Precession of isolated neutron stars — II. Magnetic fields and type II superconductivity

Abstract

We consider the physics of free precession of a rotating neutron star with an oblique magnetic field. We show that if the magnetic stresses are large enough, then there is no possibility of steady rotation, and precession is inevitable. Even if the magnetic stresses are not strong enough to prevent steady rotation, we argue that the local minimum energy state, at fixed magnetic field obliquity, is one in which the star precesses. Since the moment of inertia tensor is inherently triaxial in a magnetic star, the precession is periodic but not sinusoidal in time, in agreement with observations of PSR 1828-11. However, the problem we consider is not just precession of a triaxial body. If magnetic stresses dominate, the amplitude of the precession is not set just by the angle between the rotational angular velocity and any principal axis, which allows it to be small without suppressing oscillations of timing residuals at harmonics of the precession frequency. We argue that magnetic distortions can lead to oscillations of timing residuals of the amplitude, period and relative strength of harmonics observed in PSR 1828-11 if magnetic stresses in its core are about 200 times larger than the classical Maxwell value for its dipole field, and the stellar distortion induced by these enhanced magnetic stresses is about 100–1000 times larger than the deformation of the crust of the neutron star. Magnetic stresses this large can arise if the core is a type II superconductor or from toroidal fields ∼10¹⁴ G if the core is a normal conductor. The observations of PSR 1828-11 appear to require that the magnetic distortion of the neutron star be slightly prolate.

magnetic fields, stars: neutron, pulsars: general

1 Introduction

The convincing observation of free precession of PSR 1828-11 (Stairs, Lyne & Shemar 2000) poses challenges for theories of neutron stars. Shaham (1977, 1986) argued that vortex line pinning in the neutron star crust should prevent long-term precession. Sedrakian, Wasserman & Cordes (1999) showed that precession is still prevented if vortex lines are not pinned perfectly but vortex drag is strong. They also showed that even if vortex drag is weak, precession is damped away. Link & Cutler (2002) estimated the strength of vortex line pinning forces, and argued that PSR 1828-11 may precess at a large enough amplitude to unpin superfluid vortices in the crust. If vortex drag is small this would remove one impediment to free precession, although the free precession would still damp away eventually if it is not excited repeatedly.

Here, we consider an additional feature of radiopulsars such as PSR 1828-11, namely, that they are strongly magnetized, with magnetic axes that are at an angle to their rotation axes. Based on earlier work on magnetic stars by Mestel and collaborators (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981; see also Spitzer 1958), we argue that precession may be required– there is no equilibrium corresponding to solid-body rotation without precession for a rotating star with an oblique magnetic field. For a fluid star, though, we shall see that although the fluid must precess, the magnetic axis rotates uniformly. Although Mestel and co-workers (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) showed that hydrostatic balance also requires fluid motion in addition to the precession that affects the stellar magnetic field, these are too slow to be important observationally.

Once we also take account of the solid crust of a neutron star in addition to its fluid interior, we show that if the stellar distortions owing to the magnetic field are larger than the distortion of the crust, then steady-state rotation is very unlikely (but not necessarily impossible). If the magnetic stresses inside PSR 1828-11 are simply caused by the classical Maxwell stress tensor, evaluated with the inferred dipole magnetic field strength, then they are too weak to require precession at a period ∼1 yr. However, if the interior of the neutron star consists of a type II superconductor, the effective stress tensor is larger than for a classical magnetic field, according to Jones (1975) and Easson & Pethick (1977); as was emphasized by these authors, and Cutler (2002), the magnetic distortion is correspondingly larger. For PSR 1828-11, we estimate that the distortion that would result in a core that contains a type II superconductor can lead to precession at a period of the order of 1 yr. A sufficiently strong toroidal magnetic field (B_t∼ 10¹⁴ G) could also lead to precession at this period without type II superconductivity (see, e.g., equation 2.4 in Cutler 2002 with B_c→ 0).

Even if magnetic stresses are not strong enough to prevent steady rotation, magnetic distortion in an oblique rotator alters the physics of free precession qualitatively and quantitatively compared with what one would expect for precession owing to axisymmetric crustal distortions. Even if the crust is axisymmetric, misalignment between the symmetry axis of the crust and the magnetic field make the effective stellar moment of inertia inherently triaxial. From a phenomenological viewpoint, one manifestation of this loss of axisymmetry is that although the angular velocity of the star is a periodic function of time in the frame rotating with the crust and magnetic field, it is not a sinusoidal function of time. This feature is consistent with observations of PSR 1828-11, which reveal behaviour at several different harmonically related frequencies (Stairs et al. 2000). The relative strengths of the harmonics depend on the degree of non-axisymmetry, and, presuming an axisymmetric crust, on the distortions induced by the magnetic field. For the ordinary Maxwell stresses evaluated just with the inferred dipole field strength, the non-axisymmetry would be small, but distortions resulting from a type II superconductor, or from a normal core with a large toroidal field, could produce sufficient non-axisymmetry to lead to comparable amplitudes for at least the first few harmonics of the fundamental precession period, as is observed.

Periodic, but not sinusoidal, precession would also arise if the neutron star crust were simply non-axisymmetric even if magnetic stresses were negligible. Thus, the detection of harmonic behaviour in PSR 1828-11 cannot, by itself, be taken to be evidence for amplified magnetic stresses in its core. However, standard results for triaxial precession, which are reproduced as a byproduct of the calculations we present, show that the oscillations at harmonics of the precession frequency are smaller in amplitude than the oscillation at the fundamental frequency by powers of the precession amplitude. We shall see that this is not the case in models where magnetic stresses predominate: the amplitudes of oscillations at the precession frequency and twice the precession frequency may be comparable even at small amplitude.

Moreover, it is well known that the minimum energy state for rotation of a non-axisymmetric body is one in which the angular velocity and angular momentum are aligned with the principal axis of the body with largest eigenvalue of the moment of inertia tensor. There is no precession at all in this state. However, we shall see that even when magnetic stresses are not strong enough to require free precession, the minimum energy state for an oblique rotator does not correspond to alignment of the angular velocity with any principal axis of the effective moment of inertia tensor, provided that the magnetic obliquity is held fixed. Thus, the local minimum energy state is one in which the star precesses. We suggest that a rotating neutron star might attain this local minimum energy state on a moderately short time-scale, but only evolve towards a global energy minimum, in which the star is either an aligned or orthogonal rotator, on a longer time-scale. The local minimum energy state is one in which the star precesses, while the global minimum corresponds to steady-state rotation with the magnetic field either along or perpendicular to the angular velocity vector of the star.

We shall see that the timing residuals associated with the precession can account for observations of long-term oscillations in PSR 1828-11, but that the explanation only works if the core of the neutron star has sufficiently strong magnetic stresses that magnetic distortions are 100–1000 times larger than crustal deformations. Thus, we argue that observations of free precession in PSR 1828-11 offer evidence that the pulsar is in the regime where magnetic stresses dominate, because either its core is a type II superconductor, or, if it is a normal conductor, it possesses a very large toroidal field.

