Initial Magnetic Field Strength Research Articles

Non-ideal magnetohydrodynamics of self-gravitating filaments

Context. Filaments have been studied in detail through observations and simulations. A range of numerical works have separately investigated how chemistry and diffusion effects, as well as magnetic fields and their structure impact the gas dynamics of the filament. However, non-ideal effects have hardly been explored thus far. Aims. We investigate how non-ideal magnetohydrodynamic (MHD) effects, combined with a simplified chemical model affect the evolution and accretion of a star-forming filament. Methods. We modeled an accreting self-gravitating turbulent filament using LEMONGRAB, a one-dimensional (1D) non-ideal MHD code that includes chemistry. We explore the influence of non-ideal MHD, the orientation and strength of the magnetic field, and the cosmic ray ionization rate, on the evolution of the filament, with particular focus on the width and accretion rate. Results. We find that the filament width and the accretion rate are determined by the magnetic field properties, including the initial strength, the coupling with the gas controlled by the cosmic ray ionization rate, and the orientation of the magnetic field with respect to the accretion flow direction. Increasing the cosmic-ray ionization rate leads to a behavior closer to that of ideal MHD, reducing the magnetic pressure support and, hence, damping the accretion efficiency with a consequent broadening of the filament width. For the same reason, we obtained a narrower width and a larger accretion rate when we reduced the initial magnetic field strength. Overall, while these factors affect the final results by approximately a factor of 2, removing the non-ideal MHD effects results in a much greater variation (up to a factor of 7). Conclusions. The inclusion of non-ideal MHD effects and the cosmic-ray ionization is crucial for the study of self-gravitating filaments and in determining critical observable quantities, such as the filament width and accretion rate.

Local Quantum Uncertainty and Quantum Interferometric Power in an Anisotropic Two-Qubit System

Investigating the favorable configurations for non-classical correlations preservation has remained a hotly debated topic for the last decade. In this regard, we present a two-qubit Heisenberg spin chain system exposed to a time-dependent external magnetic field. The impact of various crucial parameters, such as initial strength and angular frequency of the external magnetic field along with the state’s purity and anisotropy of the spin-spin on the dynamical behavior of quantum correlations are considered. We utilize local quantum uncertainty (LQU) and quantum interferometric power (QIP) to investigate the dynamics of quantum correlations. We show that under the critical angular frequency of the external magnetic field and the spin-spin anisotropy, quantum correlations in the system can be successfully preserved. LQU and QIP suffer a drop as the interaction between the system and field is initiated, however, are quickly regained by the system. This tendency is confirmed by computing a measure of non-classical correlations according to the Clauser–Horne–Shimony–Holt inequality. Notably, the initial and final preserved levels of quantum correlations are only varied when variation is caused in the state’s purity.

The Galactic population of canonical pulsars

Context. Current wisdom suggests that the observed population of neutron stars are manifestations of their birth scenarios and their thermal and magnetic field evolution. Neutron stars can be observed at various wavebands as pulsars, and radio pulsars represent by far the largest population of neutron stars. Aims. In this paper, we aim to constrain the observed population of the canonical neutron star period, its magnetic field, and its spatial distribution at birth in order to understand the radio and high-energy emission processes in a pulsar magnetosphere. For this purpose we design a population synthesis method, self-consistently taking into account the secular evolution of a force-free magnetosphere and the magnetic field decay. Methods. We generated a population of pulsars and evolved them from their birth to the present time, using the force-free approximation. We assumed a given initial distribution for the spin period, surface magnetic field, and spatial Galactic location. Radio emission properties were accounted for by the polar cap geometry, whereas the gamma-ray emission was assumed to be produced within the striped wind model. Results. We find that a decaying magnetic field gives better agreement with observations compared to a constant magnetic field model. Starting from an initial mean magnetic field strength of B = 2.5 × 108 T with a characteristic decay timescale of 4.6 × 105 yr, a neutron star birth rate of 1/70 yr and a mean initial spin period of 60 ms, we find that the force-free model satisfactorily reproduces the distribution of pulsars in the P−Ṗ diagram with simulated populations of radio-loud, radio-only, and radio quiet gamma-ray pulsars similar to the observed populations.

