Rapid Cooling of Magnetized Neutron Stars

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Rapid co oling of magnetized neutron stars

1 2;3 2 1

Debades Bandyopadhyay, Somenath Chakrabarty, Prantick Dey, and Subrata Pal

Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India

Department of Physics, University of Kalyani, Kalyani 741235, India

IUCAA, P.B. 4, Ganeshkhind, Pune 411007, India

Abstract

The neutrino emissivities resulting from direct URCA pro cesses in neutron

stars are calculated in a relativistic Dirac-Hartree approach in presence of a

magnetic eld. In a quark or a hyp eron matter environment, the emissivity

due to nucleon direct URCA pro cesses is suppressed relative to that from

pure nuclear matter. In all the cases studied, the magnetic eld enhances

emissivity compared to the eld-free cases.

PACS numb ers: 26.60.+c, 21.65.+f, 12.39.Ba, 97.60.Jd

Typ eset using REVT X

E 1

Neutron stars are b orn in the aftermath of sup ernova explosions with interior temp era-

tures T 10 K, but co ol rapidly in a few seconds by predominant neutrino emission [1]

10 5 6

to T < 10 K. Neutrino co oling then dominates and lasts for t 10 10 yr and sub-

sequently photon emission takes over when T 10 K. Since the long term co oling of the

8 10

young neutron stars (T 10 10 K) pro ceeds via emission of neutrinos primarily from

matter at supranuclear densities within the core, the study of the co oling of neutron stars

by examination of neutrino emissivities may provide considerable insight into their interior

structure and comp osition.

For a long time the dominant neutrino co oling mechanism has b een the so-called standard

mo del based on the mo di ed URCA pro cesses [2,3]

(n; p)+n ! (n; p)+ p + e + ;

(n; p)+p+e ! (n; p)+ n + : (1)

The ROSAT detection [4] of thermal emission from neutron stars indicates the necessity of

faster co oling mechanism in some young neutron stars, in particular the Vela pulsar. Faster

neutrino emission than the standard mo del was prop osed by invoking pion [3,5] or kaon [6]

condensates whichhave neutrino emissivities comparable to that from the -decay of quarks

[7] in quark matter (consisting of u, d and s quarks)

d ! u + e + ; u + e ! d + : (2)

e e

A similar relation for s-quark -decaymay o ccur and is obtained by replacing d by s quark

in Eq. (2). The most p owerful energy losses, exp ected to date, are pro duced by the so-called

direct URCA mechanism involving nucleons [8]

n ! p + e + ; p + e ! n + : (3)

e e

The threshold density of this pro cess is however considerably larger than that of the mo di ed

URCA pro cess.

Observations of pulsars predict large surface magnetic eld of B 10 G [9]. In the

core the eld may b e considerably ampli ed due to ux conservation from the original weak

eld of the progenitor during its core collapse. In fact, the scalar virial theorem [10] predicts

large interior eld B 10 G or more [11], and these elds are frozen in the highly

conducting core. It has b een demonstrated [11] that when the eld B is comparable to

(c)

or ab ove a critical eld B , the energy of a charged particle changes signi cantly in the

quantum limit; the quantum e ects are most pronounced when the particle moves in the

lowest Landau level. The phase space mo di cations stemming from the strong magnetic

eld in the core are exp ected to in uence the neutrino emission rate from young neutron

stars.

In this communication weevaluate the neutrino emissivity for the nucleon direct URCA

pro cess of Eq. (3) in presence of a magnetic eld B , and demonstrate that it would lead to

more rapid co oling in the core. (A straightforward extension of the emissivity for nucleons

into the quark sector may also be obtained in a magnetic eld.) For this purp ose, we

consider a npe matter in -equilibrium within a relativistic Dirac-Hartree approach in the

linear -! - mo del [12]. 2

At the neutron star core at temp eratures well b elow the typical Fermi temp erature of

T 10 K, the nucleons and electrons participating in neutrino pro ducing pro cesses are all

degenerate (the and are free) and have their momenta close to the Fermi momenta p ,

e e F

where i = n; p; e. Since neutrino and antineutrino momenta are kT =c p , the nucleon

direct URCA pro cess is allowed by the momentum conservation when p + p p . Since

F F F

p e n

matter is very close to -equilibrium, the chemical p otentials of the constituents satisfy the

condition = + . (Henceforth we set h = c = k = 1.)

n p e

Employing the Weinb erg-Salam theory for weak interactions, the interaction Lagrangian

density for the charged current reaction (3) may b e expressed as L =(G = 2) cos l j ,

F c

int

49 3

where G ' 1:435 10 erg cm is the Fermi weak coupling constant and the Cabibb o

F c

angle. The lepton and nucleon charged weak currents are resp ectively, l = (1 )

5 2

and j = (g g ) . Here, and in other formulae to follow, the indices i =14

V A 5 1

refer to the n, , p and e, resp ectively. The vector and axial-vector coupling constants are

g = 1 and g =1:226.

