Neutron stars cannot be surrounded by vacuum because particles are expelled from the surface and accelerated in the surrounding strong electromagnetic field. This is indirectly deduced from their broad band electromagnetic spectrum for which the Crab pulsar is an archetypal example [

31]. Although an exact analytical solution for a rotating dipole in vacuum exists, known as Deutsch solution [

13], realistic magnetospheres require the presence of plasma producing charges and currents that retroact to the electromagnetic field. Because the problem is highly nonlinear, numerical simulations are compulsory. Two-dimensional neutron star magnetospheres have been computed in the force-free regime two decades ago starting with the aligned FFE case [

32] and followed several years later by the general three-dimensional oblique cases by [

33]. Since then, these results have been retrieved by several other authors using different numerical approaches like finite difference/finite volume methods [

34,

35,

36] or pseudo-spectral methods [

37,

38,

39]. Even a combined spectral/discontinuous Galerkin method has been tried including general-relativistic effects for a monopole [

40] or a dipole [

41]. Some extension to dissipative magnetospheres was undertaken by [

39,

42,

43] assuming an ad hoc prescription for the dissipation.

Here we show three models of pulsar magnetosphere for an oblique rotator with obliquity (angle between rotation axis and magnetic axis) $\chi =\{{0}^{\circ},{30}^{\circ},{60}^{\circ},{90}^{\circ}\}$, namely the vacuum, the force-free and the radiative cases. Simulations are performed in the observer inertial frame but using the corotating coordinate system leading to Maxwell equations written as Equation (15). This particular frame ensures that the solution relaxes to a time independent solution where the current sheet remains at a fixed position in space in order to ease its location for subsequent purposes. In other words, the time derivatives in Equation (15) must vanish when the solution becomes stationary.

#### 5.1. Numerical Schemes

Spectral and pseudo-spectral numerical schemes convert a system of partial differential equations (PDE) into a larger system of ordinary differential equations (ODE) much easier to integrate numerically with standard ODE integration techniques like the explicit Runge–Kutta and Adams-Bashforth schemes. See [

44] for a detailed review of these techniques. We emphasize that spectral methods do not approximate the equations of the problem but the solution itself. Therefore, the numerical problem exactly reflects the mathematical problem with the same boundary conditions which need to be properly imposed without any under or over-determinacy. Please note that finite volume/finite difference codes are prone to large (with respect to spectral codes) diffusion/dissipation and are therefore able to damp boundary conditions that are not exactly identical to the mathematical problem making it analytically an ill-posed problem (mathematically speaking not from a numerical point of view). Spectral methods are primarily dealing with expansion coefficients of the unknown quantities not their value themselves at the grid points. This expansion possesses the great advantage of removing singularities of differential operators like the gradient, the divergence and the curl in spherical coordinates along the polar axis. We use this flexibility to solve Maxwell equations in polar spherical coordinates

$(r,\theta ,\phi )$ with no special care about the polar axis. Boundary conditions on the stellar surface can thus be properly and exactly imposed as required by the original mathematical problem.

Specifically, the components of the electromagnetic field are expanded onto a real Fourier–Legendre–Chebyshev basis. The azimuthal dependence is expanded into a standard Fourier series in

$cos\left(m\phantom{\rule{0.166667em}{0ex}}\phi \right)$ and

$sin\left(m\phantom{\rule{0.166667em}{0ex}}\phi \right)$ whereas the latitude is expanded into Legendre functions

${P}_{\ell}^{m}\left(\theta \right)$ where

ℓ and

m are integers related to the spherical harmonics

${Y}_{\ell ,m}(\theta ,\phi )$ [

45]. The radial part is expanded into Chebyshev polynomials

${T}_{n}\left(x\left(r\right)\right)$ where

$r\in [{R}_{1},{R}_{2}]$ is mapped into the normalized range

$x\in [-1,1]$ by a linear transformation. The straightforward implementation of this mapping accumulates the discrete grid found from the Chebyshev-Gauss-Lobatto points unevenly near the boundary points where the resolution becomes prohibitively high. The constrain on the time step is therefore to severe. To distribute more evenly the grid points, we use the Kozloff/Tal–Ezer mapping [

46]. See also [

47] for a similar implementation of this technique for axisymmetric neutron star magnetospheres. Derivatives are computed in the Fourier–Legendre–Chebyshev space by simple algebraic operations, instead of pure function derivatives, and then transformed back to real space on grid points. The outer boundary conditions are outgoing waves with a sponge layer absorbing spurious reflections. The inner boundary conditions enforce the tangential part of the electric field and the normal component of the magnetic field at the stellar surface. To keep a mathematically well-posed problem, we employ the characteristic compatibility method described in [

44]. Time integration is performed via a standard third order Runge–Kutta scheme.

