Force-free Regime Research Articles

Simulating the shock dynamics of a neutron star accretion column

ABSTRACT Accretion on to a highly magnetized neutron star runs through a magnetospheric flow, where the plasma follows the magnetic field lines in the force-free regime. The flow entering the magnetosphere is accelerated by the gravity of the star and then abruptly decelerated in a shock located above the surface of the star. For large enough mass accretion rates, most of the radiation comes from the radiation–pressure-dominated region below the shock, known as accretion column. Though the one-dimensional, stationary structure of this flow has been studied for many years, its global dynamics was hardly ever considered before. Considering the time-dependent structure of an accretion column allows us to test the stability of the existing stationary analytic solution, as well as its possible variability modes, and check the validity of its boundary conditions. Using a conservative scheme, we perform one-dimensional time-dependent simulations of an ideal radiative MHD flow inside an aligned dipolar magnetosphere. Whenever thermal pressure locally exceeds magnetic pressure, the flow is assumed to lose mass. Position of the shock agrees well with the theoretical predictions below a limit likely associated with advection effects: if more than $2/3$ of the released power is advected with the flow, the analytic solution becomes self-inconsistent, and the column starts leaking at a finite height. Depending on the geometry, this breakdown may broaden the column, mass load the field lines, and produce radiation-driven, mildly relativistic ejecta. Evolving towards the equilibrium position, the shock front experiences damped oscillations at a frequency close to the inverse sound propagation time.

Ill-posedness of the mean-field dynamo equations with a linear electromotive force

We show that the initial-value problem for the non-relativistic magnetic dynamo equation turns out to be ill-posed in L2 norm when the electromotive force depends linearly on the magnetic field. This result implies that the increasing of magnetic energy does not necessarily come from physical amplification mechanisms, since certain magnetic modes could arbitrarily grow as wave-frequency increases, despite any dynamo process. Thus, up to this order, the theory is not suitable for astrophysical simulations. We then study the case when electromotive forces are linear in magnetic field derivatives, showing that the resulting system has a well-posed problem. Finally, we apply the well-posed theory to the force-free regime, for which we find bounds for the corresponding magnetic energy analyzing the evolution of the magnetic helicity.

Exploring the Energy-dependent Radiation Properties in Dissipative Magnetospheres with Fermi Pulsars

Abstract The equatorial current sheets outside the light cylinder (LC) are thought to be promising sites of high-energy emission based on the results of recent numerical simulations. We explore the pulsar light curves and energy spectra by computing the curvature radiation based on the FIDO magnetospheres. The FIDO magnetospheres with a near force-free regime inside the LC and a finite but high conductivity outside the LC are constructed using a spectral algorithm. The pulsar high-energy emission properties are explored by integrating the trajectories of the test particles under the influence of both the accelerating electric field and the curvature radiation losses. As an application, we compare the predicted energy-dependent light curves and energy spectra with those of the Crab and Vela pulsars published in the Fermi 2PC catalog. We find that the observed characteristics of the light curves and energy spectra from the Crab and Vela pulsars can be well reproduced by the FIDO model.

Alfvén Wave Mode Conversion in Pulsar Magnetospheres

Abstract The radio emission anomaly coincident with the 2016 glitch of the Vela pulsar may be caused by a starquake that launches Alfvén waves into the magnetosphere, disturbing the original radio emitting region. To quantify the lifetime of the Alfvén waves, we investigate a possible energy loss mechanism, the conversion of Alfvén waves into fast magnetosonic waves. Using axisymmetric force-free simulations, we follow the propagation of Alfvén waves launched from the stellar surface with small amplitude into the closed zone of a force-free dipolar pulsar magnetosphere. We observe mode conversion happening in the ideal force-free regime. The conversion efficiency during the first passage of the Alfvén wave through the equator can be large, for waves that reach large amplitudes as they travel away from the star, or propagate on the field lines passing close to the Y point. However, the conversion efficiency is reduced due to dephasing on subsequent passages and considerable Alfvén power on the closed field lines remains. Thus while some leakage into the fast mode happens, we need detailed understanding of the original quenching in order to say whether mode conversion alone can lead to reactivation of the pulsar on a short timescale.

Off-centred force-free neutron star magnetospheres

ABSTRACT Neutron star electromagnetic activity produces pairs that fill their magnetosphere represented to the zeroth order by the force-free approximation. Neither dissipation nor acceleration nor radiation from charged particles is expected from this simplified model. So far, only centred dipole magnetic fields have been studied in this limit. In this paper, we explore the consequences of a rotating off-centred dipole on the force-free magnetosphere, showing the new magnetic field geometry, its spin-down luminosity, and the electromagnetic kick and torque felt by the neutron star. Solutions are obtained by time-dependent numerical simulations of the force-free regime using our pseudo-spectral code written in spherical coordinates. Our results are also compared to known analytical expressions found for the off-centred vacuum dipole by an expansion to lowest order in the parameter ϵ = d/R, where d is the displacement of the dipole from the stellar centre and R the neutron star radius. The presence of a force-free plasma enhances the spin-down luminosity and the electromagnetic kick and torque with respect to a centred force-free dipole. The impact on isolated and binary neutron stars is revised in light of these new results.

