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Register a new userImpact of the Hall effect in star formation : improving the angular momentum conservation
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We present here a minor modification of our numerical implementation of the Hall effect for the 2D Riemann solver used in Constrained Transport schemes, as described in Marchand et al. (2018). In the previous work, the tests showed that the angular momentum was not conserved during protostellar collapse simulations, with significant impact. By removing the whistler waves speed from the characteristic speeds of non-magnetic variables in the 1D Riemann solver, we are able to improve the angular momentum conservation in our test-case by one order of magnitude, while keeping the second-order numerical convergence of the scheme. We also reproduce the simulations of Tsukamoto et al. (2015) with consistent resistivities, the three non-ideal MHD effects and initial rotation, and agree with their results. In this case, the violation of angular momentum conservation is negligible in regard to the total angular momentum and the angular momentum of the disk.
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Magnetic fields play an important role in star formation by regulating the removal of angular momentum from collapsing molecular cloud cores. Hall diffusion is known to be important to the magnetic field behaviour at many of the intermediate densities and field strengths encountered during the gravitational collapse of molecular cloud cores into protostars, and yet its role in the star formation process is not well-studied. We present a semianalytic self-similar model of the collapse of rotating isothermal molecular cloud cores with both Hall and ambipolar diffusion, and similarity solutions that demonstrate the profound influence of the Hall effect on the dynamics of collapse. The solutions show that the size and sign of the Hall parameter can change the size of the protostellar disc by up to an order of magnitude and the protostellar accretion rate by fifty per cent when the ratio of the Hall to ambipolar diffusivities is varied between -0.5 <= eta_H / eta_A <= 0.2. These changes depend upon the orientation of the magnetic field with respect to the axis of rotation and create a preferred handedness to the solutions that could be observed in protostellar cores using next-generation instruments such as ALMA. Hall diffusion also determines the strength and position of the shocks that bound the pseudo and rotationally-supported discs, and can introduce subshocks that further slow accretion onto the protostar. In cores that are not initially rotating Hall diffusion can even induce rotation, which could give rise to disc formation and resolve the magnetic braking catastrophe. The Hall effect clearly influences the dynamics of gravitational collapse and its role in controlling the magnetic braking and radial diffusion of the field merits further exploration in numerical simulations of star formation.
Magnetic fields play an important role in star formation by regulating the removal of angular momentum from collapsing molecular cloud cores. Hall diffusion is known to be important to the magnetic field behaviour at many of the intermediate densities and field strengths encountered during the gravitational collapse of molecular cloud cores into protostars, and yet its role in the star formation process is not well-studied. This thesis describes a semianalytic self-similar model of the collapse of rotating isothermal molecular cloud cores with both Hall and ambipolar diffusion, presenting similarity solutions that demonstrate that the Hall effect has a profound influence on the dynamics of collapse. ... Hall diffusion also determines the strength of the magnetic diffusion and centrifugal shocks that bound the pseudo and rotationally-supported discs, and can introduce subshocks that further slow accretion onto the protostar. In cores that are not initially rotating Hall diffusion can even induce rotation, which could give rise to disc formation and resolve the magnetic braking catastrophe. The Hall effect clearly influences the dynamics of gravitational collapse and its role in controlling the magnetic braking and radial diffusion of the field would be worth exploring in future numerical simulations of star formation.
Suppose a classical electron is confined to move in the $xy$ plane under the influence of a constant magnetic field in the positive $z$ direction. It then traverses a circular orbit with a fixed positive angular momentum $L_z$ with respect to the center of its orbit. It is an underappreciated fact that the quantum wave functions of electrons in the ground state (the so-called lowest Landau level) have an azimuthal dependence $propto exp(-imphi) $ with $mgeq 0$, seemingly in contradiction with the classical electron having positive angular momentum. We show here that the gauge-independent meaning of that quantum number $m$ is not angular momentum, but that it quantizes the distance of the center of the electrons orbit from the origin, and that the physical angular momentum of the electron is positive and independent of $m$ in the lowest Landau levels. We note that some textbooks and some of the original literature on the fractional quantum Hall effect do find wave functions that have the seemingly correct azimuthal form $proptoexp(+imphi)$ but only on account of changing a sign (e.g., by confusing different conventions) somewhere on the way to that result.
Smoothed particle hydrodynamics (SPH) is positioned as having ideal conservation properties. When properly implemented, conservation of total mass, energy, and both linear and angular momentum is guaranteed exactly, up to machine precision. This is particularly important for some applications in computational astrophysics, such as binary dynamics, mergers, and accretion of compact objects (neutron stars, black holes, and white dwarfs). However, in astrophysical applications that require the inclusion of gravity, calculating pairwise particle interactions becomes prohibitively expensive. In the Fast Multipole Method (FMM), they are, therefore, replaced with symmetric interactions between distant clusters of particles (contained in the tree nodes) Although such an algorithm is linear momentum-conserving, it introduces spurious torques that violate conservation of angular momentum. We present a modification of FMM that is free of spurious torques and conserves angular momentum explicitly. The new method has practically no computational overhead compared to the standard FMM.
Exact conservation of the angular momentum is worked out for an elastic medium with spins. The intrinsic anharmonicity of the elastic theory is shown to be crucial for conserving the total momentum. As a result, any spin-lattice dynamics inevitably involves multiphonon processes and interaction between phonons. This makes transitions between spin states in a solid fundamentally different from transitions between atomic states in vacuum governed by linear electrodynamics. Consequences for using solid-state spins as qubits are discussed.
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