Thermal evolution and magnetic decay of neutron stars

Pergamon

Chin. Astron. Astrophys. Vol. 21, No. 3, pp. 277-284, 1997

A translation of Acta Astrophys. Sin. Vol. 17, No. 2, 113-120, pp. 1997 0 1997 Elsevier Science B.V. All rights reserved

Printed in Great Britain

027% 1062197 $32.00 + 0.00

PII: SO2751062(97)00037-4

Thermal evolution and magnetic decay of

neutron stars t

DA1 Zi-gao LU Tan

Department of Astronomy, Nanjing University, Nanjing 210008

Abstract We study the interaction among thermal, rotational and magnetic

evolutions of neutron stars. The following self-consistent model is considered:

the star spins down due to magnetic dipole radiation, the interior heats up, the

magnetic field decays due to ohmic dissipation in the crust. We show that the

field decay enhances the heating effects, which in turn slows down the field decay.

Thus, the thermal, rotational and magnetic evolutions may be interdependent.

The evolutions moreover depend on the initial conditions. From radio and X-ray

observations we may be able to place certain constraints on the age of the pulsar,

the initial magnetic field and the period.

Key words: neutron stars-compact matter-magnetic field

1. INTRODUCTION

The thermal evolution of neutron stars has always been a research topic of great interest,

the main reason being that the evolution is closely related to the properties of the compact

matter in the star’s interiorf’). Some X-ray satellites (for example, Einstein, EXOSAT

and ROSAT) may already have detected thermal radiation from several pulsars (e.g., Vcla,

Geminga, PSR0656+14, PSR 1055-52)[2]. Tl reoretically, several heating effects have been

found that will affect the cooling of the star. First, in the course of spin-down, interaction

between normal matter in the inner shell and the neutron superfluid generates friction and

converts part of the rotational energy of the superfluid into heat [3 I. In the last few years

some researchers[4-6] have considered this particular heating effect and investigated the

thermal evolution of the star. Their results showed that even a small heating rate can raise

the internal temperature during the photon cooling stage. Second, during the spin-down,

matter in the interior will depart from chemical equilibrium, so will store chemical energy,

and some reactions will convert the stored chemical energy into heat. The resultsf7s] showed

this effect to be appreciable for fast rotating, old neutron stars with weak steady magnetic

t Supported by National Nsturd Science Foundation

Received 199602-12; revised w&on 1996-12-10

277

278 DA1 Zi-gao & LU Tan

fields. Here we need to emphasize that both heating effects are determined by the spin-down

rate.

After the discovery of pulsars, their spin-down has been taken as evidence for the

following point of view: strong magnetic fields exist on the surface of pulsars and their

rotational energy is lost through magnetic dipole radiation. Thus, the rotational evolution

of isolated pulsars must be determined by the evolution of their magnetic fields.

Magnetic field evolution has become an important topic of increasing interest in recent

years. As pointed out by Phimley and Kulkarm *[‘I, when discussing the observed facts, there

is still no consensus on the question whether or not the magnetic fields of isolated neutron

stars decay significantly. Theoretically, the evolution of the magnetic field is closely related

to its origin. One possible origin is that the field is generated by thermoelectric instability in

the shell after the birth of the star [lo*lll. If so, the field will evolve through ohmic decay in

the shell. Numerical studies have shown [12-151 that the field does not decay exponentially,

and that its evolution is related to the history of cooling of the star.

Summarizing, it seems that the thermal, rotational and magnetic evolutions of a ncu-

tron may not be independent of one another, rather, the three may influence one auother.

For example, friction in the differentially rotating portion of the shell and departure from

chemical equilibrium of matter in the interior can heat the star and so alter its thermal

evolution. These heating effects and the spin-down rate are directly affected by the CVO-

lution of the magnetic field, while the latter is affected by the thermal evolution through

the ohmic dissipation mechanism. In the past, researchers have either colllplctely separated

the three evolutions in their studies, or consitlcrcd only the interaction bctwcen two of the

three. In this paper, for the first time, WC are putting all three together in our study of their

interactions.