Inertial oscillation modes of an inclined dipolar magnetosphere as a source of band-limited noise in X-ray pulsars

ABSTRACT Magnetic fields of strongly magnetized stars can trap conducting matter due to frozen-in condition. In the force-free regime, the motion of the matter along the field lines may be considered in the ‘bead on a wire’ approximation. Such a motion, if gravity and centrifugal forces are taken into account, has equilibrium points, some of which are stable. In most cases, stability is possible in about several per cent of the possible locations. Corresponding oscillation frequencies span the range from zero to $\sqrt{3}$ of the spin frequency. We suggest that this variability mode may be excited in some X-ray pulsars during the outbursts and create the peaked broad-band noise component near the break frequency in the power density spectrum, as well as produce some of the quasi-periodic oscillation features in this frequency range. Existence of this variability does not require any changes in mass accretion rate and involves only a small amount of matter infiltrating from the disc and magnetic flow due to interchange instabilities.

Radiative pulsar magnetospheres: aligned rotator

ABSTRACT Force-free neutron star magnetospheres are nowadays well known and found through numerical simulations. Even extension to general relativity has recently been achieved. However, those solutions are by definition dissipationless, meaning that the star is unable to accelerate particles and let them radiate any photon. Interestingly, the force-free model has no free parameter however it must be superseded by a dissipative mechanism within the plasma. In this Letter, we investigate the magnetosphere electrodynamics for particles moving in the radiation reaction regime, using the limit where acceleration is fully balanced by radiation, also called Aristotelian dynamics. An Ohm’s law is derived, from which the dissipation rate is controlled by a one parameter family of solutions depending on the pair multiplicity κ. The spatial extension of the dissipation zone is found self-consistently from the simulations. We show that the radiative magnetosphere of an aligned rotator tends to the force-free regime whenever the pair multiplicity becomes moderately large, κ ≫ 1. However, for low multiplicity, a substantial fraction of the spin-down energy goes into particle acceleration and radiation in addition to the Poynting flux, the latter remaining only dominant for large multiplicities. We show that the work done on the plasma occurs predominantly in the equatorial current sheet right outside the light-cylinder.

Modeling Gamma-Ray Light Curves with More Realistic Pulsar Magnetospheres

Abstract We study the gamma-ray emission patterns and light curves in dissipative pulsar magnetospheres. We produce the gamma-ray light curves by using the geometric method and the particle trajectory method. For the geometric method, assuming the gamma-ray emission originates in a finite-width layer along the last closed lines, we generate the gamma-ray light curves based on the uniform emissivity along the magnetic field lines in the comoving frame. For the particle trajectory method, we consider the spatial distribution of conductivity σ by assuming a very high conductivity within the light cylinder (LC) and a finite conductivity outside the LC. Assuming that all the γ-ray emission originates in the outer magnetosphere outside the LC, we generate the gamma-ray light curves by computing realistic particle trajectories and Lorentz factors, taking into account both the accelerating electric field and curvature radiation loss. Further, we compare the modeling light curves to the observed light curves at >0.1 GeV energies for the Vela pulsar. Our results show that the magnetosphere with the low σ value is preferred for the geometric method. However, the magnetosphere with a near force-free regime within the LC and a high σ value outside the LC is favored for the particle trajectory method. It is noted that the particle trajectory method uses the parallel electric fields that are self-consistent with the magnetic fields of the magnetosphere. We suggest that the results from the particle trajectory method are better supported on the physical ground.

Centrifugal acceleration in the isotropic photon field

In this paper, we study centrifugal acceleration of particles moving along a prescribed rotating curved trajectories. We consider the physical system embedded in an isotropic photon field and study the influence of the photon drag force on the acceleration process. For this purpose, we study three major configurations of the field lines: the straight line, the Archimede spiral and the dipolar field line configuration. By analyzing dynamics of particles sliding along the field lines in the equatorial plane, we have found several interesting features of motion. In particular, it has been shown that for rectilinear field lines, the particles reach the light cylinder (area where the linear velocity of rotation exactly equals the speed of light) zone relatively slowly for bigger drag forces. Considering the Archimedes’ spiral, we have found that in cases when the field lines lag behind the rotation, the particles achieve the force-free regime of dynamics regardless of the drag force. Unlike this scenario, when the spiral is oriented in an opposite direction, the particles do not reach the force free regime, but tend to stable equilibrium locations.