MHD Simulations of Dense Core Collision

We investigated the effect of magnetic fields on the collision process between dense molecular cores. We performed three-dimensional magnetohydrodynamic simulations of collisions between two self-gravitating cores using the Enzo adaptive mesh refinement code. The core was modeled as a stable isothermal Bonnor–Ebert (BE) sphere immersed in uniform magnetic fields. Collisions were characterized by the offset parameter b, Mach number of the initial core , magnetic field strength B 0, and angle θ between the initial magnetic field and collision axis. For head-on (b = 0) collisions, one protostar was formed in the compressed layer. The higher the magnetic field strength, the lower the accretion rate. For models with b = 0 and θ = 90°, the accretion rate was more dependent on the initial magnetic field strength compared with b = 0 and θ = 0° models. For off-center (b = 1) collisions, the higher specific angular momentum increased; therefore, the gas motion was complicated. In models with b = 1 and , the number of protostars and gas motion highly depended on B 0 and θ. For models with b = 1 and , no significant shock-compressed layer was formed and star formation was not triggered.

Linear stability of an impulsively accelerated density interface in an ideal two-fluid plasma

We investigate the linear evolution of the Richtmyer–Meshkov instability (RMI) in the framework of an ideal two-fluid plasma model. The two-fluid plasma equations of motion are separated into a base state and a set of linearized equations governing the evolution of the perturbations. Different coupling regimes between the charged species are distinguished based on a non-dimensional Debye length parameter dD,0. When dD,0 is large, the coupling between ions and electrons is sufficiently small that the induced Lorentz force is very weak and the two species evolve as two separate fluids. When dD,0 is small, the coupling is strong and the induced Lorentz force is strong enough that the difference between state of ions and electrons is rapidly decreased by the force. As a consequence, the ions and electrons are tightly coupled and evolve like one fluid. The temporal dynamics is divided into two phases: an early phase wherein electron precursor waves are prevalent and a post-ion shock-interface interaction phase wherein the RMI manifests itself. We also examine the effect of an initially applied magnetic field in the streamwise direction characterized by the non-dimensional parameter β0. For a short duration after the ion shock-interface interaction, the growth rate is similar for different initial magnetic field strengths. Time progresses the suppression of the instability because the magnetic field is observed. The growth rate shows oscillations with a frequency that is related to the ion or electron cyclotron frequency. The instability is suppressed due to the oscillation of vorticity on the interface caused by the perturbed Lorentz force.

The Spheromak Tilting and How it Affects Modeling Coronal Mass Ejections

Abstract Spheromak-type flux ropes are increasingly used for modeling coronal mass ejections (CMEs). Many models aim at accurately reconstructing the magnetic field topology of CMEs, considering its importance in assessing their impact on modern technology and human activities in space and on the ground. However, so far there is little discussion about how the details of the magnetic structure of a spheromak affect its evolution through the ambient field in the modeling domain and what impact this has on the accuracy of magnetic field topology predictions. If the spheromak has its axis of symmetry (geometric axis) at an angle with respect to the direction of the ambient field, then the spheromak starts rotating so that its symmetry axis finally aligns with the ambient field. When using the spheromak in space weather forecasting models, this tilting can happen already during insertion and significantly affects the results. In this paper, we highlight this issue previously not examined in the field of space weather and we estimate the angle by which the spheromak rotates under different conditions. To do this, we generated simple purely radial ambient magnetic field topologies (weak/strong, positive/negative) and inserted spheromaks with varying initial speed, tilt, and magnetic helicity sign. We employ different physical and geometric criteria to locate the magnetic center of mass and axis of symmetry of the spheromak. We confirm that spheromaks rotate in all investigated conditions and their direction and angle of rotation depend on the spheromak’s initial properties and ambient magnetic field strength and orientation.

Supermassive Star Formation in Magnetized Atomic-cooling Gas Clouds: Enhanced Accretion, Intermittent Fragmentation, and Continuous Mergers

Abstract The origin of supermassive black holes (with ≳109 M ⊙) in the early universe (redshift z ∼ 7) remains poorly understood. Gravitational collapse of a massive primordial gas cloud is a promising initial process, but theoretical studies have difficulty growing the black hole fast enough. We focus on the magnetic effects on star formation that occurs in an atomic-cooling gas cloud. Using a set of three-dimensional magnetohydrodynamic simulations, we investigate the star formation process in the magnetized atomic-cooling gas cloud with different initial magnetic field strengths. Our simulations show that the primordial magnetic seed field can be quickly amplified during the early accretion phase after the first protostar formation. The strong magnetic field efficiently extracts angular momentum from accreting gas and increases the accretion rate, which results in the high fragmentation rate in the gravitationally unstable disk region. On the other hand, the coalescence rate of fragments is also enhanced by the angular momentum transfer due to the magnetic effects. Almost all the fragments coalesce to the primary star, so the mass growth rate of the massive star increases due to the magnetic effects. We conclude that the magnetic effects support the direct collapse scenario of supermassive star formation.

