Low mass strange stars and the compact star 1E1207.4-5209 in the Field Correlator Method
aa r X i v : . [ nu c l - t h ] A ug Low mass strange stars and thecompact star 1E1207.4-5209
F. I. M. Pereira ∗ Observat´orio Nacional, MCTI, Rua Gal. Jos´e Cristino 77, 20921-400 Rio de Janeiro RJ, Brazil (Dated: September 1, 2020)We study the anomalous mass defects in the first (ascendant) branch of stellarsequences of static strange stars. We employ the nonperturbative equation of statederived in the framework of the Field Correlator Method to describe the hydrostaticequilibrium of the strange matter. The large distance static q ¯ q potential V andthe gluon condensate G are the model parameters that characterize the equation ofstate.An attempt is made to determine, from the surface gravitational redshift mea-surements, the ratio ( P / E ) C at the center of strange stars. For V = 0 and G > ∼ .
035 GeV , it is shown that ( P / E ) C ≃ .
262 and the corresponding redshift z S ≃ .
47 are limiting values, at the maximum mass of the stellar sequence . Asa direct application of our study, we try to determine the values of V and G from astrophysical observations of the compact star 1E1207.4-5209. After two at-tempts to obtain the model parameters, our findings show that V = 0 . ± .
10 GeVand G = 0 . ± .
001 GeV in the first attempt and V = 0 . ± .
085 GeV and G = 0 . ± . in the second attempt. The corresponding ratios ( P / E ) C are 0 . +0 . . and 0 . ± .
028 , respectively. As a consequence of these high valuesof the model parameters, the respective anomalous mass defect of 1E1207.4-5209 are | ∆ M | ≃ . × erg and | ∆ M | ≃ . × erg . ∗ fl[email protected], fi[email protected] I. INTRODUCTION
The possibility of anomalous mass defects in compact stars goes back to the works ofV. A. Ambartsumyan, G. S. Saakyan and Yu. L. Vartanyam in Refs. [1–5]. Anomalousmass defects would occurs at internal stellar densities many times greater than nuclear( ε ≃ .
14 GeV fm − ). Such stellar configurations, in the presence of external perturbations,would undergo transitions of explosive character, from a metastable state to a stable state,with great amounts of liberated energy. The authors considered the superdense stellarmatter made of a degenerate gas comprising neutron, protons, hyperons and electrons,at zero temperature. The baryons being made of quarks, it would be natural to expectunbound quarks to exist in the interior of hyperdense stars. The possibility of hypotheticalcompact stars made of pure quark matter was then considered by N. Itoh in Ref. [6]. Sincethe Bodmer-Terazawa-Witten conjecture in Refs. [7–9] the existence of the strange quarkmatter (SQM), made of an equal number of up, down and strange quarks, has been subjectof a lot of theoretical studies, experimental investigations in terrestrial laboratories, andobservational studies of astrophysical phenomena.The properties of SQM in the phase diagram, at small temperatures and large densities,were not completely known due to the nonperturbative character of quantum chromodynam-ics (QCD). Withing this scenario, the MIT Bag Model became one of the most successfulphenomenological models for quark confinement to treat the cold matter at finite chemi-cal potentials and describe the properties of strange stars (see Refs. [10–12]). However, themodel has the disadvantage in that the quarks are free particles inside the bag that simulatesthe confinement; but for larger distances, when the confining forces becomes important, itdoes not naturally include the confinement from first principles. On the other hand, addi-tional complications appear when magnetic field is taken into account. In a recent work,the SQM in strong magnetic field was considered by the authors of Ref. [13] by using theRichardson potential [14] in which the asymptotic freedom and confinement are built in.Despite the highly nonlinear character of QCD, the inherent difficulties were overcomeby the more recent appearance of the nonperturbative equation of state derived, from firstprinciples, in the framework of the Field Correlator Method (FCM) in Ref. [15]. The greatadvantage of the model is that it covers the entire phase diagram plane from high temper-atures and low densities to low temperatures and