Appendix A: Surface magnetic field

In order to model the actual surface magnetic field by superposition of the star-centered global dipole $\vec{d}$ and the crust-anchored dipole moment $\vec{m}$, let us consider the general geometrical situation presented in Fig. 1. The resultant surface magnetic field is

$$\vec{B}_{\rm s}=\vec{B}_{\rm d}+\vec{B}_{\rm m},$$ (A.1)

where

$$\vec{B}_{\rm d}=\left( \frac{2d\cos \theta }{r^{3}},~\frac{d\sin \theta }{% r^{3}},~0\right),$$ (A.2)

and

$$\vec{B}_{\rm m}=\frac{3\left( \vec{r}-\vec{r}_{\rm s}\right) ... ...ert^{2}}{\left\vert \vec{r}-\vec{r}% _{\rm s}\right\vert^{5}},$$ (A.3)

where the spherical components of $\vec{B}_{\rm m}$ are explicitly given in Eqs. (A.5). The global magnetic dipole moment $d=(1/2)B_{\rm p}R^{3}$, where $% B_{\rm p}=6.4\times 10^{19}(P\cdot \dot{P})^{1/2}$ G is the dipole component at the pole derived from the pulsar spin-down rate, and the crust-anchored local dipole moment $m=(1/2)B_{\rm m}\Delta R^{3}$, where $\Delta R\sim 0.05R< L$ and $L \sim 10^5$ cm is the characteristic crust dimension (R=10⁶ cm). One can see that a local anomaly can significantly influence the surface magnetic field $(B_{\rm m}\sim B_{\rm d})$if $m/d\ga 10^{-4}$. We use the star-centered spherical coordinates with z axis directed along the global magnetic dipole moment. Thus, $\vec{r}_{\rm s}=(r_{\rm s},\theta_{\rm r},\phi_{\rm r})$ and $\vec{m}=(m,\theta_{\rm m},\phi_{\rm m})$.

The equations of the line of force of the vector field $\vec{B}(B_{\rm r},B_\theta,B_\phi)$ have a well known form in spherical co-ordinates

$$\frac{{\rm d}\theta }{{\rm d}r}=\frac{B_{\theta }}{rB_{\rm r}... ...hi }{{\rm d}r}=\frac{% B_{\phi }}{r\sin \theta B_{\rm r}}\cdot$$ (A.4)

The solution of these equations, with the initial conditions $\theta \left( r=R\right) =\theta _{0}$ and $\phi \left( r=R\right) =\phi _{0}$ determining a given field line at the surface, describe the parametric equation of the field lines. The magnetic field $\vec{B}_{\rm m}$ (Eq. (A.3)), as well as the dipolar magnetic filed $\vec{B}_{\rm d}$ (Eq. (A.2)), can be written in terms of spherical components

 

$$B_{\rm r}^{\rm m} -\frac{1}{D^{2.5}}\left( 3Tr_{\rm r}^{\rm s}-3Tr+Dm_{\rm r}\right),$$ =

$$B_{\theta }^{\rm m} -\frac{1}{D^{2.5}}\left( 3Tr_{\theta }^{\rm s}+Dm_{\theta }\right),$$ =

$$B_{\phi }^{\rm m} -\frac{1}{D^{2.5}}\left( 3Tr_{\phi }^{\rm s}+Dm_{\phi }\right),$$ = (A.5)

where

$$D =r_{\rm s}^{2}+r^{2} -2r_{\rm s}r\left( \sin \theta_{\rm r}~ \s... ...\cos \left( \phi -\phi_{\rm r}\right) +\cos \theta_{\rm r}~\cos \theta \right),$$

$${\rm and} ~T =m_{\rm r}r -\left( m_{\rm r}r_{\rm r}^{\rm s}+m_{\theta }r_{\theta }^{\rm s}+m_{\phi }r_{\phi }^{\rm s}\right).$$

According to the geometry presented in Fig. (1), the components of the radius-vector of the origin of the star-anchored local dipole moments can be written as

$$r_{\rm r}^{\rm s} r_{\rm s}\left( \sin \theta_{\rm r}~\sin \theta ~\cos \left( \phi -\phi_{\rm r}\right) +\cos \theta_{\rm r}~\cos \theta \right),$$ =

$$r_{\theta }^{\rm s} r_{\rm s}\left( \sin \theta_{\rm r}~\cos \theta ~\cos \left( \phi -\phi_{\rm r}\right) -\cos \theta_{\rm r}~\sin \theta \right),$$ =

$$r_{\phi }^{\rm s} -r_{\rm s}\sin \theta_{\rm r}~\sin \left( \phi -\phi_{\rm r}\right),$$ =

and the components of the local dipole moment are

$$m_{\rm r} m\left( \sin \theta _{\rm m}~\sin \theta ~\cos \left( \phi -\phi _{\rm m}\right) +\cos \theta _{\rm m}~\cos \theta \right),$$ =

$$m_{\theta } m\left( \sin \theta _{\rm m}~\cos \theta ~\cos \left( \phi -\phi _{\rm m}\right) -\cos \theta _{\rm m}~\sin \theta \right),$$ =

$$m_{\phi } -m\sin \theta _{\rm m}~\sin \left( \phi -\phi _{\rm m}\right).$$ =

We now define the boundary of the open field lines by setting $\vec{B}= \vec{B}_{\rm d}$ at an altitude $r_{\rm d}=50R$, where the magnetic field should appear like a pure dipole (the numerical results are quite insensitive to the choice of $r_{\rm d}$ as long as it is few times R; in fact, already at r=5R the ratio $B_{\rm m}/B_{\rm d}\sim 10^{-4}$ for $m \approx 10^{-4}~d$). Then we solve numerically the system of differential equations

