Anisotropic spheres via embedding approach in \mathcal{R}+β\mathcal{R}^{2} gravity with matter coupling
aa r X i v : . [ g r- q c ] J a n Anisotropic spheres via embedding approach in R + β R gravity with matter coupling G. Mustafa, ∗ Xia Tie-Cheng, † Mushtaq Ahmad, ‡ and M. Farasat Shamir § Department of Mathematics, Shanghai University,Shanghai, 200444, Shanghai, People’s Republic of China Department of Mathematics, Shanghai University,Shanghai, 200444, Shanghai, People’s Republic of China. National University of Computer and Emerging Sciences,Chiniot-Faisalabad Campus, Pakistan. National University of Computer and Emerging Sciences,Lahore Campus, Pakistan.
The manifesto of the current article is to investigate the compact anisotropic matter profiles inthe context of one of the modified gravitational theories, known as f ( R , T ) gravity, where R is aRicci Scalar and T is the trace of the energy-momentum tensor. To achieve the desired goal, wecapitalized on the spherical symmetric space–time and utilized the embedding class-1 solution viaKarmarkar’s condition in modeling the matter profiles. To calculate the unidentified constraints,Schwarzschild exterior solution along with experimental statistics of three different stars LMC X-4,Cen X-3, and EXO 1785-248 are taken under consideration. For the evaluation of the dynamicalequations, a unique model f ( R , T ) = R + β R + λ T has been considered, with β and λ being thereal constants. Different physical aspects have been exploited with the help of modified dynamicalequations. Conclusively, all the stars under observations are realistic, stable, and are free from allsingularities. Keywords : Anisotropic spheres; f ( R , T ) gravity; Compact stars; Embedding Class I. I. INTRODUCTION
Late time evolution of stellar configurations, triggered by an immense gravitational pull has been anticipated toa great extent in the field of astrophysics and the modified gravitational theories. It expedites the examinationof diverse attributes regarding the gravitating source by physical phenomena. Baade and Zwicky [1] forecast theinception of highly dense stellar objects inaugurating the debate that a supernova might be revolutionized into ahighly dense star. This reality came into existence when exceptionally magnetized as well as rotating neutronsstars were detected. Therefore, a fundamental shift regarding normal stars to compact stars came into existence.By the newly discovered concept, the normal stars shifted into an extensive range, such as quark stars, neutronstars, gravastars, dark stars, and finally black holes. The actuality of the extensive range of these stars led theresearchers to curiosity, regarding the formation of these stars. The stellar death of a normal star occurs, that iswhen the nuclear fusion reactions cease to act and burn all of their nuclear fuel results in the formation of newcompact stars. The newly formed compact stars are primarily distinguished from the normal stars in two ways.Since all the fuel has been utilized by the star, hence the star cannot sustain against the gravitational collapse due tothermal pressure. Analogous to that, the white dwarf is stabilized due to strong degenerate electron pressure, whilethe neutron star is stabilized due to degenerate neutron pressure. Whereas black holes are entirely the collapsedremnants, therefore there is neither a thermal pressure nor a degenerate pressure sufficient enough to repressthe centripetal pull of gravity; as a result, it leads towards the gravitational singularities and the event horizon.The formed compact stellar remnants consist of huge density and relatively small radii in contrast to the normal stars.The intention to investigate the physically stable models, leads us to an analytical approach regarding theEinstein field equations. One of the essential tools is to adopt the embedding class I space-time which transforms afour dimensional manifold into a Euclidean space of higher dimension. The conversion of curved embedding classspace–time into higher dimensional space–time is substantial to develop exact new models in the field of astrophysics.The class I embedding condition leads towards a differential equation in the framework of spherically symmetric ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]; [email protected] space–time which connects the gravitational potentials i.e., g rr and g tt , the condition is also recognized as theKarmarkar condition [2]. The Karmarkar’s condition appears to be very influential in exploring new solutions for theastrophysical models. Schlia [3] was the pioneer in developing the Karmarkar condition for a spherically symmetricspace–time. The embedding theorem based on the isometrics has been presented by Nash [4]. Maurya et al. [5]-[12]were the first explorers in the aspect of applying the embedding approach to the anisotropic matter configurations.After the new dawn of general relativity (GR), f ( R ) theory is considered to be quite a fascinating tool for theamplification of GR. Further, many researchers presented different versions of this theory, which were also veryprosperous in diverse fields. The recent extension of this theory is regarded as f ( R , T ) gravity, which was presentedby Harko et al. [13]. The f ( R , T ) theory has been the center of attention by many analysts and consequentlymany intriguing cosmological aspects have been