V A

The emissivity due to the antineutrino emission pro cess in presence of a uniform magnetic

eld B along z-axis when b oth the electrons and protons are Landau quantized is given by

Z Z Z Z Z Z

3 3

qB L =2 p qB L =2 p

m x m x

F F

p e

Vd p Vd p L dp L dp L dp L dp

1 2 y 3y y 4y z 3z z 4z

" (B )=2

3 3

(2 ) (2 ) 2 2 2 2

qB L =2 qB L =2 p p

m x m x

F F

p e

max

X X

E W f (p )[1 f (p )][1 f (p )] ; (4)

2 fi 1 3 4

where and are resp ectively the maximum number of Landau levels p opulated for

max

protons and electrons. The prefactor 2 takes into account the neutron spin degeneracy. The

p (E ; p ) are the 4-momenta and E the antineutrino energy. The functions f (E ) denote

i i i 2 i

the Fermi-Dirac functions for the ith particle. The transition rate per unit volume due to

the antineutrino emission pro cess may be derived from Fermi's golden rule and is given by

W = hjM j i=(tV ). Here t represents time and V = L L L the normalization volume.

fi fi x y z

jM j is the squared matrix element and the symbol hi denotes an averaging over initial

spins and a sum over nal spins. The matrix element for the V A interaction is given by

M = (X ) (g g ) (X ) (X ) (1 ) (X ) : (5) d X

fi V A 5 3 5 4

1 2

In presence of a uniform magnetic eld B , the normalized proton wave function is

(X ) = 1= L L exp (iE t + ip y + ip z ) f (x), where f (x) is the 4-

3 y z 3 3y 3z p ;p p ;p

3y 3z 3y 3z

comp onent spinor solution [11]. The form of the spinor in a magnetic eld (see Ref. [11])

restricts the analytical evaluation of the neutrino emissivity to elds strong enough so as to

p opulate only the ground state for electrons and protons, i.e. = =0. The only p ositive

energy spinor for protons in the chiral representation is then [11,13]

0 1

E + p

B C

f (x)=N B I C (x); (6)

=0 =0 ;p

p ;p

3y 3z

@ A

0 3

1=2 H 2

) is the e ective where N = 1= 2E (E + p ), and E = E U = (p + m

=0 3z 3

3 3

3 0;p 3z

relativistic Hartree energy. The function I (x) is similar in form as in Ref. [11]. The

=0;p

nucleon e ective and rest masses are resp ectively, m and m = m = m = 939 MeV. In

n p

presence of the magnetic eld, the wave functions for free electrons (X ) have the same

form as those for protons, but with m and E for protons replaced by the bare mass m and

3 e

kinetic energy for electrons, resp ectively. The neutrons and neutrinos/antineutrinos b eing

una ected by B , have plane wave functions.

Using these wave functions, it is straightforward to calculate the transition rate p er unit

volume and is given by

" #

2 2 2

G 1 (p p ) +(p +p )

1x 2x 3y 4y

W = exp

E E E E V L L 2qB

2 4 y z m

1 3

h i

2 2 2 2 2

(g + g ) (p p )(p p )+(g g ) (p p )(p p ) (g g )m (p p )

V A 1 2 3 4 V A 1 4 3 2 4 2

V A

(E E E E ) (p p p p ) (p p p p ): (7)

1 2 3 4 1y 2y 3y 4y 1z 2z 3z 4z

Substituting Eq. (7) in the expression (4) for emissivity, and by the change of variable

(p + p ) ! p , the integration over dp can b e p erformed to yield a factor qB L . The

3y 4y 3y 4y m x

rest of the integrals of Eq. (4) can then b e p erformed in the standard manner [7]. Electron

capture gives the same emissivity as neutron decay, although in neutrinos, and thus the

total emissivity (relativistically) for the direct URCA pro cess in nuclear matter (NM) in a

magnetic eld B is " (B )=2" (B ) i.e.

m m m

URCA

! !