high densities. We studied the generalaspects of (normal and anomalous) mass defects of strange stars within the FCM approachin Ref. [16] . We carried out the calculation of the mass defects specially at the maximummasses of the stellar sequences.In the present article, we study the anomalous mass defects of nonrotating strange stars,without magnetic field and crust (to be considered, in the FCM approach, in future workswhich are now in progress), within the same lines of our previous investigation. We start thework by studying the anomalous mass defects in the first (ascendant) branches of the stellarsequences, which also includes the region of low mass stars. Then, we use the solutions ofthe Tolmann-Oppenheimer-Volkov (TOV) equations for the case of constant energy density(see Ref.: [17]) to guide our investigation. In this case, the ratio pressure-to-energy density, P C / E C , at the center of a star can be expressed in terms of the surface or gravitationalredshift (henceforth called redshift) of a radiation emitted at a given frequency from the starsurface. We extend this simple idea to find an analogous description for the general case ofstars with non-constant energy density profiles. The new description is used to determinatethe QCD parameters V and G from the astrophysical observation of the compact star1E1207.4-5204.The present paper is organized as follows. In Sec. II we introduce the FCM main equationsto be used in our calculation. In Sec. III we present the equations needed to describe thestellar configurations: mass defects, hydrostatic equilibrium equations, and the constantenergy density solution of TOV equations. Sec. IV is devoted to the non-constant energydensity profile inside strange stars. In Sec. V, we made an attempt to estimate the modelparameters V and G from the observations of the compact star 1E1207.4-5209 . Sec. VI isdedicated to the final remarks. II. THE NONPERTURBATIVE EQUATION OF STATE AT ZEROTEMPERATURE
Let us summarize the main equations concerning the FCM thermodynamics of quarks(see Refs. [16, 18, 19] for a more detailed description). The main parameters of the non-perturbative equation of state are the large distance static q ¯ q potential V and the gluoncondensate G . For constant V , the pressure, energy density and number density of a (oneflavor) quark gas at T = 0 are given by p SLAq = N c π ( k q q k q + m q − m q (cid:20) k q q k q + m q − m q ln (cid:18) k q + p k q + m q m q (cid:19)(cid:21)) , (1) ε SLAq = N c π ( k q q k q + m q + m q (cid:20) k q q k q + m q − m q ln (cid:18) k q + p k q + m q m q (cid:19)(cid:21) + V k q ) (2)and n SLAq = N c π k q , (3)where k q = q ( µ q − V / − m q , ( q = u , d , s) , (4) N c = 3 is the color number, and m u = 5 MeV, m d = 7 MeV and m s = 150 MeV are thequark masses. The additional term ( V / k q / q ¯ q potential V .The total pressure and energy density, including electrons are given by p = X q = u,d,s p SLAq − ∆ | ε vac | + p e , (5) ε = X q = u,d,s ε SLAq + ∆ | ε vac | + ε e , (6)where ∆ | ε vac | = 11 − N f
32 ∆ G , (7)is the vacuum energy density difference between confined and deconfined phases (which fromnow on will be called vacuum energy density), N f is the number of flavors, and ∆ G ≃ G isthe difference between the values of the gluon condensate G at T < T c and T > T c ( T c is thecritical temperature) prescribed by lattice calculations in the Refs. [20, 21]. The numericalequivalence for the MIT Bag Model is made by taking ∆ | ε vac | = B and V = 0. However, wemust have in mind that ∆ | ε vac | is a nonperturbative quantity. The corresponding equationsfor the degenerate electron gas are obtained from Eqs. (1)-(3) by making the changes: N c → V → µ q → µ e and m q → m e . III. STELLAR CONFIGURATIONS