$$\frac{{\rm d}\theta }{{\rm d}r}=\frac{B_{\theta }^{\rm d}+B_{... ..._{\rm r}^{\rm d}+B_{\rm r}^{\rm m}\right) }\equiv \Theta _{1},$$ (A.6)

$$\frac{{\rm d}\phi }{{\rm d}r}=\frac{B_{\phi }^{\rm m}}{r\left... ...r}^{\rm d}+B_{\rm r}^{m}\right) \sin \theta }\equiv \Phi _{1},$$ (A.7)

obtained from Eq. (A.4) by substitutions $B_{\rm r}=B_{\rm r}^{\rm d}+B_{\rm r}^{\rm m}$, $B_{\theta}=B_{\theta}^{\rm d}+B_{\theta}^{\rm m}$ and $% B_{\phi}=B_{\phi}^{\rm d}+B_{\phi}^{\rm m}$ (Eqs. (A.2) and (A.5)), with the initial conditions $\vec{B}_{\rm m}=0$defined at r=50R and trace each field line from altitude $r_{\rm d}$ (where $\vec{B}_{\rm m}(r_{\rm d},\theta,\phi)\equiv 0$) down to the stellar surface (<tex2html_comment_mark> r=R). Here we defined two functions $\Theta_1$ and $\Phi_1$ to be used later in Eqs. (A.10)-(A.12).

The curvature $\Re=1/\rho_{\rm c}$ of the field lines (where $\rho _{\rm c}$ is the radius of curvature presented for various cases in Figs. 2-8) is calculated as

$$\Re =\left( \frac{{\rm d}s}{{\rm d}r}\right) ^{-3}\left\vert ... ...{\rm d}r}\frac{{\rm d}^{2}s}{{\rm d}r^{2}}\right) \right\vert,$$ (A.8)

where

$$\frac{{\rm d}s}{{\rm d}r}=\sqrt{\left( 1+r^{2}\Theta_1^{2}+r^{2}\Phi_1^{2}\sin ^{2}\theta \right)}.$$ (A.9)

Thus, the curvature can be written in the form

$$\Re =\left( S_{1}\right) ^{-3}\left( J_{1}^{2}+J_{2}^{2}+J_{3}^{2}\right) ^{1/2},$$ (A.10)

where

$$X_{2}S_{1}-X_{1}S_{2},\;\;\; J_{2} = Y_{2}S_{1}-Y_{1}S_{2},$$                         J₁ =

J₃ = Z₂S₁-Z₁S₂

$$\sin\theta\cos\phi +r\Theta_{1}\cos \theta \cos \phi -r\Phi_{1}\sin \theta \sin \phi ,$$ X₁ =

$$\sin \theta \sin \phi +r\Theta_{1}\cos \theta \sin \phi +r\Phi_{1}\sin \theta \cos \phi ,$$ Y₁ =

$$\cos \theta -r\Theta _{1}\sin \theta ,$$ Z₁ =

$$\left( 2\Theta_{1}+r\Theta_{2}\right) \cos \theta \cos \phi-\left( 2\Phi_{1}+r\Phi _{2}\right) \sin \theta \sin \phi$$ X₂ =

$$-r\left( \Theta_{1}^{2}+\Phi_{1}^{2}\right) \sin \theta \cos \phi -2r\Theta_{1}\Phi _{1}\cos \theta \sin \phi ,$$

$$\left( 2\Theta_{1} + r\Theta_{2}\right) \cos \theta \sin \phi+\left( 2\Phi _{1}+r\Phi _{2}\right) \sin \theta \cos \phi$$ Y₂ =

$$-r\left(\Theta_{1}^{2}+\Phi_{1}^{2}\right) \sin \theta \sin \phi +2r\Theta_{1}\Phi _{1}\cos \theta \cos \phi ,$$

$$-\Theta_{1}\sin \theta -\Theta_{1}\sin \theta - r\Theta_{2}\sin \theta -r \Theta_{1}^{2}\cos \theta ,$$ Z₂ =

$$\sqrt{ 1+r^{2}\Theta_{1}^{2}+r^{2}\Phi_{1}^{2}\sin^{2}\theta } ,$$ S₁ =

$$S_{1}^{-1}\left( r\Theta_{1}^{2}+r^{2}\Theta_{1}\Theta _{2}+r\Phi_{1}^{2}\sin ^{2}\theta \right.$$ S₂ =

$$\; \;\;\;\;\;\;\;+\left. r^{2}\Phi_{1}\Phi_{2}\sin^{2}\theta + r^{2}\Theta_{1}\Phi_{1}^{2}\sin \theta \cos \theta \right) ,$$

$$\Theta_{2} \equiv \frac{{\rm d}\Theta_{1}}{{\rm d}r},\;\;\;\; \Phi_{2}\equiv\frac{% {\rm d}\Phi_{1}}{{\rm d}r}\cdot$$ (A.11)

In the two-dimensional case Eq. (A.8) can be simplified to the following form

 

$$\Re =\left( r\frac{{\rm d}^{2}\theta }{{\rm d}r^{2}}+2\frac{{\rm ... ...left( 1+r^{2}\left( \frac{{\rm d}\theta }{{\rm d}r}\right) ^{2}\right) ^{-1.5}.$$ (A.12)

Acknowledgements

This paper is supported in part by the KBN Grant 2 P03D 008 19 of the Polish State Committee for Scientific Research. We are grateful to Dr. U. Geppert for very helpful discussions. We also thank E. Gil and G. Melikidze Jr. for technical help. DM would like to thank the Institute of Astronomy, University of Zielona Góra, for support and hospitality during his visit to the institute, where this and the accompanying paper were started.