ELSEVIER Nuclear Physics A 615 (1997) 516-536

N U C L E A R PHYSICS A

Spin polarised nuclear matter and its application to neutron stars

V.S. U m a M a h e s w a r i a, D . N . Basu a, J .N . D e a, S . K . S a m a d d a r b a Variable Energy Cyclotron Centre, 1/AE Bidhan Nagar, Calcutta 700064, India b Saha Institute of Nuclear Physics, 1/AF, Bidhan Nagar, Calcutta 700064, India

Received 19 March 1996; revised 19 December 1996

Abstract

An equation of state (EOS) of nuclear matter with explicit inclusion of a spin-isospin dependent force is constructed from a finite range, momentum and density dependent effective interaction. This EOS is found to be in good agreement with those obtained from more sophisticated models for unpolarised nuclear matter. Introducing spin degrees of freedom, it is found that it is possible for neutron matter to undergo a ferromagnetic transition at densities realisable in the core of neutron stars. The maximum mass and the surface magnetic field of the neutron star can be fairly explained in this model. Since finding quark matter rather than hadronic matter at the core of neutron stars is a possibility, the proposed EOS is also applied to the study of hybrid stars. It is found using the bag model picture that one can in principle describe both the mass as well as the surface magnetic field of hybrid stars satisfactorily. (~) 1997 Elsevier Science B.V.

PACS: 97.60.Jd; 21.65.+f Keywords: Neutron stars; Equation of state; Ferromagnetic phase in nuclear matter; Hybrid stars

1. Introduction

Since the pioneering works of Baade and Zwicky [ 1 ] and Oppenheimer and Volkoff

[2] about half a century ago, and the identification of pulsars as rotating neutron stars

around the year 1968 [3] , several studies have been made to unravel the mystery o f the

structure o f the neutron stars. It is a well accepted fact that the density inside a neutron

star varies from the surface to the core by about 15 orders of magnitude. Understanding

the structure of such complex objects requires an accurate knowledge of the equation

of state (EOS) of neutron star matter in the different density regions. The extremely

low density domain and the subnuclear region can be well described by the EOS given

by Feynman-Met ropo l i s -Te l le r (FMT) [4] and Baym-Peth ick-Suther land (BPS) [5] ,

0375-9474/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0375-9474(97)00002-X

VS. Uma Maheswari et al./Nuclear Physics A 615 (1997) 516-536 517

respectively. The EOS of the dense nuclear matter is still riddled with some uncertainties.

A consistent EOS should describe the compressional properties [6] of nuclei near the ground state and also of the hot and dense nuclear material that is created in energetic nuclear collisions [7]. It should also answer such important questions as whether a star at the later stage of its life explodes or not [8] and how the neutron star is born. Renewed interest in the rapid cooling of neutron stars [9,10] by the direct URCA process has also an immediate bearing on the nuclear EOS.

In the non-relativistic framework, EOS of nuclear matter have been constructed with

phenomenological effective interactions or from realistic interactions with different de-

grees of sophistication [11-16]. Most of these EOS can explain the properties of neutron stars like their mass, moment of inertia etc. well within the observational limits.

However, in regard to the magnetic properties of neutron stars, till now no acceptable explanation exists for the origin of the rather large magnetic field (~1012 G) at the

surface. A recent analysis [ 17] of binary millisecond pulsars suggests that a permanent component of this magnetic field could exist, sustained by a spontaneous magnetised phase inside the neutron star. Attempts have been made to explain the presence of the

magnetic field by means of a ferromagnetic transition. In the framework of the Hartree-Fock theory, employing hard and also soft core

potentials, Pfarr [ 18] does not get such a ferromagnetic transition. A similar conclusion

is reached by Forseth and Ostgaard [ 19] who made the calculations in the lowest order

constrained variational method of Pandharipande using soft core potentials; however, with a hard core potential a transition to the ferromagnetic state was seen to occur at

--~ 30p0, where P0 is the normal nuclear matter density. In a relativistic o" + to Hartree- Fock approach, a ferromagnetic transition is also predicted by Niembro et al. [20] but at a density which is too high. However, in an improved model [21] with inclusion of rr and p mesons in addition to o- + to, a ferromagnetic transition is seen to occur

at a comparatively much lower density, p ,-~ 3.5p0, but the incompressibility of normal nuclear matter is found to be too high (,-~450 MeV). It would therefore be interesting to know whether a ferromagnetic phase transition is possible at a density realisable in neutron star matter with an EOS with firmer grounds in the experimental realities of

normal nuclear matter and finite nuclei. This is the primary motivation of this work. There is a strong possibility that at the core densities of neutron stars, there is a phase

transition from nuclear matter to quark matter [22-25]. About a decade ago Witten conjectured [26] that the strange quark matter (SQM) might be the absolute ground state of hadronic matter, i.e. the mass energy per baryon may be less than 930 MeV. If this is true, then the possibility that the pulsars are rotating strange quark stars may not be ruled out. Even if SQM is not the absolute ground state (i.e. at densities less than the hadron-quark transition density the nuclear matter is energetically favoured than SQM), one may still find hybrid stars having quark cores with nucleon envelopes. Since our understanding of the confinement/deconfinement of quarks is far from complete, all the aforementioned possibilities are only speculative. One should also keep in mind that the theoretical framework used to study the phase transition in general is phenomenological

and simplistic in nature.