Stellar configurations are calculated by numerical integration of the hydrostatic equilib-rium equations of Tolman-Oppenheimer-Volkov [17, 22] (we come back to these equationsin Sec. III B). Of special importance is the total gravitational mass of a compact star (see[23] for details), M = Z R ε ( r ) dv ( r ) , (8)where dv ( r ) = 4 πr dr , which is the mass that governs the Keplerian orbital motion of thedistant gravitating bodies around it, as measured by external observers. The baryonic mass(also called rest mass) of a star is M A = m A N A , where m A is the mass per baryon of thebaryonic specie A , with the number of baryons given by N A = Z R n A ( r ) e λ ( r ) dv ( r ) , (9)where e λ ( r ) = [1 − GM ( r ) / ( c r )] − / is the spatial function of the metric; M ( r ) is the masswithin a sphere of radius r , and n A = 13 X q = u,d,s n SLAq = 13 ( n SLAu + n SLAd + n SLAs ) (10)is the equivalent baryon number density. The baryonic mass is the mass that the star wouldhave if its baryon content were dispersed at infinity, with zero kinetic energy. In the case ofstrange stars (because of the quark confinement), N A is the equivalent number of baryons(not quarks). We here assume m A = m n as in our previous works. A. Mass defects
With the masses M and M A known, we are in a position to calculate the mass defect(which in our notation is minus the binding energy E b defined in Refs. [22, 23]) is given by∆ M = M A − M . (11)It corresponds to the energy released to aggregate from infinity the dispersed baryonicmatter. A stellar configuration is stable if ∆ M > We here follow the notation according to Refs. [1, 2, 4, 24, 25] if ∆ M < dM ( r ) = ε dv ( r ) (12) dN A ( r ) = n A ( r ) e λ ( r ) dv ( r ) (13) dM A ( r ) = m A dN A ( r ) . (14)Then we find that dM ( r ) dN A ( r ) = ε ( r ) n A ( r ) e − λ ( r ) = m ( r ) r − Gc M ( r ) r . (15)We call m ( r ) ≡ ε ( r ) /n A ( r ) the mass-energy per baryon inside the star. On the surface ofthe star (by taking into account that e − λ R = e φ R at r = R ) Eq. (15) becomes dM ( r ) dN A ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = R = m R e φ R = m R r − Gc MR , (16)where φ is the temporal function of the metric, m R ≡ m ( R ) and M ≡ M ( R ) given byEq. (8).In Fig. 1, the panel (a) shows the general features of the stellar sequence for the givenvalues of the parameters V and G . The label 3 (in the first branch) indicates an intermedi-ate point around 0 . M ⊙ in the low mass region, where the mass defect is anomalous (morevisible in panels (b) and (c)). In panel (b), the slope of the M vs. N A curve at the point given by Eq. (16) and that of the M A vs. N A curve are parallel, which means that m R = m A e − φ R = m A (1 + z S ) , (17)which is a redshift relation connecting the mass per baryon at the surface of the star withthe baryonic mass m A taken as a reference mass by a distant observer. Once m A is given,we obtain m R by measuring z S . On the other hand, inside the star, the energy density ε ( r ), the baryonic number density n A ( r ), and the baryonic mass per baryon m ( r ) decreasefrom the center to the surface of the star. Thus, the inequalities m C > m R > m A (where m C ≡ m (0)) hold in the star interior. In other words, there is anomalous mass defect when m ( r ) > m A , ∀ r ∈ [0 , R ]. If m R → m A then the point goes to the origin of the M vs. N A plot in panel(b) indicating the absence of the anomalous mass defect when m R = m A , inthe first branch of the stellar sequences, as shown in Refs. [1–4, 23]. Moreover, for a givenEoS characterized by the model parameters V and G , m R is constant for all stars alongthe corresponding stellar sequence. The slope m R e φ R in Eq. (16) evolves along the stellarsequence being greater than m A at the origin of the sequence and less than m A everywhereabove the point . The mass defect | ∆ M | is maximum at the point , as we see by simpleinspection in panel (c).We have calculated for different values of V and G the maximum values of | ∆ M | at the point by searching for points where Eq. (17) holds. The results are depicted inFig. 2. Each plot starts from the origin (∆ M = 0) and ends at the maximum mass valueof ∆ M calculated in Ref. [16] where the point is located also satisfying Eq. (17) . In thelow mass region connected by the dotted line we have | ∆ M | around 1 × erg. For thesake of comparison, the energies liberated in type Ia Supernovae originated by white dwarfexplosions are of the order of (1 − × erg ; and in the supernova 1987A the total energyof the observed neutrinos was found to be around ∼ × erg . In this respect, explosionsof strange stars with anomalous mass defects would be a possibility to be considered. B. The Tolmann-Oppenheimer-Volkov equations
