Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State
JJanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd
International Journal of Modern Physics D © World Scientific Publishing Company
DISTINCT CLASSES OF COMPACT STARS BASED ONGEOMETRICALLY DEDUCED EQUATIONS OF STATE
A. C. Khunt
Department of Physics, Sardar Patel University,Vallabh Vidyanagar-388 120, Gujarat, INDIA,[email protected]
V. O. Thomas
Department of Mathematics, Faculty of Science,The Maharaja Sayajirao University of Baroda,Vadodara – 390 001, Gujarat, [email protected]
P. C. Vinodkumar
Department of Physics, Sardar Patel University,Vallabh Vidyanagar-388 120, Gujarat, INDIA,[email protected]
Received Day Month YearRevised Day Month YearWe have computed the properties of compact objects like neutron stars based on equationof state (EOS) deduced from a core-envelope model of superdense stars. Such superdensestars have been studied by solving the Einstein’s equation based on pseudo-spheroidaland spherically symmetric space-time geometry. The computed star properties are com-pared with those obtained based on nuclear matter equations of state. From the mass-radius ( M − R ) relationship obtained here, we are able to classify compact stars in threecategories: (i) highly compact self -bound stars that represents exotic matter composi-tions with radius lying below 9 km (ii) normal neutron stars with radius between 9 to 12km and (iii) soft matter neutron stars having radius lying between 12 to 20 km. Otherproperties such as Keplerian frequency, surface gravity and surface gravitational redshiftare also computed for all the three types. The present work would be useful for the studyof highly compact neutron like stars having exotic matter compositions. Keywords : neutron star; core-envelope model; dense matter equation of state.PACS numbers:
1. Introduction
Neutron stars are one of the densest objects in the observable universe. It representsstate of matter with highest densities. As such, they are valuable laboratories forthe study of dense matter. Such studies include interplay between various disciplineslike general relativity, high-energy astrophysics, nuclear and particle physics etc.
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Neutron stars have masses of about (1 ∼ (cid:12) ). These stars with masses about 1.2M (cid:12) have central densities more than normal nuclear matter density and radius of theorder of 10 km. The average mass density ρ of the neutron star is approximately 10 g cm − , which is about 3 times the nuclear saturation density ρ n = 2 . × g cm − and at the core ρ > ρ n . The magnetic field of such a compact stars lies between10 -10 G and possess gravitational field 2 × cm s − times stronger thanthat of earth’s gravitational fields. The structure of these stars can be consideredhaving an outer and an inner crust. The envelope (outer crust) matter consists ofatomic nuclei (ions) and electrons. The thickness of envelope is few hundred meters.The inner crust occurs at a density of 4 × g cm − which consists of electrons,free neutrons and neutron-rich atomic nuclei. The thickness of this crust is typicallyabout few kilometers. The outer crust envelopes the inner crust, which expandsfrom the neutron drip density to a transition density ρ tr ∼ . × g cm − . Andbeyond the transition density one enters the core , where all atomic nuclei have beenmelt down into their components, neutrons and protons. Caused by the high densityand Fermi pressure, the core might also contain more massive baryon resonancesor possibly a gas of free up, down and strange quarks. Ultimately, π and K mesonscondensates may be found there too. All these dissimilar internal structure leadto different physical equation of state and hence contrasting mass-radius (M-R)relations.In view of our inadequate knowledge of the equation of state of matter at ex-tremely high densities, when matter density of ultra dense spherical objects is muchhigher than nuclear saturation density ( ρ > ρ n ), it is difficult to have proper eluci-dation of matter in the form of an equation of state and quantitative calculations forthe structure of neutron stars become obscure. A methodical valuation on the struc-ture and properties of neutron stars can be found in