518 V.S. Uma Maheswari et al./Nuclear Physics A 615 (1997) 516-536

In view of all the above, we would like to investigate in this paper the following: Firstly, whether there exists an EOS which consistently describes the nuclear matter and finite nuclear properties, as well as predict a ferromagnetic transition at a density realisable in the interior of neutron stars. Secondly, whether the same EOS which predicts a ferromagnetic transition permits a hadron-quark phase transition and thereby the formation of a hybrid star. And finally, using such an EOS, whether we can consistently explain the structural properties such as mass, moment of inertia as well as the presence of the magnetic field at the surface within the same model.

2. Theoretical f ramework

In the following, we briefly outline the procedure to obtain the nuclear equation of state in a non-relativistic framework and discuss its merits and limitations.

2.1. Equation of state

The phenomenological momentum and density dependent finite range interaction em- ployed here to obtain the equation of state is a modified version of Seyler-Blanchard interaction [27]. To treat spin-polarised isospin asymmetric nuclear matter, the interac- tion has been generalised to include the spin-isospin dependent channel explicitly. The interaction between two nucleons with separation r and relative momentum p is given by

[ p2 ]e-r /a Ueff(r,p,p) =-CTs 1 - - - ~ - d Z ( p l ( r i ) + p2(r2)) n r/a ' ( I )

where a is the range and b defines the strength of the repulsion in the momentum dependence of the interaction. The parameters d and n are measures of the strength of the density dependence, and pl and p2 are the densities at the sites of the two interacting nucleons. The subscripts ~- and s in the strength parameter C~s refer to the likeness l and the unlikeness u in the isotopic spin and spin of the two nucleons respectively; for example, Clt refers to interactions between two neutrons or protons with parallel spins, Ctu refers to that between neutrons or protons with opposite spin etc. The energy per nucleon E and the pressure P in the mean-field approximation can then be expressed as [27]

E= 1 ~.J3/207~s) -p ~s P~S [ ,/2(ms)

2 J3/2(m~)

q ' s

- - ( 1 - m $ s V ) s ) + ~ ,

+ V~s + l bz ( 1 - dZ(2p)") Vr', + V~,] .

(2)

(3)

Here, Jk(r/) are the Fermi integrals, ~ and V2s are the single-particle and the rear- rangement potentials, V)s is the coefficient of the quadratic momentum dependent term

VS. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536 519

in the potential and defines the effective mass m* s, T the temperature, and r/ is the

fugacity given by 71~s = (l~rs - ~ - V~s)/T. For the unpolarised nuclear matter ( N M ) ,

the expressions for V~ s etc are given in Ref. [27] . It is straightforward to extend these

to the case of polarised nuclear matter and are as given below:

a s

V °, = - 4 e r a 3 (1 - d2(2po) n) [CttPr, s + Ctupr,-s + Cutp-r,.,. + Cuup-r,-s]

87-r2a 3 ( . T .5 /2 j . . + b - - - ~ C u ( 2 m r , s ) 3/2(rlr, s) + Clu(2m~,_,.T)S/zJ3/2(rl~.,-~)

+Cut ( 2m*-r, sT) 5/2 J3/2 ( rl-r,s) + Cuu (2m*--r,_ s T) 5/2J3/2 ( 77-r,-s ) } ,

VIs = 47ra3b2 [Cup~-,s + Cl.pr,-s + Cutp-r.s + Cuup-r,-s] ,

V~. = 47ra3 dZn( 2po) n-I

X E [CllPT"t's' "~ CluP'rt'-s' -~- CulP-r' ,s ' "~ CuuP-r ' , - s ' ] p'r',s', ,gt ,St

mrs = + 2 . (4)

One usually defines the neutron and proton spin excess parameters (spin asymmetry)

an = (PnT -- P n l ) / P , ap = (Ppl -- Pp+)/P, (5)

where

p = pn + pp = (Pnr + Pnt) + (PpT + Ppl) (6)

is the number density. We then define the proton fraction as x = pp/p. It is related to

the isospin asymmetry parameter X as

X = (1 - 2x) = (Pn - Pp) /P . (7)

We also define the spin excess parameter as Y = an + a t, and the spin- isospin excess

parameter as Z = an - ap. One can then express the energy per nucleon E / A of the

NM at zero temperature as

E / A = Ev + E x X 2 + EYY 2 + E z Z 2 , (8)

where terms higher than those quadratic in X, Y and Z are neglected. Here Ev is the

volume energy of the symmetric nuclear matter, taken as -16 .1 MeV and Ex is the

usual symmetry ( i sospin) energy, taken to be 33.4 MeV. The quantities Er and Ez are the spin and the sp in- i sospin symmetry energies of the NM, respectively. Their values

are uncertain to some extent. The values of Er and Ez are taken in Refs. [28,29]

as Er = 31.5 MeV and Ez = 36.5 MeV. The generalised hydrodynamical model of

Uberall [30] , which gives ( E z / E x ) U2 ~- 1.1 fixes Ez for a given value of Ex. Though

520 VS. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

in microscopic calculations [31] it is seen that Er ~- Ez, its value is relatively more uncertain. We have therefore performed calculations with different values of Er and

found that the observed maximum neutron star mass and the surface magnetic field are sensitive to the choice of the value of Er. The above-mentioned observables are best

explained with Er around 15 MeV and therefore we present the following results with Er taken as 15 MeV.

