precession of isolated neutron stars. i. effects of imperfect pinning

THE ASTROPHYSICAL JOURNAL, 524 :341È360, 1999 October 101999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

PRECESSION OF ISOLATED NEUTRON STARS. I. EFFECTS OF IMPERFECT PINNING

ARMEN SEDRAKIAN, IRA WASSERMAN, AND JAMES M. CORDES

Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853Received 1997 December 22 ; accepted 1998 December 28

ABSTRACTWe consider the precession of isolated neutron stars in which superÑuid is not pinned to the stellar

crust perfectly. In the case of perfect pinning, Shaham showed that there are no slowly oscillatory, long-lived modes. When the assumption of perfect pinning is relaxed, new modes are found that can be longlived but are expected to be damped rather than oscillatory, unless the drag force on moving superÑuidvortex lines has a substantial component perpendicular to the direction of relative motion. The responseof a neutron star to external torques, such as the spin-down torque, is also treated. We Ðnd that whencomputing the response of a star to perturbations, assuming perfect coupling of superÑuid to normalmatter from the start can miss some e†ects.Subject headings : dense matter È stars : neutron

1. INTRODUCTION

Radio pulsars can be exceptionally stable clocks. The predictability of pulse arrival times has made possible precision testsof general relativity (e.g., Taylor et al. 1992) and the discovery of the Ðrst extrasolar planetary systems (Wolszczan & Frail1992). However, imperfections in the rotation of most pulsars have been monitored for some time, most notably glitches (e.g.,Boynton et al. 1969 ; Radhakrishnan & Manchester 1969 ; Backus, Taylor, & Damashek 1982 ; Downs 1982 ; Demianski &Proszynski 1983 ; Manchester et al. 1983 ; Lyne 1987 ; Lyne & Pritchard 1987 ; Cordes, Downs, & Krause-Polstor† 1988 ;McKenna & Lyne 1990 ; Hamilton et al. 1989 ; McCulloch et al. 1990 ; Flanagan 1990, 1993 ; Lyne, Smith, & Pritchard 1992 ;Shemar & Lyne 1996) and timing noise (e.g., Boynton et al. 1972 ; Groth 1975 ; Cordes & Helfand 1980 ; Cordes & Downs1985 ; DÏAlessandro et al. 1995). These deviations from stable spin convey information about the internal structure anddynamics of neutron stars. For example, the long-time healing of the spin frequencies and spin-down rates of glitching pulsarsmay be explained if the convulsions are due to sudden unpinning of superÑuid vortex lines that were glued to nuclei in thepulsarÏs crust before the glitch, migrate as a consequence of the glitch, and repin to other nuclei in its aftermath (e.g., Anderson& Itoh 1975 ; Alpar et al. 1981, 1984a, 1984b, 1993 ; Link, Epstein, & Baym 1993).

A small number of neutron stars also exhibit long-term cyclical but not precisely oscillatory variations in their spin. Aparticularly well-known case is the Crab pulsar, whose phase residuals (after careful Ðtting that accounts for spin-down andglitches) vary systematically, with a peak-to-peak range of order ^10 ms and a characteristic cycle duration of about 20months. (Lyne, Pritchard, & Smith 1988). After its Christmas 1988 glitch, the Vela pulsar showedÈafter accounting forexponential recovery from the glitchÈdamped oscillatory phase residuals with a period of order 25 days ; evidence foroscillations in the frequency derivative of the pulsar both before and after the glitch with a period of ““ a few tens of days ÏÏ wasalso reported (McCulloch et al. 1990). Evidence for long-term variations (correlation times D100 days) in the pulse shape ofthe Vela pulsar has also been found in data spanning approximately 4 yr (Blaskiewicz 1992 ; Cordes 1993). A principal-component analysis of the pulse shape of PSR 1642[03 (Blaskiewicz 1992 ; Blaskiewicz & Cordes 1997, in preparation) yieldsevidence for cyclical pulse shape variations with a period of about 1000 days ; long-term variations on a similar characteristictimescale are also seen in the timing residuals for this pulsar (Cordes 1993). Finally, although Her X-1 is an accreting X-raypulsar, not an isolated radio pulsar, it has a well-known 35 day cycle on which it appears and disappears ; observed variationsin pulse shape over the cycle suggest that it is related to periodic variations in the rotation of the neutron star (e.g., etTru" mperal. 1986 ; Alpar & 1987).O# gelman

Soon after the discovery of radio pulsars, it was suggested that long-term variations in their spin could result from freeprecession (e.g., Davis & Goldstein 1970 ; Goldreich 1970 ; Ruderman 1970 ; Brecher 1972 ; Pines, Pethick, & Lamb 1973 ;Pines & Shaham 1972, 1974 ; Lamb et al. 1975 ; Jones 1976) and that pulsar arcanae such as drifting subpulses might be relatedto precessional e†ects. If a neutron star were a rigid or semirigid solid (see Pines & Shaham 1972, 1974), it would precess witha period of order P/v, where P is the spin period of the star and v is the fractional di†erence between its principal moments ofinertia ; would imply precession periods yr, where P(s) is the spin period in seconds. However, once it becamev [ 10~7 ZP(s)apparent that superÑuid vortex lines pin to the crust of a neutron star, Shaham (1977) demonstrated that slow, persistentprecession is impossible. Pinned superÑuid alters the e†ective asymmetry of the star to where is the moment of inertiaI

p/I, I

pof the pinned superÑuid and I the moment of inertia of the star (or stellar crust, depending on various coupling parameters) ;since or even larger, precession is very fast. Moreover, Shaham (1977) demonstrated that precession would decayI

p/I D 10~2

rapidly for some estimates of the coupling timescale between crust and core (e.g., Alpar & Sauls 1988 ; Sedrakian & Sedrakian1995).

Perhaps because of ShahamÏs (1977) pessimistic conclusions, there has been relatively little theoretical work on thelong-term variability of pulsar spins. Some argue that vortex line pinning does not occur, making slow precession possible(e.g., Jones 1988 for the Crab pulsar), but it may be hard to support the viewpoint that pinning is completely absent in the faceof physical arguments to the contrary (e.g., Alpar et al. 1984a ; Epstein & Baym 1988 ; Link & Epstein 1991). Notwithstanding

341

342 SEDRAKIAN, WASSERMAN, & CORDES Vol. 524

the pessimism of theorists, the data demand an explanation. Ruderman (1997) has mentioned long-term variations in pulsarspin rates as one of the outstanding unresolved problems of neutron star physics.

This paper is the Ðrst of two that attack this problem. Our purpose in this paper is threefold : First, we revisit the argumentsput forward by Shaham (1977) with an eye toward identifying possible loopholes. Although it will become apparent that thereare several di†erent possibilities, we concentrate here on ShahamÏs assumption that superÑuid pins perfectly to crustal nuclei.(Some of the other loopholes will be considered in a subsequent paper.) We develop the formalism for doing so in ° 2.2 andsolve the equations governing the spin dynamics of a neutron star in succeeding sections to varying degrees of complexity andrealism. As expected, we do Ðnd modes in addition to those found by Shaham (1977), but we also argue that none of thesemodes is likely to be a long-period, slowly damped oscillation.

Second, we examine what can happen in a multicomponent star, in which some regions contain pinned superÑuid andothers unpinned superÑuid. One might think that if some parts of a star are capable of slow oscillations of the spinÈeitherprecession or long-period Ñuid modes, such as Tkachenko modesÈthen there could be an observable signature of thesemodes in the detectable pulsar spin rate. However, we demonstrate that this situation is highly unlikely, for even if suchregions do exist inside actual pulsars, persistent long-period oscillations in those domains are only possible if the coupling tothe crust, where superÑuid is pinned e†ectively, is very weak ; but under such conditions, the crust is almost una†ected by theslow oscillations, which hardly manifest themselves in the crustal spin rate. Third, we begin a general examination of thee†ects of external torquesÈsuch as the spin-down torqueÈon the spin dynamics. For this purpose, we derive explicitexpressions for the response of the various components of a neutron star to rather general time-dependent torques. Ourtreatment of this problem shows that the limit of perfect coupling must be taken carefully when the response to externaltorques is needed, because the additional modes that appear when pinning is imperfect contribute to the response and cannotbe ignored.

As will become apparent in the succeeding sections, we do not believe that the long-term cyclic variability detected in thespins of some pulsars can be accounted for by free precession and that it is not likely to be due to forced precession either.However, we do believe that this paper begins to elucidate the complexity of the behavior of neutron star spin and clariÐes theconditions that must be met for precession to occur, even if those conditions are not likely to be realized.

2. OVERVIEW

2.1. Pinned SuperÑuid Suppresses Precession (Shaham 1977)Shaham (1977) showed that pinned crustal superÑuid dramatically alters the physics of precession. Let us review his

argument brieÑy. Consider a three-component neutron star that consists of (1) a rigid crust rotating at angular velocity )cr ;(2) pinned crustal superÑuid, whose angular momentum is independent of time in the frame rotating with the crust ; and (3)L

pa core (super)Ñuid rotating at angular velocity As seen in the inertial frame,)c.

Ic

dXc

dt] d(I

crÆ Xcr)

dt] Xcr ] L

p\ 0 , (1)

if there are no external torques, where is the moment of inertia of the core Ñuid and is the moment of inertia tensor of theIc

Icrcrust. We assume that the moment of inertia tensor of the core Ñuid is always of the form (d is the unit tensor)

Ic\ (I

c[ *I

c)d ] *I

c

A3XŒcXŒ

c[ d

2B

, (2)

so that its angular momentum This amounts to assuming that the core Ñuid adjusts its shape instantane-Lc\ I

cÆ X

c\ I

cX

c.

ously to an oblate spheroid Ñattened along its direction of rotation. We shall discuss this assumption more fully in asubsequent publication.

Introducing a dissipative torque that seeks to enforce corotation between the crust and the core, we get the coupledequations

d(Icr

Æ Xcr)dt

] Xcr ] Lp

\ [K(Xcr [ Xc) , (3)

Ic

dXc

dt\ K(Xcr [ X

c) , (4)

where K is a constant. These equations have a rich set of Ðxed points (where time derivatives vanish) depending on theorientation of relative to the principal axes of The full set of Ðxed points, and their possible observable signiÐcance, willL

pIcr

.be discussed completely elsewhere ; here we focus on the particularly simpleÈbut far from generalÈsituation in which isL

palong one of the principal axes of In that circumstance, the Ðxed-point solution is withIcr

. Xcr \ Xc\ X, X p L

p.

Perturbations about this Ðxed point are studied most easily in the frame corotating with the crust, where is independentIcrof time. For deÐniteness, let us suppose that the principal moments of inertia of the crust are and, to parallelI1 \ I2 \ I3Shaham (1977) as closely as possible, suppose that the Ðxed point corresponds to rotation about the 3-axis at angular velocity

Then the linearized equations areX \ eü 3 ).

