On the Angular Momentum Extraction from the Rotation Powered Pulsars

Rotation powered pulsars (RPPs), by definition, release rotational energy at a rate of $\dot{E}=-\Im\Omega_{*}\dot{\Omega}_{*}$, where $\Im$, $\Omega_{*}$, and $\dot{\Omega}_{*}$ are the moment of inertia, angular velocity and its time derivative of the neutron star. The associated angular momentum $\dot{L}=-\Im\dot{\Omega}_{*}$ is released at the same time. Therefore the ratio $\dot{E}/\dot{L}=\Omega_{*}$ is unique. This relation is usually not explicitly incorporated in theoretical modellings; it is justified on the basis of the fact that the relation is automatically satisfied as long as the system is solved self-consistently. In reality, however, the magnetosphere of RPPs is so complicated that no models have ever treated the whole system self-consistently. Instead, some specific part of the magnetosphere or neutron-star interior has been always a subject of theoretical modelling, and accordingly the relation between the energy and angular momentum $\dot{E}/\dot{L}=\Omega_{*}$ cannot be examined. Any of the past modelling works can be incompatible with the energy-angular-momentum relation. In this paper, we investigate two cases, the polar cap model and the current closure problem inside the neutron star with Ohmic dissipation, in which little angular momentum is lost.

As for the first case, the polar cap model, rotational energy is extracted through photons while hardly any angular momentum is carried off by these photons. Specifically when a curvature gamma-photon with energy $\epsilon$ is emitted from the polar caps, the angular momentum that it carries off is at most $\sim\epsilon R_{*}/c$, whereas the value implied by the energy-angular-momentum relation is $\epsilon/\Omega_{*}=\epsilon R_{L}/c$, where $R_{*}$ is the radius of the neutron star and $R_{L}=c/\Omega_{*}$ is the light cylinder radius, i.e., the favoured length of the lever arm is $R_{L}$, which is much larger than $R_{*}$. In terms of classical electromagnetism, it is worth noting that the wave zone of the magnetic dipole radiation originates around the light cylinder. In this case, the length of lever arm is $R_{L}$, regarding the wave front is not spherical but spirals in such a way that $\dot{E}/\dot{L}=\Omega_{*}$ holds. Since the energy-angular-momentum relation is required for the whole system, even if the polar caps do not emit sufficient angular momentum ($R_{*}\ll R_{L}$), a part of the outer magnetosphere must emit photons with a lever arm length larger than $R_{L}$ to compensate the insufficiency or equivalently electromagnetic waves are modified beyond the light cylinder. This may lead to a hypothesis that the polar cap emission will not make any significant effect on the magnetospheric structure. However, the observed radio luminosity becomes comparable with the spin-down luminosity for old pulsars (Wu et al., 2020). This fact implies that the polar cap luminosity can contribute to a significant fraction of $\dot{E}$, and that the inefficiency of the angular momentum loss may cause some crucial effect on the outer magnetosphere.

As for the second case, the Ohmic dissipation inside the star, the magnetospheric current goes out of and comes back to the star, thereby conveying the rotation energy of the star away through the magnetosphere. The magnetospheric current runs inside the star to make the circuit closed. This current generates the braking torque. Note that the term “magnetospheric current inside the star” does not include the current generated through the magneto-thermal evolution of the neutron star but means the current that is connected to the magnetosphere. If the Ohmic dissipation takes place on the magnetospheric current inside the star, a part of the rotational energy is converted to heat, which is eventually radiated away, while there is no loss of angular momentum from the system. This effect was not taken into account in the previous works of this subject (Beskin et al., 2015; Karageorgopoulos, Gourgouliatos & Contopoulos, 2019) which considered how the magnetospheric current ran inside the star.

In this paper, we investigate the above-mentioned two cases and discuss what would happen when the rotational energy is lost but not the angular momentum. Especially, for the second case, we demonstrate how the magnetospheric current inside the star is determined.

In this paper, we consider the obliquely rotating magnetosphere in a steady state. In this case, the electric field is written in the form of

without loss of generality (Mestel, 1999), where ${\mbox{\boldmath$u$}_{\rm c}}=\mbox{\boldmath$\Omega$}_{*}\times\mbox{\boldmath$r$}$ is the corotation vector, $B$ is the magnetic field, and $\Phi$ is a scalar potential called the “non-corotational electric potential”. The first term of the equation is the corotational electric field ${\mbox{\boldmath$E$}}_{\rm c}=-{\mbox{\boldmath$u$}_{\rm c}}\times{\mbox{\boldmath$B$}}/c$. The field-aligned electric field is given by $E_{\parallel}=-\hat{{\mbox{\boldmath$B$}}}\cdot\nabla\Phi$, where $\hat{{\mbox{\boldmath$B$}}}={\mbox{\boldmath$B$}}/|{\mbox{\boldmath$B$}}|$ is the unit vector along the magnetic field. Some useful formulae under the steadiness condition are given in the appendix.

