Hydrodynamical Collapse of Neutron Stars due to Hadron-Quark Phase Transition
aa r X i v : . [ g r- q c ] O c t Hydrodynamical Collapse of Neutron Stars due toHadron-Quark Phase Transition
Guilherme F. Marranghello , Jos´e Carlos N. de Araujo and OswaldoD. Miranda Universidade Federal do Pampa, Bag´e/RS, Brazil Instituto Nacional de Pesquisas Espaciais, S˜ao Jos´e dos Campos/SP, BrazilE-mail: [email protected]
Abstract.
We present studies of the collapse of neutron stars that undergo a hadron-quarkphase transition. A spherical Lagrangian hydrodynamic code has been written. As initialcondition we take different neutron star configurations taking into account its density, energydensity and pressure distribution. The phase transition is imposed at different evolution times.We have found that a significant amount of matter on the surface can be ejected while theremaining star rings in the fundamental and first pressure modes.
1. Introduction
Black holes (
BHs ) and neutron stars (
N Ss ) are certainly two major potential sources ofgravitational waves (
GW s ). Unlike
BHs , whose gravitational waveforms are specified essentiallyby their masses and angular momenta, the characteristics of the gravitational emission from
N Ss depend on the properties of the nuclear matter.When nuclear matter is compressed to a sufficiently high density, it turns into uniform three-flavor ( u , d , and s ) strange quark matter (SQM), since it is expected that SQM may be morestable than nuclear matter. The deconfined quark matter appears when the density is so highthat the nucleons are “touching” each other. At this point, when the number density of nucleons n ∼ O (1 fm − ), the quarks lose their correlation with individual nucleons and appear in adeconfined phase. Since the density required for this to happen is not much higher than nuclearmatter density (0 .
16 fm − ), the dense cores of neutron stars are the most likely places wherethe phase transition to quark matter may occur astrophysically. It should be noted that strangequarks (in a confined phase) could already exist in neutron stars with a hyperon core[1].In principle, the existence of a thin crust of normal matter is possible at the surface of astrange star. On the other hand, if SQM is metastable at zero pressure, so that it is relativelymore stable than nuclear matter only because of the high pressure in the cores of neutron stars,then the final products of the phase transition would be hybrid stars, which consist of quarkmatter cores surrounded by normal matter outer parts.The above arguments were made in [2] and those authors used polytropic equation of state todescribe the complex nuclear matter structure. We proceed similar calculations with a realisticequation of state based on a field theoretical model. We also make use of a spherical Lagrangianhydrodynamical code instead of the Eulerian model used in [2]. . Hydrodynamical Collapse The spherical Lagrangian hydrodynamic code divides the star into several concentric shells. Thisnumber of shells is the optimal combination for the variable time step and the number of shellsthat gives the shortest processing time. Our results show that 1600 shells are enough to modelsatisfactorily all the stars here studied. To verify if the number of shells is an optimal choicewe ran a couple of models with 6400 shells, and we see that the results are the same as for themodels with 1600 shells. The shocks that appear are treated with the inclusion of the artificialviscosity of Richmeyer and Morton [4].The hydrodynamic equations that describe the dynamics of neutron stars are: ∂ρ∂t + 1 r ∂∂r ( r ρv ) = 0 , (1)the mass conservation equation written in spherical coordinates; where ρ is the density, r theradial coordinate and v the velocity. In our code the continuity equation is calculated directlyfrom the grid. The specific volumes of the Lagrangian shells give the density of the star.The equation of motion is given by: D~vDt + 1 ρ ∇ P + ∇ φ = 0 (2)where the velocity is ~v ; the pressure of the nuclear matter is given by the model described inthe next section. The gravitational potential is φ . The energy equation is written as: DEDt = Pρ DρDt (3)where E is the thermal energy per gram.