In terms of the strength parameter C~s, the volume and symmetry energies are written in the form

(A a a A/(C/cu - -n - n C C flu IEx-p~/(6m) I

- B C - B C |E~,_p2F/(6m ) 1 . (9)

- B C C - B Cuu \ E z - p 2 / ( 6 m ) /

Here PF is the Fermi momentum of the one-component nuclear matter corresponding to the density p of the polarised NM, A = ~( f l - iS), B = t~(fl - 20t5/9), C = c~(fl - 10t5/9), t~ = 8zt2a3p3/(3h3), fl = 1 - d2(2p) n and t5 = 6p2F/(5b2). For a fixed

value of n, the parameters Crs, a, b and d are then determined by reproducing Ev, Ex, Er, Ez, the saturation density of normal nuclear matter (P0 = 0.1533 fm-3) , the surface

energy coefficient (as = 18.0 MeV), and the energy dependence of the real part of the nucleon-nucleus optical potential. The parameter n is determined by reproducing the breathing-mode energies [32].

The parameters of the interaction (with Ex = 33.4 MeV, Er = 15.0 MeV and Ez = 36.5 MeV) are listed below:

Cll = 30.2 MeV, a = 0.625 fm, Ctu = 566.8 MeV, b = 927.5 MeV/c, Cut = 1314.9 MeV, d = 0.879 fm 3n/2,

Cuu = 440.8 MeV, n = 1/6.

With the above value of the parameter n, the incompressibility of symmetric nuclear matter is K = 240 MeV. It may be mentioned that with different choices of Ex, Er and Ez the values of the strength parameters change; all the other parameters including K remain unaltered.

2.2. Merits and limitations

It has been tested that the above interaction reproduces quite well the ground state binding energies, root mean square charge radii, charge distributions and giant monopole resonance energies for a host of even-even nuclei ranging from 160 to very heavy systems. Interactions of this type have been used before with great success by Myers and Swiatecki [33] in the context of nuclear mass formula. We have also seen that for symmetric nuclear matter, our results agree extremely well with those calculated in a variational approach by Friedman and Pandharipande (FP) [ 12] with UI4 "1-TNI

VS. Uma Maheswari et al./Nuclear Physics A 615 (1997) 516-536 521

160

120

~- a0

u.l

/-.0

~n:O.O

B - J

1/-.÷ UVII

-p

1 0 0.2 0./- 0.6 0.8

p( fm -3 ]

Fig. 1. The energy per particle for pure neutron matter as a function of density is plotted for polarised energy minimised neutron matter and for unpolarised neutron matter. The calculated results of Fried- man-Pandharipande [12], Wiringa et al. with UVI4+UVII interaction [16] and with Bethe-Johnson po- tential [34] are also shown.

interaction in the density range ½P0 ~< P ~< 2p0. However, for unpolarised pure neutron matter, the energies calculated with our interaction are somewhat higher compared to the FP energies, particularly at higher densities. The entropy per particle for neutron

matter calculated with our interaction at different temperatures agrees extremely well

with the corresponding FP results. In Fig. 1, the energy per particle for neutron matter is displayed as a function of density at zero temperature. For comparison, the FP energies [12] and those obtained with the Bethe-Johnson (BJ) potential [34] in a sophisticated correlated basis function approach are also displayed. The BJ curve is very close to ours for neutron matter. In the figure, we have also displayed the energy per nucleon for neutron matter as obtained by Wiringa et al. [ 16] in a variational approach with U V I 4 + U V I I interaction. It is satisfying to note that our results are very

similar to those obtained from this recent sophisticated calculation. This good agreement between our calculations and those reported in Refs. [ 12,16,34] suggests that the present interaction can be extrapolated with some confidence to neutron matter or to nuclear matter with large isospin asymmetry at high densities. It can also be mentioned that such an interaction satisfies the Landau-Migdal stability criteria [ 35].

All the well-known non-relativistic nuclear equations of state suffer from lack of causality at high densities. The velocity of sound in nuclear matter then becomes su- perluminal. For unpolarised nuclear matter the effective interaction used by us is no

522 V.S. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

1,0

0,8

0,6

~- O,Z

0.,2

. . . . I ' ' ' y l . . . . I . . . .

0.0 . . . . I , , , , I . . . . 1 . . . .

0 5 10 15 20

P/Po Fig. 2. Velocity of sound us obtained in units of c for t-equilibrated neutron star matter taking or, = 0.0 and 1.0 is plotted as a function of the density ratio P/Po, where/90 = 0.1533 fm -3 is the symmetric nuclear matter saturation density. For EMNM the velocity of sound is discontinuous at A and B (see text).

exception. In Fig. 2, we have plotted the velocity of sound in units of c as a function of

the ratio p/po in the case of t-equil ibrated neutron matter taking an = 0 and 1. It can

be seen that for fully spin polarised state, the EOS becomes softer and the velocity of

sound us does not become acausal even at density p/Po as high as 25.

3. Ferromagnetic phase transition

It would be interesting to investigate whether the nuclear EOS discussed in the previous section, in addition to the consistent description of the nuclear matter and

finite nuclear properties predicts a ferromagnetic transition at densities meaningful in the context of neutron stars.