We review the arguments given by Mestel and collaborators (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) in Section 2. In Section 3, we consider the conditions under which precession of an oblique rotator is inevitable. In Section 4, we consider the modified precession problem for an axisymmetric crust and a misaligned magnetic field. There we argue that the local minimum energy state at a given angular momentum is one in which the star precesses, provided that the magnetic field is not either along or perpendicular to the symmetry axis of the crust. There, we also review classical results for free precession of triaxial bodies; we shall find that there are three distinct cases of interest for a star where magnetic distortions are important. We present limiting results for the timing residuals expected in this model in Section 5, and obtain approximate results for the limit in which crustal distortions dominate in Section 5.1, and for the opposite limit in which magnetic distortions dominate in Section 5.2.

Here, our main purpose is to present arguments that an oblique rotator must precess. In Section 5.3, though, we present a brief application of the model to observations of PSR 1828-11. There we argue that only models in which magnetic distortions dominate can account for all of the observed features of the long-term periodic timing residuals from this pulsar. We also suggest that the data favour prolate rather than oblate magnetic distortions.

Several appendices present cumbersome mathematical details required to derive (and verify!) analytic results presented in the text.

2 Review of Structure of Fluid Equilibria with Oblique Magnetic Fields

Let us begin by reviewing the theory of rotating stars with oblique magnetic fields developed by Mestel and collaborators (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981; see also Spitzer 1958). Consider a (fluid) star with angular momentum . In the absence of a magnetic field, the angular velocity of the star would be . Let us work in a reference system rotating at a rate , where ω is unknown, but must be Ω₀ to lowest order in small quantities. Assume that the magnetic field is axisymmetric about an axis that is fixed in this frame; we can verify that this works out correctly to lowest order later.

Solve for the density field of the star under the assumption that it is stationary; Mestel and co-workers (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) discuss the sizes of time-dependent correction terms, which are smaller than any we retain here. The result is a superposition of three components assuming slow rotation and weak magnetic fields:  

1

where ρ₀(r) is the spherical profile of the undistorted star, and and are the distortions owing to rotation and magnetic fields, axisymmetric about and , respectively. In absolute magnitude, the rotational distortion is of the order of ω²R³/GM for a star of radius R and mass M, and the magnetic distortion is of the order of BHR⁴/GM², where H=B for ordinary magnetic fields, but H corresponds to the first critical field strength in a type II superconductor (H_c1∼ 10¹⁵ G), as discussed by Jones (1975), Easson & Pethick (1977) and Cutler (2002).

The moment of inertia tensor corresponding to equation (1) is of the form  

2

where  

3

Let Ω=L/I+δΩ≡Ω₀+δΩ; then the angular momentum of the star is  

4

which implies that  

5

We can decompose this into components along and perpendicular to , i.e. , where  

6

δΩ_⊥= 0 only if is either parallel to or perpendicular to . Since Euler's equations imply a time-independent Ω only if L and Ω are parallel, in general a rotating star with an oblique magnetic field must precess.

To find the angular velocity of the reference system, we require that the magnetic axis rotates with angular velocity ; thus  

7

In this reference system, matter precesses about with angular velocity  

8

However, even though the fluid precesses, the magnetic field does not, so we do not expect any observable effects to arise from the precession.

3 Conditions Under which Steady Rotation is Impossible and Precession is Inevitable

We can look at the results of Mestel and co-workers (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) in a slightly different way, which is identical to the same order of approximation, but differs in approach slightly. Work in a reference system corotating with the matter.¹ The perfect conductivity condition demands that the magnetic axis is fixed in this reference system. Instead of equation (1), suppose that  

9

in this case, the moment of inertia tensor (formerly equation 2) becomes  

10

where the various factors are defined just as in equation (3). The angular momentum is therefore  

11

The condition for a time-independent Ω is L∥Ω. This condition is satisfied only if is either parallel to or perpendicular to . The star precesses at an angular frequency ω_p as before. However, the magnetic axis rotates at a uniform angular velocity, because, from equation (11) 

12

where  

13

which is independent of time.

The second approach makes contact with the Euler problem clearer. Angular momentum conservation is consistent with time-independent rotation as long as the angular velocity vector is aligned with one of the principal axes of the moment of inertia tensor. For anisotropic density distribution, the moment of inertia tensor is of the form  

14

where Tr(ΔI_ij) = 0. Even though ||δI_ij||≪I₀ for small distortions, the principal axes of I_ij are the principal axes of ΔI_ij. For the magnetic fluid, equations (9) and (10) imply that  

15

For there to be a principal axis of Δ_ij along Ω,  

16

where Λ is the associated eigenvalue. For this form of Δ_ij, the condition becomes  

17

which is only true in general provided that either Ω is along or Ω is perpendicular to , neither of which will be the case generally.

Next, consider what happens when we consider a neutron star model that consists of a rigid solid and a fluid, with an oblique magnetic field. In that case, equation (10) is generalized to  

18

where C_ij is the trace-free part of the moment of inertia of the crust.² The angular momentum of the star is therefore  

19

A steady state is only possible if the star rotates along a principal axis of  

20

However, for this to be true, we must have  

21

where Λ is the associated eigenvalue. The components of this equation perpendicular to Ω are  

22

Equation (17) must have a solution (with ) in order for there to be no precession.

To understand the significance of equation (17), first review the argument against precession when magnetic fields are ignored. The moment of inertia tensor of the solid is not known observationally for any neutron star, so we can specify C_ij freely. Since it is a trace-free symmetric tensor, C_ij has five independent components. We only need to choose two of these (the orientations of one of the three orthogonal principal axes) to obtain a steady state. We can make these choices without considering the magnitude of the distortion of the solid, since only the directions of the eigenvectors matter for finding a steady state. Moreover, from a physical viewpoint, we expect that at a fixed L the energy associated with rotation is minimized if L (and hence Ω) is along the principal axis with the largest eigenvalue. If a neutron star were a perfectly rigid solid, then it could precess non-dissipatively forever, without attaining the minimum energy state. An imperfectly rigid body, with finite bulk and shear moduli, supports shearing motions that can lead to dissipation. For neutron stars, the frictional interaction of the crust with various fluid and superfluid components are more important, and cause the precession to damp away (e.g. Sedrakian et al. 1999). An observer in the inertial frame would see the crust align its largest principal axis with the fixed angular momentum vector of the star.