3. Neutron Star vs. Equation of State
In our investigation we consider a phenomenological QHD Lagrangian density[5, 6, 7] whichcontains the fundamental baryon octet, the scalar meson fields σ, f and the vector meson fields ω, ρ, φ . Additionally, it contains the lightest charged leptons ( e and µ ) to allow for chargeneutrality, L = X B = N,Y [ ¯ ψ B iγ µ ∂ µ ψ B − ¯ ψ B ( M B − g ⋆σ B σ − g ⋆σ ∗ B σ ∗ ) ψ B − g ⋆ω B ¯ ψ B γ µ ψ B ω µ − g ⋆φ B ¯ ψ B γ µ ψ B φ µ − g ⋆̺ B ¯ ψ B γ µ τ · ̺ µ ψ B ] − φ µν φ µν + 12 m φ φ µ φ µ − ω µν ω µν + 12 m ω ω µ ω µ − ̺ µν · ̺ µν + 12 m ̺ ̺ µ · ̺ µ + 12 ( ∂ µ σ∂ µ σ − m σ σ )+ 12 ( ∂ µ σ ∗ ∂ µ σ ∗ − m ∗ σ σ ∗ ) + X l ¯ ψ l [ i γ µ ∂ µ − m l ] ψ l (4)where the parameterized coupling constants [5] of the theory are defined as g ⋆σ B ≡ m ∗ Bα g σ B , g ⋆σ ∗ B ≡ m ∗ Bα g σ ∗ B g ⋆ω B ≡ m ∗ Bβ g ω B g ⋆φ B ≡ m ∗ Bβ g φ B , g ⋆̺ B ≡ m ∗ Bγ g ̺ B (5)with m ∗ Bn = g σ B σ + g σ ∗ B σ ∗ n M B ! − n , n = α, β, γ . (6)or the quark matter the MIT Lagrangian [8] was adopted. Here we have assumed thatthe phase transition is of first order. This last possibility is not favored energetically when theinterface tension between quarks and hadrons is important [9, 10]. Thus, the physical conditionsat deconfinement were estimated from Gibbs criteria.
4. Numerical Results and Conclusions
We have obtained some important results which seems to be in complete agreement with thoseobtained by [2]. The equation of state is shown in Fig.1 for three different parameter α whichleads to different hadron-quark phase transition pressure and density. Three stages of evolutionare shown in Fig.2. The star radius shrinks and the central density is increasing during thecollapse. In Fig.3 one can find the density evolution of each shell. The oscillation of the innershells is clear in the final stage of evolution as it is the ejection of the external shells. The fouriertransform is shown in Fig.4. This result shows a peak very close to the one shown by [2], at3.1kHz, which shall correspond to the fundamental mode as well as other peaks, correspondingto the first pressure modes. It is also easy to find that most of the energy is released in thefundamental mode, as always claimed. In Fig.5 the radius and density evolution are plottedagainst time in this same figure.Finally, after a future detection, one can try to fit the nuclear matter parameters to obtain thefrequency of the detection, remembering that, if the frequency falls on the 3.0-3.4kHz bandwidth,the Brazilian Schenberg antenna will be able to identify the direction of the source, as shown in[3]. Acknowledgments
We would like to thank CNPq and Fapesp for financial support.
References [1] G. F. Marranghello, C. A. Z. Vasconcellos, J. A. de Freitas Pacheco 2002 PR
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CQG C63 P r e ss u r e Figure 1.
Equation of state with hadron-quark phase transition for three different α -parameter. Figure 2.
The collapsed star structure at three different instants during the compression. -6 -3 Figure 3.
Density shells evolution from collapse to oscillation modes. Time is expressed inseconds.
Figure 4.
Fourier transform of the oscillations. Frequency is written in kHz. R ( k m )
0 5 10 15 20 25 30 35 1 10 100 1000 10000
Figure 5.
Density evolution with mass ejection. Time is plotted in ms and the colors are thelogarithm of energy density in g/cm3