2 The precession

Consider a rigid, biaxial rotating star with the principle axes  $\mbox{\boldmath$e$ }_1, \mbox{\boldmath$e$ }_2, \mbox{\boldmath$e$ }_3$ and corresponding principle moment of inertia $I_1=I_2\neq I_3$. The star's angular momentum $\vec{L}$ is misaligned to the symmetry axis  $\mbox{\boldmath$e$ }_3$ by a wobble angle $\theta$, i.e. $\vec{L}\cdot \mbox{\boldmath$e$ }_3=L\cos\theta$. In general, we assume that the stellar magnetic field is dipolar and changes with time as

$$\vec{m}=m_0\sin\chi f_1(t)\mbox{\boldmath$e$ }_1+m_0\sin\chi... ...\boldmath$e$ }_2 + m_0 \cos\chi f_3(t) \mbox{\boldmath$e$ }_3,$$ (1)

where m₀ is the average value of $\vert\vec{m}\vert$ over a period $2\pi/\omega_p$ and $\chi$ is the angle between $\vec{m}$ and  $\mbox{\boldmath$e$ }_3$. The functions f₁(t), f₂(t) and f₃(t) are arbitrary functions in time and will be determined later by using the data. The equations of motion in the corotating frame are

$${\vec I}\cdot\frac{{\rm d}\mbox{\boldmath$\omega$ }}{{\rm d}... ...mega$ }\cdot\vec{m}) (\mbox{\boldmath$\omega$ }\times\vec{m}),$$ (2)

where R is the average radius of the star. The first term of the magnetic torque, $\vec{T}_{\rm ff}$, is due to the far-field radiation and has components both parallel and perpendicular to the spin axis. It is responsible for spinning down the star. The second term, $\vec{T}_{\rm nf}$, represents the near-field radiation torque and is exactly perpendicular to the spin axis. It has no contribution to the energy/angular momentum transfer from the star. This torque does affect the wobble angle and spin rate of a freely precessing star. Following Link & Epstein (2001), for small wobble angle $\theta\simeq 3^\circ$, suggested by the observed pulse shape variations of PSR B1828-11 over one precession period, and for small oblateness $\epsilon\simeq 10^{-8}$, we have $(\omega_0\tau_{\rm ff})^{-1}\ll (\omega_0\tau_{\rm nf})^{-1} < \epsilon\theta \ll \theta \ll 1$. Here $\omega_0$ is the angular frequency of the star, $\tau_{\rm nf}$ ($\sim$10⁴ yr), and $\tau_{\rm ff}$ ($\sim$10⁸ yr) are the corresponding near- and far-field radiation torque time scales, respectively. So up to the first order of $\theta$, the magnetic torques in the RHS of Eq. (2) can be neglected. Therefore in this order, the angular velocity vector  $\mbox{\boldmath$\omega$ }$ precesses freely around the star's symmetry axis as

$$\mbox{\boldmath$\omega$ }(t)\simeq \theta\omega_0\cos(\omega... ... p})\mbox{\boldmath$e$ }_2 + \omega_0 \mbox{\boldmath$e$ }_3,$$ (3)

with the precession frequency $\omega_{\rm p}=\epsilon\omega_0$, and a constant phase  $\beta_{\rm p}$. For the case of PSR B1828-11, observations suggest that $\epsilon \simeq 4.7\times 10^{-9}$(Stairs et al. 2000). Then $\omega_{\rm p}\simeq 7.29\times 10^{-8}$ Hz or equivalently $P_{\rm pre}=2\pi/\omega_{\rm p}\simeq 997$ d.