)0 cr,1 ]CAI3 [ I2I1

B] Lp

I1 )D

)cr,2 \ [ KI1 ) ()cr,1 [ )

c,1) , (5)

No. 1, 1999 PRECESSION OF ISOLATED NEUTRON STARS. I. 343

)0 cr,2 [CAI3 [ I1I2

B] Lp

I2 )D

)cr,1 \ [ KI2 ) ()cr,2 [ )

c,2) , (6)

)0 cr,3 \ [ KI3 ) ()cr,3 [ )

c,3) , (7)

)0c,1 ] )cr,2 [ )

c,2 \ KIc) ()cr,1 [ )

c,1) , (8)

)0c,2 ] )

c,1 [ )cr,1 \ KIc) ()cr,2 [ )

c,2) , (9)

)0c,3 \ K

Ic) ()cr,3 [ )

c,3) . (10)

In these equations, where d*F/dt is the time derivative of any vector F as seen in the frame rotating with theF0 \ )~1d*F/dt,crust. It is clear that the perturbations along the 3-axis decouple from those along other axes and decay exponentially with acharacteristic rate

()q)~1 \ K)A 1

I3] 1

Ic

B; (11)

this coupling time has been estimated by Alpar & Sauls (1988), according to whom )q/2n B 400È104 (and the relaxation ofthe electron distribution function is due to the scattering o† the neutron vortex magnetization), and by Sedrakian &Sedrakian (1995), who Ðnd that )q/2n is rather sensitive to the mass density o and spans the range )q/2n D (102È108)P(s) foro D (1.6È3) ] 1014 g cm~3 (here the electron scattering is o† the proton vortex clusters coupled to the neutron vortex lattice).In the latter case the relaxation time spans a wide density range with a large gradient near the crust-core interface because ofan exponential dependence of the size of the cluster on the proton e†ective mass. [See also Sedrakian et al. 1995 ; Sedrakian &Cordes 1998 ; for the decay of precession the e†ective coupling rate c is a weighted average of the range found by Sedrakian &Sedrakian 1995, implying an e†ective coupling time closer to the smallest values ; we adopt )q/2n D 100P(s) for numericalestimates below.]

The remaining four equations have normal modes proportional to exp (p)t) 4 exp (p/), where / \ )t is pulse phase. It isstraightforward to solve for the modes of a triaxial star, but the basic result can be derived under the assumption ofaxisymmetry, (We have solved the corresponding triaxial problem, and there are no qualitatively di†erent modes forI2 \ I1.slowly rotating neutron stars.) If we deÐne

p \AI3 [ I1I1

B] Lp

I1 ) , c \ K)A 1

I1] 1

Ic

B\ I3(I1 ] Ic)

)qI1(I3 ] Ic)

, (12)

then

)0 cr,1 ] p)cr,2 \ [ cIc

I1 ] Ic

()cr,1 [ )c,1) , (13)

)0 cr,2 [ p)cr,1 \ [ cIc

I1 ] Ic

()cr,2 [ )c,2) , (14)

)0c,1 ] )cr,2 [ )

c,2 \ cI1I1 ] I

c()cr,1 [ )

c,1) , (15)

)0c,2 ] )

c,1 [ )cr,1 \ cI1I1 ] I

c()cr,2 [ )

c,2) . (16)

If c \ 0, so the crust and core are uncoupled, then there are modes with p \ ^ip, which correspond to independentprecession of but at a frequency that is much larger than the conventional Euler frequency for reasonable values ofXcr(where is the moment of inertia of pinned superÑuid). The remaining modes with p \ ^i are an artifact ofL

p/I1 ) 4 I

p/I1 I

pworking in the frame that corotates with the crust and corresponds to Ðxed in the inertial frame of reference.XcWhen the modes are damped, as was discussed by Bondi & Gold (1954) in the context of the rotation of the Earthc D 0,

(without considering pinned superÑuid, of course !). The characteristic equation is fourth order in p, but we expect the roots tocome in complex conjugate pairs, so we can reduce the characteristic equation to second order by introducing the complexangular velocities

)cr(`) \ )cr,1 ] i)cr,2 , )c(`) \ )

c,1 ] i)c,2 , (17)

which satisfy the equations

)0 cr(`) [ ip)cr(`) \ [ cIc

I1 ] Ic

[)cr(`) [ )c(`)] , (18)


)0c(`) [ i)

c(`) ] i)cr(`) \ cI1

I1 ] Ic

[)cr(`) [ )c(`)] . (19)

Substituting we Ðnd[)cr(`), )c(`)] P exp (p/)

p2 ] p[c ] i(1 [ p)] ] pA

1 [ icI1I1 ] I

c

B\ 0 . (20)

The normal modes of the fourth-order system are the two solutions to this quadratic equation and their complex conjugates.Although we can solve the second-order characteristic equation exactly, it is more instructive to Ðnd approximate solutions

valid for small and large crust-core coupling. For small values of c, we rewrite the characteristic equation as

(p ] i)(p [ ip) ] cA

p [ ipI1I1 ] I

c

B\ 0 ; (21)

this form separates terms of zeroth and Ðrst order in c explicitly. To Ðrst order in c, the solutions are

pd\ [i [ c[1 ] pI1/(I1 ] I

c)]

1 ] p , (22)

pp\ ip [ cpI

c(I1 ] I

c)(1 ] p)

. (23)

For large values of c, we rewrite the characteristic equation as

p [ ip I1I1 ] I

c] c~1[p2 ] ip(1 [ p) ] p] \ 0 ; (24)

this form is useful for expanding in powers of c~1. In this case, the solutions to Ðrst order in c~1 are

pd\ [c [ i

A1 [ pI

cIc] I1

B, (25)

pp\ ipI1

I1 ] Ic[ pI

cc(I1 ] I

c)A

1 ] pI1I1 ] I

c

B. (26)

In each case, represents damping of the angular velocity di†erence between crust and core and is the precessing mode.pd

ppFor the coupling times estimated by, for example, Alpar & Sauls (1988) or Sedrakian & Sedrakian (1995), the small c limit is

the relevant one. Since the precession period is far smaller than for a rigid body, approximately spinIp/I1 ? I3/I1 [ 1, I1/I

pperiods. Moreover, the wobble damps away, lasting Dc~1 precession periods : c~1 B 400È104 according to Alpar & Sauls(1988), and a reasonable estimate for the e†ective coupling is c~1 D 100P(s) for Sedrakian & Sedrakian (1995). Even if c werelarge, the precession period would be short, although it would be lengthened by a factor of relative to the small c1 ] I

c/I1limit, implying a cycle spin periods long. The precession would persist for approximately precession(I

c] I1)/I

pI1/I

c)q

periods in this limit. Since the crust-core coupling time must exceed the light travel time across the star, q [ R/c B 0.03 msand the damping time for the precession must be precession periods, where P is the rotation period in seconds.[5000(I1/I

c)P

In neither limit is the precession either long period or persistent. From this pessimistic result, one concludes that freeprecession cannot account for the cyclical behavior seen in the long-time monitoring of some pulsars. Moreover, to explainthe data, one must invoke an excitation mechanism that acts relatively continuously, since it must Ðght the tendency forneutron star wobbles to decay rapidly. The characteristic cycle timescales of order months to years observed for these pulsarsmust reÑect the underlying processes responsible for the continuous excitations.

2.2. Imperfect PinningIn demonstrating that persistent, long-period precession is impossible for neutron stars with pinned superÑuid, Shaham

(1977) assumed perfect pinning. In actuality, superÑuid vortex lines will not pin to crustal nuclei absolutely. One purpose ofthis paper is to see whether there are new oscillatory modes that emerge when pinning is assumed to be strong but not perfect.

To study this problem, we adopt a somewhat idealized approach. In actuality, the pinning of crustal superÑuid is a highlyinhomogeneous process involving the interaction of individual vortex lines and crustal nuclei. This coupling is modeled bye†ective potentials highly localized around discrete pinning sites in the vortex creep picture (e.g., Anderson & Itoh 1975 ;Alpar et al. 1984a ; Link & Epstein 1991 ; Link et al. 1993) and by scattering of particles by and Kelvin excitation of movingvortex lines not pinned to crustal nuclei (e.g., Epstein & Baym 1992 ; Jones 1991, 1992). In our calculations, we use smoothedhydrodynamical equations to describe the coupling between the superÑuid and normal components of the crust macroscopi-cally, using the formalism developed by Khalatnikov (1965, ° 16). This formulation of the problem is linked most naturally toa picture in which superÑuid vortex lines experience drag forces as they move through a smooth medium of normal Ñuid butalso may be applied directly in the vortex creep picture in the linear approximation (i.e., when the di†erence between theangular velocities of the superÑuid and normal Ñuid are sufficiently small).

ShahamÏs results are recovered in the limit of perfect coupling, that is, when the coefficients of mutual friction are inÐnite.We can explore whether qualitatively new modes appear when the mutual friction is strong but pinning is not perfect. As weshall see, no new slowly damped, long-period modes arise.


General formulae for the mutual friction force are given by Khalatnikov (1965, ° 16). If the superÑuid vorticity is deÐned tobe where is the superÑuid velocity, then the net force per unit volume acting on the superÑuid is$ Â ¿

s, ¿

sf \ [x Â ($ Â jm) Ô b@o

sx Â u [ bo

sm Â (x Â u) ] c@o

smx Æ u , (27)

where is the superÑuid mass density, l \ x/ o x o , and, for a normal Ñuid velocityos

¿n,

u 4 ¿n[ ¿

s[ 1

os

$ Â jm ; (28)

where i is the quantum of circulation per vortex line, d is the e†ective intervortex separation, and m isj \ (osi/4n) ln (d/m),

the coherence length. The mutual friction force is deÐned to be The parameters b and c@ must be positivef ] x Â ($ Â jm)/os.

for the rate of energy dissipation resulting from f to be greater than zero locally.Qualitatively, the terms in f involving l arise from the bending of vortex lines and will be neglected here. Of the remaining

contributions to f, the two proportional to b and b@ are perpendicular to x, whereas the one proportional to c@ is along x ; thelatter is expected to be small, and we neglect it too. With these simpliÐcations, the form for f used in this paper is

f \ [b@osx Â (¿

n[ ¿

s) [ bo

sm Â [x Â (¿

n[ ¿

s)] . (29)

For getting a qualitative feeling for the relative sizes of the phenomenological quantities b and b@, we use a di†erentparameterization for the strength of the mutual friction force, based on the idea of vortex drag. The equation for the superÑuidvelocity including mutual friction is

L¿s

Lt] ¿

sÆ $¿

s\ [$(k ] /) ] f

os

, (30)

where k is the chemical potential and / the gravitational potential ; taking the curl of this equation gives

Lx

Lt\ $ Â

A¿sÂ x ] f

os

B. (31)

If f \ 0, then the superÑuid vortex lines commove with the superÑuid, but, in general, the vortex lines have a di†erent velocity,and¿

LD ¿

s,

Lx

Lt\ $ Â (¿

LÂ x) ; (32)

from the form for f given in equation (29), we can read o†

¿L

\ ¿s] b@(¿

n[ ¿

s) ] bm ] (¿

n[ ¿

s) \ ¿

n] (b@ [ 1)(¿

n[ ¿

s) ] bm ] (¿

n[ ¿

s) . (33)

Only the components of perpendicular to m contribute to as can be seen from the original expression for f. Clearly,¿n[ ¿

s¿L,

vortex lines commove with the superÑuid if o b@ o and b are both small and commove with the normal Ñuid if o b@ [ 1 o and bare small. (If superÑuid rotates faster than normal Ñuid, vortices move slowly outward relative to normal Ñuid for b small.)The motion of a vortex is found by balancing the Magnus force due to superÑuid streaming past the line and any other forcesit experiences ; for our purposes, the latter are drag forces perpendicular to the line, so the equation of motion is

osim ] (¿

L[ ¿

s) ] F

d\ 0 . (34)

If the drag force per length on a vortex is

Fd\ [g(¿

L[ ¿

n) [ g@m ] (¿

L[ ¿

n) , (35)

then the vortex line velocity is

¿L

\ ¿s] [g2 [ g@(o

si [ g@)](¿

n[ ¿

s)

(osi [ g@)2 ] g2 ] gio

sm ] (¿

n[ ¿

s)

(osi [ g@)2 ] g2 , (36)

from which we infer the relations

b \ gosi

(osi [ g@)2 ] g2 , (37)

b@ \ 1 [ osi(o

si [ g@)

(osi [ g@)2 ] g2 . (38)

These results relate the drag coefficients g and g@ with the parameters b and b@ appearing in the mutual friction force.In microscopic models for mutual friction developed so far, the coefficient g@, which determines the magnitude of the drag

force perpendicular to the motion of a vortex line through the normal Ñuid, is negligible. If g@ \ 0, then equations (37) and (38)


simplify to

b \ gosi

(osi)2 ] g2 , (39)

b@ \ 1 [ (osi)2

(osi)2 ] g2 . (40)

From these relationships, we Ðnd that when vortex lines are dragged e†ectively and tend to follow the normal Ñuidg ? osi

closely ; in that limit, and When the drag is weak, vortex lines tend to follow theb B osi/g 1 [ b@ B (o

si/g)2 B b2. g > o

si

superÑuid and and As we shall see below, this means that the dissipative torque arising fromb B g/osi b@ B (g/o

si)2 B b2.

mutual friction is much larger than the nondissipative torque when the drag is either very strong or very weak ; the twotorques are only comparable when When the situation becomes more complicated, as equations (37) andg/o

si D 1. g@ D 0,

(38) involve two nondimensional parameters, and If we assume that g@ > g, then in the strong damping limit,g/os/i g@/o

si.

as before and if but In the weakly coupled domain,b B osi/g 1 [ b@ B (o

si/g)2 g@ > o

si, 1 [ b@ B [g@o

si/g B [(g@/g)b.

as before, but b@ B b2 only when and instead when As we shallb B g/osi g@/g > g/o

si, b@ B [g@/o

si B [(g@/g)b g@/g ? o

si.

see, these results will make the existence of long-term oscillatory modes problematic when g@ > g provided that at least part ofthe crust is coupled strongly to the crustal superÑuid. For the situation turns out to be more favorable for the survivalg@ Z gof oscillatory modes. In that case, and in the limit of strong coupling andb B go

si/g@2 1 [ b@ B [o

si/g@ B [(g@/g)b

and in the limit of weak coupling.b B g/osi b@ B [g@/o

si B [(g@/g)b

The torque that results from mutual friction is

N \Pd3r r Â f (r) 4 Nb ] Nb{ , (41)

where, from equation (29),

Nb \ [Pd3r bo

sr Â Mm Â [x Â (¿

n[ ¿

s)]N , (42)

Nb{ \ [Pd3r b@o

sr Â [x Â (¿

n[ ¿

s)] . (43)

We restrict ourselves to a uniformly rotating normal Ñuid, but the analogous restriction to uniformly rotating superÑuid isdynamically inconsistent unless b and b@ are independent of position. Consequently, we imagine that the star can be dividedinto ““ shells ÏÏ in which b and b@ are independent of position and the superÑuid rotates uniformly. In these shells, ¿

s\ X

sÂ r

and with and independent of r ; this also implies that In succeeding sections, we consider stars¿n\ X

nÂ r, X

nX

sx \ 2X

s.

with one and two superÑuid shells. These examples suffice to illustrate the complex behavior that may arise in a real neutronstar, where b and b@ vary continuously.

For uniform rotation, equations (42) and (43) become

Nb \ XsÂ Tb Æ (XŒ

sÂ X

n) ] )

s(X

s[ X

n) Æ [Tb [ dTr(Tb)] ; (44)

Nb{ \ [(Xs[ X

n) Â Tb{ Æ X

s, (45)

where

Tb 4 2bP

d3rosrr , (46)

Tb{ 4 2b@P

d3rosrr , (47)

d is the unit tensor, and is the trace of It is easy to show that and so thatTr(Tb) Tb. (Xs[ X

n) Æ Nb{ \ 0 (X

s[ X

n) Æ Nb \ 0,

that is, mutual friction torques are ultimately dissipative.(Xs[ X

n) Æ N \ 0,

The tensors and can be rather complicated in general. Even in uniformly rotating superÑuid shells, the superÑuidTb Tb{density may be slightly anisotropic, principally as a result of rotational Ñattening perpendicular to which is timeo

s(r) X

s,

varying and not aligned with any of the principal axes of the crust in general. However, we shall neglect these complications,since the magnitudes of the anisotropies in and are expected to be small for slowly rotating neutron stars, which weTb Tb{focus on here. Accordingly, we approximate

Tb \ Isbeff d , (48)

Tb{ \ Isbeff@ d , (49)

where is the moment of inertia of the superÑuid, and and are suitably averaged b and b@ ; henceforth, we drop theIs

beff beff@subscript ““ e†. ÏÏ With these expressions for and the mutual friction torques simplify toTb Tb{

Nb \ [Isb)

s(X

s[ X

n) Æ (d ] XŒ

sXŒ

s) ; (50)


Nb{ \ Isb@(X

nÂ X

s) . (51)

We shall devote much of the remainder of this paper to examining the consequences of torques of this form. Here we neglectother torques that could be important, such as gravitational torques (both Newtonian and post-Newtonian) or Ñuid torquesarising from boundary conditions, and ignore the various complications in and alluded to above. Some of these issuesTb Tb{will be discussed in a subsequent publication.

3. TWO-COMPONENT STAR

Implicit in the review of Shaham (1977) presented in ° 2.1 was a treatment of the two-component system consisting of therigid crust and pinned crustal superÑuid. This was the c \ 0 limit in which the crust and core decouple entirely. In that case,we found that the crust precesses at a frequency p (see eq. [12]) under the additional assumption of axisymmetry ; for c 4 0,this mode is undamped. The three-component model discussed in ° 2.1 also reduces to a two-component system when c ] O,in which case the core and crust are coupled perfectly and must corotate. Although this limit is not realistic (see discussion ofthe maximum possible c in the penultimate paragraph of ° 2.1), it also leads to undamped precession at a frequency

Here we examine how imperfect pinning alters these results and introduces both damping and new modes.pIc/(I1 ] I

c).

3.1. Free Precession ReexaminedThe coupled equations for the angular momenta of the crust and crustal superÑuid are

d(Icr

Æ Xcr)dt

\ [Nb [ Nb{ \ Isb)

s(X

s[ Xcr) Æ (d ] XŒ

sXŒ

s) ] I

sb@(X

sÂ Xcr) (52)

Is

dXs

dt\ Nb ] Nb{ \ [I

sb)

s(X

s[ Xcr) Æ (d ] XŒ

sXŒ

s) ] I

sb@(Xcr Â X

s) , (53)

where we have substituted for in equations (50) and (51) and assumed that the angular momentum of the superÑuid isXcr Xnwhich is tantamount to assuming that the moment of inertia tensor of the superÑuid is of the formI

sX

s,

Is\ (I

s[ *I

s)d ] *I

s

A3XŒsXŒ

s[ d

2B

. (54)

Notice that if b@ \ 1 and b \ 0, these equations reduce to

d(Icr

Æ Xcr)dt

] Is(Xcr Â X

s) \ 0 , (55)

which is equivalent to equation (1) with the contribution from the core component omitted, and

dXs

dt\ Xcr Â X

s, (56)

which implies that is Ðxed in the reference frame that rotates with the superÑuid. This is the limit of perfect pinning andXsresults in undamped precession at the frequency p.

When and there are additional modes. Let us work in the frame rotating with the crust, in which case (recallb@ D 1 b D 0,that d*F/dt is the time derivative of f in this frame)

Icr

Æ d*Xcrdt

] Xcr Â (Icr

Æ Xcr) \ Isb)

s(X

s[ Xcr) Æ (d ] XŒ

sXŒ

s) ] I

sb@(X

sÂ X

cr) , (57)

d*Xs

dt] (1 [ b@)(Xcr Â X

s) \ [b)

s(X

s[ Xcr) Æ (d ] XŒ

sXŒ

s) . (58)

If we project equations (57) and (58) along the principal axes of the crust, and linearize around the Ðxed point at whichand we ÐndXcr \ X

s\ X X p eü 3

)0 cr,1 ]CAI3 [ I2I1

B] Isb@

I1

D)cr,2 [ I

sb@

I1)

s,2 \ [ Isb

I1()cr,1 [ )

s,1) , (59)

)0 cr,2 [CAI3 [ I1I2

B] Isb@

I2

D)cr,1 ] I

sb@

I2)

s,1 \ [ Isb

I2()cr,2 [ )

s,2) , (60)

)0 cr,3 \ [2Isb

I3()cr,3 [ )

s,3) , (61)

)0s,1 [ (1 [ b@)()

s,2 [ )cr,2) \ [b()s,1 [ )cr,1) , (62)

)0s,2 ] (1 [ b@)()

s,1 [ )cr,1) \ [b()s,2 [ )cr,2) , (63)


)0s,3 \ [2b()

s,3 [ )cr,3) . (64)

As before, the evolution of the perturbations along the 3-axis decouple from those along the other axes and decay exponen-tially ; the rate of decay is The remaining equations imply a fourth-order characteristic equation if we search for2b(1 ] I

s/I3).

modes Pexp (p/).