457 p

2 2

NM 2 2

" (B )= +(g g ) 1 cos G cos (qB ) (g + g ) 1

m V A 14 c m V A

URCA F

5040

p n

# " #

2 2

(p + p ) p

F F

n p

p e

2 2 6

(g g ) exp T ; (8)

V A

2qB p p

m F F

p e

n p

2 2 1=2 2 2 2

where = (p + m ) and cos =(p +p p )=2p p . The threshold factor is

14 F F

n e

i i F F F

n e p

= (p + p p ), where (x) = 1 for x > 0 and zero otherwise. For B = 0, the

F F F m

p e n

relativistic expression for the neutrino emissivity from the nucleon direct URCA pro cess is

457

NM 2 2

" (B =0)= cos G cos (g + g ) 1

m 34 c V A

URCA F

10080

! #

p m

2 2 6

+(g g ) 1 cos (g g ) T : (9)

V A 14 e

V A n p

n n p

It was shown [7] that quark matter (QM), if present, the -decay (i.e. direct URCA

pro cess) of d quarks is kinematically allowed through reaction (2) if nite mass (and/or

quark-quark interaction) is incorp orated. The relativistic expression of the neutrino emis-

sivity for the direct URCA pro cess involving u and d quarks for B = 0 and without

quark-quark interaction is given by [7]

457

2 2 6

" (B =0)= G cos (1 cos ) T ; (10)

m c 34 d u e

URCA F

840

in the usual notation [7]. The emissivity for the decay of s quark for B = 0 is similar

to Eq. (10) with cos replaced by sin . The emissivity for the decay of free d quark

c c 4

in a magnetic eld B may be obtained from Eq. (8) by substituting g = g = 1 with

m V A

! , ! , and multiplying a color factor 3 for d quark:

d u

n p

! " #

2 2

(p + p ) p

457 p

F F

d u e F

u e

2 2 6

exp " (B )= G cos (qB ) 1 T : (11)

m c m

URCA F

420 2qB p p

u m F F

u e

Similar expression is obtained for s quark, but is Cabibb o suppressed. The decay of d and

s quarks is feasible if they satisfy the resp ective inequality conditions p p p

F F F

u e

p + p and p p p p + p .

F F F F F F F

u e u e s u e

To estimate numerically the various neutrino emissivities for the direct URCA pro cesses

with and without magnetic eld in a neutron star, we describ e the nuclear matter and

electrons within the relativistic Hartree approach in the linear -! - mo del [11,13]. The

values for the dimensionless coupling constants for the , ! and mesons are adopted from

Ref. [14] which are determined by repro ducing the nuclear matter prop erties at a saturation

density of n =0:16 fm . The variation of magnetic eld with density n from surface to

0 b

center of the star is parametrized by the form [13]

surf

B (n =n )=B + B [1 exp f (n =n ) g] ; (12)

m b 0 0 b 0

where the parameters are chosen to be =10 and =6. The maximum eld prevailing

18 surf 8

at the center is taken as B =510 G and the surface eld is B ' 10 G. The number

of Landau levels p opulated for a given sp ecies is determined by the B and n [11].

m b

In Fig. 1, we show the neutrino emissivity as a function of baryon density at B = 0

(see Eq. (9)) for the direct URCA pro cess in nuclear matter (denoted by NM) at an interior

temp erature T = 10 K. Due to momentum conservation, the threshold density at which

this pro cess o ccurs, is at n =0:346 fm . The variation of emissivity with n in presence of

t b

magnetic eld B (see Eq. (8)) as seen in the gure may be explained as follows: At very

low densities n 0:35 0:73 the eld B (as given in Eq. (12)) is rather small 10

b m

G, and consequently a large number of Landau levels are p opulated. This gives essentially

eld-free results. At densities n 0:75 fm , the eld is strong enough to p opulate only the

ground levels of b oth electrons and protons [11], and would have pronounced quantization

(e)(c) 13

e ects; the critical eld for electron is B =4:414 10 G. The emissivity then rapidly

increases with density and could have values as high as 2 orders of magnitude larger

3 18

than B =0 case at n 1:2 fm . Hereafter, B saturates to a maximum of 5 10 G

m b m

so that for n > 1:2 fm , higher level states start to p opulate, and, as in the low density

situation, results in eld-free emissivityvalues. The central densities n of neutron stars with

maximum masses are also shown in Fig. 1 with (op en circles) and without (solid circles)

the magnetic eld. For B 6= 0 star, n = 1:448 fm and thus falls ab ove the kernel of

m c

enhanced emissivity leading to faster co oling compared to the eld-free case.