2.1 Timing

The observed timing behavior can be understood by considering the contribution of other torque's components in the variations of the spin rate. By multiplying $\mbox{\boldmath$\omega$ }$ in Eqs. (2) we have

$$\frac{{\rm d}\omega^2}{{\rm d}t}=\frac{2}{I_1}\left(\frac{}{... ...$ }\cdot\vec{T}- \epsilon\frac{I_1}{I_3}\omega_3 T_3\right).$$ (4)

Equation (4) shows the torque-induced variations in the spin rate of star. From Eq. (2) it is clear that the $\mbox{\boldmath$\omega$ }\cdot\vec{T}$ term does not depend on the near-field torque. It contributes to the spin rate change only through the negligible final term. Using Eqs. (1) and (2), by calculating $\mbox{\boldmath$\omega$ }\cdot\vec{T}_{\rm ff}$ and subtracting Eq. (4) from the secular spin down of the star (in the absence of precession), $-(\omega\sin^2\chi/\tau_{\rm ff}) (I_3/I_1)$, one can find the spin rate due to the far-field torque as (dropping constant terms)

 

$$\frac{\Delta\dot\omega}{\omega_0} \simeq \frac{1}{\tau_{\rm ff}}\frac{I_3}{I_1} \left[\frac{}{}\cos^2\chi f^2_3(t) \right. +\theta\sin2\chi$$

$$\times\left(\frac{}{} \cos(\omega_{\rm p}t+\beta_{\rm p}) f_1(t)f_3(t) + \sin(\omega_{\rm p}t+\beta_{\rm p}) f_2(t)f_3(t) \frac{}{}\right)$$

$$- \frac{\theta^2}{2} \sin^2\chi \left(\frac{}{} \sin^2(\omega_{\r... ...+\beta_{\rm p}) f^2_1(t) +\cos^2(\omega_{\rm p}t+\beta_{\rm p}) f_2^2(t)\right.$$

$$\left.\left. - 2 \sin(\omega_{\rm p}t+\beta_{\rm p})\cos(\omega_{\rm p}t+\beta_{\rm p})f_1(t) f_2(t) \frac{}{}\right)\frac{}{}\right].$$ (5)

Using Fourier expansion, we expand the functions f₁(t), f₂(t), and f₃(t) as follow

 

$$\sum_{n=0} a_n\cos(n\omega_{\rm d}+n\beta_{\rm d}),$$                    f₁(t) =

$$\sum_{n=0} b_n\sin(n\omega_{\rm d}+n\beta_{\rm d}),$$ f₂(t) =

$$\sum_{n=0} c_n\cos(n\omega_{\rm d}+n\beta_{\rm d}),$$ f₃(t) = (5a)

where $\omega_{\rm d}$ is frequency of the magnetic field's variation and $\beta_{\rm d}$ is constant. The coefficients a_n, b_n, and c_n will be determined by fitting the data. Equation (5) shows that the spin rate variations depend on both the precession frequency and the variation frequency of the dipole field. For the case a₀=c₀=1 and a_n=b_n-1=c_n=0 for $n\geq 1$, Eq. (5) reduces to one obtained by Link & Epstein (2001) (except by a factor I₃/I₁).

To obtain the reported spectrum of PSR B1828-11, it is enough to consider n=0 and 1 terms in Eq. (5) only. The $n\geq 2$ terms will produce the higher harmonics which will be discussed later. Simply a correct behavior of the spin rate can be found by setting c₁=0, c₀=1, and b₁=-a₁. Therefore Eq. (5) reduces to (dropping constant terms)

 

$$\frac{\Delta\dot\omega}{\omega_0} \simeq \frac{\theta}{\tau_{\rm ff}}\frac{I_3}{I_1} \left[\frac{}{} \sin2\chi\left(\frac{}{} a_0\cos(\omega_{\rm p}t+\beta_{\rm p}) \right. \right.$$

$$\left. +a_1 \cos[(\omega_{\rm p}+\omega_{\rm d})t+\beta_{\rm p} +\beta_{\rm d}] \frac{}{}\right)$$

$$-\frac{\theta}{2} \sin^2\chi \left(\frac{}{} 2a_0a_1\cos(\omega_{\rm d}t+\beta_{\rm d}) -a_0^2\cos(2\omega_{\rm p}t+2\beta_{\rm d}) \right.$$

$$\left.- 2a_0a_1\cos[(2\omega_{\rm p}+\omega_{\rm d})t +2\beta_{\rm p}+ \beta_{\rm d}] \right.$$

$$\left. \left. - a_1^2\cos[2(\omega_{\rm p} + \omega_{\rm d})t+ 2(\beta_{\rm p}+\beta_{\rm d})] \frac{}{}\right)\frac{}{}\right].$$ (6)