3.1.1. Axisymmetric CrustWhen the crust is axisymmetric, and equations (59) and (60) simplify toI1 \ I2,

)0 cr,1 ]CAI3 [ I1I1

B] Isb@

I1

D)cr,2 [ I

sb@

I1)

s,2 \ [ Isb

I1()cr,1 [ )

s,1) , (65)

)0 cr,2 [CAI3 [ I1I1

B] Isb@

I1

D)cr,1 ] I

sb@

I1)

s,1 \ [ Isb

I1()cr,2 [ )

s,2) ; (66)

these couple to equations (62) and (63), which are unchanged.As we found in ° 2.1, the fourth-order characteristic equation may be reduced to second order in this case. DeÐne

p@ \AI3 [ I1I1

B] Isb@

I14 v ] I

sb@

I1(67)

and let

)s(`) \ )

s,1 ] i)s,2 ; (68)

then we get the two coupled equations

)0 cr(`) [ ip@)cr(`) ] iIsb@

I1)

s(`) \ [I

sb

I1[)cr(`) [ )

s(`)] , (69)

)0s(`) ] i(1 [ b@)[)

s(`) [ )cr(`)] \ [b[)

s(`) [ )cr(`)] . (70)

It turns out to be convenient to use

*(`) 4 )s(`) [ )cr(`) (71)

instead of doing so yields the coupled equations)s(`) ;

)0 cr(`) [ iv)cr(`) ] Is

I1(ib@ [ b)*(`) \ 0 , (72)

*0 (`) ]GiC

1 [ b@A

1 ] Is

I1

BD] bA

1 ] Is

I1

BH*(`) ] iv)cr(`) \ 0 . (73)

Assuming that we Ðnd the relation[)cr(`), *(`)] P exp (p/)

*(`) \ [ p)cr(`)p ] i(1 [ b@) ] b (74)

and the characteristic equation

p2 ] pC

i(1 [ b@ [ p@) ] bA

1 ] Is

I1

BD] v(1 [ b@ [ ib) \ 0 . (75)

Equation (74) is useful for Ðnding the eigenvectors once equation (75) is solved ; these are needed to determine the response ofthe two spin components to external torques.

Although we can solve equation (75) exactly, it is instructive to consider the two limiting cases of weak and strong vortexdrag separately. When vortex drag is weak, b and b@ are small in magnitude, so we rewrite equation (75) as

(p [ iv)(p ] i) [ (b@ ] ib)C

ipA

1 ] Is

I1

B] vD\ 0 . (76)

The solutions to this equation to Ðrst order in the small quantities b and b@ are

pd\ [i ] (ib@ [ b)(1 ] v ] I

s/I1)

1 ] v (77)

and (the Shaham mode)

pp\ iv ] v(ib@ [ b)I

sI1(1 ] v)

. (78)


Both of these solutions damp slowly, at rates proportional to b [ 0. The second mode reduces to the conventional Eulerprecession when b \ b@ \ 0. The Ðrst mode arises because the superÑuid angular velocity would remain Ðxed in the inertialframe if b and b@ were zero but wanders slowly when the coupling is small but nonzero. (We discuss this point more fully in thecontext of three-component models ; see ° 4.1.1 below and discussion following eq. [148].) For strong vortex drag, 1 [ b@ andb are small, and we rewrite equation (75) in the form

p(p [ ip@) ] (1 [ b@)(ip ] v) ] bC

pA

1 ] Is

I1

B[ ivD\ 0 . (79)

The solutions to this equation to Ðrst order in the small quantities b and 1 [ b@ are

pd\ [v[b ] i(1 [ b@)]

p@ , (80)

pp\ ip@ [ I

s[i(1 [ b@) ] b(1 ] p@)]

I1 p@ . (81)

The Ðrst of these solutions represents a slowly damped mode with an oscillatory part that is negligible when g@ > g, implying1 [ b@ > b in this limit. (Recall discussion following eqs. [37] and [38] in ° 2.2.) The second mode corresponds to precession atp@ with slow damping : For equation (67) implies that for 1 [ b@ > 1, so the damping rate is approximatelyI

s/I1 ? v, p@ B I

s/I1in the unlikely event that then p@ B v and the damping rate is approximatelyb(1 ] I

s/I1) ; v ? I

s/I1, (I

s/I1 v)b(1 ] v).

3.1.2. Nonaxisymmetric CrustSince the neutron star crust may not be axisymmetric, it is worth checking that there are no surprises when DeÐneI1 D I2.

p1@ \AI3 [ I2I1

B] Isb@

I14 v1 ] I

sb@

I1, (82)

p2@ \AI3 [ I1I2

B] Isb@

I24 v2 ] I

sb@

I2; (83)

then equations (59) and (60) become

)0 cr,1 ] p1@ )cr,2 [ Isb@

I1)

s,2 \ [ Isb

I1()cr,1 [ )

s,1) , (84)

)0 cr,2 [ p2@ )cr,1 ] Isb@

I2)

s,1 \ [ Isb

I2()cr,2 [ )

s,2) , (85)

which must be solved along with equations (62) and (63). When we look for solutions Pexp (p/) we Ðnd the fourth-ordercharacteristic equation

0 \ p4 ] p3bA

2 ] Is

I1] I

sI2

B] p2G[b2 ] (1 [ b@)2]

A1 ] I

sI1

BA1 ] I

sI2

B[ (1 [ b@)2 Is2

I1 I2[ (1 [ b@)

AIs

I1] I

sI2

B] p1@ p2@H

] pbA

2v1 v2 ] Isv2

I1] I

sv1

I2

B] v1 v2[b2 ] (1 [ b@)2] . (86)

It is not possible to factorize the characteristic equation into the product of two second-order equations because there is noguarantee that all of the roots are simply complex conjugate pairs.

In the limit of weak coupling, expanding equation (86) up to Ðrst order in b and b@ yields

0 \ (p2 ] 1)(p2 ] v1 v2) ] bC

p3A2 ] Is

I1] I

sI2

B] pA

2v1 v2 ] Isv2

I1] I

sv1

I2

BD[ b@

Gp2C2 ] I

s(1 [ v2)

I1] I

s(1 [ v1)

I2

D] 2v1 v2H

. (87)

The approximate solutions of this form for the characteristic equation are

pd\ [i ] [2(1 [ v1 v2) ] (I

s/I1)(1 [ v2) ] (I

s/I2)(1 [ v1)](ib@ [ b)

2(1 [ v1 v2), (88)


pp\ iJv1 v2 ] ib@Jv1 v2 [(I

s/I1)(1 [ v2) ] (I

s/I2)(1 [ v1)]

2(1 [ v1 v2)

[ b[(Is/I1)v2(1 [ v1) ] (I

s/I2)v1(1 [ v2)]

2(1 [ v1 v2), (89)

and their complex conjugates. To get limiting results in the strongly coupled domain, we rewrite equation (86) in the slightlymodiÐed form

0 \ p2(p2 ] p1@ p2@ ) ] p3bA

2 ] Is

I1] I

sI2

B] p2G[b2 ] (1 [ b@)2]

A1 ] I

sI1

BA1 ] I

sI2

B[ (1 [ b@)2 Is2

I1 I2[ (1 [ b@)

AIs

I1] I

sI2

BH] pb

A2v1 v2 ] I

sv2

I1] I

sv1

I2

B] v1 v2[b2 ] (1 [ b@)2] . (90)

From equation (90), it is evident that the modes are near p2 \ 0 and To get the Ðrst-order approximation to thep2 \ [p1@ p2@ .modes with p2 \ 0 to zeroth order in b and 1 [ b@, we identify all terms in equation (90) that are potentially second order insmall quantities ; this leads to the quadratic equation

0 \ p2p1@ p2@ ] pbA

2v1 v2 ] Isv2

I1] I

sv1

I2

B] v1 v2[b2 ] (1 [ b@)2] , (91)

which has the pair of roots

pdB \ [b(v1 p2@ ] v2 p1@ )

2p1@ p2@^Cb2(v1 p2@ [ v2 p1@ )2

(2p1@ p2@ )2 [ v1 v2(1 [ b@)2p1@ p2@

D1@2. (92)

In the axisymmetric limit, these two roots reduce to the damped mode found in ° 3.1.1 and its complex conjugate, butalthough both imply damping in general, they could be purely real and di†erent in magnitude, especially since we expectp

dB

1 [ b@ to be much smaller than b in the strongly coupled regime. (This is why we could not factor eq. [86] into two quadraticequations.) It is also straightforward to expand equation (90) around the approximate root to Ðndp B iJp@1 p@2

pp\ iJp1@ p2@ [ i(1 [ b@)

2Jp1@ p2@AI

sI1

] Is

I2

B[ b2Jp1@ p2@

CIs

I1p2@ (1 ] p1@ ) ] I

sI2

p1@ (1 ] p2@ )D

(93)

to Ðrst order in the small quantities b and 1 [ [email protected] this brief foray into the modes of a triaxial star, we conclude that deviations from axisymmetry do not alter the

behavior of the precession qualitatively in either limit. The character of the damped modes may be di†erent in the strongcoupling limit for nonaxisymmetric stars, but if so, they become purely damped, with no oscillation at all. Consequently, fromhere on we specialize to axisymmetric crusts, since we do not expect to miss any important oscillatory modes.

3.2. Response to External TorquesAs we have seen, the modes of free precession for this two-component model are rapidly oscillating and/or damped. Here,

we consider the response of the system to external torques. Our analysis will reveal that the limit of perfect coupling betweenthe normal Ñuid and superÑuid must be taken with care when external torques act.

If we suppose that the crust is subject to an arbitrary time-dependent torque, then equations (72) and (73) areNcr(/),changed to

)0 cr(`) [ iv)cr(`) ] Is

I1(ib@ [ b)*(`) \ N3 cr(`)(/) , (94)

*0 (`) ]GiC

1 [ b@A

1 ] Is

I1

BD] bA

1 ] Is

I1

BH*(`) ] iv)cr(`) \ [N3 cr(`)(/) , (95)

where Apart from decaying transients, the solution to these equations isN3 cr(`) 4 I1~1(Ncr,1 ] iNcr,2).

)cr(`) \ ;a/p,d

AaP~=

Õd/@N3 cr(`)(/@) exp [pa(/ [ /@)] , (96)

*(`) \ [ ;a/p,d

pa Aapa ] i(1 [ b@) ] b

P~=

Õd/@N3 cr(`)(/@) exp [pa(/ [ /@)] , (97)

where the coefficients are

Ap\ [p

p] i(1 [ b@) ] b]

(pp[ p

d)

, Ad\ [p

d] i(1 [ b@) ] b]

(pd[ p

p)

. (98)


(Eq. [74] is useful in obtaining the Aa.)In the limit of perfect coupling between the crust and superÑuid, we take i(1 [ b@) ] b ] 0 with pd/[i(1 [ b@) ] b] \ [v/p@

constant ; taking this limit of equations (96) and (97) naively yields

)cr(`) \P~=

Õd/@N3 cr(`) exp [p

p(/ [ /@)] , (99)

with *(`) ] [)cr(`).Actually, the perfect coupling limit is a bit more subtle than the manipulations leading to equation (99). To see why,

consider the response to a time-independent torque on the crust. Then equation (96) may be integrated readily and we Ðnd

)cr(`) \ iN3 cr(`)v ; (100)

integrating equation (97) yields *(`) \ 0. These results ought to hold for any i(1 [ b@) ] b. However, equation (99), whichpurports to describe the limit of perfect coupling, i(1 [ b@) ] b 4 0, yields

)cr(`) \ [ N3 cr(`)pp

]iN3 cr(`)

p@ (101)

(assuming a small, negative real part to Which of these results is correct ?pp).