To simplify our notation let us define the dimensionless radius and mass by
X ≡ c GM ⊙ r and Z ≡ M ( r ) M ⊙ , (18)where M ( r ) is the mass within the sphere of radius r ; and GM ⊙ / c ≃ . d Z d X = η X E , (19) d P d X = − ( E + P ) ( Z + η X P ) X ( 1 − Z / X ) , (20)where η ≡ π ( GM ⊙ /c ) /M ⊙ c ≃ . GeV − . The redshift of the spectral linesemitted from the star surface is given by z S = 1 p − Z R / X R , (21)where X R ≡ X ( r = R ) and Z R ≡ Z ( X R ).As the redshift has an important role in the present work, let us explore some propertieswhich will be important in the sequel. For finite values of ε C and p C at the center of thestar we find d P ( X ) d X (cid:12)(cid:12)(cid:12)(cid:12) X =0 = 0 . (22)At the star surface where energy density is E R and P R = 0 we have1 E R d P ( X ) d X (cid:12)(cid:12)(cid:12)(cid:12) X = X R = − Z R / X R X R [ 1 − Z R / X R ]= − z S + 2 z S X R = − ( z S + 2 z S ) Z R (1 + z S ) . (23)These expressions are of general validity whatever the energy density may be constant ornot. The right hand sides of Eqs. (23) are also observables quantities directly given by themeasurements of the redshift z S and the dimensionless radius X R or mass Z R . C. Constant energy density
In the case
E ≡ E C = cte. (see Ref. [17]) we find that Z = 13 η E C X , (24) P = E C p − Z / X − p − Z R / X R p − Z R / X R − p − Z / X = E C q − η E C Z X − q − η E C Z R X R q − η E C Z R X R − q − η E C Z X . (25) Where Z ( X ) / X →
X →
The redshift is now given by z S = 1 q − η E C Z R X R . (26)By taking X = 0 in Eqs. (25), the ratio P C / E C at the center of the star is given by P C E C = z S − z S . (27)The EoS at the center of a compact star with constant energy density can be obtained bydirect measurement of the redshift of the spectral lines emitted from the star surface. Bythis way, we can use the redshift as a probe to give us the EoS at the center of a compact starin the E ≡ E C =cte. approximation. For finite P C / E C ≥
0, we note that z S <
2, accordingto Eq. (11.6.20) in Ref. [26]. Additionally, we note that the above solution for P C / E C doesnot depends on the (nuclear or strange) matter EoS. This is an interesting property to beused to test theoretical EoS models.Low mass compact stars have been commonly accepted as those stars with masses lowerthan the solar mass. They are characterized by the fact that their internal energy densityprofiles are almost constant. The mass is calculated by the Newtonian approximation M ≃ (4 π/ ε S R , where ε S is the surface energy density, in Ref. [27]. However, not all lowmass stars can be approximated by a constant internal energy density profile. For instance,for certain values of the FCM parameters V and G , the shape of the energy density maypresent a remarkable change from the center to the surface of the star. In this case, althoughwe does not have an analogous prescription to the one given in Eq. (27), it is possible toexplore the behavior of the theoretical dependence of ( P / E ) C (in the framework of the FCM)as function of the redshift to find the corresponding expression for the case of non-constantenergy density profile. IV. ( P / E ) C FOR NON-CONSTANT ENERGY DENSITY
Let us now try to build a representation to simulate a general case when the energy densityis not constant. To this end, we first generate many (theoretical) stellar sequences each onecorresponding to a different pair of parameters V and G . The plots of the ratios ( P / E ) C vs. z S , which results from the calculation, are shown in panel (a) of Fig. 3. We observe that allcurves that start very close together from the origin, in a thin bundle of lines, deviate from0the initial direction at certain points along the bundle resembling a ”cockatiel crest” at theupper parts of the figure. Moreover, the deviation points, each one corresponding to a pair( V , G ), are located at the maximum masses of the corresponding stellar sequences. Then,the lines of the ”crest” that lie in the second branches (such as the one with the point , inFig. 1) of the stellar sequences are of no interest in the present work. So, by removing thelines of the ”crest” at the maximum masses, we obtain a thin cloud of aligned points whichconverge in the low redshift region (say, z S < ∼ .