unraveled [14]-[17]. The analysis regarding isotropic matter profileof the self-gravitating system and its stability has been done by Sharif et al. [18]. Alhamzawi and Alhamzawi [19]construed the occurrence of lensing of gravitation in the context of a modified f ( R , T ) theory. Moraes et al. [20]numerically investigated the stability of the gravitational lensing by utilizing the Tolman–Oppenheimer–Volkov(TOV) equations in f ( R , T ) gravity. Das et al. [21] formulated a family of solutions by characterizing the interiorgeometry of compact stars, permitting conformal motion under the influence of f ( R , T ) gravity. Moraes et al. [22]investigated the configurations consisting of hydrostatic equilibrium along with fluids whose pressure was computedfrom equation of state (EoS) in the light of f ( R , T ) gravitational theory. Yousaf et al. [23] investigated the formationof relativistic stellar profiles in the regime of f ( R , T ) gravity by utilizing the Krori and Barura model. The studyof dense anisotropic profiles consisting of charge has been investigated by Maurya and Aurtiz [24]. In this regard,they utilized the Durgapal–Fuloria model in the context of f ( R , T ) gravity and applied gravitational decouplingutilizing geometric deformation. Waheed et al. [25] analyzed the existence of highly dense stellar configurations byutilizing Karmarkar along with the Pandey–Sharma condition. To do so, they used spherically symmetric space–timein the context of f ( R , T ) gravity. Mustafa et al. [26] analyzed the Class 1 embedding condition in the presenceof anisotropy matter profile and utilized the interior geometry of Schwarzschild along with Kohler–Chao solutionsin modified gravity. The matter configuration consisting of nuclear density of 10 gm/cc exhibits the behavior ofanisotropy i.e. p t − p r = 0 which exists due to certain factors involving magnetic flux, viscosity, phase transition, etc.In this regard, Ruderman [27] is the pioneer who argued about the anisotropy existing at the interior of the stars.Modified gravitational theories have provided an overwhelming approach in analyzing the anisotropic stellar con-figurations inheriting high matter profiles [28]-[33]. This work aims to analyze the modified f ( R , T ) gravity to devisea realistic configuration which in nature is anisotropic. For this purpose, we take into account three different matterprofiles i.e. LMC X-4, Cen X - 3, and EXO 1785–248, and apply a well-known embedding class 1 approach. Moreover,the structural aspect of anisotropic profiles has been examined by making use of spherically symmetric space–timealong with the categorical f ( R , T ) gravity model. The layout of this article is as follows: In Section 2, modified fieldequations have been formulated by utilizing the Karmarkar condition. Section 3 is to provide the matching condi-tions by considering Schwarzschild’s solution. The physical investigation has been done comprehensively in Section4. Conclusive remarks have been provided in the last Section. II. f ( R , T ) THEORY OF GRAVITY
The modified form of Einstein-Hilbert action for extended f ( R , T ) theory of gravity is defined as follows: S = 12 Z [ L m + f ( R , T )] √− gd x, (1)where L m and f ( R , T ) denote the matter Lagrangian density and a modified function, respectively. Here, R and T are known as scalar curvature and trace of the energy-momentum tensor, respectively. Now, by varying Eq.(1), weget the following modified set of equations(1 − f T ( R , T )) T µν − f T ( R , T )Θ µν = − f ( R , T ) g µν + ( R µν + ( g µν ✷ − ∇ µ ∇ ν )) f R ( R , T ) , (2)where, Θ µν = g αβ δ T µν δg µν = − g αβ ∂ L m ∂g µν ∂g αβ − T µν + g µν L m , ✷ = ∂ µ ( √− gg µν ∂ ν ) √− g , f R ( R , T ) = ∂f ( R , T ) ∂ R , f T ( R , T ) = ∂f ( R , T ) ∂ T , with ∇ , representing the covariant derivative. The energy-momentum tensor with the anisotropic matter source isdefined as T µν = ρU µ U ν + p r V µ V ν + p t ( U µ U ν − g µν − V µ V ν ) , (3)where U µ represents the vector for 4-velocity and V µ is a vector in the direction of radial pressure. Further, theexpressions, i.e., ρ, p t and p r are used to define define energy density, tangential and radial components of pressure,respectively.Using Eq.(3) in Eq.(2), we get the following set of equation: G µν = 1 f R ( R , T ) (cid:18) (1 + f T ( R , T )) T µν + ( ∇ µ ∇ ν − g µν ✷ ) f R ( R , T ) + 12 ( f ( R , T ) − R f R ( R , T )) g µν − ρg µν f T ( R , T )) . (4)We assume a static and spherically symmetric line element, which is defined as: ds = − e a ( r ) dt + e b ( r ) dr + r d Ω , (5)where the expression d Ω defines the g θθ = r dθ and g φφ = r sin θdφ components, e b ( r ) and e a ( r ) denote thegravitational components of stellar geometry. Further, we fix a quadratic model of f ( R , T ), which is defined as: f ( R , T ) = R + β × R + λ × T . (6)The considered model f ( R , T ) = R + β R + λ T involves a particular case of well-known Starobinsky model [35] withmatter coupling. It is an interesting point that, in the Starobinsky model R + β R , a maximum value of M/M ⊙ orbeyond is reached when the value of the parameter β is selected to be negative. But, this leads to an issue; specifically,the Ricci scalar performs a damped oscillation. On the other hand, the Ricci scalar smoothly decreases to zero as weapproach towards infinity for positive values of parameter β , for which the star can support a maximum mass lowerthan 2 M/M ⊙ . Now, we elaborate an eminent Karmarkar condition concisely which is the integral tool for currentstudy. The infrastructure connecting the Karmarkar condition is established on the class I space of Riemanniangeometry. A sufficient condition comprises of a second order symmetric tensor and the Riemann Christoffel tensor,given as Σ(Λ µη Λ υγ − Λ µγ Λ νη ) = R µυηγ , Λ µν ; n − Λ νη ; ν = 0 . Here ; stands for covariant derivative whereas Σ = ±