Thermal instability and multiphase gas in the simulated interstellar medium with conduction, viscosity, and magnetic fields

ABSTRACT Thermal instability plays a crucial role in the formation of multiphase structures and their dynamics in the interstellar medium, and is a leading theory for cold cloud creation in various astrophysical environments. In this paper, we use 2D simulations to investigate thermal instability under the influence of various initial conditions and physical processes. We experiment with Gaussian random field (GRF) density perturbations of different initial power spectra. We also enrol thermal conduction and physical viscosity in isotropic hydrodynamic and anisotropic magnetohydrodynamic (MHD) simulations. We find that the initial GRF spectral index α has a dramatic impact on the pure hydrodynamic development of thermal instability, influencing the size, number, and motions of clouds. Cloud fragmentation happens due to two mechanisms: tearing and contraction rebound. In the runs with isotropic conduction and viscosity, the structures and dynamics of the clouds are dominated by evaporation and condensation flows in the non-linear regime, and the flow speed is regulated by viscosity. Cloud disruptions happen as a result of the Darrieus–Landau instability. However, at very late times, all individual clouds merge into one cold structure in all hydrodynamic runs. In the MHD case, the cloud structure is determined by both the initial perturbations and the initial magnetic field strength. In high-β runs, anisotropic conduction causes dense filaments to align with the local magnetic fields and the field direction can become reoriented. Strong magnetic fields suppress cross-field contraction and cold filaments can form along or perpendicular to the initial fields.

Modeling Interplanetary Expansion and Deformation of Coronal Mass Ejections With ANTEATR‐PARADE: Sensitivity to Input Parameters

AbstractSpace weather predictions related to coronal mass ejections (CMEs) requires understanding how a CME is initiated and how its properties change as it propagates. Most predictions have been limited to the arrival time of a CME and include little to no information about the CME's internal properties. ANother Type of Ensemble Arrival Time Results‐Physics‐driven Approach to Realistic Axis Deformation and Expansion (ANTEATR‐PARADE) represents the most thorough description of the interplanetary evolution of CMEs in a highly computationally‐efficient model (Kay & Nieves‐Chinchilla, 2020, https://doi.org/10.1029/2020JA028911). presents the derivation of this model, where we have added an elliptical cross section to the original arrival time model ANTEATR and introduced internal magnetic and thermal forces that, combined with the drag, can alter the shape of the central axis and cross section. ANTEATR‐PARADE results include the transit time of CMEs, as well as the shape and size, propagation and expansion velocities, density, and magnetic field properties upon impact. We determine the dependence of each output on each of the ANTEATR‐PARADE input parameters. For a fast CME, we see that the parameterization of our thermal and magnetic models tends to be more important than the actual initial temperature or magnetic field strength. We extend to other CMEs and find that the sensitivities change with CME scale with thermal forces being more important for a weaker CME and magnetic forces being more important for a stronger CME. The most critical parameters for space weather predictions are the CME mass, the initial magnetic field strength, the adiabatic index, and the profile of the axial magnetic field strength.

A Supernova-driven, Magnetically Collimated Outflow as the Origin of the Galactic Center Radio Bubbles

