O. Benhar 1,2 - R. Rubino2
1 - INFN, Sezione di Roma, Piazzale Aldo Moro, 2, 00185 Roma, Italy
2 - Dipartimento di Fisica, Università "La Sapienza", Piazzale Aldo Moro, 2, 00185 Roma, Italy
Received 15 October 2004 / Accepted 14 December 2004
The transition from hadronic matter to quark matter in the core of neutron stars is likely to be associated with the appearance of a mixed phase, leading to a smooth variation of the star density profile. We discuss the results of a systematic study of the properties of the mixed phase on Coulomb and surface effects. A state-of-the-art nonrelativistic equation of state of nuclear matter has been used for the low density phase, while quark matter has been described within the MIT bag model, including the effect of perturbative one-gluon exchange interactions. The implications for neutron star structure are discussed.
Key words: dense matter - equation of state - stars: neutron
The possible occurrence of a core of deconfined quark matter in the interior of neutron stars has been investigated by a number of authors over the past three decades (for a recent review see, e.g., Weber 2004). Due to the complexity of the underlying dynamics, theoretical approaches to the study of hybrid stars largely rely on models to describe the equation of state (EOS) of strongly interacting matter, in both the hadronic and quark sector, as well as on a set of assumptions on the nature of the phase transition.
Most calculations have been carried out using nuclear EOS obtained from either nonrelativistic nuclear many body theory (NMBT) or relativistic mean field theory (RMFT), while the deconfined phase is usually described within the MIT bag model (Chodos et al. 1974).
In their pioneering work, Baym & Chin (1976a) employed the familiar Maxwell double tangent construction (see, e.g., Huang 1963), which amounts to assuming that the transition occurs at constant pressure. Within this picture, charge-neutral nuclear matter at energy density coexists with charge-neutral quark matter at energy density , the two phases being separated by a sharp interface.
In the early 90s Glendenning (Glendenning 1992,1997) first pointed out that the requirement that the two phases be individually charge-neutral is in fact too restrictive. In a more general scenario charged nuclear and quark matter may share a common lepton background, thus giving rise to a mixed phase extending in space over a sizable fraction of the star.
The appearance of a mixed phase strongly affects the macroscopic properties of the star. A transition at constant pressure necessarily leads to the appearance of a discontinuity in the density profile, i.e. to a star consisting of a inner core of quark matter at energy density surrounded by nuclear matter at energy density . On the other hand, the mixed phase allows for a smooth variation of the energy density, leading in turn to a smooth variation of the star density profile.
Whether the transition proceeds at constant pressure or according to Glendenning's picture depends upon i) the value of the Debye screening length, driving charge separation, and ii) the amount of electrostatic and surface energy needed for the formation of the structures of quark and nuclear matter in the mixed phase. If screening is too strong to allow for a uniform lepton density, or if the energy loss due to Coulomb and surface effects exceeds the gain in bulk energy, the standard scenario predicted by Maxwell construction turns out to be favorable.
The results of a detailed study carried out by Heiselberg et al. (1993) suggest that the mixed phase is energetically favored over a sizable density range if the surface tension is less that 70 MeV/fm2. In their calculations these authors adopted the MIT bag model (Chodos et al. 1974) for the quark matter EOS, while nuclear matter was described using a somewhat oversimplified model in which the energy-density included a quadratic compressional term and a symmetry term taken from a previous work of Lattimer et al. (1991).
Over the past decade, the availability of new nucleon-nucleon potentials, resulting from accurate fits to nucleon-nucleon scattering data, and the improvement of the computational schemes made it possible to obtain a new generation of EOS within the framework of NMBT (Akmal & Pandharipande 1997; Akmal et al. 1998). In view of the fact that NMBT is a parameter-free approach whose dynamics is strongly constrained by nuclear data and has been shown to possess a highly remarkable predictive power in theoretical studies of few nucleon systems (Pieper & Wiringa 1995), it provides a natural candidate to describe neutron star matter in the nuclear phase.
In this paper we extend the work of Heiselberg et al. (1993) carrying out a systematic study of the stabilty of the mixed phase. We adopt the state-of-the-art EOS of nuclear matter, obtained by Akmal et al. (1998) within NMBT, and analyze the dependence of the results on i) the parameters entering the MIT bag model EOS, employed to describe quark matter, and ii) the value of the surface tension, driving both Coulomb and surface effects.