To resolve the conundrum, consider a torque that turns on (or can be regarded as constant since) some time in the past, /0.Then equations (96) and (97) yield the response

)cr(`) \G iv ]C

1 ] i(1 [ b@) ] bpp

D exp [pp(/ [ /0)]

pp[ p

d[C

1 ] i(1 [ b@) ] bpd

D exp [pd(/ [ /0)]

pp[ p

d

HN3 cr(`) , (102)

*(`) \G[ exp [pp(/ [ /0)] ] exp [p

d(/ [ /0)]

pp[ p

d

HN3 cr(`) . (103)

We suppose that the damping constant associated with the precessing mode, is large enough thatpp, exp [p

p(/ [ /0)] ] 0 ;

then what we Ðnd depends on (Recall that the real parts of and are negative.) If sopd(/ [ /0). p

ppd

o Re[pd(/ [ /0)] o ? 1,

that any transient response has plenty of time to damp away between and /, then we recover equation (100) and also Ðnd/0that *(`) ] 0. On the other hand, if then we recover equation (101) and in the limit ofo Re [pd(/ [ /0)] o > 1, *(`) ] [)cr(`)

perfect coupling, i(1 [ b@) ] b ] 0, to zeroth order in To Ðrst order we also Ðnd a term that grows linearly withpd(/ [ /0).

over a sufficiently long time span, this growth would change the tilt from equation (100) to equation (101).pd(/ [ /0) ;The resolution of the apparent paradox is that there is none : equations (96) and (97) are always the correct ones to use.

What one gets in the limit of strong vortex drag depends on how the timescale on which the external torque changes compareswith the timescales inherent in the coupling of superÑuid to the normal crust. Equations (100) and (101) both have domains ofvalidity ; equation (101) is a lower bound to the steady state tilt of the rotational angular velocity away from the 3-axis in thestrong coupling limit. As long as the damping timescale associated with is short compared with any timescale associatedp

dwith changes in the external torque, however, equation (100) gives the right response. In practical terms, pulsar spin-downprovides a nearly constant torque on the crust, which can give rise to Thus, if implies decay on timescales smaller thanN3 cr(`). p

dthe pulsar spin-down time, then equation (100) describes the response of the crust, even in the strong pinning limit.We emphasize that this result could not be found from a consideration of normal modes alone ; arriving at it requires

examining the response of the star to a torque. There is therefore a subtle aspect to the limiting case considered by Shaham(1977) : While the modal frequencies he derived are correct, and his conclusions about free precession warranted, blithely usingequation (99), which would follow from the assumption of perfect pinning, instead of equations (96) and (97), is wrong even inthe limit of strong pinning.

The response of the star to torques along the 3-axis is found by solving the equations

)0 cr,3 [ 2Isb

I3*3 \ N3 cr,3(/) , (104)

*0 3 ] 2bA

1 ] Is

I3

B*3 \ [N3 cr,3(/) , (105)

where the result isN3 cr,3 \ Ncr,3/I3 ;

*3 \ [P~=

Õd/@N3 cr,3(/@) exp

C[2bA

1 ] Is

I3

B(/ [ /@)

D, (106)

)cr,3 \ I3I3 ] I

s

P~=

Õd/@N3 cr,3(/@) ] I

sIs] I3

P~=

Õd/@N3 cr,3(/@) exp

C[2bA

1 ] Is

I3

B(/ [ /@)

D. (107)


In particular, a time-independent torque results in a steady angular velocity di†erence

*3 \ [ N3 cr,32b(1 ] I

s/I3)

, (108)

while the crustal angular velocity changes linearly :

)0 cr,3 \ N3 cr,3 I3Is] I3

. (109)

The response to an impulsive torque isN3 cr,3 \ N0,3 d(/ [ /0)

)cr,3 \ N0,3MI3 ] Is

exp [[2b(1 ] Is/I3)(/ [ /0)]N

Is] I3

, (110)

*3 \ [N0,3 exp [[2b(1 ] Is/I3)(/ [ /0)] . (111)

Spin-down torques, which only change on very long timescales for all observed pulsars, provide a physical realization of a““ time-independent ÏÏ torque. Let us take the vacuum magnetic dipole torque, which is proportional to [ k ] (x Â k) (e.g.,Davis & Goldstein 1970 ; Goldreich 1970 ; Michel 1991) ; assuming a magnetic dipole moment k that is Ðxed in the frame ofthe crust, with components

k \ cos aeü 3 ] sin aeü 1 , (112)

the torque may be written (to a good Ðrst approximation) as

Nd\ N

dsin a(eü 1 cos a [ eü 3 sin a) 4 Nsd(eü 1 cot a [ eü 3) , (113)

implying and The steady state response to these torques isN3 cr(`) \ Nsd cot a/I1 N3 cr,3 \ [Nsd/I3.

)cr(`) \ iNsd cot avI1

, *(`) \ 0 , (114)

*3 \ Nsd2b(I

s] I3)

, (115)

with Two aspects of these results are especially noteworthy. First, as is well known, the superÑuid rotates)0 cr,3 \ [Nsd/I3.faster than the crust as a consequence of the spin-down torque. Second, but not so widely appreciated, the steady state ““ tilt ÏÏin the angular velocity of the crust is surprisingly large, as it is proportional to v~1. (When v \ 0, eq. [75] implies that p

d\ 0,

and grows linearly with time according to eq. [96].))cr(`)3.3. Free Precession Encore

3.3.1. Di†erent Crust and SuperÑuid Angular Velocities in Unperturbed StateThe fact that time-independent spin-down of the crust implies a di†erence between and in steady state suggests)cr,3 )

s,3that we explore free precession once again, but with a di†erent unperturbed state than we used in ° 3.1. There we assumed thatthe undisturbed star rotates with Let us consider instead what happens when and inX

s\ Xcr \ )eü 3. X

c\ )eü 3 X

s\ m)eü 3the unperturbed state. We shall not assume that o m [ 1 o must be small, although we expect this to be true.

The equations governing the precession of an axisymmetric star given this new unperturbed state are

)0 cr(`) [ ipm@ )cr(`) ] iIsb@

I1)

s(`) \ [ I

sb

I1[)cr(`) [ )

s(`)] , (116)

)0s(`) ] i(1 [ b@)[)

s(`) [ m)cr(`)] \ [b[)

s(`) [ )cr(`)] , (117)

where

pm@ 4 v ] b@Ism

I1. (118)

These equations yield the characteristic equation

p2 ] pC

i(1 [ b@ [ pm@ ) ] bA

1 ] Is

I1

BD] v(1 [ b@ [ ib) ] ibI

sI1

(1 [ m) \ 0 ; (119)

in the weakly coupled limit, the solutions are

pd\ [i ] (ib@ [ b)(1 ] v ] mI

s/I1)

1 ] v , (120)

pp\ iv ] Mivb@m [ b[v [ (m [ 1)]NI

sI1(1 ] v)

, (121)


and in the strong coupling limit,

pd\ [iv(1 [ b@) ] b[v ] I

s(m [ 1)/I1]

v ] mIs/I1

, (122)

pp\ ipm@ [ I

s[im(1 [ b@) ] b(1 ] pm@ )]

I1 pm@. (123)

These equations are hardly di†erent than their m \ 1 counterparts except for two noteworthy di†erences. First, in the weaklycoupled limit, the real part of can become positive for m [ 1 [ v, implying linear instability. However, this is not worrisomep

psince the growth time of the mode is of order the spin-down time of the star if m [ 1 \ Nsd/2b(Is] I3).

Second, in the strongly coupled limit, the real part of is enhanced for m [ 1 [ 0, implying faster damping. Otherwise, thepde†ect of is merely to renormalize the various coefficients appearing in the solutions to the characteristic equationm D 1

without introducing any qualitatively new behavior. Consequently, we shall not consider the e†ects of di†erential rotationfurther in this paper.

3.3.2. Interaction of Two Regions with Simple ModesIn °° 3.1.1 and 3.1.2, we explored the characteristic modes of precession for axisymmetric and nonaxisymmetric crusts in the

limits of weak and strong coupling to the crustal superÑuid. In general, we can imagine that there are regions of the crust towhich the superÑuid couples with di†erent strengths. These regions are presumably linked to one another by elastic forcesthat seek to enforce corotation.

To get a crude idea of what might transpire in such a situation, let us imagine dividing the crust into two components, a andb. To get a schematic feeling for the normal modes of the coupled system, suppose the angular velocities can be described bythe equations

X0a(`) [ p

aX

a(`) \ [ cI

bIa] I

b[X

a(`) [ X

b(`)] , (124)

X0b(`) [ p

bX

b(`) \ [ cI

aIa] I

b[X

b(`) [ X

a(`)] , (125)

where and This system of equations has the characteristic equationXa(`) \ X

a,1 ] iXb,1 X

b(`) \ X

b,1 ] iXb,2.

0 \ p2 [ p(pa] p

b[ c) ] p

apb[ c(p

aIa] p

bIb)

Ia] I

b. (126)

When c is small, we rewrite the above equation as

0 \ (p [ pa)(p [ p

b) ] c

Ap [ p

aIa] p

bIb

Ia] I

b

B; (127)

to Ðrst order in c, the roots are

p1 \ pa[ cI

bIa] I

b, (128)

p2 \ pb[ cI

aIa] I

b. (129)

The e†ect of weak coupling is additional damping of the modes of the individual components. When c is large, we rewrite thecharacteristic equation as

0 \ p [ paIa] p

bIb

Ia] I

b] c~1(p [ p

a)(p [ p

b) . (130)

In this limit the two roots are

p1 \ [c ] paIb] p

bIa

Ia] I

b, (131)

p2 \ paIa] p

bIb

Ia] I

b[ (p

b[ p

a)(I

a[ I

b)

c(Ia] I

b)

. (132)

In this case, represents almost pure damping at a rate close to c, whereas represents damped precession at a rate that isp1 p2approximately the sum of the precession frequencies of the individual components weighted by their moment of inertiafractions.

Consequently, if the characteristic coupling timescales among di†erent components of the crust are short, we Ðnd a meanprecession frequency weighted toward regions that comprise the bulk of the crustal moment of inertia. Only if the couplingtimes are long will the precession frequencies of individual crustal components be apparent. This indicates that even if thereare small regions of unpinned superÑuid in the crust, precession at the Euler rate appropriate to those zones may only be seen


if they do not couple to the rest of the crust efficiently, and that even if this is so, the precession will damp out eventually as aconsequence of the dissipative interaction.