2. THE MODEL

The internal structure of neutron star cau be divided into three part&“]. First, there is the

outer shell which extends from the surface to the point of neutron drop (4.3~ 10” g/cm3),

and contains electron gas and atomic nuclei in latticed point arrays. Next is the inner shell,

which extends from the point of neutron drop to near the point of nuclear density and

contains degenerate electron gas, non-degenerate neutron gas and atomic nuclei in latticed

point arrays. In this layers the free neutrons are iii the ‘So superfluid state. The vortex

lines formed by the superfluid are “nailed” to the latticed nuclei or to the space in between.

Last is the nuclear region, which contains superconducting free protons, neutrons in the 3Pz

state and electrons in the normal state. The density in the inner part of this region can be

as high as 10” g/cm 3, here the state of matter is very uncertain, it may be nuclear matter,

meson condensation or quark matter.

In the spin-down process of the star, braking caused by magnetic dipole radiation first

slows down the rotation of the charged portion of the star. The neutron superfluid in the

nuclear region because of strong coupling with the clcctrons, corotates with charged part[“l,

while in the inner shell the coupling is weaker and there is friction bctwcen the vortex liues

of the superfluid and normal matter, so gcncrating heat.

Neutron Star Evolution 279

The spin-down decreases the centrifugal force and the density of matter in the nuclear

region becomes larger. Since the state of chemical equilibrium of matter is determined by

the density, matter in the nuclear region may therefore be in chemical non-equilibrium, and

chemical energy may be stored. o-decay reactions then convert the chemical energy into

heat (note the rate of p-reactions is positively correlated with the temperature).

During the spin-down, even tiny breaks in the shell can generate heatl”] and affect the

thermal evolution of the star. Here, we shall not consider this heating process.

We assume the star to lose its thermal energy by standard neutron processes, n + n 4

n+p+e-+c, andn+p+e-+n+n+v,. We assume the thermodynamic state of the

matter can be described by two parameters, the internal temperature T and the chemical

potential difference 6~ _ pP + pL, - p,, . We assume the density in the interior to be uniform,

so these two parameters are independent of the position. The equilibrium value of proton

abundance, xes can be found from 6~ = 0.

2.1 Equation of Thermal Evolution

As in Refs. [4,7,8,1&X], we adopt the isothermal approximation, which approximation has

been shown to be quite effective for t > lo3 yr L51 Then, the thermal evolution can be written .

as dT

C,lt=I’6p-~t,+hf-fi7 . (1) where C, is the specific heat, I’61.1 is the rate of release of chemical energy (mean rate per

nucleon), cv is the neutron energy loss rate, hf is the frictional heating rate, and & is the

energy loss rate from blackbody radiation from the surface.

If we assume no interaction at all between the nucleons and electrons and complete

degeneracy, then according to formula 11.8.1 of Ref. [lG] the specific heat can be written as

Cn = 6.3~ 10-14(n/no)-2’3Ts [( 1 - xes)‘i3 + xii” + 0.35(7~/no)‘/~&~] MeV . (2)

Here, n is the number density of baryons, no N 0.16 fmw3, and Ts is the temperature in units

of 1O’K. The chemical energy released per unit time by the Urea reaction under chemical

non-equilibrium and the neutrino energy loss rate are, respectivelyl’~‘“1,

rs/l = E;o) 1468~~ + 7560~~ + 840~~ + 24~s

11513 , (3)

and 22020~~ + 5670~~ + 4202~~ + 9u8

11513 1 * (4)

where u = 6p/akT and E’,“’ is the neutrino energy loss rate at equilibrium,

6:“) = 1.4x 10-‘g~~~3(n/no)-2’3T~ MeVs-’ . (5)

Here, we have neglected the effect of superfluidity on the specific heat and the reaction rates.