1. These values signify a space-like or time-like manifold relyingon the sign considered as − or +. Now, the Karmarkar condition is defined as R R = R R + R R . (7)These Riemann tensor components are given below as follows. R = e a ( r ) (2 a ′′ ( r ) + a ′ ( r ) − a ′ ( r ) b ′ ( r ))4 , R = r sin θ ( e b ( r ) − e b ( r ) , R = rb ′ ( r )2 , R = rsin θb ′ ( r ) e a ( r ) − b ( r )2 , R = R sin θ, R = 0 , where, R = 0. A differential equation can be achieved by utilizing the Karmarkar condition using Eq. (7) as a ′ ( r ) b ′ ( r )1 − e b ( r ) − (cid:0) a ′ ( r ) b ′ ( r ) + a ′ ( r ) − (cid:0) a ′′ ( r ) + a ′ ( r ) (cid:1)(cid:1) = 0 , e b ( r ) = 1 . (8)Integration of Eq. (8) provides a connection between two main gravitational components of the space-time as follows e b ( r ) = e a ( r ) × a ′ ( r ) + 1 + K, (9)where K is a constant of integration. We choose a specific model for a g tt component which is expressed as e a ( r ) = ψ (cid:0) r ψ + 1 (cid:1) n , (10)where ψ , ψ are assumed as constants, n is an integer. By plugging Eq. (10) in Eq. (9), we get the g rr component,which is calculated as e b ( r ) = r ψ ψ (cid:0) r ψ + 1 (cid:1) n − + 1 , (11)where ψ = 4 n × ψ × ψ × K . It is mentioned here that we get realistic results for n >
2. Now, we are able tocalculate the following set of modified field equations for the anisotropic stellar configuration. ρ = 12( λ + 1)(2 λ + 1) (cid:0) r ψ ψ Υ n − + r (cid:1) (cid:18) ψ Υ − n + r ψ ψ Υ × (cid:18) − β ( λ + 1) n r ψ ( − Υ ) Υ + Υ + 4 β (3 λ + 1) n r ψ Υ − β (3 λ + 2) Υ × (cid:18) − ψ Υ Υ n +21 Υ + ψ r (cid:0) nΥ (cid:0) r ψ − (cid:1) − n (cid:0) r ψ − r ψ + 1 (cid:1) − ψ Υ n +11 Υ × (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) − nr ( − Υ ) Υ Υ − Υ − Υ + Υ (cid:19) + n r ψ Υ Υ + 2 ψ Υ n − (cid:0) β (9 λ + 5) + Υ (cid:0) (2 λ + 1) r − βλ + β ) (cid:1)(cid:1) − nΥ × (cid:18) λΥ (cid:18) r ψ ψ Υ Υ − n + r ψ ψ Υ − (cid:0) β + r (cid:1)(cid:1) + β (cid:18) λ + 2 r ψ (cid:18) Υ Υ − n + r ψ ψ Υ + Υ − Υ (cid:19)(cid:19) (cid:19)(cid:19)(cid:19) , (12) p r = − λ + 1)(2 λ + 1) r Υ (cid:18) − βrψ (cid:18) ψ Υ Υ Υ − n + r ψ ψ Υ + 4 r ψ Υ Υ + Υ + Υ (cid:19) − Υ Υ + 4 rψ × ψ Υ n − n r ψ Υ Υ + 2 nrψ Υ + (cid:0) Υ (cid:0) (2 λ + 1) r − βλ + β ) (cid:1) − β (15 λ + 7) (cid:1) (cid:18) β (cid:0) λ + 4) + 2 r ψ × (cid:18) λ + 2) ψ Υ Υ n − Υ + 4(5 λ + 6) n ( − Υ ) Υ + ψ Υ Υ Υ − n + r ψ ψ Υ + Υ − Υ (cid:19)(cid:19) − Υ × (cid:18) λr ψ ψ Υ Υ − n + r ψ ψ Υ + 4 (cid:0) ( λ + 1) r − βλ (cid:1)(cid:19) (cid:19)(cid:19) , (13) p t = ψ λ + 1)(2 λ + 1) (cid:0) r ψ ψ Υ n − + r (cid:1) (cid:18) βΥ Υ + n r ψ Υ Υ + 2 ψ Υ Υ Υ − n + r ψ ψ Υ − Υ × n ( β ( − λ + 2 r ψ λ + 7) ψ Υ Υ n − Υ + 4(3 λ + 5) n ( − Υ ) Υ − λ + 6) nr ψ (cid:0) r ψ − (cid:1) Υ − λ + 19) r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) + Υ Υ − n + r ψ ψ Υ + 12(3 λ + 2) r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) + Υ Υ ! + 8 ! + Υ Υ ! + Υ − Υ (cid:19) , (14)where Υ i , { i = 1 , ..., } , are given in the Appendix ( I ). III. COMPARISON OF EXTERIOR AND INTERIOR SOLUTION
By considering the Jebsen-Birkhoff’s theorem, the spherically symmetric vacuum solution of GR field equationsmust be asymptotically flat. In particular, the spacetime is of the form ds = εdt − ε − dr − r ( dθ + sin θdφ ) , (15)where ε = (cid:0) − Mr (cid:1) . Here, M denotes the stellar mass of the star. Now, considering the constraint p r ( r = R ǫ ) = 0 atthe boundary r = R ǫ and using metric coefficients g tt and g rr from Eq. (5) and Eq. (15), we calculate the following TABLE I: Predicted values of ψ , ψ , ψ , and λ with β = 2 and and (Radii=9.711 km, & Mass =1.29 M/M ⊙ ).LMC X-4 n ψ ψ ψ λ × − ψ , ψ , ψ , and λ with β = 2 and (Radii= 10.136 km, & Mass =1.49 M/M ⊙ ).Cen X - 3 n ψ ψ ψ λ × − expressions ψ (cid:0) ψ R ǫ + 1 (cid:1) n = 1 − MR ǫ , (16) ψ ψ R ǫ (cid:0) ψ R ǫ + 1 (cid:1) n − + 1 = (cid:18) − MR ǫ (cid:19) − , (17) nψ ψ (cid:0) ψ R ǫ + 1 (cid:1) n − = MR ǫ , (18) p r ( r = R ǫ ) = 0 . (19)Utilizing these boundary conditions from Eqs. (16-19), we get the following relations ψ = ( R ǫ − M ) (cid:16) − M nM − nR ǫ + M (cid:17) − n R ǫ , (20) ψ = MR ǫ ( nR ǫ − (2 n + 1) M ) , (21) ψ = 2 n (cid:18) − M nM − nR ǫ + M (cid:19) − n , (22) λ = λ + λ + λ + λ λ + λ + λ + λ , (23)where λ i , { i = 1 , ..., } are given in the Appendix ( II ).The estimated values of the above parameters, i.e., ψ , ψ , ψ , & λ are given in Table I , Table II and Table III . TABLE III: Predicted values of ψ , ψ , ψ , and λ with β = 2 and (Radii=8.849 km, & Mass =1.30 M/M ⊙ ).EXO 1785-248 n ψ ψ ψ λ × − IV. PHYSICAL ANALYSIS
In this section, we briefly present the results by analyzing the physical attributes along with different aspects of thestellar configurations under the acquired f ( R , T ) gravity model. In order to fulfill the purpose, experimental data ofdistinct stars i.e. LMC X-4, Cen X – 3 and EXO 1785–248 is used. All the attributes of the stellar configurations aredepicted in tabular form as well as graphically. A. Gravitational Metric Potential