In Sect. 2 we summarize the main features of the model EOS of both nuclear and quark matter, while the implementation of Gibbs conditions in the case of two chemical potentials, leading to the appearance of the mixed phase, is discussed in Sect. 3. Our main results are presented in Sect. 4, devoted to the role of Coulomb and surface energy. The implication of the appearance of the mixed phase for neutron star structure are outlined in Sect. 5. The conclusions of our work are stated in Sect. 6.
In this section we summarize the main features of the EOS employed in our work, focusing on the region of nuclear and supranuclear density ( fm-3). For the lower density region, corresponding to the outer and inner crust of the star, we have used the EOS of Baym et al. (1971) and Pethick et al. (1995), respectively. However, our results are largely unaffected by the details of the EOS at subnuclear density, as the fraction of star mass in the crust is only about 2%.
The theoretical descriptions of nuclear and quark matter are both based on the standard assumptions that the system is at zero temperature and transparent to neutrinos produced in weak interaction processes (see, e.g., Shapiro & Teukolski 1997).
Within NMBT, nuclear matter is
viewed as a collection of pointlike protons and neutrons, whose dynamics
are described by the Hamiltonian
The many-body Schrödinger equation associated with the Hamiltonian of Eq. (1) can be solved exactly, using stochastic methods, for nuclei of mass number . The resulting energies of the ground and low-lying excited states are in excellent agreement with experimental data (Wiringa & Pieper 1995). Exploiting translational invariance, accurate calculations can also be carried out for uniform nuclear matter (Wiringa et al. 1988; Akmal & Pandharipande 1997).
Akmal & Pandharipande (1997) have used cluster expansions and chain summation techniques to obtain the energy per particle of both pure neutron matter (PNM) and symmetric nuclear matter (SNM). In their approach the Argonne v18 potential of Wiringa et al. (1995) is modified to take into account the fact that NN potentials fitted to scattering data describe interactions between nucleons in their center of mass frame, in which the total momentum vanishes. Within the approach of Akmal & Pandharipande, relativistic corrections arising from the boost to a frame in which , are included up to order .
Interpolating between the PNM and SNM results of Akmal & Pandharipande (1997), Akmal et al. (1998) have determined the energy of matter with arbitrary ptoton fraction needed to obtain the EOS of -stable matter, consisting of neutrons, protons, electrons and muons. Their calculations span a range of baryon number density extending up to 8 n0, n0 = 0.16 fm-3 being the empirical saturation density of symmetric nuclear matter.
At any given value of ,
proton and lepton densities are determined
by the requirements of charge neutrality,
where P and
denote pressure and
energy density is obtained from the density dependence of the binding energy
|Figure 1: Energy per baryon of nuclear matter calculated by Akmal et al. (1998), plotted as a function of baryon number density. The dashed, dotdash and solid lines correspond to pure neutron matter, symmetrical nuclear metter and -stable matter, respectively. The box represents the equilibrium properties of symmetrical nuclear matter obtained from extrapolation of empirical data.|
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The results discussed in the following sections have been obtained using the EOS of Akmal et al. (1998), hereafter referred to as APR, to describe the nuclear phase of neutron star matter. The corresponding energy per baryon of PNM, SNM and -stable matter is displayed in Fig. 1 as a function of density.
Due to the complexity of the fundamental theory of strong interactions (Quantum Chromo-Dynamics, or QCD) a first principle description of the EOS of quark matter at high density and low temperature is out of reach of the existing computational approaches. To describe the quark matter phase, we have used the simple MIT bag model (Chodos et al. 1974), in which the main features of QCD are implemented through the assumptions that: i) quarks occur in colour neutral clusters confined to a finite region of space (the bag), whose volume is limited by the pressure of the QCD vacuum (the bag constant B), and ii) residual interactions between quarks are weak, and can be treated using low order perturbation theory.
Within the MIT bag model the thermodynamic potential
can be written
The EOS of quark matter can be obtained from the relations
linking pressure and energy density to :
The lowest order perturbative contributions to the thermodynamic potential,
with n = 0 and 1, are (see, e.g., Tamagaki & Tatsumi
For any baryon desity, quark densities are dictated by the requirements of
baryon number conservation, charge neutrality and weak equilibrium.
In the case of two flavors, in which only the light up and down quarks
are present, we have
As the baryon density increases, the d-quark chemical potential
reaches the value
being the mass of the strange quark.