1) to the constant energy density solutiongiven by Eq. (27) , as shown in panel (b).The next step is to represent the cloud of points by the interpolating curve given by (cid:18) PE (cid:19) C = y (1 − y + 2 y ) (28)where y ≡ P C / E C is the constant energy density solution given by Eq. (27). The secondand third coefficients on the right hand side of Eq. (28) were initially determined throughbest-fitting methods and then rounded in order to give an error estimate < ∼
4% (the bestwe obtained after many attempts!) at an intermediate redshift range and zero errors atzero redshift and at the maximum mass redshift, as shown in panels (c) and (d). In panel(b), the solid line shows the interpolating curve extrapolated to higher redshifts to becomevisible . Of course, this is a model dependent procedure valid for the case of the FCMnonperturbative EoS we are considering, but with an interesting quasi-model independentfeature.Panels (c) and (d) show the fractional error of ( P / E ) C as function of the redshift and thecorresponding error for z S in terms of ( P / E ) C (obtained by inverting Eq. (28)) , respectively,in a scale from zero to 100. So, by this way, we are able to use a redshift measurement as aprobe to estimate ( P / E ) C at the center of a strange star governed by the nonperturbativeFCM equation of state within the errors shown in panels (c) and (d). A direct applicationof Eq. (28) is done in Sec. V to investigate the compact source 1E1207.4-5209.Coming back to panel (a), the curves at the upper parts of the figure become more andmore closer, but never exceeding the limiting redshift z S ≃ .
51 , for stars in the secondbranches of the corresponding stellar sequences. On the other hand, the values of ( P / E ) C at the maximum masses along the bundle of curves does not exceed a certain limit whateverthe values of G may be, suggesting the existence of an upper limit for ( P / E ) C and acorresponding limit for G according to the values of V . In order to obtain the limits1for G , we considered the solutions for the cases with V = 0 (which gives the highest”crests”), and V = 0 . V = 0 the value of ( P / E ) C, c becomes constant around 0 .
262 at z S ≃ .
47 (point c in panel(b)) for G > ∼ .
035 GeV . At this point we have the ratio Z R / X R ≡ R S / (2 R ) ≃ . R S = 2 GM/c is the Schwarzchild radius of the star. The mass of the star is M ≃ . M ⊙ and its radius is R ≃ .
19 km ≃ . R S . For V = 0 . P / E ) C, d becomes constant around 0 .
24 at z S ≃ .
44 (point d in panel (b)), but for a toogreat value of G , say, > ∼ . . The corresponding mass and radius are M ≃ . M ⊙ and R ≃ .
34 km ≃ . R S .We now end this section by considering a situation given by (cid:18) PE (cid:19) C = Z R X R = 12 (cid:20) − z S ) (cid:21) . (29)Panel (b) shows the plot of the right hand side of Eq. (29) together with the plots of P C / E C and ( P / E ) C . The point a is located by solving the equation y − (cid:20) − z S ) (cid:21) = 0 , (30)from which we obtain the root z S ≃ .
39 and the corresponding ratio (cid:18) PE (cid:19) C, a ≃ . . (31)Although the point a is on the constant energy density solution of TOV equations, the valueof ( P / E ) C, a can also be obtained for certain values of V and G . As a result, the point a is on the second branches of the stellar sequences for values of the pair ( V , G ) between( V ≃ .
23 GeV , G = 0 .
001 GeV ) and ( V = 0 . , G ≃ .
012 GeV ).Analogously, by solving the equation y (1 − y + 2 y ) − (cid:20) − z S ) (cid:21) = 0 , (32)we find the root z S ≃ .
49 and the ratio (cid:18) PE (cid:19) C, b ≃ .
275 (33)corresponding to the point b . This point is located on the second branches of the stellarsequences for V = 0 and ∀ G > ∼ .