The existence of anomalies within the sphere such as geometric singularities are contemplated to be an essentialpeculiarity in the investigation of stellar spheres. In order to unravel the existence of singularities, we examine thenature of gravitational potential g tt = e a and g rr = e b at the core r = 0 of the sphere. Physical essence and enduranceof the models rely upon the gravitational metric potentials and it should be decreasing on regular intervals withinthe spherical structures. It can be observed from Fig. 1 that the metric potential with in the interior of the sphereexhibits the behavior of e b ( r =0) = 1 and e a ( r =0) = 0 which is consistent and physically valid. It can also be observedthat both of the metric potentials exhibit minimum values at the center and show non-linear increasing behaviortowards the boundary. FIG. 1: Visual representation of gravitational potentials with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) B. Energy Density and Pressure Evolutions
Prior to analysis of the anisotropy, we investigate the evolutional change of the matter profiles in connection to theenergy density ρ along with anisotropic stresses such as p r and p t . The energy density along with pr and pt exhibitsthe exceptional behavior of high density of matter configuration. The phenomenal high density is due to the strongforces of attraction which are regarded as dipole interactions and intermolecular forces. Numerical values of densityand the components of the pressure for the three compact spheres are provided in Tables IV - VI . All of the physicalattributes remain positive and appear to be finite at the core. It confirms that the current system is independentof all singularities. From the Figs. 2-4, it is evident that the matter configuration under consideration attains themaximum mass at the core and tends to zero at the boundary of the star, which depicts the high compactness of thestellar spheres. These graphical plots establish the presence of anisotropy of the compact sphere under the influenceof our f ( R , T ) model. FIG. 2: Visual representation of ρ with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ )FIG. 3: Visual representation of p r with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ )FIG. 4: Visual representation of p t with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) C. Anisotropy and Gradients
In order to model the interior geometry of the relativistic stellar configuration under current circumstances, therole of anisotropy is crucial for the compact sphere modeling and it is represented as∆ = p t − p r . (24)It depicts the information regarding the anisotropic nature of the stellar configuration. If p t > p r then the anisotropyis considered to be non negative and is drawn outwards and depicted as ∆ <
0. Whereas if p r > p t then the anisotropyturns out to be negative and this shows that anisotropy is drawn inwards. From 5 it is observed that for our ongoingstudy anisotropy remains positive, hence, directed outwards. The deviation of radial derivatives of the energy densityand pressure components , i.e., dρdr , dp r dr and dp t dr , are shown in Figs. 6-8 such that FIG. 5: Visual representation of ∆ with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) dρdr < , dp r dr < , dp t dr < . (25)It can be observed from the second order derivatives that the pressure components and energy density show themaximum value at the core r = 0, i. e., d ρdr > , d p r dr > , d p t dr > . (26) FIG. 6: Visual representation of gradient of ρ with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) FIG. 7: Visual representation of gradient of p r with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ),and n = 500( ⋆ )FIG. 8: Visual representation of gradient of p t with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) D. Energy Conditions
Energy conditions appear to be quite helpful in analyzing the realistic distribution of matter. These attributes playa decisive role to classify the exotic and normal mater distribution within the stellar model. The energy conditionshave been crucially important in debating the issues related to cosmology and astrophysics. The energy condition areclassified as
N EC : ρ > , ρ + p r ≥ , ρ + p t ≥ , DEC : ρ ≥ | p r | , ρ ≥ | p t | ,W EC : ρ − p r ≥ , ρ − p t ≥ ,SEC : ρ − p r ≥ , ρ − p t ≥ , ρ − p r − p t ≥ . (27) TABLE IV: Predicted values of physical parameters at center and boundary.LMC X-4n e a ( r =0) e b ( r =0) ρ R ( g/cm ) p r = p t ( dyne/cm ) ρ ( g/cm ) p r /ρ = p t /ρ . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . TABLE V: Predicted values of physical parameters at center and boundary.Cen X - 3 (mass =1.49
M/M ⊙ & radii= 10.136 km)n e a ( r =0) e b ( r =0) ρ R ( g/cm ) p r = p t ( dyne/cm ) ρ ( g/cm ) p r /ρ = p t /ρ . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . M/M ⊙ & Radii=8.849 km)n e a ( r =0) e b ( r =0) ρ R ( g/cm ) p r = p t ( dyne/cm ) ρ ( g/cm ) p r /ρ = p t /ρ . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . . × . × . × . Here,
N EC stands for null energy condition,
SEC for strong energy condition ,
DEC for dominant energy conditionand
W EC for week energy condition. It can be seen from Fig. 9 that all the energy bounds exhibit decreasingbehavior with the increase in radii of the compact stellar sphere.