From time to time it has also been suggested that Tkachenko modes of the core superÑuid could manifest themselves in theobserved rotation of a pulsar, which is presumably the angular velocity of its crust. If the crustal superÑuid is strongly pinned,we can regard one component, a, as the crust, with and the other as the core, with the oscillationp

a\ p \ v ] I

s/I1, p

bfrequency of the core in the zero coupling limit. Equation (129) shows that if the interaction between crust and core is weak,there is indeed a mode of oscillation close to For small but nonzero c the oscillations decay at a rate whichp

b. cI

a/(I

a] I

b),

implies a damping time of rotation periods, if we assume this would be of orderDc~1Ic/I1 I

a\ I1 > I

b\ I

c; (400È104)I

c/I1for the damping times estimated by Alpar & Sauls (1988), or approximately days to months for comparable to theI

c/I1 D 102,

estimated periods of oscillation if they can occur. Moreover, for small values of c, the e†ect of the core oscillations on thecrustal angular velocity is small : equation (124) implies that

Xa(`) \C cI

b(I

b] I

a)(p

b[ p

a) ] cI

b

DX

b(`) , (133)

which means that the amplitude of the oscillations in the angular velocity of the crust is times the amplitudeDcIb/p

a(I

b] I

a)

of the oscillations of the angular velocity of the core, assuming that Thus, even if the coupling is so weak that theo pao ? o p

bo .

damping time associated with core oscillations is very long, their observable manifestation in the angular velocity of thetdcrust is suppressed by a factor of D()t

d)~1.

4. THREE-COMPONENT STAR

The calculations presented in ° 3 are valid if the crust and crustal superÑuid are either completely decoupled from the coreof the star or coupled to it perfectly. In the limit of perfect pinning, Shaham (1977) found that precession was damped as aconsequence of imperfect coupling to the core. Here we examine a model consisting of three components, two of which aresuperÑuid components that couple directly to the rigid crust. We have in mind two possible applications, one in which thethree components are rigid crust, crustal superÑuid, and core (super)Ñuid, and another in which the three components arerigid crust and two di†erent regions of crustal superÑuid with di†erent frictional couplings to the rigid crust.

4.1. Free Precession4.1.1. Crust, Crustal SuperÑuid, and Core (Super)Fluid

We assume that the core couples directly only to the rigid component of the crust, via a torque of the form

Ncc \ [Icf)

c(X

c[ Xcr) ] I

cf@(Xcr Â X

c) , (134)

for small di†erences between the angular velocities of the crust and core, which are both nearly This form of the torque is)eü 3.analogous to equations (50) and (51), except that the dissipative torque has been assumed to be isotropic (by contrast to Nb).We have also included a nondissipative torque in unlike Shaham (1977 ; see ° 2.1) ; this contribution can be ignored byNcc,setting f@ \ 0. Below, we shall assume that f D f@, at least for keeping track of small quantities.

If we deÐne

*c(`) \ )

c,1 [ )cr,1 ] i()c,2 [ )cr,1) , (135)

then the coupled equations for the three-component star may be written in the form

)0 cr(`) [ iv)cr(`) ] Is

I1(ib@ [ b)*(`) ] I

cI1

(if@ [ f)*c(`) \ 0 , (136)

*0 (`) ]GiC

1 [ b@A

1 ] Is

I1

BD] bA

1 ] Is

I1

BH*(`) ] iv)cr(`) [ I

cI1

(if@ [ f)*c(`) \ 0 , (137)

*0c(`) ]G

iC

1 [ f@A

1 ] Ic

I1

BD] fA

1 ] Ic

I1

BH*

c(`) ] iv)cr(`) [ I

sI1

(ib@ [ b)*(`) \ 0 . (138)

Assuming modes proportional to exp (p/), we Ðnd that

*(`) \ [ p)cr(`)p ] i(1 [ b@) ] b , *

c(`) \ [ p)cr(`)

p ] i(1 [ f@) ] f . (139)

The third-order characteristic equation for this system may be written in the form

0 \ (p ] i)G

p2 ] pC

i(1 [ b@ [ p@) ] bA

1 ] Is

I1

BD] v(1 [ b@ [ ib)H

[ (if@ [ f)C

p2A1 ] Ic

I1

B] pG

iC

(1 [ b@)A

1 ] Ic

I1

B[ p@D] b

A1 ] I

s] I

cI1

BH] v(1 [ b@ [ ib)

D, (140)

where p@ is deÐned in equation (67).


When f and f@ are small, two of the solutions of equation (140) are close to the solutions and of the second-orderpd

ppequation (75) ; to Ðrst order in f and f@, the corrections to and arep

dpp

dpd\ (I

c/I1)(if@ [ f)[p

d(ip@ [ bI

s/I1) [ v(1 [ b@ [ ib)]

(i ] pd)(p

d[ p

p)

, (141)

dpp\ (I

c/I1)(if@ [ f)[p

p(ip@ [ bI

s/I1) [ v(1 [ b@ [ ib)]

(i ] pp)(p

p[ p

d)

, (142)

respectively. The third solution is a new mode near p \ [i ; to Ðrst order in small quantities it is

pd@ \ [i ] i(ib@ [ b)(if@ [ f)[1 ] v ] (I

s] I

c)/I1]

(i ] pd)(i ] p

p)

. (143)

When b and b@ are small, we use equations (77) and (78) in equations (141) and (142) to Ðnd

dpd\ I

c(if@ [ f)(v ] I

s/I1)

I1(1 ] v), (144)

dpp\ I

cv(if@ [ f)

I1(1 ] v); (145)

the new mode is

pd@ \ [i ] (if@ [ f)(1 ] v ] I

s/I1 ] I

c/I1)

1 ] v ] Is/I1

. (146)

When b and 1 [ b@ are small, on using equations (80) and (81) in equations (141) and (142) we Ðnd that is higher order in bdpdand 1 [ b@, and hence very small, while

dpp\ I

c(if@ [ f)p@

I1(1 ] p@) , (147)

pd@ \ [i ] (if@ [ f)(1 ] p@ ] I

c/I1)

1 ] p@ ; (148)

these two results are equivalent to equations (23) and (22) of ° 2.1 if we substitute p for p@ and [c for The(if@ [ f)(1 ] Ic/I1).

qualitative conclusion reached by Shaham (1977) for weak crust-core coupling is duplicated here : according to equations(145) and (147), precession damps out in precession periods irrespective of the e†ectiveness of vortex drag. We note,DI1/I

cf

however, that the mode corresponding to implies angular velocities that are nearly but not precisely Ðxed in the inertialpd@

frame of the observer ; if f@ [ f, these could complete at least one period of oscillation before decaying away (although we donot expect this to be the case generally). Such modes are also found in studies of the rotation of the Earth, where they mayarise from departures from rigid rotation of the Ñuid core conÐned by the overlying crust ; for the Earth, the result is aretrograde motion of the pole (see Lambeck 1980, ° 3.3 for a physical and historical review). Since the mode arises because thecore angular velocity remains Ðxed in the inertial frame when f and f@ are zero identically, we expect that for small but Ðnitecrust-core coupling, the mode corresponds principally to oscillations of the angular velocity of the core. From equation (139)we Ðnd

)cr(`)*

c(`) B (f@ ] if)

Ap@ ] I

cI1

BB (f@ ] if)

AIs] I

cI1

B(149)

for this mode, which decreases linearly with the frequency of oscillation, but may be substantial nevertheless if Is] I

c? I1.

For large values of o if@ [ f o , one of the roots of equation (140) is

pd@ \ (if@ [ f)

A1 ] I

cI1

B[ iA

1 [ p@Ic

Ic] I1

B, (150)

which is equivalent to equation (25) if we substitute p for p@ and [c for this root does not depend on the(if@ [ f)(1 ] Ic/I1) ;

strength of the vortex drag to lowest order in o if@ [ f o~1. When vortex drag is weak, the other two roots of equation (140) are

pd\ [i ] (ib@ [ b)(1 ] v ] I

s/I1 ] I

c/I1)

1 ] v ] Ic/I1

, (151)

pp\ iv

1 ] Ic/I1

] Isv(ib@ [ b)

I1(1 ] v ] Ic/I1)

[ Icv(if@ ] f)(1 ] v ] I

c/I1)

I1[(f@)2 ] f2](1 ] Ic/I1)3 . (152)


(Corrections to eq. [151] proportional to o if@ [ f o~1 are higher order in b and b@ and have been dropped.) In the limit ofstrong vortex drag,

pd\ [v[i(1 [ b@) ] b]

p@ , (153)

pp\ ip@

1 ] Ic/I1

[ Is

I1 p@G

i(1 [ b@) ] bC

1 ] p@ [ IsIc

I1(I1 ] Ic)DH[ I

cp@(if@ ] f)(1 ] v ] I

c/I1)

I1[(f@)2 ] f2](1 ] Ic/I1)3 . (154)

(Corrections to eq. [153] proportional to o if@ [ f o~1 are higher order in b and 1 [ b@ and have been dropped.) The correctionfor strong but imperfect crust-core coupling in equation (154) is equivalent to equation (26) if [c is substituted for (if@ [ f)(1 ] I

c/I1).

Interaction with the core of the star enhances the damping of the precessing modes in all of the limiting cases exploredabove. For very small or very large coupling between the crust and crustal superÑuid, the crust-core interactions are theprincipal cause of decay. In the strongly pinned regime, the characteristic timescales for decay are just what Shaham (1977)estimated. Imperfect pinning allows a new eigenvalue but the associated mode damps out quickly and so cannot be thep

d,

explanation for observations of persistent cyclical variations in pulsar spin rates.4.1.2. Crust Coupled to Two Di†erent Regions of Crustal SuperÑuid

The equations governing the spin dynamics of this system are the same as equations (136), (137), and (138) if we identify Xcwith the angular velocity of the second crustal superÑuid component and f and f@ with the coefficients coupling this

component to the rigid crust. Then it is clear that the characteristic equation for the normal modes of this system is stillequation (140), which we rewrite in the form

0 \ p3 [ ip2Av ] Is] I

cI1

B] (1 [ b@ [ ib)

Cip2A1 ] I

sI1

B] pA

v ] Ic

I1

BD] (1 [ f@ [ if)

Cip2A1 ] I

cI1

B] pA

v ] Is

I1

BD] (1 [ b@ [ ib)(1 [ f@ [ if)

Civ [ p

A1 ] I

s] I

cI1

BD, (155)

which exhibits symmetry under interchange of superÑuid components explicitly.This form of the characteristic equation is especially useful when both superÑuid components couple strongly to the rigid

crust. In that limit one of the roots is

pp\ ip@@ [A1 ] p@@

p@@BG

[i(1 [ b@) ] b]Is

I1] [i(1 [ f@) ] f]

Ic

I1

H(156)

to Ðrst order in small quantities, where

p@@ 4 v ] Ic] I

sI1

; (157)

the appearance of this root suggests a simple generalization to a multicomponent superÑuid, with separate moments of inertiaand coupling coefficients andI

s,j bj

bj@ :

pp\ ip@@ [A1 ] p@@

p@@B

;j

[i(1 [ bj@) ] b

j]

Is,jI1

, (158)

where

p@@ 4 v ] ;j

Is,jI1

. (159)

The other two roots are Ðrst order small to leading order ; they are approximately equal to the two roots of the quadraticequation

0 \ p2p@@ ] ipC

(1 [ b@ [ ib)A

v ] Ic

I1

B] (1 [ f@ [ if)A

v ] Is

I1

BD[ v(1 [ b@ [ ib)(1 [ f@ [ if) . (160)

When the crust is only slightly nonspherical, so v > 1, the two roots of this equation are approximately

p`

B [ [i(1 [ b@) ] b](Ic/I1) ] [i(1 [ f@) ] f](I

s/I1)

p@@ . (161)


p~ B [ v[i(1 [ b@) ] b][i(1 [ f@) ] f][i(1 [ b@) ] b](I

c/I1) ] [i(1 [ f@) ] f](I

s/I1)

. (162)

Notice that when one of the superÑuid components is coupled to the rigid crust more strongly than the other, is dominatedp`by the tighter coupled component, but the slow mode is dominated by the weaker coupled one and reduces to equationp~(80), with if and ifp@ B I

s/I1 I

so i(1 [ f@) ] f o ? I

co i(1 [ b@) ] b o p@ B I

c/I1 I

co i(1 [ b@) ] b o ? I

so i(1 [ f@) ] f o .