The effect will be discussed in Section 4 below.

The frictional heating rate in the inner shell isl’l

h, = -J&n,fM = -5.2~10-~J&M;’ MeVs-’ . (6)


where &f, is the mass of the star in units of solar masses and

is the differential angular momentum between the superfluid

J44 = I,w,,/lO” g cm3 rad s-l

with frictional interaction and

the normal matter, Is being the rotational inertia of the shell and u,, the critical angular

velocity for the superfluid vortex lines becoming “un-nailed”. Using the relation between

the surface and interior temperatures given in Ref. [20], we obtain the blackbody radiation

energy loss rate,

,?$ = 2.8~1O-~~T~.~(l - 0.3M,R,‘)-“’ MeVs-’ ,

Re being the radius of the neutron star in units of 106cm.

(7)

2.2 Equation of Rotational Evolution

If we assume the spin-down of the star to be due to magnetic dipole radiation, then

the equation governing the evolution of the angular velocity is

dS2 2R6

dt = 3c31 --fi3(t)B2(t) = -2.47x lo-l7

@B:,f13 S-2

ICJ

14s being the total rotational inertia in units of 10d5 g cm’, and Bl?, the surface magnetic

field in units of 1012 G.

2.3 Equation of Magnetic Field evolution

The evolution for an isolated neutron star is goverued by the equation of ohmic dissi-

pation, 8B -= at -;v x (‘V xB). (9)

u

c being the electric conductivity. Equation (9) was numerically solved by some researchers [13-15)

on assuming an initial magnetic field trapped in the outer shell, who found that the field

did not decay exponentially.

On the other hand, Urpin et al.[“l d erived an analytic solution of the equation for a

dipole field. These authors first wrote the conductivity in the form c = qh”T-P, h being

the height and q,a,j3 constants. Using [la], the authors obtained

Ly = 2, ,0=1, TD<T a = 712, p = 2, T,,op < T < TD

1

(10) a = 1, P=O, T<Tirn,,

Here, TD is the Debye temperature and Tilllp is the upper bouud to the temperature due to

effect of impurities on @,

TD = 3.30 x 106(Z/A)‘l’y3/”

zlmp = 1.67x 106y5/‘Q1~?/(ZA)‘~” 1 (11)

31 = hAgmp/Zm,c~, g the surface gravity, and Q is the impurity paramctcr, usually Q -

10-3[“?l. Using equation (20) of Ref. [21], we then have the equation of magnetic field

evolution,

J,‘, Tqt’)dt’ -s B(t) = “1 Tb(to)to ] ’ 112)


Bo being the field intensity at time to. In our calculation we assumed the shell to be

composed of 56Fe and a depth corresponding to p = 10gg/cm3, because the effect of the

surface layer on the field decay is rather small[“‘l.

2.4 Ancillary Equation

To solve equation (1) we must give an equation for the evolution of 6~. Accordiug to

Ref. [7], we have

where CY’ - 0.73. For a given equation of state we can evaluate the second-order derivatives

of the energy density, E,,, and E,,.

3. NUMERICAL RESULTS

We chose the equation of state AV14+UVII of Ref. [23]. Tl ie interior of a neutron star in

this state mainly emits neutrinos through the modified Urea reaction. A 1.4 Ma neutron

star in such a state has the following parameters: pe = 1.2~10’~ g/cm3, n = 0.56fn1-3,

R = 10.4 km, I = 1.148gcm2, 1, = 0.51 x10” gem’, and xeC, = 0.07. For a given initial

period and magnetic field, we can solve numerically the equations (l)-(8), (lo)-( 13). III the

calculation we took w,, = 1 rad/s, possibly the maximum value required by the observations

(see the discussion in Ref. [IS]). W e oo an initial period of Pi = 1 Ins, which would make t k

the chemical heating effect most pronounced.[q.