It has been conjectured [36,37] that in the neutron star matter, in contrast to nuclei

where the neutrons generally pair up to spin J = 0 in their ground states, the neutrons

may pair up to spin J = 1 at higher nuclear densities, thus leading to a ferromagnetic transition. To investigate this aspect, we calculate the energy per particle for the neutron matter minimising it with respect to spin polarisation or, and compare this with that calculated for unpolarised neutron matter in Fig. 1. We find that above p ~ 4.0p0 the energy of polarised matter is lower compared to that for unpolarised neutron matter. This

reflects that the neutrons that pair up to J = 0 at lower densities undergo a transition to

a spin polarised configuration as density builds up. In other words, the system prefers a

V.S. Uma Maheswari et al./Nuclear Physics A 615 (1997) 516-536 523

1.0

O.B

,_os

0./-.

0.2

oo i 0.9

f

J , I I I 0.6 0.7 0.B

p( fm -3)

Fig. 3. The value of spin polarisation parameter an obtained from minimisation of energy for pure neutron matter (full line) and fl-equilibrated neutron star matter (dashed line).

ferromagnetic state for p > 4.0p0. We denote this transition density as RFM. In Fig. 3 the

value of an obtained from minimisation of energy of pure neutron matter (solid line)

and that of fl-equilibrated neutron star matter (dash line) are displayed as a function of

density. For the latter case the ferromagnetic transition is seen to be sharper and takes

place at a somewhat higher density (PFM ~ 4.5p0). It is further observed that at the

minimum energies the proton polarization parameter a t, is zero for the fl-equilibrated

matter at all the densities studied.

Though the energy density increases, the pressure for the energy minimized spin

polarized neutron matter shows a structure around the ferromagnetic density as seen

in the EOS of van der Waals systems for liquid-gas phase transition. This is more pronounced for fl-equilibrated matter as the electron pressure decreases due to rapid

reduction o f proton fraction with density. A physically meaningful variation of pressure

with density is obtained using a Maxwell type construction in the density domain of

~4p0 to ~5.5p0 (which ensures coexistence of unpolarised and polarised matter in

chemical equilibrium). The velocity of sound is thus discontinuous in the region A to B as shown in Fig. 2.

The behaviour o f the magnetic susceptibility g of neutron matter as a function of

density also portrays the occurrence of ferromagnetic phase transition. In general, the magnetic susceptibility is defined [ 38 ] as g = a M ~all , where H is the applied magnetic

field and .A,4 = txn (NT - N I ) / V is the total magnetisation per unit volume with/zn being

the magnetic moment o f a neutron. Using the definition of a , [Eq. (5 ) ] , we rewrite

M as .A// = ~nanpn.

524 VS. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

We need to determine the optimum value of an using the energy minimisation criteria

O( E~l(p,ot~nan) /N) ~,,--~,, = O. (10)

The total energy EH/N per particle of a system of N number of neutrons in the presence of an external weak magnetic field H is

Eft(p, o~n) /N = E(p, an) /N - (l~nHan) . ( 11 )

Expanding the energy E(p, an) in powers of an up to O(a2), we get

02( E(p, an) /N) ,~,,=0 la2e2. 12) E(p, an) /N= E(p, ctn = 0 ) / N + la2n 0a] = e0 + (

Because the energy E(p, an) is symmetric in an, all the odd derivatives in the expansion

of E(p, an) vanish. (It may be said here that only in the calculation of X, we have expanded E(p, an) in powers of an, otherwise we have calculated it numerically.) It can be seen that a ferromagnetic phase becomes energetically favourable when E(p, an) < E(p, an = 0), which for small values of an also means that e2 is negative in the

o is determined by minimising the ferromagnetic phase. Then, the optimum value a n

energy given by Eq. (11) (valid only in the paramagnetic phase [38] ) as an ° = ~nH/e2. Now, we can determine X and it is given as

0 0 (13) e2

Using the effective interaction given in Eq. (1), we get

O2( E(p'an) /N) ,~,,---o e2 = c~ce 2

=--27ra3pn [ a l ( C l l - C l u ) - 2-~22/3A2(2Cll-Clu)] + 22/3p2 3m ' (14)

where Al = 1 -- d2(2pn) n and A2 = pnp3/b 3. In the limit of no interaction, e2 is simply given by the kinetic term alone, i.e. e~ ree = 22/3p2/(3m). It is then straightforward

to calculate the ratio Xfree/X, where Xfree is the magnetic susceptibility of the non- interacting neutron gas. The onset of a ferromagnetic transition is depicted by the vanishing of the ratio Xfree/X. Further, the effect of the nuclear matter incompressibility K on the ferromagnetic transition density is studied and the results are shown in Fig. 4. As the value of K is increased from 240 to 304 MeV (by increasing the density exponent n of the effective interaction given by Eq. ( l ) from 1/6 to 2 /3) , the density at which the transition takes place decreases from ~ 4.0po to ,~ 3.7p0. It is thus seen that the effective interaction given in Eq. (1) predicts a ferromagnetic phase transition at a density p ,~ 4.0p0.

