Equation of State of Neutron Stars with Junction Conditions in the Starobinsky Model
aa r X i v : . [ g r- q c ] S e p Equation of State of Neutron Stars with Junction Conditions inthe Starobinsky Model
Wei-Xiang Feng,
1, 2, ∗ Chao-Qiang Geng,
1, 2, 3, † W. F. Kao, ‡ and Ling-Wei Luo
4, 2, § Synergetic Innovation Center for Quantum Effects and Applications (SICQEA),Hunan Normal University, Changsha 410081, China Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan Institute of Physics, Chiao Tung University, Hsinchu 300, Taiwan (Dated: September 5, 2017)
Abstract
We study the Starobinsky or R model of f ( R ) = R + αR for neutron stars with the structureequations represented by the coupled differential equations and the polytropic type of the matterequation of state. The junction conditions of f ( R ) gravity are used as the boundary conditionsto match the Schwarschild solution at the surface of the star. Based on these the conditions, wedemonstrate that the coupled differential equations can be solved directly . In particular, fromthe dimensionless equation of state ¯ ρ = ¯ k ¯ p γ with ¯ k ∼ . γ ∼ .
75 and the constraint of α . . × m , we obtain the minimal mass of the NS to be around 1.44 M ⊙ . In addition,if ¯ k is larger than 5.0, the mass and radius of the NS would be smaller. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] . INTRODUCTION The astrophysical observations from the Type Ia Supernovae [1, 2], large scale struc-ture [3, 4] and baryon acoustic oscillations [5] as well as cosmic microwave background [6–8]indicate the necessity of new physics beyond the Einstein’s general relativity (GR). Themodified theories of gravity [9, 10] become more significant in order to explain the accel-erated expansion phenomenon not only inflation [11–15] in the early epoch but also darkenergy [16–21] in the recent stage of the universe. A class of alternative theories of themodification from the geometric point of view is the so-called f ( R ) gravity theories [22–25].In these theories, the Lagrangian density is modified by using an arbitrary function f ( R )instead of the scalar curvature R of the Einstein-Hilbert term. The most well-known f ( R )model is the Starobinsky or R model with f ( R ) = R + αR , originally proposed to ob-tain the quasi-de Sitter solution for inflation [15]. Furthermore, several viable f ( R ) gravitytheories [26–31] have been used to explain the cosmic acceleration problems.In order to realize the structure of a compact star, one needs to know its equation of state(EoS), which characterizes the thermodynamic relation between the density ρ , pressure p and temperature T of the dense matter. Under the adiabatic assumption, the EoS is reducedto a polytropic relation ρ = k p γ . This assumption has been discussed for the neutron stars(NSs) in the literature [32–45]. In particular, the allowed region of the polytopes has beenshown in [46].The compact relativistic star was first studied by Chandrasekhar [47], who assumedthat a white dwarf is supported only by the completely degenerate electron gas, and thenobtained the so-called Chandrasekhar limit of a white dwarf with the maximal mass of 1.44 M ⊙ . Subsequently, Oppenheimer and Volkoff [48] proposed a limit of 0 . M ⊙ of a NS byconsidering a completely degenerate neutron gas. However, this approach is inappropriatedue to the strong nuclear repulsive forces of neutrons and other strong interaction of theheavy hadrons in dense matter.In the scenario of GR, the structure of the relativistic stars is determined by EoS of matterinside the stars without an explicit constraint, whereas it is