Abstract A pair of nonthermal radio bubbles recently discovered in the inner few hundred parsecs of the Galactic center bears a close spatial association with elongated, thermal X-ray features called the X-ray chimneys. While their morphology, position, and orientation vividly point to an outflow from the Galactic center, the physical processes responsible for the outflow remain to be understood. We use 3D magnetohydrodynamic simulations to test the hypothesis that the radio bubbles/X-ray chimneys are the manifestation of an energetic outflow driven by multiple core-collapsed supernovae (SNe) in the nuclear stellar disk, where numerous massive stars are known to be present. Our simulations are run with different combinations of two main parameters, the supernova birth rate and the strength of a global magnetic field being vertically oriented with respect to the disk. The simulation results show that a hot gas outflow can naturally form and acquire a vertically elongated shape due to collimation by the magnetic pressure. In particular, the simulation with an initial magnetic field strength of 80 μG and a supernova rate of 1 kyr−1 can well reproduce the observed morphology, internal energy, and X-ray luminosity of the bubbles after an evolutionary time of 330 kyr. On the other hand, a magnetic field strength of 200 μG gives rise to an overly elongated outflow that is inconsistent with the observed bubbles. The simulations also reveal that, inside the bubbles, mutual collisions between the shock waves of individual SNe produce dense filaments of locally amplified magnetic field. Such filaments may account for a fraction of the synchrotron-emitting radio filaments known to exist in the Galactic center.

Magnetized decaying turbulence in the weakly compressible Taylor-Green vortex

Magnetohydrodynamic (MHD) turbulence affects both terrestrial and astrophysical plasmas. The properties of magnetized turbulence must be better understood to more accurately characterize these systems. This work presents ideal MHD simulations of the compressible Taylor-Green vortex under a range of initial subsonic Mach numbers and magnetic field strengths. We find that regardless of the initial field strength, the magnetic energy becomes dominant over the kinetic energy on all scales after at most several dynamical times. The spectral indices of the kinetic and magnetic energy spectra become shallower than k^{-5/3} over time and generally fluctuate. Using a shell-to-shell energy transfer analysis framework, we find that the magnetic fields facilitate a significant amount of the energy flux and that the kinetic energy cascade is suppressed. Moreover, we observe nonlocal energy transfer from the large-scale kinetic energy to intermediate and small-scale magnetic energy via magnetic tension. We conclude that even in intermittently or singularly driven weakly magnetized systems, the dynamical effects of magnetic fields cannot be neglected.

Long-term evolution of a merger-remnant neutron star in general relativistic magnetohydrodynamics: Effect of magnetic winding

Long-term ideal and resistive magnetohydrodynamics (MHD) simulations in full general relativity are performed for a massive neutron star formed as a remnant of binary neutron star mergers. Neutrino radiation transport effects are taken into account as in our previous papers. The simulation is performed in axial symmetry and without considering dynamo effects as a first step. In the ideal MHD, the differential rotation of the remnant neutron star amplifies the magnetic-field strength by the winding in the presence of a seed poloidal field until the electromagnetic energy reaches $\sim 10\%$ of the rotational kinetic energy, $E_{\rm kin}$, of the neutron star. The timescale until the maximum electromagnetic energy is reached depends on the initial magnetic-field strength and it is $\sim 1$ s for the case that the initial maximum magnetic-field strength is $\sim 10^{15}$ G. After a significant amplification of the magnetic-field strength by the winding, the magnetic braking enforces the initially differentially rotating state approximately to a rigidly rotating state. In the presence of the resistivity, the amplification is continued only for the resistive timescale, and if the maximum electromagnetic energy reached is smaller than $\sim 3\%$ of $E_{\rm kin}$, the initial differential rotation state is approximately preserved. In the present context, the post-merger mass ejection is induced primarily by the neutrino irradiation/heating and the magnetic winding effect plays only a minor role for the mass ejection.

Similarity solutions for the strong shock waves in magnetogasdynamics with the effect of monochromatic radiation

In this paper, the method of invariance of Lie group is used to find the similarity solutions of the cylindrical shock waves for one-dimensional unsteady flow in magnetogasdynamics in the presence of monochromatic radiation. The density is assumed to be non-uniform ahead of the shock. The radiation flux is assumed to move through the gas. It is also shown that how the radiation parameter, ambient density exponent, ratio of specific heats (adiabatic constant) and the initial magnetic field strength affect the profiles of the flow variables. It is found that the shock wave decays in the presence of strong magnetic field and shows reverse behavior in the case of monochromatic radiation, i.e., the shock wave grows faster in the presence of radiation. To obtain the similarity solutions and the profiles of the flow variables, numerical calculations are performed using the software package MATHEMATICA.

