Anistropic Stars: Exact Solutions and Stability Krsna Dev∗ Department of Physics, Haverford College Haverford, PA 19041, USA Marcelo Gleiser †‡ Department of Physics and Astronomy, Dartmouth College Hanover, NH 03755, USA I report on recent work concerning the existence and stability of self-gravitating spheres with anisotropic pressure. After presenting new exact solutions, Chandrasekhar’s variational formalism forradialperturbationsisgeneralizedtoanisotropicobjectsandappliedtoinvestigatetheirstability. It is shown that anisotropy can not only support stars of mass M and radius R with 2M/R≥8/9 andarbitrarilylargesurfaceredshifts,butthatstableconfigurationsexistforvaluesoftheadiabatic index γ smaller than the corresponding isotropic value. 4 0 0 2 I. INTRODUCTION n a J A common assumption in the study of stellar structure and evolution is that the interior of a star can be modeled 6 asaperfect fluid[1]. Thisperfectfluid modelnecessarilyrequiresthe pressureinthe interiorofa startobe isotropic. 2 This approach has been used extensively in the study of polytropes, including white dwarfs, and of compact objects suchasneutronstars[2]. However,theoreticaladvancesinthelastdecadesindicatethatdeviationsfromlocalisotropy 1 in the pressure, in particular at very high densities, may play an important role in determining stellar properties [3]. v The physical situations where anisotropic pressure may be relevant are very diverse. By anisotropic pressure we 6 mean that the radial component of the pressure, p (r), differs from the angular components, p (r) = p (r) ≡ p (r). 4 r θ ϕ t 5 (That pθ(r)=pϕ(r) is a direct consequence of spherical symmetry.) Of course,spherical symmetry demands both to 1 be strictly a function of the radial coordinate. Boson stars, hypothetical self-gravitating compact objects resulting 0 from the coupling of a complex scalar field to gravity, are systems where anisotropic pressure occurs naturally [4]. 4 In the interior of neutron stars pions may condense. It has been shown that due to the geometry of the π− modes, 0 anisotropic distributions of pressure could be considered to describe a pion condensed phase configuration [5]. The / h existence of solid cores and type P superfluidity may also lead to departures from isotropy within the neutron star p interior [2]. - Sincewestilldonothaveadetailedmicroscopicformulationofthepossibleanisotropicstressesemergingintheseand o r othercontexts,wetakethegeneralapproachoffindingseveralexactsolutionsrepresentingdifferentphysicalsituations, t modeled by ansatze for the anisotropy factor, p −p . Previous studies have found some exact solutions, assuming s t r a certain relations for the anisotropy factor [6]. Our goal here is two fold: first, to find new exact solutions which may : better modelrealistic situations andexplore their physicalproperties[7]; second,to investigatetheir stability against v i smallradialperturbations[8]. Forthis,wegeneralizeChandrasekhar’swell-knownvariationalapproachtoanisotropic X objects. We find that not only interesting exact solutions can be found [7], but that they may have a wider stability r range when contrasted with their isotropic counterparts [8]. a II. RELATIVISTIC SELF-GRAVITATING SPHERES: BASICS Weconsiderastaticequilibriumdistributionofmatterwhichissphericallysymmetric. InSchwarzschildcoordinates the metric can be written as ds2 =eνdt2−eλdr2−r2dθ2−r2sin2θdφ2 , (1) † InvitedcontributiontotheInternational WorkshoponAstronomyandRelativisticAstrophysics,October12-162003,Olinda,Brazil. ∗Electronicaddress: kdev@haverford.edu ‡Electronicaddress: gleiser@dartmouth.edu 2 where all functions depend only on the radial coordinate r. The most general energy-momentum tensor compatible with spherical symmetry is Tµ =diag(ρ,−p ,−p ,−p ) . (2) ν r t t We see thatisotropyis notrequiredby sphericalsymmetry; it is anaddedcondition. The Einstein fieldequations for this spacetime geometry and matter distribution are ν′ 1 1 e−λ + − =8πp ; (3) (cid:18)r r2(cid:19) r2 r 1 1 1 (ν′−λ′) e−λ ν′′− λ′ν′+ (ν′)2+ =8πp ; (4) t (cid:18)2 4 4 2r (cid:19) λ′ 1 1 eλ − + =8πρ . (5) (cid:18) r r2(cid:19) r2 Note that this is a system of 3 equations with 5 unknowns. Consequently, it is necessary to specify two equations of state, such as p =p (ρ) and p =p (ρ). r r t t It is often useful to transform the above equations into a form where the hydrodynamical properties of the system are more evident. For systems with isotropic pressure, this formulation results in the Tolman-Oppenheimer-Volkov (TOV) equation. The generalized TOV equation, including anisotropy, is dp ν′ 2 r =−(ρ+p ) + (p −p ) , (6) r t r dr 2 r with 1 m(r)+4πr3p ν′ = r, (7) 2 r(r−2m) and r m(r)= 4πr2ρdr . (8) Z 0 Taking r = R in the above expression gives us the Schwarzschild mass, M. [This implicitly assumes that ρ = 0 for r >R.] Inordertosolvetheaboveequationswemustimposeappropriateboundaryconditions. Werequirethatthesolution be regular at the origin. This imposes the condition that m(r)→0 as r →0. If p is finite at the origin then ν′ →0 r as r →0. The gradient dp /dr will be finite at r =0 if (p −p ) vanishes at least as rapidly as r when r →0. This r t r will be the case in all scenarios examined here. The radius of the star is determined by the condition p (R) = 0. It is not necessary for p (R) to vanish at the r t surface. But it is reasonable to assume that all physically interesting solutions will have p ,p ≥0 for r ≤R. r t III. EXACT SOLUTIONS In ref. [7], we obtained severalexactsolutions for different forms of the pressure anisotropy. Our solutions fall into two classes: i) ρ = constant, and ii) ρ∝1/r2. Given the limited space, we will restrict the presentation to the latter case. Interested readers should consult ref. [7] for details. Consider stars with energy density modeled as 1 a ρ= +3b , (9) 8π r2 (cid:16) (cid:17) where both a and b are constant. The choice of the values for a and b is dictated by the physicalconfigurationunder consideration. For example, a = 3/7 and b = 0, corresponds to a relativistic Fermi gas, as in the Misner-Zapolsky solution for ultradense cores of neutron stars[9]. If we take a = 3/7 and b 6= 0 then we have a relativistic Fermi gas 3 core immersed in a constant density background. For large r the constant density term dominates (r2 ≫a/3b), and c can be thought of as modeling a shell surrounding the core. We also take the pressure anisotropy to be 1 c p −p = +d , (10) t r 8π r2 (cid:16) (cid:17) with c and d constant. We found it convenient to seek solutions for the metric function ν(r) directly, rather than solving the generalized TOVequation. We thenuse the knownfunctionsλ(r) andν(r) tofind the radialandtangentialpressures. Fromeqs. 3, 4, and 5, we find ν′′ (ν′)2 λ′ 1 1 λ′ 1 + e−λ−ν′ + e−λ− + e−λ+ =8π(p −p ). (11) (cid:18) 2 4 (cid:19) (cid:18)4 2r(cid:19) (cid:18)r2 2r(cid:19) r2 t r ν Introducing a new variable y =e2, eq. 11 becomes, λ′ 1 1 λ′ 1 (y′′)e−λ−y′ + e−λ−y ( + )e−λ− =8πy(p −p ). (12) (cid:18)2 r(cid:19) (cid:20) r2 2r r2(cid:21) t r Since e−λ =1−2m(r)/r, using eq. 9 we find e−λ =1−a−br2 ≡I2(r), (13) b where we defined the function I2(x)≡1−a−bx2. When b=0, we write I2 ≡1−a. Using the expressionfor e−λ in b 0 eq. 12 and substituting for the pressure anisotropy we find br4−(1−a)r2 y′′+(1−a)ry′−(a−c−dr2)y =0 . (14) (cid:2) (cid:3) The full solution of eq. 14 with a,b,c,d 6= 0 is in [7]. Here, we will only consider solutions with b = d = 0. In this case, the total mass is M =aR/2, and exp[−λ]=1−a. Since for any static spherically-symmetric configuration we expect (2M/R) ≤1, we must have a<1. (Also, the metric coefficient g becomes infinite when a=1). crit rr Since we want to construct stars with finite radii and density in the context of anisotropic pressure, we impose boundary conditions such that p (R)=0. With b=d=0, eq. 14 reduces to an Euler-Cauchy equation, r (1−a)r2y′′−(1−a)ry′+(a−c)y =0. (15) Thesolutionsofthisequationfallintothreeclasses,dependingonthevalueofq (q isreal,q =0,orq isimaginary), where 1 (1+c−2a)2 q ≡ . (16) 1 (1−a)2 We will only show results for q real. (See ref. [7] for the other cases.) The solution for y is r 1+q r 1−q y =A +A , (17) + − R R (cid:16) (cid:17) (cid:16) (cid:17) with