Electromagnetic luminosity of the coalescence of charged black hole binaries

The observation of a possible electromagnetic counterpart by the Fermi GBM group to the aLIGO detection of the merger of a black hole binary has spawned a number of ideas about its source. Furthermore, observations of fast radio bursts (FRBs) have similarly resulted in a range of new models that might endow black hole binaries with electromagnetic signatures. In this context, even the unlikely idea that astrophysical black holes may have significant charge is worth exploring, and here we present results from the simulation of weakly charged black holes as they orbit and merge. Our simulations suggest that a black hole binary with mass comparable to that observed in GW150914 could produce the level of electromagnetic luminosity observed by Fermi GBM ($10^{49}$ ergs/s) with a non-dimensional charge of $q \equiv Q/M = 10^{-4}$ assuming good radiative efficiency. However even a charge such as this is difficult to imagine avoiding neutralization long enough for the binary to produce its electromagnetic counterpart, and so this value would likely serve simply as an upper bound. On the other hand, one can equivalently consider the black holes as having acquired a magnetic monopole charge that would be easy to maintain and would generate an identical electromagnetic signature as the electric charges. The observation of such a binary would have significant cosmological implications, not the least of which would be an explanation for the quantization of charge itself. We also study such a magnetically charged binary in the force-free regime and find it much more radiative, reducing even further the requirements to produce the counterpart.

An oblique pulsar magnetosphere with a plasma conductivity

An oblique pulsar magnetosphere with a plasma conductivity is studied by using a pseudo-spectral method. In the pseudo-spectral method, the time-dependent Maxwell equations are solved, both electric and magnetic fields are expanded in terms of the vector spherical harmonic (VSH) functions in spherical geometry and the divergencelessness of magnetic field is analytically enforced by a projection method. The pulsar magnetospheres in infinite (i. e., force-free approximation) and finite conductivities are simulated and a family of solutions that smoothly transition from the Deutsch vacuum solution to the force-free solution are obtained. The $\sin^2\alpha$ dependence of the spin-down luminosity on the magnetic inclination angle $\alpha$ in which the full electric current density are taken into account is retrieved in the force-free regime.

On the role of rotation in the outflows of the Crab pulsar

In order to study constraints imposed on kinematics of the Crab pulsar's jet, we consider motion of particles along co-rotating field lines in the magnetosphere of the Crab pulsar. It is shown that particles following the co-rotating magnetic field lines may attain velocities close to observable values. In particular, we demonstrate that if the magnetic field lines are within the light cylinder (LC), the maximum value of the velocity component parallel to the rotation axis is limited by 0.5c. This result in the context of the X-ray observations performed by Chandra X-ray Observatory seems to be quite indicative and useful to estimate the density of field lines inside the jet. Considering the three-dimensional (3D) field lines crossing the LC, we found that for explaining the force-free regime of outflows the magnetic field lines must asymptotically tend to the Archimedes spiral configuration. It is also shown that the 3D case may explain the observed jet velocity for appropriately chosen parameters of magnetic field lines.

Dissipation, energy transfer, and spin-down luminosity in 2.5D PIC simulations of the pulsar magnetosphere

We perform 2.5D axisymmetric simulations of the pulsar magnetosphere (aligned dipole rotator) using the charge conservative, relativistic, electromagnetic particle in cell code PICsar. Particle in cell codes are a powerful tool to use for studying the pulsar magnetosphere, because they can handle the force-free and vacuum limits and provide a self-consistent treatment of magnetic reconnection. In the limit of dense plasma throughout the magnetosphere, our solutions are everywhere in the force-free regime except for dissipative regions at the polar caps, in the current layers, and at the Y-point. These dissipative regions arise self-consistently, since we do not have any explicit dissipation in the code. A minimum of $ \approx 15-20\%$ of the electromagnetic spindown luminosity is transferred to the particles inside 5 light cylinder radii. However, the particles can carry as much as $\gtrsim 50 \%$ of the spindown luminosity if there is insufficient plasma in the outer magnetosphere to screen the component of electric field parallel to the magnetic field. In reality, the component of the spindown luminosity carried by the particles could be radiated as gamma rays, but high-frequency synchrotron emission would need to be implemented as a sub-grid process in our simulations and is not present for the current suite of runs. The value of the spindown luminosity in our simulations is within $\approx 10\%$ of the force-free value, and the structure of the electromagnetic fields in the magnetosphere is on the whole consistent with the force-free model.

GENERAL RELATIVISTIC SIMULATIONS OF MAGNETIZED PLASMAS AROUND MERGING SUPERMASSIVE BLACK HOLES

Coalescing supermassive black hole binaries are produced by the mergers of galaxies and are the most powerful sources of gravitational waves accessible to space-based gravitational observatories. Some such mergers may occur in the presence of matter and magnetic fields and hence generate an electromagnetic counterpart. In this Letter, we present the first general relativistic simulations of magnetized plasma around merging supermassive black holes using the general relativistic magnetohydrodynamic code Whisky. By considering different magnetic field strengths, going from non-magnetically dominated to magnetically dominated regimes, we explore how magnetic fields affect the dynamics of the plasma and the possible emission of electromagnetic signals. In particular we observe a total amplification of the magnetic field of ~2 orders of magnitude which is driven by the accretion onto the binary and that leads to much stronger electromagnetic signals, more than a factor of 10^4 larger than comparable calculations done in the force-free regime where such amplifications are not possible.