The importance of magnetic fields for the initial mass function of the first stars

ABSTRACT Magnetic fields play an important role for the formation of stars in both local and high-redshift galaxies. Recent studies of dynamo amplification in the first dark matter haloes suggest that significant magnetic fields were likely present during the formation of the first stars in the Universe at redshifts of 15 and above. In this work, we study how these magnetic fields potentially impact the initial mass function (IMF) of the first stars. We perform 200 high-resolution, three-dimensional (3D), magnetohydrodynamic (MHD) simulations of the collapse of primordial clouds with different initial turbulent magnetic field strengths as predicted from turbulent dynamo theory in the early Universe, forming more than 1100 first stars in total. We detect a strong statistical signature of suppressed fragmentation in the presence of strong magnetic fields, leading to a dramatic reduction in the number of first stars with masses low enough that they might be expected to survive to the present-day. Additionally, strong fields shift the transition point where stars go from being mostly single to mostly multiple to higher masses. However, irrespective of the field strength, individual simulations are highly chaotic, show different levels of fragmentation and clustering, and the outcome depends on the exact realization of the turbulence in the primordial clouds. While these are still idealized simulations that do not start from cosmological initial conditions, our work shows that magnetic fields play a key role for the primordial IMF, potentially even more so than for the present-day IMF.

Exploring the accretion-induced evolution of the spin period and magnetic field strength of Be/X-ray Pulsars

Based on the detected spin periods ( $P$ ) and inferred magnetic field strengths ( $B$ ) by the cyclotron resonance scattering features, we analyze the $B-P$ properties of Be/X-ray Pulsars (BeXPs). We find that the $P$ distribution of BeXPs exhibits a bimodal feature separated at $P\sim 40$ s, where the average spin period of the BeXPs with $P>40$ s ( $\langle P\rangle \sim 267$ s) is larger than that of the sources with $P<40$ s ( $\langle P\rangle \sim 10$ s) by about one magnitude of order. Meanwhile, the average magnetic field strength of the long period BeXPs ( $\langle B\rangle \sim 4.9\times 10^{12}$ G) is higher than that of the short period sources ( $\langle B\rangle \sim 2.7\times 10^{12}$ G) by a factor of $\sim 2$ . We try to explain these phenomena by the accretion-induced evolution process, and find that for the neutron star (NS) with the initial magnetic field strength of $B_{0}\sim 10^{12.2}-10^{13}$ G, when it accretes about $\Delta M\sim 10^{-6.5}\,\mathrm{M_{\odot }}$ companion matter, its spin period can shorten from $P_{0}\sim 1000$ s to $P\sim 260$ s, while its magnetic field strength decays little. Furthermore, when the NS accretes about $\Delta M\sim 10^{-5.5}\,\mathrm{M_{\odot }}$ matter, its spin period can shorten to $P\sim 10$ s, while its magnetic field strength decays by half. Finally, we also notice that as the continuing of the accretion process in Be/X-ray binary, when its NS accretes about $\sim 10^{-3}\,\mathrm{M_{\odot }}$ mater, it has the possibility to evolve to the double neutron star.

How primordial magnetic fields shrink galaxies

ABSTRACT As one of the prime contributors to the interstellar medium energy budget, magnetic fields naturally play a part in shaping the evolution of galaxies. Galactic magnetic fields can originate from strong primordial magnetic fields provided these latter remain below current observational upper limits. To understand how such magnetic fields would affect the global morphological and dynamical properties of galaxies, we use a suite of high-resolution constrained transport magnetohydrodynamic cosmological zoom simulations where we vary the initial magnetic field strength and configuration along with the prescription for stellar feedback. We find that strong primordial magnetic fields delay the onset of star formation and drain the rotational support of the galaxy, diminishing the radial size of the galactic disc and driving a higher amount of gas towards the centre. This is also reflected in mock UVJ observations by an increase in the light profile concentration of the galaxy. We explore the possible mechanisms behind such a reduction in angular momentum, focusing on magnetic braking. Finally, noticing that the effects of primordial magnetic fields are amplified in the presence of stellar feedback, we briefly discuss whether the changes we measure would also be expected for galactic magnetic fields of non-primordial origin.

