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Roles of Hyperons in Neutron Stars

Shmuel Balberg, Itamar Lichtenstadt

The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

and

Gregory. B. Cook

Center for Radiophysics and Space Research, Space Sciences Building,

Cornell University, Ithaca, NY 14853

http://arxiv.org/pdf/astro-ph/9810361.pdf

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ABSTRACT

We examine the roles the presence of hyperons in the cores of neutron stars may

play in determining global properties of these stars. The study is based on estimates

that hyperons appear in neutron star matter at about twice the nuclear saturation

density, and emphasis is placed on effects that can be attributed to the general multispecies

composition of the matter, hence being only weakly dependent on the specific

modeling of strong interactions. Our analysis indicates that hyperon formation not only

softens the equation of state but also severely constrains its values at high densities.

Correspondingly, the valid range for the maximum neutron star mass is limited to

about 1.5 − 1.8 M⊙, which is a much narrower range than available when hyperon

formation is ignored. Effects concerning neutron star radii and rotational evolution are

suggested, and we demonstrate that the effect of hyperons on the equation of state

allows a reconciliation of observed pulsar glitches with a low neutron star maximum

mass. We discuss the effects hyperons may have on neutron star cooling rates, including

recent results which indicate that hyperons may also couple to a superfluid state in high

density matter. We compare nuclear matter to matter with hyperons and show that

once hyperons accumulate in neutron star matter they reduce the likelihood of a meson

condensate, but increase the susceptibility to baryon deconfinement, which could result

in a mixed baryon-quark matter phase.

Subject headings: stars: neutron — elementary particles — equation of state — stars:

evolution

– 2 –

1. Introduction

The existence of stable matter at supernuclear densities is unique to neutron stars. Unlike

all other physical systems in nature, where the baryonic component appears in the form of atomic

nuclei, matter in the cores of neutron stars is expected to be a homogeneous mixture of hadrons and

leptons. As a result the macroscopic features of neutron stars, including some observable quantities,

have the potential to illuminate the physics of supernuclear densities. In this sense, neutron stars

serve as cosmological laboratories for hadronic physics. A specific feature of supernuclear densities

is the possibility for new hadronic degrees of freedom to appear, in addition to neutrons and protons.

One such possible degree of freedom is the formation of hyperons - strange baryons - which is the

main subject of the present work. Other possible degrees of freedom include meson condensation

and a deconfined quark phase.

While hyperons are unstable under terrestrial conditions and decay into nucleons through the

weak interaction, the equilibrium conditions in neutron stars can make the reverse process, i.e.,

the conversion of nucleons into hyperons, energetically favorable. The appearance of hyperons in

neutron stars was first suggested by Ambartsumyan & Saakyan (1960) and has since been examined

in many works. Earlier calculations include the works of Pandharipande (1971b), Bethe & Johnson

(1974) and Moszkowski (1974), which were performed by describing the nuclear force in Schr¨odinger

theory. In recent years, studies of high density matter with hyperons have been performed mainly

in the framework of field theoretical models (Glendenning 1985, Weber & Weigel 1989, Knorren,

Prakash & Ellis 1995, Schaffner & Mishustin 1996, Huber at al. 1997). For a review, see Glendenning

(1996) and Prakash et al. (1997). It was also recently demonstrated that good agreement with these

models can be attained with an effective potential model (Balberg & Gal 1997).

These recent works share a wide consensus that hyperons should appear in neutron star (cold,

beta-equilibrated, neutrino-free) matter at a density of about twice the nuclear saturation density.

This consensus is attributed to the fact that all these more modern works base their estimates of

hyperon-nucleon and hyperon-hyperon interactions on the experimental constraints inferred from

hypernuclei. The fundamental qualitative result from hypernuclei experiments is that hyperon

related interactions are similar in character and in order of magnitude to nucleon-nucleon interactions.

In a broader sense, this result indicates that in high density matter, the differences between

hyperons and nucleons will be less significant than for free particles.

The aim of the present work is to examine what roles the presence of hyperons in the cores

of neutron stars may play in determining the global properties of these stars. We place special

emphasis on effects which can be attributed to the multi-species composition of the matter while

being only weakly dependent on the details of the model used to describe the underlying strong

interactions.

We begin our survey in § 2 with a brief summary of the equilibrium conditions which determine

the formation and abundance of hyperon species in neutron star cores. A review of the widely

accepted results regarding hyperon formation in neutron stars is given in § 3. We devote § 4 to

an examination of the effect of hyperon formation on the equation of state of dense matter, and

the corresponding effects on the star’s global properties: maximum mass, mass-radius correlations,

rotation limits, and crustal sizes. In § 5 we discuss neutron star cooling rates, where hyperons

might play a decisive role. A discussion of the effects of hyperons on phase transitions which may

occur in high density matter is given in § 6. Conclusions and discussion are offered in § 7.