As one expected, the expression for observable variations in period derivative, $\Delta \dot p$, will be modified as well. The star's residual in $\dot p$ is owing to both torque variation, Eq. (5), and the geometrical effect. The later is due to the orientation of the star's angular velocity vector $\mbox{\boldmath$\omega$ }$ relative to the observer. As expected, the torque effects dominate the geometrical effects by a factor $(P_{\rm pre}^2 /\pi P_0 \tau_{\rm ff})(I_3/I_1)\sin^2\chi\simeq 100{-}1000$ for the precession period $\simeq 1000$ d, and so we neglected it here. Therefore

 

$$\Delta\dot{p} \simeq - \frac{P_0^2}{2\pi}{\Delta\dot\omega}$$

$$\simeq -{P_0\over T}\theta \left[\frac{}{} \cot\chi\left(\frac{}{} a_0\cos(\omega_{\rm p}t+\beta_{\rm p}) \right.\right.$$

$$\left. +a_1 \cos[(\omega_{\rm p}+\omega_{\rm d})t+\beta_{\rm p} +\beta_{\rm d}] \frac{}{}\right)$$

$$-\frac{\theta}{4} % \left(\frac{}{} 2a_0a_1\cos(\omega_{\rm d}t+\beta_{\rm d}) -a_0^2\cos(2\omega_{\rm p}t+2\beta_{\rm d}) \right.$$

$$\left.- 2a_0a_1\cos[(2\omega_{\rm p}+\omega_{\rm d})t +2\beta_{\rm p}+ \beta_{\rm d}] \right.$$

$$\left. \left. - a_1^2\cos[2(\omega_{\rm p} + \omega_{\rm d})t+ 2(\beta_{\rm p}+\beta_{\rm d})] \frac{}{}\right)\frac{}{}\right],$$ (7)

where $T=(\tau_{\rm ff}/2\sin^2\chi)(I_1/I_3)\simeq t_{\rm age}$ is approximately equal to the characteristic spin-down age and P₀ is the spin period of star. Equation (7) gives the period derivative residual due to far-field torque variations. Now let us set $\omega_{\rm p}+\omega_{\rm d}\simeq 2\pi/500$ d^-1 (equivalently $1/P_{\rm pre}+1/P_{\rm d}\simeq 1/500$ d^-1) in Eq. (7). By assuming the fundumental precession frequency is $\omega_{\rm p}\simeq 2\pi/500$ d^-1, i.e. $\omega_d\simeq 0$, only the 500 d and 250 d components will survive in Eq. (7), as obtained by Link & Epstein (2001), and all other harmonics are forbidden. Although, as shown by observations, both 500 d and 250 d Fourier components are dominant and have comparable amplitudes, the other Fourier components, especially the 1000 d-component, have nonzero amplitudes (Stairs et al. 2000). To get the 1000 d-component we choose $\omega_{\rm p}\simeq 2\pi/1000$ d^-1corresponding to the fundamental Fourier frequency seen in the data, so we have $\omega_{\rm d}\simeq \omega_{\rm p}\simeq 2\pi/1000$ d^-1. As is clear from Eq. (7), the fundamental period $\simeq$1000 d and its first three harmonics, $\simeq$500, 333, and 250 d are present in period derivative residual variations. We note that the latter assumption requires the magnetic field of the star changes in time with period $P_{\rm d}\simeq 1000$ d close to the precession period $P_{\rm pre}$. We will get back to this point later. Therefore, Eq. (7) reduces to (for $\beta_{\rm p}=0$ and $\beta_{\rm d}=0$)

