Using a multi domain spectral method, we investigate systematically the
general-relativistic model for axisymmetric uniformly rotating, homogeneous
fluid bodies generalizing the analytically known Maclaurin and Schwarzschild
solutions. Apart from the curves associated with these solutions and a further
curve of configurations that rotate at the mass shedding limit, two more curves
are found to border the corresponding two parameter set of solutions. One of
them is a Newtonian lens shaped sequence bifurcating from the Maclaurin
spheroid sequence, while the other one corresponds to highly relativistic
bodies with an infinite central pressure. The properties of the configuration
for which both the gravitational and the baryonic masses, moreover angular
velocity, angular momentum as well as polar red shift obtain their maximal
values are discussed in detail. In particular, by comparison with the static
Schwarzschild solution, we obtain an increase of 34.25% in the gravitational
mass. Moreover, we provide exemplarily a discussion of angular velocity and
gravitational mass on the entire solution class.Comment: 4 pages, 4 figures, 1 table, submitted to A&A, corrected eq. for W,
  W' in 3.

Ansorg

Bardeen

Bonazzola

Butterworth

Chandrasekhar

Dyson

Eriguchi

Friedman

K. Schöbel

Komatsu

Kowalewsky

Lattimer

M. Ansorg

Nozawa

Poincaré

Stergioulas

Wilson

Wong

English

arXiv

Using a multi domain spectral method, we investigate systematically the general-relativistic model for axisymmetric uniformly rotating, homogeneous fluid bodies generalizing the analytically known Maclaurin and Schwarzschild solutions. Apart from the curves associated with these solutions and a further curve of configurations that rotate at the mass shedding limit, two more curves are found to border the corresponding two parameter set of solutions. One of them is a Newtonian lens shaped sequence bifurcating from the Maclaurin spheroid sequence, while the other one corresponds to highly relativistic bodies with an infinite central pressure. The properties of the configuration for which both the gravitational and the baryonic masses, moreover angular velocity, angular momentum as well as polar red shift obtain their maximal values are discussed in detail. In particular, by comparison with the static Schwarzschild solution, we obtain an increase of 34.25% in the gravitational mass. Moreover, we provide exemplarily a discussion of angular velocity and gravitational mass on the entire solution class

Schöbel, K.

Ansorg, M.

MPG.PuRe

Maximal mass of uniformly rotating homogeneous stars in Einsteinian gravity

Ansorg, Marcus

Max Planck Society eDoc Server

Using a multi domain spectral method, we investigate systematically the
general-relativistic model for axisymmetric uniformly rotating, homogeneous
fluid bodies generalizing the analytically known Maclaurin and Schwarzschild
solutions. Apart from the curves associated with these solutions and a further
curve of configurations that rotate at the mass shedding limit, two more
curves are found to border the corresponding two parameter set of solutions.
One of them is a Newtonian lens shaped sequence bifurcating from the Maclaurin
spheroid sequence, while the other one corresponds to highly relativistic
bodies with an infinite central pressure. The properties of the configuration
for which both the gravitational and the baryonic masses, moreover angular
velocity, angular momentum as well as polar red shift obtain their maximal
values are discussed in detail. In particular, by comparison with the static
Schwarzschild solution, we obtain an increase of 34.25% in the gravitational
mass. Moreover, we provide exemplarily a discussion of angular velocity and
gravitational mass on the entire solution class

EDP Sciences OAI-PMH repository (1.2.0)