The limit in which one component couples strongly to the crust while the other couples only weakly is also potentially ofinterest, particularly in the aftermath of a pulsar glitch, in which some parts of the crustal superÑuid may decouple rapidly andrecouple only slowly if at all (e.g., Sedrakian 1995). The modes for that situation are given by equation (81) with the correctiongiven in equation (147), equation (80), and equation (148). If f@ [ f, it is possible that the mode corresponding to given byp

d@

equation (148) yields slowly damped oscillations in the angular velocity, as was discussed in ° 4.1.1.

4.2. Response to External TorquesThis section is analogous to ° 3.2, except that we need to consider three distinct external torques, acting on the crust, crustal

superÑuid and core, respectively, and the linear response of the three di†erent components to each.

4.2.1. Response to External Torques on the CrustWhen the crust is subject to an external torque Ncr(/),

)0 cr(`) [ iv)cr(`) ] Is

I1(ib@ [ b)*(`) ] I

cI1

(if@ [ f)*c(`) \ N3 cr(`)(/) , (163)

*0 (`) ]GiC

1 [ b@A

1 ] Is

I1

BD] bA

1 ] Is

I1

BH*(`) ] iv)cr(`) [ I

cI1

(if@ [ f)*c(`) \ [N3 cr(`)(/) , (164)

*0c(`) ]G

iC

1 [ f@A

1 ] Ic

I1

BD] fA

1 ] Ic

I1

BH*

c(`) ] iv)cr(`) [ I

sI1

(ib@ [ b)*(`) \ [N3 cr(`)(/) . (165)

It is straightforward to show that these equations have the particular solution

)cr(`) \ ;p,d,d@

AaP~=

Õd/@N3 cr(`)(/@) exp [pa(/ [ /@)] , (166)

*(`) \ [ ;p,d,d@

pa Aapa ] i(1 [ b@) ] b

P~=

Õd/@N3 cr(`)(/@) exp [pa(/ [ /@)] , (167)

*c(`) \ [ ;

p, d, d@

pa Aapa ] i(1 [ f@) ] f

P~=

Õd/@N3 cr(`)(/@) exp [pa(/ [ /@)] , (168)


Ap\ [p

p] i(1 [ b@) ] b][p

p] i(1 [ f@) ] f]

(pp[ p

d)(p

p[ p

d@ ) , (169)

Ad\ [p

d] i(1 [ b@) ] b][p

d] i(1 [ f@) ] f]

(pd[ p

d@ )(p

d[ p

p)

, (170)

Ad@ \ [p

d@ ] i(1 [ b@) ] b][p

d@ ] i(1 [ f@) ] f]

(pd@ [ p

d)(p

d@ [ p

p)

. (171)

Qualitatively, these are similar to what we found for the response of a two-component star, except that the responsetimescales are shorter as a consequence of crust-core coupling, which implies (perhaps signiÐcantly) enhanced decay of themodes. Notice that as the coupling between the rigid crust and crustal superÑuid becomes perfect, and the*(`) ] [)cr(`),response simpliÐes to

)cr(`) \ (pp[ p

d@ )~1P

~=

Õd/@N3 cr(`)(/@)M[p

p] i(1 [ f@) ] f] exp [p

p(/ [ /@)]

[ [pd@ ] i(1 [ f@) ] f] exp [p

d@ (/ [ /@)]N , (172)

*c(`) \ (p

p[ p

d@ )~1P

~=

Õd/@N3 cr(`)(/@)M[p

pexp [p

p(/ [ /@)] ] p

d@ exp [p

d@ (/ [ /@)]N ; (173)

these results are identical with equations (96) and (97) for the response of a two-component star, with the substitution of *c(`)

for *(`), for and i(1 [ f@) ] f for i(1 [ b@) ] b. However, their usefulness is restricted to that vary on timescalespd@ p

d, N3 cr(`)(/)

short compared with the characteristic damping time implied by the slowest decaying mode, as was discussed in ° 3.2.pd,


The response to a time-independent torque on the crust is simply just as in the two-component case, with)cr(`) \ iN3 cr(`)/v,These are the approximate responses obtained for a constant torque originating at a Ðnite time in the past*(`) \ *

c(`) \ 0. /0provided that p

d(/ [ /0) ? 1.

4.2.2. Response to Torques on the Core (Super)FluidIf the core is subject to a torque thenN

c(/)

)0 cr(`) [ iv)cr(`) ] Is

I1(ib@ [ b)*(`) ] I

cI1

(if@ [ f)*c(`) \ 0 , (174)

*0 (`) ]GiC

1 [ b@A

1 ] Is

I1

BD] bA

1 ] Is

I1

BH*(`) ] iv)cr(`) [ I

cI1

(if@ [ f)*c(`) \ 0 , (175)

*0c(`) ]G

iC

1 [ f@A

1 ] Ic

I1

BD] fA

1 ] Ic

I1

BH*

c(`) ] iv)cr(`) [ I

sI1

(ib@ [ b)*(`) \ N3c(`)(/) , (176)

where Apart from decaying transients, the solution to these equations isN3c(`) 4 I

c~1(N

c,1 ] iNc,2).

)cr(`) \ ;p,d,d@

BaP~=

Õd/@N3

c(`)(/@) exp [pa(/ [ /@)] , (177)

*(`) \ [ ;p,d,d@

pa Bapa ] i(1 [ b@) ] b

P~=

Õd/@N3

c(`)(/@) exp [pa(/ [ /@)] , (178)

*c(`) \ [ ;

p, d, d@

pa Bapa ] i(1 [ f@) ] f

P~=

Õd/@N3

c(`)(/@) exp [pa(/ [ /@)] , (179)


Bp\ [p

p] i(1 [ f@) ] f][p

d] i(1 [ f@) ] f][p

d@ ] i(1 [ f@) ] f][p

p] i(1 [ b@) ] b]

[i(1 [ f@) ] f][i(1 [ b@) ] b [ i(1 [ f@) [ f](pp[ p

d)(p

p[ p

d@ ) , (180)

Bd\ [p

p] i(1 [ f@) ] f][p

d] i(1 [ f@) ] f][p

d@ ] i(1 [ f@) ] f][p

d] i(1 [ b@) ] b]

[i(1 [ f@) ] f][i(1 [ b@) ] b [ i(1 [ f@) [ f](pd[ p

p)(p

d[ p

d@ ) , (181)

Bd@ \ [p

p] i(1 [ f@) ] f][p

d] i(1 [ f@) ] f][p

d@ ] i(1 [ f@) ] f][p

d@ ] i(1 [ b@) ] b]

[i(1 [ f@) ] f][i(1 [ b@) ] b [ i(1 [ f@) [ f](pd@ [ p

p)(p

d@ [ p

d)

. (182)

When the crust and crustal superÑuid are coupled to one another perfectly, and sopd/[i(1 [ b@) ] b] ] [v/p@ B

d] 0,

and*(`) ] [)cr(`)

)cr(`) \ [ [pp] i(1 [ f@) ] f][p

d@ ] i(1 [ f@) ] f]

[i(1 [ f@) ] f](pp[ p

d@ )

P~=

Õd/@N3

c(`)(/@)Mexp [p

p(/ [ /@)] [ exp [p

d@ (/ [ /@)]N , (183)

*c(`) \P

~=

Õd/@N3

c(`)(/@)

Gpp[p

d@ ] i(1 [ f@) ] f] exp [p

p(/ [ /@)]

[i(1 [ f@) ] f](pp[ p

d@ ) [ p

d@[p

p] i(1 [ f@) ] f] exp [p

d@ (/ [ /@)]

[i(1 [ f@) ] f](pp[ p

d@ )

H; (184)

5. CONCLUSIONS

Shaham (1977) demonstrated that when superÑuid pins perfectly to crustal nuclei, the precession period of a neutron star isshortened immensely and, moreover, the precession damps quickly as a result of weak coupling to the stellar core. One of theprincipal goals of this paper has been to examine whether there are additional modes with long periods and long dampingtimescales when the assumption of perfect coupling between crustal nuclei and superÑuid is relaxed. In fact, when the couplingis strong but imperfect, there are new modes that have very long characteristic timescales. One new mode is given by equation(80),

pd\ [ v[b ] i(1 [ b@)]

p@ ,

for an axisymmetric, two-component star, where v is the oblateness of the star and where is the moment ofp@ \ v ] Is/I1, I

sinertia of the crustal superÑuid and one of the principal moments of inertia of the crust. (Coupling of the crust to the stellarI1core hardly alters this result ; see discussion in ° 4.1.) Since b > 1, in the limit of strong vortex drag, this mode is extremely longlived ; moreover, since 1 [ b@ > 1 in this limit, the mode undergoes oscillations that are also extremely slow. The problem is


that we expect that o 1 [ b@ o D b2 > b in the strong coupling domain as long as the vortex drag coefficient g@, which governsthe strength of the drag force perpendicular to the direction of motion of the vortex through the normal Ñuid, is smallcompared with g, the analogous coefficient for the strength of the drag force antiparallel to the direction of motion (e.g., eqs.[37] and [38] and ensuing discussion). Thus, this mode is not actually oscillatory at all, for it damps before it can complete asingle cycle. In fact, for a nonaxisymmetric star, splits into two modes, with (see eq. [92])p

d

pdB \ [ b(v1 p2@ ] v2 p1@ )

2p1@ p2@^Cb2(v1 p2@ [ v2 p1@ )2

(2p1@ p2@ )2 [ v1 v2(1 [ b@)2p1@ p2@

D1@2,

both of which may be purely real and decaying.Another new mode arises in three-component models when, for example, the crustal superÑuid is coupled strongly to the

rigid crust in some regions and weakly in others, or else the crustal superÑuid is strongly coupled to the rigid crust but thesuperÑuid core is coupled to it only weakly. Under such circumstances, one solution to the three-component characteristicequation is equation (148)

pd@ \ [i ] (if@ [ f)(1 ] p@ ] I

c/I1)

1 ] p@ ,

where f@ and f are the coupling parameters between the rigid crust and the component that is barely tied to it. As wasdiscussed in ° 4.1.1, this mode can lead to a slow wandering of the pole of the neutron star as seen in the inertial referenceframe. However, the excitation amplitude is relatively small for the crustal angular velocity in this mode (e.g., eq. [149]) ;moreover, in the weak coupling domain, we expect f@ > f if g@ > g (e.g., ° 2.2), so the mode decays before completing oneoscillation.