the constants A and A fixed by boundary conditions. For the case under consideration here (b=d=0), the + − boundary conditions are dν a e−λ(R) =eν(R) =I2, and eν(R) | = . (18) 0 dr R R Applying the boundary conditions we find I 1−3I2 A = 0 + 0 and A =A (q →−q). (19) + − + 2 4qI 0 The radial pressure for this case, after substituting the expressions for A and A , is + − (3I2−1)2−4q2I4 R2q−r2q 8πp = 0 0 . (20) r r2 (cid:20)(3I2−1+2qI2)R2q+(1−3I2+2qI2)r2q(cid:21) 0 0 0 0 4 FIG. 1: Radial pressure as a function of r for ρ and (pt−pr)∝r−2 and q real. We note that the boundary conditions automatically guaranteethat p (R)=0. The radialpressureis alwaysgreater r than zero provided a<2/3 and a2 >4c(1−a). Since by definition a>0, the second condition implies c>0. Thus, this model does not allow for negative anisotropy. Further, since we are considering the case q > 0, we must impose the condition 1+c < 2a. Combining the two inequalities for a and c, we obtain, 2a−1 < c < a2/4(1−a). Since we have 0 < a < 2/3 we find that 0 < c < 1/3. We note that for the anisotropic case the maximum value of a is 2/3,correspondingto a 33%increase when comparedwith the isotropic case (a=3/7). In figure 1 we plot the radial pressure, p , as a function of the radial coordinate r, for a=3/7 and several values of c. Note that for this choice of r a, the inequality c2). Itisthusconceivablethatobjectswhichareobservedatlargeredshiftmayactually s be closer than we think due to anisotropic distortions. They may also be more stable then we think, especially if pressure anisotropy exists near the stellar core. Here, I have presentedresults only for one of the caseswe treated, starswith energy density scalingas 1/r2. These are of interest as they include ultra-relativistic equations of state used to model the cores of neutron stars. Perhaps themostimportantlessonofourstudyisthatonemustkeepanopenmindastowhetherisotropyisornotajustified assumptionto describe stellar matter. Until we havea better microscopicunderstanding of whattruly goes oninside ultra-densecompactobjects, isotropyshouldbe takenwith agrainofsalt. Especiallyifsome ofthese objects contain bosonic condensates at their cores, as is the case for several models of neutron star interiors and for a whole class of fully-anisotropic hypothetical objects known as boson stars[4]. Acknowledgments I’dliketoexpressmygratitudetothe organizersofIWARA2003,andinparticulartoHelioCoelho,SandraPrado, and Cesar Vasconcellos for their invitation and wonderful hospitality. My research is sponsored in part by a NSF grant PHY-0099543. [1] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, (The University of Chicago Press, Chicago, 1983); R. Kippenhahn and A.Weigert, Stellar Structure and Evolution, (Springer-Verlag,Berlin, 1991). [2] N. K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity, (Springer-Verlag, Berlin, 1997); H.Heiselberg and M. H.-Jensen,Phys. Rep. 328, 237 (2000). [3] M.Ruderman,Annu.Rev.Astron.Astrophys.,10,427 (1972); V.Canuto,Annu.Rev.Astron.Astrophys.,12,167(1974). [4] M. Gleiser, Phys. Rev. D38, 2376 (1988); M. Gleiser and R. Watkins, Nucl. Phys. B319, 733 (1989). For comprehensive reviews see, A.R.Liddleand M.S.Marsden, Int.J. Mod.Phys. D1101, (1992); P.Jetzer, Phys.Rep.220, 163 (1992); E. W. Mielke and F. E. Schunck, in Proceedings of 8th M. Grossmann Meeting, T. Piran (ed.), (World Scientific, Singapore, 1998). [5] R. Sawyerand D.Scalapino Phys.Rev. D7, 382 (1973). [6] R. L. Bowers and E. P. T. Liang, Ap. J. 188, 657 (1974); J. Ponce de Leon, J. Math. Phys., 28, 1114 (1987); M. Gokhroo and A.Mehra Gen. Rel. Grav., 26, 75 (1994); H. Bondi, Mon. Not. R.Astron. Soc. 259, 365 (1992); E. S.Corchero, Clas. QuantumGrav.15,3645(1998);L.Herrera,Phys.Lett.A 165,206(1992);L.HerreraandN.O.Santos,Phys.Rep.,286, 53, (1997). [7] K. Dev and M. Gleiser, Gen. Rel. Grav. 24, 1793 (2002). [8] K. Dev and M. Gleiser, Gen. Rel. Grav. 35, 1435 (2003). [9] C. Misner and H. Zapolsky, Phys.Rev.Lett., 12, 635 (1964).

Anisotropic Stars: Exact Solutions and Stability

Similar Anisotropic Stars: Exact Solutions and Stability