2, 4–10 (and references therein).Many theorists have developed theoretical models for the structure of neutron starswhich may be made up of various layers including core (inner and outer), crust(inner and outer) in which atomic nuclei are arranged into a crystal and the liq-uid ocean composed of the coulomb fluid. The central region, i.e., core containshyperons
11, 12 or quark matter. A detailed analysis of quarks core models are dis-cussed by Bordbar, Bigdell and Yazdizadeh. Alternative method to study compacthigh-density astrophysical objects is through the space-time metric of the generaltheory of relativity and solving the relevant Einstein’s equations. Such attemptsparticularly for compact object, like the neutron star exist. Thus, for the presentstudy we make use of the core-envelope model for neutron stars studied based onthe geometric approach making use of the relativistic model for these regions. Thecore-envelope model of a neutron star has different physical properties in en-velope and core regions. From this we have considered two different EOS, basedon anistropic pressure in core or envelope region. The core-envelope models studiedby Thomas, Ratanpal and Vinodkumar (TRV model) have considered anisotropicpressure in the envelope region and isotropic pressure in the core region. While inanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State another case studied by S. Gedela, N. Pant, J. Upreti and R. Pant (SNJR model) have taken both the core and envelope region as anistropic. In both the cases validsolutions of the Einstein’s equations were studied in appropriate metrics. The EOSsdeduced from these models are then used to compute the neutron star properties.Brief descriptions of these two models
16, 17 are given in the following section.
2. Relativistic core-envelope framework
Our primary focus in this paper is based on the models belonging to the core-envelope family as discussed by Thomas et al. (TRV) and Gedela et al. (SNJR).We summarize below only the relevant part of the formalism adopted for the studyof compact objects with appropriate geometric consideration. More details can befound in the earlier works
16, 17
A nonrotating spherical metric in a most general form can be expressed as ds = e ν ( r ) dt − e λ ( r ) dr − r dθ − r sin θdφ (2.1)where, r is the radial coordinate, θ is the polar angle and φ is the azimuthal angle.The right boundary condition for the metric is to match (2.1) with the Schwarchildexterior metric at the surface of the star. It is implemented as ν ( r = a ) = ln (cid:18) − GMac (cid:19) (2.2) λ ( r = a ) = − ln (cid:18) − GMac (cid:19) (2.3)Here, a and M is the radius and mass of the star.The Einstein field equation is given by R µν − R g µν = − πGc T µν (2.4)has been solved for the metric given by Eqn (2.1) for an energy momentum tensorrelevant for perfect fluid
16, 17 T µν = ( ρ + p ) u µ u ν − P g µν + π µν (2.5)where π µν denotes anistropic stress tensor give by π µν = √ S (cid:20) C µ C ν −
13 ( u µ u ν − g µν ) (cid:21) (2.6)where S = S ( r ) is the magnitude of anisotropy stress tensor and C µ = (0 , − e − λ , , have discussed core-envelope model on pseudo-spheroidal space-time with core consisting of isotropic distribution of matter and envelope withanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar anisotropic distribution of matter. While anisotropic core-envelope models by as-suming linear equation of state in the core and quadratic equation of state in theenvelope have been studied by Gedela et. al. In the following sub-sections wederive important aspects of these two models.