Next, let us consider what happens when we restore the magnetic field. Expand  

23

where C_μ are the eigenvalues and are the eigenvectors of C_ij, and rewrite equation (17) as  

24

We know that if I_B= 0, a steady state is possible irrespective of the magnitudes C_μ, and that if all C_μ= 0 there is no steady state. When I_B is non-zero, we can adjust the so that equation (19) is satisfied, just as for I_B= 0, but the adjustment depends on the magnitudes C_μ relative to I_B. In general, we would expect there to be no solution if |C_μ|≲I_B, in which case the neutron star must precess. In Section 4 we shall also see that even when I_B is not large compared with C, the rotational energy is minimized, to first order in distortions, when the angular velocity is not precisely along a principal axis of the effective moment of inertia tensor, and therefore not precisely along .

To appreciate this result better quantitatively, let us focus on the simplest special case in which the crust is axisymmetric. If is the symmetry axis of the crust, then we may write  

25

so that the largest eigenvalue of the moment of inertia of the crust is along . Then the condition for a steady state, equation (19), may be written as  

26

If equation (21) has a solution, it must have coplanar , and . Thus, let us adopt and  

27

We substitute into equation (21) to find  

28

Therefore, for a steady state, we must require  

29

which is only possible if |I_B sin 2χ|≤C. If I_B≳C, then steady-state rotation is rather unlikely, with a probability that decreases with increasing I_B/C. When I_B sin 2χ/C≤ 1, equation (24) has the solutions  

30

as can be verified by direct substitution; the choice of signs depends on specific parameter values.

In order of magnitude, 3I_B/I≡βBHR⁴/GM², where β∼ 1 is a structure constant, so that  

31

for a neutron star mass and radius M= 1.4M_1.4 M_⊙ and R= 10R₆ km, a magnetic axis inclination angle and BH= 10²⁷(BH)₂₇ G. (BH∼ 10²⁷ G for B∼ 10¹² G and H≃ 10¹⁵ G ∼H_c1– see Easson & Pethick 1977.) The period associated with ω_B is P_B= 2π/ω_B≃ 16.6P₀(s)(β cos χ)⁻¹M²_1.4R⁻⁴₆(BH)⁻¹₂₇ yr for a neutron star rotation period P₀= 1P₀(s) s. If magnetic effects are strong enough to require precession, then we should expect a precession period of the order of P_B. We consider the precession period more completely in the next section.

For PSR 1828-11, the spin period is P₀= 0.405 s and the dipole field strength deduced from the observed spin-down is B≃ 5 × 10¹² G (Stairs et al. 2000); these values would imply P_B≃ 1.35(β cos χ)⁻¹[5/(BH)₂₇] yr, or about 492(β cos χ)⁻¹(BH₂₇/5) d, similar to the observed period for β cos χ∼ 1. Note that the precession period would be far longer for BH→B², by a factor of about 200. Thus, if magnetic effects are the reason for the observed ‘precession’ then ether the neutron star interior must be a type II superconductor (Jones 1975; Easson & Pethick 1977; Cutler 2002), or there must be a substantial toroidal field in the core (e.g. equation 2.4 in Cutler 2002 with B_c→ 0).

4 Modified Euler Problem

4.1 Basic equations, principal axes and eigenvalues

Next, let us consider the modified Euler problem. To allow an analytic treatment, continue to assume that the crust is axisymmetric. Although this is a simplification, the effective moment of inertia of the star will still be triaxial because of the misaligned distortion introduced by the magnetic field, except for special orientations.

The Euler equation in the rotating frame of reference is  

32

where the effective moment of inertia tensor is  

33

and d^⋆/dt is the time derivative in the rotating frame. The two vectors, and define a plane, and one of the eigenvectors of I^eff_ij is along the unit vector perpendicular to that plane, , and has an eigenvalue I₂=I₀− 2I_Ω+I_B−C. The other two eigenvectors lie in the – plane. As in the previous section, take and define by equation (22). Then the other two eigenvectors, are³ 

34

where  

35

The eigenvalues associated with these eigenvectors are I_±=I₂+δI_±, where  

36

In Appendix A we show that for I_B > 0, I₊ > I₂ > I₋ and for I_B <0, I₊ > I₋ > I₂.

Approximate results are derived for |I_B|≪C in Appendix B and for |I_B|≫C in Appendix C. When |I_B|≪C, the eigenvectors are nearly aligned with the principal axes of C_ij, and the eigenvalues are nearly the eigenvalues of C_ij, apart from small corrections ∼|I_B/C|. For |I_B|≫C, the eigenvectors are nearly along and perpendicular to and the eigenvalues are almost determined by the magnetic distortions alone, apart from corrections ∼|C/I_B|. We shall argue below that the case I_B <0, |I_B/C|≫ 1 may be especially relevant to observations of PSR 1828-11.

4.2 Solution of the Euler equations

4.2.1 Basic equations and minimum energy state

The modified Euler equations are simply what one finds in general for a triaxial system,  

37

As is well known, equations (32) conserve both the magnitude of L and the rotational energy,  

38

Note that the magnitude of the angular velocity is not conserved, so that in actuality I_Ω is not independent of time. However, we will assume that the variation is slow enough, and of small enough amplitude, that its effect is only higher order than any others we consider here.

The condition for steady rotation, equation (24), can be rederived from equation (30) under the assumption that is along either or , with χ <π/2 or χ > π/2, respectively. We are interested in finding the rotational energy of a rotating star in its minimum energy state, which may or may not be one in which the star precesses. To be specific, let χ be the angle between and when they are coplanar with (and therefore with ). Let us determine θ from the requirement that E is minimum for a given L². Thus, let us consider the quantity 2E/L² at the epoch when , and are coplanar. If we define  

39

then we find  

40

where the approximation holds for Δ≪ 1, and  

41

Assuming that θ≤π/2 we find  

42

To first order in Δ, the energy is minimized when is maximized. Differentiating with respect to the two angles θ and χ implies  

43

The absolute maximum value of is at θ=χ= 0 for I_B <0, and θ=χ=π/2 for I_B > 0. In this global minimum energy configuration, there is no precession, and the star is either an aligned or orthogonal rotator, depending on the sign of I_B. This is the global minimum energy state irrespective of the ratio of |I_B|/C.

The global minimum energy state may only be achieved slowly, as considerable dissipation may be required for the magnetic axis to become either aligned or orthogonal. On a shorter time-scale, we may expect that the star seeks a local energy minimum that can be achieved via internal dissipative processes such as mutual friction and crust–core coupling (e.g. Sedrakian et al. (1999)). The local minimum may involve adjustment of both θ and χ, but since there is much less energy involved in the rotation of the crustal distortion (∼CΩ²≃ 4 × 10³⁷C₃₆P⁻²₀ erg for a crustal distortion C≃ 10⁻⁹I= 10³⁶I₄₅ g cm²) than in the tilted magnetic field (∼BHR³∼ 10⁴⁵B₁₂H₁₅R³₆ erg), we expect θ to adjust more quickly than χ. At fixed χ, the local minimum energy state is θ=χ, and is a state in which the star precesses. At the instant when the symmetry axis of the crust, magnetic axis and angular velocity vector are coplanar, the angular velocity vector is along the symmetry axis of the crust in this state. For small values of |I_B|/C, this local minimum energy state may be the relevant one even though a lower-energy state is possible in which the star does not precess, provided that the evolution of χ towards either 0 or π/2 is slower than the dissipation time-scale needed to adjust θ→χ at a given χ≠ 0 or π/2. Note that if instead χ can evolve faster than θ→χ the second of equations (38), which is identical to equation (24), would imply that, at a given θ, the minimum energy state corresponds to steady-state rotation.