In previous calculations in the non-relativistic formalism using realistic interactions, one usually does not find such a ferromagnetic transition [ 18,19] or if there is such

VS. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536 525

2.0

i/ \"~ 1.s ;, K =3o

N \,,N,x,

\\\ 0.0

0 1 2 3

Fig. 4. Magnetic susceptibility X of pure neutron matter calculated for two values of nuclear matter in- compressibility K is shown as a function of the density p, where ,l'tiee is the magnetic susceptibility of non-interacting neutron gas.

a possibil i ty, it occurs at densities [ 19] not realisable in the context of neutron stars.

In a relativistic framework, one finds that the simple o - + o) model [20] does not

suggest any such transition. However, with an improved model [21] including other

mesons l ike p and ~" in addit ion to or ÷ (o, one finds a transition at about p ~ 3.5p0.

But, the incompressibi l i ty of the normal nuclear matter obtained in this model is too

high (,--,450 MeV) . In contrast, our nuclear interaction that predicts a ferromagnetic

transition at a similar density yields a value of incompressibil i ty that is close to the one

well-accepted [6] .

4. Structural properties of neutron stars

In this section, we explore the various static properties of neutron stars such as proton

fraction, mass, size and moment of inertia using the proposed equation of state. We

further study the influence o f spin polarisation on these observables.

4.1. Beta-equilibrium proton fraction

In recent years, attention has been drawn to the direct URCA process in neutron stars

which may be the primary mechanism for its rapid cooling. This can, however, occur only

when the beta-equil ibr ium proton fraction x in the star is ~>0.11, where only electrons

5 2 6 V.S. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

0.10 X

Z o

o t U-

z 0 . 5 o

0 n."

I1

0.0

C( n

- - 0 . 0 . . . . EMNM

ii

, I I ,

0.5 1.0 1.5

,P( fm -3 )

Fig. 5. Beta-equilibrium proton fraction x as a function of density of neutron star matter. Results are shown for both an = 0 (full line) and for EMNM (dashed line) with inclusion of e - + / z - and e - alone.

are considered, and ~>0.148, if both electrons and muons are considered. It would be interesting to know whether spin polarisation favours or disfavours direct URCA process. In our study, the lepton energy per particle EL(p, x) is given by the relativistic, ideal

Fermi-gas expression [39]; in addition to e - , / x - are also considered as and when they a

are energetically favoured. At beta-equilibrium, one has ~x (E (p , x )+EL(X) ) = O, where E(p, x) is the baryonic energy per particle including the rest masses. In Fig. 5,

the beta-equilibrium proton fraction thus obtained in the neutron star matter is displayed as a function of the baryon density invoking the condition of charge neutrality. The full curves correspond to unpolarised matter. It is seen that the proton fraction x shows a peaked structure against density. When only e - are considered, x increases as density increases, reaches a maximum value of about 0.085 at p ~_ 3po, and then decreases to very low values at higher densities. With the inclusion o f / z - , the structure of the curve remains almost the same; however, it lies higher than that of the former case, at all densities. The peak value is then about 0.11. The dashed lines in Fig. 5 correspond

to proton fraction for B-equilibrated polarised star matter when energy minimisation with respect to neutron polarisation o~n and proton polarisation tep (subsequently this is referred to as EMNM) is taken into consideration. The proton fraction is seen to fall very sharply beyond the ferromagnetic transition density. It may be noted that the present EOS in use does not favour direct URCA process, since the proton fraction is always below the critical value. Introduction of spin polarisation does not help direct URCA process. It may be remarked that various calculations give different conclusions [ 10] regarding the direct URCA process. It may also be mentioned that inclusion of exotic processes like pion condensation or kaon condensation in dense neutron star matter may enhance the proton fraction thereby favouring direct URCA process [40], but occurrence of such exotic phenomena is still very much unsettled [9].

V.S. Urea Maheswari et al . /Nuclear Physics A 615 (1997) 516-536 527

2.0

1.5 O

"~2.0

2.5

EMNM . . . . .

0,0 I I i 0 5 I0 15 20

Fig. 6. The neutron star mass is plotted as a function of central density for unpolarised matter and for EMNM.

4.2. Mass and size of neutron stars

We now determine the structure of neutron stars using a composite EOS, i.e. FMT, BPS, Baym-Bethe-Pethick (BBP) [ 11 ] and the present interaction with progressively

increasing densities. Then, the total mass and the size of the neutron star can be obtained by solving the general relativistic Tolman-Oppenheimer-Volkoff (TOV) equation

dP(r______~) = _ G [e ( r ) + P ( r ) ] [m(r)c 2 + 47rr3p(r)] (15)

d r c a r z [1 - 2Gm(r) /rc 2] '

where r

m( r)c 2 = / e( rl)d3r '. (16) 1 1

o

The quantities e(r) and P(r) are the energy density and pressure at a radial distance r

from the centre, and are given by the equation of state. The mass of the star contained

within a distance r is given by m(r) . The size of the star is determined by the boundary condition P(R) = 0 and the total mass M of the star integrated up to the surface R is given by M = re(R). The single integration constant needed to solve the TOV equation

is Pc, the pressure at the center of the star calculated at a given central density Pc. The mass functions of the star thus obtained as a function of its central density are

shown in Fig. 6 for unpolarised star matter and EMNM. The radii, central densities and surface redshifts corresponding to the maximum m a s s Mmax configuration are tabulated in Table 1 for these two cases, The surface redshift zs is defined [ 11 ] as