The TRV core-envelope model
It has been shown that core and envelope regions consist of different physical fea-tures. They have chosen ansatz for a pseudo-spheroidal geometry of spacetime tosolved the Einstein’s equations. According to their metric, potential for pseudo-spheroidal geometry is expressed as e λ ( r ) = 1 + K r R r R (2.7)where K and R are geometric variables.The energy momentum tensor components (2.5) with anisotropic stress tensor π µν has non-vanishing components T = ρ, T = − (cid:18) p + 2 S √ (cid:19) , T = T = − (cid:18) p − p √ (cid:19) . (2.8)The magnitude of anistropic stress is give by S = p r − p ⊥ √ S ( r ) = 0 for 0 ≤ r ≤ R C and S ( r ) (cid:54) = 0 for R C ≤ r ≤ R E (2.10)where R C refers to the core boundary radius and R E corresponds to the envelopeboundary radius which is the same as the radius of the star ( a ) under consideration.Making use of these conditions with Eqn. (2.1), (2.4), (2.7) and (2.8), the Einsteinfield equations give the equations for density and pressureAccordingly, the density distribution ( core and envelope region ) is expressed as ρ = 18 πR (cid:20) r R (cid:21)(cid:20) r R (cid:21) − . (2.11)where R is a geometrical parameter. Equation (2.11) provides the density distribu-tion in core and envelope region by using boundary condition for 0 ≤ r ≤ R C forcore and R C ≤ r ≤ R E for envelope region.The radial and transverse pressure in the envelope region is given byanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State πp E = C (cid:113) r R (cid:0) r R (cid:1) + DR (cid:0) r R (cid:1) (cid:0) C (cid:113) r R + D (cid:1) , (2.12)8 πp E ⊥ = 8 πP E − r R (cid:0) − r R (cid:1) R (cid:0) r R (cid:1) . (2.13)and anisotropy S has expression8 π √ S = r R (cid:0) − r R (cid:1) R (cid:0) r R (cid:1) . (2.14)The constants C and D are given by C = − (cid:18) a R (cid:19) − , (2.15) D = 12 (cid:114) a R (cid:18) a R (cid:19)(cid:18) a R (cid:19) − . (2.16)The radial pressure in the core region is given by8 πp C = A (cid:113) r R + B (cid:20)(cid:113) r R L ( r ) + √ (cid:113) r R (cid:21) R (cid:0) r R (cid:1)(cid:20) A + (cid:113) r R + B (cid:18)(cid:113) r R L ( r ) − √ (cid:113) r R (cid:19)(cid:21) (2.17)where L ( r ) = ln (cid:18) √ (cid:114) r R + (cid:114) r R (cid:19) . where A and B are given by A = [5 √ − √ √ L ( R c ) − √ . C + √ [5 √ √ √ L ( R c ) − √ . D , (2.18) B = √ [3 √ C − D ] . (2.19)Equation (2.11) implies that the matter density at the center is explicitly relatedwith geometrical variable R asanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar R = (cid:115) λ πρ ( a ) , λ = ρ ( a ) ρ (0) = 1 + a R (1 + 2 a R ) (2.20)We have plotted the graph of pressure against density in the TRV model and dis-played by solid curve in Fig. 1 for density variation parameter λ = 0 .
01. The bestfit for the pressure-density curve is found to be in the quadratic from p = ρ + αρ + βρ (2.21)where ρ = − . × − , α = 406 and β = 1 .
69. It has been shown as a dottedcurve in Figure 1. It can be shown that the model reveals quadratic equation ofstate for different choices of the density variation parameter λ . Fig. 1. (Color online) The radial pressure and density are given by Thomas et al. (given inunits of km − ), is plotted with solid curve. The dashed curve corresponds to the fitted curve with α = 406, β = 1 .
