculations5.1. Mass Function and Compactness
The mass function within the radius r is defined as [10](38)mr=∫0r4πr2ρdr=2π∫0rr22Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr28πeAr31+ξdr
From Figure 19, we have seen that at origin the mass function is regular (i.e., m(r)→0 when r→0) and monotonic increasing with respect to radius (r). We have also evaluated the values of mass for a few strange stars from our model to compare these values with observational data (see Table 2).
Calculated values of mass, energy density, and pressure from our model.
Compact Stars
Mass standard data (in km)
Mass from model (in km)
ρ0(gm/cc)
ρR(gm/cc)
p0(dyne/cm2)
SAXJ1808.4-3658(SS1)
2.116625
2.0868
1.996428×10-12
1.000531×10-12
-1.001789×1012
4U1820-30
3.31875
3.34265
1.997923×10-12
1.000286×10-12
-1.001040×1012
VelaX-12
2.61075
2.61043
1.998555×10-12
1.000253×10-12
-1.000723×1012
PSRJ1614-2230
2.90575
2.91837
1.998451×10-12
1.000239×10-12
-1.000775×1012
This figure represents the variation of m(r) versus r (km) for the strange star taking β=1, β=2, and β=3.
The compactness of the star is defined by u(r) [10] in the form of(39)ur=mrr=2πr∫0rr22Bβ+3Cβr+3Aβr+256ζπ2e2Ar31+ξ+2Bβ+3Cβr+3Aβr28πeAr31+ξdrWe have plotted the corresponding function given by Figure 20.
This figure represents the variation of u(r) versus r (km) for the strange star taking β=1, β=2, and β=3.
5.2. Relation between Mass and Radius
In this section we discuss the mass radius relation of the strange stars. From [57], twice the maximum allowable ratio of mass to the radius for an astrophysical object is always less than 8/9 (2M/R<8/9) whereas the factor M/R is called “compactification factor". From Table 3, we find that the calculated values corresponding to our model lie in the expected range [34]. Compactification factor for strange star always lies between 1/4 and 1/2. The calculated values of the compactification factor of the strange stars from our model are compatible with the condition (see Table 3).
Calculated values of the desired parameters of our model.
Compact Stars
M/R (standard data)
M/R from model
2M/R<8/9
ρ0/ρR
zs
SAXJ1808.4-3658(SS1)
0.299381
0.295163
0.590325
1.995368
0.562358
4U1820-30
0.331875
0.334265
0.66853
1.997352
0.736912
VelaX-12
0.266134
0.261304
0.522609
1.998050
0.447314
PSRJ1614-2230
0.282112
0.283337
0.566674
1.997973
0.519122
5.3. Surface Redshift
The redshift function can be defined as [10, 47, 49, 51](40)zs=11-2mr/r-1,where m(r) has been obtained from (38). According to Bohmer and Harko, the surface redshift should be ≤5 for an anisotropic star in the presence of a cosmological constant [58]. We calculate the maximum surface redshift from our model in Table 3. Now, it is clear that our model for strange stars obeys the relation zS≤5 though the cosmological constant is absent in our model which is quite reasonable.
6. Discussions
This paper has given out the anisotropic strange star model in f(T) gravity with modified Chaplygin gas. Using the diagonal tetrad field we have obtained the equations of motion where we have solved the unknown function f(T) as βT+β1, β and β1 being constants. Then we have solved the differential equation of energy density from where we have found the value of energy density (22) of it. With the help of this energy density, we have found out radial pressure ensuring this model as a quintessence dark energy candidate from Figures 1, 2, 3, and 4. We have also noticed that both the energy density (ρ) and radial pressure (pr) are monotonic decreasing function with respect to r and they have maximum value at r=1.46 by Figures 6–9. Figure 5 shows that the transverse pressure is decreasing with the rise of r. We have calculated anisotropic factor to see whether there exists gravitational attractive force or repulsive force for the strange star and we have studied from Figure 10 that there exists attractive force as well as repulsive gravitational force with different values of β. Here, the square of energy is monotonic decreasing with the increment of radial coordinate given by Figure 11. From Figures 12, 4, 13, and 14 we have concluded that all energy conditions are satisfied for our proposed model.
Using matching condition, the unknown parameters A, B, and C have been calculated for the different strange stars from our model which is given by Table 1. By stability analysis given on the basic of Figures 15–18, we have observed that 0<vsr2,vst2≤1, vsr2>vst2, and |vst2-vsr2|≤1 always. Finally, we have ensured that our model is potentially stable.