The energy of quark matter can then be lowered turning d-quarks into
In presence of three flavors, Eqs.(18)-(20) become
As quarks are confined and not observable as individual particles, their masses are not directly measurable and must be inferred from hadron properties. The Particle Data Group (Hagiwara et al. 2000) report masses of a few MeV for up and down quarks and 60 to 170 MeV for the strange quark. We have set mu=md=0 and MeV for the up, down and strange quark, respectively. In the density region relevant to our work heavier quarks do not play a role.
The strong coupling constant
can be obtained from the renormalization
group equation, yielding
The values of the bag constant resulting from fits of the hadron spectrum range between 57 MeV/fm3, with MeV, (De Grand et al. 1975) and 350 MeV/fm3, with MeV (Carlson et al. 1983). However, the requirement that the deconfinement transition does not occur at density n0 constrains B to be larger than 120-150 MeV/fm3, and lattice results suggest a value of 210 MeV/fm3(Satz 1982). In order to gauge the dependence of the results on the value of B, we have carried out our calculations setting B = 120 and 200 MeV/fm3.
|Figure 2: Energy density of neutral quark matter in weak equilibrium as a function of baryon number density. The solid and dashed lines have been obtained setting and B = 200 and 120 MeV/fm3, respectively, while the dashdot line corresponds to and B = 200 MeV/fm3. The quark masses are mu = md = 0, MeV.|
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Figure 2 shows the energy density of neutral quark matter in weak equilibrium as a function of baryon density for different values of Band . The solid and dashed lines have been obtained setting and B = 200 and 120 MeV/fm3, respectively, while the dashdot line corresponds to and B = 200 MeV/fm3. Comparison between the dotdash line and those corresponding to suggests that, contrary to what is stated by many authors (see, e.g., Steiner et al. 2000), perturbative gluon exchange, whose inclusion produces a sizable change of slope, cannot be simulated by adjusting the value of the bag constant and must be explicitly taken into account.
Early works on the possible occurrence of quark matter in neutron stars (e.g. Baym & Chin 1976a) were based on the assumption that nuclear and quark matter were both charge neutral. As a consequence, the transition was described using the Maxwell construction (see, e.g., Huang 1963) and the resulting picture of the star consisted of a quark matter core surrounded by a mantle of nuclear matter, the two phases being separated by a sharp interface.
Glendenning (1992,1997) pointed out that this assumption is too restrictive. More generally, the transition can proceed through the formation of a mixed phase of charged nuclear and quark matter, global neutrality being guaranteed by a uniform lepton background.
Equilibrium between charged phases of nuclear
matter (NM) and quark matter (QM) at T=0 requires the fulfillment of
Gibbs conditions (see, e.g., Huang 1963)
The above equations imply that, for any pressure ,
the projection of the surfaces
defines two curves, whose intersection corresponds to the equilibrium
values of the chemical potentials.
As the chemical potentials determine the charge densities of the two phases,
the volume fraction occupied by quark matter, ,
can then be obtained
exploiting the requirement of global neutrality
Requiring that the two phases be individually neutral, as in the pioneering work
of Baym & Chin (1976a), reduces the number
of chemical potentials to one, thus leading to the equilibrium conditions
|Figure 3: Isobars MeV/fm3 ( upper panel) and 350 MeV/fm3 ( lower panel) obtained using the APR EOS of nuclear matter (solid lines) and the MIT bag model of quark matter, with and B = 200 MeV/fm3 (dashed lines). The intersections determine the values of the chemical potentials corresponding to equilibrium of the two phases according to Gibbs rules.|
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In the present work the intersection between the surfaces describing the
pressure of nuclear
and quark matter has been determined numerically, choosing as independent
variables, instead of
the proton and neutron chemical
In nuclear matter they
are simply related to the lepton chemical potential through the
In quark matter the chemical potentials
of up and down quarks can be obtained from
The phase transition between nuclear and quark matter, obtained setting B = 200 MeV/fm3 and , is illustrated in Fig. 4. Dashed and dotdash lines show the dependence on of the energy density of charge neutral nuclear and quark matter in weak equilibrium, respectively, while the solid line corresponds to the mixed phase. The latter turns out to be the ground state of neutron star matter at densities fm-3.