69 and ρ = − . × − . For a density variation ( λ = 0 . anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State The SNJR core-envelope model
In the second case of core-envelope anisotropic model of Gedela et al., two distinctEOSs for core and envelope region are proposed. For the core region (0 ≤ r ≤ R c ),here a linear EOS as given below is used. p C = (0 . ρ − (7 . × − ) (2.22)The numerical values appeared in equation (2.22) are the same as given in. Theexpressions of density and pressure for core region are given by ρ C = c ( br − π ( br + 1) (2.23) p C = cα ( br − π ( br + 1) − β (2.24)where c , b , α and β are constants whose numerical values are − . − ,0 . − , 0 . − and 0 . × − km − , respectively. For enveloperegion ( R C ≤ r ≤ R E ), they have considered quadratic EOS in the form p E = κρ − γ (2.25)where κ and γ are constants whose numerical values are 108 km − and 1 . × − km − , respectively. Further, the density and pressure profile in the envelope region are given by ρ E = a ( br − π ( br − (2.26) p E = a κ ( br − π ( br + 1) − γ (2.27)anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar
Fig. 2. (Color online) Variation of a pressure P in (km − ) with respect to a density ρ in (km − ). Figure based on Eqn. (2.22) and (2.25) with κ =108 km − and γ =1 . × − km − . An important feature of both of these core-envelope models (TRV and SNJR) is thatthey have the stable equilibrium under hydrostatic configuration. Theoretical studyof the relativistic core-envelope model using paraboloidal spacetime by Ratanpaland Sharma have shown that paraboloidal geometry also admit quadratic equa-tion of state. Other EOSs that we have considered in the present work for com-parison include those considering different physical compositions of nuclear matterreported by. The different models used in this study are listed in Table 1.A very crucial feature of the equation of state is the causal limit( a sound signal can-not propagate faster than the speed of light, ν s = dp/dρ ≤ c ). In both cases basedon the geometrical models (TRV and SNJR) the causality condition is satisfied.In particular, Thomas et al. have studied the causality limit for different densityvariables. The computed speed of sound ( ν s ) versus radius as shown in Figure 3.Both the cases clearly indicate the validity of causality condition.anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State ν s , in unit of the speed of light, c, as a function of radiuscalculated for the TRV (Red solid line) equation of state (for λ = 0 .
01) and SNJR (Blue dashedline) equation of state.
3. Compact Star Structure : Static Equilibrium configurations
It is vital to explore static and spherical symmetrical gravity sources in general rel-ativity, especially when it comes to internal structure of compact objects. For sim-plicity , we consider only nonrotating, spherically symmetric stars. The geometryinside the star is described by the familiar Tolman–Oppenheimer–Volkoff (TOV)equation, which is valid for a perfect fluid. The equation of state is all that isrequired to solve the TOV equations. For static, spherically symmetric stars in hy-drostatic equilibrium, the TOV equations may be written as a pair of first-orderdifferential equations. The calculation of neutron star structure is obtained by nu-merically integrating the Tolman-Oppenheimer-Volkoff equation dPdr = − Gm ( r ) ρ ( r ) r (cid:0) P ( r ) ρ ( r ) c (cid:1)(cid:2) πr P ( r ) m ( r ) c (cid:3) − Gm ( r ) rc , (3.1) dm ( r ) dr = 4 πr ρ ( r ) . (3.2)anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar
Here P is the radial pressure, ρ is the mass density, r is the radial distance measuredfrom the center, and m ( r ) is the enclosed mass from the center r = 0 where P = P c and ρ = ρ c to a radial distance r . In the present work we have fixed the centraldensity for both geometrical models at ρ c = 1 . × g cm − . The seven nuclearEOSs with a fixed central density at ρ c = 1 . × g cm − , equations (3.1) and (3.2)are integrated numerically to determine the global structure (e.g. radius and mass)of a neutron star. To begin with, the density close to the center of the compact staris assumed to be homogeneous, with the density ρ = ρ c , the radius r = 0 . πρ c r /
3. Equations (3.1) and (3.2) are integrated numerically from r = 0 . P ( a ) = 0). The total mass of the star is then given by M = m ( a ).Using the data files provided by ¨Ozel et al., we have re-ploted pressure-densityprofile corresponds to all the model EOS’s listed in Table 1 along with geometricEOS’s of TRV and SNJR. It can be seen that these EOS’s distinctly differ fromeach other. Fig. 4. (Color online) Geometrical EOS TRV( wine short dash line) and SNJR (olive solid line)compared to the selected nuclear EOS’s ( ALF1 (black solid line), APR (red dash line), BKS19(green dot line), ENG (blue dash dot line), SLy (magenta dash dot dot line), WWF1 (navy shortdash line) and SQM1 (purple dot line). Details of these EOSs are listed in Table 1. anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd
Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State Mass-Radius Relation
In this work, we have considered two general relativity inspired equations of stateand compared with seven different nuclear equations of states as listed in Table 1.