In Table 2, with the help of energy density (22) and radial pressure (23) we have calculated the numerical values of the mass of the different strange stars from our model to show the closeness of these values with the observational data. Also, we have obtained the values of central and surface density and central pressure for the above-mentioned strange stars from our model which have been calculated in Table 2. From Table 3, we have observed that twice the compactification factor are always less than <8/9 and maximum values of the surface redshift function are always less than 5. So, our proposed model is completely rational.
Figures 19, 20, and 21 represent the plots of mass function, compactness, and surface redshift function taking the values of A, B, and C from Table 1 and ξ=2 and ζ=1.
This figure represents the variation of zs versus r (km) for the strange star taking β=0.009, β=0.010, and β=0.011.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
PerlmutterS.AlderingG.GoldhaberG.Measurements of Ω and Λ from 42 high-redshift supernovae19995172, article no. 56510.1086/307221Zbl1368.85002SpergelD. N.BeanR.DoréO.NoltaM. R.BennettC. L.DunkleyJ.HinshawG.JarosikN.KomatsuE.PageL.Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology20071702, article no. 377HawkinsE.MaddoxS.ColeS.LahavO.The 2dF Galaxy Redshift Survey: correlation functions, peculiar velocities and the matter density of the Universe20033461789610.1111/j.1365-2966.2003.07144.xEisensteinD. J.ZehaviI.HoggD. W.ScoccimarroR.BlantonM. R.NicholR. C.ScrantonR.SeoH.-J.TegmarkM.ZhengZ.Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies20056332, article no. 560MazurP.MottolaE.Gravitational Condensate Stars: An Alternative to Black Holeshttps://arxiv.org/abs/gr-qc/0109035, 2011MazurP.MottolaE.Gravitational Vacuum Condensate Stars2004101269545955010.1073/pnas.0407076101UsmaniA.RahamanF.RayS.NandiK.KuhfittigP. K.RakibS.HasanZ.Charged gravastars admitting conformal motion2011701438839210.1016/j.physletb.2011.06.001RahamanF.RayS.UsmaniA. A.IslamS.The(2+1)-dimensional gravastars20127073-431932210.1016/j.physletb.2011.12.065MR2874215RahamanF.UsmaniA. A.RayS.IslamS.The (2+1) -dimensional charged gravastars20127171-31510.1016/j.physletb.2012.09.010MR2989132BharP.Higher dimensional charged gravastar admitting conformal motion2014354245746210.1007/s10509-014-2120-72-s2.0-84919341429de FeliceA.TsujikawaS.f(R) Theories2010131, article no. 310.12942/lrr-2010-3Zbl1215.83005DurrerR.MaartensR.Ruiz-LapuenteP.2010Cambridge University PressMR2656427NojiriS.OdintsovS. D.Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models20115052-45914410.1016/j.physrep.2011.04.001HarkoT.LoboF. S. N.NojiriS.OdintsovS. D.f(R,T) gravity201184202402010.1103/PhysRevD.84.0240202-s2.0-80051705048CognolaG.ElizaldeE.NojiriS.OdintsovS. D.ZerbiniS.Dark energy in modified Gauss-Bonnet gravity: Late-time acceleration and the hierarchy problem200673808400710.1103/PhysRevD.73.105013BengocheaG. R.FerraroR.Dark torsion as the cosmic speed-up2009791212401910.1103/PhysRevD.79.124019LinderE. V.Einstein's other gravity and the acceleration of the Universe2010811212730110.1103/PhysRevD.81.127301MyrzakulovR.Accelerating universe from F(T) gravity201171, article no. 175210.1140/epjc/s10052-011-1752-9BambaK.GengC.-Q.LeeC.-C.LuoL.-W.Equation of state for dark energy in f(t) gravity201120111, article no. 02110.1088/1475-7516/2011/01/021WuP.YuH.f(T) models with phantom divide line crossing2011712, article no. 155210.1140/epjc/s10052-011-1797-9CameraS.CardoneV. F.RadicellaN.Detectability of torsion gravity via galaxy clustering and cosmic shear measurements201489808352010.1103/PhysRevD.89.083520WillC. M.The Confrontation between General Relativity and Experiment200691, article no. 3DentJ. B.DuttaS.SaridakisE. N.f(T) gravity mimicking dynamical dark energy. Background and perturbation analysis2011201101, article no. 009MiaoR.-X.LiM.MiaoY.