035 GeV . The differences between the values of ( P / E ) C z S at the points b and c are about (4-5) % ; and the differences for the masses, radii and Z R / X R are < ∼ (1-3) % . Then, for observations with error bars > ∼ (4-5) % we can assume that b ≃ c ( c is at the maximum mass) to estimate, from z S measurements, whether an observedstar is near its maximum value ( P / E ) C, c . V. THE COMPACT STAR 1E1207.4-5209
The possible SQM composition of the compact star 1E1207.4-5209 was considered inRef. [28]. In a mass-radius relation investigation in Ref. [29], the mass, radius and redshift of1E1207.4-5209 were determined (by independent methods) as being M = 0 . ± . M ⊙ , R = 4 . ± . z S = 0 . − . V and G from the given astrophysical data.Because the mass, radius and redshift are related by Eq. (21) we have two independentquantities to determine the parameters. As the redshift measurement is given only within arange, we try to obtain the parameters by fallowing two steps.First, we take as the central value of the redshift the one calculated from the values ofthe above mass and radius plus their respective error bars, by using Eq. (21). Then, we hereassume z S = 0 . +0 . − . to be used in our calculation. However, the measurements of M and R , which appear in the ratio M/R , are not suficient to discriminate the values of V and G .Two compact stars with different masses and radii, but with the same ratio M/R , have thesame redshifts. For instance, this would be the case of two stars with the same redshift, oneon the ( P / E ) C curve and the other on the upper ”crest”, as we can observe in panels (c) and(d) of Fig. 3 . To avoid this ambiguity, we have attempted to explore the right hand side ofEq. (23), by saying that even when the redshifts of two different stars are the same, their radiiand/or masses are not. Unfortunately, the values of [( dp/dr ) /ε ] r = R are practically constant(around − . − ), with a variation of ∼ . V ∈ [0 , .
5] GeV and G ∈ [0 . , . , as wesee in the V vs. G plot depicted by the solid line in panel (b) of Fig. 5. In order to narrow thesearch to get better results, we use the additional observable ( P / E ) C = 0 . +0 . − . (indicatedby the cross A in panel (a)) obtained from Eq. (28) to determine the values of the parameters3 V and G . The error bars were determined by the use of standard methods of data analysisin Refs. [30, 31]. As a result (the cross A in panel (b)) we obtained V = 0 . ± .
11 GeV and G = 0 . ± .
001 GeV for the compact star 1E1207.4-5209 . Then, according to thesepredictions, the compact star 1E1207.4-5209 is characterized by the central pressure P C ≃ .
12 GeV fm − and energy density E C ≃ . − ≃ ε (where ε ≃ .
141 GeV fm − is the nuclear energy density); the mass per baryon m C ≃ .
78 GeV ≃ . m A at r = 0 and m R ≃ .
75 GeV ≃ . m A at r = R . As a consequence of the high values of V and G , thepredicted anomalous mass defect is | ∆ M | ≃ . × erg .Second, let us consider the the redshift range 0.12 - 0.23 from which we take the redshift z S = 0 . ± .
055 at the central point and the respective additional observable ( P / E ) C =0 . ± .
028 (indicated by the cross B in panel (a)) obtained from Eq. (28). The dashed linein panel (b) has the same meaning as the solid line in first step. By an analogous procedureto narrow our search we obtained V = 0 . ± .
085 GeV and G = 0 . ± . .In this case, the compact star 1E1207.4-5209 becomes characterized by the central pressure P C ≃ .
17 GeV fm − , energy density E C ≃ .
95 GeV fm − ≃ ε ; the mass per baryon m C ≃ .
81 GeV ≃ . m A at r = 0 and m R ≃ .
77 GeV ≃ . m A at r = R . Thecorresponding anomalous mass defect is | ∆ M | ≃ . × erg .In both steps considered above, it is a remarkable feature of the FCM that the determi-nation of the q ¯ q interaction potential from astrophysical observations is in good agreementwith V = 0 . G are ∼
20% above G = 0 . (predicted in Ref. [15]) in the first step and above ∼ | ∆ M | are not in contradictionwith the results shown in Fig. 2 . The star 1E1207.4-5209 is not at the maximum mass norat the point of the stellar sequence corresponding to the above values of V and G . VI. FINAL REMARKS