Maximal mass of uniformly rotating homogeneous stars 
	in Einsteinian gravity

A&A 405, 405–408 (2003)DOI: 10.1051/0004-6361:20030634c© ESO 2003Astronomy&AstrophysicsMaximal mass of uniformly rotating homogeneous starsin Einsteinian gravityK. Scho¨bel and M. AnsorgTheoretisch-Physikalisches Institut, Friedrich-Schiller-Universita¨t Jena, Max Wien Platz 1, 07743 Jena, GermanyReceived 6 February 2003 / Accepted 22 April 2003Abstract. Using a multi domain spectral method, we investigate systematically the general-relativistic model for axisymmetricuniformly rotating, homogeneous fluid bodies generalizing the analytically known Maclaurin and Schwarzschild solutions.Apart from the curves associated with these solutions and a further curve of configurations that rotate at the mass sheddinglimit, two more curves are found to border the corresponding two parameter set of solutions. One of them is a Newtonianlens shaped sequence bifurcating from the Maclaurin spheroid sequence, while the other one corresponds to highly relativisticbodies with an infinite central pressure. The properties of the configuration for which both the gravitational and the baryonicmasses, moreover angular velocity, angular momentum as well as polar red shift obtain their maximal values are discussed indetail. In particular, by comparison with the static Schwarzschild solution, we obtain an increase of 34.25% in the gravitationalmass. Moreover, we provide exemplarily a discussion of angular velocity and gravitational mass on the entire solution class.Key words. stars: rotation – gravitation – relativity – methods: numerical1. IntroductionVarious numerical methods have been developed to investigaterelativistic rotating models for extraordinarily compact astro-physical objects like neutron stars (Wilson 1972; Bonazzola& Schneider 1974; Friedman et al. 1986, 1989; Komatsuet al. 1989a, 1989b; Lattimer et al. 1990; Neugebauer &Herold 1992; Herold & Neugebauer 1992; Bonazzola et al.1993, 1998; Eriguchi et al. 1994; Stergioulas & Friedman 1995;Nozawa et al. 1998). For reviews see Friedman (1998) andStergioulas (1998).Although at present more realistic equations of state are ex-plored, we consider in this paper only homogeneous and uni-formly rotating star models. This is interesting and importantfor subsequent computations, because the Newtonian and thestatic cases are well understood, in particular the analyticallyknown Maclaurin and Schwarzschild solutions. These mod-els were first studied by Butterworth & Ipser (1975, 1976),who found that, in addition to the analytic solutions, they arebounded by a sequence of configurations rotating at the massshedding limit1. The investigation of such limiting configura-tions is instructive since certain physical quantities reach max-imal values there. With regard to the astronomical observa-tions and a reliable identification of black holes, one is mostinterested in an upper bound on the gravitational mass andSend offprint requests to: K. Scho¨bel,e-mail: K.Schoebel@tpi.uni-jena.de1 Due to uniform rotation, a shedding of matter sets in when cen-trifugal forces balance gravity at the equator. Then a cusp at the star’sequatorial rim appears.especially on how it is aected by rotation. Butterworth & Ipserfor example estimated a 30% increase for fixed central pressureowing to rotation.In the present paper we completed their studies, findingtwo further limiting curves, in particular a sequence of starswith infinite central pressure and a sequence of Newtonianlens-shaped configurations that bifurcates from the Maclaurinspheroids before ending in a mass shedding limit (Bardeen1971; Ansorg et al. 2003b). All five limiting curves were foundto circumscribe entirely the general relativistic solution forhomogeneous star models that are continuously joined to thestatic Schwarzschild solution – hereafter called the “general-ized Schwarzschild solution” 2. This was done using the re-cently developed AKM method (Ansorg et al. 2002, 2003c),which allows one to solve the Einstein equations to high accu-racy even for critical configurations. In particular we are ableto determine to high precision the extreme configuration pos-sessing both a mass shed and infinite central pressure. At thispoint several physical quantities reach their global maxima andwe provide explicit values for these.In what follows we use units in which the speed of light andNewton’s constant of gravitation assume the value 1.2 This name (and not for example “generalized Maclaurin solu-tion”) was chosen since every possible Schwarzschild solution is con-tained within this class of solutions. In contrast, only those Maclaurinspheroids with axis ratios above 0.17126 are contained in this class(see Sect. 4).406 K. Scho¨bel and M. Ansorg: Maximal mass of uniformly rotating homogeneous stars in Einsteinian gravity2. Metric tensor and field equationsThe line element of a stationary, axisymmetric and asymptot-ically flat spacetime describing a uniformly rotating perfectfluid body can be cast into the formds2 = e−2U[e2k(dρ2 + dζ2) +W2dϕ2]− e2U(adϕ + dt)2 .The corresponding Lewis-Papapetrou