Table 1. Nuclear and Geometrical equations of state used for the construction of models ofgeneral relativistic static neutron starsLabel EOS Composition and model Reference1 ALF1 nuclear plus quark matter (MIT Bag Model) Alford et al. (2005) npeµ , variational theory, Nijmegen NNplus Urbana NNN potential Akmal et al. (1998) µ effective nucleon energy functional Douchin andHaensel et al. (2001) The composition and model used for all these equation of state and their respectivebibliographic references are also listed in Table 1. Making use of these equations ofstate, we obtained the mass-radius relationship for a compact star.The mass-radius relations obtained with the help of nuclear equations of state ofdifferent compositions are compared with the geometrical equations of state andare plotted in Fig. 4. These plots reiterate the fact that nuclear and geometricalequations of state manifest three distinct types of compact stars. The first one cor-responds to the two cases represented by the models 7 and 8 of Table 1 , the secondone corresponds to the models (1 to 6) largely represented by the nuclear matterEOSs and third type corresponds to the model (9) represented by the geometricmodel (SNJR). In all the three cases the maximum masses correspond to stablestructure varies from 1.4 to 2.3 M (cid:12) , while the radius at their maximum masseslie 8 - 9 kms in the case of the first category, 9 - 12 kms in the cases of (secondcategory) and beyond 12 km in the case of the third category. The central densityanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar at maximum mass obtained here for the stable configurations are listed in Table 2.
Fig. 5. (Color online) Neutron star mass as a function of radii for pure nuclear matter EOSs vs.geometrical EOSs. The labels are explained in Table 1.
The M-R diagram obtained from the two geometrically deduced models behavedifferently. We found that TRV equation of state resulted into the mass-radiuscurve similar to the one obtained for strange quark matter stars (SQM1, label-7). The monotonically increasing mass with radius ( M ∝ a ) is expected forthe class of ultra compact objects which are self-bound. The surface density ofstrange star is roughly fourteen orders of magnitude larger than the surface densityof normal neutron stars. The TRV model gives a stable configuration in the sameorders of magnitude, with the surface density, ρ s ≈ × g cm − . Thus, it is anappropriate geometrical model for the study of ultra compact stars having exoticmatter composition.The isotropic fluid distribution in the core part of the TRV model is justified if thecore matter distribution is of quarks or strange matter, governed by MIT bag model.Further, the envelope with anisotropic fluid distribution can be viewed as due tohadronization to baryonic matter. Thus the TRV model prediction fit well with thatanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State of the strange quark matter stars with its maximum mass, ( M max = 1 . (cid:12) ) andradius, 8.76 km. The SNJR model that predicts the third category in which EOShas the linear behaviour inside the core and quadratic behaviour at the envelopehas resulted into the M-R diagram different from all other cases. Its M-R curve isbroader as compared to all other cases studied here. And its density is much lowerthan that of normal neutron like stars. Recent observations of binary neutron-starmergers (GW170817) have reported an estimation for the radius of the neutron starin the range 10.6 to 11.5 kilometers. , Keplerian frequency (rotation frequency of neutron star)
The Kepler frequency expresses the balance of centrifugal and gravitational forceon a particle on equatorial plane at the surface of a star. It is expressed asΩ c = (cid:114) Ma , (3.3)where the subscript c denotes classical symmetry of the centrifugal and gravitationalforces, which is the Newtonian expression for the Kepler angular velocity. Thisequation do to not hold in General Relativity, but as it turn out, it holds to verygood accuracy if the right side is multiplied by a prefactor( C ). It has been shownby J. M. Lattimer, et al., Haensel et al. and B. Haskell et al. that the numericalvalue of the Keplerian frequency, namely the maximum rotational frequency ofa neutron star accounting for the effects of general relativity, deformation , andindependent on the EOS, can be well fitted from the simple formulaΩ K ≈ C (cid:18) MM (cid:12) (cid:19) / (cid:18)
10 km a (cid:19) / Hz , (3.4)providing the neutron star mass is not very close to the maximum stable value, M and a are the mass and the radius of the nonrotating star respectively. The constant C of Eq.3.4 are given by B. Haskell et al. For the self bound compact stars it isgiven as 1 .