These considerations also hold true for large values of |I_B|/C, except that, in that regime, equation (24) shows that steady rotation is only possible when χ is already within ∼C/|I_B| of either 0 or π/2. For general χ, the lowest-energy state that can be attained is one in which θ=χ. Thus, for large values of |I_B|/C, we would expect the star to precess even if χ relaxes faster than θ, provided that the relaxation time-scale is still relatively long, so that χ remains far from 0 or π/2 for time-spans much longer than the precession period. Thus, we would expect precession to be likelier in pulsars with large |I_B|/C. Below, we shall see that the timing data on PSR 1828-11 can be accounted for most easily if |I_B|/C≳ 100.

A detailed treatment of the internal torques acting in a multicomponent, differentially rotating, magnetized neutron star with strong internal magnetic stresses is needed to evaluate the different relaxation time-scales and assess whether the star evolves along a sequence of local minimum energy states with θ=χ until it finds its global energy minimum. (Such a model would amount to extending the multicomponent calculations of Sedrakian et al. 1999 to include magnetic field effects.) Here, we tentatively assume that the star will seek the local minimum energy state on relatively short time-scales, and only tend towards the global minimum on time-scales that may exceed the spin-down time-scale for the star (e.g. Goldreich 1970). However, we shall not restrict our calculations of timing residuals to θ=χ even when C > |I_B|.

Note that in the local minimum energy configuration, need not be very small, although it vanishes in the minimum energy state for I_B→ 0. For small values of I_B/C,  

44

where the last result assumes θ→χ. Thus, for small I_B, we find small but non-zero in the minimum energy state. For |I_B/C cos 2θ|≫ 1, . On the other hand, for |I_B/C cos 2θ|≫ 1, or −cos χ, depending on the sign of I_B.

When I_B≡ 0, θ=χ minimizes exactly, and we find, as usual, , corresponding to an angular velocity aligned with the principal axis with the largest moment of inertia. When I_B≠ 0, θ=χ does not correspond to exact alignment of the angular velocity and the principal axis with the largest moment of inertia. This is because the eigenvalues of I^eff_ij depend on θ, so Δ depends on θ. If Δ were independent of θ, then the energy would be minimized for , the largest possible value of . The value of  

45

is only exactly one when either or ; the angle between and is more generally.

When I_B≪C, the local minimum energy state corresponds to an angle of approximately between and , and an angle between and .

4.2.2 Non-linear solution: I_B > 0

The complete, non-linear solution to equations (32) is given in Landau & Lifshitz (1969, Section 37), in terms of elliptic functions. We will have to consider the two cases I_B > 0 and I_B <0 separately. Here, we consider I_B > 0, which implies I₊ > I₂ > I₋ according to equation (A2) and the ensuing discussion. Adapting the solution in Landau & Lifshitz to this situation (and our notation) we have (see their equations 37.8–37.12)⁴ 

46

where , , , sn (τ) is defined by  

47

with  

48

The motion is periodic, with a dimensionless period 4K(k²), where  

49

Equations (41) and (43) also imply that  

50

which is independent of time up to terms ∼Δ², thus validating the approximation of time-independent I_Ω used above.

One of the distinguishing features of the timing model we will develop below is that we shall not demand that be small. In particular, we shall see that when |I_B|≫C, will not be small in general, but the observable effect of the precession on pulse arrival times could still be small. This is because in the limit where magnetic distortions are far larger than crustal distortions the star tends to precess about its magnetic axis. If there were no crust at all, as in the magnetic fluid stars considered by Mestel and collaborators (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981), the star would precess exactly around its magnetic axis, which would therefore rotate uniformly. The crust breaks this symmetry, and allows the precession to be observable. We shall see this emerge in some detail when we consider timing residuals in Section 5.2.

For the opposite case, where the crustal deformations dominate, the precession amplitude is set by , according to equations (41). In that case, we also see that equation (43) shows that . Thus, if the precession amplitude is small, so is k² as long as e² is not very large. Since k² governs the importance of oscillations at harmonics of the fundamental precession frequency, we see that a small amplitude will imply oscillations predominantly at the fundamental if crustal deformations dominate. (This will also be true for the solution given in Section 4.2.3 for I_B <0, where equation 53 will also imply that oscillations at harmonics of the precession frequency are suppressed for small precession amplitude.) This is not the case when magnetic deformations dominate.

The better-known results for free precession of an axisymmetric star are recovered for Δ−Δ₀= 0 =k². However, note that in the triaxial case, k² need not be small. We have already noted that does not have to be small, even in the local minimum energy state, and from equations (34) and the definition of Δ₀ we see that  

51

which is not necessarily small either. For 0 <I_B/C≪ 1, we note that k²≃I³_B sin ²2χ sin ²χ/4C³ to lowest order in I_B/C in the local minimum energy state, so k²≪ 1 in this case. However, for I_B≫C, we shall see that k²∼I_B/C≫ 1, and in general, since Δ₀ <0 for I_B > C, it is possible for k² to exceed one rather generally in that regime. The solution to the Euler problem is still given in terms of elliptic equations when k² > 1, but we have to make the replacements  

52

in equations (41), and dn²(kτ) = 1 −k⁻² sn²(kτ). The explicit solution for k² > 1 is  

53

where is the sign of , which may be positive or negative, , and  

54

This somewhat unfamiliar case is not treated in Landau & Lifshitz, but is found easily from the equations given there, and corresponds simply to the transformations in equations (47). Physically, it turns out to be important for a substantially prolate figure, which is what happens when I_B is large and positive.

An important difference between axisymmetric and triaxial precession is that even though both are periodic, only the axisymmetric case is precisely sinusoidal. The functions cn (τ) and sn (τ) contain all odd harmonics of πτ/2K(k²), whereas the function dn (τ) contains all even harmonics. When k² is small, the expansions are dominated by their leading terms, but, as we have seen, the general case does not demand small values of k². The key parameter in the expansions is  

55

where  

56

(Abramowitz & Stegun, equations 16.1.1 and 16.23.1–16.23.3). Since q(k²) <1 by its definition, only the first few harmonics ought to be prominent in the solution, but as long as k² is not especially small compared with unity, the amplitudes of the first few harmonics ought to be roughly comparable. This feature is consistent with the observed properties of PSR 1828-11 (Stairs et al. 2000). If this interpretation of the observations is correct, then the fundamental precession period must be 500 d.