coordinates (ρ, ζ, ϕ, t)are uniquely defined if we require continuity of the metric co-ecients and their first derivatives at the body’s surface. In acomoving frame, for which the metric assumes the same formwith metric functions U 0, k0, W0 and a0, the relativistic Eulerequation can easily be integrated to determine the pressure p.For constant mass-energy density µ this results inp = µ(eV0−U0 − 1), (1)where V0 is the constant surface value of U0.Taking into account asymptotic flatness, boundary and tran-sition conditions at the surface and regularity along the rota-tional axis (ρ = 0), the interior and exterior field equations forma complete set of equations to be solved, which is done by ap-plying the AKM method. For a comprehensive discussion ofthis multi domain spectral method see Ansorg et al. (2003c).3. Known static and Newtonian limits3.1. Schwarzschild solutionSolving Einstein’s equation for a static homogeneous staryields the famous Schwarzschild metric which reads in theabove coordinateseU =1 − M/(2r)1 + M/(2r) ek = 1 −(M2r)2W = ekρ a = 0for the exterior (r  √ρ2 + ζ2  R) andeU0=12[3 1 − M/(2R)1 + M/(2R) −1 − Mr2/(2R3)1 + Mr2/(2R3)]ek0=[1 + M/(2R)]31 + Mr2/(2R3) eU0 W0 = ek0ρ a0 = 0for the interior region (0  r  R). Here M is the gravita-tional mass and R denotes the coordinate radius given implic-itly throughM = µ4pi3 R3(1 + M2R)6For any physically relevant solution the pressure (1) has to re-main finite, which is fulfilled for R > M. This imposes an upperbound on the mass, namelyM <49√3piµ= 0.14477 . . . 1pµ3.2. Maclaurin spheroidsIn Newtonian gravity the problem of self gravitating rotatingideal fluids requires solving the Poisson equation for the body’sgravitational field while satisfying the Euler-Lagrange equa-tion governing its motion as an ideal fluid. This leads to a freeboundary problem. A particular solution for uniform rotationand constant mass density µ can be found by assuming thesurface to be a spheroid. One obtains the so called Maclaurinspheroids, parametrized here by focal distance ρ0 and the ra-tio rp/re between polar radius rp and equatorial radius re.Having computed the gravitational field, the Euler-Lagrange equation is seen to be satisfied for a squared angularvelocityΩ2 = 2piµξ[(3ξ2 + 1) arccotξ − 3ξ], ξ  r2er2p− 1− 12(bottom solid curve in Fig. 1). This relation holds independentof ρ0.Note that Maclaurin spheroids exist for every rp/re 2 [0, 1]and ρ0 2 [0,1[ , thus comprising a two parameter solution witharbitrary mass for fixed µ.3.3. First Newtonian lens sequenceOn the Maclaurin curve, an infinite series of points corre-sponding to axisymmetric secular instabilities occurs, begin-ning at rp/re = 0.17126, and accumulating in the Maclaurindisk limit rp/re ! 0 (Chandrasekhar 1967; Bardeen 1971).They are bifurcation points of further Newtonian sequences andcorrespond to singular post-Newtonian corrections (see Petro2003). The first one of these sequences is comprised of twosegments that depart from the first bifurcation point. One seg-ment proceeds towards the Dyson rings (Dyson 1892, 1893;Wong 1974; Kowalewsky 1885; Poincare´ 1885a, 1885b, 1885c;Eriguchi & Sugimoto 1981) whereas the other one ends in amass shedding limit (Bardeen 1971; Ansorg et al. 2003b). Thelatter we will refer to as the “first Newtonian lens sequence”motivated by the shape of the corresponding bodies.4. Generalized Schwarzschild solutionFor a systematic investigation of the general case the choiceof parameters is not restricted by the numerical approach.Nevertheless it is convenient to take non-ambiguous param-eters that are restricted to a compact interval (here [0, 1]) insuch a way that the endpoints represent limiting configurations.Corresponding to the limits found, we selected the followingmagnitudes:– Mass shed parameter (as defined in Ansorg et al. 2003c)β  − r2er2pd(ζ2s )d(ρ2)∣∣∣∣∣∣ρ=re= − rer2plimρ!reζsdζsdρ ,where ζs(ρ) is the function describing the surface shape.Maclaurin and Schwarzschild bodies are characterizedby β = 1 and the mass shed limit is fixed by β = 0.K. Scho¨bel and M. Ansorg: Maximal mass of uniformly rotating homogeneous stars in Einsteinian gravity 407Fig. 1. Squared angular velocity on the generalized Schwarzschildsolution. Dotted lines indicate curves of constant central pressure(equally spaced in p˜c = pc/(µ + pc), pc being the central pressureand µ the constant mass density). Ergo-toroids appear for configura-tions situated above the dashed line.