[ 2aMl-1; Zs = 1 Rc 2 j - 1 . (17)

528 V.S. Urea Maheswari et aL /Nuclear Physics A 615 (1997) 516-536

Table 1 Values of the mass Mrnax, size R, central density pc and surface redshifi Zs obtained in the case of poladsed and unpolarised neutron stars are shown. The tabulated values correspond to the maximum mass configuration

an Mmax/M® R (km) pc/po zs

0.0 2.03 10.24 7.63 0.555 EMNM 1.69 1 ! .60 5.75 0.326

1.0

4o ', ~ 0.0

\ E

0.2 ~,

o.o . ' . ' \ 2.0 4.0 6.0 8.0

F(r) / f (o)

Fig. 7. The integrated mass upto radius r is plotted as a function of the density at radius r for unpolarised matter and for EMNM.

Values o f Mmax/M® are also given in the same table. It can be seen that the maxi-

mum mass and the corresponding central density and surface redshift Zs decrease with

inclusion of polarisation. However, it has been observed that for the same mass of the

star, the central densi ty is considerably larger and the radius smaller with inclusion o f

polarisation. This is because of the fact that the spin polarised neutron star matter is

more compressible than the unpolarised one. The measured mass [41] of 4U0900-40,

( 1.85 -4-0.3)M e , poss ibly provides the lower limit of the maximum mass of the neutron

star. Our calculated values are well within this limit.

We have also studied the sensitivity o f the mass distribution m(r ) to spin polarisation,

where m ( r ) denotes the total mass contained within a given radial distance r from the

centre. Fig. 7 displays m(r) /Mmax as a function of the density p ( r ) at that point r,

where Mmax corresponds to the maximum mass configuration for the relevant case. The

zero o f the abscissa refers to the surface of the neutron star; the points where the mass

functions m ( r ) meet the abscissa refer to the centre of the star.

v.s. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

4.3. Moment of inertia of neutron stars

529

The moment of inertia of neutron stars is calculated by assuming the star to be rotating slowly with an uniform angular velocity /2 [42]. The angular velocity 6J(r) of a point in the star measured with respect to the angular velocity of the local inertial frame is determined by the equation

where

2Gm(r).) 1/2 j = e -~b(r) 1 rc 2 j (19)

The function q~(r) is constrained by the condition

e4°(r)tz(r ) = constant = / z (R) ~/1 2GM Rc 2 , (20)

where the chemical potential/x(r) is defined as

e(r) + P(r) ~ ( r ) - (21)

p(r)

Using these relations, Eq. (18) can be solved subject to the boundary conditions that 6~(r) is regular as r ,0 , and &(r) ) g'/as r ) o~. Then moment of inertia of the star can be calculated using the definition I = J/12, where the total angular momentum

J is given as

c 2 d& J = - ~ R4 (--~r ) r=R . (22)

Values of I thus obtained for unpolarised neutron star matter and for EMNM are plotted as a function of the central density in Fig. 8. It can be seen that with the inclusion of

polarisation, the values of 1 are reduced when the central densities are larger than the ferromagnetic transition density.

We have calculated the important structural properties of neutron stars and studied their dependence on the spin polarisation. It is found that the properties obtained here are well within the observational limits. Motivated by this success in explaining the structural properties of neutron stars, it would be interesting to see how far one can account for the surface magnetic field. Our present study suggests that the neutron star matter at densities p > 4.5p0 is ferromagnetic. The magnetic field at the surface of the neutron star due to the presence of the ferromagnetic core can be estimated using the relation [39] H ~ O t n t . t n N / R 3, where N is the number of neutrons in the ferromagnetic phase and R is the star radius. We have from our calculations N ~ 2 × 1054, a~n ~'~ 1

and R ----- 11.6 km for the maximum mass configuration. With the I /zn I = 1.91 nuclear magneton, we then obtain H ~ 1013 G. The surface magnetic field is thus somewhat

1 d [r4jdgo ] 4dj r4dr ~ + r~rr 6~ =0 , (18)

530 V.S. Uma Maheswari et aL /Nuclear Physics A 615 (1997) 516-536

' ' ' ' i ' ' ' ' I i , , i

1.0

O.B

0.6

I I c

~' 0.~

0.2

i i i ! I i i J , I , . , ,

0 5 I0 15

p/e0 Fig. 8. Moment of inertia obtained for unpolarised matter and for EMNM is shown as a function of density pdpo.

overestimated compared to the observed value of ~ 1012 G [39]. The ferromagnetic transition density increases with reduction of Ey and thus the surface magnetic field is very sensitive to the choice of Er. For example if Er is chosen as 31.4 MeV as in Refs. [28,29] the value of surface magnetic field increases to a value as high as

1016 G. One could thus arrive at the observed value of the surface magnetic field by fine-tuning Er.

5. Hybrid stars

At the very high densities in the core of neutron stars a strong possibility of a phase transition from hadronic matter to quark matter cannot be ruled out. Hybrid stars, i.e. quark cores with nucleon envelopes, may then become a reality. To understand the structural properties of such stars, in the following we construct fl-equilibrated, electrically neutral quark matter equation of state.