Here let us assume at the moment that the star is a perfect conductor satisfying the ideal-magneto-hydrodynamic (MHD) condition, ${\mbox{\boldmath$E$}}+{\mbox{\boldmath$v$}}\times{\mbox{\boldmath$B$}}/c=0$, and rotates rigidly with ${\mbox{\boldmath$v$}}={\mbox{\boldmath$u$}_{\rm c}}$, where $v$ indicates the velocity field. Then the electric field is ${\mbox{\boldmath$E$}}_{\rm c}$ and $\Phi=\mbox{constant}=0$ inside the star. If the surrounding space is filled with plasma satisfying the ideal-MHD condition, then

holds so that ${\mbox{\boldmath$B$}}\cdot\nabla\Phi=0$, meaning that the value $\Phi=0$ on the stellar surface propagates into the magnetosphere along $B$. Thus we have the corotation electric field and the iso-rotation

in the magnetosphere, where $\kappa$ is a scalar function.

The torque density in the star is given by $({\mbox{\boldmath$u$}_{\rm c}}/\Omega_{*})\cdot({\mbox{\boldmath$j$}}\times{\mbox{\boldmath$B$}}/c)$, and therefore the net torque on the star is

where $j$ is the current density of the magnetospheric current, and ${\cal{V}}_{*}$ is the stellar volume. The rotational power released from the star is thus given by $\Omega_{*}N$. The Poynting theorem

where

reduces to

in the steady condition. Combining (4) and (7) with the assumption ${\mbox{\boldmath$E$}}={\mbox{\boldmath$E$}}_{\rm c}$, we have the rotation power represented by the surface integral over the stellar surface ${{\cal{S}}}_{*}$:

This indicates that the rotational energy flowing out to the magnetosphere is accounted by the Poynting flux on the stellar surface.

We investigate the influence of the polar cap acceleration on the global torque balance of the magnetosphere.

We use a simple model as described below and illustrated in Fig. 1. Suppose a closed surface ${\cal{S}}$ surround the neutron star. The ideal-MHD condition is satisfied inside the surface ${\cal{S}}$ except for a particle-acceleration region along open field lines (the grey region in the figure). The region between ${\cal{S}}$ and ${{\cal{S}}}_{*}$ may be called the “inner magnetosphere”, provided that ${\cal{S}}$ is within the light cylinder. The region beyond ${\cal{S}}$ may be called the “outer magnetosphere”, where the ideal-MHD condition is assumed to hold. The outer magnetosphere possesses a field-aligned outflow, the pulsar wind.

Immediately above the stellar surface ${{\cal{S}}}_{*}$ and below the acceleration region, $\Phi=0$, ${\mbox{\boldmath$E$}}={\mbox{\boldmath$E$}}_{\rm c}$ and the flow obeys the relation ${\mbox{\boldmath$v$}}={\mbox{\boldmath$u$}_{\rm c}}+\kappa{\mbox{\boldmath$B$}}$. In the acceleration region, by contrast, there is a field-aligned electric field $E_{\parallel}=-\hat{{\mbox{\boldmath$B$}}}\cdot\nabla\Phi\not=0$. Even if the ideal-MHD condition is recovered beyond the acceleration region, variation of the non-corotational electric potential causes drift motions:

This may be attributed to the sub-pulse drift motion (Szary et al., 2020).

The equation of motion for the particles in the inner magnetosphere may be

where $\mbox{\boldmath$F$}_{\rm ext}$ represents any non-electromagnetic forces, the subscript “s” indicates the particle species, and $e_{\rm s}$, $n_{\rm s}$, ${\mbox{\boldmath$v$}}_{\rm s}$, and $\gamma_{\rm s}$ are respectively the charge, number density, velocity, and the Lorentz factor. For the reaction force by the curvature radiation we have

where $R_{\rm c}$ is the the radius of curvature. The work done by the electromagnetic force per unit time is given by the left-hand side of ${\mbox{\boldmath$v$}}_{\rm s}\cdot$(10) for each particle species, so that in total we have

where ${\cal{V}}_{\rm in}$ indicates the volume between the surface ${\cal{S}}$ and ${{\cal{S}}}_{*}$. The value $\dot{E}_{\rm in}$ is the energy carried off by photons and/or particles from the inner magnetosphere.