4.2.3 Non-linear solution:I_B <0

For I_B <0, the solution is analogous to what was given in Section 4.2.2, except that, according to equation (A4) and the ensuing discussion, I₊ > I₋ > I₂. Instead of equations (41) we have  

57

and instead of equation (43) we have  

58

The definitions of sn (τ), cn (τ) and dn (τ) are the same as before. The magnitude of the angular velocity is now  

59

Equation (54) implies that |Ω| is independent of time up to terms ∼Δ², just as we found from equation (45) for I_B > 0. In this case, k² <1, so these solutions suffice.

4.3 Effect of the spin-down torque

The solutions given in Sections 4.2.2 and 4.2.3 are for free precession. Radiopulsars spin-down as a result of electromagnetic radiation, on a time-scale that is long compared with the precession periods of interest: for PSR 1828-11, t_sd≃ 1.1 × 10⁵ yr. A careful examination of the Euler equations with spin-down included shows that the precession is described by the torque-free solutions up to small corrections (just as was found by Link & Epstein 2001). We also note here that the quantity only changes on a still longer time-scale, ∼t_sd/Δ. However, sinusoidal variations of the spin-down torque result from the precession, since the angle between and Ω varies with time. The amplitude of these variations need not be small, and the associated timing residuals can dominate (Cordes 1993). In fact, we shall see that they are dominant, just as was found by Link & Epstein (2001) for precession of an axisymmetric neutron star.

For vacuum magnetic dipole radiation,  

60

where K is a constant to sufficient accuracy, and the magnitude of the angular velocity changes at a rate  

61

where Ω₀ is the angular frequency at some reference time t= 0; in the absence of precession, ζ= 0. We use the conventional definition of the spin-down time as . We can use the solutions to the Euler problem found above to find the time-dependent ζ case by case:  

62

for, respectively, I_B > 0, k² <1, , and I_B <0. Here, we do not restrict the solution to small values of , and equations (56) and (57) imply oscillations of the spin-down torque at both even and odd harmonics of the precession frequency. (There are also evidently zero-frequency corrections but these can always be combined with sin ²χ and factored out.) Note that for simple triaxial precession with a small tilt of the angular velocity away from the principal axis with maximum moment of inertia, the amplitude of the oscillations at 2ω_p would be smaller, by a factor , than the amplitude of the oscillations at ω_p. For the model developed here, can become substantial, particularly if I_B <0 and |I_B|≫C.

5 Pulse Arrival Times

To determine pulse arrival times, we need to represent in the inertial frame of reference of the observer. We can define the angular momentum vector L, which is conserved apart from spin-down in this reference frame, to lie along the z-axis, and we can further choose to place the observer in the x–z plane. Pulses arrive when (actually, only half of the solutions correspond to pulse arrival — the other half might be an interpulse or else unobserved). The general problem is addressed in Appendix D, where the three different types of solutions are treated separately in Appendices D1 and D2 for positive and negative I_B, respectively. Approximate solutions are also derived in those appendices (equations D17, D24 and D32), but those results only apply when the star is nearly axisymmetric and the precession amplitude is very small.

Here, we shall investigate two different nearly axisymmetric cases, |I_B/C|≪ 1 and |I_B/C|≫ 1. For |I_B|≪C, it will turn out that , and the results of Appendices D1 and D2 will be directly applicable. However, for |I_B|≫C, we have already mentioned that need not be small (see, for example, discussion following equation 36).

5.1 Pulse arrival times for |I_B|≪C

When |I_B|≪C, we can apply the results in equations (D17) and (D32) directly. Using the approximations given in Appendix B, and accounting for pulsar spin-down using the results of Section 4.3, we find that the oscillating parts of the phase residuals are  

63

for I_B > 0, and  

64

for I_B <0, where  

65

and , which is zero in the local minimum energy configuration (see Appendix B). Thus, to lowest order in the (presumed) small quantity , phase residuals oscillate only at ω_p. Oscillations at higher harmonics, such as 2ω_p, have amplitudes that are smaller by additional factors of . It is possible for these to be comparable in magnitude to the terms retained in equations (58) and (59), but only if is very small i.e. if , or . This was also found by Link & Epstein (2001), who required χ to be very close to π/2 in order for their axisymmetric precession model to account for the observed precession of PSR 1828-11 (see also Rezania 2003). Here, we also note that , where e² represents the deviation of the star from axisymmetry. Thus, non-axisymmetric effects alone cannot introduce substantial harmonic structure in the phase residuals for small either.

The importance of spin-down in the timing residuals is measured by the non-dimensional parameter  

66

where P_p= 1000P_p,1000, d is the precession period, P₀ is the pulsar period in seconds and t_sd= 10⁵t_sd,5 yr. For PSR 1828-11, we have Γ_sd≃ 844, so the pulsar spin-down dominates the oscillatory terms.

5.2 Pulse arrival times for |I_B|≫C

When |I_B|≫C, the moment of inertia tensor is once again approximately axisymmetric, but neither nor has to be small. Thus, the expansions in Appendices D1 and D2 are not applicable, and we shall have to solve the timing equation in a different way. In doing so, equations (C2) and (C3) will prove to be useful. To keep the notation compact, we define the non-dimensional parameter .

Let us consider the two possible sign choices separately. For I_B > 0, we use equation (D20) to evaluate . The problem is simplified since, from equation (C2), . Consequently, to first order in , we find  

67

so pulses arrive when  

68

mapping from phase to time implies  

69

Taking account of pulsar spin-down we find that the oscillatory part of the timing residuals is  

70

Note that the phase residuals vanish as even though the star still precesses. Moreover, there are oscillations at both ω_p and 2ω_p, for which the amplitudes may be comparable, in agreement with observations of PSR 1828-11.

For I_B <0, we find, to order , we find that pulses arrive when  

71

and taking account of spin-down results in oscillating timing residuals  

72

Once again, this involves oscillations at both ω_p and 2ω_p which can have comparable magnitudes. We see again that as , the oscillatory phase residuals disappear.

5.3 Application to PSR 1828-11

For PSR 1828-11, timing residuals appear to oscillate at both ω_p and 2ω_p with similar amplitude. The results of Section 5.1 show that this situation is incompatible with small values of if |I_B|≪C. Moreover, we note that the results of Section 5.1 continue to hold as |I_B|→ 0, so we see that equal amplitudes at ω_p and 2ω_p cannot arise from a model without magnetic stresses, but with a triaxial crust, unless there is some fine-tuning of parameters (as in the axisymmetric model of Link & Epstein (2001), which requires χ to be very close to π/2).

Thus, we focus on the strongly magnetic case, |I_B|≫C. In this case, equations (65) and (67) show that it is possible for the phase residuals to oscillate with comparable amplitudes. The difference between the large and small |I_B|/C limits is that for small |I_B|/C, the amplitude of the observed timing residuals is determined solely by , but at large |I_B|/C the amplitude is determined by C/|I_B| primarily. Thus, in contrast to what we found for |I_B|≪C, small amplitudes need not suppress the oscillations at 2ω_p.