In the present work, we addressed the question of anomalous mass defects of low massstrange stars in the framework of the Field Correlator Method (FCM). Redshift measure-ments played an important role in the determination of the model parameters from as-trophysical observations. In the case of the constant energy density solution of Tolmann-Oppenheimer-Volkov equations, the ratio ( P / E ) C at the center of a compact star was an4important observable quantity determined directly by redshift measurements. It tells us howthe equation of state at r = 0 is.In the general case, when the energy density is not constant, we verified that the plots of( P / E ) C vs. redshift, for different values of the model parameters V e G , are concentrated ina thin region with a quasi linear behavior, ranging from the origin to the maximum masses inthe first branches of the stellar configurations. This fact enabled us to build a representationfor the ratio ( P / E ) C in terms of the redshift similar to the one for the case of constant energydensity. A remarkable feature of the approach is that the ratio ( P / E ) C as function of theredshift present the lowest values with respect to other models of nuclear matter. Forinstance, as illustrated in Fig. 6 for the Walecka nuclear mean field theory in Refs. [33, 34],our preliminary calculations have shown that the values of ( P / E ) C ∀ z S ∈ [0 , . g σ /m σ and g ω /m ω of the nuclear mean filed theory the curves of ( P / E ) C (in the redshift range we are considering) are not concentrated in a line of points as they arefor the MFC. An interesting task to be considered in future works is the investigation of thebehavior of the ratio ( P / E ) C provided by other models of nuclear matter in the frameworkof mean field theories. Our attempt to determine ratio ( P / E ) C in terms of the redshift ismodel dependent, but with an almost model independent feature. It is important to verifyin future works whether this feature remains valid for other approaches used to describeSQM. ACKNOWLEDGMENTS
This work was done with the support provided by the Minist´erio da Ciˆencia , Tecnologiae Inova¸c˜ao (MCTI). [1] V. A. Ambartsumyan, G. S. Saakyan, Sov. Astron -AJ 4 (1960) 187.[2] V. A. Ambartsumyan, G. S. Saakyan, Sov. Astron -AJ 5 (1962) 601.[3] V. A. Ambartsumyan, G. S. Saakyan, Sov. Astron -AJ 5 (1962) 779.[4] G. S. Saakyan, Yu. L. Vartanyan, Sov. Astron.-AJ 8 (1964) 147.[5] V. A. Ambartsumyan, Problemes de la Cosmogonie Contemporaine, Editions MIR, Moscou, Book Company, New York, 1969.[31] P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences,McGraw-Hill Higher Education, New York, 2003.[32] O.Kaczmarek, F. Zantov, arXiv:hep-lat/0506019.[33] J. D. Walecka, Ann. Phys. (1974) 491.[34] B. D. Serot, J.D. Walecka, Advances in Nuclear Physics , Plenum Press, 1986. FIGURE CAPTIONFig. 1 -
For the given values of G (in GeV units) and V (in GeV units): panel (a),gravitational mass M and baryonic mass M A as function of the central density; panel (b),gravitational mass M and baryonic mass M A as function of the baryonic number N A ; panel(c), mass defect as function of the gravitational mass M . Labels and indicate the pointsat which M = M A and where the solid curve and dashed curve cross itself, in panel (b). Fig. 2 -
The mass defect ∆ M as function of G for different values of V between V = 0and V = 0 . V (in GeV units). Solid lines :∆ M at the point obtained by Eq. (17). Short dashed lines : ∆ M at the maximum massesof the stellar configurations (cf. Fig. 4 in Ref. [16]). Dotted lines : ∆ M at the point of thelow stellar masses ranging from M/M ⊙ ≃ . V = 0 to M/M ⊙ ≃ . V = 0 . Long dashed lines : only connecting points of ∆ M when the point is at the maximummass. Fig. 3 -
Panel (a): for the given ranges of G (in GeV units) and V (in GeV units),the ratio ( P / E ) C at r = 0 as function of the redshift. The values of V increase from topto bottom. The values of G increase from bottom to top. Panel (b): as in panel (a), butfor values of ( P / E ) C ending at the maximum masses of the stellar configurations. Shortdashed line : P C / E C corresponding to the energy solution of the TOV equations given byEq. (27) . Solid line : the interpolating curve given by Eq. (28) . Panel (c): the fractional error δ ( P / E ) C / ( P / E ) C as function of the redshift. Panel (d): as in panel (c), but for δz S /z S . Fig. 4 -
As in panel (b) of figure 3, but including the Z R / X R plot (long dashed line).The label a corresponds to Eq. (31) and the label b corresponds to Eq. (33). The label c indicates the upper bound ( P / E ) C, c ≃ .
262 and the corresponding redshift z S ≃ .
47 .Panel (b): for increasing values of G , the ”constancies” of ( P / E ) C at b , c and d (not visiblein the scale of the figure). Fig. 5 -
Panel (a): as in panel (b) of Fig. 3, but without the cloud of points. Thecross A indicates the value of ( P / E ) C = 0 . +0 . − . calculated by Eq. (28) corresponding tothe redshift z S = 0 . +0 . − . . The cross B indicates the value of ( P / E ) C = 0 . ± . z S = 0 . ± .