The neutrino energy losses from direct URCA pro cesses of quark matter comp osed of

free u, d and s quarks and e are estimated in the bag mo del. The current masses of the

quarks are taken as m = 5 MeV, m = 10 MeV and m = 150 MeV, and the bag constant

u d s

as B = 250 MeV fm . In Fig. 1, we display the neutrino emissivity from the -decay

of d and s quarks at B = 0 (see Eq. (10)) in quark matter (denoted by QM). The d

n . At densities n ' 0:85 fm quark -decay reactions are kinematically allowed if n

0 b b

and ab ove when s quark decay is allowed, the emissivity is increased to ab out an order of 5

magnitude. This is caused by the large s quark mass which allows the momenta of the free

particles to deviate appreciably from collinearity which tends to increase the matrix element.

It was, however, shown [7] that by the inclusion of quark-quark interaction, the neutrino

emissivities from d and s quark -decay are comparable in magnitude. The emissivities

for the quark direct URCA pro cesses in presence of the magnetic eld (see Eq. (11)),

remain virtually unaltered from the eld-free case due to the p opulation of a large number

NM 3

of levels in all the quark sp ecies. In either case, it is found that " =" 10 .

URCA URCA

T=109 K

29 NM

NM(HY) )

1 28 -

s NM -3

27 NM(HY) (erg cm

ε 26 NQP 10 NQP(HY)

Log 25 QM

Mmax/MO. NM 1.778(1.650) 24 NQP 1.610(1.487) NM(HY) 1.532(1.396) NQP(HY) 1.520(1.380) 23 0.00.51.01.52.02.53.03.5 -3

nb (fm )

FIG. 1. The neutrino emissivities as a function of baryon density from the direct URCA pro cess

8 18

for a magnetic eld B = 0 (solid line) and for B =10 510 G (dashed line) for: nucleons

m m

in nuclear matter (NM); quarks in quark matter (QM); anucleon-quark phase transition (NQP);

nucleons in nuclear matter with hyp erons (NM(HY)); a nuclear matter with hyp erons to quark

phase transition (NQP(HY)). The maximum masses of the stars M with these various comp o-

max

8 18

sitions are given for B = 0 and those in the parentheses are for B = 10 5 10 G. The

m m

corresp onding central densities are indicated by solid and op en circles, resp ectively.

In a realistic situation, if quarks at all exist, a star with increasing density from the surface

to the center would have a pure nucleon phase at the inner crust and core with a p ossible pure

quark phase at the center and a mixed nucleon-quark phase (NQP) in b etween. The mixed

phase of nucleons and quarks is describ ed following Glendenning [14]. The conditions of

global charge neutrality and baryon numb er conservation are imp osed through the relations

n q n

Q +(1 )Q =0 and n = n +(1 )n , where represents the fractional volume

b b

o ccupied by the hadron phase. Furthermore, the mixed phase satis es the Gibbs' phase

n q

rules: = 2 + and P = P . The neutrino energy loss rate in this phase is given

p u d 6

NQP QM

by " = " +(1 )" . The neutrino emissivities for the nucleon-quark phase

URCA URCA URCA

transition are shown in Fig. 1 (denoted by NQP). For B = 0 case, with the app earance of

the quarks at n = 0:533 fm , the emissivity decreases from the corresp onding NM case.

Apart from the reduced emissivity of the quark phase (which b eing, however small at large

), the reduction in the chemical p otentials of the nucleons and electrons resulting from

the requirement of the global charge neutrality and baryon numb er conservation conditions

in the mixed phase, primarily causes the decrease in emissivity. For stars with B 6= 0,

the emissivities in the mixed phase are enhanced, particularly in the regime dominated by

nucleons (i.e. for > 0:5). The central densities of maximum mass stars fall within the

mixed phase, and consequently such stars would have faster co oling than pure quark stars.

The maximum mass NQP stars with and without magnetic eld, however, havemuch smaller

emissivities than that of the corresp onding NM stars, while NQP stars with B have nearly

identical co oling as that of eld-free NM stars even though their maximum masses are very

distinct.

Since quark matter furnishes b oth baryon number and negative charge, intuitively, the

trends exhibited by the emissivities for NQP stars may be anticipated by invoking strange

baryons, namely hyp erons ('s, 's and 's). The -equilibrium conditions then generalize

to = b q , where b and q are the baryon number and charge for the ith particle.

i i n i e i i

Since the hyp erons are more massive than the protons, the e ect of the magnetic eld on

their direct URCA pro cesses is negligible. Because of the large uncertainties in the hyp eron-

nucleon interactions even at nuclear density, for a conservative estimate of the emissivities

we set the nucleon-meson and hyp eron-meson coupling constants equal. Furthermore, the

critical density for nucleon direct URCA pro cess is nearly identical to the hyp eron threshold

density in the relativistic mean eld mo del, and the emissivities from the hyp eron direct

URCA pro cesses are ab out 5-100 times less than that from the nucleons [15]. Therefore, we

shall present emissivity vs n results only for the nucleon direct URCA pro cess in presence

of hyp erons. This is shown in Fig. 1 and denoted by NM(HY). With the app earance of

hyp erons, the reduction in the chemical p otentials of the nucleons and electrons required

by the baryon number conservation and charge neutrality condition causes a substantial

reduction of the emissivity compared to that from NM. In fact, with increasing density

when hyp eron abundances grow rapidly, the emissivities gradually decrease. For B 6= 0,

only the ground Landau levels for e and p are p opulated over a considerable density range

in this matter. Consequently, the emissivities with B in NM(HY) stars are signi cantly

larger than that for the corresp onding eld-free stars.

