Critical phenomena in neutron stars: II

We consider the head-on collision of equal-mass neutron stars boosted towards each other and we study the behaviour of such systems near the threshold of black-hole formation. In particular, we confirm the previous findings by Jin K J et al (2007 Phys. Rev. Lett 98 131101) that a type I critical phenomenon can be observed by fine-tuning the initial mass of the two neutron stars. At the same time, we argue against the interpretation that the critical solution is not a perturbed spherical star and show instead that the metastable star corresponds to a (perturbed) equilibrium solution on the unstable branch of the equilibrium configurations. As a result, the head-on collision of two neutron stars near the critical threshold can be seen as a transition in the space of configurations from an initial stable solution over to a critical metastable one which can either migrate to a stable solution or collapse to a black hole. The critical exponent for this process shows a fine structure which was already observed in the case of the critical collapse of scalar fields but never before for perfect fluids. PACS numbers: 04.25.Dm, 04.40.Dg, 04.70.Bw, 95.30.Lz, 97.60.Jd 1. Motivation and introduction In 1993, Choptuik [2] considered the one-parameter families of solutions, S[P ], of the Einstein–Klein–Gordon equations for a massless scalar field in spherical symmetry, such that for every P > P , S[P ] contains a black hole and for every P < P , S[P ] is a solution not containing singularities. He studied numerically the behaviour of S[P ] as P → P and found that the critical solution, S[P ], is universal, in the sense that it is approached by all nearly critical solutions regardless of the particular family of initial data considered. He also found that S[P ] exhibit discrete self-similarity and that, for supercritical solutions (P > P ), the mass of the black hole satisfies MBH = c|P −P |γ , with γ being a universal constant, i.e. not depending on the particular family of initial data. 0264-9381/10/235016+16$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1 Class. Quantum Grav. 27 (2010) 235016 T Kellerman et al After Choptuik’s seminal work, similar transitions were discovered for a wide range of systems, including massive scalar fields and ultra-relativistic fluids, see [3] for a recent review. All these phenomena have the common property that, as P approaches P , S[P ] approaches a universal solutionS[P ] and that all the physical quantities ofS[P ] depend only on |P−P |. In analogy with critical phase transitions in statistical mechanics, these transitions in gravitational collapse were later classified as ‘type I’ critical phenomena, with static or periodic critical solutions and discontinuous transitions in the vicinity of the critical point, or ‘type II’ critical phenomena, with self-similar critical solutions and continuous transitions in the vicinity of the critical solution [3]. The study of critical phenomena in neutron star (NS) collapse started with the work by Evans and Coleman [4] on radiation fluids and was later extended to more general ultrarelativistic equations of state (EOS) [5, 6] and ideal-gas EOS [7–9]. In all these studies the collapse was triggered using strong perturbations and a type II critical phenomenon was found. Type I critical phenomenon in the collapse of unstable configuration under very small perturbations was instead studied only very recently [10, 11]. The first study of critical phenomena, the head-on collision of NSs, was instead carried out by Jin and Suen in 2007 [1]. In particular, they considered a series of families of equal mass NSs, modelled with an ideal-gas EOS, boosted towards each other and varied the mass of the stars, their separation, velocity and the polytropic index in the EOS. In this way they could observe a critical phenomenon of type I near the threshold of black-hole formation, with the putative solution being a nonlinearly oscillating star. In a successive work [12], they performed similar simulations but considering the head-on collision of Gaussian distributions of matter. Also in this case they found the appearance of type I critical behaviour, but also performed a perturbative analysis of the initial distributions of matter and of the merged object. Because of the considerable difference found in the eigenfrequencies in the two cases, they concluded that the critical solution does not represent a system near equilibrium and in particular not a perturbed Tolmann–Oppenheimer–Volkoff (TOV) solution [1]. In this paper we study the dynamics of the head-on collision of two equal-mass NSs using a setup which is as similar as possible to the one considered in [1]. While we confirm that the merged object exhibits a type I critical behaviour, we also argue against the conclusion that the critical solution cannot be described in terms of the equilibrium solution. Indeed, we show that, in analogy with what found in [11], the critical solution is effectively a perturbed unstable solution of the TOV equations. Our analysis also considers a fine structure of the scaling relation of type I critical phenomena and we show that it exhibits oscillations in a similar way to the one studied in the context of scalar-field critical collapse [13, 14]. The remainder of this paper is organized as follows. In section 2 we describe the numerical settings of the simulations, the properties of the used initial data and we define some of the quantities that we analysed. In section 3 we show in details our results, while section 4 is dedicated to conclusions and discussion. Finally, in the appendix, we presented a study of the compactness of the metastable solution and a comparison with its ‘hoop radius’. The work reported is closely related to the analysis carried in a companion paper (hereafter paper I) about critical phenomena in linearly unstable nonrotating NS models [11]. Because of the logical affinity between the two works we will often refer the reader to the results obtained in [11]. We use a spacelike signature (−,+,+,+), with Greek indices running from 0 to 3, Latin indices from 1 to 3 and the standard convention for the summation over repeated indices. Unless explicitly stated, all the quantities are expressed in the system of dimensionless units in which c = G = M = 1.