The equation of state of a three-flavour quark matter consisting u, d, and s quarks is obtained here using the phenomenological MIT bag model [43]. The total kinetic energy density of a system of non-interacting, relativistic quarks of flavour r and mass mr is given as

3 (mrc2)4 [ V/ ( ~ ) ] e r - - 8 ~ 2 (hC)---------- T - Xr l + x 2 r ( l + 2 x 2 ) - - l n Xr+ (23)

V.S. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536 531

where x, = p~F/(m~c), and p~- is the Fermi momentum and is related to the quark

number density p, of a given flavour as p,I- = h('/T2pr) 1/3. The densities pertaining to the three flavours can be expressed in terms of the total quark number density pq and the asymmetry parameters t3. d and 8us as

p . = ( p q / 3 ) [1 - - 6ud - - t3us] ,

P d = ( p q / 3 ) [1 + 26ud - - t3us ] ,

Ps = (pq /3) [ 1 -- ¢~ud ~- 26 .s] , (24)

where 6.d = (Pd -- Pu)/Pq, 6us = (Ps -- Pu)/Pq, a n d pq = Pu + Pd q- Ps. Similarly, t h e

energy density eL pertaining to a system of relativistic non-interacting electron gas can be calculated [39].

We then determine the equilibrium composition of the quark matter subject to the fl-equilibrium condition

]'Zd -- /U'u = ]-'£e a n d /Zd = # s , (25)

and the charge neutrality condition,

1 Pe = -~ (2pu - Pd - Ps) . (26)

Using Eq. (24) one obtains Pe = --(pq/3)(¢~ud + 6us). Similarly, we can express the chemical potentials/z., /Zd, and /Zs in terms of the three quantities pq, ¢~ud, and &.~ as follows:

I'~u ~,,apu I -- apq

= ( O e q ~ p,,p, . = O~eq -1"- _ _ _ _

l + ~ u e OEq 1 + ~us Oeq

Pq 06ud pq 06us"

1 - ~3.d OEq 6us Oeq

pq OSud pq 06us'

6ud Oeq 1 --6us Oeq

Pq O~,d -}- - - - - ' Pq a~us (27)

where eq = ~-~ e~ + B is the total quark energy density and B is the bag parameter. Using these expressions, the fl-equilibrium conditions can be rewritten as

Oeq Oeq 2 ~geq 1 Oeq 06.,l 06.s = O, - - ~ + - - =/xe, (28) Pq O~ud Pq 06us

= /p2 C2q_ m2eC4. Thus, for a given baryon density Pb where /z~ V F,e = pq/3, the three

quantities Pe, 6.d, and Sus are fixed by Eqs. (26) and (28). Subsequently, the equation of state is completely described by the total energy density eQM and the pressure PQM of the system calculated for a given Pb using the definitions

eQM = ~ er(Pq, ¢3ud, ~us) q- eL(Pq, 6ud, 6us) + B, T

t~EQM PQM = Pq Opq EQM. (29)

532 V.S. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

1.3

1.2

1.0 / / O.M

/ 1 - - - - EMNM

0 9 . i i i i i i i i r i

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.L 1.6

Pb (f m-31

Fig. 9. The total energy per baryon of quark matter (QM) calculated using the bag model picture is compared with the energies per baryon of EMNM with B 1/4 = 165 MeV.

It may be mentioned that in our present study we have omitted the lowest order quark- quark interaction terms as it can be effectively absorbed into the bag constant [44]. The masses of the quarks are taken as mu = 5 MeV, ma = 10 MeV, and ms = 250 MeV.

To know whether there is a phase transition from the nuclear matter to quark matter,

we compare the total energies per baryon obtained in the two phases. The EOS corre-

sponding to NM is given by Eqs. (2), (3). For B-equilibrated NM, these equations are supplemented with the leptonic contributions. The EOS of QM is given by Eq. (29). In

Figs. 9 and 10 we show the energy per baryon as a function of the baryon density for two values of bag constant, B 1/4 = 165 MeV and B 1/4 = 185 MeV, respectively. These

are compared with the curves obtained for EMNM in the same figures. The baryon

density at which the EMNM curve intersects with the QM one is denoted by RHQ. It can be seen in Fig. 9 that for B 1/4 = 165 MeV the unpolarised NM is energetically

favoured upto a density pb -~ 0.55 fm -3. As Pb is increased further, the quark matter

is found to be the lowest energy state. Therefore, there is no region of spin polarised NM in this particular case. On the other hand, for B 1/4 = 185 MeV it can be seen that

for densities within the values PFM ~ 0.70 fm -3 and PHQ ~ 1.50 fm -3, the polarised NM is the state of lowest energy. Thus, in this case (Fig. 10), one has a spin polarised region sandwiched between a quark matter core (pb/> PHQ) and an unpolarised nuclear matter envelope (Pb ~< PV'M). Futher, we also study the acceptable bag parameter domain in which we might have a spin polarised region for a few values of ms. In Fig. 11, we display PHQ as a function of bag parameter for ms = 150, 200 and 250 MeV. The horizontal dashed line represents PFM. Only for regions above this line one can have spin polarised matter sandwiched between an unpolarised NM envelope and a QM core. It can be seen that for allowed values of B 1/4 and ms, both the presence and non-presence

VS. Uma Maheswari et al./Nuclear Physics A 615 (1997) 516-536 5 3 3

A >~

<

:E

i , f i , i i i

1./. B T/4 : 185 MeV . .