|Figure 4: Dashed and dotdash lines show the energy density of charge neutral nuclear and quark matter in weak equilibrium, respectively. The bag model parameters have been set to B = 200 MeV/fm3 and . The solid line corresponds to the mixed phase, obtained from Gibbs equilibrium conditions.|
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The dependence of our results on the MIT bag model parameters can be gauged from the upper panel of Fig. 5. It clearly appears that a lower value of the bag constant, corresponding to a softer quark matter EOS, leads to the appearance of the mixed phase at lower density. Keeping and setting B = 120 MeV/fm3 one finds that the mixed phase is energetically favored in the range fm-3. An even larger effect, illustrated by the lower panel of Fig. 5 is obtained with B = 120 MeV/fm3 and , i.e. neglecting perturbative gluon exchange altogether. For this case we also show the results obtained from the Maxwell construction, leading to the coexistence of charge-neutral nuclear matter at fm-3 and charge-neutral quark matter at fm-3. This cohexistence region is to be compared to the region of stability of the mixed phase, corresponding to fm-3.
|Figure 5: As in Fig. 4, but with the EOS of quark matter obtained using different values of the MIT bag model parameters. Upper panel: B = 120 MeV/fm3 and ; lower panel: B = 120 MeV/fm3 and . The straight line in the lower panel is the double tangent resulting from the Maxwell construction, while the open circles show the extrema of the coexistence region.|
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The results of the previous section suggest that, irrespective of the details of the EOS, the transition from nuclear to quark matter proceed through the formation of a mixed phase. However, two issues relevant to both the appearance and the stability of the mixed phase, not taken into account in the discussion of Sect. 3, need to be further analyzed.
Consider a mixed phase consisting of droplets of quark matter immersed in -stable nuclear matter, global charge neutrality being guaranteed by a lepton background. This picture is obvioulsy based on the assumption that the appearance of the charged droplets do not significantly affect the space distribution of the leptons, i.e. that the Debye screening length is large compared to both the typical size of the droplets and their separation distance (Heiselberg et al. 1993; Heiselberg & Hjort-Jensen 2000). If this condition is not satisfied the lepton background is distorted in such a way as to screen electrostastic interactions.
The estimates of reported by Heiselberg et al. (1993) suggest that screening effects can be disregarded if the structures appearing in the mixed phase of quark and nuclear matters have typical size and separation distance of 10 fm. The results of our calculations, that will be discussed discussed later in this section, show that this appears indeed to be the case.
The second issue deserving consideration is the stability of the mixed phase, i.e. the question of whether or not its energy is lower than the energy of the coexisting phases of nuclear and quark matter.
Formation of a spherical droplet of quark matter requires the energy
The energy density needed for the formation of the structures appearing in the
mixed phase has been obtained by Ravenhall et al. (1983) in the
context of a study of matter in the neutron star inner crust. It can be written in
the concise form
both surface and Coulomb energies vanish, and the energy density
of the mixed phase is given by Eq. (29), while for
The value of the surface tension at the interface between nuclear and quark matter is not known. It has been estimated using the MIT bag model and ignoring gluon exchange (Berger & Jaffe 1987, 1991). Assuming that a strange quark has mass of 150 MeV, Berger & Jaffe predict MeV/fm2. To quantitatively investigate the stability of the mixed phase, we have calculated for different values of , ranging from 2 MeV/fm2 to 10 MeV/fm2.
For any given value of the baryon number density , the energy density of Eqs. (46)-(47) has been calculated using the nuclear and quark matter densities determined according to the procedure described in Sect. 3 and carrying out a numerical minimization with respect to the value of the dimensionality parameter d. As increases, the resulting values of d change initially from 3 to 2 and 1 and then again to 2 and finally to 3. For example, in the case illustrated by Fig. 6, and corresponding to MeV/fm2, we find that spherical droplets of quark matter () appear at fm-3 and turn into rods () and slabs () at and 1.2 fm-3, respectively. For larger densities, quark matter becomes the dominant phase (i.e. ): at and 1.7 fm-3 the mixed phase features rods () and droplets () of nuclear matter that eventually dissolve in the quark matter background.
|Figure 6: The solid lines correspond to the difference (see Eq. (48)), evaluated for , 5 and 2 MeV/fm3. The dashed line shows the difference between the energy density resulting from the Maxwell conatruction and . The arrows mark the limits of the cohexistence region. Nuclear and quark matter are described by the APR EOS and the MIT bag model EOS, with and B = 200 MeV/fm3, respectively.|
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|Figure 7: Same as in Fig. 6, but with and B = 120 MeV/fm3. The solid lines show to the difference (see Eq. (48)), evaluated for and 2 MeV/fm3, respectively.|
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Our results are summarized in Figs. 6 and 7, that correspond to different choices of the MIT bag model parameters. The solid lines show the dependence of the calculated for different values of the surface tension . The dashed line represents the difference , where is the energy density obtained from Maxwell construction. For any given value of the surface tension, the mixed phase is favorable if the corresponding solid line lies below the dashed line.