Thus, it appears likely that although there are new, possibly long-lived modes for a neutron star with strong but imperfectcoupling between superÑuid and rigid crust, these modes are not principally oscillatory as long as g@ > g. Only if there areregions in the star where this inequality is reversed somehow could damped oscillations occur.

There may be regions of weak coupling between crustal superÑuid and nuclei interspersed among regions of strongcoupling. If so these regions could, if tied to the strong coupling regions tenuously, undergo nearly independent oscillationswith both long cycle times and insigniÐcant damping. Moreover, there could be regions of the core that may undergolong-period oscillations if detached from the crust. However, in both cases, the e†ect of nearly independent, slow andpersistent oscillations on the portion of the crust where superÑuid vortex lines are pinned would be minimal, tending to zeroin the limit of complete independence. Thus, if slow, persistent oscillations can occur somewhere in the star, the chances thatone can know about them from observations of the rotation rate of that part of the crust where superÑuid is pinned stronglyare remote. In the opposite limit, in which all regions of the star are coupled to one another closely, the observed frequenciesare averages weighted by moment of inertia and tend to be dominated by regions of high frequency and/or large moment ofinertia.

Under the combined action of external and internal torques, the angular velocity of the crust tends to tilt away fromalignment with its principal axes. If the external torque is time independent or only varies on a very long timescale thenultimately the tilt angle approaches a constant value where is the value of the constant external torque,h D o Nex o /vIcr )2, Nexis the typical moment of inertia of the crust, and v is the crustal oblateness. If is the spin-down torque, where I isIcr Nex \ I)0the moment of inertia of the star, then the steady state tilt angle is where I is the total moment of inertia ofD[(I/vIcr))0 /)2,the star. Even though where is the spin-down timescale, is very small[ )0 /)2 D ()t

ds)~1, tsd [B5 ] 10~9P(s)/tsd(yr)] I/vIcrmay be very large and h could be nonnegligible. An amusing side e†ect of this tilt is that even an axisymmetric neutron star

could be a source of gravitational radiation, with an amplitude that can be determined from observables (e.g., spin-downtimescale, period), quantities that can be inferred observationally with varying degrees of conÐdence (e.g., distance) andtheoretically determined parameters (e.g., total moment of inertia) but do not depend on the oblateness v. Unfortunately, theimplied wave amplitudes (strain amplitude are well below the projected capabilities of the advanced LIGO.h [ 10~30)

One key assumption behind this estimate of the steady state tilt angle is contained in the italicized word ultimately in theprevious paragraph. As was discussed in ° 3.2, the asymptotic value of h is only attained if the slowest damped mode of the stardecays in a time short compared with the timescales on which the external torque varies. Practically speaking, this means thatif the crustal nuclei and superÑuid are closely pinned, then the steady state tilt is approached on the damping time of (givenp

dby eq. [80]) ; if this is short compared with the spin-down timescale, then the asymptotic value of h is reached. Otherwise, thetilt could be smallerÈsomewhere between and time variable. As was pointed out in °D[(I/p@Icr))0 /)2 [ (I/vIcr))0 /)2Èand3.2, the correct steady state tilt angle, and the evolution toward that angle, could not be found from Shaham (1977), whereperfect coupling between crust and crustal superÑuid was assumed. The crux of the solution is in the timescales implied by theimperfection of the pinning.

We began this paper by proposing to examine perturbations about a particular Ðxed point of the equations governing therotational dynamics of a neutron star, the one corresponding to equal angular velocities of all components lined up along oneof the principal axes of the crust (the one with largest moment of inertia eigenvalue). We have only wavered from this programbrieÑy, in ° 3.3.1, where we considered perturbations about a state in which the rigid crust and crustal superÑuid have parallelangular velocities with slightly di†erent magnitudes. However, in spite of the constancy of our approach, we have uncoveredsome hints that it may be unrealistic, for when external torques are taken into account, the correct Ðxed points may involvetilts away from principal axes. In a sequel to this paper, we shall investigate the implications of time-dependent andtime-independent tilts due to external torques, as well as to internal torques we have neglected here.

360 SEDRAKIAN, WASSERMAN, & CORDES

This research was supported in part by NSF grants AST-93-15375 and AST-95-30397 and NASA grant NAG5-3097. A. S.gratefully acknowledges the support of the Max Kade Foundation.

REFERENCESAlpar, M. A., Anderson, P. W., Pines, D., & Shaham, J. 1981, ApJ, 249, L29ÈÈÈ. 1984a, ApJ, 276, 325ÈÈÈ. 1984b, ApJ, 276, 791Alpar, M. A., Chau, H. F., Cheng, K. S., & Pines, D. 1993, ApJ, 409, 345Alpar, M. A., & H. 1987, A&A, 185, 196O# gelman,Alpar, M. A., & Sauls, J. 1988, ApJ, 327, 723Anderson, P. W., & Itoh, N. 1975, Nature, 256, 25Backus, P. R., Taylor, J. H., & Damashek, M. 1982, ApJ, 255, L63Blaskiewicz, M. 1992, Ph.D. thesis, Cornell Univ.Blaskiewicz, M., & Cordes, J. M. 1997, in preparationBondi, H., & Gold, T. 1954, MNRAS, 115, 41Boynton, P. E., Groth, E. J., Hutchinson, D. P., Nanos, G. P., Partridge,

R. B., & Wilkinson, D. T. 1972, ApJ, 175, 217Boynton, P. E., Groth, E. J., Partridge, P. B., & Wilkinson, D. T. 1969, IAU

Circ. 2179Brecher, K. 1972, Nature, 239, 325Cordes, J. M. 1993, in ASP Conf. Ser. 36, Planets around Pulsars, ed. J. A.

Phillips, S. E. Thorsett, & S. R. Kulkarni (San Francisco : ASP), 43Cordes, J. M., & Downs, G. S. 1985, ApJS, 59, 343Cordes, J. M., Downs, G. S., & Krause-Polstor†, J. 1988, ApJ, 330, 835Cordes, J. M., & Helfand, D. J. 1980, ApJ, 239, 640DÏAlessandro, F. D., McCulloch, P. M., Hamilton, P. A., & Deshpande,

A. A. 1995, MNRAS, 277, 1033Davis, L., & Goldstein, M. 1970, ApJ, 159, L81Demianski, M., & Proszynski, M. 1983, MNRAS, 202, 437Downs, G. S. 1982, ApJ, 257, L67Epstein, R. I., & Baym, G. 1988, ApJ, 328, 680ÈÈÈ. 1992, ApJ, 387, 276Flanagan, C. S. 1990, Nature, 345, 416ÈÈÈ. 1993, MNRAS, 260, 643Goldreich, P. 1970, ApJ, 160, L11Groth, E. J. 1975, ApJS, 29, 443Hamilton, P. A., King, E. A., McConnell, D., & McCulloch, P. M. 1989,

IAU Circ. 3729Jones, P. B. 1976, Ap&SS, 33, 215ÈÈÈ. 1988, MNRAS, 235, 545ÈÈÈ. 1991, ApJ, 373, 208ÈÈÈ. 1992, MNRAS, 257, 501

Khalatnikov, I. M. 1965, An Introduction to the Theory of SuperÑuidity(New York : Benjamin), 93È104

Lamb, D. Q., Lamb, F. K., Pines, D., & Shaham, J. 1975, ApJ, 198, L21Lambeck, K. 1980, The EarthÏs Variable Rotation (Cambridge : Cambridge

Univ. Press), 37È39Link, B., & Epstein, R. I. 1991,Link, B., Epstein, R. I., & Baym, G. 1993, ApJ, 403, 285Lyne, A. G. 1987, Nature, 326, 569Lyne, A. G., & Pritchard, R. S. 1987, MNRAS, 229, 223Lyne, A. G., Pritchard, R. S., & Smith, F. G. 1988, MNRAS, 233, 667Lyne, A. G., Smith, F. G., & Pritchard, R. S. 1992, Nature, 359, 706Manchester, R. N., Newton, L. M., Hamilton, P. A., & Goss, W. M. 1983,

MNRAS, 202, 269McCulloch, P. M., Hamilton, P. A., McConnell, D., & King, E. A. 1990,

Nature, 346, 822McKenna, J., & Lyne, A. G. 1990, Nature, 343, 349Michel, F. C. 1991, Theory of Neutron Star Magnetospheres (Chicago :

Univ. Chicago Press)Pines, D., & Shaham, J. 1972, Phys. Earth and Planetary Interiors, 6, 103ÈÈÈ. 1974, Nature, 248, 483Pines, D., Pethick, C. J., & Lamb, F. K. 1973, Ann. NY Acad. Sci., 224, 237Radhakrishnan, V., & Manchester, R. N. 1969, Nature, 222, 228Ruderman, M. A. 1970, Nature Phys. Sci., 225, 838ÈÈÈ. 1997, in Unsolved Problems in Astrophysics, ed. J. N. Bahcall &

J. P. Ostriker (Princeton : Princeton Univ. Press), 283Sedrakian, A. D. 1995, MNRAS, 277, 225Sedrakian, A. D., & Cordes, J. M. 1998, preprint (astro-ph 9806042)Sedrakian, A. D., & Sedrakian, D. M. 1995, ApJ, 447, 305Sedrakian, A. D., Sedrakian, D. M., Cordes, J. M., & Terzian, Y. 1995, ApJ,

447, 324Shaham, J. 1977, ApJ, 214, 251Shemar, S. L., & Lyne, A. G. 1996, MNRAS, 282, 677Taylor, J. H., Wolszczan, A., Damour, T., & Weisberg, J. M. 1992, Nature,

355, 132J., Kahabka, P., H., Pietsch, W., & Voges, W. 1986,Tru" mper, O# gelman,

ApJ, 300, L63Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145

precession of isolated neutron stars. i. effects of imperfect pinning

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