expected that the f ( R ) theoriesdo provide some constraints with singularity problems [49–51]. The relativistic stars in themodified gravities have been studied in the literature [38–45, 52–69]. It has been argued thatthe compact relativistic stars are difficult to exist due to the curvature scalar R divergence2nside the star in f ( R ) [52]. However, the realistic EoS in the Starobinsky’s dark energymodel [27] has been constructed in Ref. [53], in which R does not diverge inside the star, sothat the relativistic stars could occur in f ( R ). The pure geometric study is formulated inRef. [44], which imposes the junction conditions in f ( R ) [70, 71] as the additional conditionsto solve the coupled structure equations and obtain the final result indirectly .In this study, we consider the R model by performing the calculation only in the Jordan frame. In our discussions, we solve the coupled structure equations by the junction conditions approach directly rather than the perturbation methods [39, 42, 55, 58] . We show thatthe NSs can exist in the R model under the polytrope assumption of EoS. The possible dimensionless EoS ¯ ρ ∼ . p . is concluded by the analysis of the various values of thedimensionless parameter ¯ α in the R model, where the bars represent the dimensionlessquantities. The theoretical constraint on the coefficient α of the R term in the model isgiven by α . . × m . By applying the resultant EoS and critical value of α , the minimal mass of the NSs is obtained about 1.44 M ⊙ which is the same as the Chandrasekharlimit of the white dwarf [47]. For a fixed parameter α , we observe that the mass and theradius get larger when k decreases, while the maximal value of ¯ k = 5 . R model. In Sec. III, we analyze the model parameter α and explore its reasonable value fromthe typical units in the neutron star system. We discuss our result of EoS under the specificchoice of the initial conditions. Finally, we give conclusions in Sec. IV. II. SPHERICALLY SYMMETRIC SOLUTION OF THE R MODEL
The action of the f ( R ) theories with matter is given by S = 12 κ Z d x √− g f ( R ) + S m , (2.1)with κ = 8 π and the conventional units of G = c = 1. By the variation with respect to themetric g µν , we have the modified Einstein equations f ′ R µν − f g µν − ( ∇ µ ∇ ν − g µν (cid:3) ) f ′ = κ T µν , (2.2) Some other non-perturbative methods have been addressed in Refs. [72–74] T µν the energy-momentum tensor and (cid:3) = g µν ∇ µ ∇ ν the D’Alembertian operator. Inaddition, “ ′ ” in this paper denotes the differentiation with respect to its argument, e.g. f ′ ( R ) = df ( R ) /dR . We will focus on the Starobinsky or R model with the function of theLagrangian density f ( R ) = R + αR . (2.3)As a result, we obtain the following field equation G µν (1 + 2 αR ) + α g µν R − α ( ∇ µ ∇ ν − g µν (cid:3) ) R = κ T µν (2.4a)with G µν = R µν − (1 / R g µν the Einstein tensor. Consequently, the trace equation reads − R + 6 α (cid:3) R = κ T . (2.4b)In order to study the system of a compact star, we will study the solution with an ansatzgiven by the static spherical symmetric metric ds = − e r ) dt + e r ) dr + r d Ω , (2.5)where d Ω = dθ + sin θ dϕ and exp(2Λ( r )) = (1 − m ( r ) /r ) − with m ( r ) the massfunction characterizing the mass enclosed within the radius r . In GR, m ( r ) = R r π ¯ r ρ (¯ r ) d ¯ r with ρ (¯ r ) the density function. For the radius of the star r s , m ( r s ) = M can be identified asthe total mass in the Newtonian limit . In the R model, the mass function should be modifiedwith some correction terms. However, it cannot be integrated by the density function ρ directly. The function Φ( r ) can be regarded as the effective relativistic gravitationalpotential. Subsequently, we can obtain the Einstein tensor from (2.5), given by G tt = − r e ddr (cid:18) r ( e − − (cid:19) = 2 r e m ′ , (2.6a) G rr = − r e (1 − e − ) + 2 r Φ ′ , (2.6b) G θθ = r (cid:18) Φ ′′ + Φ ′ − Φ ′ Λ ′ + 1 r (Φ ′ − Λ ′ ) (cid:19) e − , (2.6c) G ϕϕ = sin θ G θθ . (2.6d) A. Coupled Differential Equations