The component along the rotation axis of the torque (the transferred angular momentum per unit time) is given by $({\mbox{\boldmath$u$}_{\rm c}}/\Omega_{*})\cdot$(10):

where (1) is used in the last manipulation and $q=\sum_{\rm s}e_{\rm s}n_{\rm s}$ is the charge density. Subtracting $\Omega_{*}$ times (15) from (13) yields

where $j_{\parallel}={\mbox{\boldmath$j$}}\cdot\hat{{\mbox{\boldmath$B$}}}$, the approximation in the last step is based on the facts that the ratio of the microscopic to macroscopic time scales $|j_{\perp}/j_{\parallel}|=|{\mbox{\boldmath$j$}}\times\hat{{\mbox{\boldmath$B$}}}|/|{\mbox{\boldmath$j$}}\cdot\hat{{\mbox{\boldmath$B$}}}|\approx(c/R_{\rm c})/(e_{\rm s}B/\gamma_{\rm s}m_{\rm s}c)$ is negligible, and $|{\mbox{\boldmath$u$}_{\rm c}}|\ll c$ holds in the inner magnetosphere. The derived equation indicates that no significant amount of the angular momentum is lost from the inner magnetosphere.

In our simple model, the outer magnetosphere is treated in the one-fluid approximation with the ideal-MHD condition. The equation of motion is

From ${\mbox{\boldmath$v$}}\cdot$ (18) and $({\mbox{\boldmath$u$}_{\rm c}}/\Omega_{*})\cdot$ (18), it is straightforward to obtain the energy and angular momentum conservation laws:

where

where ${\mbox{\boldmath$w$}}={\mbox{\boldmath$v$}}-{\mbox{\boldmath$u$}_{\rm c}}$. The first terms of the expressions above are the contribution from the particles, while the last two terms are from the electromagnetic fields. The energy-angular momentum outflow through the surface ${\cal{S}}$ can be evaluated with the last two terms, namely

These are used in part for the wind acceleration, and the unused parts of them propagate to the termination shock in a form of Poynting energy. As is expected, (23), (24) and (16) are combined to yield

The torque balance is satisfied in the whole system. Although we are not concerned with what actually happens in the outer magnetosphere, it is certain that the outer magnetosphere carries angular momentum so efficiently as to achieve the torque balance of the whole system. This fact together with the source of Poynting flux being provided by braking torque, (8), is included in classical electromagnetism (Beskin, 2010). If we define the effective angular velocity $\Omega_{\rm out}$ of the outer magnetosphere as $\Omega_{out}=\dot{E}_{\rm out}/\dot{L}_{\rm out}$, we obtain the relation

This indicates that the effective light-cylinder radius $c/\Omega_{\rm out}$ is larger than $R_{L}$. In other words, the lever arm of the angular momentum carried away from the outer magnetosphere becomes larger to compensate the inefficient angular momentum loss from the inner magnetosphere; the body inside the surface ${\cal{S}}$ is effectively a slow rotator. In this case, if photons are emitted in the outer magnetosphere, the emission site must be beyond the light cylinder. Alternatively, the Poynting vector in the wind would tilt to satisfy $|\mbox{\boldmath$r$}\times\mbox{\boldmath$S$}|/|\mbox{\boldmath$S$}|>R_{L}$. What exactly happens in the outer magnetosphere is discussed in §6.

If Ohmic dissipation is present on the magnetospheric current which makes a circuit inside the star, then rotational energy is consumed without any loss of angular momentum. Let us consider this case in this section.

For the present discussion we employ a two-component (proton-electron) plasma with frictional force¹¹1The proton-electron model should be sufficient for the present discussion although inclusion of heavier-ion components would make the model closer to the reality.. The equations of motion can be written as

where the left-hand sides represent the Lagrangian time derivative, the subscripts indicate the particle species, and $\mbox{\boldmath$F$}_{\rm f}$ is the frictional force which is given by

where $\nu_{\rm coll}$ is the effective collision frequency. A set of equations (27) and (28) is reduced to a familiar set of the one-fluid equation of motion and Ohm’s law:

where the electric conductivity $\sigma=e^{2}n_{\rm e}/m_{\rm e}\nu_{\rm coll}$ is introduced and the approximation is based on ${\mbox{\boldmath$v$}}=(m_{\rm p}n_{\rm p}{\mbox{\boldmath$v$}}_{\rm p}+m_{\rm e}n_{\rm e}{\mbox{\boldmath$v$}}_{\rm e})/(m_{\rm p}n_{\rm p}+m_{\rm e}n_{\rm e})\approx{\mbox{\boldmath$v$}}_{\rm p}$, $q=e(n_{\rm p}-n_{\rm e})$, ${\mbox{\boldmath$j$}}=e(n_{\rm p}{\mbox{\boldmath$v$}}_{\rm p}-n_{\rm e}{\mbox{\boldmath$v$}}_{\rm e})$ and $q/en_{e}\ll 1$, whereas we ignore the Hall term, which is very small for the magnetospheric current (see § 6.2 for discussion). Using Ohm’s law is equivalent to using the two-component model with friction.

For the energy conservation, adding (27)$\cdot{\mbox{\boldmath$v$}}_{\rm p}$ and (28)$\cdot{\mbox{\boldmath$v$}}_{\rm e}$, we have

where the quasi-neutrality condition $n_{e}=n_{p}$ has been applied, and ${\mbox{\boldmath$j$}}=en_{\rm e}({\mbox{\boldmath$v$}}_{\rm p}-{\mbox{\boldmath$v$}}_{\rm e})$ in the last manipulation. Here we are concerned with only the rotational motion and set ${\mbox{\boldmath$v$}}_{\rm p}=\Omega r\mbox{\boldmath$e$}_{\varphi}$, where $r$ is the distance from the rotation axis and $\Omega$ is the angular velocity as a function of the position, whereas $\Omega_{*}$ is used for the angular velocity on the stellar surface. We should note that $\Omega_{*}$ determines the ratio of the energy and the angular momentum which flows out from the star. Ignoring the electron inertia, we obtain

This is integrated over the volume of the star to give

where

is the moment of inertia of the star, and the operator $\langle\ \rangle$ indicates taking the average. Equation (34) implies that some part of the rotational energy goes into Joule heating and the rest is the source of the Poynting flux which flows into the magnetosphere; the volume integral of ${\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$E$}}$ is the surface integral of $S$ on the stellar surface (equations (4) and (8)).

As for the component along the rotation axis of the angular momentum, adding (27)$\cdot({\mbox{\boldmath$u$}_{\rm c}}/\Omega_{*})$ and (28)$\cdot({\mbox{\boldmath$u$}_{\rm c}}/\Omega_{*})$, we have

where the frictional terms cancel each other out, following the action-reaction principle, and the Ohmic term has no effect on the angular momentum. Equation (36) is rewritten as, with ${\mbox{\boldmath$E$}}=-{\mbox{\boldmath$u$}_{\rm c}}\times{\mbox{\boldmath$B$}}/c-\nabla\Phi$ and $\nabla\cdot({\mbox{\boldmath$j$}}-q{\mbox{\boldmath$u$}_{\rm c}})=0$,

The second term of the right-hand side is integrated over the star to give zero because $\Phi=0$ on the surface of the star:

Consequently, we obtain

This means that all the lost angular momentum goes into the magnetosphere.

The ratio of the net losses of the energy and angular momentum is calculated to be

Therefore, the stellar interior rotates faster than the surface when Ohmic dissipation takes place. The main conclusion of this section is that in determining the magnetospheric current inside the star one must treat the dynamics of the interior matter, particularly the angular momentum transfer.

In the previous section, we have found that the magnetospheric current inside the star is determined not by Ohm’s law but by magneto-hydrodynamics. An illustrative example is given in this section.

Since Ohmic dissipation is very small in reality, the ideal-MHD condition may be appropriate in the stellar interior. In this case, we need not consider the differential rotation caused by Ohmic heating . In the following discussion we assume the rigid rotation in the form of ${\mbox{\boldmath$v$}}=\mbox{\boldmath$r$}\times\Omega(t)\mbox{\boldmath$e$}_{z}$. The basic equations are

where $\rho$ is the mass density, $\Phi_{g}$ is the gravitational potential, and $\Pi$ is the stress tensor. The magnetic field $B$ and the associated current $j$ include the components that are “intrinsic” to the star, i.e., those determined from the star structure and evolution. We write them in the from of the intrinsic term plus the magnetospheric term: ${\mbox{\boldmath$B$}}={\mbox{\boldmath$B$}}_{0}+{\mbox{\boldmath$B$}}^{\prime}$ and ${\mbox{\boldmath$j$}}={\mbox{\boldmath$j$}}_{0}+{\mbox{\boldmath$j$}}^{\prime}$. The expansion is

Note that the the magnetospheric components are much smaller than the intrinsic ones ($|{\mbox{\boldmath$B$}}^{\prime}|/|{\mbox{\boldmath$B$}}_{0}|\sim(R_{*}/R_{L})^{3/2}$). The first term ${\mbox{\boldmath$j$}}_{0}\times{\mbox{\boldmath$B$}}_{0}/c$ is not related to the spin-down. The dominant braking force is due to the term ${\mbox{\boldmath$j$}}^{\prime}\times{\mbox{\boldmath$B$}}_{0}/c$. For the axisymmetric cases, this term is the only force in the azimuthal direction. We ignore the last term.

Because we have specified the velocity field in terms of $\Omega(t)$ and have introduced elasticity, the set of equations (41) and (42) have a solution for any given magnetospheric current ${\mbox{\boldmath$j$}}^{\prime}$. Then we cannot find a unique solution for the magnetospheric current distribution. However, if the stress is very high, the material will have viscous motion and/or cracks to reduce the elastic energy $E_{\rm el}$. The spin-down torque is too small to break the latices. Hence, minimization of $E_{\rm el}$ will be achieved only through tiny non-elastic change in the material, and thus the current distribution will change to the state in which $E_{\rm el}$ is minimized. We take the following steps: (i) solving (41) for a trial ${\mbox{\boldmath$j$}}^{\prime}$, (ii) calculating $E_{\rm el}$ from the obtained displacement $u_{\varphi}$, (iii) varying ${\mbox{\boldmath$j$}}^{\prime}$ and then recalculating $E_{\rm el}$, and (iv) finding ${\mbox{\boldmath$j$}}^{\prime}$ in the searched parameter space so as to minimize $E_{\rm el}$.

For simplicity, we assume axisymmetry about and uniform background magnetic fields along the rotation axis, ${\mbox{\boldmath$B$}}_{0}=B_{0}\mbox{\boldmath$e$}_{z}$. The star is assumed to be a uniform elastic body. The azimuthal component of the equation of motion is

where we use the cylindrical coordinates $(r,\varphi,z)$, for which the unit vectors are denoted by $\mbox{\boldmath$e$}_{r}$, $\mbox{\boldmath$e$}_{\varphi}$ and $\mbox{\boldmath$e$}_{z}$, $u_{\varphi}$ is the displacement in the azimuthal direction, $F_{\varphi}=({\mbox{\boldmath$j$}}^{\prime}\times{\mbox{\boldmath$B$}}_{0})\cdot\mbox{\boldmath$e$}_{\varphi}/c$ is the electromagnetic force generated by the magnetospheric current, and $\mu_{\rm s}$ is the shear modulus which is uniform. We may use the spherical polar coordinates $(R,\theta,\varphi)$ occasionally.

The magnetospheric current inside the star is purely poloidal and is represented by, with the use of the current stream function $I=rB^{\prime}_{\varphi}c/2$,

It is notable that the contours of $I$ give the stream lines of the current. The value of $I$ on the star surface is denoted by $I_{*}(s)$, which is a function of the arc length from the pole along the meridian $s=R_{*}\theta$. The function $I_{*}(s)$ is treated as the boundary condition of the present model, connecting the star and magnetosphere. Note that $I_{*}(s)$ gives the net current running through the surface area with an angular range of $0\leq\theta\leq s/R_{*}$.

The azimuthal component of the electromagnetic force is

The net torque on the star is straightforwardly obtained with surface integration to be

Once $I_{*}(s)$ has been given as a boundary condition, $\dot{\Omega}=N/\Im$ is determined.

The shear stress is given by

The elastic energy is

If $I$ is specified, (44) is easily solved numerically, and the elastic energy is obtained.