FIG. 9: Evolution of energy bounds with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) E. Analysis of Stability
The stability of the stellar configuration plays a decisive role in analyzing the consistency of the acquired model.Many analytical discussions have been done in order to find the stability of the matter configuration but Herrera’scracking conception emerged to be very effective [34]. The radial and tangential speed of sound are defined as v sr = dp r dρ , and v st = dp t dρ . (28)For the conservation of causality condition, components of the speed of sound must be with the bounds of the interval[0 ,
1] i.e 0 ≤ v r and v t ≤
1. It can be observed from the Figs. 10-11 that the condition i.e 0 ≤ v r and v t ≤ − ≤ | v t − v r | ≤ ≤ | v t − v r | ≤ FIG. 10: Visual representation of v r and v t with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ )FIG. 11: Visual representation of | v t − v r | and | v r − v t | with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) F. Equilibrium Analysis for Modified f ( R , T ) Gravity
In this section, we will analyze the equilibrium condition by considering the stability of the acquired solution of thethree different stellar configuration. For the purpose, we make use of the TOV equation [36]-[41]2∆ r − dp r dr − a ′ ρ + p r ) + λ (cid:16) − dp r dr − dp t dr + 3 dρdr (cid:17) λ + 1) = 0 . (29)The above equation characterizes the necessary and sufficient condition for the hydrostatic-equilibrium. It comprisesof four different forces F g = a ′ ρ + p r ) , F h = dp r dr , F a = 2∆ r , F e = λ (cid:16) − dp r dr − dp t dr + 3 dρdr (cid:17) λ + 1) . (30) • F a represents the anisotropy force. • F h represents the hydrostatic force. • F g represents the gravitational force.2 • F e represents the extra force.Consequently, the T OV equation can also be written as F g + F h + F a + F e = 0. From the attained graph as shownin Fig. 12, it is deduced that all of the forces sum up to neutralize the total effect, and this confirms the existence ofthe stable stellar structures. FIG. 12: Visual representation of F g , F h , F a , and F e with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) G. Evolution of Adiabatic Index
For the energy density, EoS stiffness can be better described by the adiabatic index. The stability of relativisticas well as the non-relativistic stellar structures can be explained through the adiabatic index. The concept of thedynamical stability via the radial adiabatic index was presented by Chandrasekhar [42]. This was further utilized bymany authors [43]-[48]. For the system to be dynamically stable, the adiabatic index must go beyond 4 /
3. Adiabaticindex corresponding to the radial stress is given asΓ r = ρ + p r p r ( dp r dρ ) = ρ + p r p r v r . (31)One of the quite fascinating fact of the above equation is that the stability of the Newtonian matter configurationis achieved when Γ r > /
3. While if Γ r = 4 / r < / r = 43 + ( ρ i p ri | p ′ ri | r + 43 p ti − p ri | p ′ ri | r ) . (32)From the Fig. 13, the graphical behavior of Γ r with respect to increasing radii can be observed. It is noted that Γ r shows the monotonically increasing conduct for all the stellar spheres and Γ r is always greater than 4 /
3. Hence, Γ r is consistent for the stability of our model in f ( R , T ) gravity. H. Equation of state
The evolution of the emergence of compact stars can be determined by
EoS of matter. Moreover,
EoS has a strongimpact on the conditions of nucleosynthesis. Therefore,
EoS is a vital tool in many astrophysical simulations. The
EoS is considered to be a ratio of the pressure terms p r and p t with density. The components of EoS for the studyof stellar configuration are i.e., ω r and ω t and are mathematically connected as ω r = p r ρ , ω t = p t ρ . (33)From Fig. 14, it can be observed that with the increase in radii, the components of the EoS shows monotonicallydecreasing behavior and are always less than 1. Moreover, the positive nature is observed for both of the componentsof
EoS i.e. ω r and ω t with in the matter configuration. The accomplishment of the condition i.e. 0 ≤ ω r and ω t < FIG. 13: Evolution of Γ r with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ )FIG. 14: Evolution of energy EoS with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) I. Compactness factor and Surface Redshift
For the existence of any matter configuration mass function, compactness factor along with surface redshift functionare contemplated to be an essential constituent. The fundamental relation for the mass function is given as m ( r ) = Z R πρr dr. (34) FIG. 15: Evolution of m ( r ) with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) FIG. 16: Evolution of µ ( r ) with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ )FIG. 17: Evolution of z s with n = 3( ⋆ ), n = 5( ⋆ ), n = 10( ⋆ ), n = 20( ⋆ ), n = 50( ⋆ ), n = 100( ⋆ ), and n = 500( ⋆ ) The generalize compactness factor i.e. µ ( r ) is represented as µ ( r ) = mr = 1 r Z R πρr dr. (35)The strong intermolecular interaction forces with in the stellar matter configuration and its corresponding EoS canbe characterized by the term surface redshift i.e., Z s . The generalized relation is Z s = 1( ψ ( ψ r + 1) n ) / − . (36)Figs. 15-17 represent the evolution of redshift function along with compactness factor corresponding to the increasingradii. It can be observed that the surface redshift is always less than 5 i.e., 5 i.e. Z s ≤ .
30 i.e., µ ( r ) ≤ .
30. All of the mentioned functions are positive throughout the configuration.Hence, our models are stable.
V. CONCLUSION
The manifesto of the current study is to identify the realistic and stable configuration for the stellar sphere in themodified f ( R , T ) gravitational theory. For the analysis of the stellar matter configuration, a viable model f ( R , T ) = R + β R + λ T is considered along with spherically symmetric space–time. In order to achieve the current objective,observational data of three compact spheres i.e. LMC X-4, Cen X-3 and EXO 1785–248 inheriting an anisotropicmatter distribution has been utilized. This realistic range for masses of stars under this study is 1.29 to 1.5 solar mass.In this context, the considered models LMC X-4 (mass 1.29 M/M ⊙ ), Cen X-3 (mass 1.49 M/M ⊙ ) and EXO 1785-248(mass 1.29 M/M ⊙ ) are with in suitable range under this study. Moreover, the radii are also with in prescribed range8.849 km to 10.136 km. In general, the study is valid for other models of stars and those within the given ranges of5mass and radius under this study. Embedding class 1 condition is used to find the potential i.e. g rr by consideringthe primary potential as i.e g tt = e a ( r ) = χ (cid:0) r χ + 1 (cid:1) K . To find the unknown constraints, Schwarzschild’s exteriorsolution has been utilized. All the obtained results can be summarized as: • Metric potentials : The presence of singularities within the stellar configuration is an essential topic, worthy ofdebate. The stability of the stellar matter configuration depends on it. Therefore, the aspects of metric potentialplay a decisive role. The graphical behavior of Fig. 1 depicts that the fundamental condition i.e. e b ( r =0) = 1and e a ( r =0) = 0 has been encompassed by the gravitational potential. Increasing attribute of the potentials isalso observed throughout the configuration. Therefore, the potentials are free from any singularity and so is ourmodel. • Energy density and stress constraints : From the Figs. 2-4 the evolution of density and components of stressesi.e., p r and p t can be observed. The density along with stress components show decreasing behavior and are non-negative throughout the configuration. The peak value is accomplished at the core while decreasing evolutionis observed with the increase in radii and it tends to 0 towards the boundary. • Anisotropy and gradients : Fig. 5 depicts anisotropy behavior for our current configuration. It is noted that p t = p r and p t > p r , therefore, △ > dρdr < dp r dr < dp t dr <
0. Since the non-positive behavior of all gradients alongwith their vanishing attribute at r = 0 is observed, therefore, our stellar configuration is stable. • Energy bounds : From the Figs. 2-4 and Fig. 9 the characteristic behavior of functions ρ , p r , p t , ρ − p r , ρ − p t , ρ − p r − p t are depicted. Decreasing characteristic behavior is observed with the increase in radii which furtherfulfills the bounds of energy such as ( N EC ), (
SEC ), (
DEC ) and (
W EC ). Hence the matter profile is realisticand viable. • Causality analysis
The behavior of the constraints of sound speed is depicted in Figs. 10 and Fig. 11. Thedecreasing attribute is observed for both the components of the speed and it is shown that they are alwayswithin the limits i.e., 0 ≤ v r ≤ ≤ v t ≤
1. The fulfillment of Aberu condition is also observed i.e., − ≤ | v t − v r | ≤
0. For the current model v t > v r . Therefore, fulfillment of all the conditions confirms theviability of the stellar sphere. • Equilibrium and
EoS analysis : The balancing nature of all the forces, i.e., F a , F h and F g is depicted in Fig.12. As all these forces add up to 0 and balance the effect of each other, therefore, the equilibrium condition issatisfied. From Fig. 14, the attributes of the parameters of EoS are observed. It is concluded that constraintsi.e., ω r and ω t of EoS are positive in the interior of stellar profile and are in the stability bounds of 0 ≤ ω r and ω t < • Adiabatic index stability analysis : The behavior of adiabatic index Γ r can be seen in Fig. 13 showing thatΓ r > /
3. It also depicts the positive and decreasing nature, justifying the effectiveness of our system in theframework of f ( R , T ) theory. • Redshift, mass function and compactness factor : Fig. 15-17 exhibit the compactness factor, mass function alongwith gravitational redshift. It can be seen that both µ ( r ) ≤ .