Spectral methods are known to converge to the exact solution faster than finite difference or finite volume schemes for sufficiently smooth problems without discontinuities. They require less resolution for the same accuracy [

48]. Because spectral methods rely on Fourier-like series expansions, they are also sensitive to the Gibbs phenomenon [

45], spoiling the solution with overshoot possibly leading to unphysical quantities like negative densities or pressures. In our strong electromagnetic field limit; however, no positivity constrain is required for the unknown field. However, in order to stabilize the algorithm, tending to put more and more energy into small scales because of the Gibbs effects, we need to filter the highest frequencies by applying for instance an exponential filter damping the highest order coefficients in the Fourier–Legendre–Chebyshev expansion. Eventually, we check a posteriori that the simulation has converged to the desired solution to good accuracy by performing a resolution analysis, meaning that increasing by a factor two the grid resolution in each direction, the solution does not significantly change. We found that for the simulations shown below, a resolution

${N}_{r}\times {N}_{\theta}\times {N}_{\phi}=257\times 32\times 64$ already gave reasonable results. We checked on a few cases that increasing by a factor 2 the resolution in all directions did not change the results (but drastically increased the computational time on a single core). Consequently, we adopted a resolution of

${N}_{r}\times {N}_{\theta}\times {N}_{\phi}=257\times 32\times 64$ for accurate and converged results. In the special case of an aligned rotator, the Gibbs phenomenon is strongest. We had to resort to higher resolution of

${N}_{r}\times {N}_{\theta}\times {N}_{\phi}=513\times 64\times 1$ for accurate and converged results.

#### 5.3. Radiative Magnetospheres

For the radiative magnetosphere, we introduce an additional free parameter represented by the pair multiplicity factor

$\kappa $ such that the electric current derived from the Aristotelian electrodynamics becomes

It is decomposed into a

$\mathbf{E}\wedge \mathbf{B}$ drift similar to force-free but without the additional constraint

$E<c\phantom{\rule{0.166667em}{0ex}}B$ and a part along

$\mathbf{E}$ and

$\mathbf{B}$ which reduces in the drift frame to a motion along the common direction of

${\mathbf{E}}^{\prime}$ and

${\mathbf{B}}^{\prime}$.

Figure 1 shows some field lines in the equatorial plane for the orthogonal rotator with

$\chi ={90}^{\circ}$ in the radiative regime with pair multiplicity

$\kappa =\{0,1,2,5\}$. In the most dissipative case corresponding to

$\kappa =0$, field lines cross the current sheet at smaller distances compared to less dissipative cases with

$\kappa =2$ or

$\kappa =5$.

Figure 2 shows the associated radial dependence of the Poynting flux for force-free and radiative cases with

$\kappa =\{0,1,2,5\}$. The radiative magnetosphere dissipates a small fraction of the Poynting flux into particle acceleration and radiation, most efficiently when

$\kappa =0$, corresponding to a charge separated plasma. Increasing the pair multiplicity factor

$\kappa $ to higher values shifts the radiative model towards the force-free limit. In the aligned case, the decrease in Poynting flux is abrupt right at the light cylinder. It is most prominent for

$\kappa =0$. However, due to the intrinsic dissipation of our algorithm, even in the FFE case there some Poynting flux dissipation is observed. This is due to the infinitely thin current sheet with discontinuous toroidal magnetic field that is smeared by our spectral methods (Gibbs phenomenon). The situation improves for oblique cases as the displacement current takes over some fraction of the electric current within the sheet.

In

Figure 3 we show the Poynting flux crossing the light cylinder for force-free and radiative cases with

$\kappa =\{0,1,2,5\}$ and depending on the inclination angle

$\chi $. All cases can be fitted with a single formal expression summarized as

with different coefficients depending on the regime considered. The fitted values extracted from the numerical simulations are listed in

Table 1. The most dissipative case

$\kappa =0$ slightly decreases the Poynting flux for the aligned rotator already inside the light cylinder. The decrease is accurately quantified by the fitting parameter

a. The FFE normalized Poynting flux is 1.42 whereas for the radiative

$\kappa =0$ case it is 1.36. The fitting parameter

b seems less dependent to the regime considered. The aligned rotator also shows the most prominent gradual decrease in the Poynting flux with respect to distance. Dissipation starts at the light cylinder but goes on at several light cylinder radii. For oblique rotators, the slope of this radial decrease slowly diminishes, becoming negligible for the orthogonal rotator.