-G.Violation of the first law of black hole thermodynamics in f(T) gravity2011201111, article no. 033WangT.Static solutions with spherical symmetry in f(T) theories201184202404210.1103/physrevd.84.024042DaoudaM. H.RodriguesM. E.HoundjoM. J. S.New static solutions in f(T) theory20117111, article no. 181710.1140/epjc/s10052-011-1817-9BöhmerC. G.MussaA.TamaniniN.Existence of relativistic stars in f(T) gravity2011282424502010.1088/0264-9381/28/24/245020Zbl1232.83064CapozzielloS.GonzálezP. A.SaridakisdE. N.Va'squezY.Exact charged black-hole solutions in D-dimensional f (T) gravity: torsion vs curvature analysis201320132, article no. 039SharifM.RaniS.Wormhole solutions in f(T) gravity with noncommutative geometry20138812123501SharifM.RaniS.f(T) gravity and static wormhole solutions20142927145013710.1142/S0217732314501375Zbl1297.83042SharifM.RaniS.Charged noncommutative wormhole solutions in f(T) gravity201412910, article no. 23710.1140/epjp/i2014-14237-5SharifM.RaniS.Dynamical Instability of Expansion-Free Collapse in f(T) Gravity2015548252425422-s2.0-84887521005WittenE.Cosmic separation of phases1984302, article no. 27210.1103/physrevd.30.313JotaniaK.TikekarR.Paraboloidal space–times and relativistic models of strange stars200615811751182HerreraL.SantosN. O.Local anisotropy in self-gravitating systems199728625313010.1016/s0370-1573(96)00042-7MR1459087AbbasG.Effects of electromagnetic field on the collapse and expansion of anisotropic gravitating source2014352295596110.1007/s10509-014-1989-52-s2.0-84904469590AbbasG.Collapse and expansion of anisotropic plane symmetric source20143501307311AbbasG.Phantom energy accretion onto a black hole in Hořava-Lifshitz gravity201457460460710.1007/s11433-013-5306-zAbbasG.Cardy-Verlinde Formula of Noncommutative Schwarzschild Black Hole20142014430625610.1155/2014/782631AbbasG.SabiullahU.Geodesic study of regular Hayward black hole2014352276977410.1007/s10509-014-1912-0MakM. K.HarkoT.Quark stars admitting a one-parameter group of conformal motions2004131149156KroriK. D.BaruaJ.A singularity-free solution for a charged fluid sphere in general relativity197584, article no. 50810.1088/0305-4470/8/1/015KalamM.RahamanF.RayS.HosseinSk. M.KararI.NaskarJ.Anisotropic strange star with de Sitter spacetime20127212, article no. 224810.1140/epjc/s10052-012-2183-yKalamM.RahamanF.HosseinSk. M.RayS.Central density dependent anisotropic compact stars201373, article no. 263810.1140/epjc/s10052-013-2638-9DelidumanC.YapışkanB.Absence of Relativistic Stars in f(T) Gravityhttps://arxiv.org/abs/1103.2225, 2015AbbasG.NazeerS.MerajM. A.Cylindrically symmetric models of anisotropic compact stars2014354244945510.1007/s10509-014-2110-9AbbasG.KanwalA.ZubairM.Anisotropic compact stars in f(T) gravity20153572, article no. 10910.1007/s10509-015-2337-0AbbasG.MomeniD.Aamir AliM.MyrzakulovR.QaisarS.Anisotropic compact stars in f(G) gravity20153572, article no. 15810.1007/s10509-015-2392-6AbbasG.QaisarS.MerajM. A.Anisotropic strange quintessence stars in f(T) gravity20153572, article no. 15610.1007/s10509-015-2389-1AbbasG.ZubairM.MustafaG.Anisotropic strange quintessence stars in f(R) gravity20153582, article no. 2610.1007/s10509-015-2426-0AbbasG.QaisarS.JawadA.Strange stars in f(T) gravity with MIT bag model20153592, article no. 5710.1007/s10509-015-2509-ySotiriouT. P.FaraoniV.f (R) theories of gravity2010821, article no. 45110.1103/RevModPhys.82.451MR2629610BambaK.CapozzielloS.NojiriS.OdintsovS. D.Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests2012342115522810.1007/s10509-012-1181-8Zbl1314.83037BenaoumH. B.Modified Chaplygin Gas Cosmology2012201212357802HerreraL.Cracking of self-gravitating compact objects1992165320621010.1016/0375-9601(92)90036-lAndréassonH.Sharp Bounds on the Critical Stability Radius for Relativistic Charged Spheres2009288271573010.1007/s00220-008-0690-3BuchdahlH. A.General relativistic fluid spheres19591164, article no. 102710.1103/PhysRev.116.1027Zbl0092.20802BohmerC. G.HarkoT.Bounds on the basic physical parameters for anisotropic compact general relativistic objects20062322, article no. 647910.1088/0264-9381/23/22/023MR2272016