– p˜c  pc/(µ + pc), where pc is the central pressure. Thusp˜c = 0 stands for the Newtonian limit where mass andhence pressure vanish and p˜c = 1 for the limiting configu-rations with infinite central pressure.Additionally we will use the ratio rp/re of polar to equatorialcoordinate radius that is 1 only for Schwarzschild solutions.Our numerical analysis revealed that the generalizedSchwarzschild solution is entirely bounded by the followingfive limiting curves (joined in the order listed):– the sequence of Schwarzschild solutions(rp/re = 1, β = 1, p˜c 2 [0, 1]);– the Maclaurin sequence from the sphere to the first axisym-metric bifurcation point(rp/re 2 [0.171, 1], β = 1, p˜c = 0);– the first Newtonian lens sequence(rp/re 2 [0.171, 0.192], β 2 [0, 1], p˜c = 0);– a sequence of configurations rotating at the mass sheddinglimit (rp/re 2 [0.192, 0.573], β = 0, p˜c 2 [0, 1]);– a sequence of configurations with infinite central pressure(rp/re 2 [0.573, 1], β 2 [0, 1], p˜c = 1).This makes it possible to determine maximal values of all in-teresting physical quantities (see Sect. 5). Moreover, we cannow state that on the generalized Schwarzschild solution noquasistationary transition to a Kerr black hole is possible (incontrast to what was found for the relativistic Dyson rings byAnsorg et al. 2003a) and that the surface remains convex in ρ-ζ-coordinates.Figure 1 depicts the (squared) angular velocity versusthe radius ratio rp/re. This is the completion of results ofButterworth & Ipser (1975, 1976)3. It shows that the configura-tion with maximal angular velocity rotates at the mass sheddinglimit and possesses infinite central pressure. The magnificationin Fig. 2 demonstrates that the mass shedding curve does not3 Note that in contrast to our work they used proper radial distancesand kept the baryonic mass constant.Fig. 2. Magnification of Fig. 1 in the vicinity of the first axisymmetricbifurcation point (C). Values for the first Newtonian lens sequencewere taken from Ansorg et al. (2003b). Points (A) and (C) correspondto the cross sections thus labelled in Fig. 6 ibid.Fig. 3. Gravitational mass on the generalized Schwarzschild solution.The small interval on the rp/re-axis, emphasized by a thick line,represents the first Newtonian lens sequence. Dotted lines indicatecurves of constant central pressure (equally spaced in p˜c, see Fig. 1).Ergo-toroids appear for configurations situated above the dashed line.terminate at the first axisymmetric bifurcation point (C), as wasconjectured by Butterworth and Ipser. Instead it is linked to theMaclaurin spheroids at this point via the first Newtonian lenssequence.The evolution of the gravitational mass for fixed centralpressure is seen in Fig. 3 and reveals a 34.25% increase in themaximal mass with respect to the static case. The global max-imum is again found for the same configuration as for angularvelocity. In both figures we included the line above which ergo-regions appear. Interestingly, this line corresponds roughly toone of constant mass.Other physical quantities as baryonic mass, angular mo-mentum, polar red shift and circumferential radius show a sim-ilar behaviour: for fixed mass shed parameter or radius ratiothey increase with increasing central pressure. Likewise theyincrease with decreasing mass shed parameter or decreasing408 K. Scho¨bel and M. Ansorg: Maximal mass of uniformly rotating homogeneous stars in Einsteinian gravityTable 1. Properties of the maximal mass configuration. Only validdigits are given. An asterisk indicates that the corresponding quantityhas a global maximum there.Physical quantity valueGravitational mass M = 0.19435 µ−1/2 Baryonic mass M0 = 0.27316 µ−1/2 Angular velocity Ω = 1.8822 µ1/2 Angular momentum J = 0.03637 µ−1 Polar radius rp = 0.04856 µ−1/2Equatorial radius re = 0.08475 µ−1/2Radius ratio rp/re = 0.5730Circumferential radius Rcirc = 0.41538 µ−1/2 Polar red shift Zp = 7.378 Fig. 4. Meridional coss section (solid line) and ergo-region (dashedline) of the maximal mass configuration. Axes are scaled identically.radius ratio if the central pressure is kept constant. So a globalmaximum for each of them is obtained on the common edge ofmass shed and infinite central pressure curves, correspondingthus to a very special limit star.5. Maximal mass configurationIn Table 1 masses and other quantities are listed for this ex-treme configuration. Because it resides on two critical curveswe notice a loss in accuracy that was somewhat recoveredby an extrapolation to an infinite approximation order m ofthe AKM method (cf. Ansorg et al. 2003c). Observe also theunexpectedly high value for the red shift Zp of a photon emit-ted at one of the poles.Figure 4 shows a meridional cross section of this config-uration including the border of the ergo-region. As in generalfor diverging central pressure, the ergo-toroid degenerates bypinching together in the center.We see the above work as a first step in the direction of theinvestigation of more realistic equations of state with the AKMmethod. The results, including the relation to relativistic ringsolutions and further aspects like stability, will be publishedelsewhere.Acknowledgements. We would like to thank Prof. R. Meinel andD. Petro for valuable discussions and helpful advice. This work wassupported by the Deutsche Forschungsgemeinschaft (DFG projectsME 1820/1–3 and SFB/TR 7 – B1).ReferencesAnsorg, M., Kleinwa¨chter, A., & Meinel, R. 2002, A&A, 381, L49Ansorg, M., Kleinwa¨chter, A., & Meinel, R. 2003a, ApJ, 582, L87Ansorg, M., Kleinwa¨chter, A., & Meinel, R. 2003b, MNRAS, 339,515Ansorg, M., Kleinwa¨chter, A., & Meinel, R. 2003c, A&A, 405, 711Bardeen, J. M. 1971, ApJ, 167, 425Bonazzola, S., Gourgoulhon, E., & Marck, J. A. 1998, Phys. Rev. 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