30 and m ( r ) show increasing behavior. Apart fromthis, Z s shows the decreasing attribute and Z s ≤ f ( R , T ) gravity [23, 25, 26, 49, 50].6 Appendix (I) Υ = r ψ + 1 , Υ = 48 β (3 λ + 2) r ψ ψ (cid:0) ( n − r ψ + 1 (cid:1) (cid:0) Υ − n + r ψ ψ Υ (cid:1) , Υ = Υ (cid:0) ( n − r ψ − n + 4 (cid:1) ,Υ = Υ (cid:0) ( n − n − r ψ + 5 n − (cid:1) , Υ = r ψ ψ Υ n + Υ , Υ = r ψ (cid:0) ( n − n − r ψ + 9 n − (cid:1) + 6 ,Υ = r ψ (cid:0) ( n − n + 1)(2 n − r ψ + 3( n ( n + 4) − r ψ + 6 n − (cid:1) + 3 ,Υ = r ψ (cid:0) ( n − r ψ − n + 9 (cid:1) + 3 , Υ = 2 βr ψ ψ (cid:0) ( n − r ψ + 1 (cid:1) (cid:16) λ − λ +2) nr ψ ( − Υ ) Υ + 24 (cid:17)(cid:0) Υ − n + r ψ ψ Υ (cid:1) ,Υ = λ + 8 β (3 λ + 2) ψ ψ Υ n − (cid:0) Υ Υ − r ψ ψ Υ Υ n (cid:1) Υ + λr ψ ψ Υ n − ,Υ = 4 βn r ψ (cid:18) rψ ψ (6 λr + r ) ( ( n − r ψ +1 ) Υ − n + r ψ ψ Υ − λ + 1) (cid:19) Υ ,Υ = 4 ψ (cid:0) ( n − r ψ + 1 (cid:1) (cid:18)(cid:0) (2 λ + 1) r − βλ (cid:1) (cid:0) r ψ ψ Υ n − + 1 (cid:1) + β (cid:18) − λ + 2 Υ × (3 λ + 2) r ψ ( − n × (cid:0) r ψ − (cid:1) + 6 nr ψ (cid:0) r ψ − (cid:1) − ψ Υ n +11 (cid:0) Υ Υ − r ψ ψ Υ Υ n (cid:1) Υ ! − !! ,Υ = 8 β (3 λ + 2) + 2 λr (cid:0) r ψ ψ Υ n − + 1 (cid:1) + 2 βrψ (8 − λ ) rψ (cid:0) ( n − r ψ + 1 (cid:1) Υ − n + r ψ ψ Υ − λ + 11) r ψ ψ (cid:0) ( n − r ψ + 1 (cid:1) (cid:0) Υ − n + r ψ ψ Υ (cid:1) + 2(3 λ + 4) rψ Υ Υ n − Υ − n ( − Υ ) (6 λr + r ) Υ ! ,Υ = r ψ ψ Υ n − + 1 , Υ = ( n − r ψ + 1 , Υ = Υ Υ − r ψ ψ Υ Υ n ,Υ = − λ + 1) ψ Υ Υ n − Υ + 4(3 λ − n ( − Υ ) Υ − λ + 2) r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) ,Υ = 4(3 λ + 4) nr ψ (cid:0) r ψ − (cid:1) Υ + 4(3 λ + 2) r ψ ψ Υ n − (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) Υ ,Υ = 8(9 λ + 8) r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) + ψ Υ (cid:18) λ + 6) nr ψ ( − Υ ) Υ + (3 λ + 2) (cid:18) r ψ ψ Υ Υ n − Υ + 4 (cid:19)(cid:19) ,Υ = 8 β (3 λ + 1) n r ψ Υ + 8 βn r ψ (cid:16) λ + rψ ψ Υ ( r − λr ) Υ − n + r ψ ψ Υ (cid:17) Υ + λ (cid:18) nr ψ ( − Υ ) Υ + 24 βrψ ψ Υ Υ − n + r ψ ψ Υ (cid:19) ,Υ = − λ + 1) n r ψ ( − Υ ) Υ − λ + 4) ψ Υ Υ n − Υ + 24 λr ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) ,Υ = r ψ ψ Υ (cid:18) λ ( (11 n +24) r ψ +(48 − n ) r ψ +24 ) Υ + 24 (cid:19)(cid:0) Υ − n + r ψ ψ Υ (cid:1) − n ( − Υ ) (cid:16) λ + λr ψ ψ Υ Υ n − Υ + 4 (cid:17) Υ ,Υ = 4( λ + 1) nΥ (cid:0) r ψ − (cid:1) + λ (cid:0) nr ψ − nr ψ Υ + 3 nΥ + 2 ψ Υ n +11 (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) Υ ! , Υ = 2 r ψ (cid:16) − λψ Υ Υ n +11 Υ + 4 n (cid:0) r ψ − (cid:1) + 6 λnr ψ (cid:0) r ψ − (cid:1)(cid:17) Υ − λ,Υ = β r ψ − ( λ − ψ Υ Υ n − Υ − λ − n ( − Υ ) Υ + (7 λ − r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) + 4( λ + 3) ψ Υ Υ − n + r ψ ψ Υ (cid:19) − λ (cid:19) + 2 λr Υ ,Υ = 8( λ − r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) + 4( λ − nr ψ (cid:0) r ψ − (cid:1) Υ − λr ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) ,Υ = 4 λr ψ ψ Υ n − (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) Υ ,Υ = 68 λ + 2 r ψ (cid:16) λψ Υ Υ n Υ + (13 λ − n ( − Υ ) (cid:17) Υ + 32 ,Υ = 4 β ( λ + 1) n r ψ Υ − β (3 λ + 2) nr ψ ψ ( − Υ ) Υ Υ n − ( r ψ ( ψ Υ n + 2) + r ψ + 1) − Υ − λ + 1) nr ( − Υ ) (cid:0) r ψ ( ψ Υ n + 2) + r ψ + 1 (cid:1) Υ ,Υ = 4 βψ Υ n − (cid:0) λ + ( λ + 1) r ψ ψ Υ n − + 8 (cid:1) − βn r ψ (cid:16) λ +3) r ψ ψ Υ Υ − n + r ψ ψ Υ − (cid:17) Υ ,Υ = (7 λ + 5) n r ψ ( − Υ ) + 2 λψ Υ Υ n +21 Υ + 2 n ( − Υ ) Υ (cid:18) λ + 2(3 λ + 2) r ψ ψ Υ Υ n − Υ + 2 (cid:19) + 2 r ψ × (cid:0) λ + 3) nΥ (cid:0) r ψ − (cid:1) − (3 λ + 2) (cid:0) nr ψ − nr ψ Υ + 3 nΥ + 2 ψ Υ n +11 (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) Υ !! ,Υ = 4 β (9 λ + 7) + 4 βr ψ (cid:16) λ + 7) n ( − Υ ) Υ + (3 λ + 2) (cid:16) ψ Υ Υ n +11 Υ − nr ψ (cid:0) r ψ − (cid:1)(cid:17)(cid:17) Υ − Υ × (cid:0) β ( λ + 1) + (2 λ + 1) r (cid:1) ,Υ = − β (3 λ + 2) + 4 βr ψ − (7 λ + 6) ψ Υ Υ n − Υ − λ + 3) n ( − Υ ) Υ + (21 λ + 17) r ψ ψ Υ (cid:0) Υ − n + r ψ ψ Υ (cid:1) − λ + 4) ψ Υ Υ − n + r ψ ψ Υ (cid:19) + 2( λ + 1) r Υ , Υ = 2( λ + 1) r ψ ψ Υ Υ − n + r ψ ψ Υ − (cid:0) β ( λ + 1) + r (cid:1) ,Υ = 4(3 λ + 2) r ψ ψ Υ n − (cid:0) r ψ ψ ( Υ Υ ) Υ n − ψ Υ Υ n +21 + ( n − Υ Υ (cid:1) ,Υ = ψ Υ r ψ (cid:16) (5 λ + 4)( − n ) ( − Υ ) − (3 λ +2) ψ Υ Υ n Υ (cid:17) Υ − λ + 6) . Appendix (II) λ = − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n + r ψ ψ Υ n + 8 r ψ ψ Υ n + 28 r ψ ψ Υ n + 56 r ψ ψ Υ n + 70 r ψ ψ Υ n +56 r ψ ψ Υ n + 28 r ψ ψ Υ n + 8 r ψ ψ Υ n + ψ Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n + 4 r ψ ψ Υ n + 24 r ψ ψ Υ n + 60 r ψ ψ Υ n +80 r ψ ψ Υ n + 60 r ψ ψ Υ n + 24 r ψ ψ Υ n + 4 r ψ ψ Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n − ψ nψ r Υ n + 6 r ψ ψ Υ n + 24 r ψ ψ Υ n + 36 r ψ ψ Υ n + 24 r ψ ψ Υ n +6 r ψ ψ Υ n + 4 r ψ ψ Υ n − nψ r − nψ r − nψ r − nψ r − nψ r − nψ r − nψ r − nψ r − nψ r − n,λ = 2 βn r ψ + 12 βn r ψ + 30 βn r ψ + 40 βn r ψ + 30 βn r ψ + 12 βn r ψ + 2 βn r ψ + 4 βn r ψ +20 βn r ψ + 36 βn r ψ + 20 βn r ψ − βn r ψ − βn r ψ − βn r ψ + 76 βn r ψ ψ Υ n − βn r ψ − βn ψ + 18 βn r ψ + 128 βn r ψ + 368 βn r ψ + 528 βn r ψ + 340 βn r ψ − βn r × ψ − βn r ψ + 660 βn r ψ ψ Υ n − βn r ψ + 130 βn r ψ ψ Υ n + 80 βnψ − βψ ψ Υ n + 144 βnψ × ψ Υ n + 16 βnr ψ + 192 βnr ψ + 896 βnr ψ + 2240 βnr ψ − nr ψ ψ Υ n + 4 r ψ ψ Υ n + 3360 × βnr ψ − nr ψ ψ Υ n + 8 r ψ ψ Υ n + r ψ ψ Υ n + 3136 βnr ψ − βr ψ ψ Υ n + 1792 βnr ψ − βr ψ ψ Υ n + 1840 βnr ψ ψ Υ n + 576 βnr ψ − βr ψ ψ Υ n + 800 βnr ψ ψ Υ n λ = − ψ ψ r βΥ n − ψ ψ r βΥ n + 18 n r βψ ψ Υ n + 36 n r βψ ψ Υ n + 18 n r βψ ψ Υ n + 32 r β × ψ ψ Υ n + 72 n r βψ ψ Υ n + 220 n r βψ ψ Υ n + 212 n r βψ ψ Υ n + 80 nr βψ ψ Υ n + 32 r βψ ψ Υ n +108 n r βψ ψ Υ n + 520 n r βψ ψ Υ n + 798 n r βψ ψ Υ n + 544 nr βψ ψ Υ n + 72 n r βψ ψ Υ n + 600 × n r βψ ψ Υ n + 1432 n r βψ ψ Υ n + 1520 nr βψ ψ Υ n + 18 n r βψ ψ Υ n + 340 n r βψ ψ Υ n + 1358 n r β × ψ ψ Υ n + 2240 nr βψ ψ Υ n + 16 n r βψ ψ Υ n + 192 n r βψ ψ Υ n + 440 n r βψ ψ Υ n + 240 nr βψ × ψ Υ n + 80 n r βψ ψ Υ n + 424 n r βψ ψ Υ n + 464 nr