055 . Panel (b): the solidand dashed lines show the first attempts to determine the model parameters V and G fromthe mass, radius and redshift measurements provided by the observations of the compactstar 1E1207.4-5209. The crosses A and B indicate the final results of our narrowed searches8for V and G . For comparison, the dashed line at V = 0 . Fig. 6 -
The ratios pressure to energy density at r = 0: (a) Solid lines , P C / E C and( P / E ) C given by Eqs. (27) and (28) ; (b) Short dashed line , for the nuclear mean field theory,with the coupling constants ( g σ /m σ ) = 11 .
798 fm and ( g ω /m ω ) = 8 .
653 fm fixed to givethe bind energy E b = − .
75 MeV and k F = 1 .
42 fm − ; (c) Long dashed line , as in (b)but for the arbitrarily chosen values of ( g σ /m σ ) = 15 . and ( g ω /m ω ) = 12 . , asfunctions of the surface redshift.9 ,FIG. 1. For the given values of G (in GeV units) and V (in GeV units): panel (a), gravitationalmass M and baryonic mass M A as function of the central density; panel (b), gravitational mass M and baryonic mass M A as function of the baryonic number N A ; panel (c), mass defect as functionof the gravitational mass M . Labels and indicate the points at which M = M A and where thesolid curve and dashed curve cross itself, in panel (b). FIG. 2. The mass defect ∆ M as function of G for different values of V between V = 0 and V = 0 . V (in GeV units). Solid lines : ∆ M atthe point obtained by Eq. (17). Short dashed lines : ∆ M at the maximum masses of the stellarconfigurations (cf. Fig. 4 in Ref. [16]). Dotted lines : ∆ M at the point of the low stellar massesranging from M/M ⊙ ≃ . V = 0 to M/M ⊙ ≃ . V = 0 . Long dashed lines : onlyconnecting points of ∆ M when the point is at the maximum mass. ,,FIG. 3. Panel (a): for the given ranges of G (in GeV units) and V (in GeV units), the ratio( P / E ) C at r = 0 as function of the redshift. The values of V increase from top to bottom. Thevalues of G increase from bottom to top. Panel (b): as in panel (a), but for values of ( P / E ) C endingat the maximum masses of the stellar configurations. Short dashed line : P C / E C corresponding tothe energy solution of the TOV equations given by Eq. (27) . Solid line : the interpolating curvegiven by Eq. (28) . Panel (c): the fractional error δ ( P / E ) C / ( P / E ) C as function of the redshift.Panel (d): as in panel (c), but for δz S /z S . ,FIG. 4. As in panel (b) of figure 3, but including the Z R / X R plot (long dashed line). The label a corresponds to Eq. (31) and the label b corresponds to Eq. (33). The label c indicates the upperbound ( P / E ) C, c ≃ .
262 and the corresponding redshift z S ≃ .
47 . Panel (b): for increasing valuesof G , the ”constancies” of ( P / E ) C at b , c and d (not visible in the scale of the figure). ,FIG. 5. Panel (a): as in panel (b) of Fig. 3, but without the cloud of points. The cross A indicates the value of ( P / E ) C = 0 . +0 . − . calculated by Eq. (28) corresponding to the redshift z S = 0 . +0 . − . . The cross B indicates the value of ( P / E ) C = 0 . ± .
028 calculated by Eq. (28)corresponding to the redshift z S = 0 . ± .
055 . Panel (b): the solid and dashed lines show thefirst attempts to determine the model parameters V and G from the mass, radius and redshiftmeasurements provided by the observations of the compact star 1E1207.4-5209. The crosses A and B indicate the final results of our narrowed searches for V and G . For comparison, the dashedline at V = 0 . FIG. 6. The ratios pressure to energy density at r = 0: (a) Solid lines , P C / E C and ( P / E ) C given by Eqs. (27) and (28) ; (b) Short dashed line , for the nuclear mean field theory, with thecoupling constants ( g σ /m σ ) = 11 .
798 fm and ( g ω /m ω ) = 8 .
653 fm fixed to give the bind energy E b = − .
75 MeV and k F = 1 .
42 fm − ; (c) Long dashed line , as in (b) but for the arbitrarilychosen values: ( g σ /m σ ) = 15 . and ( g ω /m ω ) = 12 .2