– 3 –

2. Equilibrium Conditions for Hyperon Formation Neutron Stars

In the following discussion we assume that the cores of neutron stars are composed of a mixture

of baryons and leptons in full beta equilibrium (thus ignoring possible meson condensation and a

deconfined quark phase - these issues will be picked up again in § 6). The procedure for solving

the equilibrium composition of such matter has been describes in many works (see e.g., Glendenning

(1996) and Prakash et al. (1997) and references therein), and in essence requires chemical

equilibrium of all weak processes of the type

B1 → B2 + ℓ + ¯νℓ

; B2 + ℓ → B1 + νℓ

, (1)

where B1 and B2 are baryons, ℓ is a lepton (electron or muon), and ν (¯ν) is its corresponding

neutrino (anti-neutrino). Charge conservation is implied in all processes, determining the legitimate

combinations of baryons which may couple together in such reactions.

Imposing all the conditions for chemical equilibrium yields the ground state composition of

beta-equilibrated high density matter. The equilibrium composition of such matter at any given

baryon density, ρB, is described by the relative fraction of each species of baryons xBi ≡ ρBi

/ρB

and leptons xℓ ≡ρℓ/ρB.

Evolved neutron stars can be assumed to be transparent to neutrinos on any relevant time scale

so that neutrinos are absent and µν = µν¯ = 0. All equilibrium conditions may then be summarized

by a single generic equation

µi = µn − qiµe , (2)

where µi and qi are, respectively, the chemical potential and electric charge of baryon species i,

µn is the neutron chemical potential, and µe is the electron chemical potential. Note that in the

absence of neutrinos, equilibrium requires µe =µµ. The neutron and electron chemical potentials are

constrained by the requirements of a constant total baryon number and electric charge neutrality,

X

i

xBi = 1 ; X

i

qixBi +

X

ℓ

qℓxℓ = 0 . (3)

The temperature range of evolved neutron stars is typically much lower than the relevant

chemical potentials of baryons and leptons at supernuclear densities. Neutron star matter is thus

commonly approximated as having zero temperature, so that the equilibrium composition and

other thermodynamic properties depend on density alone. Solving the equilibrium compositions

for a given equation of state (EOS) at various baryon densities yields the energy density and pressure

which enable the calculation of global neutron star properties.

3. Hyperon Formation in Neutron Stars

In this section we review the principal results of recent studies regarding hyperon formation in

neutron stars. The masses, along with the strangeness and isospin, of nucleons and hyperons are

given in Tab. 1. The electric charge and isospin combine in determining the exact conditions for

each hyperon species to appear in the matter. Since nuclear matter has an excess of positive charge

and negative isospin, negative charge and positive isospin are favorable along with a lower mass

for hyperon formation, and it is generally a combination of the three that determines the baryon

density at which each hyperon species appears. A quantitative examination requires, of course,

– 4 –

modeling of high density interactions. We begin with a brief discussion of the current experimental

and theoretical basis used in recent studies that have examined hyperon formation in neutron stars.

3.1. Experimental and Theoretical Background

The properties of high density matter chiefly depend on the nature of the strong interactions.

Quantitative analysis of the composition and physical state of neutron star matter are currently

complicated by the large uncertainties regarding strong interactions, both in terms of the difficulties

in their theoretical description and from the limited relevant experimental data. None the less,

progress in both experiment and theory have provided the basis for several recent studies of the

composition of high density matter, and in particular suggests it will include various hyperon

species.

Experimental data from nuclei set some constraints on various physical quantities of nuclear

matter at the nuclear saturation density, ρ0 = 0.16 fm−3

. Important quantities are the bulk binding

energy, the symmetry energy of non-symmetric matter (i.e., different numbers of neutrons and

protons), the nucleon effective mass in a nuclear medium, and a reasonable constraint on the compression

modulus of symmetric nuclear matter. However, at present, little can be deduced regarding

properties of matter at higher densities. Heavy ion collisions have been able to provide some information

regarding higher density nuclear matter, but the extrapolation of these experiments to

neutron star matter is questionable since they deal with hot non-equilibrated matter.

Relevant data for hyperon-nucleon and hyperon-hyperon interactions is more scarce, and relies

mainly on hypernuclei experiments (for a review of hypernuclei experiments, see Chrien & Dover

(1989), Gibson & Hungerford (1995)). In these experiments a single hyperon is formed in a nucleus,

and its binding energy is deduced from the energetics of the reaction (typically meson scattering

such as X(K−, π−)X).