Fig. 1 shows, for different initial magnetic fields, the evolutiou of the internal tempera-

ture under the two heating effects, frictional and chemical (solid lines). For comparison we

show also the thermal evolution curves for a fixed magnetic field (dashed lines), considered

in Refs. [5-81. Fig. 1 shows that, the smaller the initial field, the more pronounced is the

effect of heating and that evolution of the magnetic field enhances the importance of the

heating effect, greatly increasing the internal temperature a.9 compared to the case of no field

evolution. Fig. 2 shows the corresponding curves for the surface temperature. It is obvious

that an evolving magnetic field will make the neutron star more ea..ily observed during the

photon cooling stage (t > 106yr).

Fig. 3 shows the evolution of the magnetic field for an initial field of 10” G. The solid,

dashed, dotted and chained lines are the results, respectively, for (i) with both heating effects,

(ii) with frictional heating only, (iii) with chemical heating only, and (iv) with neither. The

figure shows that the magnetic field evolution is dependent on the thermal history. In the

case of no heating, the initial decay during t < lo6 yr is followed by a platform, which is then

followed by a second rapid decay for t > 10’ yr. This is in agreement with the numerical

results given in Refs. [14,15]. I n contrast, when one or both heating effects arc present, the

field does not decay rapidly after t > 10” yr.


Fig. 1 Evolution of the internal temperature for initial magnetic fields

B; = lOlo, lo”, 1012, 1013, 1014 G from top down. The solid lines are for decaying fields, the

dotted lines, fixed fields. The chained line includes neither field decay nor heating effect

-2.0 - .\ '\ -1 '\

-2.5 - \ '\

_3,0_ I I I I I I I I 1 A -

4 6 8

Fig. 2 Evolution of the surface temperature. Fig. 3 Evolution of the surface magnetic field

Meaning of the lines same as Fig. 1 with and without heating effects. See text


4. DISCUSSION

This paper differs from previous papers that include consideration of both thermal

and rotational evolutions in the following respects. 1) Refs. [5-8, 181 when considering the

thermal evolution including heating effect did not involve the evolution of the magnetic field.

2) Shibazaki and Lamb141 investigated the effect of an exponentially decaying magnetic

field on the thermal evolution. Our view is that magnetic field decay through rotational

evolution affects the thermal evolution (by means of heating effects) while the thermal

evolution through ohmic dissipation mechanism in the shell in turn affects the evolution of

the magnetic field. The model of the present paper is self-consistent.

We have already considered cooling of the neutron star using the modified Urea reaction

as the neutrino energy loss mechanism. However, when the proton abuudance in the nuclear

matter exceeds ll.l%, direct Urea reaction can proceed in the interior of the starI’.‘]. When

the neutron star matter is described by the relativistic mean field theory, such direct Urea

reactions may not appear inside a 1.4Ma neutron starIr51. Besides the existence of states

with a large proton abundance, neutron star matter can be in states of meson (r or K)

condensation or quark matter. Such states will lead to faster cooling of the star, and a

faster cooling may give rise to slower magnetic decay I151. Hence the interaction between fast

cooling and magnetic decay will not be pronounced.

We have also neglected the effect of the superfluid on the specific heat and the reaction

rates. As discussed in Ref. 171, such effects may be large. For temperatures far below the

critical temperature for normal mater to become superfluid, the specific heat and reaction

rates may decrease appreciably. These changes may lead to even more important chemical

heating effect and even slower magnetic decay (during the photon cooling stage).

If, like the ohmic dissipation in the shell, the magnetic field of a pulsar decays signif-

icantly in a time span of a few million years, then its real age must be far smaller than

the spin-down time scale. In such a case, it is incorrect to write the age of the pulsar as

jfl/2Ql. Certain parameters of pulsars are available from observations. For example, from

radio observations we can get the period, the period rate and the surface magnetic fieldI’Gl.

Also, from X-ray observations we can obtain the surface temperatures of certain pulsars 121.