For PSR 1828-11, we also know that Γ_sd≃ 844 ≫ 1, so let us approximate the oscillatory timing residuals further as  

73

The parameter  

74

governs the relative strengths of the two harmonics in the timing residuals. If the precession is in the local minimum energy state, θ=χ, then u= 1 for χ≃ 76°.

The timing residuals in equation (68) vary between different minimum and maximum values. For I_B > 0, Ω₀Δt_osc is minimum when , provided that |tan χ tan θ| <1; for χ=θ, this is so as long as χ > 63°. Presuming this to be so, the minimum value is  

75

The maximum is at , where we find  

76

The ratio of maximum to minimum timing residual is  

77

for u= 1, this ratio is 16/9 ≃ 1.8, and the ratio is 2 for u= 2. Observationally, the timing residuals for PSR 1828-11 appear skewed towards positive values, with a maximum about twice the magnitude of the minimum (see fig. 1 in Stairs et al. (2000)). Similarly, for I_B <0, the maximum value of Ω₀Δt_osc occurs when sin τ=−u/4, and the minimum occurs when sin τ= 1; in this case, the ratio of minimum to maximum values is  

78

for u= 1, this ratio is 9/16 ≃ 0.56, and the ratio is 2 for either u≃ 16.9 or u≃−0.89. To the extent that we expect u > 0 (manifestly so for χ=θ), and u∼ 1 (but not ≫1) the observed residual arrival times appear to favour I_B > 0, i.e. prolate magnetic distortions.

Fig. 1 illustrates the residuals in the arrival times, Δt (dotted, left-hand panel), period derivatives (solid, dotted) and period ΔP (solid, right-hand panel) for θ=χ and u= 5 for the prolate (upper) and oblate (lower) cases. The period and period derivatives are computed by differentiating Ω₀Δt_osc with respect to time:  

79

The agreement between these evaluations and the results plotted in fig. 2 of Stairs et al. (2000) is good, superficially, for the prolate model. Better agreement is seen for the variable period and period derivative than for the arrival times themselves; this may have been expected (e.g. Cordes 1993).

Figure 1.

Results of evaluating the oscillating residual arrival time Δt and its first two derivatives ΔP and for u= 5 and χ=θ. The top panels are for prolate models and the bottom ones for oblate models. The left-hand panels show Δt (dotted) and (solid), and the right-hand panels show ΔP. The units of Δt are , the units of ΔP are (approximately for PSR 1828-11), and the units of are (approximately for PSR 1828-11).

Using the results plotted in Fig. 1, we can estimate the magnitude of and the angles involved, even though these results do not constitute a true fit to the data, but just a plausible model. The curves for and ΔP resemble the observational results better, so let us focus on those. Fig. 1 was prepared for χ=θ and u= 5, which corresponds to χ=θ= 60.8°. From fig. 2 in Stairs et al. (2000), we see that the maximum values of ΔP and are about 1 ns and 0.2 × 10⁻¹⁵ for PSR 1828-11. For the prolate model, which resembles the observations better, Fig. 1 implies a peak value of ns for ΔP and for . We therefore estimate from ΔP and from . Since we have not attempted true curve fits (i.e. by varying the parameters u, θ and ) we regard this as acceptable agreement, provisionally.

6 Discussion

Here, we have extended previous studies of precession of neutron stars to incorporate the effects of oblique magnetic fields. We have shown that if the magnetic stresses are large enough, then steady rotation is unlikely, and the neutron star must precess. Moreover, even when the magnetic stresses are relatively weak, so that steady rotation is possible irrespective of the obliquity of the magnetic field, the local minimum energy state is not generally a steady state. Thus, even in this case, the neutron star will precess. We argued, in Section 4, that the minimum energy, precessing state is a local energy minimum that applies at fixed angle between the magnetic and rotational axes. On a longer time-scale, we would expect the star to seek its global energy minimum, which should correspond to either aligned or perpendicular magnetic and rotation axes, and no precession. We might expect short time-scale dissipative effects to drive the system towards its local minimum, and that the global minimum is only achieved on a somewhat longer time-scale, perhaps as a result of electromagnetic spin-down torques (e.g. Goldreich 1970).

The effective moment of inertia tensor of a neutron star with an inclined magnetic field is inherently triaxial. Consequently, the precession is periodic but not sinusoidal in time. In general, the solution for the rotational angular velocity of the star can be expanded in a Fourier series involving harmonics of the precession frequency. We have shown that at least the first few terms in such an expansion can have comparable magnitudes provided that the interior magnetic stresses are not very small.

The condition that magnetic stresses play an important role is that the magnetic-induced distortions are comparable to or larger than the distortions of the stellar crust. For precession periods of the order of years, the implied magnetic stresses exceed those expected from the classical Maxwell stress tensor, evaluated using the inferred dipole magnetic field strength, by a couple of orders of magnitude. However, if the interior of a neutron star contains a type II superconductor, or else is a normal conductor but possesses large toroidal magnetic fields, the magnetic stresses are larger, and the implied distortions can be of the right order of magnitude (Jones 1975; Easson & Pethick 1977; Cutler 2002). Thus, the observation of neutron star precession can be taken as indirect evidence for enhanced magnetic stresses, owing to either type II superconductivity or large toroidal fields.

We postpone a detailed application of the ideas set forth here to PSR 1828-11 to another paper (Akgun, Epstein & Wasserman, in preparation). However, in Section 5.3 we argued that only a model with |I_B|≫C can lead to time residuals that oscillate with comparable amplitude at both ω_p and 2ω_p. In this case, the amplitude of the observed time residuals is set by the dimensionless ratio , not by the tilt of the angular velocity vector away from any principal axis, which may be large.

In contrast, if the stellar distortions associated with magnetic stresses are smaller than those associated with the crust, then equations (58) and (59) show that the timing residuals oscillate predominantly at ω_p. Oscillations at 2ω_p would be down by factors of , where is given by equation (60), and is ∼|I_B|/C in the local minimum energy configuration. Moreover, we argued, in Sections 5.1 and 5.3, that precession of a triaxial crust alone would probably not, at small precession amplitude, be capable of producing oscillations of comparable magnitude at both ω_p and 2ω_p, because the precession amplitude is proportional to and the oscillations at harmonics of ω_p are suppressed by factors of . Thus, a solution in which the crustal distortion is responsible for precession is unlikely to explain the data on PSR 1828-11. The model in which magnetic stresses dominate is not merely precession of a triaxial body because small-amplitude phase residuals can arise even when is not small; the ratio C/|I_B|≪ 1 determines the amplitude of the phase residuals in this case.