Distortion of Magnetic Fields in a Starless Core. VI. Application of Flux Freezing Model and Core Formation of FeSt 1–457

Abstract Observational data for the hourglass-like magnetic field toward the starless dense core FeSt 1–457 were compared with a flux freezing magnetic field model. Fitting of the observed plane-of-sky magnetic field using the flux freezing model gave a residual angle dispersion comparable to the results based on a simple 3D parabolic model. The best-fit parameters for the flux freezing model were a line-of-sight magnetic inclination angle of γ mag = 35° ± 15° and a core center to ambient (background) density contrast of ρ c/ρ bkg = 75. The initial density for core formation (ρ 0) was estimated to be ρ c/75 = 4670 cm−3, which is about one order of magnitude higher than the expected density (∼300 cm−3) for the interclump medium of the Pipe Nebula. FeSt 1–457 is likely to have been formed from the accumulation of relatively dense gas, and the relatively dense background column density of A V ≃ 5 mag supports this scenario. The initial radius (core formation radius) R 0 and the initial magnetic field strength B 0 were obtained to be 0.15 pc (1.64R) and 10.8–14.6 μG, respectively. We found that the initial density ρ 0 is consistent with the mean density of the nearly critical magnetized filament with magnetic field strength B 0 and radius R 0. The relatively dense initial condition for core formation can be naturally understood if the origin of the core is the fragmentation of magnetized filaments.

There is no magnetic braking catastrophe: low-mass star cluster and protostellar disc formation with non-ideal magnetohydrodynamics

Abstract We present results from the first radiation non-ideal magnetohydrodynamics (MHD) simulations of low-mass star cluster formation that resolve the fragmentation process down to the opacity limit. We model 50 M⊙ turbulent clouds initially threaded by a uniform magnetic field with strengths of 3, 5 10, and 20 times the critical mass-to-magnetic flux ratio, and at each strength, we model both an ideal and non-ideal (including Ohmic resistivity, ambipolar diffusion, and the Hall effect) MHD cloud. Turbulence and magnetic fields shape the large-scale structure of the cloud, and similar structures form regardless of whether ideal or non-ideal MHD is employed. At high densities (106 ≲ nH ≲ 1011 cm−3), all models have a similar magnetic field strength versus density relation, suggesting that the field strength in dense cores is independent of the large-scale environment. Albeit with limited statistics, we find no evidence for the dependence of the initial mass function on the initial magnetic field strength, however, the star formation rate decreases for models with increasing initial field strengths; the exception is the strongest field case where collapse occurs primarily along field lines. Protostellar discs with radii ≳ 20 au form in all models, suggesting that disc formation is dependent on the gas turbulence rather than on magnetic field strength. We find no evidence for the magnetic braking catastrophe, and find that magnetic fields do not hinder the formation of protostellar discs.

Study on the action law of key influencing factors of magnetic field strength of magnetic filter material

In recent years, the application of magnetic field technology in the field of water treatment has been relatively mature. The common ways of adding magnetic field are external magnetic field and internal magnetic powder. In order to solve the problem of uneven distribution of magnetic field caused by external mode and easy loss of magnetic powder caused by internal injection mode, this paper studied a preparation method of magnetic filter material used in biological filter tank, and studied the key influencing factors such as composition, particle size, magnetization time and times that affect magnetic field strength of filter material. The results show that the nanometer Fe3O4 content has a great influence on the magnetic field strength of the magnetic filter material. When the content is 16.6%, the magnetic field strength of the filter material is the best. The larger the particle size of the filter material is, the less the magnetic structure of the surface magnetization vector is not collinear, and the higher the internal coercivity is, the higher the magnetic field strength of the filter material is. The magnetic field intensity of the filter material increased gradually with the extension of magnetization time, and basically became saturated after 60min. Repeated magnetization can improve the initial magnetic field strength of the filter material, but has little effect on the stable magnetic field strength after magnetic recession.

Particle acceleration mechanism due to interaction between one-dimensional fast plasma flow and perpendicular magnetic field

We have numerically investigated the behavior of one-dimensional fast plasma flow in a perpendicular magnetic field for understanding the particle acceleration mechanisms of collisionless shocks in the non-relativistic region. Electromagnetic hybrid particle-in-cell simulation was performed for simulating pulsed-power-driven fast plasma flow. Numerical results showed that accelerated particles are generated in the plasma flow after passing through the perpendicular magnetic field. The gradient of the compressed magnetic field due to the propagation of the plasma flow affects the acceleration of charged particles. The maximum energy of accelerated particles depends on the initial peak strength of the perpendicular magnetic field.