15 KHz and for other gravitationally bound neutron stars it is given as1 .
08 KHz.The deduction of Ω K generally requires the calculation of rotating general rela-tivistic configurations. Nevertheless , Haensel et al. (2009) have shown to a gooddegree of accuracy that the mass-shedding frequency Ω K,max can be determined bythe EOS-independent empirical formula as given in Eq.(3.4). On the other hand, itallows to determine Ω K using the mass and radius of the nonrotating star.The calculated Keplerian frequency based on the mass-radius relations obtained us-ing all the nine equations of state are shown in Fig. 6. Here we found that Keplerianfrequency corresponds to TRV and SQM1 are similar with higher values of Ω K (14-18 KHz). While other cases Ω K varies from 2 KHz to 18 KHz. The results ofanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar
Fig. 6. (Color online) Kepler frequency, Ω k , as a function of neutron star mass using the twodifferent classes of EOS ( nuclear and geometrical ) Keplerian frequency for the maximum mass of stable stars are shown in Table 2 forall the nine models.
Surface Gravity
The surface gravity of neutron stars denoted by g s (i.e., the acceleration due togravity as measured on the surface), is an important parameter for the study ofneutron star atmospheres. The upper bound of the surface gravity for neutronstars with various baryonic EOSs is studied by Bejger et al. (2004). The surfacegravity of neutron star is many orders of magnitude larger than that of other stars;it is ∼ times stronger than gravity at the Earth surface, and 10 times largerthan that for the white dwarfs.The expression for g s is given by : g s = GMa √ − x GR (3.5)anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State Here, x GR = 2 GM/ac = r g /a , where r g is the Schwarzschild radius. The impor-tance of relativistic effects for a neutron star mass M and radius a is characterizedby the compactness parameter r g /a . Usually for a neutron star with M = 1 . (cid:12) and Radius is about 10 km, surface gravity becomes ( g s ) = 2.43 × cm s − . Inconsequence it is suitable to measure g s in units of 10 cm s − and is representedas g s, ≡ g s / (10 cm s − ). The computed surface gravity, g s, for all the casesstudied here are shown in Fig. 8 against mass expressed in M (cid:12) . The numericallyvalues of g s, correspond to maximum stable mass of the star are also listed inTable 2. Fig. 7. (Color online) Plots of g s, versus gravitational mass M . Surface gravity in the units of10 cm s − . It is found that for M = 1 . (cid:12) , g s, ranges from 1.43 to 2.8 and for M ≈ . (cid:12) the surface gravity lies between 1.88 to 4.38 . The nuclear EOSs (labeled : 1 and 6)with an exotic quarks phase have relatively low g s,max . A similar situation occursfor the SNJR EOS that gives lowest value of surface gravity. The only reason SNJREOS have low surface gravity is that they have a greater radius compared to otherEOSs. The TRV EOS (labeled 8 ) yields g s,max similar to BKS19 and SLy EOSs.anuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar
Their values of g s, max ) range from 4.10 to 4.60. Gravitational Redshift of Neuron Star
In general relativity the ratio of the emitted wavelength λ e at the surface of anonrotating neutron star to the observed wavelength λ received at radial coordinate r , is given by λ e /λ = [ g tt ( a ) /g tt ( r )] / . From this the definition of gravitationalredshift, z ≡ ( λ − λe ) /λe from the surface of the neutron star as measured by adistant observer ( g tt ( r ) → −
1) is given by z = | − g tt ( a ) | − / − (cid:18) − GMac (cid:19) − / − g tt = − e λ ( r ) = − (1 − GM/c a ) is the metric components. For agiven EOSs the maximum value z maxsurf increase with increase of M max . Neutron starsof M ≥ M (cid:12) are expected to have sizable z surf . The computed values of z surf for allthe cases studied here are listed in Table 2. The computed values of z surf are foundto lie between 0 . . Fig. 8. (Color online) Gravitational redshift at the neutron star surface as a function of thestellar gravitational mass for the nine considered EOS models.