For a precessing star in which the distortion is primarily caused by magnetic stresses, the precession period is  

80

where the value given is for PSR 1828-11. This is similar to the observed period, about 500 d, provided that β cos χ is not very small. Thus, if χ∼ 60° the timing residuals could oscillate with about the right period, amplitude and relative importance of the fundamental precession period and its first harmonic. In this case, we note that the condition that there is no steady state, can be satisfied: for χ= 60°, for example, , whereas we estimated that –0.01 in Section 5.3. This possible explanation of the timing residuals for PSR 1828-11 only works if the magnetic stresses in this star are ∼200 times larger than would be indicated by its dipole magnetic field. Thus, we conclude that either the interior is a type II superconductor or it is a normal conductor with a toroidal field of strength ∼10¹⁴ G. Otherwise, the expected magnetic stresses are far smaller than is needed for this solution to apply.

We also noted, in Section 5.3, that the ratio of the minimum and maximum timing residuals, and the shape of the variation of ΔP and seen in PSR 1828-11 appear to favour a model in which I_B > 0, so that magnetic distortions are prolate. Prolate distortions would arise naturally from the stresses caused by a toroidal field, with or without type II superconductivity (Cutler 2002), but may also result if magnetic flux tubes have been transported outward in the core and accumulate at its outer boundary (Ruderman, Zhu & Chen 1998; Ruderman & Chen 1999), which could ‘pinch’ the interior.

Crustal distortions are still needed in order for the precessional amplitude to be non-zero. In fact, to be more precise, the precessional amplitude depends on the component of the moment of inertia tensor of the crust that is not symmetric about the magnetic axis. This can be seen directly from equations (65) and (67), which show that the oscillating time residuals vanish as , where θ is the angle between the magnetic axis and, in this axisymmetric distortion model, the symmetry axis of the relevant crustal deformation, . Thus, although all neutron stars precess as a consequence of their magnetic stresses for |I_B|≫C in the picture advanced here, only those with sufficiently large non-aligned crustal deformations would have discernible oscillations of their timing residuals. In this sense, PSR 1828-11 may be special.

Although we have treated the basic physics of precession of an oblique rotator in some detail, we have not treated several effects that might play significant roles. We have not explicitly included either vortex line pinning or vortex drag. Link & Cutler (2002) have argued that vortex lines can unpin globally at large enough precession amplitude. For |I_B|≫C, as is required to explain the timing residuals in PSR 1828-11, the precession amplitude is ≃sin χ or cos χ (depending on the sign of I_B), which is not small, so global unpinning is expected. Thus, we may expect that vortex lines are unpinned in neutron stars with magnetic fields that are strong enough to have |I_B|≫C except for χ very close to either zero or π/2, depending on the sign of I_B. For these, vortex line drag, if weak enough, may simply serve to bring the neutron star superfluid into corotation with its crust, and drive the rotating star towards its local minimum energy state. For large |I_B|/C, we have seen that precession is required, so weak vortex drag or other forms of dissipation need not prevent precession. For small values of |I_B|/C, the precession amplitude is given by equation (60) and can be very small; in the local minimum energy state, equation (60) implies . In this case, it is possible that precession cannot overcome pinning forces, as discussed in Link & Cutler (2002), and so precession is fast and rapidly damped (Shaham 1977; Sedrakian et al. 1999).

Theories of pulsar glitches involve the pinning, unpinning and repinning of crustal superfluid vortex lines (e.g. Anderson & Itoh 1975; Alpar et al. 1981; Alpar et al. 1984a; Alpar et al. 1984b; Alpar et al. 1993; Link, Epstein & Baym 1993). As we have seen, for large |I_B|/C, precession amplitudes are large, and vortices may be expected to unpin, but it is possible for vortex lines to remain pinned in neutron stars with C≫|I_B|. Thus, there could be a dichotomy between pulsars that glitch (C≫|I_B|) and those that precess (C≪|I_B|). If, during the course of a glitch, all crustal superfluid vortices were to unpin, then the star might precess briefly. Perhaps that explains the detection of damped, quasisinusoidal timing residuals in the Vela pulsar after and perhaps before its Christmas 1988 glitch (McCulloch et al. 1990).

We have kept the problem of precession of an oblique rotator as simple as possible by considering what happens when the magnetic field is axisymmetric about some axis, and the crustal distortions are also axisymmetric, but about a different axis. More realistically, both of these simplifying assumptions are likely to be violated. Most likely, the crust is not axisymmetric. When magnetic stresses dominate, we do not expect including intrinsic crustal asymmetry to alter the results found here qualitatively, since the effective moment of inertia is already triaxial here. Triaxiality of the crust, in the limit of rather small crustal distortions, would simply rotate the principal axes slightly. Furthermore, the magnetic field may have a more complicated structure than we have assumed. A substantial quadrupolar component would presumably render the contribution to the inertia tensor from magnetic stresses alone triaxial. We shall consider these complications elsewhere.

Although we have included the spin-down torque in our evaluations of timing residuals, we did not include near-zone electromagnetic torques (Good & Ng 1985; Melatos 1997, 1999, 2000). The principal effect of such torques would be to renormalize the moment of inertia tensor of the star. Near-zone torques can play a role similar to the magnetic distortions considered here, but are smaller by a factor of ∼(H/B)(Rc²/GM), which is non-negligible even if H=B. However, we note here that the large magnetic distortions we propose would presumably apply to the spin-down of magnetars and anomalous X-ray pulsars, in much the same fashion as proposed by Melatos (1999, 2000). We shall pursue this idea elsewhere.

We have also ignored motions of the fluid and crust of the star apart from rigid rotation. Mestel and collaborators (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) have pointed out that the equilibrium in a fluid star with an oblique magnetic field must involve fluid motions with velocities ∼(Ω²R³/GM)ω_pR. These distort the stellar magnetic field by a fractional amount ∼Ω²R³/GM over the precession period 2π/ω_p (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981). Similar motions might arise in the crust, but with magnitudes ∼(Ω²R²/c²_t)ω_pR, where c_t is the speed of sound for transverse waves. The expected amplitude of the resulting magnetic wander is ∼Ω²R²/c²_t. Although slow, these displacements cause the magnetic field of the star to oscillate about its undisturbed, axisymmetric state and might influence the long-term behaviour of the observed spin-down. We shall investigate whether there is any long-term observational signature of these motions elsewhere. In addition, it is likely that magnetic and rotational deformations of the core must also deform the crust, as they exert pressure on its inner boundary. We would expect the magnetic deformation of the core to promote crustal deformation symmetric about , which would not lead to observable precession, but the rotation-induced deformation need not be symmetric about , and should be substantial. An important question left unanswered here is whether the crustal distortion, C_ij, that leads to detectable precession is relatively steady, or is simply owing to a seismic fluctuation. We shall explore the important issue of crustal deformations elsewhere.