1.3

1.2

1.1 / " / ~ (iN . . . . EMNM /

s

1.0 /" I /

I I I I I I I I I

0"90.0 0.2 0./. 0.6 0.8 1.0 1.2 1./~ 1.6 1.6 2.0

Pb (fro-3)

Fig. 10. Same as Fig. 9, but for B 1/4 = 185 MeV.

1.0

0.8

?" 0.6 E o

0.z.

0.2

:." . "

- . " ' ...' ...'

..-" °.. ..---== - - _ .,:: . . . . .-.~

..-" . - - . o . " . . ' " • " . . ' "

... '" . . . . " . . . . ' "

. t I" '" . - ' " I . - ' "

. . . "

. " . . ' " . . . "

...'" . ..... .~ . - " I - - - - , M s = 1 5 0 M e V *'" ....... - ......... """ M, =200 Id eV ~'.~ ...... ~ * Ms =250 M eV

I I I I I I I

00150 160 170 180 190 BV~CMeV)

Fig. 11. The hadron quark transition density PHQ is plotted as a function of the bag parameter B for three values of strange quark mass, ms = 150, 200 and 250 MeV. The horizontal dashed line represents the ferromagnetic transition density PFM.

of a polarised region are possible. Thus, it is clear from the above discussions that the allowed range of B and m, values is not able to decide upon the presence/non-presence of a spin polarised region in the hybrid star. It would be interesting to know whether the observational limits on the mass of pulsars would help in finding an answer,

In view of this, we explore the structural properties of the hybrid stars choosing three particular values of the bag parameter B while keeping the strange quark mass ms

534 VS. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536

Table 2 Values of the mass Mmax, size R, central density pc and surface redshift Zs obtained in the case of hybrid stars using three values of bag parameter B are shown. The strange quark mass ms --- 250 MeV. The tabulated values correspond to the maximum mass configuration

B j/4 (MeV) Mmax/M® R (km) Pc/Po Zs

165 1.54 10.54 7.76 0.327 175 1.68 11.58 5.87 0.324 185 1.69 11.60 5.75 0.326

B 1/4 = 165 MeV 175 MeV 185 MeV

~ U N M ~ ~ p N ~ [ ~ Q . M Fig. 12. Schematic representation of the three configurations considered in our study of hybrid stars with B 1/4 = 165, 175, and 185 MeV and ms = 250 MeV.

fixed at 250 MeV. The three different configurations corresponding to the three sets of

MIT bag model parameters are shown schematically in Fig. 12 for the maximum mass

configuration. It is clear from the figure that for a fixed value of ms, one can fine-tune

the value o f the bag parameter B such that the ferromagnetic shell (dot ted region) is of

the right thickness which reproduces the surface magnetic field ,~ 10 j2 G, the observed

value. The mass, size, central density and surface redshift were then calculated for the

three values of B. The results obtained pertaining to the maximum mass configuration

are given in Table 2. It is seen that the observables corresponding to EMNM in Table 1

are identical with those presented in the last row of Table 2. This is because of the fact

that for B 1/4 = 185 MeV, the neutron star is purely hadronic for the maximum mass

configuration. From Table 2 one also sees that maximum mass of the neutron star are

well within acceptable limits and therefore from the present study one cannot decide

whether there would be spin polarised matter within the hybrid star or not. However, we

would l ike to mention that using the proposed equation of state and the MIT bag model

with acceptable parameters (B 1/4 ,~ 175 MeV, ms = 250 MeV) , one can in principle

describe both the structural propert ies as well as the surface magnetic field satisfactorily.

6. Summary

To summarise, we have constructed a nuclear equation of state from a finite range

momentum and density dependent interaction and then applied it to investigate some

propert ies o f spin polarised nuclear matter with particular reference to the neutron star

vs. Urea Maheswari et al./Nuclear Physics A 615 (1997) 516-536 535

matter. The parameters of the interaction have a firm basis in the well known properties

of nuclear matter and of finite nuclei. Extrapolating this interaction to neutron matter

and to higher densities, it is found that the present equation of state agrees well with

those obtained from more sophisticated calculations.

Introducing spin degrees of freedom, it is seen that at density p ~ 4.0p0, the pure

neutron matter undergoes a ferromagnetic phase transition. This aspect is demonstrated

by studying the density behaviour of the total energy (Fig. 1 ) and the magnetic suscep-

tibility (Fig. 4). To the best of our knowledge, this is the first non-relativistic calculation

that gives a ferromagnetic transition at a density well realisable in the neutron star core.

The proposed equation of state is then applied to the investigation of the structure of

neutron stars. With inclusion of polarisation, the maximum mass and the corresponding

central density decreases. However, for the same neutron star mass, the central density

increases with inclusion of polarisation as the EOS becomes softer. The maximum mass

of stars obtained from calculations with inclusion of polarisation are found to be well

in agreement with the observations. The observed surface magnetic field can also be

explained by fine-tuning the parameter Er.

There is a good possibility that one finds quark matter rather than spin polarised

nuclear matter at the core of stars. We therefore investigated this plausibility using the

MIT bag model. We found that using the proposed EOS and the MIT bag model, one

can in principle obtain the maximum mass and the surface magnetic field of hybrid stars

well within the acceptable limits.

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