βψ ψ Υ n + 144 n r βψ ψ Υ n + 392 nr βψ ψ Υ n +120 nr βψ ψ Υ n − ψ ψ r βΥ n − ψ ψ r βΥ n − ψ ψ r βΥ n − ψ ψ r βΥ n − ψ ψ r βΥ n − ψ ψ βΥ n λ = 16 βn r ψ ψ Υ n + 32 βn r ψ ψ Υ n + 32 βn r ψ ψ Υ n + 144 βn r ψ ψ Υ n + 24 βn r ψ ψ Υ n + 184 × βn r ψ ψ Υ n + 24 βn r ψ ψ Υ n + 16 βn r ψ ψ Υ n − βn r ψ ψ Υ n + 8 βnr ψ ψ Υ n − βr × ψ ψ Υ n + 56 βnr ψ ψ Υ n + 24 βnr ψ ψ Υ n − βr ψ ψ Υ n + 56 βnr ψ ψ Υ n − βr ψ ψ Υ n − βr ψ ψ Υ n + 8 βnr ψ ψ Υ n − βr ψ ψ Υ n − βr ψ ψ Υ n − βr ψ ψ Υ n − βr ψ ψ Υ n − βnr ψ ψ Υ n − βr ψ ψ Υ n − βr ψ ψ Υ n λ = − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ ψ r Υ n + 11 nr ψ ψ Υ n + 71 nr ψ ψ Υ n + 195 nr ψ ψ Υ n + 295 × nr ψ ψ Υ n + 265 nr ψ ψ Υ n + 141 nr ψ ψ Υ n + 41 nr ψ ψ Υ n + 5 nr ψ ψ Υ n − ψ Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ ψ r Υ n − ψ ψ r Υ n + 114 × nr ψ ψ Υ n + 51 nr ψ ψ Υ n + 9 nr ψ ψ Υ n − n ψ r − n ψ r − n ψ r − n ψ r − n ψ × r + 3 nr ψ − n ψ r + 25 nr ψ + 92 nr ψ − n ψ r + 196 nr ψ + 266 nr ψ − n ψ r +238 nr ψ + 140 nr ψ − n ψ r + 52 nr ψ + n + 11 nr ψ λ = − ψ nψ βΥ n + 200 βψ ψ Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n + 15 nr ψ ψ Υ n + 69 nr ψ ψ Υ n + 126 nr ψ ψ Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ n ψ r Υ n − ψ ψ r Υ n − ψ ψ r Υ n + 9 nr × ψ ψ Υ n + 25 nr ψ ψ Υ n + 23 nr ψ ψ Υ n + 7 nr ψ ψ Υ n − ψ ψ r Υ n − ψ ψ r Υ n − ψ ψ r Υ n +2 nr ψ ψ Υ n + 2 nr ψ ψ Υ n − ψ ψ r Υ n + 84 n r βψ + 560 n r βψ + 700 n r βψ + 420 n r βψ +124 n r βψ − n ψ r β − n ψ r β − nψ r β − n ψ r β − n ψ r β − n ψ r β − n × ψ r β − n ψ r β − n ψ r β − n ψ r β − nψ r β − n ψ r β − n ψ r β − n × ψ r β − n ψ r β − n ψ r β − nψ r β − n ψ r β − n ψ r β − nψ r β − n ψ r β − n ψ r β − nψ r β − n ψ r β − nψ r β − nψ β + 14 n βψ λ = − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ ψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − × ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ n × ψ r βΥ n − ψ nψ r βΥ n + 2 n r βψ ψ Υ n + 88 nr βψ ψ Υ n + 8 n r βψ ψ Υ n + 56 r βψ ψ Υ n +12 n r βψ ψ Υ n + 1800 r βψ ψ Υ n + 8 n r βψ ψ Υ n + 5080 r βψ ψ Υ n + 2 n r βψ ψ Υ n + 56 n r βψ ψ × Υ n + 6520 r βψ ψ Υ n + 40 n r βψ ψ Υ n + 4392 r βψ ψ Υ n + 34 n r βψ ψ Υ n + 1496 r βψ ψ Υ n + 3454 r × βψ ψ Υ n + 1220 r βψ ψ Υ n + 106 βψ ψ Υ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ nψ r βΥ n λ = 60 n r βψ ψ Υ n + 114 r βψ ψ Υ n + 772 r βψ ψ Υ n + 2534 r βψ ψ Υ n + 4216 r βψ ψ Υ n − ψ n × ψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ n × ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ n × ψ r βΥ n − ψ n ψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n + 126 r βψ ψ Υ n + 528 r β × ψ ψ Υ n + 16 n r βψ ψ Υ n + 924 r βψ ψ Υ n + 52 nr βψ ψ Υ n + 800 r βψ ψ Υ n + 278 r βψ ψ Υ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ n ψ r βΥ n − ψ nψ r βΥ n − ψ nψ r βΥ n + 42 r β × ψ ψ Υ n + 108 r βψ ψ Υ n + 66 r βψ ψ Υ n + 6 r βψ ψ Υ n References [1] Baade and Zwicky., PNAS , 1241 (2015).[2] K.R. 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