In all regimes, the electromagnetic fluxes are very similar while inside the light cylinder. The discrepancies occur outside the light cylinder, in regions where the electric field is dominant and not fully screened by the plasma because of the too low pair multiplicity. A corotative ideal and dissipationlessness magnetosphere inside the star is therefore a good approximation, whereas outside, efficient dissipation sets in right at the light cylinder, around the current sheet.

Figure 4 shows a summary of the Poynting flux crossing the light cylinder (larger markers) and crossing a sphere of radius

$4\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{L}}$ (smaller markers) for oblique rotators in force-free and radiative regimes. The dissipation going on at large distances is most visible for the aligned rotator with green triangles.

Some fraction of the electromagnetic flux goes into particle acceleration and radiation. Quantitatively, this dissipation of the electromagnetic energy is computed as a work done on the plasma such that

This dissipation rate, for

$\kappa =\{0,1,2,5\}$, is shown in

Figure 5 on a log scale. It shows the location of largest dissipation for an orthogonal rotator according to the dissipation rate controlled by

$\kappa $. Poynting flux goes into particle acceleration and radiation mainly outside the light cylinder along the current sheet starting from the Y-point. We expect therefore gamma-rays to be produced along this sheet, emitting pulses at the neutron star rotation frequency. Such models have already been put forward and known as the striped wind. See for instance [

50] for the production of high-energy emission and [

51] for demonstrating the pulsation.

We conclude that energy conversion occurs mainly around the current sheet. Within the light cylinder, the electric field is always less than the magnetic field

$E<c\phantom{\rule{0.166667em}{0ex}}B$. Therefore, the force-free condition can be maintained without resorting to artificial damping of

$\mathbf{E}$. However, dissipation sets in right at the light cylinder, where magnetic field lines start to cross the light cylinder. The dissipation region follows a spiral pattern with decreasing amplitude with distance from the star. These new simulations offer for the first time a fully self-consistent description of a dissipative and radiative magnetosphere, where feedback between plasma flow, particle radiation and electromagnetic field is included. Please note that emission occurs only along the current sheet outside the light cylinder. This conclusion supports the idea of the striped wind model introduced by [

52] and by [

53]. It also explains pulsed high-energy emission from gamma-ray pulsars as demonstrated by [

51,

54,

55].

Contrary to the vacuum case, by construction, particles cannot move faster than the speed of light, even if the corotating velocity Equation (

22) is used. This is because the magnetic field is now sufficiently bent to counterbalance the effect of adding the corotation velocity given by Equation (

14). Note also that radiative magnetospheres presented in our study do not tend to the vacuum solution when the pair multiplicity vanishes

$\kappa =0$ because inside the light cylinder we enforce force-free conditions by construction.

Dissipative losses in the current sheet, also called striped wind, have also been proposed by other authors. For instance, ref. [

56] found a new standard solution for the aligned rotator, free of separatrix current layer within the light cylinder. Dissipation occurs only in the equatorial current sheet where acceleration and radiation of particle is allowed. They found an increase of 23% of the spindown with respect to

${L}_{\perp}$, 40% of which goes into the current sheet dissipation. In our solutions, we found a spindown increase from 36% to 42% depending on the pair multiplicity, see

Table 1. The crux of the matter is the microphysical description of this current sheet that conditions the whole magnetospheric solution. To prescribe the electric current in this sheet, ref. [

56] assumed a null-like current everywhere, a prescription which is questionable. Moreover, 60% of the magnetic flux crossing the light cylinder opens up to infinity. It is not clear how this percentage is controlled by the solution. A better solution would get all magnetic flux dissipated sooner or later in the equatorial current sheet. Ref. [

57] used another approach, performing Particle In Cell (PIC) simulations of pulsar magnetospheres. Here the sensitive parameter is the unconstrained pair injection rate, from the surface or from the whole magnetosphere. The stationary solution crucially depends on this injection mechanisms, going from an electrosphere to an almost force-free magnetosphere. They found that less than 15% of the Poynting flux is dissipated within

$2\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{L}}$. It is not clear how much additional decrease is expected if the solution would have been computed to larger distances. A partial answer is given in [

58] where the dissipation is as high as 35% at

$5\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{L}}$. Comparing both models is difficult because they are not performed with the same set up. The most critical variable being the pair multiplicity which is not fixed by the user and not easily controlled. We showed that

$\kappa $ strongly affects the asymptotic large distance dissipation in the axisymmetric case. These different approaches can only be reconciled in light of the pair content within the magnetosphere.

To summarize, all results performed with different numerical codes and different assumptions demonstrated that the magnetosphere relaxes automatically to a state where corotation with the star is enforce by the electric current prescription. However, while this picture is simple and easily understood, nothing forbids solutions with differentially rotating plasmas. Such solutions are discussed in the next section.