Considering a static perfect fluid with the energy-momentum tensor T µν = ( ρ + p ) u µ u ν + pg µν with u µ , ρ , and p denoting the 4-velocity, the density , and the pressure of the fluid4espectively. The ν = r component of the conservation equation ∇ µ T µν = 0 givesΦ ′ = − p ′ ρ + p . (2.7)In addition, we can obtain the identityΛ ′ = r m ′ − mr ( r − m ) (2.8)via the definition of Λ( r ). In the local rest frame, u t = − e Φ and u i = 0 with i = r, θ and ϕ the spatial coordinates, we have T tt = ρ e , T rr = p e , T θθ = p r , T ϕϕ = p r sin θ , (2.9)due to u µ u µ = −
1. In addition, we can obtain the following identities for convenience (cid:3) R = e − (cid:18) R ′′ + (cid:18) Φ ′ − Λ ′ + 2 r (cid:19) R ′ (cid:19) , (2.10) ∇ t ∇ t R = − e − Λ) Φ ′ R ′ , (2.11) ∇ r ∇ r R = R ′′ − Λ ′ R ′ . (2.12)Consequently, with the metric given by (2.5) and the energy momentum tensor given by thestatic perfect fluid, we can write the field equations as a set of differential equations m ′ = r αR ) (cid:18) πρ + 48 πp + R (2 + 3 αR ) (cid:19) − α (16 πpr + 4 m (1 + 2 αR ) − αr R − αR ′ r ( r − m )) R ′ αR )(1 + 2 αR + αrR ′ ) , (2.13a) p ′ = − ( ρ + p )(16 πpr + 4 m (1 + 2 αR ) − α r R − α rR ′ ( r − m ))4 r (1 + 2 αR + α rR ′ )( r − m ) , (2.13b) R ′′ = − (8 π ( ρ − p ) − R ) r + 12( r − m ) αR ′ αr ( r − m ) + r ((1 + 3 αR ) R + 16 πρ ) R ′ r − m )(1 + 2 αR ) + 2 αR ′ (1 + 2 αR ) . (2.13c)Note that we have written m ′ = m ′ ( R, R ′ , p, m ), p ′ = p ′ ( R, R ′ , p, m ) and R ′′ = R ′′ ( R, R ′ , p, m )as algebraic functionals of R, R ′ , p, m . Here Eq. (2.13a) is derived from the tt -componentand the trace equation of the field equation (2.4a). In addition, Eq. (2.13b) is derived fromthe rr -component of the field equation (2.4a), and is also known as the modified Tolman-Oppenheimer-Volkoff (mTOV) equation [48]. Finally, Eq. (2.13c) is derived from the traceequation (2.4b). 5or the perfect fluid, we assume the EoS is polytrope , i.e., ρ = k p γ . (2.14)In order to simplify the calculations, we can choose the typical values r ∗ , m ∗ , p ∗ , ρ ∗ and R ∗ for the compact star system and express r ≡ xr ∗ , m ≡ ¯ mm ∗ , p ≡ ¯ pp ∗ , ρ ≡ ¯ ρρ ∗ , R ≡ ¯ RR ∗ and α ≡ ¯ αα ∗ ≡ ¯ α (1 /R ∗ ) in terms of the dimensionless quantities x , ¯ m , ¯ p , ¯ ρ , ¯ R and ¯ α , while thederivatives of p , m and R can be written as p ′ = ¯ p ′ ( p ∗ /r ∗ ), m ′ = ¯ m ′ ( m ∗ /r ∗ ), R ′ = ¯ R ′ ( R ∗ /r ∗ )and R ′′ = ¯ R ′′ ( R ∗ /r ∗ ), respectively, where the prime of the dimensionless quantities denotesthe derivative with respect to x . The polytropic type of EoS in terms of the dimensionlessquantities can be given as ¯ ρ = ¯ k ¯ p γ with ¯ k = kρ − ∗ p γ ∗ . Since we are interested in the NSs inthe R model, it is convenient for us to define the following typical values in SI units, m ∗ ≡ M ⊙ = 1 . × kg ,r ∗ ≡ m = 10 km ,ρ ∗ ≡ Neutron mass(Neutron Compton wavelength) ∼ kg / m ,p ∗ = ρ ∗ = 8 . × Pa = 8 . × kg m − s − ,R ∗ = ρ ∗ = 7 . × − m − = 7 . × − km − . According to the typical units, we can rewrite (2.13a), (2.13b) and (2.13c) as the dimen-sionless equations:¯ m ′ = x α ¯ R ) (cid:18) π ¯ ρ + 48 π ¯ p + ¯ R (2 + 3 ¯ α ¯ R ) (cid:19)(cid:18) ρ ∗ r ∗ m ∗ (cid:19) − ¯ α ((16 π ¯ p − ¯ α ¯ R ) x ( R ∗ r ∗ ) + 4 ¯ m (1 + 2 ¯ α ¯ R )( m ∗ r ∗ ) − αx ( x − m ( m ∗ r ∗ )) ¯ R ′ ) ¯ R ′ α ¯ R )(1 + 2 ¯ α ¯ R + ¯ αx ¯ R ′ )( m ∗ r ∗ ) , (2.15a)¯ p ′ = − ( ¯ ρ + ¯ p )( x (16 π ¯ p − ¯ α ¯ R )( R ∗ r ∗ ) + 4 ¯ m (1 + 2 ¯ α ¯ R )( m ∗ r ∗ ) − x ¯ α ¯ R ′ ( x − m ( m ∗ r ∗ )))4 x ( x − m ( m ∗ r ∗ ))(1 + 2 ¯ α ¯ R + ¯ αx ¯ R ′ ) , (2.15b)¯ R ′′ = − x (8 π ( ¯ ρ − p ) − ¯ R )( R ∗ r ∗ ) + 12( x − ¯ m ( m ∗ r ∗ )) ¯ α ¯ R ′ αx ( x − m ( m ∗ r ∗ ))+ x ((1 + 3 ¯ α ¯ R ) + 16 π ¯ ρ ) ¯ R ′ ( R ∗ r ∗ )( x − m ( m ∗ r ∗ ))(1 + 2 ¯ α ¯ R ) + 2 ¯ α ¯ R ′ α ¯ R , (2.15c)respectively. The dimensionless parameters m ∗ /r ∗ = 0 . R ∗ r ∗ = 0 . ρ = ¯ k ¯ p γ . B. Boundary Conditions
In GR, the Birkhoff’s theorem states that the spherically symmetric vacuum solutionmust be given by the Schwarzschild metric. On the other hand, even though the absenceof the Birkhoff’s theorem in f ( R ) theories might lead to the non-uniqueness of this vacuumsolution, the Schwarzschild metric can serve as a vacuum solution in f ( R ) under somecircumstances. It has been shown that the conditions of R = 0 with f (0) = 0 and f ′ (0) = 0for the existence of the Schwarzschild metric are satisfied in the Starobinsky model [75]. Asa result, we introduce the Schwarzschild vacuum solution for the exterior region. In this way,we can obtain the mass and radius of the star from the Schwarzschild metric once (2.15) issolved with proper boundary conditions.In the following, we consider the star without thin shells. In order to match the solutionat the surface of the star , we use the Schwarzschild solution for the exterior region ( r > M ) ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r d Ω , (2.16)where ˜ M is the mass parameter in GR. The junction conditions for the f ( R ) theories shouldbe more restrictive as discussed in Refs. [44, 70, 71]. The first and the second fundamentalforms of the conditions are [ h µν ] = 0 and [ K µν ] = 0, respectively, where [ ] denotes the jumpat the boundary surface of the star. We can identify ˜ M with M = m ( r s ) only when thefirst fundamental form matches. However, there are two additional conditions for the scalarcurvature across the surface [44], given by [ R ] = 0 , (2.17a)[ ∇ µ R ] = 0 . (2.17b)In our assumption with the static and spherically symmetric metric, the curvature R isonly a function of r . By matching of the second fundamental form to make the pressurevanishing at the boundary surface [70], the boundary conditions are reduced to R ( r s ) = 0, R ′ ( r s ) = 0 and p ( r s ) = 0. Inside the star, we have to determine the boundary conditions atthe center of the star. There are two first-order and one second-order differential equations7n Eq. (2.13). Hence, only four boundary conditions are required to solve these coupledordinary differential equations. To satisfy the regularity conditions at the center of the star,we must have m (0) = 0, p ′ (0) = 0, ρ ′ (0) = 0 and R ′ (0) = 0 [44], in which two of themare redundant. According to Eq. (2.13b), p ′ (0) = 0 is automatically satisfied as long as m (0) = 0 and R ′ (0) = 0 as r →