The results of Fig. 6, corresponding to B=200 MeV/fm3 and , show that the mixed phase, while being always the lowest energy phase for MeV/fm2, becomes energetically unfavorable at some densities for MeV/fm2. For MeV/fm2 coexistence of charge neutral phases of nuclear and quark matter turn out to favorable over the whole density range.
To gauge the dependence upon the MIT bag model parameters we have repeated the calculations setting B=120 MeV/fm3 and . The results of Fig. 7 show that for in the range 2-5 MeV/fm2 the mixed phase is energetically favorable over a density region larger than the coexistence region predicted by the Maxwell construction.
Finally, we return to the problem of the comparison between the Debye screening length and the typical size of the structures appearing in the mixed phase. Our results suggest that the condition outlined in the work of Heiselberg et al. (1993) are indeed fulfilled. For example, in the case B=200 MeV/fm3 and we find that in the region of , corresponding to formation of droplets of quark matter, the droplets radius given by Eq. (44) is 2-3 fm.
Plugging the EOS
into the Tolman Oppenheimer
Volkoff (TOV) equations (Tolman 1934; Oppenheimer & Volkoff 1939)
The occurrence of the transition to quark matter makes the EOS softer, thus leading to a lower value of the maximum mass. In Fig. 8 we compare the mass-central energy density relations obtained using the APR EOS only to that obtained allowing for a transition to quark matter with and B= 120 and 200 MeV/fm3. The transition is described according to Gibbs conditions, neglecting surface and Coulomb effects. We find for the star made of nuclear matter oly and 1.89 and 2.03 for the hybrid stars corresponding to B= 120 and 200 MeV/fm3, respectively.
In Fig. 8 we also compare the curves obtained setting B= 120 MeV/fm3 and and adopting either the Gibbs or Maxwell picture. Whether the phase transition proceeds through the appearance of a mixed phase or charge-neutral coexisting phases does not appear to significantly affect the mass-central energy density relation. On the other hand, neglecting perturbative gluon exchange results in a rather low maximum mass, , barely compatible with the measured neutron star masses.
|Figure 8: Relation between neutron star mass and central energy density for different EOS. Dotdash line: pure nuclear matter (APR EOS); dotted lines: nuclear matter (APR EOS) and quark matter (MIT bag model with B= 120 and 200 MeV/fm3 and ). The phase transition is described according to Gibbs rules. Solid line: same as the dotted line, but with B= 120 MeV/fm3 and ; dashed line: same as the solid line, but with the phase transition described using Maxwell construction.|
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The neutron star mass-radius relations associated with the curves of Fig. 8, displayed in Fig. 9, show that in this case using the Maxwell construction instead of Gibbs rules produces a visible effect. All EOS predict the existence of stable star configurations with masses in the range allowed by observation (Thorsett & Chakrabarty 1999; Qaintrell et al. 2003), as well as a M(R) relation compatible with that resulting from the gravitational red shift measurement of Cottam et al. (2002).
|Figure 9: Mass radius relation for different EOS. The meaning of the curves is the same as in Fig. 8. The horizontal lines correspond to the observational limits on neutron star mass, whereas the third straight line is the mass-radius relation resulting from the gravitational redshift measurement of Cottam et al. (2002).|
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Figure 10 shows that different descriptions of the phase transition lead to remarkably different star density profiles. While in the presence of the mixed phase the density is a smooth function of the distance from the star center, the Maxwell construction leads to the appearance of a disontinuity. For comparison, we also show the profile of a star of the same mass, 1.4 , made of pure nuclear matter described by the APR EOS.