Finally, we emphasize that any neutron star with strong enough core magnetic stresses ought to precess, but we may not be able to detect their precession because their magnetic axes can still rotate more or less uniformly. This is because, at small values of C/|I_B|, the neutron star precesses almost exactly about its magnetic axis, which therefore rotates almost uniformly as seen in the inertial frame. Although the precession may not be detectable readily from timing residuals for most pulsars, gravitational radiation amplitudes would be larger than would arise without enhanced internal magnetic stresses (e.g. Cutler & Thorne 2002; Cutler 2002). The distortions required for PSR 1828-11 are still smaller than would be needed for detection by LIGO, even if it were spinning faster (Brady et al. 1998). If there are young, highly magnetized neutron stars rotating rapidly, they would be the brightest emitters of gravitational radiation. Such objects have been hypothesized to be the sources of the highest-energy cosmic rays (Blasi, Epstein & Olinto 2000; Arons 2002).

Acknowledgments

Partial support for this work was provided by a grant from IGPP at LANL. I thank T. Akgun, J. Cordes, R. Epstein and B. Link for comments.

1

Mestel and co-workers (Mestel & Takhar 1972; Mestel et al. 1981; Nittmann & Wood 1981) show that maintaining hydrostatic balance also requires additional motions of the fluid. We shall discuss this feature in more detail below.

2

More precisely, C_ij is the portion of the moment of inertia tensor of the crust that is not aligned with the magnetic distortions. Any crustal distortions symmetric about would just renormalize I_B.

3

The following results imply that , and , so that these axes define a right-handed coordinate system.

4

There is actually a sign ambiguity in the solution given by Landau & Lifshitz, which we resolve by always choosing , rather than . The Euler equations (27) require that if we choose , then we should also choose the sign of the coefficient of Ω₂ to be the same as the sign of the coefficient of Ω₋.

Appendix

Appendix A: Inequalities Among Eigenvalues

For I_B > 0, we can rewrite equation (31) in the form  

(A1)

from which it follows that  

(A2)

Equation (A2) implies that δI₊≥ 3(C−I_B) > 0 and δI₋ <0 if I_B <C, and δI₊ > 0 and δI₋ <3(C−I_B) <0 if I_B > C. Thus, if I_B > 0, then δI₊ > 0 and δI₋ <0 irrespective of whether I_B > C or I_B <C. Thus, for I_B > 0, I₊ > I₂ > I₋.

For I_B <0, we can rewrite equation (31) in the form  

(A3)

from which it follows that  

(A4)

Equation (A4) implies that δI₊ > 3C if |I_B| <C and δI₊ > 3|I_B| if |I_B| > C. Thus, we see that for I_B <0, I₊ > I₋ > I₂.

Appendix B: Approximate Results for |I_B|≪C

For |I_B|≪C, we can expand these results to find  

(B1)

Using equation (B1) we also find that  

(B2)

where and .

Appendix C: Approximate Results for |I_B|≫C

For |I_B|≫C, we find  

(C1)

where s_B≡I_B/|I_B| is the sign of I_B. For I_B > 0 these results imply  

(C2)

and for I_B <0 the same results imply  

(C3)

Appendix D: Timing Solution

To find pulse arrival times, we need to determine the motion of in the inertial reference frame of the observer. To do this, we need the Euler angle rotation from the rotating frame of reference to the inertial frame; for an arbitrary vector V this is (see, e.g., Goldstein equation 4-47)  

(D1)

where α, φ, ψ are the Euler angles defined in fig. 47 of Landau & Lifshitz, Section 35, except that, to avoid confusion with our definition of θ as the angle between and , we label their Euler angle θ as α. We assume that L is along the direction in the inertial frame. We can then determine the two angles α and ψ from the equations  

(D2)

the third Euler angle φ is not determined by these relations, but can be found from  

(D3)

The choice of axes in the rotating frame of reference is somewhat arbitrary, and we shall make three different choices below, as the situation demands.

D1 Pulse arrival times for I_B > 0

I_B > 0 and k² <1

For I_B > 0 and k² <1, we choose , and ; then equations (D1) imply (Landau & Lifshitz equation 37.15)  

(D4)

and (Landau & Lifshitz, equation 37.16)  

(D5)

where  

(D6)

measures the non-axisymmetry. Note that in perfect axisymmetry, Ω_z and dφ/dt are independent of time, another difference between the magnetic case, which is inherently triaxial, and precession with an axisymmetric crust. When the star is nearly axisymmetric,  

(D7)

thus, dφ/dt oscillates at twice the precession frequency in this limit. Associated with the time development of dφ/dt would also be variability of the pulsar spin-down, at even harmonics of the precession frequency. The amplitude of the main variation would be of the order of ∼e²(P₀/P_p)²; successive harmonics would be smaller by powers of ∼e². For PSR 1828-11, we would have e²(P₀/P_p)²∼ 10⁻¹⁶e². For comparison, electromagnetic spin-down produces oscillations with an amplitude , where τ_sd is the spin-down time-scale, which is considerably larger (Link & Epstein 2001).

Applying equations (D1) to , and using equations (D4) we find  

(D8)

No approximations have been made in equation (D8); in fact, there is also no explicit reference to magnetic distortions here, and so these results apply to triaxial stars in general. We note here that  

(D9)

and in the local minimum energy state we can take θ≃χ to lowest order in distortions. When C is large compared with I_B, these reduce to and .

Pulse arrival times are found from the condition that if we assume that the observer is in the x–z plane. Define  

(D10)

then, the pulse arrival times are the solution to  

(D11)

Let  

(D12)

then pulses arrive when  

(D13)

When , pulses arrive at η= (2n+ 1/2)π, so let η= (2n+ 1/2)π+δ to find  

(D14)

We combine equation (D14) with equation (D7) and  

(D15)

to find that pulses arrive when  

(D16)

This result does not invoke any approximation (except that when accounting for the time dependence of L in this fashion there is a tacit assumption that the spin-down time-scale is slow compared with the precession time-scale).

For small e² and , pulses arrive when  

(D17)

to first order in , neglecting terms of the order of and smaller.

I_B > 0 and k² > 1

For k² > 1, we choose , and . (The sign choice is required to guarantee that this is a right-handed coordinate system.) We then find that  

(D18)

where  

(D19)

defining as before,  

(D20)

where are given by equation (D9). Pulse arrival times are found by solving . For this case, we define  

(D21)

then pulses arrive when  

(D22)

where and  

(D23)

When and are both small, pulses arrive when  

(D24)

D2 Pulse arrival times for I_B <0

For I_B <0, and , and equation (D4) is replaced by  

(D25)

and equation (D5) is replaced by  

(D26)

In this case, we define  

(D27)

then  

(D28)

where are given by equation (D9). Pulse arrival

times are determined from the condition . Let  

(D29)

then  

(D30)

If we take φ(0) = 0 then we obtain  

(D31)

where . For small e² and ,  

(D32)