0. In addition, ρ and p are related by EoS in (2.14), leadingto p ′ (0) = 0 and ρ ′ (0) = 0, so that only conditions m (0) = 0 and R ′ (0) = 0 are left.Consequently, we have three boundary conditions at the surface and two boundary onesat the center written in the dimensionless forms, given by¯ R ( x s ) = 0 , ¯ R ′ ( x s ) = 0 , ¯ p ( x s ) = 0 , ¯ m (0) = 0 , ¯ R ′ (0) = 0 . (2.18)These boundary conditions are referred to as the Schwarzschild boundary conditions . Math-ematically, since there are four undetermined integration constants c , c , c and c in (2.15),only four in (2.18) are enough to solve it. However, these integration constants shouldbe associated with the model parameter α and ( γ, ¯ k ) in the EoS. The fifth one in (2.18)can be used to constrain the parameter space of ( α, γ, ¯ k ). For example, if we choose¯ m (0) = ¯ R ′ (0) = ¯ p ( x s ) = ¯ R ( x s ) = 0, then we have to determine whether ¯ R ′ ( α, γ, ¯ k ; x ) | x = x s satisfies ¯ R ′ ( x s ) = 0 for fixed values of α , γ and ¯ k .According to the mTOV equation in (2.13) and conservation equation in (2.7), we have d Φ dr = 16 πpr + 4 m (1 + 2 αR ) − αr R − αr ( r − m ) R ′ r (1 + 2 αR + αrR ′ )( r − m ) . (2.19)In the region outside of the star ( r ≥ r s ), the pressure and scalar curvature as well as thederivative of the scalar curvature should be continuous, resulting in p ( r ) = 0, R ( r ) = 0 and R ′ ( r ) = 0 by (2.18). It can be checked that the exterior solution of (2.19) coincides with theSchwarzschild solution e r ) = 1 − M /r . III. ANALYSIS AND RESULTSA. Determination of α In principle, Eq. (2.13) can be regarded as the GR results with αR as the modificationterm. For example, Eq. (2.13b) corresponds to the TOV equation in GR [48] when α → m ′ and R ′′ equations in GR for (2.13a) and (2.13c) with α → m ′ = 4 πr ρ − r (8 π ( ρ − p ) − R )6(1 + 2 αR ) − αRr αR ) (32 πρ − R ) − α (16 πpr + 4 m (1 + 2 αR ) − αr R − αr ( r − m ) R ′ ) R ′ r (1 + 2 αR )(1 + 2 αR + αrR ′ ) ≡ πr ρ eff , (3.1)where ρ eff = ρ − π ( ρ − p ) − R π (1 + 2 αR ) − αR π (1 + 2 αR ) (32 πρ − R ) − α (16 πpr + 4 m (1 + 2 αR ) − αr R − αr ( r − m ) R ′ ) R ′ πr (1 + 2 αR )(1 + 2 αR + αrR ′ ) . (3.2)In the limits of α → R → π ( ρ − p ), we have m ′ → πr ρ , which is the same asresult in GR.However, in the numerical analysis, there are problems of choosing α for the system. Onone hand, the main numerical difficulty arises from (2.13c), in which R ′′ = − (8 π ( ρ − p ) − R ) r αr ( r − m ) + 12( r − m ) R ′ r ( r − m ) + r ( R + 16 πρ ) R ′ r − m ) (3.3)by taking α →
0. Furthermore, we have the boundary conditions R ′ (0) = 0 and m (0) = 0as r →
0, and obtain R ′′ ≡ R ′′ | r → = − π ( ρ − p ) − R α (cid:12)(cid:12)(cid:12)(cid:12) r → , (3.4)which implies the singularity of R ′′ as α → p (0) and R (0). On the other hand, we would liketo discuss the upper bound for ¯ α . In the dimensionless form x = r/r ∗ , (2.15a) with (3.1)and (3.4) in x → m ′ = x (cid:18) π ¯ ρ − π ( ¯ ρ − p ) − ¯ R α ¯ R ) − ¯ α ¯ R α ¯ R ) (32 π ¯ ρ − ¯ R ) (cid:19)(cid:18) ρ ∗ r ∗ m ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x → , (3.5)and ¯ R ′′ ≡ ¯ R ′′ | x → = − π ( ¯ ρ − p ) − ¯ R α ( R ∗ r ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) x → (3.6)respectively, where ( ρ ∗ r ∗ ) /m ∗ = 0 . R ∗ r ∗ = ρ ∗ r ∗ = 7 . × − , which character-izes the compactness of a star. In order to determine the proper value of ¯ α , we use (3.6) torewrite (3.5) as¯ m ′ = 4 πx ¯ ρ (cid:18) ρ ∗ r ∗ m ∗ (cid:19) + (cid:18) ¯ αx α ¯ R (cid:19)(cid:20) ¯ R ′′ (cid:18) r ∗ m ∗ (cid:19) − ¯ R (cid:18) π ¯ ρ − ¯ R (cid:19)(cid:18) ρ ∗ r ∗ m ∗ (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x → . (3.7)9ince ¯ R is convex upward around x = 0, we expect that ¯ R ′′ ≤
0. Then, we have 8 π ( ¯ ρ − p ) − ¯ R ≥ α ≥