Since the present work tells us that the thermal, rotational and magnetic evolutions not

only affect one another, they are also dependent on the initial conditions, it may be possible

to derive from radio and X-ray observations, certain constraints on the age, and the initial

magnetic field and period of pulsars.

5. CONCLUSION

We have studied the mutual influence among the thermal, rotational and magnetic

field evolutions of neutron stars. For this we have constructed a self-consistent model as

follows: the neutron star spins down because of magnetic dipole radiation, this gcneratcs

frictional heating and chemical heating in the interior, and the magnetic field of the star

evolves through ohmic dissipation mechanism in the shell. We found that the magnetic

decay enhances the importance of the heating processes, and conversely, heating cffccts slow


down the magnetic decay. Thus we get the result that the thermal, rotational and magnetic

field evolutions in neutron stars may not be independent. Moreover, these evolutions are

dependent on the initial conditions, so it may be possible to derive certain constraint on the

age and the initial magnetic field and period of pulsars from radio and X-ray observations.

ACKNOWLEDGMENT We thank Dr K. S. Cheng of Hong Kong University for useful

discussion.

111

121

131

[41

[51

161

(71

181 t91

WI

WI

P21

P31

1141

1151

@I

1171

W31 P91 PO1 WI PI I231

P4l

P51

WI

References

Tsuruta S., In: M. A. Alpar, 0. Kilziloglu, J. vau Paradijs et al., eds., The Lives of the

Neutron Stars, Dordrecht: Kluwer, 1995

~gehnau H., In: M A. Alpar U. Kilziloglu, J. vau Parsdijs et al., eds., The Lives of the

Neutron Stars, Dordrecht: Kluwer, 1995

Alpar M. A., Anderson P. W., Pines D. et al., ApJ, 1984, 276, 325

Shibazaki N., Lamb F. K., ApJ, 1989, 346, 808

Umeda H., Nomoto K., Tsuruta S. et al., ApJ, 1994, 431, 309

Van Riper K. A., Link B., Epstein IX., ApJ, 1995, 448, 294

Reisenegger A., ApJ, 1995, 442, 749

Cheng K. S., Dai Z. G., ApJ, 1996, 468, 819

Phinney E. S., Kulkarni S. It., ARA&A, 1994, 32, 591

Blandford R. D., Applegate J. H., Hemquist L., MNRAS, 1983, 204, 1025

Romani R. W., Nature, 1990, 347, 741

Sang Y., Chanmugam G., ApJ, 1987, 323, L61

Sang Y., Chanmugam G., ApJ, 1990, 363, 597

Urpin V. A., Levshakov S. A., Yakovlev D. G., MNRAS, 1986, 219, 703

Urpin V. A., Van Riper K. A., ApJ, 1993, 411, L87

Shapiro S. L., Teukolsky S. A., Black Holes, White Dwarfs, and Neutron Stars: The Physics

of Compact Objects, New York: Wiley, 1983

Alpar M. A., Langer S. A., Sad J. A., ApJ, 1984, 282, 533

Cheng K. S., Chau W. Y., Zhang J. L. et al., ApJ, 1992, 396, 135

Haensel P., A&A, 1992, 262, 131

Gudmundsson E. H., Pethick C. J., Epstein R. I. et al., ApJ, 1983, 272, 286

Urpin V. A., Chanmugam G., Sang Y., ApJ, 1994, 433, 780

Flowers E., Ruderman M. A., ApJ, 1977, 215, 302

Wirings R. B., Fiks V., Fabrocini A., Phys. Rev., 1988, C38, 1010

Lattimer J. M. et al., Phys. Rev. Lett., 1991, 66, 2701

Cheng K. S., Dai Z. G., Yao. C. C., ApJ, 1996, 464, 348

Taylor J. H., Manchester R. N., Lyne A. G., ApJS, 1993, 88, 529

Documents

Thermal evolution and magnetic decay of neutron stars