From the listed values of x GR in Table 2, we found that all the models studied heresatisfy the Buchdahl inequality, a ≥ (9 / r g = (9 / GM/c which is stricter thananuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd Distinct Classes of Compact Stars Based On Geometrically Deduced Equations of State the Schwarzschild bound. A consequence of this is that the gravitational redshiftshould satisfy z ≤
2. The precise upper bound on the surface redshift for neutronstar is z surf = 0 .
851 for subluminal EOSs. In the present study, we found that z surf for all the nine cases computed here lie much below the upper bound for thegravitational redshift.
4. Results and Discussion
We have computed several properties of a compact star like neutron star, usingnuclear and geometrically deduced equations of state. We have used geometricalequations of state from the core envelope model, that describes different propertiesof the physics in the core and envelope region. Many similarities and dissimilaritiesare observed from the properties computed based on the geometrical EOSs and thenuclear EOSs.
Table 2. Calculated properties of nonrotating neutron star modelsLabel M max ( M (cid:12) ) a max (km) ρ c (10 gm cm − ) Ω k (10 s − ) z surf g s, (cm s − ) x GR In Table 2 we have listed computed properties of such a compact star with allthe different types of EOSs. Like, maximum mass ( M max ), stellar radius ( a max )correspond to the maximum mass , central density ( ρ c ), Keplerian frequency (Ω k )correspond to the maximum mass for the stable structure of the star, gravitationalredshift ( z surf ), surface gravity ( g s ) and compactness parameter ( x GR ). We com-pared all these properties with the properties obtained from geometrically deducedequations states. The properties obtained from TRV equation of state are in goodaccordance with the properties obtained from other nuclear matter based models.While the parameters obtained using the SNJR model are quite different from oth-ers except for z surf and ( x GR ). The central density that yields the maximum massof ≈ (cid:12) in the case of SNJR is very low and the radius is about 12 km. It isalso reflected in the low values of the surface gravity. It is noticed that mass-radiusconfiguration as shown in Fig. 5 obtained from geometrical models will be pertinentfor divergent class of compact stars. Particularly, the pseudo-spheroidal spacetimeof TRV model seemed to describe the ultra dense compact stars like the strangeanuary 5, 2021 2:2 WSPC/INSTRUCTION FILE ws-ijmpd A. C. Khunt, V. O. Thomas and P. C. Vinodkumar self-bound stars. The spacetime geometry adopted for the SNJR model representslow density neutron like star, where radius lie between 12 ≤ R ≤
20 kms.The mass-radius diagram in Fig. 5 clearly classify the nature of compact stars inthree categories :(i) highly compact self-bound stars represented by the TRV Modeland SQM1 model with exotic matter compositions (ii) the normal neutron starswith nuclear matter EOS and (iii) the ultra soft compact stars represented by theSNJR geometrical model. At the end, we are able to identify a correspondencebetween the geometric description with the structure of the matter distribution incompact objects like a strange star. To summarize, we have been able to classifyneutron like compact stars in three distinct types each one having different internalstructures. We hope that TRV model for compact neutron like stars will be usefulfor the study of superdense self-bound stars having exotic matter compositions.
5. Acknowledgements
We would like to thank Feryal ¨Ozel for kindly providing us with EOS table for thenuclear matter. Further , we thank Dr. B. S. Ratanpal for many helpful discussions.
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