Allowing now baryon to quark phase transition, the emissivity displayed in Fig. 1 (de-

noted by NQP(HY)) for B =0 is larger than that from the NQP matter. This is caused

by the delayed app earance of quarks in hyp eron rich matter, so that the the total emissivity

is primarily dominated by the nucleons. In presence of the eld, the total emissivity of

NQP(HY) matter is ab out an order of magnitude larger for the maximum mass star and

therefore leads to faster co oling compared to the corresp onding eld-free star.

Within the non-relativistic (but interacting) approximation for the sp eci c heat c and

neutrino emissivity, the time for the center of a NM star to co ol by the direct URCA pro cess

to a temp erature T at B =0 may be estimated to b e t = (c =" )dT 10T s.

9 m v

URCA

In contrast, for B 6= 0, the NM star's center co ols faster with t 0:5T s. By invoking

quarks and/or hyp erons, the decrease in emissivity is much more compared to that of the 7

sp eci c heat resulting in slow co oling of the stars center. The typical time scale asso ciated

with the propagation of thermal signals through the outer core and the crust to the surface

b efore the sudden temp erature drop is quite high 1 to 100 yr, dep ending on the crustal

comp osition and relative sizes of the crust and the core and thus up on the equation of state.

Therefore, it seems to be quite dicult to distinguish observationally from the e ects of

direct URCA pro cess, the interior constitution of a star.

Throughout our discussion we have assumed that the electron is the only lepton. If the

triangle inequality p + p p is satis ed then nucleon direct URCA pro cess with muons

F F F

p n

will o ccur; the threshold density for this pro cess is higher than electrons since m >m . The

-equilibrium condition = moreover implies that the emissivity for URCA pro cess with

muons is same as that for the corresp onding pro cess with electrons. In the present mo del the

nucleon direct URCA pro cesses are not p ermitted at densities n < 0:34 fm . In this outer

core region, the dominant neutrino emission pro cess are then the mo di ed URCA pro cesses

of reaction (1) for which the emissivity is smaller by a factor (T=T ) than the direct

URCA pro cesses.

8 10

At certain densities and temp erature T < T 10 10 K, the nucleon sup er u-

idity may set in. The sp eci c heat and direct URCA rate are then reduced by a factor

exp (=T ), where is the larger of the neutron and proton gaps. The mo di ed URCA

rates are, however, reduced by a factor exp (2=T ). In presence of a magnetic eld, the

sup er uid protons are b elieved to form a typ e I I sup erconductor in the outer core within the

density range 0:7n

0 b 0

15 16

are resp ectively H 10 G and H 3 10 G [16]. With the choice of variation of

c1 c2

14 16

B with n (see Eq. (12)), the eld is 10 10 G at the bulk of the outer core and could

m b

form a sup erconducting region at T

c m

Gisinthe normal state without sup erconductivity.

In conclusion, the neutrino emissivities for all the cases studied here are found to be

dramatically enhanced in a magnetic eld compared to that from the non-magnetized stars.

However, for certain stars unambiguous determination of the interior constituents may be

dicult. There can be stars with the same comp osition, as for example the NM stars in a

magnetic eld but with slightly di erent masses of 1:55M and 1:60M having their central

3 3

densities at 0.72 fm and 0.88 fm residing b elow and within the kernel of fast co oling

resp ectively, and thus have completely di erent emissivities. On the other hand, a NM

star with B = 0 and a NQP star in a magnetic eld, though p ossessing di erent interior

comp ositions, have nearly identical emissivities. It is also found that when pure nuclear

matter is injected with nonleptonic negative charges, namely hyp erons and quarks, the

emissivities turn out to be smaller than that from the nuclear matter. It has b een already

demonstrated [17] that nonleptonic negative charges cause a softening of the equation of

state. We thus arrive at a general result that when matter contains nonleptonic negative

charges, the maximum masses of the stars are smaller with a suppression of the neutrino

emissivity than that of the pure nuclear matter with and without a magnetic eld. 8

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