0, which can be seen from (3.6). We can choose ¯ α . ( m ∗ /r ∗ )( R ∗ r ∗ ) = 0 . m ′ & πx ¯ ρ (cid:18) ρ ∗ r ∗ m ∗ (cid:19) + (cid:18) x α ¯ R (cid:19)(cid:20) ¯ R ′′ − ¯ R (cid:18) π ¯ ρ − ¯ R (cid:19) ( R ∗ r ∗ ) (cid:21) ( R ∗ r ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) x → . (3.8)The last two terms in the square bracket represent the first-order and second-order cor-rections in the R ∗ r ∗ unit, respectively. Therefore, we derive α = ¯ α/R ∗ . . × m . In addition, several constraints on α from the observational data have been de-rived in [55, 58, 76]. Moreover, Gravity Probe B [77] gives α . × m ; the precessionmeasurement of the pulsar B in the PSR J0737-3039 system [78] yields α . . × m ;and the strong magnetic NS [55, 58] results in α . m . Furthermore, it has been shownthat the ghost-free condition f ′′ ( R ) ≥ α >
0. Here, it should be noted thatonly within the condition ¯ R → π ( ¯ ρ − p ) can we have a finite ¯ R ′′ in the limit ¯ α →
0. Thiscondition assures that the R model is consistent with GR in α → B. Numerical Results
By using the Runge-Kutta 4th-order (RK4) procedure, Eq. (2.15) can be solved by choos-ing ¯ p (0) and ¯ R (0) as the central values with boundary conditions ¯ m (0) = ¯ R ′ (0) = ¯ p ( x s ) = 0.We can obtain ¯ R ( x s ) and ¯ R ′ ( x s ) by applying random values of ¯ p (0) and ¯ R (0) numerically.In terms of the problem of (3.4), we have to find out the appropriate values of ¯ p (0) and¯ R (0) to satisfy ¯ R ( x s ) = 0 and ¯ R ′ ( x s ) = 0, which maintain the Schwarzschild boundaryconditions (2.18).The parameters ¯ k and γ affect the behaviors of the coupled equations (2.15) as well asthe boundary values at the surface. Clearly, they can be determined once our boundaryconditions are fixed in the numerical calculations. All the results are given in the typicalunits m ∗ , r ∗ , ρ ∗ , p ∗ , and R ∗ as defined in Sec. IIA. We look for the reasonable EoS for¯ α = 0 .
01 and 0 . α = 0). For simplicity, we keepthe high pressure at the center of the star to be ¯ p (0) = 1 initially. Then, we end up thecalculation with ¯ p ( x s ) = 10 − at the surface of the star, corresponding to the density at the m ∗ = M ⊙ r ∗ = 10 km ABLE I. The results of the radius x s = r s /r ∗ and mass ¯ M with the polytropic exponent γ andcentral Ricci curvature ¯ R (0) for the R model respect to the various ¯ α with the fixed centralpressure ¯ p (0) = 1 and polytropic constant ¯ k = 5 . α x s ¯ M γ ¯ R (0)0.01 1.999 1.444 0.7525000000 8.950.0005 2.477 1.557 0.7503553926 35.00GR ( α = 0) 2.297 1.672 0.7503553926 16 π bottom of the NS’s outer crust around 10 ∼ kg m − . We keep ¯ k = 5 . γ and ¯ R (0) in order to satisfy ¯ R ( x s ) = 0 and ¯ R ′ ( x s ) = 0. The results aregiven in the TABLE I. From this table, we find that for a smaller ¯ α , ¯ R (0) is larger , and thesame goes for ¯ M , which are the generic feature of the model. The behaviors of the growing¯ α and decreasing ¯ M have been also discussed in Ref. [74] with the realistic EoS instead ofthe polytropic one in this study. We note that the different choices of ¯ k will be shown inTABLE II. -60-50-40-30-20-10 0 10 20 30 40 0 0.5 1 1.5 2 2.5-60-50-40-30-20-10 0 10 20 30 40 R R ´ r [10 km] R R´ (a) -10 0 10 20 30 40 50 60 0 0.5 1 1.5 2 2.5-10 0 10 20 30 40 50 60 T R r [10 km] R - TR - RGR - T (b) FIG. 1. (color online) (a) The curvature scalar ¯ R (dotted line) and the derivative of the curvaturescalar ¯ R ′ (solid line) of the radial coordinate r in the unit of 10 km and (b) the profiles of thecurvature ¯ R of the R model (dotted line) and the negative trace of the energy momentum tensor¯ T := 8 π (¯ ρ − p ) of the R model (solid line) and GR (dotted long-dashed line), where ¯ k = 5 . α = 0 . ρ r [10 km] α = 0.01 α = 0.0005 GR (a) m / M O · r [10 km] α = 0.01 α = 0.0005 GR Chandra (b) FIG. 2. (color online) (a) The density ¯ ρ and (b) mass m as functions of the radial coordinate r in the unit of 10 km with ¯ k = 5 .