There exists a large body of data for single Λ-hypernuclei, which clearly shows bound states of

a Λ hyperon in a nuclear medium. Millener, Dover & Gal (1988) used the nuclear mass dependence

of Λ levels in hypernuclei to derive the density dependence of the binding energy of a Λ hyperon in at density ρ0 to be about −28 MeV, which is about one third of the equivalent value for a nucleon

in symmetric nuclear matter. The data from Σ-hypernuclei are more problematic (see below). A

few emulsion events that have been attributed to Ξ-hypernuclei seem to suggest an attractive Ξ

potential in a nuclear medium, somewhat weaker than the Λ−nuclear matter potential.

A few measured events have been attributed to the formation of double Λ hypernuclei, where two

Λ’s have been captured in a single nucleus. The decay of these hypernuclei suggests an attractive Λ−

Λ interaction potential of 4−5 MeV (Bodmer & Usmani 1987), somewhat less than the corresponding

nucleon-nucleon value of 6−7 MeV. This value of the Λ−Λ interaction is often used as the baseline

for assuming a common hyperon-hyperon potential, corresponding to a well depth for a single

hyperon in isospin-symmetric hyperon matter of -40 MeV. While this value should be taken with

a large uncertainty, the typical results regarding hyperon formation in neutron stars are generally

insensitive to the exact choice for the hyperon-hyperon interaction, as discussed below.

We emphasize again that the experimental data is far from comprehensive, and great uncertainties

still remain in the modeling of baryonic interactions. This is especially true regarding densities

– 5 –

greater than ρ0, where the importance of many body forces increases. Three body interactions are

used in some nuclear matter models (Wiringa, Fiks & Fabrocini 1988, Akmal, Pandharipande &

Ravenhall 1998). Many-body forces for hyperons are currently difficult to constrain from experiment

(Bodmer & Usmani 1988), although some attempts have been made on the basis of light

hypernuclei (Gibson & Hungerford 1995). Indeed, field theoretical models include a repulsive component

in the two-body interactions through the exchange of vector mesons, rather than introduce

explicit many body terms. We note that the effective equation used here is also compatible with

theoretical estimates of ΛNN forces through the repulsive terms it includes (Millener, et al. 1988).

In spite of these significant uncertainties, the qualitative conclusion that can be drawn from

hypernuclei is that hyperon-related interactions are similar both in character and in order of magnitude

to the nucleon-nucleon interactions. Thus nuclear matter models can be reasonably generalized

to include hyperons as well. In recent years this has been performed mainly with relativistic theoretical

field models, where the meson fields are explicitly included in an effective Lagrangian. A

commonly used approximation is the relativistic mean field (RMF) model following Serot & Walecka

(1980), and implemented first for multi-species matter by Glendenning (1985), and more recently

by Knorren et al. (1995) and Schaffner & Mishustin (1996) (see the recent review by Glendenning

(1996)). A related approach is the relativistic Hartree-Fock (RHF) method that is solved with

relativistic Green’s functions (Weber & Weigel 1989, Huber at al. 1997). Balberg & Gal (1997)

demonstrated that the quantitative results of field theoretical calculations can be reproduced by

an effective potential model.

The results of these works provide a wide consensus regarding the principal features of hyperon

formation in neutron star matter. This consensus is a direct consequence of incorporating

experimental data on hypernuclei (Balberg & Gal 1997). These principal features are discussed

below.

3.2. Estimates for Hyperon Formation in Neutron Stars

Hyperons can form in neutron star cores when the nucleon chemical potentials grow large

enough to compensate for the mass differences between nucleons and hyperons, while the threshold

for the appearance of the hyperons is tuned by their interactions. The general trend in recent

studies of neutron star matter is that hyperons begin to appear at a density of about ρB = 2ρ0,

and that by ρB ≈ 3ρ0 hyperons sustain a significant fraction of the total baryon population. An

example of the estimates for hyperon formation in neutron star matter, as found in many works, is

displayed in Fig. 1. The equilibrium compositions - relative particle fractions xi - are plotted as a

function of the baryon density, ρB. These compositions were calculated with case 2 of the effective

equation of state detailed in the appendix, which is similar to model δ = γ =

5

3

of Balberg & Gal

(1997). Figure 1a presents the equilibrium compositions for the “classic” case of nuclear matter,

nuclear matter. In particular, they estimate the potential depth of a Λ hyperon in nuclear matter