 

$$\Delta\dot{p} \simeq - {P_0\over T}\theta \left[\frac{}{} \cot\chi\left(\frac{}{} a_0\cos(2\pi t/1000)+ a_1\cos(2\pi t/500) \frac{}{}\right)\right.$$

$$\left. +\frac{\theta}{4}\left(\frac{}{} 2a_0a_1\cos(6\pi t/1000) + a_1^2 \cos(2\pi t/250) \frac{}{}\right)\frac{}{}\right],$$ (8)

where t measured in days. Here we ignore the 1000 d and 500 d contributions to the $\theta^2$ order. For $\vert a_0\vert\ll \vert a_1\vert$, the 250 d-component will be comparable to the 500 d-component if we have $(a_1\theta/4)\tan\chi>1$, or $\tan\chi>4/(a_1\theta)$. For a small $\theta$ ($a_1\geq 1$), one finds that the magnetic dipole moment must be nearly orthogonal to the symmetry axis $\mbox{\boldmath$e$ }_3$. Hence for $\chi>89^\circ$ and $\omega_{\rm p}+\omega_{\rm d}\simeq 2\pi/500$ d^-1, the most dominant terms are the second and fourth harmonics, 500 d and 250 d, in good agreement with the observed data. Since the proposed inclination angle between the star's spin axis and the magnetic field's symmetry axis is nearly a right angle, $\chi>89^\circ$, one may consider the rotation of dipole vector as a magnetic poles reversal, see Sect. 3 for more discussion.

In Fig. 1, by using Eq. (8) we fit the observed $\Delta p$ and $\Delta \dot p$ data (Stairs et al. 2000) with a precession period of $P_{\rm pre}\simeq 1015$ d, a wobble angle $\theta=3\hbox{$.\!\!^\circ$ }2$, and the inclination angle $\chi=89^\circ$ between the magnetic dipole and star's symmetry axis. The Fourier expansion coefficients are a₀=.01 and a₁=1. We note that our fit is indistinguishable from the one obtained by Link & Epstein (2001).

Figure 1: Timing data for PSR B1828-11 (from Stairs et al. 2000). The top panel shows period residual $\Delta p$ (in ns) relative to the star's secular spin-down. The bottom panel gives the time derivative $\Delta \dot p$ (in units of 10-15). The solid curves are the fit explained in the text.

It is interesting to note that Eq. (8) includes 1000, 500, 333, and 250 d Fourier components. Further, by considering $n\geq 2$ terms in Eq. (5), one can find the higher harmonics in  $\Delta \dot p$. These terms were missing in the Link & Epstein's model. In Table 1 we compare the time-varying magnetic field model with one suggested by Link & Epstein (2001), and the observations made by Stairs et al. (2000).

 

Table 1: In this table we compare the time-varying magnetic field model for PSR B1828-11 with one suggested by Link & Epstein (2001) and the observed data reported by Stairs et al. (2000). P₀, $P_{\rm pre}$, $\epsilon =P_0/P_{\rm pre}$, $\theta$, and $\chi$ are the star's period, precession period, star's oblateness, wobble angle, and field's inclination angle, respectively.

Data/Model	$P_{\rm pre}$ (d)	$\epsilon$	$\theta$	$\chi$
Data	$\simeq$ 1000,  500,  250,  167	$4.7 \times 10^{-9}$	$\simeq$$3^\circ$	>$89^\circ$
Time-varying mag. model	$\simeq$1000a, all harmonics	$4.7 \times 10^{-9}$	$\simeq$$3^\circ$	>$89^\circ$
Link & Epstein's model	$\simeq$500a, $250^{\rm b}$	$9 \times 10^{-9}$	$\simeq$$3^\circ$	>$89^\circ$

aFundamental.
bHarmonic.