0, where the solid and dotted lines indicate the R model with¯ α = 0 .
01 and 0 . M , radius x s and Ricci curvature ¯ R (0) at the center for differentvalues of the polytropic constant ¯ k with ¯ α = 0 .
01 and γ ∼ .
75 in the R model.¯ k M x s R (0) 8.95 8.40 7.54 6.15 4.93 In FIG. 1, we illustrate the deviation of the interior region of the star in the R modelfrom GR. The profiles of the scalar curvature ¯ R and its derivative ¯ R ′ are shown in FIG. 1a.Clearly, these two quantities satisfy the boundary conditions ¯ R ( x s ) = 0 and ¯ R ′ ( x s ) = 0.The results of the negative trace of the energy momentum tensor in GR and the R modelin the interior of the NSs are displayed in FIG. 1b, illustrating similar behaviors. However,the conduct of the scalar curvature in the R model is different from that of GR with R = − πT = κ ( ρ − p ) due to the R term.For the density ¯ ρ and mass function m/M ⊙ profiles of the star, we exhibit ¯ k = 5 . α = 0 .
01 and 0 . R model12 m / M O · r [10 km] k = 5.0 k = 4.5 k = 4.0 k = 3.5 k = 3.2 Chandra FIG. 3. The mass m as a function of radial coordinate r with ¯ α = 0 .
01 and γ ∼ .
75, where thevalue 1.44 is the Chandrasekhar (Chandra) limit (long-dashed line). from GR is small in FIG. 2a, whereas that of the resultant mass is large in FIG. 2b. The end-points of the curves in FIG. 2b correspond to the mass ¯ M and radius x s shown in TABLEI. In the R model, the mass function in (3.1) is deviated from GR due to the geometriceffect of the effective density ρ eff .In TABLE I, the mass of the NS exceeds the Chandrasekhar limit (1 . M ⊙ ) of the whitedwarf [47]. Note that if the collapsing process is supplied only by gravity, the Chandrasekharlimit could be considered as a lower bound of the mass for a star whose ultimate destiny isa NS or black hole.According to our analysis in the R model, which allows a lighter NS than that in GR asshown in FIG. 2b. Furthermore, from the upper limit α . . × m , we find thatthe minimal mass of the NS is around 1.44 M ⊙ for γ ∼ .
75 and ¯ k = 5 . α equal to the critical value ¯ α = 0 .
01, we can analyze the properties of thethe NS in the minimal mass condition. The profiles of the mass function m of the radialcoordinate r with γ ∼ .
75, ¯ k = 5 .
0, 4.5, 4.0, 3.5 and 3 . M and radius x s of the NSs and their corresponding Riccicurvature ¯ R (0) at the center. From the table, we observe that the mass becomes larger as ¯ k gets smaller, whereas ¯ R (0) becomes smaller. We note that the case of ¯ k = 3 . ρ > p for ordinary matter inside the NS has been excluded in our discussion. On the other hand,we expect that the mass of the NS is not smaller than the Chandrasekhar limit and thevalue ¯ k can not be larger than 5.0. The reasonable maximal value of ¯ k can be determinedas 5.0. IV. CONCLUSIONS
We have addressed the R model on a compact star, especially on the NS through thejunction conditions. We have solved the mTOV equation rather than the perturbationmethod in the literature. In order to satisfy the junction conditions (Schwarzschild con-ditions), the central pressure p (0) and Ricci scalar R (0) should be well-selected. In f ( R )gravity, more specifically, the R model, the parameters k and γ in the polytropic EoS canbe constrained by p (0) and R (0) due to the coupled structure equations. With the junctioncondition, in particular, we have shown that there exists the solution of EoS ¯ ρ = ¯ k ¯ p γ with¯ k ∼ . γ ∼ . α = 1 . × m , we have obtained the minimal mass of the NS.Under ¯ ρ = 5 . p . , the typical value of the NS mass is around 1.44 M ⊙ . We have shownthat ¯ k has the maximal value of ¯ k = 5 .
0. In our discussion, we have only considered the ghost-free f ( R ) theories ( α > α intoaccount under the polytrope assumption in Ref. [42]. For α >
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CKNOWLEDGMENTS
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