We need to provide some trial current function. In this study, we employ for it simple cubic forms with a few parameters, which are determined so that $E_{\rm el}$ is minimized. For $I_{*}(s)$, a simple analytic function

is assumed, where $s_{\rm pc}=R_{*}\theta_{\rm pc}$ with the polar cap co-latitude $\theta_{\rm pc}=\sqrt{R_{*}\Omega_{*}/c}$ (Fig. 2), and $I_{0}$ is the peak value, meaning the total circulating current. In the following numerical illustration , we use a slightly large value of $\theta_{\rm pc}=(R_{L}/R_{*})^{-1/2}=0.2913$ (corresponding to a 1-msec pulsar) to reduce numerical load. The current function within the star $I(r,z)$ describes how current runs inside the star. We additionally introduce a simple cubic function $K$ with the parameter $z_{2}$ above which one half of the current $I_{0}$ closes:

with

where $z_{\rm surf}=\sqrt{R_{*}^{2}-r^{2}}$ is the $z$-value on the surface for a given $r$ (right panel of Fig. 2). Note that ${K}(0)=0$, ${K}(1/2)=1/2$, ${K}(1)=1$, and ${K}(\zeta(z=z_{2}))=1/2$.

Although how deep the magnetospheric current penetrates into the star is parameterized by $z_{2}$, one may need to restrict the current in a layer $R_{1}<R<R_{*}$, say the crust. If this is the case, we use a diminishing factor of the Fermi-Dirac type:

where $\Delta R$ is the thickness for the transition. If not, $K_{\rm cut}=1$. Incorporating this factor $K_{\rm cut}$, we have

where $R=\sqrt{r^{2}+z^{2}}$, and $s=R_{*}\arcsin(r/R_{*})$. Some examples of $I(r,z)$ are shown in Fig. 3. As $z_{2}$ is decreased (see panels (a), (b) and (c), of Fig. 3 in this order), the current runs deeper along the field lines. If the diminishing factor is applied, the current is restricted above $R_{1}$ (panel (d)).

For the given current, we calculate $F_{\varphi}$ and $\dot{\Omega}$ from (46) and (47). Fig. 4(a) shows the shear moment $(F_{\varphi}-\rho\dot{\Omega}r)r^{2}$ for the case of $z_{2}=0.5z_{\rm surf}$. It shows that the inertial force in $+\varphi$ direction dominates in the equatorial region, whereas the electromagnetic braking force in $-\varphi$ direction does in the region $0\leq r\leq R_{*}\sin\theta_{\rm pc}$, which is the polar cap magnetic flux.

With the specified $F_{\varphi}$ and $\dot{\Omega}$, we solve (44) numerically to obtain the displacement $u_{\varphi}$. The obtained displacement is shows in Fig. 4(b). As is expected, the inner ($r\la 0.5$) and outer ( $r\ga 0.5$) regions are deformed in $+\varphi$ and $-\varphi$ directions, respectively. In the boundary region between the two, the elastic energy is large as shown in Fig. 4(c). The numerical calculation is done in the non-dimensional form, where all the formulae are the same except $c=1$. The force density, displacement and $E_{\rm el}$ are respectively in units of $F_{0}=B_{0}I_{0}/cR_{*}^{2}$, $F_{0}R_{*}^{2}/\mu_{\rm s}$, and $\dot{E}_{0}\Omega_{*}^{-1}$, where the unit of the power is $\dot{E}_{0}=(\mu_{\rm m}^{2}\Omega_{*}^{4}/c^{3})(2R_{L}/R_{*})$, and $\mu_{\rm m}$ is the magnetic moment of the star.

We then obtain a series of solutions for different $z_{2}$ and find that the elastic energy is minimized at $z_{2}\sim 0.4z_{\rm surf}$ as shown in Fig. 5(a). This result is plausible, i.e., the elastic energy is decreased if the braking force is distributed over the whole neutron star to diminish the inertial force. Conversely, if the braking force is localized in a small region near the surface region ($z_{2}\sim 1$) or the core region ($z_{2}\sim 0$), then the elastic energy is increased. If the current is restricted in a layer above $R_{1}$, the thinner the current is restricted, the larger the elastic energy is (Fig. 5(b)).

In addition to this, if the current penetrates out of the polar cap magnetic flux tube ($r>R_{*}\sin\theta_{\rm pc}$), the electromagnetic force works in the opposite direction to the braking force ($+\varphi$ direction), and the elastic energy increases. As a result, the magnetospheric current inside the star is restricted in the polar cap magnetic flux tube. Thus, the configuration of ${\mbox{\boldmath$B$}}_{0}$, especially that of the polar cap magnetic flux plays essential role to relax the elastic energy of the star. This point is discussed in § 6.2.

This work is supported by KAKENHI 18H01246 (SS, SK), 19K14712 and 21H01078 (SK). The authors thank the anonymous referee for comments which led to improvements in the manuscript.

The data underlying this article will be shared on reasonable request to the corresponding author.