The discontinuous behavior can be easily understood noting that TOV Eqs. (49) and (50) require that the pressure P(r) be a monotonically decreasing function. It follows that if the pressure is the same for two different values of density, as in the phase transition in the Maxwell construction, they must necessarily correspond to the same value of r.
|Figure 10: Density profiles of a neutron star of mass 1.4 resulting from different models. Dashed line: APR EOS and MIT bag model EOS with and B = 120 MeV/fm3. Phase transition described allowing for the presence of the mixed phase. Solid line: same as the solid line, but with the phase transition described according to Maxwell construction. Dotdash line: pure nuclear matter with APR EOS.|
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In the coexisting phases scenario, the transition only takes place in stars whose central density exceeds the density of the quark matter phase. These star configurations turn out to be marginally stable, their mass being close to the maximum mass. For example, setting B= 200 MeV/fm3 and we find that the transition only occurs in stars having mass 2.0 . The radius of the quark matter core is small (1 Km), while the density jump is large, going from g/cm3 to g/cm3. These results are to be compared with those obtained in the mixed phase scenario when Coulomb and surface effects are neglected. In this case there is no jump and the density varies smoothly. At the center of a star of mass 2.0 , corresponding to energy-density g/cm3 the volume fraction occupied by quark matter reaches %.
We have carried out a study of the transition from nuclear matter to quark matter in the inner core of neutron stars, aimed at assessing whether the appearance of a mixed phase is energetically favorable and how the emerging picture depends upon the parameters entering the MIT bag model EOS.
In order to minimize the uncertainty associated with the description of the nuclear matter phase, we have adopted a EOS obtained from an ab initio calculation, based on a dynamical model stongly contrained by experimental data and not involving any adjustable parameters (Akmal et al. 1998).
Our results show that the effect of perturbative gluon exchange on the MIT bag model EOS is large and cannot be accounted for adjusting the value of the bag constant B. Neglecting interactions between quarks leads to a considerable softening of the EOS, resulting in a drastically lower transition density.
The fact that the setting lead to a much softer EOS is reflected by the rather small value of the maximum neutron star mass, barely exceeding the canonical value of 1.4 .
The stability of the mixed phase turns out to be strongly affected by surface and Coulomb effects. Using the softer quark matter EOS ( , B=120 MeV/fm3) we find that even a very small value of the surface tension, MeV/fm3, produces a narrowing of the density region spanned by the mixed phase. With the harder EOS ( , B=200 MeV/fm3) the coexistence of neutral phases of nuclear and quark matter is energetically favored at all densities for MeV/fm3.
Comparison with the results of Heiselberg et al. suggests that surface and Coulomb effects become larger when a realistic EOS is employed to describe the nuclear matter phase.
While the mass-central density relation appears to be largely unaffected by the occurrence of the mixed phase, its effect can be clearly seen in the M(R) curve. However, as measurements of neutron star radii are plagued by large uncertainties, this feature is not likely to be exploitable to extract clearcut information from observations.
The most striking difference between the Maxwell and Gibbs picture of the phase transition appears in the neutron star density profile, which in the case of transition at constant pressure exhibits a sharp discontinuity.
The presence of a density jump is known to affect neutron star dynamics, leading to the appearance of a class of nonradial oscillation modes, called g-modes, associated with emission of gravitational radiation.
Early investigations of the g-modes focused on the discontinuities produced by the changes of chemical composition in the low density region of the neutron star crust, corresponding to a fractional distance from the surface 10% (Finn 1987; McDermott 1990). These studies have been recently extended to the case of g-modes produced by a discontinuity located at much larger density and involving a much larger density jump, such as those associated with the transition to quark matter (Miniutti et al. 2003). Based on the results of calculations carried out using a simple polytropic EOS, Miniutti et al. (2003) argue that a simultaneous measurements of the frequencies of the fundamental f-mode and the g-mode would provide information on both size and location of the discontinuity.
Although it is unlikely that the first generation of laser interferometric antennae will detect gravitational waves emitted by an oscillating neutron star, the new detectors currently under investigation (see, e.g., the EURO proposal (2000)) are expected to be much more sensitive at the relevant frequencies above 1-2 kHz. Hopefully, information on the neutron star matter EOS and the nature of the transition to quark matter may be provided by gravitational wave astronomy.
The authors are deeply indebted to Ignazio Bombaci, Adelchi Fabrocini, Stefano Fantoni, Valeria Ferrari and Vijay Pandharipande for a number of usefuls discussions on issues related to the subjet of this paper.